Lesson #2 - Augsburg College

CHALLENGE MATH FOR 5TH GRADERS, FIRST EDITION 

Published at Carleton College in Northfield, Minnesota. Copyright © 2008. 

Please contact Deanna Haunsperger, Carleton College Professor of Mathematics, at dhaunspe@carleton.edu to 

purchase a copy. 

2

Special Thanks 

First and foremost, we’d like to thank our advisor, Deanna Haunsperger, for the guidance and assistance she 

provided throughout the project. We’d also like to thank April Ostermann for bringing us into her fifth grade 

class and for facilitating the teaching aspect of the project. We are also very grateful for the inspiration that 

Robert and Ellen Kaplan provided through their visit to Carleton College. Finally, the printing process would 

not have been possible without the generous help of the Carleton College Dean’s Office and the Carleton College 

Department of Mathematics. 

3

4 

Table of Contents 

An Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Hi and Welcome! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Where are Fifth Graders Developmentally? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Where are Fifth Graders Academically? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Why Teach Challenge Math? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Gender and Multiculturalism in Math Education . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Teaching Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

About the Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

How to Use This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson Headers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Quick Icons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Supplements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

IMPORTANT: Read This! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Geometry: A Shapely Approach to Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #1: Introduction to Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #2: Angle Sums of a Triangle: The Magic Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #3: Congruent Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #4: The Pythagorean Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #5: Pythagorean Triples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #6: Pythagorean Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #7: Circles and Pi (π) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Probability: What are the Chances?! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #8: Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #9: Dice Probabilities I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #10: Dice Probabilities II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #11: Coins, Dice, and Playing Cards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #12: The Monty Hall Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Estimation and Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #13: Estimating the Volume of the Gym . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #14: How Many 5th Graders Can Fit in the Gym? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Combinatorics: How Many Ways Can You... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #15: Football Scores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #16: Combinatorial Theory Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #17: Cracking the Postal Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #18: The Catalan Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Again and Again and Again... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #19: Fractals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #20: Conway’s Game of Life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

7 

7 

7 

7 

8 

9 

9 

10 

12 

12 

13 

14 

14 

14 

14 

14 

16 

17 

22 

26 

32 

35 

39 

45 

50 

51 

54 

58 

61 

65 

69 

70 

73 

77 

78 

82 

85 

89 

95 

96 

104

Algebra: Solve for X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #21: Order of Operations or Challenge Math’s Biggest Problem . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #22: Solving for a Variable using the Golden Rule of Algebra . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #23: Puzzle Worksheets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Graphing: Plot the Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #24: Graphing Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #25: Jumping Jacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #26: Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #27: Graphing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #28: Manipulating Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

An Introduction to Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #29: Set Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #30: Infinite Sets and Greek Punishments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #31: The Infinite Hotel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Fun with Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #32: Negative Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #33: Introduction to Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #34: Patterned Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #35: Fun with the Fibonacci Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #36: Introduction to Phi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #37: Prime Numbers and the Ulam Spiral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Different Bases: Beyond Base 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #38: Binary in Computer Science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #39: Binary Number System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #40: Exploding Dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #41: Jeopardy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Connecting Dots and Coloring Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #42: The Four Color Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #43: The Seven Bridges Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #44: The Three Utilities Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Topology: Coffee Cups and Doughnuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #45: Playdough and Mobius Strips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #46: Euler Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Thinking Puzzles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #47: A Pico Fermi Bagel Exploration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #48: A Pico Fermi Bagel Challenge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #49: Secret Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Lesson #50: Playing Games: A Lesson in Strategy and Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Space Fillers and Mathematical Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

111 

112 

117 

127 

138 

139 

144 

148 

152 

157 

162 

163 

167 

170 

175 

176 

179 

183 

187 

196 

201 

207 

208 

212 

215 

221 

224 

225 

236 

240 

246 

247 

251 

257 

258 

262 

265 

269 

273 

279 

5

An Introduction 

Hi and Welcome! 

Hi! We spent one year teaching Challenge Math to fifth graders in Northfield, Minnesota, and we created 

this curriculum to provide you with fun lessons that you can use when you teach Challenge Math. The lessons 

are organized by topic, and we recommend that you spend two to four weeks on a topic before moving on to the 

next one. We’ve provided you with more lessons than it would be possible to teach in a year, so feel free to pick 

out your favorites. 

We were all math majors at Carleton College, so we think that math is pretty cool. Math, though, is not just 

fun at the college level; it’s fun at the elementary and middle school levels as well. Math teaches students how 

to use their imaginations to solve problems, and thus encourages them to think creatively. Math also teaches students 

how to organize and explain their thoughts, which is a skill that they can apply to all areas of their lives. 

It’s true that math can be hard and sometimes frustrating for students, but as long as the lesson remains fun, with 

your guidance your students will work past their frustrations and achieve success! 

Where are Fifth Graders Developmentally? 

Fifth graders are in a period of transition. From approximately the ages of 6-12, children develop the ability 

to think about concrete events, which they can physically see and manipulate. By the end of this developmental 

stage, children have the ability to observe more than one quality of an object at a time and can classify objects 

according to multiple characteristics. They should also understand the concept of conservation by the end of this 

stage. A classic example of conservation is that the volume of a short, fat glass can be the same as the volume 

of a tall thin glass, even though one of the glasses may look bigger. Fifth graders should be able to tell you how 

the taller glass compensates for the shorter one. They should have also developed a sense of identity (the idea 

that if nothing has been added to or subtracted from an initial quantity, the initial amount will remain the same). 

Finally, 5th graders should have developed a sense of reversibility; they can explain how one effect can reverse 

or negate another, such as addition negating subtraction or multiplication negating division. 

These breakthroughs are very important because they are necessary for the development of mathematical 

reasoning. Many adults never progress beyond this stage of development because the next stage can require 

a bit of difficult intellectual digging. Challenge Math provides the perfect opportunity to help students begin 

this next stage, which is commonly referred to as the formal thinking stage. In this stage, students develop the 

mental processes necessary for thinking about abstract events, which are not visible or tangible. Thinking abstractly 

can prove to be much more difficult than thinking about concrete objects or events. It requires a mental 

cataloguing of ideas that must be practiced and encouraged. The ability to think abstractly is important, if not 

necessary, for success in all areas of education. Developing this ability can begin with opening one’s mind to 

the possibility that there are multiple ways to view a problem and approach it. Challenge Math is a great setting 

for developing these new ways of thinking because it provides opportunities to discover solutions through 

multiple avenues. 

Where are Fifth Graders Academically? 

Within any group of students, a range of mathematical ability is inevitable. In a class of thirty students, it is 

likely that there are three or more broad categories of ability, such as under-achieving, over-achieving, and one 

ore more categories between these two. But what constitutes standard achievement? 

7

The National Council of Teachers of Mathematics (NCTM) is the largest organization in the world that focuses 

on mathematics education. It has developed a series of standards for mathematics learning that is largely 

accepted by the United States Department of Education. The Principles and Standards for School Mathematics 

are detailed and wide-ranging. To give you an idea of how detailed they are, let it suffice to say that an outline 

of just the 5th grade standards would be far too long for this introduction. For more information about NCTM 

and to read the Standards for yourself, visit NCTM’s website: http://www.ntcm.org. In addition to the NCTM 

Standards, an outline of the mathematical concepts that Northfield Public Schools expect 5th graders to know is 

available from your child’s teacher. 

• To provide a strong math education 

8 

Why Teach Challenge Math? 

Math education begins at the elementary level, and developing a strong mathematical foundation is critical 

so that students can continue to have success in math as they continue their education. Challenge Math aims 

to help students to develop this strong foundation by teaching them different ways to ask questions and solve 

problems. 

• To show students that math is fun 

In Challenge Math, students learn to associate math with something enjoyable, like thinking games or exploration, 

rather than with something boring like memorizing multiplication tables. 

• To reinforce past and future learning through exposure to new activities 

Challenge Math exposes students to ideas that they might otherwise never come across and introduces 

them to ideas that they will learn about in more depth in subsequent grades. Though Challenge Math is not just 

forward-looking; it also reinforces ideas that students have already learned in the classroom. If a student sees an 

idea that they learned in fourth grade implemented in a fifth grade Challenge Math lesson, they are more likely 

to remember that idea in the future. 

• To give students something to look forward to each week 

Students really do look forward to Challenge Math every week. Teaching Challenge Math can be very rewarding, 

for there’s nothing better than watching a student’s face as they figures out the answer to a hard problem. 

• To provide a happy and healthy school environment in which every student can and will succeed 

As a Challenge Math teacher, you are a role model for the students. They look up to you, and you have the 

power lift up those students who struggle. We’re glad that you’re in the position of a role model for the students, 

because it’s important for students to have strong, positive, and encouraging adults in their lives. To give 

students a positive Challenge Math experience, we’d recommend that you bring a positive attitude and lots of 

energy to your lessons. The students will respond to your excitement, and your enthusiasm will be contagious. 

We’d encourage you to think of yourself as a guide rather than as a teacher. As a guide, you will lead the students 

through many fun math explorations and encourage their creative ideas along the way without giving 

them answers to memorize.

Gender and Multiculturalism in Math Education 

In the history of teaching and learning math there have been some disturbing gaps in achievement between 

students of different demographics. The first important achievement gap to recognize as a math teacher is between 

boys and girls. Statistics show blatantly that women are under-represented in the field of mathematics in 

our society; you can tell by entering a male-dominated college classroom, and the problem only worsens as you 

move to a graduate level classroom. Most argue that girls are socialized away from the math and science fields, 

beginning as early on in school as kindergarten. A girl is encouraged to be artistic and emotional and pursue 

careers working with people, while a boy is socialized to be pragmatic and objective—qualities that lead very 

nicely into science and math. 

This gender gap plays out in schools nation-wide and our Northfield schools are no exception. Some have 

suggested that we should be doing more to set girls up for success in math. A program like Challenge Math is 

an ideal environment in which to do so. By placing them in a Challenge Math group, teachers are recognizing 

and validating the mathematical abilities that many of our girls have. These students like math, and by leading 

your own Challenge Math group with the gender achievement gap in mind, you have the opportunity to nurture 

and reinforce girls’ confidence as mathematicians so that they can be proud, not embarrassed, or their math 

smarts and set an example for younger students. 

The racial achievement gap is in many ways like the gender gap, but it quickly becomes more complicated. 

The phrase ‘achievement gap’ originally referred to the comparison of academic achievement between black 

students and white students. Nationally it was and is still confirmed that black students perform less well in 

all areas of school than white students. This discrepancy is even more severe in mathematics, and now it extends 

to all minority students, not just black students. With immigration to the U.S. growing exponentially and 

students for whom English is a second language being in a majority of elementary classrooms, the achievement 

gap occurs due to a compilation of different ‘minority’ labels regarding race, class, and culture. Northfield 

schools like Bridgewater, Sibley, and Greenvale are a perfect example both of the increasing Latino immigrant 

population and of the achievement gap between minorities and white students in math and in general. 

Thinking about both achievement gaps may leave you feeling discouraged but we feel strongly that, on the 

contrary, these gaps can be successfully addressed and countered in Challenge Math. Because Challenge Math 

teachers are volunteers, we have a fresh outside perspective not tainted by a given student’s reputation among 

teachers or history at school from previous years. If you believe that every student has a different ideal learning 

style, and when they’re given the opportunity to learn in that way, they can succeed, then you can single-handedly 

erase the achievement gaps in one small group of students. 

You might not believe it at first, but you can easily make success a part of each student’s Challenge Math 

experiences. If Jake is good at handwriting, make him the data recorder and he can participate in a way that 

makes him feel competent and confident. If Suzy doesn’t like brainstorms because she’s often the last to shout 

out answers, make a rule that no one can speak twice until everyone speaks once. You will quickly become 

an adult upon whom students rely for praise, and that’s great! Students who suffer at the bottom end of the 

achievement gaps can and will thrive in the informal small group setting of Challenge Math. 

Teaching Methods 

The field of mathematics instruction has changed dramatically in recent history from a system that asks, 

“What is the right answer?” to a system that asks, “Why is the answer right?” This shift has taken hold at various 

levels of instruction, but it is not yet universal. Many students still think that math is boring, and some 

9

students even fear it. Memorization is still a very important part of the curriculum. Not all memorization is bad 

by nature, but when given no supporting explanation, students will not learn the reasons behind the memorized 

facts. Also, the act of memorization is monotonous and rarely excites the students. 

This year of Challenge Math presents you with a unique opportunity to inspire a group of fifth graders. 

Throughout these lessons, your goal is not to replace their classroom curriculum but rather to supplement it 

with explorations into various areas of mathematics. Perhaps the most important gift that a good mathematical 

education can give to a student is the ability to logically approach a problem with confidence. Decoding the 

question and systematically searching for an asnwer is the basis for all proof-based math. While you will not 

be rigorously proving theorems with your students, Challenge Math provides students with the opportunity to 

discover creative approaches to problem solving. The lessons provided in this curriculum will encourage the 

students to think outside of the box and will help them feel the excitement of reaching conclusions on their own. 

Ultimately, these lessons give the students the tools to ask, ‘Why?” and then provides activities that will help 

them determine the answers. 

The “Math Circle” style of teaching, created by Robert and Ellen Kaplan, heavily influences the pedagogical 

ideals present in this curriculum. The fundamental idea behind this style of teaching is that the students should 

discover the math on their own. This presents many challenges for the teacher, whose natural instinct is to tell 

the students the solutions to the problems. Even for those implementing this style of teaching, it is still difficult 

to steer the class in the right direction without directly handing the students the answers. This is not to say that 

you should play no role in the class except to give the students a push at the beginning, but it’s important that 

you act more as an guide toward discovery than a bestower of truth. Have faith that your students will surprise 

you by thinking through problems and working to find the answers. 

The lessons in this curriculum often direct you to “ask the students” a leading question or to “discuss” an 

interesting solution, but we recognize that discussion may not always be feasible. If your class dynamic isn’t 

right for discussion, it’s okay to move past these prompts. Just remember that students learn best when given the 

opportunity to approach the challenges of each lesson in their own ways. 

For a Challenge Math lesson to be successful, the students must want to be there. Engaged students will 

always appreciate a well-planned lesson, but a student who isn’t engaged can bring the rest of the group down, 

making it hard for you to teach. Also, the engagement level of a student does not at all reflect their interest or 

mathematical ability. As a teacher, it is your responsibility to reach out to all students and to understand that 

each student needs a different kind of encouragement. The following lessons have the potential not only to teach 

the desired subject matter, but also to teach the process of learning. Since it encourages creative problem solving, 

ultimately the students will take away both concrete mathematical knowledge and also a new way of thinking. 

10 

About the Authors 

Gabe Hart: 

As a double major in math and computer science, I live for the satisfaction of solving problems. I know of 

no greater satisfaction than that which comes after searching for a difficult solution and then finding the flash of 

insight needed to fit all the seemingly unrelated pieces together. My particular academic interest are computer 

graphics and digital vision which use mathematical techniques to simulate the visual world and to extract information 

from real-world visual data. If put to the right use, I think these fields have a great deal of potential to 

benefit society. I will be headed to the University of North Carolina at Chapel Hill next year to pursue an M.A. 

in Computer Science. Outside of the classroom, I keep myself busy playing Ultimate Frisbee for the Carleton 

Ultimate Team. We’ve attended the national tournament three out of the last four years and hope to do so again

this year. For less competitive recreation, I love to mountain bike and have developed a strong interest in nature 

photography. The process of making this book has been very rewarding to me because I love the feeling of 

teaching something and then watching the students really grasp it and get excited about it. Hopefully these lessons 

will help students and teachers alike find that excitement that comes when all the pieces of the puzzle click 

into place. 

Alissa Pajer: 

I like math because I like the critical and creative thinking necessary to solve hard problems. Specifically, 

my favorite field in mathematics is analysis, which studies the rigor behind calculus. Aside from math, I like to 

read literature and essays; I have a long list of texts and authors that I want to read. I feel most at home when 

I am in the outdoors, and when I graduate I’m moving to Boulder, Colorado where I plan to continue running, 

hiking, swimming, and cycling. Also, next year I plan to start a business writing comic-style math, history, and 

philosophy textbooks for middle school and high school students. My favorite philosophical topic is consciousness, 

and some of my favorite memories are of profound discussions I’ve had with my friends. I enjoyed making 

this book because it allowed me to share my love of mathematics with you and with more kids than I could 

ever teach myself. 

Melissa Schwartau: 

I want to teach because so many people regard the study of mathematics dubiously, and I believe I can help 

young people learn math without furthering its negative stereotype. I also want to teach because I sympathize 

with students who do not have a natural affinity for math and want to ease their minds. Math does not have to 

be difficult. It can be fun and rewarding, especially when knowledge of it is gained through exploratory learning 

and discovery. Leading Challenge Math this year and writing lesson plans for parents has been an invaluable 

experience and has shaped my personal teaching philosophy. My plans for after graduation include: working 

in Naknek, AK for the salmon season in June, July, and August; returning to Northfield to student teach 

at NHS in the fall to complete my certification to teach 5-12th grade math in Minnesota; and being Carleton 

College’s 5th-year math intern for the 2008-09 school year. 

My work on this project is dedicated to my brother Ben (1988-2001) and to my parents, Pam and Bill 

Schwartau. I cannot thank them enough for their investment in my education and continuous supply of unconditional 

love and support. Better parents simply do not exist. I would also like to thank two of the women who 

contributed significantly to my love of learning and desire to teach are still teaching at the school I attended 

from 7th through 12th grades: Lori Durant (who taught me 7th grade English) and Lynn Fryberger (who taught 

me Geometry, Algebra II, and Calculus). Finally, I would like to extend a (gigantic) special thank you to Deanna 

Haunsperger, the true originator of this project. Without her, none of this would have been possible. Thanks 

everybody! 

Lily Thiboutot: 

I was raised in Northfield, MN and have attended Prairie Creek Community School, Sibley Elementary, 

Northfield Middle School, Northfield High School, and Carleton College. I have loved math all along the way. 

My favorite part of math has always been when I have those Aha! moments, and that’s why creating lessons that 

facilitate those same moments for 5th graders has been my Challenge Math mission all year. One of my favorite 

college courses in Math was an introductory Combinatorial Theory class, and I ambitiously sought to bring 

its topics to my 5th grade audience this fall. Even though I had spent the majority of the ten-week class focused 

on the sequence of numbers named Catalan Numbers, teaching this sequence back to my 5th grade audience 

made for a totally new wave of Aha! moments. My hope is that as you use this book you will find yourself 

reading these lessons and others, venturing out from what’s familiar and saying Aha! from time to time. 

11

My interest in teaching and working with kids began five years ago when I was an Americorps member in 

Boston. I have since worked with kids in various capacities, including volunteer work at People Serving People 

in downtown Minneapolis and supervising the brand new PLUS afterschool enrichment program for 4th and 

5th graders at Bridgewater and Greenvale Elementaries. With my degree in Math and a concentration in Educational 

Studies, I will enter the real world when I begin as a Service Learning Coordinator at a public K-8 school 

in Minneapolis next year. My long term goal is to be a 5th grade teacher. 

12 

How to Use This Book 

Lesson Headers 

At the beginning of each lesson you’ll notice a colorful header. This header serves many functions, all of 

which are outlined in this section. Following is an example of a typical lesson header. 

Quick Icons Title Space 

Other Information 

Lesson Title 

Approximate lesson length: x days 

Follows lesson # __ 

Precedes lesson # __ 

Fits well with lesson # __ 

Quick Icons (see next section for more information): 

High Energy 

Individual Work 

Group Work 

Difficulty Rating (one, two or three stars) 

Title Space: 

The lesson number followed by the title of the lesson will appear here. 

Unit Header: 

Unit Header 

Unit 

Background Color

The name of this lesson’s unit goes here. 

Other Information: 

Approximate lesson length: Some lessons can be stretched over multiple days, so this field indicates about 

how long the lesson will take. Depending on your group, a lesson may require fewer or more days. 

Follows lesson #_: Many of the lessons are designed to be taught in sequence, so this field indicates the 

lesson number that should directly precede the current one. This field will not be included if the lesson can be 

taught on its own. 

Precedes lesson #_: This field indicates the lesson that should follow the current one. If no lesson follows, 

then this field will not be included 

Fits well with lesson #_: Many of the lessons include similar topics, so this field indicates which lessons fit 

well with the current one but need not be taught directly before or after. 

Background Color: The headers are color-coded to correspond with their unit. 

Quick Icons 

In working with fifth graders, we noticed that there are differences in group dynamics. Some groups are loud 

while others are quiet. Some groups thrive on challenges while others occasionally need an easier lesson to help 

build their confidence. To address these observations, we classified each lesson into a few helpful categories. 

High Energy 

These lessons are great for high-energy groups because they include activities that ask the students move 

around. If your group has trouble sitting still, or enjoys more active lessons, these lessons may be good choices. 

Individual Work and Group Work 

While teaching, we noticed that some groups work well together, while other groups work best as individuals 

who then come together in the end to discuss what they learned. Thus, to help you distinguish which lessons 

may be best for your group, each lesson in this curriculum has either an Individual Work or a Group Work icon. 

Group Work lessons encourage and often require the students to work with each other to solve the problems 

presented. Some Group Work lessons involve the entire group, while others may break the whole group into two 

or three smaller groups. Individual Work lessons provide problems that each student can work on alone. In some 

Individual Work lessons, the students will all work on the same problem, while in others each student will solve 

a unique problem. 

Difficulty Rating 

At the top of every lesson you’ll also see either one, two, or three stars, which indicate the Difficulty Rating 

of the lesson. If a lesson has one star, it is one of the easier lessons in the book. Easy lessons will still require 

the students to think creatively, but they may cover material with which the student has previous experience or 

may require little abstract thought. If a lesson has three stars, it is one of the more difficult lessons in the book. 

Three-starred lessons explore more complex topics with which the students may have no experience. Often it 

will take your students more than one Challenge Math session to fully grasp the concepts in a difficult lesson. 

Lessons with two stars fall somewhere in the middle; they may be easy for some groups and more challenging 

for others. It is also possible that you could tailor the difficulty of a two-starred lesson to make it easier or 

13

harder. Yet despite the stars, know that all the lessons in this book are planned for fifth graders, and thus no lesson 

is ever too hard for the students to understand. The students will simply take more time to understand some 

concepts than others. 

14 

Notes 

At the end of each lesson you will find a blank sheet titled “Notes”. This page is for you. Once you have 

taught a lesson, we invite and encourage you to write down any comments you have about that lesson. These 

comments can include but are not limited to: problems you encountered when teaching the lesson, suggestions 

for others who might teach the lesson, students’ questions that you were unable to answer, questions of your 

own, or anything interesting that came up when you taught the lesson. Don’t hold back! At the end of the year, 

you’ll leave this book with your group’s fifth grade teacher so that future Challenge Math teachers can teach the 

same lessons and benefit from your notes. Ultimately, this belongs to all the students who have learned from it, 

and all the teachers who have taught from it. We want you to leave your mark and join us in the ongoing creation 

of an invaluable tool that volunteers like yourself will use for years to come. 

Supplements 

Some lessons include supplements at the end. These supplements are often crucial for the lesson and 

will help make your teaching more effective. To prepare to teach your lesson, make a photocopy of the 

supplement(s) for each student. Sometimes it can be helpful to have a couple extra copies on hand in case a student 

needs another one. Then, un the body of each lesson, you’ll find prompts for when it might be a good time 

to pass the supplements out to the group. 

Further Reading 

All the information you need to teach any given lesson is supplied in the lesson itself (unless otherwise 

noted), but sometimes it could be helpful to learn more about a topic. For this reason, we provided a Further 

Reading section at the end of each lesson. Theses website will have more information about the subject of the 

lesson and many will include links to related topics. If your students are particularly interested in a certain topic, 

you could use these links to learn more. Also, sometimes the links will have great java applets and animations 

that you could use while teaching the lesson. 

Glossary 

As you read through the lessons, you’ll see that there are some words in bold. We provided a definition of 

all the bolded words in the Glossary in the back of this book so that you can have easy access to the meanings 

of some of the more mathematical terminology used in the lessons. 

IMPORTANT: Read This! 

If you read nothing else in the Intro Section, please read the following key points and keep them in mind as 

you prepare to lead Challenge Math.

1. Read a lesson before you teach it. 

You might even want to read it two or three times. It is likely that you haven’t been exposed to all of the 

ideas in these lessons, so don’t assume that you’ll be able to teach one without first reading through it completely. 

Also, don’t choose not to teach a lesson just because it has a difficulty rating of three stars; if it looks interesting 

to you, read it first and then decide if you think it is too difficult. Some of the lessons that are the most 

fun are the ones rated to be the most difficult. If you can’t bring this book home with you, it may be a good idea 

to make a copy of a lesson you want to teach a week before you teach it and bring it home to read. Remember: 

You want to be as comfortable with the material as possible so that you can lead the best lesson you can. 

2. Encourage exploration and discovery learning. 

Part of the inspiration for this project came from Robert and Ellen Kaplan’s Math Circle. Their approach 

to teaching “is to pose questions and let congenial conversation take over.” (This quotation was taken from 

their website: .) Most of the lessons in this book were written with this philosophy 

in mind. Since discovering a concept for oneself provides a greater potential for retaining knowledge of that 

concept, we believe it is important that you act as the students’ “guide by the side” rather than the “sage on the 

stage” whenever possible. 

3. You won’t know all the answers… 

…and that’s a good thing! It is important for math students to know that there isn’t always an answer within 

immediate reach. When a student asks you a question to which you don’t know the answer, admit it and then 

ask them what they think the answer might be. How might they go about finding an answer? Are they sure that 

there is an answer? Above all, don’t get flustered. 

4. Have fun and be positive. 

Challenge Math provides students with the chance to see how much fun math can be. If you enjoy yourself 

while you are teaching, your students will notice your enthusiasm and as a result will have a more rewarding 

Challenge Math experience. 

15

16 

Geometry: A Shapely Approach to Math

Lesson #1: Introduction to Triangles 

Approximate lesson length: 2-3 days 

Lesson Objectives: 

• Explore the triangle inequality 

• Understand why the area of a triangle = 1/2 × base × height 

Materials: 

• Paper and pencil 

• Graph paper 

• Ruler 

• Scissors 

Part 1: What is a Triangle? 

Geometry: A Shapely Approach to Math 

Let’s start with the most basic question regarding triangles: What is a triangle? Ask your students this question 

and let them brainstorm different definitions. Encourage them to be specific with their definitions. For example, 

if a students says that a triangle has three sides, draw three lines that don’t connect and ask the student 

why this is not a triangle. One possible definition of a triangle is that it is a three-sided figure with each line 

crossing the other two lines at two different points. 

Now ask your students to each draw a triangle, and then to look at the lengths of the sides of their triangles. 

What do they notice if they compare the length of one side compared to the sum of the lengths of the other two 

sides? Ask your students to draw a triangle in which the sum of the lengths of two of the sides is less than the 

length of the third side. (This is impossible.) Your students will find that the third side is simply too long to ever 

connect with the other two sides. Encourage them to use a ruler so they know they are measuring exactly. For 

example, below the student would begin with drawing a side of length 3 connected to a side of length 4. Then, 

their task would be to draw a third side of length greater than 7. But as we can see in the illustration, the side 

drawn of length 7 is too long to make a triangle. 

Next ask your students if it’s possible to draw a triangle such that the sum of the lengths of two sides equals 

the length of the third side. Ask your students to draw a picture of what a triangle like this would look like. Following 

is an example of a picture your students might draw. There is only one line in this “triangle.” Note that 

17


the first two sides drawn of lengths 3 and 4 form a straight line, which has length 3 + 4 = 7 and is therefore the 

third side. 

So the drawing above isn’t really a triangle, since it has only one side and no area. Now ask your group if 

it’s possible to draw a triangle in which any side is less than the sum of the other two. (This is always possible.) 

Traditionally, this is written as c ≤ a + b, where a, b, and c each represent the length of a side of a triangle. 

Another way you could explain this to your students is the following: Imagine you’re walking down the 

road, but are in a hurry. If you take the dirt path across the grass (and cut off the corner), then the distance you’ll 

walk will be shorter than the distance you would walk if you continued walking on the road and then turned 

right at the intersection. 

Part 2: Area of a Triangle 

Begin by asking your students to define area. What is area? It’s a measurement of how big or small a surface 

is. Ask your students if they know what the area of a rectangle is. They will most likely know to calculate 

this area by multiplying the base by the height. Now ask them how they would calculate the area of a triangle. 

(The formula for the area of a triangle is the length of the base of the triangle multiplied by the height of the 

triangle, all divided by two.) Now let’s figure out why this is! 

First, start by looking at a triangle. Just as we need to know the length and width of a rectangle to determine 

its area, we need to know some dimensions of a triangle to determine its area. The base of the triangle is which- 

18


ever side you draw on the bottom of the paper. Thus, if you rotate the same triangle, each of the three sides 

could be the base, depending on how it’s oriented. Now ask your students what the height of the triangle is. The 

height is the distance from the point opposite the base to the base. Thus, the height of a triangle may change 

depending on which side is the base. Ask your students to draw three of the same triangle, and then mark the 

height when the base is each of the three sides. In the diagrams below, the height is represented by the dashed 

line. Notice that the dashed line intersects the base at a right angle. As illustrated by the drawing on the right, 

you may need to extend the base with a dashed line in order to construct the height. Just remember that the 

length of the base is still the length your original line segment. 

Now that we know the base and the height of the triangle, how do we use these two numbers to calculate the 

area of the triangle? The formula, as your students may know, is base multiplied by height, divided by two. But 

how can we see this in a picture? To give your students a hint, ask them if they can turn the triangle into a shape 

that they know the area of, like a rectangle. 

There are a couple different ways to turn a triangle into a rectangle. Let’s say we begin with the triangle created 

by going from point A to B to C. Then draw the height of the triangle, which is the line from B to D, and 

two lines parallel to line BD coming up from points A and C. Finally, connect these two lines by making another 

new line EF, which is parallel to line AC. Our original triangle ABC now consists of two triangles: ABD 

and CBD. 

First let’s look at triangle ABD. We see that this triangle is half of the rectangle AEBD. We know the area of 

the rectangle AEBD is base times height, which is AD × EA. And since triangle ABD is half that rectangle, its 

area is the length of AD (its base) multiplied by the length of BD (its height) divided by two! Now we can do 

the same thing for the rectangle BFDC. In conclusion, we can see that the area of triangle ABC is half the area 

of rectangle AEFC, and thus we have shown how to calculate the area of a triangle. 

19


Ask your students to calculate the area of the same triangle using each of the three sides as a base. Should 

all the answers be the same? (Yes, they should, since a given triangle has the same area no matter how you rotate 

it on the page.) 

Now there is also a second way to show that the area of a triangle is base times height divided by two. 

Again, begin with triangle ABC. First make it into parallelogram ABDC, as shown in the diagram below. Note 

that triangle ABC is identical to triangle DCB. Thus we now have a shape ABDC that has twice the area of our 

original triangle. So how do we calculate the area of a parallelogram? It turns out that it has the same forumla 

as the area of a rectangle: base times height. To see this, you can create a rectangle BDEF and see that triangle 

ABF is the same as triangle CDE. 

20 

It may be helpful to ask your students to cut along some or all of the lines in the last two figures and then re- 

Further Reading: 

• http://en.wikipedia.org/wiki/Triangle_inequality 

• http://en.wikipedia.org/wiki/Triangle

Notes: 


21


22 

Lesson #2: Angle Sums of a Triangle: The Magic Number 



• Know how to draw and measure an angle 

• State a hypothesis on the sum of the three interior angles of a triangle 

• Test and revise their hypothesis based on triangles drawn and measured during Challenge Math 

Materials: 


• One white board and marker 

• Protractor 

Begin by drawing an angle on the white board and asking the students to talk about what they see. 

Wait for them to name it, so don’t use the word angle yourself. When they identify it, urge them to 

find other vocabulary they might have heard by asking: What can we do with angles? Can we reproduce 

them? Can we measure them? (Yes! We learned that in class, you use a...what’s it called? A protractor!) 

If no one volunteers the words obtuse and acute, put the protractor on the angle you drew and comment 

on how the drawn line is either bigger than or smaller than the perpendicular line that marks 90 degrees 

on the protractor. What words are there for this? Finally, what’s the measurement of this angle? Read the 

protractor as a group and make sure everyone agrees on the measurement. 

At this point there are two directions this lesson could follow. If your students have no trouble reading 

the protractor, skip this paragraph and continue with the next. My students didn’t know how to use a protractor. 

They had been introduced to them in class and asked to use them, but had a considerable amount 

of disagreement about the correct way to measure an angle. If you have a group of students at a similar 

stage it would be well worth it to spend a full lesson exploring the use of protractors by posing questions 

such as the following:

• 

• 

• 

• 


What part of the protractor would you expect to use to measure an obtuse angle? And an acute one? 

(The left side? The right side? These answers will differ depending on which direction the angle is 

facing.) 

What does it mean for one angle to be ‘bigger’ than another? (Its measurement, in degrees, is larger.) 

Does an angle measurement tell us anything about the length of the lines which form the angle? 

(No!) 

Draw two angles with the same measurement (as best you can!) but make one with lines only an inch 

long, and the other with the lines extended to seven or eight inches. Are these two angles different? 

Now you’re ready to explore how much the degrees in a triangle add up to. Before doing any measuring, 

ask students to comment on how much the three angles of a triangle add up to. (More than 90! Isn’t it 360? 

Wait, if they were all obtuse angles than it would be have to be bigger than 3 x 90 because there are three 

angles in a triangle. That’s crazy, how can you make a triangle out of three obtuse angles? Can you make 

one with three acute angles?) Keep in mind that we want students to learn through discovery, so be sure 

never to say the answer is 180 degrees. Maybe suggest that you bet there’s one magic number all triangles 

have in common. Draw five or six triangles on the white board and ask the group: Don’t you think if you 

add up the angles in each triangle they will all be about the same? 

Draw a triangle on the white board and tell students the goal is to find that magic number. Ask for suggestions 

on how to find it. Continue with questions or suggestions until students articulate that we could do 

a bunch of examples and collect data. They should each make a table on their own paper to record the angle 

sum measurements of a number of triangles. Draw a bunch of different triangles, one at a time, on the white 

board. Let each student have a turn at using the protractor to measure the angles inside the triangle, write 

them down and add them up. Along the way, look for opportunities to guide students toward articulating 

facts they observe about triangles and angles and, if and when they make these statements, validate them and 

write them down. For example, one of my students exclaimed, “It’s impossible for a triangle to have three 

obtuse angles!” while his classmate noticed early on that “an angle that’s 180 degrees looks like a straight 

line.” 

When there are about ten minutes left stop the data collection. Now your group’s job is to find the magic 

number, if there is one. Is the data similar? What do you notice about it? (All the numbers are bigger 

than 100! This one doesn’t really fit because it’s 155 and all the others are between 170 and 190!) Discuss 

and debate what the magic number might be. If students are frustrated by the fact that their data includes 

a bunch of slightly different numbers and only a couple that are the same, you may want to talk about the 

human error involved in measuring angles. Convince your students that it would be impossible to measure 

each angle with complete accuracy. You can do so by referencing a time during the lesson that two students 

looked at the same angle with the same protractor and disagreed about what number to use as the measurement. 

When time is up, if students haven’t come to consensus about the magic number, tell them that they 

should ask everyone they know between now and the next lesson. They can also do their own research online 

or in an older sibling’s math book to confirm or deny their hypothesis. The magic n umber is 180; every 

triangle has three angles which add up to 180 degrees. If the students discover this during the lesson and are 

confident of it by the end, tell them you bet they’re right and the best way to check is to ask everyone they 

know and see if anyone else thinks it’s 180. 

23


24 


• www.mathsteacher.com.au/year7/ ch09_polygons/02_anglesum/sum.htm 

• www.algebralab.org/lessons/lesson. aspx?file=Geometry_AnglesSumPolygons.xml 

• http://www.amblesideprimary.com/ambleweb/mentalmaths/angleshapes.html (How to use a pro- 

tractor - an interactive website.) 

• www.mathsisfun.com/geometry/protractor-using.html (Another protractor how-to website.)

Notes: 


25


26 

Lesson #3: Congruent Triangles 

Approximate lesson length: 1 day 

Follows lesson #2 

Fits well with lesson #4 


• Deduce which and how much information is needed about a triangle in order to construct an exact 

replicate (ie: lengths of the sides, measures of the angles) 

• Learn that SAS (side-angle-side), ASA (angle-side-angle), and SSS (side-side-side) work 

• Understand why AAA (angle-angle-angle) does not work 

Materials: 

• Writing utensils 

• Unlined paper 

• Rulers 

• Protractors 

• Copies of Pre-made Triangles (Supplement 1) 

• Copies of Information Slips (Supplement 2) 

• Scissors 

It is important that everybody has a good understanding of how to refer to the sides and angles of a 

particular triangle. Explain that we can designate the sides of a triangle as A, B, and C, with the angles as 

a, b, and c. Note that an angle and the side directly across from it are designated by the same letter (sides 

in upper case and angles in lower case). 

b 

C 

A 

When you hand out triangles, have the students write down the side and angle names on the triangle 

itself as shown above (a, b, c for the angles and A, B, C for the sides directly across from each respective 

angle). Then pose these questions, but do not give away the answers, because these are the two questions 

around which this lesson is centered: 

• What is the fewest number of sides and/or angles that I need to know in order to construct a triangle 

congruent to yours? (You only need a total of three pieces of information!) 

• Does it matter which sides/angles you tell me? (Yes!) 

a 

B 

c


The following activity is aimed at helping the students discover these answers through trial and error. 

First, have the group cut out the pre-made triangles and Information Slips. To begin the activity, have students 

pair off. Then have one student in each pair pick a triangle while the other takes an Information Slip. 

The student with the Information Slip should then ask his or her partner for pieces of information. 

* These can be any combination of angles and sides, but be specific. For example: “Give me 

angle b, angle c, side A, and side C,” rather than “Give me two angles and two sidese.” You may 

want to emphasize the importance of keeping track of the angle and side names. (We named them 

for a reason!) 

The student with the triangle should then take the appropriate measurements and tell his or her partner, 

who can then write them down on the Information Slip. Now the task is for the student with the Information 

Slip to try to create (draw) a congruent triangle with the specified information. Remember: sometimes this 

will be possible and sometimes it will not. It all depends on how much and which pieces of information a 

student has about a triangle. It may be worthwhile to have both students in each pair take a triangle and an 

Information Slip. This way, both can be trying to construct each other’s triangle at the same time, and the 

group will have twice as many trials. 

The following discussion questions may be appropriate to ask during the activity, or they may be more 

appropriate for after the activity. Use your judgment. The goal of asking these questions is to guide discovery 

and help students summarize their findings. Here are the questions: 

• Is it necessary to have all six pieces of information (i.e.: all three angle and all three side measure- 

ments) in order to construct a congruent triangle? 

* Why do you think this is? 

• Was anybody able to do it with five pieces of information? Four? Three? Two? 

* Why do you think this is? 

• Of the people who tried using three pieces of information, were you always able to successfully 

construct a congruent triangle? (Hopefully someone answers no!) 

* What happens if the three pieces of information are all angle measures? If nobody has 

tried this, have the group try it and see what happens. (It is impossible to construct a congru- 

ent triangle knowing only the angle measures. These triangles will have the same shape but 

not necessarily be the same size. These are called similar triangles. There are pic- 

tures of examples of these at the end of the lesson.) 

* What happens if the three pieces of information are all side lengths? Again, if nobody has 

tried this, do so. (This, perhaps surprisingly, will produce a congruent triangle! Given three 

lengths, there is only one way for them to fit together and make a triangle. Cool!) 

• What combinations of three pieces of information do not work? 

* For a specific combination that doesn’t work, have one student who tried this combination 

explain in his or her own words why this is. 

• What combinations of pieces of information do work? (Below are some definitions that help us talk 

about these combinations. Refer to the pictures at the end of the lesson for examples of each.) 

* S-A-S stands for “side-angle-side” and means that if you tell me the lengths of two sides 

and the measure of the angle between them, I can construct a triangle congruent to yours. 

* A-S-A stands for “angle-side-angle” and means that the measures of two angles and the 

length of the side in between them is also sufficient information. 

* S-S-S, or “side-side-side” also works. That is, if you tell me the lengths of all three sides 

(but don’t tell me any of the angle measures!), I can construct a congruent triangle. 

• Why was it important to name the sides and the angles? 

27


To conclude, ask the questions from the beginning of the lesson again. (What is the fewest number of 

sides and/or angles that I need to know in order to construct a triangle congruent to yours? Does it matter 

which sides/angles you tell me?) The group should be able to answer these questions at this point. One of 

the most important points to make is that it does matter which sides and which angles you know the measurements 

of. For example, S-A-S requires that the angle you know be the angle between the two sides that you 

know. (Similar for A-S-A.) Ask a couple of students to explain their answers to these questions in their own 

words. 

Here are the examples of similar triangles to provide a visual for understanding why A-A-A (“angle-angle-angle”) 

does not work: 


• http://nlvm.usu.edu/en/nav/frames_asid_165_g_2_t_3.html?open=instructions 

• http://www.amblesideprimary.com/ambleweb/mentalmaths/angleshapes.html (This page has multiple 

activities that require the use of a protractor that you control with your mouse. If you’ve forgotten 

how to use a protractor, go to this site and practice at home or at the school for a few minutes before 

teaching this lesson.) 

28 

40° 

80° 

60° 

40° 

80° 

60° 

40° 

80° 

60°

Supplement 1: Pre-made Triangles 


29

Geometry: A Shapely Approach to Math Supplement 2: Information Slips 

30 

Information Slip 

Angle a = _____ Side A = _____ 

Angle b = _____ Side B = _____ 

Angle c = _____ Side C = _____ 


Angle a = _____ Side A = _____ 

Angle b = _____ Side B = _____ 

Angle c = _____ Side C = _____ 


Angle a = _____ Side A = _____ 

Angle b = _____ Side B = _____ 

Angle c = _____ Side C = _____ 


Angle a = _____ Side A = _____ 

Angle b = _____ Side B = _____ 

Angle c = _____ Side C = _____ 


Angle a = _____ Side A = _____ 

Angle b = _____ Side B = _____ 

Angle c = _____ Side C = _____ 


Angle a = _____ Side A = _____ 

Angle b = _____ Side B = _____ 

Angle c = _____ Side C = _____

Notes: 


31


32 

Lesson #4: The Pythagorean Theorem 


Precedes lesson #6 



• Discover the Pythagorean Theorem for right triangles (a 2 + b 2 = c 2 ) 

Materials: 


• Rulers 

• Scissors 

c 

b 

The Pythagorean Theorem is one of the most important principles of geometry. It sates that, for any 

right triangle (one with a 90 o angle in it), the squares of the two smaller sides (a 2 + b 2 ), when added together 

will always be equal to the square of the larger side (c 2 ). Before starting the lesson, make sure the 

students understand why triangles are so important to the study of geometry (any shape that is made up of 

straight sides can be represented as a collection of triangles). 

Furthermore, let them know that any triangle that doesn’t have a right angle in it can be broken down 

into two triangles that do. 

With this in mind, they should be able to see why it’s important to know things about right triangles. 

The idea of this lesson is to have each student draw out a triangle with a right angle in it, cut out 

squares to represent a 2 , b 2 , and c 2 , and then cut a 2 and b 2 up into smaller parts so that they fit into c 2 . They 

a


should start by drawing a right triangle (making sure that the long side is less than 8 inches). 


• http://en.wikipedia.org/wiki/Pythagorean_theorem 

c 

b 

Next the students should measure the length of each side and cut out three squares, one with side length 

a, one with side length b, and one with side length c. Finally, they should cut the two smaller squares up 

so that they fit into the larger one. (This might get a bit messy if students find that they need to cut their 

squares into very small pieces.) This should work, or come close to it - depending on accuracy, every time! 

From this exercise they should be able to see that a 2 + b 2 = c 2 

If there’s time left over, ask the group to think about triangles that don’t have right angles in them. They 

can try doing the same experiment using non-right triangles and find that it doesn’t work. 

a 

33


34 

Notes:

Lesson #5: Pythagorean Triples 



Fits well with lesson #3 and #4 


• Discover the Pythagorean Theorem: a 2 + b 2 = c 2 

• Learn about and be able to describe Pythagorean Triples 

Materials: 


• Rulers 

• Calculators 

• Sticks/straws 

• Scissors 


Have students pick their own two numbers to represent a and b. These numbers should be somewhere 

between 5 and 15. You should also pick two numbers. Then show them this equation: 

a 2 + b 2 = c 2 

Next (and this is the hardest part), show the students how to compute c. This may require a discussion 

about squares, square roots, and the “Golden Rule of Algebra” (whatever you do to one side of an equation 

you must do to the other). If this seems beyond the scope of understanding, it would be reasonable to sim- 

ply state as a given that . However you get there, you want to eventually show the group how 

to compute using your values of a and b as an example. Depending on the type of calculator, 

you may have to do this calculation in parts. In other words, you may have to compute a 2 , then clear the 

screen and compute b 2 , then clear the screen again and add those two numbers, and finally take the square 

root of that sum. Refer to the end of the lesson for some examples of this computation. 

Once everybody has computed their c, have students write down their values for a, b, and c so they 

don’t forget them. (You may want to emphasize that it is important not to mix up the numbers.) Then have 

the students measure and cut straws or sticks to the lengths of these values (in centimeters). It might be 

good to have them somehow mark which straws are a, b, and c (remember, we don’t want to mix up which 

numbers are which). Finally, have them put the sticks/straws together to form a triangle. (If you’ve done 

the Lesson #3: Congruent Triangles, you might recall that there is only one way to make a triangle if you 

know the lengths of all three sides.) 

Lay out all of the triangles where everybody can see them. Pose the question: What do you notice 

about all of the triangles? (They all have right angles!) Is there any pattern as to where a, b, and c are with 

respect to the right angle? (a and b are the sides that form the right angle, and c is always the longest side.) 

35


Try this again, using a = 3 and b = 4, or with a = 5 and b = 12, or any multiple of these (ex: a = 6 and b = 

8). What do you notice when you compute c? (It’s an integer!) These sets of three integers that represent the 

lengths of the sides of a right triangle are called Pythagorean Triples (for example [a = 3, b = 4, c = 5] and [a = 

5, b = 12, c = 13] and [a = 8, b = 15, c = 17]). You might want to give students time to try these out to convince 

themselves. There are relatively very few of these (only 16 with c < 100), which is pretty cool! 

If you get through the lesson quickly and have some time to spare, you might ask this question: Why are all 

multiples of Pythagorean Triples also Pythagorean Triples? (Hint: If you’ve done Lesson #3: Congruent Triangles, 

think about similar triangles.) 

Another activity you may want to do as part of the lesson is the following. Again, have students choose an 

a and b. Then, before computing c, have the students draw two lines that are the lengths of their a and b and 

that meet at a 90° angle. A good way to do this might be to use the corner of a piece of paper as a template for 

the 90° angle. Once this is done, each student should have something that looks like this (if a = 3 and b = 4, for 

example): 

36 

3 

4 

Then, have each student connect the two lines so as to form a triangle. Measure this new line and write 

down this value. Call it x. Now, using a calculator if necessary, compute c using the formula . 

What do you find? What, in fact, does everyone find? (That x = c.) Below is the continuation of my example 

above: 

3 

4 

x = 5 

Then I computed c using our equation and 

found out that c also equals 5. So x = c. If 

everyone in the group measures accurately 

and computes c correctly, they should also 

find that their x and c are equal. 

Using a dotted line, I connected 

the endpoints to form 

a triangle. Then I measured 

my new (dotted) line and 

found out that its length is 5. 

So I say x = 5.

Here are some more examples of how to compute c, given a and b: 


• http://en.wikipedia.org/wiki/Pythagorean_triple 


37


38 

Notes:

Lesson #6: Pythagorean Proofs 

Approximate lesson length: 2 days 

Follows lesson #4 or #5 

Fits well with lesson #4 or #5 


• Prove the Pythagorean Theorem for 3-4-5 triangles 

• Prove the Pythagorean Theorem for all right triangles 

Materials: 

• Pencil and paper 

• Scissors 

Part 1: 3-4-5 Triangles 


Now that your students understand the Pythagorean Theorem, the object of this lesson is to prove 

that it is true. First, ask the group how to define a proof. One possible way they could define a proof is to 

explain that it is a logical set of arguments that show how something is absolutely true. So to prove the 

Pythagorean Theorem, means to show that it is always true, no matter what. 

First, we’ll begin with a proof that the Pythagorean Theorem is true for triangles with side lengths 3, 4, 

and 5. We can see this is true because 3 2 + 4 2 = 5 2 , but here we are going to prove that with an illustration. 

Begin by presenting the students with the illustration below. Then, even though they already know that if a 

right triangle has sides of length 3 and 4, the third side must be 5, ask them if they can use the illustration 

to show that the length of this side is absolutely 5. 

I will now outline a proof using the above illustration. (It should be noted that the characters in the illustration 

above have nothing to do with this lesson and are not necessary for understanding this proof. We 

39


borrowed this image from the Web, and the students thought they were cool so we kept them.) The students 

may not approach the proof in the way that I will, but encourage their ideas. Also, if they suggest a proof 

that you immediately see won’t work, you could ask them to continue on with their logic, and eventually 

they should come to a contradiction. 

So the proof: Begin by looking at each of the four smaller rectangles, one of which is highlighted in red 

below. This red rectangle has an area of 3 × 4 = 12. Thus, each of the two triangles created by the diagonal 

through the highlighted rectangle has area 12 / 2 = 6, since this diagonal divides the area of the red rectangle 

in half. 

Now notice the four central triangles, one of which is highlighted in blue in the following diagram. Each 

of those triangles has an area of 6, as we just showed, and there are 4 of them, so the total area of those triangles 

is 6 × 4 = 24. 

Newt let’s consider the large square that is tilted sideways, highlighted in green below. Since the four 

triangles that make up this square have a total area of 24, and since the single square highlighted in yellow in 

middle has an area of 1, the total area of this green square is 25. 

40


Since we know it is a square, each of its sides must be 5, since 5 × 5 = 25. Thus, if we look at one of the 

triangles highlighted in blue earlier, we see that one of its sides is 3, the other 4, and we have just shown that the 

third side must have length 5! 

Part 2: Other Right Triangles 

We begin with the right triangle below, with sides a, b, and c. We give these sides names with letters since 

we don’t know how long they are. Thus, to prove the Pythagorean Theorem, we need to show that a 2 + b 2 = c 2 . A 

proof is outlined below. 

For this proof, we can use a similar illustration as before to show that, for all right triangles the sum of the 

squares of the two legs (a 2 + b 2 ) equals the square of the hypotenuse (c 2 ). 

41


From this diagram, we know that the area of the large, tilted square is c 2 . Next we will move around two of 

the four triangles in this illustration to prove our theorem. It may be helpful for your students to cut out the two 

triangles with arrows in them and the small square in the center because the next step is to shift each of these 

triangles down to their new locations, outlined with dotted lines. 

Now if we look at this same figure, we can see that the area of the red square is a^2 and the area of the blue 

square is b^2. We already know that the area of the largest square with side length c is c^2. And the red square 

and blue square include exactly all four triangles from the original illustration, and the small square in the center, 

which is exactly what the large square of area c^2 includes: four triangles and the small square. Therefore 

their areas are equal, so we see that c 2 = a 2 + b 2 , and we have proven the Pythagorean Theorem! 

42


To conclude the lesson, ask your students to recall their earlier definition of a proof. Have the activities they 

just completed changed this definition? If so, then how? If they had to explain a proof to someone who knows 

nothing about mathematics, what would they say? 


• http://en.wikipedia.org/wiki/Pythagorean_theorem 

• http://www.cut-the-knot.org/pythagoras/index.shtml 

43


Notes: 

44


• Learn how to refer to different parts of a circle 

• Discover that pi (π) is the same for all circles 

Materials: 


• Ruler and yardstick 

• String 

• Calculator 

• Compass (to draw circles with) 

• Scissors (for cutting string) 

Lesson #7: Circles and Pi (π) 



Begin by asking each student to draw a circle with their compass. Now have them look at their circles. Ask 

them what different ways they can think of to measure their circle? Let them explore all the ideas they think of; 

there are no wrong answers here. 

Here are some examples of ways to measure circles that they might suggest: 

Measure the length of the perimeter of their circle - 

• This is called the circumference of a circle. In 

mathematics it is denoted by the letter c. For example, if a circle has a circumference of 6, that is written 

as c = 6. 

45


46 

• Measure the distance from the center of the circle to a point on the edge - Ask the students if this distance 

is the same no matter where on the circle you draw the line to. (It is.) This distance is the radius 

of the circle and is denoted by the letter r. So if the radius is 4 you’d write r = 4. Ask the students if they 

can think of a way to describe the center of the circle in relation to all the points on the circle. One way 

is to say that the center of a circle is defined as the point equidistant from all points on the circle. 

Measure the distance across the circle from one point on the circle to another point on the circle - 

• Any 

line connecting two points on a circle is called a chord. Different chords can have different lengths in 

the same circle. A chord that passes through the center of a circle is called the diameter of the circle. 

The diameter is denoted by the letter d. What relationship does the diameter have to the radius? (The 

diameter is twice the radius, but help the students figure this out for themselves.) And how can we represent 

this relationship using r and d? We can write d = 2 × r. To help the students see this, they can write 

it out. For example, let’s say that d = 10 and r = 5. Thus 10 = 2 × 5.


Also, note that the diameter is the longest chord in a circle. Thus the longest line connecting two points on a 

circle always goes through the center of the circle! 

At this point, ask the kids if it is possible to draw a straight line that only touches a circle at one point. (Yes, 

it is possible!) This line is called a tangent line to a circle. 

Now ask your students if it is possible to draw a line through three points on the circle? (This is imossible 

if the line is straight, but if it’s curved it can cross the circle at infinitely many points!) Have your kids explain 

their responses. 

Now, to solidify these ideas, have the kids draw a few circles of different radii with a compass. Then, for 

each circle, ask them to measure the radius, diameter, and circumference. For each circle, they should write r 

=, d =, and c = and then fill in the blanks after each letter. Encourage them to measure as carefully as possible. 

They can use a string to measure the circumference by laying the string around the circle on the paper and then 

measuring the length of the string with a ruler. This will teach them that each circle has a different radius, diameter, 

and circumference, and thus that the letters r, d, and c stand for different numbers when they correspond 

with different circles. Also ask them to draw a tangent line for each circle. 

Ask the students if they can draw two cirlces, both with a radius of 3 but with two different circumferences? 

This is impossible, as they will figure out. It is also impossible to draw two circles with the same circumference 

but different radii. This is because, as the radius of a circle increases, so does its circumference, and it turns out 

that the circumference and radius have a special relationship. When you divide the circumference of some circle 

by the radius of the same circle, you will always get the same answer no matter how big or how small your 

circle! The answer you get is pi (pronounced like apple pie). The mathematical symbol used for pi is π, and its 

numerical value to five decimal places is 3.14159. Thus, using only symbols and letters, we can write π = c / d. 

Before you explain all this to your students, help them to discover it on their own! For each circle that they 

drew earlier, ask them to calculate (with a calculator) the value of the circumference divided by the diameter. 

When they are done, they should compare the values they have calculated. Then, they can also compare their 

values with the other students in the group. Although their values will not be perfectly precise due to inevitably 

imperfect measuring, all their values should be close to 3.14. 

47


Now explain to them that the circumference divided by the radius is the same for all circles, and that this 

value is called π. You might have them explain this to each other in their own words, because it can be a challenge 

to understand. 

With this knowledge, give them this puzzle: The radius of a circle is 10. What is the circumference of the 

circle? If they need a hint, remind them of the formula π = c / d. Thus, since the diameter equals 10 and π equals 

about 3.14, we know two of the three parts of the formula. We know that 3.14 = c / 10. So what does c equal? 

That is, what number, divided by 10 equals 3.14? Another way to write the formula π = c / d is c = π × d. Thus 

we can write: c = π × 10 = 31.4. 

What about if the radius equals 12? Or 5? And what happens if you know the length of the diameter, but 

don’t know the circumference? Well, since the formula π = c / d can also be written as π × d = c (as we saw earlier) 

and as d = c / π, we only need to know either the circumference or the diameter to figure out the other. 

For a challenge, you could pose this question to your students: What value must the diameter have if the circumference 

is pi? The answer is 1. Why? Because, if we let c = π, then we can replace c with π in the equation d 

= c / π. Thus d = π / π = 1. 

To enforce this knowledge take a walk around the school with the group in search of circular objects. When 

you find one measure its circumference. Then, using the formula the students have just learned, calculate the 

value of its diameter. Then, actually measure the length of the diameter and compare this length with the value 

you calculated with the formula. The two values should be almost the same! You could also measure the diameters 

of objects you find, then calculate their circumferences with the formula, and then actually measure the circumference 

and compare your results. Although to do this, you may need a very long piece of string, depending 

on the size of the circular objects you find around school. 

A neat property of π that you might want to mention is that its digits go on forever and never repeat themselves. 

The website lists the first million digits of π. The students 

may really enjoy seeing the digits of π in print, so you could print of a few pages and give each kid a page 

to take home. 


• http://en.wikipedia.org/wiki/Circle 

• http://en.wikipedia.org/wiki/Pi 

48

Notes: 


49

50 

Probability: What are the Chances?!

Lesson #8: Combinations 




Probability: What are the Chances?! 


• Define event and outcomes 

• Discover a way to count the total possible combinations of rolling a die twice or three times 

• When rolling two dice, find the probability of rolling a 5 and then a 6 

Materials: 


Begin by asking each student to write down in their own words the definition of probability. Ask everyone 

to share and discuss their responses. Pose the question: What’s the probability of rolling a six when 

you roll 1 die? And a four? Two? One? (It’s 1 out of 6!) Why? (Because there’s six possible numbers 

and they’re all different!) Continue the discussion until you find students are making this comment confidently 

and understand what the probability of rolling a given number 1-6 is and why. At this point tell 

them that they already know the strategy for calculating any probability: Find out how many total possible 

outcomes there are, then count how many of the possible outcomes count as the event. 

Write this strategy on your own white board or paper as you talk it through with them. Ask the students 

to describe in their own words what they think ‘outcomes’ means and what ‘event’ means. They 

should be able to realize that in this case there are 6 total possible outcomes: rolling a 1, rolling a 2, rolling 

a 3, rolling a 4, rolling a 5, and rolling a 6. The event “rolling a six” only happens in one of the outcomes. 

Following the strategy, the probability of rolling a six is 1/6. 

Once the group is comfortable conceptualizing how we calculate the probability of rolling any single 

given number, pose this more difficult question: What is the probability of rolling a 5 on the first roll and 

then a 6 on the second roll? Open up a discussion about how we might go about figuring this out. 

To figure this out, we need to know how many possible combinations there are of rolling a die twice. 

Let the ordered pairs (A,B) represent rolling A first and B second. So for example (1,1) means I rolled a 1 

first and a 1 second. (2,4) means I rolled a 2 first and a 4 second. Students may come up with their own 

notation for this concept, but if they don’t, suggesting this method may be helpful to them. 

Ask the students to write down all the possible ways to roll a die twice. When I taught this lesson, every 

student began by writing down lots of random combinations. This led me to ask the question: Is there 

any way we can write down all of these combinations so that we can keep track of which ones we’ve accounted 

for and which ones we still need? Ultimately, making a table like the following is what emerged: 

51


After everybody makes their own table, pose the question again: What is the probability of rolling a 5 

and then a 6? It should now be easy to see that out of the 36 possible combinations, there is only one way to 

roll (5,6) so the answer is 1/36. 

Now pose the question: If you roll a die three times, what is the probability of rolling (2,2,2)? How do 

we begin to figure this out? We need to know how many possible ways there are to roll a die three times. Is 

there some way we can use what we know about the possible ways to roll two dice to figure out how many 

ways there are to roll three dice? Yes! There is! If this isn’t immediately obvious to the students (which it 

probably won’t be), point out that every time you roll three dice, you must roll two dice first. We know all 

of the ways to roll two dice. Therefore, we know all of the ways to roll the first two out of three dice. If we 

wanted to make a table like before, what would it look like? 

The basic idea here is that each of the ordered pairs above has six possibilities for the third roll. For 

example, if you first roll (1,1) then all of the possible outcomes after the third roll are: 

52 

(1,1,1) (1,1,2) (1,1,3) (1,1,4) (1,1,5) (1,1,6) 

Therefore, there are (36 ways to roll the first two dice) × (6 ways to roll the third die) = 36 × 6 = 216 

possible ways to roll three dice. Therefore, the probability of rolling (2,2,2) is 1/216. 

If it seems like students are understanding what is going on, ask them what the probability of rolling the 

same number all three times is. One thing we will need to know is how many ways are there to roll the same 

number three times. There are six ways to do this: (1,1,1), (2,2,2), . . . , (6,6,6). Therefore, the probability of 

this happening is 6/216. 

This might be easier for the students to see in the case of rolling just two dice, since they have a table 

with all the possibilities written out. This way, they can circle each way to roll the same number twice: 

(1,1), (2,2), (3,3), (4,4), (5,5), & (6,6). Therefore there are 6 ways to roll double-digits out of 36 total possibilities, 

so the probability is 6/36 = 1/6. Now try applying this idea to the case of rolling three dice. 


• http://www.studyzone.org/testprep/math4/d/combinationl.cfm

Notes: 


53


54 

Lesson #9: Dice Probabilities I 


Follows lesson #8 



• Define probability in their own words 

• Discover that the probability of one event OR another event means to add the probability of each 

together 

• Discover that the probability of one event AND another event means to multiply both of the indi- 

vidual probabilities 

Materials: 

• Two dice for each student, preferably in two different colors 

Recalling Lesson #8: Combinations, pose the question: What’s the probability of rolling doubles when 

you roll two dice? Work to find this probability using the same strategy as before, asking each student to 

write down all the possible outcomes when you roll two dice. Encourage each student to develop a chart 

or table on their paper/white board where they can write down all the possible outcomes. Here is an example: 

Ask students to compare tables/charts with their neighbor when they’ve finished to make sure they’ve 

thought of all the possibilities. There are 36 possible outcomes when you roll two dice, and once the students 

have them all written down, it should be easy as a group to count the number of doubles there are in 

the chart/table and see that the probability of rolling doubles is 6/36. For an extra challenge, ask the students 

to reduce this fraction and they’ll see that this probability is 1/6, the same as the probability of rolling 

a six when you roll just one die.


Now that you’re working with the list/table/chart of 36 outcomes, you can explore the probabilities of 

other events such as getting a sum of five on the two dice, rolling at least one 2 (in other words, rolling one 

2 OR two 2s), or getting a sum of 5 on the two dice AND rolling a two. Give one example of an event and 

find the probability of that event as a group. Then let each student think of an event on their own and challenge 

the group to compute the probabilities. While students are making up their own events, you should 

pay close attention to whether they are using AND or OR in their event. Some may not be as obvious, as in 

the case above where the “OR” event is described as “rolling at least one 2.” 

Here is an illustration of the outcomes that apply to the events from above: 

Rolling at least one 2 (one 2 OR two 2s): 11/3 Rolling a sum of 5 AND rolling a 2: 2/36 

If a student has already come up with an AND or OR event use it as an example to point out how many 

more complicated probabilities contain the words AND and OR, and these words are very important in probability. 

Have each student make up an AND or OR event and calculate the probabilities of these as a group. 

You should approach these as a process and break them down into smaller more doable steps. Here is an 

example? 

Student-created event: Rolling two dice whose product is more than 15 

1. Re-write the event using AND or OR: rolling two dice whose product is 16 OR 17 OR 18 OR 19 OR 

20 OR 21 OR 22 OR 23, etc. up to the highest possible product 36. 

2. Count the number of outcomes whose product is 16. The answer is one, it’s the double (4.4) 

3. Count the number of outcomes whose product is 17. There are none. 

4. Count the number of outcomes whose product is 18. The answer is two, the pairs (3,6) and (6,3) 

5. Repeat this for each possible product up to 36. 

6. Add all your answers to get the number of outcomes that are included in the event “rolling two dice 

whose product is more than 15.” 

7. This number out of 36 is your final probability. 

Ask the students if anyone can think of a different strategy for computing this probability. They may 

come up with the idea that you could count all the pairs whose product is less than 15 OR equal to 15. Then 

you’d know that however many pairs remain is the number whose product is more than 15, and you have 

your probability out of 36. 

55


As an ending note, point out to your students that probability can be particularly fun because, as you see 

from the examples, there is always more than one way to compute probabilities! 

56 


• http://en.wikipedia.org/wiki/Probability

Notes: 


57


58 

Lesson #10: Dice Probabilities II 




• Discover how conditional probability works with one and two dice 

• Answer the questions, “If I roll an x with my first die, what is the probability of getting a sum of y 

when I roll the second die?” and “If I rolled a sum of y, what is the probability that one of the dice is a 

x?” 

Materials: 

• Two dice 

• Paper and Pencil 

The first part of the lesson is to talk about the probabilities associated with two dice. Start by discussing 

the probability of rolling any number between 1 and 6 on a single die (1/6 probability of each). Next 

ask them what they think the probability will be of rolling some number between 2 and 12 when rolling 

two dice (i.e.: rolling a sum of 2, a sum of 3, a sum of 4, etc.). After they’ve thought about this and written 

down some ideas, show them how to make a chart with all the possible outcomes. It should look something 

like this (keep in mind that these numbers represent the sum of the two dice: 

Die #2 

Die #1 

Have the students count the possible unique rolls. There are 36, so the probability of rolling any one of 

them is 1/36. (This is because, for example, rolling a 2 and then a 3 counts as one possible outcome, and 

rolling a 3 and then a 2 counts as another possible outcome.) Each white box in the table above represents 

one unique outcome. 

To find the probability of rolling any one sum, just count the number of entries there are in the chart for 

that sum and divide it by 36. (If you have taught Lesson #8: Combinations, recall that this is dividing the 

number of rolls that count as the event by the number of total possible outcomes.) For example, since we 

can see that there are five unique ways to roll a sum of 6, the probability of rolling a sum of 6 is 5/36. 

Once the students have a handle on this, pick some number x between 1 and 6 and another number


• http://en.wikipedia.org/wiki/Probability 


y between 2 and 12 and ask the question: If I rolled an x on the first die, what is the probability of getting 

a sum of y when I roll the second die?” Have the students experiment with this until they will realize: 1) 

Sometimes it won’t be possible to get the desired sum given a small first roll. It’s impossible, for example, 

to get a sum of 12 if your first roll is a 1. 2) If it is possible to get the desired sum, then the conditional 

probability is 1/6. This is the case because once the first die is rolled, and you are only looking at the 

second die and for any number x, there is only one other number that can be added to it to get the sum of 

y. Since each face has a probability of 1/6 of being rolled, this leaves a probability of 1/6 of getting y for a 

sum. 

Once they’ve got a hang of this, the next question is: If I roll a sum of y, what is the probability that one 

of the dice is an x? To solve this, use the chart from the beginning of this lesson and look for all the sums 

equal to y. Find out how many of these sums have an x in them (either as Die #1 or as Die #2). The general 

purpose of this lesson is to use dice to explore some conditional probability situations, so any other questions 

you can think of will work just fine. Here are some examples of questions you might ask: 

• 

• 

• 

If your sum is an even number, what is the probability of one of the dice being a 2? (Answer: 5/18) 

If your sum is 4, what is the probability that you rolled two 2s? (Answer: 1/3) 

If your first roll is a 2, what’s the probability that the sum will be even? (Answer: 3/6 or 1/2) 

59


60 

Notes:

Lesson #11: Coins, Dice, and Playing Cards 


Fits well with lesson #8 and #9 



• Understand and be able to explain the difference between independent events and dependent events 

Materials: 


• Pennies or other small counters 

• Dice 

• Deck of playing cards 

Part 1: Coins 

To begin, ask the students to write down 50 imaginary flips of a two-sided coin (without actually flipping a 

coin). When they’re done they should have a 50-item list with either “heads” or “tails” as each item. Now ask 

them to actually flip a coin 50 times and to write down the results. Ask them to compare their two lists, and then 

encourage them to discuss their findings with each other. 

They may find that when they actually flipped the coin, sometimes they got six, seven, eight, or more heads 

or tails in a row. But on their list of imaginary flips most students list no more than four or five heads or tails in 

a row. 

When students make their imaginary lists they often assume that if they flipped heads one time, then they 

are less likely to flip heads the next time, but this isn’t true. Each flip of a a coin is an independent event. This 

means that all previous flips do not influence the probability of flipping a head or a tail this time. In this lesson, 

we are going to investigate this idea further. 

Ask your students what the probability of flipping tails is. The answer: 1/2 or 50%. Now ask your students, 

what the probability of flipping tails is if the previous flip was tails. The answer is still 1/2, but the students may 

think it is less than one half. You could exaggerate the situation by asking them what the probability is of flipping 

tails after flipping twenty tails in a row. Again, the answer is still 1/2. Why is this? 

There are a couple ways to explain this, but first we’ll do an activity so the students can see it for themselves. 

To begin with, have each student come up a hypothesis explaining the probability of flipping another 

heads after they just flipped one. An example of a student’s hypothesis could be: I think that after flipping a head 

there is a 25% chance of flipping another head. After each student thinks up their hypothesis, ask them how they 

might test their hypotheses. 

Depending on the hypotheses, one way to test them is this: Have the students look at their list of heads and 

tails from when they actually flipped the coin. Now find all the times heads was flipped and write down what 

61


they flipped after flipping heads. They should see that, after flipping heads, they flipped another heads half the 

time and tails the other half. It is possible that only 50 flips is not enough to see this, so they could flip the coin 

100 times initially or simply add their lists together to produce one big list of actual flips. 

Now that the students have seen for themselves that it is true that there is a probability of ½ of flipping both 

a head or a tail no matter the previous flips, ask one or two students to explain this to the group in their own 

words. 

Here a couple ways to explain why each time you flip a coin, it is an independent event. The first is to point 

out that the coin doesn’t know that it’s been flipped before. The coin itself only has two sides, and always has 

two sides no matter if it was flipped before or not. Another way to explain this is by, instead of flipping one 

coin ten times in a row, say, flip ten different coins each one time. Just because you flipped a head on one coin 

doesn’t influence the probability of flipping a head on another coin. Finally, you could also pose the question to 

the students: What if someone flipped this coin five years ago? Would that influence how it flips today? The answer 

is no, and if they can understand that, then they may better be able to understand why it also doesn’t matter 

that someone flipped the coin five minutes ago. The probability of flipping heads is still ½. 

Part 2: Dice 

Now we will look at the same idea of independent events, now using dice instead of coins. Here I am assuming 

that your dice are six-sided with the numbers 1-6, with one number on each side. Ask your students 

what the probability of rolling a 3 is. What about a 2? The answer: the probability of rolling each side is 1/6. 

The ultimate aim of using the dice is to show the students that, just as with the coin, each roll of the die is an 

independent event. That is, even if you’ve rolled a 5 three time in a row, the probability of rolling another 5 is 

still 1/6. 

If they don’t see this right away, or struggle making the connection between flipping a coin and rolling a die, 

you can repeat the strategies explained in Part 1 with the die. That is, the students could roll a die 50 times, and 

analyze how many times they rolled, for example, a 5 after rolling a 6. For this study it will be useful to combine 

all your students lists of rolls since there is a lower probability of rolling each number to begin with. 

Part 3: Playing Cards 

Now that your students understand independent events, we will look at dependent events. A dependent 

event is characterized by the fact that its outcome is influenced by previous events. To illustrate the difference 

between independent and dependent events, consider the following scenarios. 

Drawing a red card from a full deck with replacement is an independent event. The probability of drawing 

a red card out of a full deck of cards is 1/2 since half the cards are red and the other half are black. If you then 

put that drawn card back into the deck (this is called replacement) and draw again, the probability of drawing a 

red card is still 1/2 because the deck still consists half of black and half of red cards (since you replaced the one 

you took out). 

Drawing a red card from a deck of cards without replacement is a dependent event. Let’s say that you 

draw a red card from a deck of cards and then leave that card out of the deck (that is, you do not replace it). 

What’s the probability now, with one red card removed, of drawing a red card? 

To answer the question, first ask your students, with a full deck of cards, what the probability of drawing a 

red card is. They may say 1/2, but they may also say 26/52, which is equivalent but maybe a more helpful way 

62


of looking at the problem. It may be helpful to think of this in terms of the number of ways the event can occur 

divided by the total number of possible outcomes, especially if you have taught lesson #8: Combinations and 

have this vocabulary available. 

Now, with a red card removed, what is the probability of drawing another red card? First, how many cards 

are left in the deck? There are 51. And of those 51 cards, how many are black? There are 26 black cards, since 

we haven’t removed any of them. And how many cards are red? There are only 25 red cards, since we have 

remove one of them. So there are 25 ways to draw a red card (the event) out of 51 total possible cards to draw 

(outcomes), which gives us a probability of 25/51. 

Now have the students compare the numbers 25/51 and 26/51. Which number is greater? (26/51 is greater.) 

What does this mean? It means that there is a higher probability of picking a black card than a red card when 

one red card is removed from the deck. 

Remember that in the beginning there was a 26/52 probability of drawing either a black or a red card. So 

now that we have drawn one red card and kept it out of the deck, we see that there is a higher probability of next 

drawing a black card! This is why drawing cards from a deck without replacement is a dependent event. That 

is, the probability of drawing a red card from a deck that has had another card drawn from it previously (without 

replacement) depends on what color that other card was. 

To reinforce this idea, you can do similar examples with suit, number, and face cards versus non-face cards. 

Just remember that the key to understanding dependent events is that, each time you go to draw a card from the 

deck, you must count the number of cards with a given characteristic, say suit, and then divide this number by 

the total number of cards left in the deck. 

Now that your students understand dependent events, they may look back on the independent event of flipping 

a coin and understand it even better. They can see that each time they flip a coin, nothing is lost. That is, 

after each coin flip, the coin still has two sides and thus the probability of flipping a head or a tail is always ½. 

As a final exercise, you may want to ask one or two students to explain the difference between independent and 

dependent events in their own words. 


• 

• 

63


Notes: 

64

Lesson #12: The Monty Hall Problem 




• Understand the Monty Hall Problem and why it is paradoxical 

• Be able to explain the problem and its solution to another person 

Materials: 


• Small candies 

• 3 “doors” (manila folders stood on end work well) 


Begin by explaining the game (taken from http://en.wikipedia.org/wiki/Monty_hall_problem): 

Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door 

is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s 

behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do 

you want to pick door No. 2?” Is it to your advantage to switch your choice? 

Ask the students what they think about the question. (First of all, do they understand the question? 

Make sure everybody is understands what is being asked before proceding.) Most people’s reaction is that, 

whether you switch or not, there is a 50% chance of winning. This, however, is not the case. (Do not tell 

the group this yet! Instead, ask them for their intuitions and their reasoning behind what they think.) In 

order to see this, play the game lots of times with you as the host. 

Set up the three doors. Behind one, put a piece of candy. Have students take turns being the contestant. 

As the host, your job is to always reveal a door with nothing behind it after the student’s initial guess. 

(Since there are two doors with nothing behind them, it will always be possible to do this.) Have the 

students develop a system for keeping track of wins/losses when switching/not switching. When I did this, 

we made a table like so: 

Switch Don’t Switch 

Win Lose Win Lose 

65


After each simulation of the game, have the student who played the contestant put a tally mark in the 

appropriate column. I found that allowing the students who are not playing the game to watch what happens 

behind the doors (without giving the contestant any hints!) helps them realize what is going on. If one or 

more students figures out what is going on, give each student the opportunity to explain it to their classmates 

in his or her own words. 

After at least twenty trials, look at the results. There should be significantly more wins in the Switch column 

than in the No Switch column. 

66 

* It should be noted that it is possible that this will not happen. If you see that there are more wins in 

the No Switch column, there are a couple of things you can do. The first is to continue playing until 

you have more wins in the Switch column. According to the laws of probability, with enough simulations 

of the game, this should eventually happen. Another thing you could do is to point out that 

winning when you don’t switch is equivalent to losing when you switch. (A good challenge for the 

students would be to have them verbalize why this is true.) This way, you can make your table with 

only two columns. Now you should be able to see that the Don’t switch and lose or Switch and win 

column has significantly more tally marks than the other column. 

Don’t switch and win Don’t switch and lose 

or or 

Switch and lose Switch and win 

Have the students discuss what they see. If someone previously thought that changing or not makes no 

difference, have they changed their mind? Ask them to explain why (or why not). Once everybody in the 

group seems to have a firm belief about whether changing doors or not matters, reveal to them the fact that 

there is a difference. If a contestant chooses to change doors, the probability of that person winning is 2/3. 

If a contestant chooses not to change doors, the probability of that person winning is 1/3. 

The following is an explanation of this seemingly paradoxical fact. This explanation has two parts: the 

first follows a player who will not switch doors and the second part follows a player who will switch doors. 

Part 1: If a player goes into the game knowing she is going to switch doors, the probability of winning is 

2/3. Here is why. Imagine you choose a door knowing that you are going to switch when given the option. 

The host then removes a door with nothing behind it. This means that if you initially choose the door with 

the candy behind it, you will definitely switch to the other door that has nothing behind it. If you initially 

choose a door with a goat behind it, the host removes the other goat, so when you switch you will definitely 

switch to the door with the candy behind it. So, if you know you are going to switch, you want to initially 

pick a door with nothing behind it. The odds of doing this are 2/3, since two of the doors have nothing behind 

them at the start of the game. 

Part 2: Now imagine you choose a door knowing that you are going to stay with your choice once the 

host has removed a door. Since you won’t switch, you will win only if you initially choose the door with


candy behind it, and you will lose if you initially choose a door with nothing behind it. The odds of choosing 

the door with candy behind it are 1/3. Therefore, the probability of winning when you switch is 2/3 and the 

probability of winning when you don’t switch is 1/3. 

Please do not attempt to teach this lesson in one day. There is a lot going on here and there are a lot of 

good opportunities for discussion. You may just want to pose the problem and play the game a bunch of 

times on the first day. Then on the second day you can ask one or more students to recall the problem and 

explain it to the group, discuss what students recall happening, and then discuss why this happens. Here are 

some good questions that can keep discussion going if your group catches on quickly: 

• Is it possible to win three times in a row if you never switch? (Yes.) Why or why not? (Hint: This 

question is similar to asking whether or not you can flip a coin and get Heads three times in a row.) 

• If you win ten times in a row, are you less likely to win the next time? (No.) Why or why not? 

(Similarly, if you flip a coin and get Heads ten times in a row, the next time you flip the coin there 

will still be a 50% chance you will get Heads and a 50% chance you will get Tails. You may recall 

this from Lesson #11: Coins, Dice, and Playing Cards.) 

• Why does playing the game more times create more accurate data? 


• http://en.wikipedia.org/wiki/Monty_Hall_problem (This Wikipedia article contains multiple expla 

nations of this problem, which you might want to read before teaching this lesson. The section 

titled “Increasing the number of doors” may be one of the more helpful ones.) 

67


Notes: 

68

Estimation and Approximation 

69


70 

Lesson #13: Estimating the Volume of the Gym 




• Define estimation in your own words 

• Measure the volume of the school gym using the unit of measurement“average 5th grader height 

cubed” 

• Understand that the average of multiple measurements provides a more accurate measurement 

Materials: 


Begin this lesson with a prompt: How do you define estimation in your own words? Have students 

write a definition down and then share their answers. Look for words like guess, guestimate and educated 

guess in their responses. Explain that today we’ll be estimating the volume of the school gym. First 

review that volume of a rectangular prism equals length times width times height. To estimate the volume, 

then, does it work to estimate each of the three measurements length, width and height? If students say 

yes, move on. If they aren’t sure, lead a discussion and/or do a small example to illustrate that it works to 

estimate each dimension separately and use those estimates to compute an estimated volume. 

Now you are ready to estimate each of these measurements. If we go into the gym without any materials 

besides our bodies, what units can we use to measure these things? (Nothing! We need rulers or yardsticks!) 

It looks like we’ll have to estimate the measurements, and we want to do it without wasting time 

going back to the classroom to get yardsticks. What if we measure it by laying head to toe across the entire 

gym floor? What will be our unit of measurement then? 

Discuss until the group agrees that this will give a measurement of the length and width of the gym in 

the unit of ‘5th graders height.’ If the group wants they can record each members’ height and actually find 

the average in inches or feet at the end of the lesson, but for now you only need to name the unit, not to 

know what height it actually translates to. Challenge the kids to organize a way to get the measurements 

as a group. (A group of students I worked with decided to lay down head-to-toe starting with one student 

lying perpendicular to the wall, their feet touching it. Once the final student in the group lies down, the 

first one gets up and goes to the other end to lay down, then the second person moves to the end of the line, 

etc.) 

Depending on time, have them take two or three measurements of each distance (length and width). 

Ask them why they should take multiple measurements, and if they expect them all to be the same or


slightly different. They will likely agree that they should take two or three measurements and get an average 

of them all to be more accurate. 

The length and width of the gym can be found relatively easily by lying on the floor head to toe and 

counting the number of bodies, but what about the height of the gym? Let students debate and decide 

amongst themselves how to estimate this third and final measurement. They may want to have one student 

stand against the wall and the rest of the group stand as far away as possible. From this vantage point, they 

can take turns eyeing the height of the gym by envisioning this student’s height replicated again and again 

until the ceiling is reached. Someone might suggest the pencil method, which you can read about online 

from the further reading section. 

My group noticed the gym wall was make of bricks which were visible enough to count from floor to 

ceiling. They estimated a 5th grader’s height in bricks by having a few of them stand straight against the 

wall, counting the number of bricks it took to get from their feet to the top of their head, and averaging those 

measurements. The height of the gym, then, was the total number of bricks from floor to ceiling divided by 

the average height in bricks. 

Whatever method they choose, encourage students to do as before and get multiple measures then take 

the average. When you have all three measurements, sit down together as a group and pose the final question: 

What is the volume of the school gym? 

After performing the calculation (encourage them to do it by hand and then check with a calculator if 

you have one!) listen carefully to whether or not they state the answer with units. If they do not say the 

unit, push them by asking “1400 what? Feet cubed? Yards cubed? Miles cubed?” If they do say “1400 

fifth graders cubed” ask them to define what the unit of 5th grader means. A really excellent way to explore 

this idea more is to pose the following questions: What would the length of the gym be in the unit of kindergarteners? 

How about the unit of our shoes? Will the measurement be the same in Kindergarteners? Will 

it be bigger? Ultimately you want them to use their estimating skills to make a comparison between 5th 

graders and the other units. They may decide that the average kindergartener is half the height of the average 

5th grader; then the length of the gym would be twice as much in kindergarteners. And about how many 

shoes do they think equal the height of a 5th grader? If it takes ten shoe lengths to equal the length of a 5th 

grader, how different will the length of the gym be in shoes? 

To conclude, ask the students to revisit their definitions from the beginning of your lesson. Did anyone 

write about units? Would they change their definitions now that they’ve estimated the volume of the gym? 

Let a couple students share their reflections, and compliment them on the strengths of their definitions. 


• http://www.inquiry.net/outdoor/skills/b-p/estimation.htm (An estimation tutorial with explanation 

of the pencil method.) 

71


72 

Notes:


Lesson #14: How Many 5th Graders Can Fit in the Gym? 




• Apply estimation skills to answer a new but only slightly different question 

• Estimate how many 5th graders can fit in the school gym 

• Articulate in your own words why the answer to this question is different from the 

answer to the volume question from the previous lesson 

Materials: 


This lesson is a direct result of some confusion that occurred while I taught Lesson #13: Estimating 

the Volume of the Gym to a group of twelve 5th graders. Pose the question “How many 5th graders can 

you possibly fit in the school gym?” and let the students discuss this for a while. Hopefully one or more of 

them will recognize that the answer is not as simple as the volume of the gym in 5th graders. Help them to 

convince the others of this. The questions differ because, to answer this one, we should actually visualize 

the gym packed full of 5th graders, more or less in layers, and then estimate how many 5th graders can fit 

in each layer, followed by how many layers can fit from the ceiling to the floor. The measurements needed 

for this lesson are not all in the same unit. We’ll need to introduce two new units: “5th grader breadths” 

(distance from back to chest) and “5th grader widths” (distance from shoulder to shoulder). 

For one dimension of the gym floor you’ll still use the height of the average 5th grader are your measurement 

unit. To do this, have the kids line themselves up head to toe on the floor. 

73


While they’re lying there, ask them to visualize how many of these rows of kids can fit on the gym floor. 

How can we estimate the number of these rows? (We can count the number of kids who can lay shoulderto-shoulder 

on the floor from this side to the other side!) Encourage them to do this and, as in the previous 

lesson, take the average of two or three measurements of this distance. Come back together as a group and 

review what you’ve found so far: How many 5th graders does it take to cover the whole gym floor? 

At this point ask: How do we use this answer to find out how many 5th graders can fit in the whole gym, 

not just across the floor? The idea you should lead them toward is that the maximum number of 5th graders 

you could fit in the gym will be the number of 5th graders who can fit on the floor (the ‘first layer’) multiplied 

by the number of layers we can fit between the floor and the ceiling. To do this, we need to estimate 

the number of layers. 

74


How can we do this? (We need to know how many of us can fit from floor to ceiling if we lay on top of 

each other on our backs!) Help the students decide the best way to estimate this measurement. They may 

need to do it by eyeing the height of the gym and simply reasoning or guessing how many of them it would 

take stacking themselves like pancakes from floor to ceiling. Maybe the walls of the gym are constructed 

with bricks or patterned wall paper and the students can find a relationship between the size of the bricks 

and the average ‘width’ of their waists. Whatever they decide, make sure they can explain why it’s the 

most accurate they can get. 

With the final measurement of the gym’s height, do the calculations as a group to find the total number 

of 5th graders that can fit in the gym: length (in head-to-toes) × width (in shoulder-to-shoulders) × height 

(in ‘kid widths’). Ask the following questions and have each student respond as a conclusion to the lesson: 

1. Did you expect this number to be bigger or smaller than the volume of the gym in 5th grade height? 

Why? 

2. What’s your estimate of how many kindergarteners can fit in the gym? 

3. Explain the first thing you would do to find out how many kindergarteners can fit in the gym. 

the Catalan Sequence, which begins 1, 2, 5, 14 and then jumps to much higher numbers which we 

will not take the time to illustrate in Challenge Math! 


• http://en.wikipedia.org/wiki/Estimation 

75


76 

Notes:

Combinatorics: How Many Ways Can You... 

77


78 

Lesson #15: Football Scores 



• Use the different ways of scoring in football to explore how numbers can be combined 

• Find ways, using football scores, to make all numbers between 2 and 20 

• Alternately, find as many ways to make 20 (or some other number) as possible 

Materials: 


• Copies of Supplement 1 

Combinatorics studies the different ways that numbers can be arranged and combined. This lesson uses 

a popular example (American Football) to look at some fairly complicated ways of combining numbers 

that involve both a limited set of possible scoring methods and rules that restrict the order in which these 

scores can be reached. The goal is to start with 2 points (the lowest possible score in football) and find a 

way to make every score up to 20 (or higher if you want, but after 20 it becomes pretty repetitive). There 

are a number of different ways to score in football, and some of them can only happen in certain sequences: 

− Touchdown (TD): 6 points 

− Extra Point (EP): 1 point (can only be scored directly after a touchdown) 

− 2-Point Conversion (2P): 2 points (can only be scored directly after a touchdown) 

− Field Goal (FG): 3 points 

− Touch Back (TB): 2 points 

Here are three examples of ways to score 10: 

− [TD(6) + EP(1)] + FG(3) 

− FG(3) + FG(3) + TB(2) + TB(2) 

− TB(2) + TB(2) + TB(2) + TB(2) + TB(2) 

The ultimate goal of this lesson is for the students to get a feel for combining numbers in different ways 

to get the same results. 

This lesson can be particularly engaging if your students are football fans. Start off by seeing if the 

students can tell you what the different ways of scoring are in football. Allow a little time for any interested 

parties to talk about their favorite teams in order to let the interest grow. Next, ask them what the lowest 

possible score in football is (besides 0). Ask them if they’ve ever walked in part way through a game and


seen a really strange score. Finally, ask them if there are any scores they think can’t possibly be reached 

in football. These are primer questions to get the students thinking along the right lines, so leave time for 

discussion of each of them if they are engaged. 

Next, ask the students if they think there are any scores that you can never reach in football (you can actually 

make any score you want except 1). Put them in pairs and, starting with a score of 2 and working up 

to 20, have each pair try to write down possible ways to reach each score. Have them check their answers 

periodically with you to make sure they’re using the scoring correctly. If a student comes up with a particularly 

unusual way to create a score, make a note of it and have them share it with the group. 

Once all the pairs have completed their score sheets, come back together as a group and compile a 

master list of scores. You should see that for any desired score, there are numerous ways to reach it, and the 

higher you get, the more different ways there are. 

An alternate activity for any students that has trouble focusing on filling out the list of scores is to give 

them a fairly hight score (20 works well) and have them try to write down all the possible ways of reaching 

that score. 

For a further challenge, ask the students if they think this same exercise can be done without Touch 

Backs (they really aren’t a very common way of scoring in football anyway). Are there any scores that you 

can no longer make? 


• http://en.wikipedia.org/wiki/American_football#Scoring 

79

Combinatorics: How Many Ways Can You... Supplement 1: Master 

1) _______________________________________________________ 

2) _______________________________________________________ 

3) _______________________________________________________ 

4) _______________________________________________________ 

5) _______________________________________________________ 

6) _______________________________________________________ 

7) _______________________________________________________ 

8) _______________________________________________________ 

9) _______________________________________________________ 

10) ______________________________________________________ 

11) ______________________________________________________ 

12) ______________________________________________________ 

13) ______________________________________________________ 

14) ______________________________________________________ 

15) ______________________________________________________ 

16) ______________________________________________________ 

17) ______________________________________________________ 

18) ______________________________________________________ 

19) ______________________________________________________ 

20) ______________________________________________________ 

80

Notes: 


81


82 

Lesson #16: Combinatorial Theory Basics 


Precedes lesson #17 and #18 


• Learn to compute the ‘number of ways’ to rearrange sets of 3 or 4 distinct objects 

• Learn the ‘factorial’ notation 

• Explore different applications of factorials in basic counting problems 

Materials: 

• Whiteboards and dry-erase markers 

Start with the question: How many ways can 3 students line up at the door? Another way to say this is: 

How many different line orders are there for 3 students lining up at the door? Let some discussion go on 

about whether they think it’s a lot or not very many, and then encourage them to actually work it out with 3 

students (let’s call them A, B and C) from the group as the example, and the rest writing down the different 

ways (this can be done either standing up or on paper). You’ll get a list of these six triplets: 

ABC, ACB, BAC, BCA, CAB, CBA 

From here you should ask how many ways there are if a specific student (let’s say A) is always first 

in line. Count the triplets that start with that student, there will be 2: ABC and ACB. Now you can ask 

how many different choices there are for the student who gets to be first in line. Since there are 3 students 

lining up and each one of them could be chosen to be leader, the answer is 3. For each of the 3 line-leader 

possibilities, there are 2 line orders that are possible (you’ll want to reference the written list you came up 

with to convince yourselves of this). This means to calculate the total ways for 3 students to line up we 

simply do 3 × 2 = 6. 

Here’s the big step: take one of the 6 possible 3-person lines on the list you already wrote (ABC, for 

example) and see what happens when you insert a 4th person (D) into the line. Notice that ‘inserting’ a 4th 

person preserves the original order of A before B and B before C though it may be interrupted by the D, as 

if person D budged in line: 

ABCD, ABDC, ADBC, or DABC 

Since we’ve only used ABC above, there are 5 remaining 3 person lines: ACB, BAC, BCA, CAB, and 

CBA. Assign each of these to one or two students. Tell them to find and write down all the possible ways 

a 4th person, D, can budge into their 3-person line (for the group with CBA, for example, there are these 

4 ways: CBAD, CBDA, CDBA, DCBA). Each student/pair will end up with four ways for D to budge in. 

Since you have six 3-person lines and 4 different ways for D to budge into each of them, you’ve found 4 × 

6 lines in total. Ask the students, Do you think we’ve found all the possible ways to make a 4-person line?


If they say no, challenge them to come up with one that isn’t already written down. If they say yes ask them 

why they’re so sure. (Well we started with all the possible ways for 3 people to line up, and then we found 

all the possible ways the 4th person could budge into each of those lines, so there’s nothing left to find!) 

At this point you want to pause and carefully re-cap the work you’ve just done, explaining it in terms of 

multiplication problems in the following way: to find out how many 3-person lines were possible, what multiplication 

problem did we do? (3 × 2, 3 for the number of possible line leaders, times 2 for each of the lines 

that start with a given leader.) To find out the possible 4-person lines, what multiplication problem did we 

just do? (4 × 6, 4 for the number of ways for the 4th person to budge, times 6 for the total number of 3-person 

lines.) Point out to students that the way we got 6 for the number of 3-person lines was by multiplying 

3 (the number of possible line leaders) by 2 (the number of lines that start with a given leader). Make sure 

all students see and agree with you as you re-write the 6 in 4 × 6 on your white board to get 4 × 3 × 2 for the 

total number of 4-person lines. To recap, we had 3 × 2 for 3-person lines, 4 × 3 × 2 for 4-person lines. Does 

anyone have a guess for 5-person lines? (5 × 4 × 3 × 2) And 6? (6 × 5 × 4 × 3 × 2) What about the number 

of ways for the whole class to line up? (25 × 24 × 23 × 22 × 21 ×.....all the way down to 4 × 3 × 2) Tell 

the students that there’s actually a name for this process so we don’t have to say ALL the numbers from 25 

down to 1. The word for this is factorial, and the notation is an exclamation point. For example, 6! is said 

“six factorial” and means 6 × 5 × 4 × 3 × 2 × 1. 

Now pose a similar question: how many ways are there to rearrange the letters in the word MATH? Let 

them work on this for a while, they may want to start writing out the possibilities but encourage them to find 

a quicker way or at least make a guess at a quicker way. Ask these questions to get them going in the right 

direction. 

1. How many ways can you rearrange just the letter M? (only one way) 

2. How many ways can you rearrange the two letters MA? (two ways, the M can go before the A or after 

it: MA and AM) 

3. How many ways can you rearrange the letters MAT? (six ways, found by inserting the T in all possible 

spots of MA and then doing the same for AM: TMA, MTA, MAT, TAM, ATM AMT) 

4. Does anyone see a pattern developing here? 

The answer is that there are 24 new ‘words’ to be formed out of the 4 letters in MATH. 

MATH, MAHT, MTHA, MTAH, MHTA, MHAT, ATHM, ATMH, AMTH, AMHT, AHTM, AHMT, TMAH, 

TMHA, TAHM, TAMH, THMA, THAM, HATM, HAMT, HTMA, HTAM, HMAT, HMTA 

Is 24 a factorial? It should be fresh in their minds from the last example that 24 is 4! or 4 × 3 × 2. Wait 

for students to see the connection between this example and the example of 4 people lining up. Ask students 

to talk about how the problems are similar. (It’s like M, A, T and H are people’s names and we’re making 

them into lines!) You can conclude that factorials hold for another example, too. 

If your group is enthusiastic about factorials, you should definitely look at the next lesson which uses 

factorials to crack the postal code! 


• http://en.wikipedia.org/wiki/Permutation 

• http://www.mathagonyaunt.co.uk/STATISTICS/ESP/Perms_combs.html 

83


84 

Notes:

Lesson #17: Cracking the Postal Code 






• Learn to compute the ‘number of ways’ to rearrange sets of 3 or 4 objects with repeated objects in 

the group 

• Apply their knowledge of combinatorics to a real-life example 

Materials: 

• Whiteboards and markers for all 

• 5 or 6 envelopes that have been through the mail, preferably arriving at places with different zip 

codes 


Start the lesson with the question: How many different words can we make from the four letters in the 

word BEEP? (Spelling real words doesn’t matter, we’re going for different arrangements of the letters.) 

Students may jump to the conclusion, from the combinatorial theory lesson, that the answer is 4! (read 

“four factorial”) or 24 ways. Ask them to write them all out and see if there are 24 different words. 

This could take a bit of time, but no matter how many they list, there will be at most twelve distinct 

words: 

BEEP, BEPE, BPEE, EBEP, EBPE, EEBP, EPEB, EEPB, EPBE, PEEB, PEBE, and PBEE 

Why isn’t the answer 24? How is this question different from the number of words you can make 

out of the 4 letters in MATH? (Beep has two E’s, but there aren’t any double letters in MATH!) Indeed, 

if we were to color one of the E’s blue and the other one green we could then see the 24 distinct words. 

BE(green)E(blue)P is different from BE(blue)E(green)P and so on, but we know that in writing words, one 

letter E is no different from the next. 

Show students that the answer can be found by first computing 4! as expected, but then you need to 

account for the double letters. You do this by dividing by 2!, since there are two Es. Let students do the 

calculation and agree that it comes out to 12, exactly the number of ‘words’ listed above (4! / 2!=(4 × 3 × 

2) / (2 × 1) = 24 / 2 = 12). Using this approach, how many words can we make from XYYY? The answer 

is 4, which can be found by doing 4! and then dividing it by 3! for the repeated Ys (4! / 3! = 24 / 6 = 4). 

You can check your answer by writing them out: 

XYYY, YXYY, YYXY, YYYX 

Now how many words can you make out of XXYYY? This one’s a little bit trickier because you have 

85


a double X and a triple Y. The answer is maybe as you’d expect: do 5! (5 × 4 × 3 ×2 = 120) and then divide 

it by 3! × 2! (3 ×2 ×1 ×2 ×1 = 12). This gives you 120 divided by 12, which is ten. Indeed, there are only 

ten possible words to make out of two X’s and three Y’s: 

86 

XXYYY, XYXYY, XYYXY, XYYYX, YXXYY, YXYXY, YXYYX, YYXYX, YYXXY, YYYXX 

Now you’re ready to apply this to an example that’s similar but doesn’t use letters and words. The question 

is, How many combinations of three short lines and two long lines are there? 

If students immediately try to write out the combinations, challenge them to think through the problem 

using the skills we just learned. After they come to the answer using factorials and calculations (5! / 3! × 

2! = 120 / 12 = 10), let them try to write out all ten of the possible combinations. Ask them why they think 

this works the same way as the letters example. How are the short and long lines like the letters? (There are 

repeats! There are three of one and two of the other, just like XXYYY) What if we represented the short lines 

with an S and the long ones with an L? Then we’d have the question of arranging two of one letter and three 

of another letter into as many different ‘words’ as possible. Sound familiar? 

At this point distribute an envelope to each of them and ask them to find one of the ten possible combinations 

somewhere on the front of the letter. 

They’ll see right away that their short and long lines make up the stamped code (somewhat resembling a 

barcode) at the bottom of the envelope. Each group of three short lines and two long lines corresponds to a 

digit between 0 and 9. Hand out copies of the Supplement and ask them to work in partners to make a chart 

or table which shows which digit corresponds to each 5-line combination. 

After everyone has a chart and agrees on the digit assignments, refer back to the envelopes and check 

that the codes at the bottom turn out to be the zip code written in the address on the envelope. Congratulations, 

you’ve cracked the postal code! 


• http://mdm4u1.wetpaint.com/page/4.3+Permutations+with+Some+Identical+Elements?t=anon

Supplement 1 


87


88 

Notes:

Lesson #18: The Catalan Numbers 



Fits well with lesson #33, #34, and #35 



• Define sequence in their own words 

• Discover the unique Catalan Sequence through hands-on investigation of various sets counted by 

the Catalan numbers 

Materials: 

• Copies of the Student Worksheet supplements for each student 

• One copy of the Parents’ Guide supplement 

Begin with a discussion on: What is a sequence? Ask students to give an example of a sequence 

(1,2,3,4,5,6,... And there’s 2,4,6,8,10,12,14,...) for you to write down. Write down a handful of sequences 

they come up with, and if they don’t offer something like these add them to the list and make sure they 

agree that they’re sequences: 

1,-2,3,-4,5,-6,7,-8,... 1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7,... 1, 0, 1, 0, 1,... 4, 6, 1/3, -5, 101, 2,.. 

Confirm that a sequence is an ordered list of objects. Ask them to identify which of the sequences you 

wrote down have patterns and which don’t. Patterned sequences are those with a rule you follow or a pattern 

that develops as you go from one number to the next. Ask the group: Do you think mathematicians 

would rather study patterned sequences or random lists of numbers? They should agree that patterned 

sequences are the ones most interesting to study. You should ask them if they’ve ever seen this sequence: 

1,1,2,3,5,8,13,21,34... 

If they don’t recognize it as the Fibonacci sequence, don’t call it that, but simply ask if they think 

there might be a pattern to it. The pattern is that any number is the sum of the previous two numbers in the 

sequence, but give the students time to figure this out on their own. 

At this point you should suggest that there might be a sequence that is in fact a patterned sequence 

even if you can’t look at it and see the pattern or rule automatically. The Catalan Numbers are a perfect 

example! A mathematician named Eugène Charles Catalan from the 19th century spent a lot of his life 

looking at a list of numbers and trying to see the rule that would tell him what number comes next, and he 

finally got it after years of thinking. You’ll be able to explore a few slightly different ways the rule can be 

illustrated using the supplement that follows the lesson. 

Hand out a copy of the Student Worksheets to each student (keep a copy of the Parent’s Guide for your 

89


own study and reference!) and decide as a group which page to tackle first: Triangulation of polygons, Catalan 

paths, or Staircases. Let the group work independently, but make sure to talk aloud about what you see 

students doing, and check in with individuals as they go. If some students are going pretty fast and finding 

the various ways to complete the tasks, pair them up with others who are struggling to find each of the ways 

to do whatever task is described and instruct them to help each other out. Make sure that you spend some 

time looking at the parent’s guide to the worksheet supplement attached and understand the task for each 

page. Have it on hand during the lesson, because it can get confusing and difficult to help a student when 

they’ve found all but two of the 14 triangulations of a hexagon! 

The discovery you should convince yourself of before teaching this lesson is this: the answers to each of 

the successive questions on a given page follow the same pattern. That sequence of numbers is in fact the 

Catalan sequence, which begins 1, 2, 5, 14 and then jumps to much higher numbers which we will not take 

the time to illustrate in Challenge Math! 

After completing each page make a point of highlighting the common results: for each task (polygon triangulation, 

staircases and Catalan paths) there’s 1 way to do problem one, 2 ways to do the second, 5 ways 

to do the third, and 14 ways to do the fourth! After you’ve done two of the three pages, stop and ask students 

what numerical answers they expect to see on the third. When you’ve done them all, they’ll see that 

each set of tasks produces the same set of numbers. Do they think this is a coincidence or is it happening for 

a reason? If there’s a reason, can they articulate what it is? Let this discussion go on as long as you like, but 

at some point ask them if they have any guesses about what number comes next. After each student shares, 

ask them to explain the reasoning behind their guess. At the end of your lesson time, show them the first ten 

or twenty numbers in the Catalan sequence, written below, and explain that after 14 it gets really really hard 

to draw that many Catalan paths, polygon triangulations, staircases, etc. to keep illustrating the sequence 

The beginnings of the Catalan sequence: 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 

742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 

91482563640, 343059613650, 1289904147324, 4861946401452, ... 

90 

For those of you who are curious, the recursive function looks like this: 


• http://en.wikipedia.org/wiki/Catalan_number

Supplement 1: Parents’ Guide 


The Catalan Numbers: A Parents’ Guide 

1. Triangulation of polygons. To triangulate a shape with more than three sides, draw a line that connects 

two of the vertices. Continue connecting vertices until there are no more lines you can draw that 

do not cross each other. The shape will be split into triangles. To see the Catalan Numbers emerge, you 

must start with a triangle and then progress through the sequence by triangulating a square, then a pentagon, 

hexagon, etc. 

Number of ways to triangulate a triangle: 1 

Number of ways to triangulate a square: 2 

Number of ways to triangulate a pentagon: 5 

Number of ways to triangulate a hexagon: 14 

Number of ways to triangulate a heptagon: 42 

91

Combinatorics: How Many Ways Can You... Supplement 1: Parents’ Guide 

2. Catalan paths. A Catalan path is a way to get from the bottom left hand corner of a grid to the top right 

hand corner using only moves up and to the right. In addition, you CANNOT CROSS THE DIAGONAL 

between those two corners. The first Catalan number, 1, is not illustrated here but your students should be 

able to convince you that there is only one way—first right and then up—to get from bottom left to top right 

in a 1x1 grid. The next catalan number is the number of Catalan paths possible on a 2x2 grid. The next in 

the sequence is represented by the number of Catalan paths on a 3x3 grid, and so forth. 

Number of different Catalan paths on a 2x2 grid: 2 




92

Supplement 1: Parents’ Guide 


3. Staircases. For each staircase shape the task is to find the number of possible ways to divide the shape 

into a number of rectangles equal to the number of steps it has. For example, you must divide the 2-step 

staircases into 2 rectangles, while the 3-step staircases must be divided into 3 rectangles. As the number of 

steps increases from 2 to 3 to 4 and so on, the Catalan numbers will emerge. (Note: students may need to 

be reminded that a square is, indeed, a rectangle too!) 

Number of different ways to divide a 2-step staircase into two rectangles: 2 

Number of different ways to divide a 3-step staircase into three rectangles: 5 

Number of different ways to divide a 4-step staircase into four rectangles: 14 

Number of different ways to divide a 5-step staircase into five rectangles: 42 

93


Notes: 

94

Again and Again and Again... 

95


96 

Lesson #19: Fractals 



• Understand self-similarity 

• Understand iteration 

• Answer the question: Is it possible for a closed figure to have a finite area but an infinite perimeter? 

Materials: 


• Ruler 


The goal of this lesson is to encourage students to discover the wonders of fractals on their own. This lesson 

is organized so that it explains all of the background you’ll need to know to teach it rather than telling you 

exactly how to teach it. Although there is some guidance later in the lesson, the most important thing for you to 

remember is that this lesson is all about self-discovery. 

Fractals are everywhere in nature. Ferns, clouds, mountains, cauliflower, and coastlines are just a few examples. 

So what are fractals? One way to describe them is that they are self-similar shapes that are created by 

infinite iterations. But what exactly does that mean? 

If an object is self-similar, this means that on different levels of magnification the object looks the same. 

For example, a fern is self-similar because one leaf of the fern looks just like the whole fern. Also, a piece of 

broccoli is self-similar because one tiny piece broccoli looks like the larger piece of broccoli that it came from. 

Following are three images of a fern. The first image is the entire fern. Then the next image is a part of the 

original fern rotated, and the third image is a part of the second image, again rotated. Notice how the two smaller 

images look remarkably similar to the original fern. This is an illustration of self-similarity.


Iteration occurs when you repeat a process over and over again, taking the output of the last iteration as 

your new input. For example, begin with a line. Then remove the middle third of the line to create two shorter 

lines. Then remove the middle third of each of the two new lines to get a total of four lines. The first four sages 

of this process are shown below. 

Original line segment (Stage 1) 

Stage 2 

Stage 3 

Stage 4 

If you continue this process forever, always erasing the middle third of each line segment, you create something 

called the Cantor Set. Each time you remove the middle thirds from each line segment (i.e.: each time you 

complete a stage), you complete an iteration. After an infinite number of iterations, the Cantor Set consists of 

only points. (The points that remain after an infinite number of iterations is usually referred to as Cantor Dust.) 

The Cantor Set is also self-similar because, as you can see from the picture below, the smallest circled pair of 

lines is the same as the two larger circled pairs of lines (except for the fact that it is smaller). Refer to the following 

picture if you have trouble visualizing or understanding the concept of self-similarity. 

97


Thus, recalling our definition of a fractal, after an infinite number of iterations the Cantor Set is self-similar 

so it is a fractal! Starting with a line segment, what other fractals can you make? What happens if, instead of 

removing part of the line, you add a new line somewhere? And what happens if, instead of starting with just a 

line, you start with a triangle or a square? 

To explore what happens when we create a fractal starting with a triangle, we will begin by looking at the 

perimeter of the United States. First, measure the perimeter of the United States in eight inch intervals. Then 

measure it in shorter and shorter intervals. What happens to the length of the perimeter as we use shorter and 

shorter intervals to measure it with? The length of the perimeter increases! What will happen if we use 1/2 inch 

interval? What about 1/100 inch interval? As the size of the interval decreases, the length of the perimeter increases. 

So what happens if we use an infinitely small interval to measure the length of the perimeter? If we had 

a map that could provide detailed magnifications infinitely, we would find that the perimeter has an infinite measurement. 

So it turns out that it is possible to create a fractal with an infinite perimeter! 

The Von Koch Snowflake is an example of a fractal with infinite perimeter, but don’t tell your students this 

yet. First they’ll draw the fractal and see for themselves how long they think the perimeter is. To create the Von 

Koch Snowflake, begin with an equilateral triangle. Then replace the middle third of each line segment with two 

line segments each equal in length to one third of the original line segment. Below are the first four iterations of 

the Von Koch Snowflake. 

98 

After an infinite number of iterations, the Snowflake looks like this:


So how long is the perimeter of the Von Koch Snowflake? The following is an explanation of how to calculate 

this length. Assume that the perimeter of the original figure (the equilateral triangle) is 1. Perform the first 

iteration, focusing your attention to just one of the sides. When you iterate, each side becomes 4 segments. Notice 

that the length of the side before you iterated was equal to three of these segments. So you took three thirds 

and, through iteration, created four thirds. In other words, you increased the length of that side by one third. 

Since this is what happens to every side of the snowflake within each stage of iteration, you’re really increasing 

the entire perimeter by one third. We can write this as an equation: The length of the perimeter after 

one iterations equals (the length of the perimeter before iteration) plus (one third the length of the perimeter before 

iteration), or P n+1 = P n + (1/3)P n = (4/3)P n , where P n stands for the perimeter of the snowflake before iteration 

and P n+1 stands for the perimeter of the snowflake after iteration. 

Make a table to compute the length of the perimeter after each iteration and notice how we can write the perimeter 

of the snowflake as a power of (4/3). Also notice that the perimeter, then, can be written as (4/3) n , where 

n = the number of iterations you’ve done: 

After two iterations, we again increase the length of each line segment by 1/3, and thus also increase the 

perimeter of the triangle by 1/3. After two iterations, there are 48 line segments, each with length 1/27. Thus the 

perimeter is 48/27, which equals 16/9. Continuing this process, we find a pattern: the length of the perimeter 

equals (4/3)n, where n = the number of iterations. Thus, as the number of iterations goes to infinity, (4/3)n also 

goes to infinity, and the perimeter is infinitely long! 

But the area of the Von Koch Snowflake is finite. To see this, draw a circle around the initial triangle, and 

see that the area of the Snowflake never gets larger than the area of the circle. Since the circle has finite area, the 

Snowflake must also have finite area. 

One question that may come up during this lesson is: How many sides does the snowflake have in each 

stage? The following is an explanation of how to calculate the number of sides the Von Koch Snowflake has 

after n iterations. (It is purely supplemental information, and you may or may not want to include this in your 

lesson.) To begin understanding, count the number of sides in the first 3 iterations (n = 3). (Assume that the 

starting figure (n = 0) is a triangle.) If you decide to teach this as part of the lesson, you might challenge the 

students to come up with ways to count the number of sides in the n = 2 and n = 3 stages without counting each 

and every one. Make a table (we’re going to add on to it, so don’t skip this part): 

n =0 3 sides 

n = 1 12 sides 

n = 2 48 sides 

n = 3 192 sides 

Do you notice anything? If it isn’t immediately obvious, take a couple of minutes to look for a pattern. 

Hopefully you’ll find that the number of sides increases by a multiple of 4 with each iteration. Add this information 

to the table like so: 

99


100 

n =0 3 

n = 1 12 = 3 × 4 

n = 2 48 = 12 × 4 

n = 3 192 = 48 × 4 

So if we wanted to write an equation for the number of sides after the first iteration, we could write 3 × 4 = 

12. Then if we wanted to do this for the second iteration, we could just add another multiple of 4 to the equation: 

3 × 4 × 4 = 48. (This is the same as 12 × 4, right? Have the students check this if they’re skeptical.) Similarly 

for the third iteration, we’d have 3 × 4 × 4 × 4 = 192. (Which is the same as 48 × 4, right?) 

Now explain that the rest of the explanation involves exponents. Ask if anyone in the group can explain 

what exponents are. For example, what does it mean to say “2 to the 5th power,” written 2 5 ? (It means 2 × 2 × 

2 × 2 × 2, or 2 multiplied by itself 5 times.) With this in mind, ask the students if they think it would be okay to 

write 3 × 4 × 4 = 48 as 3 × 4 2 = 48. (If you’ve taught Lesson #21: Order of Operations, you might want to recall 

PEMDAS. Follow the rules of order of operations and you will find that it is okay to write it this way!) Similarly, 

we can rewrite 3 × 4 × 4 × 4 as 3 × 4 3 = 192! Add this information to your table: 

n =0 3 3 3 × 4 0 

n = 1 12 = 3 × 4 3 × 4 3 × 4 1 

n = 2 48 = 12 × 4 3 × 4 × 4 3 × 4 2 

n = 3 192 = 48 × 4 3 × 4 × 4 × 4 3 × 4 3 

It seems that we have found a formula for figuring out how many sides there are in the Von Koch Snowflake 

after n iterations! (After n iterations, there are 3 × 4 n line segments.) Don’t worry if your group doesn’t develop 

a sound understanding; this is pretty tough stuff! Also, note that this is not a good “exploratory learning” opportunity, 

so don’t feel pressured to help the students to develop this formula on their own. 

Another fractal to create is the Sierpinski Triangle. Begin again with an equilateral triangle. Then remove an 

upside down middle triangle from each black triangle. After iterating infinitely, the end result in the Sierpinski 

Triangle. 

After an infinite number of iterations, the Sierpinski Triangle looks approximately like this:


You may want to note that this is only an approximation because to truly see the Sierpinski Triangle would 

require being able to see an impossible amount of detail. 

When constructing this with your students, it is easier to begin with the outline of an equilateral triangle, and 

then to simply draw the outline of each upside down middle triangle instead of making that triangle white. 

What other fractals can your students create? You can have them work individually or in pairs, and then ask 

them to present the fractal they created to the class. Ask them to explain why it is a fractal, that is, to explain 

why and how it is self-similar and what iterative process they used to create it. It’s likely that your students will 

create their own fractals based on the Sierpinski Trianlge or the Von Koch Snowflake. For example, they might 

add three line segments to the outside of the Snowflake instead of two, or might draw squares instead of triangles 

to create a Sierpinski Triangle-like fractal. Challenge your students to create a fractal that uses elements of 

both the Sierpinski Triangle and the Von Koch Snowflake, and see what they come up with! 

Another way to approach this lesson is with computers. Before even talking with your students about fractals, 

have them search online for images of the Sierpinski Triangle. Then, once they’ve each found a large image 

of the Triangle, have them draw it on paper. Encourage them not to copy it line for line, but to draw it with the 

patterns and repetition inherent in the Triangle in mind. Also, as long as they stay focused, you could let them 

search for other fractals as well. 


• http://en.wikipedia.org/wiki/Fractals 

• http://en.wikipedia.org/wiki/Cantor_set 

• http://en.wikipedia.org/wiki/Koch_snowflake 

• http://en.wikipedia.org/wiki/Sierpinski_triangle (some great animations!) 

101

Again and Again and Again... Supplement 1: Map of the USA 

102

Notes: 


103



• Understand the rules of the game 

• Discover patterns 

104 

Lesson #20: Conway’s Game of Life 


Materials: 


• Round plastic counters (available in the classroom) or pennies 

• Copies of all Supplements (and scissors to cut out the grids if you’d like) 

John Conway’s Game of Life is an artificial life simulation that occurs on an infinite two-dimensional grid 

of squares. Each square in the grid is called a cell. Cells can be colored either black or white. Black cells are 

“live,” and white cells are “dead.” Each cell has eight neighbors, which are the eight cells that surround it. For 

example, the live (black) cell marked with a white square below has three live neighbors and five dead (white) 

neighbors. (All of the neighbors of the black cell marked with a white square are marked with gray dots.) 

Although the grid above has only twenty-five squares, imagine that the grid continues forever in all directions 

and that each cell on the border also has eight neighbors. 

To play the Game of Life, create a grid with penny-sized cells that you can lay flat on a table. I’d recommend 

that the grid be at least forty by forty squares so that each student can have their own bit of board space. 

To make a cell live, place a penny, bean, or any other small but inedible object on that cell. (I wouldn’t recommend 

using M&M’s, for example, because by the end of your lesson they will all have disappeared.) To make a 

cell dead, simply leave that cell empty. 

To play the Game of Life, begin by setting up an initial configuration of live and dead cells. (A small-ish


number of live cells - 3 to 5 - is a good idea.) This initial configuration of cells is called the seed and is Generation 

Zero in the game. To advance from one generation to the next, we apply a specific set of rules simultaneously 

to each cell. This set of rules never changes, and thus once we create our seed, we simply follow the rules 

and observe what happens. The rules are as follows: 

1) A live cell with fewer than two live neighbors dies (because it’s lonely). 

2) A live cell with two or three live neighbors stays alive. 

3) A live cell with more than three live neighbors dies (because it’s overcrowded). 

4) A dead cell with exactly three live neighbors comes to life. 

5) A dead cell without exactly three live neighbors stays dead. 

For example, in the grid below the two live cells marked with white circles each have exactly one live 

neighbors. Therefore by rule 1) each of those cells will die in the next generation. The live cell marked with the 

white square has exactly two live neighbors, and thus by rule 3) it will stay alive in the next generation. And the 

dead cell marked with the black square has exactly three live neighbors, so by rule 4) it will come to life in the 

next generation. All other dead cells remain dead by rule 5). Below we see generations zero and one. 

Generation Zero Generation One 

In Generation One, both live cells have exactly one live neighbor, so they both will die in the next generation. 

No dead cells have exactly three live neighbors, so all dead cells remain dead. Thus Generation Two is the 

entirely dead grid, as seen below. No cells can come to life in an entirely dead grid, and thus will remain in this 

state for all future generations. 

Generation Two 

Ask your students if they think it possible for a configuration of live and dead cells to remain the same for- 

105


ever. When a configuration remains the same for all future generations, it is called a still life. Ask the students 

to find examples of still lifes with the pennies on the board. On a separate sheet of graph paper, the students can 

work together to create a list of all the still lifes that they find. Two examples of still lifes, called the Block and 

the Boat, follow: 

106 

Block Boat 

A second type of pattern occurs when a group of cells repeats itself. These patterns are called oscillators. 

The Blinker is a two-phase oscillator, meaning that it consists of two different configurations of cells. In Generation 

Zero, the middle cell in the middle row has two live neighbors, so it remains alive. The other two live cells 

each have one live neighbor, so they die. The middle cells in both the top and bottom rows each have exactly 

three live neighbors, so they come alive in the next generation. Then, when we apply the rules to Generation 

One, Generation Two is the same as Generation Zero, and thus this pattern oscillates between the two configurations 

as seen below. 

Blinker – Generation Zero Blinker – Generation One Blinker – Generation Two 

Another example of an oscillator is the Toad. The Toad is also a two-phase oscillator. 

Toad – Generation Zero Toad – Generation One Toad – Generation Two 

After you’ve discussed oscillators, give out copies of all the Supplements to the students. Have each student 

write down the rules for reference and then let them work on some of the initial configurations provided, placing 

pennies or counters on the blank grids to represent live cells in the forward generations. 

A third type of pattern occurs when a group of cells moves across the board forever. This is called a space-

ship. The most famous example of a spaceship is the Glider. 


Glider – Generation Zero Glider – Generation One Glider – Generation Two 

Glider – Generation Three Glider – Generation Four 

After four transformations, the same initial configuration shifts down one cell and to the right one cell. In 

this way, the Glider moves across the plane forever. When constructing the glider with the students, it is helpful 

to keep each generation set up on the board and to construct the next generation next to it. That way, since the 

rules apply to all cells simultaneously, the students can refer to the previous configuration when constructing the 

next one. 

For a separate lesson, you can ask the students to create their own rules. What happens if they make it harder 

for cells to die? What happens if they make it easier for cells to come alive? Is it possible to create a set of rules 

for which the entire board can be alive? These questions are just prompts. Let the students fully experiment with 

the rules they create. There are no right or wrong rules, and every set of rules they create will yield interesting 

results. 

It’s helpful to see the Game of Life in action on a large scale. To see this, visit the website 

. If you want to show this website to your students, you will 

need to email your students’ teacher before the day that you want to do so because the computers at the school 

have limited access to websites (especially ones with game-like Java applets). Inform him or her of what you 

want to do, and hopefully he or she will be able to coordinate with the school’s “Web Master” and allow access 

to this website. I really encourage doing this - this website is amazing because it allows you to see hundreds of 

iterations occur in a matter of seconds! 


• 

• 

107

Again and Again and Again... Supplement 1: Grids and Initial Configurations 

108

Supplement 2: Grids and Initial Configurations 


109


Notes: 

110

Algebra: Solve for X 

111


112 

Lesson #21: Order of Operations 

or Challenge Math’s Biggest Problem 


Precedes lesson #22 


• Learn the order of operations 

• Write their own mnemonic device for PEMDAS 

• Have substantial practice performing operations (parentheses, exponents, multiplication, division, 

addition, and subtraction) in the correct order 

Materials: 

• Whiteboards and markers for all (including yourself) 

• One LARGE piece of paper (ask a teacher or administrative assistant where the school keeps 

butcher paper for signs and banners. Cut a piece that’s about 10 feet long.) 

• Regular markers in different colors, one per student 

Most 5th graders will be taught the Order of Operations in their math class at some point during the 

year. This lesson is designed to be taught in conjunction with class work on the subject, so make sure you 

speak with the students or their teacher about timing, and don’t teach the lesson before the class covers 

order of operations. 

Begin the lesson by talking casually about this thing called “order of operations.” Has anyone heard of 

it? What is it? What are “operations” in math? Can someone list all of the operations? Are you supposed 

to do a certain operation first? Which one? And, finally, what is the order of the operations? 

Use this opening to take mental notes about what the group is confident about and what they are still 

shaky on. If they need it, review the order of operations on a whiteboard by writing the letters PEMDAS 

in a vertical column on your whiteboard. This is the order, top to bottom, that operations must be performed 

in:


Each of these letters stands for an operation. Which of these can the group fill in? If they cannot come 

up with everything that’s okay. Teach them the missing parts and then review the order of operations by 

having a volunteer recite them. 

At this point ask the students to think about what an operation does to a number. (It changes it! Multiplication 

makes it bigger…unless you multiply by a negative number (see lesson #_) or a fraction. Subtracting 

makes it smaller and adding makes it bigger! …unless you subtract a negative number or add a negative 

number, then it’s the opposite. Exponents make numbers bigger! Parentheses put problems in groups that 

you have to deal with first before going outside the parentheses.) 

The main point you should highlight in their responses is the fact that all the operations change a number. 

Ask students to think about why the order in which you make different changes to a number matters. 

Let them think about this for a bit without shouting out. Tell them if they think there’s a problem with doing 

the operations in a random order they should think of an example that illustrates the problem. Here is an 

example you can show them on your whiteboard using only two operations, multiplication and additon: 

2 + 4 × 3 = ? 

One way to do this is addition first, then multiplication: 2 + 4 = 6, 6 × 3 = 18. If we switch the order of 

addition and multiplication, we get a different number: 4 × 3 = 12, 2 + 12 = 14. Another example is 

2 + 4 2 = ? 

Adding and then taking to the second power gives us 2 + 4 = 6, 6 2 = 6 × 6 = 36. Whereas squaring the 4 

and then adding 2 gives us 4 2 = 16, 2 + 16 = 18. Again, there are different answers based on which order we 

perform the operations. This is why it’s important for everyone to know the correct order and to memorize 

it. The letters for the six operations are PEMDAS; many people use the phrase ‘Please Excuse My Dear 

Aunt Sally’ to remember their order. Challenge each of your students to come up with their own creative 

saying using the PEMDAS letters and they can use it to remember the order of operations! 

Now it’s time for the big sheet of paper. Lay it out on the floor and have the students take turns writing 

Challenge Math’s Longest Problem. This will be one long order of operations problem, so they’ll need to 

make sure there’s lots of variety in the operations they use. Encourage them especially to use parentheses 

and exponents since those are what most students will forget. The problem they write should fill the entire 

top of the paper from short side to short side, and will look something like the following: 

Original problem 

4 2 + 5 × 2 × (13 – 12) – 6 × 5 + ((24 / 6) × 2 3 + 7) + (20 / (7 – 3)) × 2 2 + 6 × 2 × (7 + 2) / 2 

113


Direct students to take turns ‘slimming down’ this problem. The first student to take a turn will have the job 

of rewriting the problem underneath the original without the innermost set of parentheses. Notice whether or 

not the student keeps a pair of parentheses around these numbers once they’re simplified, for example 20 / (4) 

vs. 20 / 4, and make it clear that parentheses around a single number is just that number itself since there are no 

more operations left inside the parentheses. 

114 

Step 1: Simplify the innermost parentheses. 

4 2 + 5 × 2 × 1 – 6 × 5 + (4 × 2 3 + 7) + (20 / 4) × 2 2 + 6 × 2 × 9 / 2 

Notice that there are parentheses remaining. The next student should continue with the parentheses, because 

order of operations says they come before anything else. Notice, however, that nested inside one set of parentheses 

is a problem involving 2 3 and multiplication and addition. Ask students what we do to get the number 

that goes inside this pair of parentheses. Follow the order of operations! By order of operations, the first thing 

to calculate is 2 3 , which is 8. 

Step 2: Simplify the remaining parentheses. 

4 2 + 5 × 2 × 1 – 6 × 5 + ((4) × 8 + 7) + 5 × 2 2 + 6 × 2 × 9 / 2 

Next, do the multiplication inside the parentheses to get: 

4 2 + 5 × 2 × 1 – 6 × 5 + (32 + 7) + 5 × 2 2 + 6 × 2 × 9 / 2 

Finally, do the addition . 

4 2 + 5 × 2 × 1 – 6 × 5 + 39 + 5 × 2 2 + 6 × 2 × 9 / 2 

Step 3: now that parentheses are gone, take care of all the exponents. 

16 + 5 × 2 × 1 – 6 × 5 + 39 + 5 × 4 + 6 × 2 × 9/2 

Step 4: Multiplication is next. 

16 + 10– 30 + 39 + 20 + 12 × 9/2 

More multiplication… 

16 + 10– 30 + 39 + 20 + 108/2 

Step 5: Division 

16 + 10 – 30 + 39 + 20 + 54 

Step 6: Addition 

26 – 30 + 59 + 54 

Keep adding… 

26 – 89 + 54 

Almost there! 

26 – 143 

Step 6: Subtraction 

-117 

One important note is that addition and subtraction are really the same operation since subtraction is the


same thing as adding a negative number. This means that the A and S in PEMDAS are actually performed at 

the same time. The same goes for multiplication and division since dividing by a number is the same thing as 

multiplying by the fraction 1 over that number. 

Throughout this whole process, make sure to keep all students in your group watching the writer of the moment 

and checking all their calculations to make sure they agree. When you’ve finished one giant problem you 

may have time to do another and keep practicing, or you may need to stop there. Feel free to ask the students’ 

teacher if Challenge Math’s Longest Problem can be displayed on the wall in class or saved for future classroom 

lessons on order of operations! 


• http://www.mathgoodies.com/lessons/vol7/order_operations .html (This is a good one for illustrat- 

ing the steps in solving a problem using order of operations.) 

• http://www.math.com/school/subject2/lessons/S2U1L2GL.html (This is a nice interactive tutorial.) 

115


Notes: 

116


Lesson #22: Solving for a Variable using the Golden Rule of Algebra 


Follows lesson #21 




• Solve for a variable in equations involving: addition and subtraction, multiplication and division, square 

roots and exponents 

Materials: 


• Copies of Supplements 1-4 

Begin by writing 5 = 7 on piece of paper. Show it to your students, and see what they say. Is it true? Can 

they make it be true? Ask them to explain what they say. If they say that it is wrong, ask them why? What about 

1 = 2, or 4 = 9? 

Once they are certain that, for example, 4 is the only number that equals 4, write down some simple addition 

or subtraction equations. For example, you could write: 1 + 3 = 4, 5 - 2 = 3, and 2 + 5 = 7. Now you can 

pose the question to your students: What does the “equals” sign mean? Ask them each to describe it in their own 

words. One way to describe it is to say that the numbers on each side of the equals sign have the same value. 

Let’s begin with an equation 4 = 4, which we know is true. Each equation has two sides, one side to the 

left and the other to the right of the equals sign. Ask your students to add a number to one side of the equation. 

Let’s say that one student adds 2 to the left side, which yields the equation 4 + 2 = 4. Is this equation true? The 

answer is no, because 6 does not equal 4. What if we subtract a number from one side of the equation? Let’s say 

we subtract 3 from the right side, which yields the equation 4 = 4 – 3. Is this true? No, it’s not, because 4 does 

not equal 1. 

Now, ask the group to add 1 to both sides of the equation 4 = 4. This yields the equation 4 + 1 = 4 + 1. Is 

that equation still true? It is, because 4 + 1 = 5, and 5 = 5. What if we subtract a number from both sides? Let’s 

subtract 2. This yields 4 – 2 = 4 – 2, which is true since both sides equal 2. Reveal to students that there is a 

Golden Rule of Algebra and ask them what they think it is. The answer: They must either add the same number 

or subtract the same number from both sides of the equation. If they add or subtract to only one side, the equation 

does not stay true. The real Golden Rule is that whatever you do to one side of the equation you must also 

do to the other side. 

Now let’s look at a trickier example. Let’s begin with 3 + 2 = 5. What happens if we add 4 to both sides? We 

get the equation 3 + 2 + 4 = 5 + 4. Well, we know that on the left side 3 + 2 + 4 = 9 and on the right side 5 + 4 = 

9, and since 9 = 9, we know that adding 4 to both sides of the equal sign keep the equation true. What if we subtract 

4 from both sides of the equation? Will the equation still be true? The answer: Yes, it will. 

117


Part 1: Addition and Subtraction 

Now let’s look at an equation with a blank space in it. It is the students’ goal to fill in the blank space. Let’s 

take 3 + = 8. The students may see immediately that they should put the number 5 in the blank space to make 

the equation true, and this is correct. Tell them that it is their goal to, from this equation, be able to write = 

5. Ask them for suggestions on how to turn the equation 3 + = 8 into something that looks like = 5. If they 

need a hint, ask them how to get rid of the 3 that is on the same side of the equation as the . Remember before 

when we subtracted the same number from both sides? Well, what if we try that now? Here the students may 

suggest that they subtract 3 from both sides, which would yield the equation: 3 + – 3 = 8 – 3. Since 3 – 3 = 

0, the left hand side becomes + 0, which equals . And the 8 – 3 on the right side equals 5. Thus we can simplify 

this as = 5, which is exactly what the students saw from the beginning. 

The trick here is to figure out how to “undo” addition and subtraction. Let’s say that we have 3 + 5 = 8. We 

want to rewrite that equation as 3 equals something. So how do we get the 3 alone on the left side? Subtract 5 

from both sides, which yields 3 + 5 – 5 = 8 – 5. That equation become 3 + 0 = 8 - 5 which is the same as 3 = 8 – 

5. Before moving on, ask the students to recall the original equation and compare it with this final result: 

118 

3 + 5 = 8 became 3 = 8 - 5 

Now you can give each student their own problems from the Supplement (or make up your own!). Each 

time the students should add or subtract something from both sides of the equation in order to get alone on 

one side. 

Traditionally in algebra, we use a variable to represent an unknown number, rather than a “blank space.” At 

this point in the lesson, once your students are comfortable with the idea of a “blank space,” you can explain 

to them that blank spaces are actually variables. So what is a variable? It is simply a letter used to represent an 

unknown number. For the remainder of the lesson, we will use the letter “x” to represent a variable. You can explain 

to your students that writing an “x” is the same as writing the symbol “ ” like we did before. You can now 

hand out Supplement 1. 

Part 2: Multiplication and Division 

Before we begin multiplication and division with variables, let’s look back to the beginning of the lesson. 

We already know that 4 = 4. What happens if we multiply both sides of the equation by 2? We get a new equation 

that reads 4 × 2 = 4 × 2. Since 4 × 2 = 8, and 8 = 8, we know that this equation is true. Does it work for division 

as well? Let’s say we begin with 6 = 6. Now if we divide each side by 3 we get the equation 6 / 3 

= 6 / 3. And since 6 / 3 = 2, we see that 2 = 2. 

Let’s begin by looking at the equation 3 × 6 = 18, which we know is true. If we want to get the 3 alone on 

the left side, what can we do? Can we divide by a number? The answer: Divide both sides by the number 3. This 

yields: 3× 6 / 6 = 18 / 6. Thus, since 6 / 6 = 1, we have 3 × 1 = 18 / 6. Since 3 × 1 = 3, we see that 3 = 18 

/ 6. In this way, division and multiplication “undo” each other. 

Now for some algebra. Let’s begin with the equation 6 × x = 18. The students may see that x = 3, but ask 

them to show you why. Their goal, just like in the previous example, is to get the x alone on the left side. Last 

time they divided both sides by 6, so what happens if they try this again? They get 6 × x / 6 = 18 / 6, which can 

be rewritten as x = 18 / 6 = 3. You can now hand out Supplement 2.

Part 3: Square Roots and Exponents 


Square roots and exponents “undo” each other just like multiplication and division “undo” each other and 

addition and subtraction “undo” each other. For example, if we begin with and then square it, we see that 

the square undoes the square root because we end up with: . Now let’s begin with 5 2 . 

How do we “undo” the square? We simply take the square root of 5 2 : . Ask your students to explain 

this in their own words. They might say that if you square and take the square root of a number, you get that 

numbe back. Ask the group if it matters if they square the number or take the square root of the number first. 

The answer: It doesn’t matter, as we see in the example with the number 5. The first time we took the square 

root and then square it, and the second time we square it and then took the square root, and both times we ended 

up with the number 5. 

Let’s first look at an example without a variable. Say we have 4 2 = 16. If we want to write this equation with 

the 4 alone on one side of the equation, what do we do? The answer: Take the square root of both sides! (Remember 

the Golden Rule that says we always must apply the same opperation to both sides of an equation.) We 

then have the equation 4 

€ 

2 = 16 , which we can see is true! But what does 16 equal? To help your students 

see the answer to this question, you could ask: What does the square of negative four equal? Written out, this 

looks like: (-4) 

€ 

2 . The answer: (-4) 2 = 16 as well! (Recall that a negative times a negative always equals a positive.) 

With this in mind, ask your students again what 16 equals. The answer: Well, there are two answers... 4 

and -4. Why is this? This is because 4 × 4 = 42 = 16 and -4 × -4 = (-4) 2 = 16. So the square root of any positive 

number always has two answers, where one answer is negative and the other is positive. 

€ 

Next, as a brief but important aside, ask the group what the square root of negative seven is. The answer: 

The answer doesn’t exist! Since a negative number square is a positive number, and a positive number square is 

also a positive number, there is no number squared that equals a negative number. (There are special numbers 

called “complex numbers” which take into account the square roots of negative numbers, but we will not discuss 

those here. For more information on them, see < http://en.wikipedia.org/wiki/Complex_number>.) 

Say we have: 25 = 5, which your students should agree is true. Pose the question: How do we eliminate 

the square root? The students may suggest that you square both sides, which is the correct answer. If they struggle 

to see this, you could remind them that squares “undo” square roots. Similarly, if you have the equation: 5 

€ 

2 = 

25 and want to eliminate the square, simply take the square root of both sides. 

Now let’s try a problem with a variable. Let’s say we want to solve the equation: = 3. What do we 

do? We square both sides, which then shows us that x = 3 2 = 9. And what if we know that x 2 = 49? We take the 

square root of both sides, so we see that x = 7 and x = -7. You can now hand out Supplement 3. 

Part 4: Everything! 

This is the trickiest part of the lesson, because it involves solving for a variable in equations where it is not 

always obvious operation to apply to both sides. Before you begin this section, review with your students Lesson 

#21 on order of operations. Now let’s look at an example of an equation with addition, multiplication, and 

a square: 2 × x 2 + 7 = 39. Pose the question: How do we solve this equation for x? Then let the students try and 

work through it. If they arrive at an answer, ask them to substitute it back into the original equation to see if the 

equation stays true. If the equation stays true, then they have found the correct answer! If not, then they should 

go back through their work and see where they made a mistake. 

119


So how do we solve this equation? We begin with addition and subtraction. What can we add or subtract to 

this equation to make it more simple? The answer: We can subtract 7 from both sides, which yields the equation 

2 × x 2 + 7 - 7 = 39 - 7, which can be simplified to read 2 × x 2 = 32. Next, we divide both sides of the equation by 

2, which results in the equation x 2 = 16. Lastly, we take the square root of both sides, which tells us that x = 4 

and x = -4. Notice that, to solve this equation for x, we used the order of operations backwards. We did addition 

and subtraction first, then multiplicatoin and division, and finally exponents. You can now hand out Supplement 

4. 

As a conclusion to this lesson, ask your students to recall the Golden Rule of Algebra. Ask them to explain it 

to each other in their own words. In their explanations, look for them to say that the Golden Rule says to do the 

same thing to both sides. Then pose the question: Why must be do the same thing to both sides of an equation? 

The answer: So that both sides of the equation remain equal. 


• 

120

Supplement 1: Addition and Subtraction 

1) 3 + x = 12 

2) x – 5 = -4 

3) x + 2 = -21 

4) 3 – 2 = x + 1 

5) 4 = x – 18 

6) 23 = 76 + x 

7) x – 7 = -145 

8) 2 + x = 131 

9) -5 = x – 23 

10) 89 = x + 520 


121

Algebra: Solve for X Supplement 2: Multiplication and Division 

11) 3 × x = 57 

12) x / 4 = 3 

13) 8 = -2 × x 

14) x × 5 = 45 

15) 6 × x = -72 

16) -14 = x / 2 

17) -8 × x = 64 

18) x / 12 = 180 

19) 20 / x = -10 

20) -x = 32 

122

Supplement 3: Square Roots and Exponents 

21) = 25 

22) 16 = x 2 

23) 12 = 

24) x 2 = 9 

25) 225 = x 2 

26) = -7 

27) 81 = x 2 

28) -10 = 

29) 6 = 

30) x 2 = 0 


123

Algebra: Solve for X Supplement 4: Everything! 

31) (3 × x) + 7 = 16 

32) 12 = (7 + x) / 3 

33) 

34) 151 = x 2 + 7 

35) (12 – x) × 5 = 100 

36) 17 = (x / -2) – 3 

37) 

38) 

39) (x + 15) 2 = 9 

40) – 3 = 1 

124

Supplement 5: Teacher Answer Key 

1) x = 9 

2) x = 1 

3) x = -23 

4) x = 0 

5) x = 14 

6) x = -53 

7) x = -138 

8) x = 129 

9) x = 18 

10) x = -431 

11) x = 19 

12) x = 12 

13) x = -4 

14) x = 9 

15) x = -12 

16) x = -28 

17) x = 8 

18) x = 2,160 

19) x = -2 

20) x = 32 

21) x = 5 

22) x = 4, x = -4 

23) x = 144 

24) x = 3, x = -3 

25) x = 15, x = -15 

26) x = 49 

27) x = 3, x = -3 

28) x = 100, x = -100 

29) x = 36 

30) x = 0 

31) x = 3 

32) x = 29 

33) x = 32 

34) x = 12, x = -12 

35) x = -8 

36) x = -40 

37) x = 3 

38) x = 7, x = -7 

39) x = -12, x = -18 

40) x = 16 


125


Notes: 

126

Lesson #23: Puzzle Worksheets 




• Work with various types of algebra problems 

• Understand the processes needed to solve the harder problems 

Materials: 


• Copies of Supplements 2 - 9 (Supplement 1 is the Teacher’s Worksheet Answer Key) 


Fifth graders will have varying degrees of familiarity with basic algebra, so the worksheets 

start with very simple problems and become increasingly difficult towards the bottom. Most 5th 

graders are pretty good at looking at a problem and figuring out the answer, but don’t understand 

the concept of systematically doing the same thing to both sides using the Golden Rule of Algebra 

(see the previous lesson), to isolate the variable. An example of isolating the variable looks 

like: 

(x - 4) × 12 = 24 

((x - 4) × 12) / 12 = 24 / 12 Divide both sides by 12 to remove the 12 from the left side 

x - 4 = 2 

(x - 4) + 4 = 2 + 4 Add 4 to each side to remove the 4 from the left side 

x = 6 Now x is isolated on the left 

Some of the problems on the worksheets can be figured out just by looking (even the ones that 

require order of operations), but for those that can’t be intuited, the students may get pretty lost. 

Be prepared to review the Golden Rule of Algebra and to carefully explain isolating the variable 

on an individual basis to those sho are struggling. 

The puzzle works pretty simply. Each sheet has a number from 1 through 6 in the upper left 

corner corresponding to its position in the final solution. The solution to each question on each 

sheet corresponds to a letter in the alphabet (or a space if x = 0). Each student must solve all 6 

questions on his or her worksheet and match them with their corresponding letters. Once done, 

the whole puzzle can be put together by taking the six letters from Worksheet 1 and writing them 

down in order from 1 through 6, then the letters from Worksheet 2 in order, then 3, etc... until all 

36 characters are full. The puzzle should read “Challenge math is fun plus education” 


• http://en.wikipedia.org/wiki/Algebra (see Elementary Algebra) 

127


128 

(1) 

1) x + 4 = 7 (A: x = 3) 

2) x / 2 = 4 (A: x = 8) 

3) (7 × x) + 1 = 8 (A: x = 1) 

4) (x / 2) - 4 = 2 (A: x = 12) 

5) 24 / x = 2 (A: x = 12) 

6) (75 - x) / 10 = 7 (A: x = 5) 

(2) 

1) 18 - x = 4 (A: x = 14) 

2) 7 × x = 49 (A: x = 7) 

3) 35 / x = 7 (A: x = 5) 

4) (77 + x) / 7 = 11 (A: x = 0) 

5) x + 7 = 20 (A: x = 13) 

6) (22 - x) / 7 = 3 (A: x = 1) 

(3) 

1) 25 + x = 45 (A: x = 20) 

2) 64 / x = 8 (A: x = 8) 

3) x / 12 = 0 (A: x = 0) 

4) 6 × x = 54 (A: x = 9) 

5) 38 / x = 2 (A: x = 19) 

6) ((x + 27) / 27) - 1 = 0 (A: x = 0) 

Supplement 1: Teacher’s Worksheet Answer Key 

(4) 

1) x + 4 = 10 (A: x = 6) 

2) x / 3 = 7 (A: x = 21) 

3) 2 × x = 28 (A: x = 14) 

4) (24 × x) + 15 = 15 (A: x = 0) 

5) 4 2 = x (A: x = 16) 

6) (36 - x) / 4 = 6 (A: x = 12) 

(5) 

1) 22 - x = 1 (A: x = 21) 

2) 1 + x = 20 (A: x = 9) 

3) (3 + x) × 4 = 12 (A: x = 0) 

4) 8 × x = 40 (A: x = 5) 

5) x 2 - 2 = 14 (A: x = 4) 

6) (x / 3) + 5 = 12 (A: x = 21) 

(6) 

1) 2 + x = 5 (A: x = 3) 

2) 29 - x = 28 (A: x = 1) 

3) 6 × x = 120 (A: x = 20) 

4) √ x = 3 (A: x = 9) 

5) x / 3 + 2 = 7 (A: x = 15) 

6) ((x / 2) - 2) × 2 = 10 (A: x = 14)

Supplement 2: Solution Sheet 

1 2 

3 4 5 6 


Fill in the Blanks! 

129

Algebra: Solve for X Supplement 3: Worksheet 1 

Algebra Puzzle 

Worksheet 1 

1) x + 4 = 7 

2) x / 2 = 4 

130 

x = ____ 

x = ____ 

3) (7 × x) + 1 = 8 

x = ____ 

4) (x / 2) - 4 = 2 

x = ____ 

5) 24 / x = 2 

x = ____ 

6) (75 - x) / 10 = 7 

x = ____

Supplement 4: Worksheet 2 


Worksheet 2 

1) 18 - x = 4 

x = ____ 

2) 7 × x = 49 

x = ____ 

3) 35 / x = 7 

x = ____ 

4) (77 + x) / 7 = 11 

x = ____ 

5) x + 7 = 20 

x = ____ 

6) (22 - x) / 7 = 3 

x = ____ 


131



Worksheet 3 

1) 25 + x = 45 

132 

x = ____ 

2) 64 / x = 8 

x = ____ 

3) x / 12 = 0 

x = ____ 

4) 6 × x = 54 

x = ____ 

5) 38 / x = 2 

x = ____ 

6) ((x + 27) / 27) - 1 = 0 

x = ____



Worksheet 4 

1) x + 4 = 10 

2) x / 3 = 7 

x = ____ 

x = ____ 

3) 2 × x = 28 

x = ____ 

4) (24 × x) + 15 = 15 

5) 4 2 = x 

x = ____ 

x = ____ 

6) (36 - x) / 4 = 6 

x = ____ 


133



Worksheet 5 

1) 22 - x = 1 

134 

x = ____ 

2) 1 + x = 20 

x = ____ 

3) (3 + x) × 4 = 12 

x = ____ 

4) 8 × x = 40 

x = ____ 

5) x 2 - 2 = 14 

x = ____ 

6) (x / 3) + 5 = 12 

x = ____



Worksheet 6 

1) 2 + x = 5 

x = ____ 

2) 29 - x = 28 

x = ____ 

3) 6 × x = 120 

4) √ x = 3 

x = ____ 

x = ____ 

5) x / 3 + 2 = 7 

x = ____ 

6) ((x / 2) - 2) × 2 = 10 

x = ____ 


135

Algebra: Solve for X Supplement 9: Code Key 

Code Key 

0 -> space 

1 -> A 

2 -> B 

3 -> C 

4 -> D 

5 -> E 

6 -> F 

7 -> G 

8 -> H 

9 -> I 

10 -> J 

11 -> K 

12 -> L 

13 -> M 

14 -> N 

15 -> O 

16 -> P 

17 -> Q 

18 -> R 

19 -> S 

20 -> T 

21 -> U 

22 -> V 

23 -> W 

24 -> X 

25 -> Y 

26 -> Z 

136

Notes: 


137

138 

Graphing: Plot the Points


• Understand pie charts 

• Understand bar graphs 

• Learn the Cartesian coordinate system 

Materials: 


• Calculator 

Lesson #24: Graphing Data 


Precedes lesson #25, #26, and #27 

Graphing: Plot the Points 

To begin this lesson, ask the students how they would define data. Let them discuss it, and then tell them 

that they are going to collect some data. First, with their predominant hand, ask them to write the alphabet 

over and over again as fast as the can but so the letters are still legible. Have them write for one minute 

straight and then ask them to stop writing when the minute is up. Then do the same thing with their other 

hand. Next, ask them to count how many letters they wrote with their right hands and how many with their 

left hands. Does the number of letters written correspond with their handedness? The students may compete 

to see who has written more letters, but it doesn’t matter who wrote more. 

The number of letters each student wrote with each hand is called a data set. Now that they have collected 

the data, ask them again how they would define data. When the students define “data” before doing this exercise, 

they may only say that data is a group of numbers or information. After this exercise, encourage them to 

understand data as something from which they can get information and determine meaning. 

With this data in hand, the students’ goal is to find ways to represent this data. Begin by asking them for 

ideas. You can keep the question broad so that they can brainstorm ideas. Following are three ways the students 

could represent this data, but they are by no means the only ways. If the students begin to describe one 

of the ways explained below, feel free to reference it or explain to them that what their describing has a name. 

But if they think of entirely different ways to represent their data, go with it and see what happens! 

Method 1: Bar Graph 

A bar graph is a chart with rectangular bars whose lengths are proportional to the values that they represent. 

For example, each student can make a bar graph of their own data, where one bar represents the number 

of letters written with their left hand and another with their right hand. 

Ask the group how they could make a single bar chart that represents all of their data. (To do this they 

could, for example, add up their data like they did to make the pie chart, and make a bar graph with two bars.) 

Here is an example of a bar graph: 

139


Method 2: Pie Chart 

If the students struggle with ideas, you could suggest that they draw a picture. A pie chart is an example of 

a picture they could draw. Pie charts are circular graphs divided into pie-shaped pieces that represent the amount 

or frequency of an event. 

For example the students could make a pie chart with two sectors. They could add together the data from 

each student to determine the total number of letters written with left hands and the total number of letters written 

with right hands in the entire group. One piece of pie would represent each sector. Following is an example 

of such a pie chart. 

140 

To make it easier to figure out the size of each piece of pie, you could do a mini-lesson on percentages. Let’s


say that in total, the students wrote 136 letters with their right hands and 83 letters with their left hands. Ask the 

students what percentage of letters written were written by right hands. The answer: 136 letters divided by the 

total number of letters written, which is 136 + 83 = 219. Thus the percentage of letters written by left hands is 

136 / 219, and the percentage of letters written by right hands is 83 / 219. A completely filled-in pie chart represents 

100%, so given these new percentages the students can divide up the pie chart. 

What other ways can the group represent the data using a pie chart? 

Method 3: Cartesian Coordinate System 

The Cartesian coordinate system is a bit more difficult to discover without any help, so you may need to 

guide your students towards its discovery. A Cartesian coordinate plane consist of two number lines that are perpendicular 

to each other and intersect at zero. These lines are called axes. The horizontal line corresponds with 

the x variables, and thus is called the x-axis. The vertical line corresponds with the y variables, and thus is called 

the y-axis. 

Each point in the plane has an x-value and a y-value associated with it. Let’s say we’d like to plot the point 

x = 3 and y = -2. First, draw a vertical line through the number 3 on the x-axis. Then draw a horizontal line 

through the number -2 on the y-axis. Where those two lines meet is the point x = 3 and y = -2. We write this 

point as the point (3, -2), where the first number is the parentheses refers to the x-value and the second number 

refers to the y-value. So the generic way to write a point is (x, y). Ask your students to plot a few other points, 

such as (2, 3), (0, 0), (-3, 1), and (-1.5, -2.5). Below is a Cartesian coordinate graph with those points plotted. 

Now ask the students how they could use the Cartesian coordinate plane to represent their data. One way 

would be to plot the number of left-hand letters on the y-axis and the number of right-hand letters on the x-axis 

141


for each student. For example, let’s say that one student wrote 71 letters with her right hand and 29 letters with 

her left hand. These numbers would then correspond to the point (71, 29). The students could then each plot 

their point on the same graph. 

142 

After the students have each plotted their point on the same graph, pose the following questions: 

• Where do the points lie on the graph? Do they lie closer to the x-axis or the y-axis? What does this mean? 

(The majority of the points will lie closer to the x-axis. This is because most students in you group will be righthanded 

and thus will write more letters with their right hands than their left hands.) 

• What does it mean if a point has the same x and y values? (This means that the student wrote the same 

number of letters with both his right and left hand.) 


• http://en.wikipedia.org/wiki/Bar_chart 

• http://en.wikipedia.org/wiki/Pie_chart 

• http://en.wikipedia.org/wiki/Cartesian_coordinate_system

Notes: 


143


144 

Lesson #25: Jumping Jacks 




• Practice collecting data 

• Use Cartesian graphs to help understand the data 

Materials: 


• Stop watch or timer 


• Rulers 

Cartesian graphs are one of the most powerful tools in mathematics, and they can be used in many 

different contexts. The most immediate use of graphing is data analysis, which is a really good place to 

start understanding how graphs work. For this lesson, the students will practice collecting some data that 

works with two variables (in this case time and number of jumping jacks), plotting the data, and then figuring 

out what they can learn from it. 

Part 1: Jumping Jacks 

To start things off, take the students outside or somewhere else where they can be active and loud. The 

goal of this part is to have the students collect data about how many jumping jacks they can do in a certain 

amount of time. Put the students in pairs and have one from each pair be the jumper and one be the counter 

to start. Start with 10 seconds, having the jumper perform jumping jacks (make sure they’re real ones 

and not just trying to fit as many in as possible) and the counter count how many the jumper does. Have 

the students write the results down in a chart like this: 

time (seconds) jumping jacks 

10 11 

Next, have the jumper and counter switch and jump for 10 seconds again. Repeat this for 20, 30, and 

40 seconds so that each person has a number of jumping jacks associated with each time value on their 

own chart. The resulting charts should look something like this:

Part 2: Graph It 


• http://en.wikipedia.org/wiki/Point_plotting 

time (seconds) jumping jacks 

10 11 

20 20 

30 28 

40 35 


Head back inside. Have each student set up a pair of axes on their paper (if not using graph paper, make 

sure that the intervals are evenly spaced). Have them label the x-axis “Time” and the y-axis “Number of 

Jumping Jacks.” Next, have each student plot the points from their own data. Now look at the graphs and 

ask the students if they can learn anything from the graphs. Most likely, the students’ graphs will slope off 

and not stay in a straight line. This shows that the students got tired as they had to do jumping jacks for 

longer time periods. See if the students can discover on their own, and make sure to leave time to talk about 

this at the end of the lesson. 

145

Graphing: Plot the Points Supplement 1 

146

Notes: 


147


148 

Lesson #26: Functions 

Approximate lesson length: 1 




• Understand the idea that a function is something that takes input, does something with it and 

produces output 

• Learn about functions in math 

Materials: 

• Pencil 

• Copies Supplement 1 

• Rulers 

Functions appear in many different scientific disciplines. At the most basic level, a function is essentially 

a black box that takes input, performs some set of operations on that input, and returns the result as 

output. In math, the inputs are primarily numbers, the operations are mathematical operations (addition, 

subtraction, multiplication, division, exponents, etc...), and the output is the result of the calculations. A 

mathematical function is usually assigned a variable name (canonically f) and is written like this: 

f(x) = 2 × x + 1 

In this example, the input is the variable x (indicated in the parentheses after f), and the operations are 

to multiply x by 2 and then add 1. For input x = 1, you replace x with 1 in the expression f(x) = 2 × x + 1. 

First, f multiplies 1 by 2 and then adds 1 resulting in f(1) = 3. This lesson starts by using a simple game to 

introduce the students to the concept of a function and then moves on to looking at an example of a mathematical 

function. 

Part 1: The Function Game 

The lesson starts with this simple game to introduce the students to the idea of a function. You, the 

instructor, will be a human function. Think of an action or type of behavior (for example folding arms) 

that your function will produce. Assign one student to be the guesser, then go around a corner so that the 

guesser can’t see you. Have each non-guessing student come around the corner and then tell them to perform 

your action or behavior when they walk back out. The guesser’s job is to figure out what your function 

does. Here are some examples of functions to try: 

- the folded arms function 

- the sitting down function 

- the silence function 

- the crazy function (for my group it ammounted to running around with arms flailing!) 

- the somersault function


For an added bit of exploration, try making your function take two pieces of input (people) and having 

them perform some sort of action together as the output. For example, the holding hands function. Now 

propose to the students that you can represent different mathematical operations using these human functions. 

Give the example of the “plus Billy” function where every input goes around the corner alone and 

comes back with Billy (a group member) alongside them as the output. Ask for volunteers who think they 

have a good representation of functions involving subtraction and multiplication. While you can’t actually 

split apart any of your human inputs, talk hypothetically about what a “divide by two” function would look 

like. Using this game to illustrate mathematical functions allows for a smooth transition into part 2. 

Part 2: On Paper 

Start with a simple function like f(x) = 2 × x + 1 or f(x) = x 2 . Have them compute several sample outputs 

for given inputs (start with positive whole numbers like 1, 2, 3, etc...) and put them in a chart like this one: 


• http://en.wikipedia.org/wiki/Function_(mathematics) 

x 

1 

2 

3 

f(x) = x 2 

1 

4 

9 

Once they have computed these sample points, have the students graph them. When graphing the points, 

it is important to realize that the students will be used to plotting points with x and y coordinates. When 

working with functions, f(x) is just another way of writing y, so the y-axis becomes the f(x)-axis, and each 

point has an x coordinate and an f(x) coordinate instead of an x-coordinate and a y-coordinate. For a more 

thorough explanation of graphing equations, see the next lesson. If there is extra time, give them several 

functions and have them graph them on the same set of axes to see how different functions can compare to 

each other. 

149

Graphing: Plot the Points Supplement 1 

150

Notes: 


151


152 

Lesson #27: Graphing Equations 






• Understand the idea of a coordinate 

• Set up rules (that are actually equations in disguise) to graph points 

Materials: 



When we have a variable in an equation, there is only one solution for that variable. But if there are two 

variables in an equation, say x and y, then for each value of x there is one associated solution for y. 

A variable can be thought of as a blank space in an equation that can be filled in with a number. Variables 

often are represented as the letter x or the letter y, but any non-numerical symbol will work. An example of an 

equation with one variable is: 

5 + x = 7 

How many different numbers can x equal in the equation? The answer is that x can only equal one number, 

and in this case x = 2 because 5 + 2 = 7. What if we have an equation with two variables such as: 

y = x -3 

How many different numbers can x and y equal in the equation? The answer: There are infinitely many 

possible solutions. This is because x and y depend on each other. For example, if x = 10, then we have the 

equation y = 10 - 3. And based on this equation, we can see that y = 7. But what if we let x = 8 instead of 10. 

Then we have the equation y = 8 - 3. And thus now y = 5. So the value of y depends on the value of x; that is, 

as x changes value, so does y. Encourage the students to experiment with multiple different equations and lots 

of different values for x and y. 

To look at the solutions of an equation with two variables, we graph the equation in a Cartesian coordinate 

plane. (For a refresher on how to graph points, see Lesson #24.) To graph an equation on a Cartesian coordinate 

plane we simply graph a few points that satisfy the equation, and then draw an estimate of a line that 

connects those points. To get started graphing, first let’s look at the equation: 

y = x + 1 

Have your students make a chart with four columns. One column has values of x, another has the values x 

+ 1 = y, the third has just the value of y, and the fourth writes out the coordinate in the form (x, y). Encourage


your students to make the chart for both positive and negative values of x. Here is an example of chart a student 

could make: 

Now have the students plot the points (x, y) from their chart onto a graph. Following is an example of 

what the graph looks like that corresponds to the previous chart. Ask the students if they should connect the 

points in the graph or not? (They should.) To see why, ask them to plot a point in between two points they 

have already plotted. For example, they could plot the point where x = 0.5. Since y = x + 1, in this case y = 

0.5 + 1 = 1.5. If the students plot the point (0.5, 1.5), they will see that it lies on the same line as all the other 

points they have already plotted. Thus, between every two points, there is another point they could plot using 

the equation y = x + 1. 

153


Now, ask the students to each think up their own equation with two variables. Have them make a chart for 

their new equation and graph it in the same way as before. 


• 

• 

154 

Some equations you could suggest to graph are: 

y = x 

y = -x 

y = 2x + 1 

y = -2x - 3 

y = x 2 

Encourage the students to make predictions about what the graph will look like, first by looking just a the 

equation, and then by looking at the values on their charts.

Supplement 1: Graph Paper 


155


Notes: 

156 

Graphing: Plot the Points

Lesson #28: Manipulating Graphs 




• Discover how to manipulate functions to change the appearance of their graphs 

• Practice with graphing functions 

Materials: 

• Sidewalk and chalk (if nice outside) or sheets of butcher paper and markers 



The goal of this lesson is to expose the students to the correlation between a function and its corresponding 

Cartesian graph. It uses some of the most common families of functions as examples to explore 

how to shift a graph around on the plane, expand and contract it, or flip it over. 

In lesson #[Functions] we learned about what a function is, and in lesson #[graphing equations] we 

learned about how to graph equations. The trick to graphing functions is to combine the ideas in these two 

lessons. In lesson #[graphing equations] we looked at graphing equations of the form y = something and 

then graphing a point as an x-coordinate and a y-coordinate. Functions work the same way except that they 

are of the form f(x) = something. We can graph functions the same way we do equations by substituting 

f(x) for y. This means that instead of a y-axis, we will have an f(x) axis, and instead of an x-coordinate and 

a y-coordinate, we will have an x-coordinate and an f(x)-coordinate. A simpler way to think about this is 

that f(x) is essentially interchangeable with y. 

Part 1: x and x 2 

Begin by plotting the graphs of the two basic functions. If the weather is nice, it can be fun to do this 

outside on the sidewalk with chalk, otherwise use sheets of butcher paper and markers. Draw two large 

sets of axes, forming all four quadrants with each, and label the horizontal axis with x and the vertical axis 

with f(x). Mark the axes with evenly spaced hatch marks that go from -10 to 10 on both the horizontal and 

vertical axes. Above the first write the function f(x) = x and above the other write f(x) = x 2 . Next, make a 

value chart for each one like this: 

x f(x) 

157


Starting with f(x) = x 2 , ask the students to give you sample numbers for x and have them compute the 

corresponding f(x). Write these results down in the table. Once they they are comfortable choosing an input 

and computing the output, designate one student as the writer. Go around the circle giving each student 

some input in the range -3 to 3 and having them compute the corresponding f(x). Have the writer plot the 

points as the others calculate them. Once they have computed the outputs for all x values in the range -3 to 3 

and plotted the corresponding points, connect the points in a smooth curve. Repeat the same process with a 

different writer and the function f(x) = x on the other set of axes. Throughout the lesson, switch writers each 

time you make a new graph. The resulting graphs should look roughly like these: 

Part 2: Flip them over 

158 

f(x) = x 2 f(x) = x 

For the rest of the graphs in the lesson, use the same process as before to make them: 

- Make a value chart 

- Go around the circle and have students compute the outputs for given inputs 

- Once they are comfortable with the function, choose one to be the writer 

- Go around the circle, giving each student an x value that can be plotted (isn’t too big) and having 

them compute the resulting f(x) value while the writer plots the point. 

- When all the points have been plotted, connect them with a smooth curve (if all the points form a 

straight line, connect them with straight lines) 

The first manipulation to try is to make the functions negative. Using the same sets of axes, repeat the 

plotting process for the functions f(x) = -x 2 and f(x) = -x (for clarity, these functions can be written f(x) = -(x 2 ) 

and f(x) = -(x) if the students aren’t familiar with order of operations). The resulting graphs should look 

like these:

f(x) = -x 2 f(x) = -x 3 

Notice that in both cases, making the function negative flips the graph over the x-axis. 

Part 3: Shifting on the y-axis 


The next transformation to explore is what happenes when you add or subtract a constant number from 

the function. For these examples, use f(x) = x 2 + a where a is a constant. Start off by posing the question: 

What if I wanted to move the whole graph up on the y-axis by 1? Give them time to try to figure out how to 

change the function in order to shift it up or down on the y-axis. You might have a student that just gets it, or 

you might not. Either way, give them time to grapple with the question then do examples with f(x) = x 2 + 1 

and f(x) = x 2 - 1. The results should look like these plots: 

f(x) = x 2 + 1 f(x) = x 2 – 1 

159


Part 4: Shifting on the x-axis 

Shifting on the y-axis is pretty straightforward: just add and subtract to shift up and down. Shifting on 

the x-axis, however, is a little trickier. To shift left by 1, f(x) = x 2 becomes f(x) = (x + 1) 2 , and to shift right it 

becomes f(x) = (x – 1) 2 . This may seem backwards since you are adding to go in the negative direction and 

subtracting to go in the positive direction, but it makes more sense if you think about it in terms of the function’s 

input and output. If you know that an input of 2 will give you an output of 4 with f(x) = x 2 , but you 

want an input of 1 to give you an output of 4 (shifted 1 to the left), what do you have to do to the desired 

input of 1 to be able to run it through f(x) = x 2 and get an output of 4? You have to add 1 to it to make it 2. 

When you incorporate this into your new function, it comes out looking like f(x) = (x + 1) 3 . The same sort of 

logic applies when trying to shift to the right. Do two examples of this with the students, one with f(x) = (x - 

1) 2 and the other with f(x) = (x + 1) 2 . They should look like these plots: 

Part 5: Extras 


• http://en.wikipedia.org/wiki/Function_(mathematics) 

• http://en.wikipedia.org/wiki/Cartesian_coordinate 

160 

f(x) = (x – 1) 2 f(x) = (x + 1) 2 

If you have extra time, or want to stretch this lesson out, you can also do examples that combine multiple 

transformations. For example, if you wanted to shift f(x) = x 2 up on the y-axis by 1 and left on the x-axis by 

2, the resulting function would be f(x) = (x + 2) 2 + 1. You can also look into the scaling transformation. This 

transformation takes a function and stretches it or compresses it. If you wanted to compress the graph of f(x) 

= x 2 by a factor of 2, the result would be f(x) = 1/2 × x 2 .

Notes: 


161

162 

An Introduction to Sets

Lesson #29: Set Basics 



An Introduction to Sets 


• Explain the concepts of Union and Intersection 

• Explain that if there exists a one-to-one matching between two sets they must be the same size 

Materials: 

• Chairs (one fewer than number of group members) and an open space to play a game 

• Yarn 

• White board or other visible writing surface 

• Paper and pencil for yourself to record the results of the game 

Set theory is a structured way of looking at sets of things. A set is simply a collection of zero or more 

objects. These objects can have traits in common or be totally unrelated. The basic notation for a set uses 

curly brackets like this: 

{x, y, z, a, dog, 1, 23, -9.87} 

The two most basic concepts in set theory are the ideas of union and intersection. These are both 

ways of combining sets. The union of two sets, A and B, is the set containing all elements that appear in 

either A or B. Union is denoted with a U symbol like this: 

{1, 2, 3, 4} U {4, 5, 6, 7} = {1, 2, 3, 4, 5, 6, 7} 

In picture form, the union of two sets looks like ths: 

Notice that even though 4 appears in both A and B in this example, it only appears once in the answer. 

The intersection of two sets, A and B, is the set containing all elements that appear in both A and B. Intersection 

is denoted with a symbol like this: 

U 

{1, 2, 3, 4} {3, 4, 5, 6} = {3, 4} 

U 

163


164 

In picture form, the intersection of two sets looks like this: 

Two more important notes about sets are that the order of the elements does not matter, so the set {1, 2, 

3} is considered to be the same as the set {2, 3, 1} and that sets don’t contain repeat elements. 

One aspect of set theory that mathematicians are particularly interested in is the size of sets. This seems 

fairly trivial when dealing with relatively small sets, but when they become very large and even infinite, 

knowing how to find the size of sets can provide some very interesting results. If you are trying to find the 

size of some set, one common sense approach is to compare it to sets that you do know the size of. This 

can be done by matching elements in the known set to elements in the unknown set. If there is exactly one 

element in the know set for each of the elements in the unknown set, then it’s easy to see that the sets are 

the same size. This is called finding a one-to-one matching between the two sets. An example might look 

something like this: 

Part 1: Fun and Games 

known: size = 26 unknown: size = ? 

{1, 2, 3, 4, ..., 26} {a, b, c, d, ..., z} 

{1, 2, 3, 4, ..., 26} 

{a, b, c, d, ..., z} 

unknown: size = 26 

To start off, use this basic gym game as a way to get the extra energy out and to collect some good data. 

Set up a circle of chairs so that there is one fewer chair than there are students. Every student gets a chair 

except for one who starts in the middle. The student in the middle says something that applies to some or 

all of the students in the circle (for example, “is wearing jeans”), and any student to which it applies has to 

get up and find a new chair. The student in the middle also tries to claim a chair so when all is said and done 

there will still be someone new in the middle. The game continues for as long as you want it to (15 minutes 

is probably a good bet). For each category that the students use, write down the category and all the students 

to which it applies. When you finish playing, you should have a nice group of sets to use for examples. 

Some example sets might be: 

Wearing Blue Jeans = {Jim, Jane, Joe} 

Has Brown Eyes = {Jim, Jan} 

Has Green Hair = {}

Part 2: Tying the sets together 


• http://en.wikipedia.org/wiki/Set_theory 


Now that you have collected the sets, choose a few of them that demonstrate the ideas of union and 

intersection well. For union, find two sets that have mostly different students in them (one or two overlaps 

could be good, though). Have the students in the first set stand up and tie them loosely together with yarn. 

Next have the students in the second set stand up And tie a piece of yarn around them. Next, put a big piece 

of yarn around all the standing students to demonstrate the union of the two sets. To demonstrate intersection, 

use the same idea. Have two sets stand up and tie each together with yarn. There should be at least one 

person that is shared between the two sets (in the intersection). This person, or these people, should get tied 

into both groups to demonstrate the intersection. Next, present the idea of set size and a one-to-one matching. 

Choose two sets that are the same size (preferably ones that have no intersection and use all the students in 

the group), have them stand up, and tie each one together. Now have them stand facing each other and have 

each member of the set hold hands with someone from the other set (or just stand across from their partner if 

they don’t want to hold hands). Everyone should be able to hold someone else’s hand. This is clearly because 

the sets are the same size. Next, use two groups that are different sizes and do the same thing to show 

them that someone has to get left out meaning that there isn’t a one-to-one matching. Again, this is clearly 

because the sets are different sizes. Do as many of these examples as you want until the students have a good 

handle on union, intersection, and one-to-one matching. 

Part 3: Sets in real life 

Once the students have figured out how the basic ideas work, ask them to think of sets in the real world. 

Some popular examples are playing cards, letters in the alphabet, countries, and states. Using the examples 

that they come up with, ask them more questions about union, intersection, and one-to-one matchings. Can 

they make a one-to-one matching between the clubs and the spades? Can they make more than one? What 

is in the intersection of Asia and Europe? These sorts of questions will help them see how set theory can 

apply to real life. If you get a chance, try to suggest some infinite sets for them to think about. The easiest 

examples are the counting numbers (0, 1, 2, 3, 4, ...), the rationals (fractions), and the real numbers (all 

decimals). 

165


Notes: 

166

Lesson #30: Infinite Sets and Greek Punishments 






• Present two examples of infinite sets in Greek Mythology 

• Show that infinite sets, even if one seems to be growing more quickly, can be the same size 

Materials: 

• White board (or other visible writing surface) 

• Scratch paper and pencil (don’t give it to the students until after the stories) 

The basic idea behind this lesson is to present two infinite sets that on first glance seem like they might 

be different size and show, using the idea of a one-to-one matching from the last lesson, that they are actually 

the same size in the long run. 

Part 1: The myths 

Start the lesson by telling these two stories to the students. They are both examples from Greek Mythology 

where the main character gets punished to repeat some task for all eternity. While telling the 

stories, emphasize the infinite nature of the punishments. 

Prometheus: Prometheus was a Titan in ancient Greek mythology who was in charge of creating the 

human race. He was one of the only Titans that Zeus (king of the gods) trusted, so he was also entrusted 

with the task of guarding the sacred element of fire so that none but the gods could use it. As the creator 

of the human race, however, Prometheus liked humans too much, so he disobeyed Zeus and taught them 

about fire. When Zeus found out, he was furious, and he punished Prometheus very harshly. Prometheus 

was to spend the rest of eternity in Hades strapped to a giant boulder. Periodically an eagle would land on 

him and peck out his liver. Since Prometheus was a Titan, however, his liver would just grow back, so the 

eagle would come back and peck out the new one. This cycle was to go on for all eternity. 

Sisyphus (pronounced like sissy fuss): Sisyphus was a powerful king in ancient Greece who was know 

for being deceitful and treacherous. In order to cheat death, Sisyphus told his wife that when he died she 

should not make the usual sacrifices that show respect for the dead. When he did eventually die, Sisyphus 

convinced Persephone, queen of the underworld, that his wife was being neglectful and that he had to go 

back and set her straight. She released him from Hades on the condition that when he had confronted his 

wife he was to return. Sisyphus, however, did not honor this deal and once out of Hades, never planned 

to go back. Eventually, however, the gods caught up to him and he was taken back and given a very sever 

punishment. He was to spend the rest of eternity rolling a large boulder up a hill only to have it roll back 

down to the bottom once he reached the top. 

167


The made up part: It just so happened that Prometheus and Sisyphus were right next to each other in 

Hades. Prometheus noticed Sisyphus rolling his rock up the hill and started wondering how many times Sisyphus 

got to the top in between times that his liver got pecked out. It turned out that Sisyphus was able to 

roll the rock up the hill three times for every one time that Prometheus’ liver grew back. The next question 

that Prometheus asked himself was, who would end up performing their punishment more often in the long 

run. 

Part 2: The math 

After telling these stories to the students, pose Prometheus’ last question to them: which punishment gets 

performed more times in the long run? The two conflicting answers are: 


• http://en.wikipedia.org/wiki/Set_theory 

• http://en.wikipedia.org/wiki/Prometheus 

• http://en.wikipedia.org/wiki/Sisyphus 

168 

- Sisyphus does more repetitions because he does his three times for every one of Prometheus’ 

- It doesn’t matter since they’re both being performed for ever. 

Suggest to the students that they look at the punishments as sets. Write the two sets on the board like 

this: 

Prometheus: 1 2 3 ... 

Sisyphus: 1 2 3 4 5 6 7 8 9 ... 

It’s easy to see that you can make a one-to-three matching between these two sets by matching each 

of Prometheus’ new livers with the three times that Sisyphus rolls the rock up the hill while it is growing. 

Unfortunately that doesn’t actually tell you anything about the relative sizes of them. Ask the students if 

they can find a way to make a one-to-one matching. With luck, they will discover it on their own, but you 

may have to coach them along. The trick is to just match the 1 in the Prometheus set to the 1 in the Sisyphus 

set, the 2 in P with the 2 in S, the 3 in P with the 3 in S, etc... Since both sets are infinite, you will always be 

able to make such a matching, and therefore both sets are the same size. 

If you have extra time, start the students thinking about other infinite sets. Ask them if they think the 

counting numbers (0, 1, 2, 3, 4, ...) are the same size as the integers (... -3, -2, -1, 0, 1, 2, 3, ...). The answer 

is yes, and to show this, just match the even counting numbers with the positive integers and the odd counting 

numbers with the negative integers. 

0 1 2 3 4 5 6 7 8 

0 -1 1 -2 2 -3 3 -4 4

Notes: 


169


170 

Lesson #31: The Infinite Hotel 




• Explore several scenarios related to infinite sets 

• Explain how different infinite sets can be the same size 

• Understand that infinity is not a number in the conventional sense 

Materials: 

• The story (Supplement 1) 

• Whiteboard and dry-erase marker 

• OR Pencil and paper 

The idea behind this lesson is to read the following story, pausing every time the clerk is presented 

with a new problem. Each time a new problem comes up, ask the students to help the clerk figure out 

what to do. The story contains good explanations of the solutions, so use them to help the students along 

as needed. The students may need help getting into the right mindset on the first one, and the last one is 

particularly difficult. It can also help to draw a picture of the hotel (or the first part of it) and illustrate the 

solutions to each problem. Some of the language and humor will be a bit over the heads of the students, so 

it’s a good idea to paraphrase and skip sections in order to tailor the story to your group. 


• http://scidiv.bcc.ctc.edu/Math/infiniteHotel.html (This is the source of the story.)

The Story of the 

HOTEL AD INFINITUM 

by B. David Stacy 


This story is not true (in the sense of being real ), for certainly there is no such thing as an infinite hotel . 

What I have done is taken an idea of David Hilbert’s [1862-1943] and put it in a context that students would 

enjoy, so rest assured that the mathematics is perfectly valid. The point of the story is that the concept of 

infinity is a very strange and abstract thing. So if you’ll just play along with me, I’ll tell you about a very 

weird night I had one time while I was in college, working at the Hotel Ad Infinitum ... 

I arrived at work that night, ready to relieve the desk clerk who worked before me on Friday nights. He 

told me the most unbelievable thing: the hotel was full! Perhaps I should describe the place to you. It was 

just one great big long hallway; there was a door at the entrance, and when you walked in, the desk was at 

the left. Then the hall opened before you, endlessly. Along the left hand side of the hall were all the odd 

numbered rooms {1, 3, 5, 7, 9, ...} and at the right side were the even numbered rooms {2, 4, 6, 8, ...}. The 

hallway went on and on, on and on forever! It was hard to imagine that the place was full, but he assured me 

that it was. I should have known right then and there that something strange was going on, but I had an exam 

coming up, so I sat down, pulled out my calculus book, and started studying. 

in. 

A little after one o’clock, a huge stretch limo pulled into the parking lot. A chauffeur got out, and walked 

“Howdy, I need a room for the night; my boss is sleepy; he had a hard game tonight.” 

“Baseball player?” That figured; even in those days salaries were out-of-sight! But I told him that the 

place was full: “That’s what the sign says, right?” 

“Wrong; back in a minute.” He went out to the limo, popped open the trunk, and pulled out a little package 

about the size of a loaf of banana nut bread; it turned out it was a different kind of bread all together! He 

brought it in, set it on the desk and slid off its velvet cover and--lo and behold--it was a gold brick! 

We’d been studying compound interest in one of my classes, and I knew that the student loans I was taking 

out were going to cost me a LOT more than I was getting from them. My eyes widened with amazement. 

I looked up at the driver, who was smiling as he said, “So, you think there’s something we can work out?” 

*** Pause and pose the question *** 

You bet there was! I immediately grabbed the intercom, and announced to all the guests: “Please excuse 

the interruption, but if you’re in Room N , would you kindly move to Room N+1 ?” 

So, the guy in Room1 went to Room 2, the couple in Room 2 went to Room 3, et cetera. It was a mad 

flurry of rushing folks, dashing across the infinitely long hallway at the Hotel Ad Infinitum ... amazing! 

Please note that no one lost out, because there was no end to the hallway, and when everyone was settled, 

there was no one in Room1, right? So the baseball guy took Room 1, I took the gold brick, and proceeded 

to write my letter of resignation. Incidentally, this must mean that infinity plus one equals infinity, because 

171


I took an infinite number of guests, added the baseball player, and put them all up in the Hotel Ad Infinitum. 

Amazing, isn’t it? I was blown away, but the real weirdness had not yet begun. 

While I was trying to figure out how to turn my gold brick into normal money, I heard a tremendous rattling 

sound, looked out into the parking lot, and suddenly there appeared a beat up old VW van, smoke pouring 

out of its engine, a little trail of oil following it. The driver turned it off (though it kept running for a bit, 

sputtering and clicking and gasping) and ran into the hotel. Looking a little wild-eyed, he exclaimed that they 

needed rooms for the night. They? Rooms? 

172 

“Sir, did you notice the NO VACANCY sign outside, all lit up, bright, flashing neon?” 

He talked on for quite a bit, got confused a few times, but I managed to sort out the story. Seems Dylan 

was playing nearby, and the van outside was carrying an infinite number of Dylan freaks, all ready to catch 

their man in action. Actually, he was my man too, so I was very interested. We talked awhile and it turned out 

that he had an extra ticket. I was wondering where he had gotten an infinite number plus one Dylan tickets, 

but I figured what-the-hey? Anyway, he offered to lay the ticket on me if only I could put ‘em up for the 

night. 

*** Pause and pose the question *** 

I was ready to quit anyway, so I figured why not, and jumped back on the intercom and announced, “Ah, 

sorry to interrupt again, folks, but we have an emergency here, and if you’re in Room N would you please 

move to Room 2N.” 

So the baseball player went to Room 2, the guy in Room 2 went to Room 4, the couple in Room 3 went 

to Room 6, and so on. Again, no one was put out on the street, since--as you may have guessed--the Hotel Ad 

Infinitum had no back door! When that was over, all of the original guests, along with the baseball guy, were 

all on the right-hand side of the hotel, in the even-numbered rooms (of which there are an infinite number) 

and that left all the odd-numbered rooms vacant. So, I put the Dylan freaks into the odd-numbered rooms, 

which was sort of appropriate, I suspect! I guess this means that infinity plus infinity equals infinity, since I 

added an infinite number of Dylan freaks and put them in an infinite hotel which was already full!?!? Wait a 

minute ... 

So there I was, my gold brick, resignation letter, and Dylan ticket in hand, staring at the clock, counting 

down to my new-found freedom, when all of a sudden--oh no, how could this be--a caravan of buses pulled 

in, an infinite number of buses, and on each bus, an infinite number of people! An infinite number of infinities! 

What was happening, as I was soon to find out, was that there was to be an ecumenical council of all 

the galaxy’s religions, and every single religion had sent its own busload, loaded with an infinite number of 

its faithful! Yes, I was seriously in trouble on this one! Naturally, the driver of the first bus jumped out, came 

bounding in, and requested “a few” rooms for the night ... uh huh! Sure, an infinite number of infinities, it 

was clear to me, clear as mud! Of course, I reminded him about the sign and how we were full and all, and 

he smiled and began asking about the Dylan freaks, about the baseball player (how he knew I had no idea) 

and then started to remind me of the story of Mary and Joseph trying to get a room at the inn, and suddenly it 

occurred to me that with an infinite number of religions being represented here (all the religions of the galaxy) 

that one of them, no doubt, was the “right” one, and that it would not be wise to go down as the guy who 

wouldn’t give them a room for the night and sent them to the manger. I mean, did you ever wonder about that 

guy that sent Mary and Joseph to the manger? I wonder how he’s doing? 

*** Pause and pose the question ***


Well, in one of my courses, we’d been studying prime numbers {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ...} 

and how Euclid, nearly two-thousand years ago, proved that the list of prime numbers is infinitely long. So 

I got the following idea: I got back on the intercom (last time, I promise) and asked the current guests, “If 

you’re in room N, please move to room 2N.” 

Thus, the people in the hotel at that time went to rooms 2, 4, 8, 16, 32, 64, .... Then I went outside and explained 

my plan to the first few bus drivers, and asked them to pass it on. Here’s the plan: each bus received 

its own prime number, starting at three. So the first bus was 3’s, the second 5’s, the third 7’s, the fourth 11’;s 

and so on, one prime for each bus. Then, as for the people on the bus, they all received powers of those 

primes. For example, the first bus was assigned rooms 3, 9, 27, 81, 243, and so on, powers of 3 (the N-th person 

on the bus was assigned room 3N). The next bus was assigned powers of five, so they had rooms 5, 25, 

125, 625, et cetera, the N-th person being assigned room 5N. There were infinitely many primes, one for each 

bus, and an infinity of powers of each prime, so everyone had his own room! It took me a while to explain 

the scheme to everyone--there were a few of them that were math atheists, and it was rough going once or 

twice, but they all finally settled in. 

Then, as I was going over the register, I noticed that no one was in room 6 (= 2x3), nor in room 10 (= 

2x5), nor in any room whose number was a product of two or more different primes, since these rooms were 

not powers of a single prime, and hence had no bus assigned to them. A quick calculation showed that there 

were, in fact, an infinite number of vacancies! Incredible! I had taken an infinite hotel that was full, added an 

infinite number of infinities, and when all was done, I still had an infinite number of vacancies! 

The point of all this is that infinity is NOT a number, and--though there is a subject called “transfinite 

arithmetic”--you can’t think in terms of doing ordinary arithmetic with infinity. The best way to think about 

it, is that infinity is a property that some sets possess. Richard Dedekind defined an infinite set to be one 

which could be put in one-to-one correspondence with a proper subset of itself. It is this strange property that 

I have played with in the telling of my weird tale. 

Incidentally, you might like to know that the hotel closed shortly after that night. Seems there were a lot 

of lawsuits and stuff, and the last I heard, lawyers -- the number of which is growing without bound -- were 

convening there. Maybe they’ll all be trapped forever, and they won’t be bothering common folks any more. 

173


Notes: 

174

Fun with Numbers 

175


176 

Lesson #32: Negative Numbers 


Fits well with lesson #22 and #27 


• Understand a visual explanation of negative numbers and how they work with the operations 

addition, subtraction, and multiplication. 

• Practice basic mathematical operations with negative numbers 

Materials: 

• The numbers -10, -9, -8, ..., -1, 0, 1, ... 9, 10 written big and bold, each on a separate piece of paper 


Negative numbers are a fundamental piece of mathematics and a basic component needed for many of 

the lessons in this book. Students unfamiliar with them, however, can be really confused by the fact that a 

negative times a negative is a positive and that subtracting a negative number is really like adding a positive 

version of that number. 

Part 1: Adding and Subtracting Negatives 

For the setup, find a nice long wall and lay the numbered sheets of paper out along it to form a large 

number line (make sure they are evenly spaced). Give the students an example of adding two positive 

numbers (make sure that they don’t add up to more than 10). Have one student stand on the first number 

in the problem and have a second student walk the appropriate number of steps to the left or right to represent 

the addition/subtraction of the second number. To illustrate the problem 5 + 5 one student will stand 

on 5 and a second student will walk five to the right to get to 10. Next, give them an example of adding a 

negative number to a positive number and have students stand on each one. Explain that adding a negative 

number is just like taking away (subtracting) the positive version. On the number line this translates to 

walking to the left. 

Do a few examples like this until the students get the hang of it. For each example, after demonstrating 

with the students standing on the number line, do the problem on paper so that they get the hang of doing 

them that way too. Some students will want to crunch numbers, and others might benefit more from drawing 

the number line on their own paper. Both approaches are great! 

After addition, repeat the same idea with subtraction. Explain that subtracting a negative number 

means taking away a negative quantity so it is the same as adding a positive quantity. To illustrate the 

problem 3 - (-5) the second student will walk five to the right to get to 8.

Part 2: Multiplication 


• http://en.wikipedia.org/wiki/Negative_numbers 


After subtraction move on to multiplication. The most important part here is to show them that multiplying 

any number by -1 flips it over 0 on the number line like this: 

×-1 

X X 

-10 -5 0 5 10 

At this point propose to the group that any negative number can be written as -1 times the positive version. 

To help them believe this, show for example -7 = -1 × 7. Similarly, -54 = -1 × 54. Once the students 

get the hang of this, it’s easy to see that a positive times a negative is a negative while a negative times 

a negative is a positive. This can then be expanded to see that when dealing with exponents, a negative 

number raised to an even power will always be even, while a negative number raised to an odd power will 

always be odd. This lesson is most effective if you mix the visual examples of the human number line with 

paper and pencil examples. 

177


Notes: 

178

Lesson #33: Introduction to Sequences 


Precedes lesson #34, #35, and #36 


• Define sequence 

• Create examples of sequences 

• Identify and construct the Fibonacci sequence 

Materials: 



Using this exact phrasing, ask the group to write down a sequence of numbers. Each student should 

make his or her own sequence. If asked for more specific instructions, try to keep restrictions to a minimum. 

In other words, encourage them to define the word for themselves; don’t impose any guidelines. 

For example, if a student asks how long the sequence should be, respond by saying that there is no limit on 

how short or how long it can be. 

Once everybody is satisfied with their sequence (give them a minute or two), put all of the sequences in 

a place where everybody can see them. Now pose these questions in order to lead a discussion about what 

students think a sequence is: 

• Does everybody agree that these are all sequences? 

* If yes, why? (There are no correct or incorrect answers here!) 

* If no, why not? 

* If somebody thinks that one of the sequences is not a sequence, ask them to explain 

why they think that. 

• Did everybody use the same criteria for creating a sequence? 

* Did anybody generate a list of random numbers? 

* Did anybody generate a list of numbers based on a pattern? 

* Are there different types of patterns? 

• Does order matter? In other words, if I take somebody’s sequence and switch the second number 

with the fifth number, is it still a sequence? Why or why not? If it is still a sequence, is it the same 

sequence or is it a new one? (Still no right or wrong answers!) 

Now it’s time to propose a formal definition: A sequence is an ordered list of objects. Let’s break this 

down. In this definition, ordered means that order matters. For example, the sequence 1, 2, 3, 4 is different 

from the sequence 1, 3, 4, 2. Both are sequences with the same four objects, but they are not the same 

sequence. Moving the objects (or elements) around changes the sequence. Objects in our definition just 

179


means numbers, since students were asked to write down a sequence of numbers. It is possible to have sequences 

whose objects are not numbers, such as sequences of variables like a, b, c, d, e, f, g, h. 

Now pose these questions, keeping in mind the above definition (a sequence is an ordered list of objects): 

180 

• According to this new definition, how long can sequences be? (The definition says nothing about 

length, so sequences can be however long or short you need or want them to be!) 

* Here are some more definitions that help us talk about different kinds of sequences based 

on the number of elements in them: 

Finite sequences have a finite number of elements. They do not continue on forever. 

Most, if not all, of the examples that students generated will probably be finite se- 

quences. All of the examples given above are finite sequences. 

Infinite sequences have an infinite number of elements. For example 1, 2, 3, 4, 5, ... 

and 3, 5, 7, 9, 11, . . . are infinite sequences. We use the three dots at the end to indi- 

cate that a sequence continues forever. Mathematicians are generally more 

interested in infinite sequences than finite ones. 

• Can a number appear in a sequence more than once? (Yes! Again, there’s nothing saying one can’t. 

For example 0, 1, 0, 1, 0 is a sequence under our definition. Note that it is also a finite sequence.) 

• Are lists of random numbers sequences? (Sure!) 

* However, lists of random numbers are not very interesting. For this reason, when math- 

ematicians talk about sequences, they are usually referring to a sequence of numbers that 

follows some kind of pattern. These patterned sequences are the ones that interest us! 

Now ask each student to make another sequence, but this time require that it follow some pattern. Encourage 

creative patterns. They can be finite or infinite, simple or complicated. Once everybody has their 

sequence, have everybody look at all of them again and talk about the different patterns people used. Hopefully 

there will be at least a couple different types of patterns, but in case there aren’t, here are some examples 

of sequences that follow different types of patterns: 

2, 6, 2, 6, 2, 6 (Flip-flop between two numbers.) 

1, -1, 1, -1, 1, -1, . . . (Flip-flop OR multiply by -1 every time.) 

0, 2, 4, 6, 8, 10, . . . (Add 2 each time.) 

1, 1/2, 1/4, 1/8, 1/16, 1/32, . . . (Divide by 2 each time.) 

3, 6, 12, 24, 48, . . . (Multiply by 2 each time.) 

Now that we’ve got a good understanding about what sequences are and which types of sequences are 

perhaps more interesting than others, write this sequence out for the group: 

1, 1, 2, 3, 5, 8, 13, 21, 34, 56, 90, 146, . . .


Ask the question: Does anybody spot the pattern? (Except for the first two, every element in this sequence 

is obtained by adding the preceding two elements.) This is the Fibonacci sequence! It occurs in 

nature, art, the human body, etc. as can be explored and demonstrated in a separate lesson. 

*Note: It may be that one or more students are familiar with the Fibonacci sequence. If a student 

produces this sequence as an example before you offer it up, you might just ask them to hang tight for a 

while. Then, once you’re ready to introduce it, ask him or her to explain what the sequence is (how he or 

she constructed it). Or you could break with the order of the lesson and talk about the Fibonacci sequence 

before you talk about other types of sequences. Go with the flow! Do whatever you feel suits the situation 

and whatever you’re comfortable with. Above all, encourage exploration and discussion! 


• http://mathworld.wolfram.com/FibonacciNumber.html 

181


Notes: 

182


• Find patterns in sequences 

• Define and explore converging sequences 

• Define and explore diverging sequences 

Materials: 


Part 1: Pattern Detectives 

Lesson #34: Patterned Sequences 




Recall the definition of a sequence to the group (A sequence is an ordered list of objects) and ask students 

to share some things they remember from Introduction to Sequences. In this lesson we will first practice finding 

the patterns in various sequence examples. Below are examples of sequences with their patterns explained in 

parentheses at the end. You may use any of these or create your own 

1, 2, 3, 4, 5, 6, 7, 8, … (add 1) 

0, -2, -4, -6, -8, -10 … (subtract 2) 

3, 6, 9, 12, 15, 18, … (add 3) 

1, 2, 4, 7, 11, 16, 22, … (add 1, add 2, add 3, add 4, add 5, …) 

1, 1, 2, 3, 5, 8, 13, 21, … (the Fibonacci sequence: add previous two numbers together) 

1, 1, 1, 3, 5, 9, 17, 31, 57, … (add previous three numbers together) 

1, 3, 2, 4, 3, 5, 4, 6, 5, … (add 2, subtract 1, add 2, subtract 1…) 

10, 5, 2.5, 1.25, 0.625, … (divide by 2) 

1, 2, 4, 8, 16, 32, 64, … (multiply by 2) 

1, -2, 4, -8, 16, -32, ... (multiply by -2) 

1, 1, 2, 6, 24, 120, 720, 5040, … (multiply by 1, multiply by 2, multiply by 3, …) 

With a certain patterned sequnce in mind, begin by writing the first two numbers of the sequence down on 

paper and have the students guess what the pattern is based on these first two numbers. There will be many different 

patterns that they can guess with only two numbers. For example, if the first two numbers are 1, 2 the students 

could guess that you add 1 to get the next number or that you multiply by 2 to get the next number. Both 

of these suggestions are correct. Then reveal one more number at a time, and have the students guess the pattern 

after each number. As you write down more numbers, the students will see that some of their hypotheses were 

incorrect, and eventually they will see what the true pattern is. 

Now ask your students what strategies they used to determine the sequence. Many students will explain that 

they used a trial and error strategy, which means that they thought of a pattern, and then checked to see if it actually 

worked. Other strategies include seeing if the numbers are increasing or decreasing. If they are increasing, 

183


addition or multiplication might be involved in the pattern. If they are decreasing, subtraction or division might 

be involved. What other observations can they make about sequences that might help figure out the pattern? 

Next ask each student to think up their own sequence and write down the first few numbers in it. Once each 

student writes down the beginning of their sequence, each student can take a turn showing their sequence to 

the group and having the other group members guess what the pattern is. After determining the pattern behind 

each sequence, the group can then discuss the strategies that they each used to figure them out. Collect these 

sequences in a pile and put them aside for later in the lesson. 

Part 2: Convergence and Divergence 

Now we’ll look at some characteristics of sequences. Ask the group what they think it means to say that 

a sequence converges. If they don’t know what the word converges means, you could paraphrase it to mean 

“comes together” or “heads toward.” In math, we say that a sequence converges to a certain number if the terms 

in the sequence get closer and closer to that certain number forever, or if they reach that number and then stay 

there forever. 

For example, the sequence 1, 1/2, 1/4, 1/8, 1/16, 1/32, 1/64, … converges to 0 since the numbers get closer 

and closer to zero. One way to see this is to draw a number line with the numbers 0 and 1. Have the students 

mark out each subsequent number in the series. So they’d begin by making tick marks at 1, 1/2, 1/4, 1/8, etc. 

Ask them if the tick marks will ever go past 0? (The answer is no.) Will the numbers ever reach 0? (The answer 

is also no.) The students can think about it in the following way: as you are always dividing the remaining line 

segment between the previous tick and 0 in half, so there is always space between each tick mark and 0. 

184 

0 1/16 1/8 1/4 1/2 1 

The students may struggle to see this since with pencil and paper, there are only so many tick marks you can 

draw before they start to overlap and look like they touch 0. But then you could suggest that the students draw 

a blown-up image of, say, the part of the line from 1/16 to 0, and they could see that you could always magnify 

the image to make more room for more tick marks to fit between the last tick mark and 0. 

0 1/256 1/128 1/64 1/32 1/16 

1, 1/2, 1/4, 1/8, ... is an example of a sequence that converges to 0. Ask the students if they can think of any 

other sequences that converge. Some other examples are: 

1, 1, 1, 1, 1, 1, 1, 1, ... (sequence of ones; converges to 1) 

1, 1/3, 1/9, 1/27, 1/81, … (divide by increasing powers of 3; converges to 0) 

1, ½, 1/3, ¼, 1/5, 1/6, 1/7, 1/8, … (add 1 to the denominator; converges to 0) 

What about a divergent sequence? Ask your students what they think that means. If there’s no response 

give them this hint: divergent is the opposite of convergent. Together with your group define a divergent sequence 

as one that does not converge, but instead gets larger and larger without bound. A divergent sequence 

thus approaches positive or negative infinity, or both. 

The Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, … ) is an example of a divergent sequence since the numbers 

in that sequence get larger without approaching any other number. To see this visually, the students could


draw a number line and, like before, mark out each number in the sequence with a tick mark. They will see that 

soon their tick marks will run off the page. Thus the sequence diverges. Some other examples are: 

1, 1, 1, 3, 5, 9, 17, 31, 57, … (add previous three numbers together; diverges to infinity) 

1, 3, 2, 4, 3, 5, 4, 6, 5, … (add 2, subtract 1, add 2, subtract 1, etc.; diverges to infinity) 

0, -2, -4, -6, -8, -10 … (subtract 2; diverges to negative infinity) 

3, 6, 9, 12, 15, 18, … (add 3; diverges to infinity) 

1, 2, 4, 7, 11, 16, 22, … (add 1, add 2, add 3, add 4, add 5, …) 

1, -2, 4, -8, 16, -32, ... (multiply by -2; converges to both positive and negative infinity) 

Ask the group if they believe that every patterned sequence either converges or diverges. In fact, there are 

some which don’t! The flip-flopping sequences 1, -1, 1, -1, 1, -1, ... and 0, 1, 0, 1, 0, 1, ... are both examples. 

Pass out the students’ patterned sequences which you collected previously and ask the students to classify each 

of them as either convergent or divergent. Did anyone construct a sequence that was divergent to both negative 

and positive infinity? Did anyone construct a sequence that was neither convergent nor divergent? 

Part 3: A Fun Closing Activity 

On a lighter and slightly less mathematical note, another neat sequence is the “Look-and-say Sequence.” 

Tell your students what the sequence is called, and then show it to them to see if they can figure out the pattern 

based on the title. The sequence is: 

1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, ... 

To generate the next number in the sequence, simply say aloud the number(s) in the previous term. For example, 

the first “1” is said aloud as “one 1” which you then write as “11” to make the next term. Then the “11” 

is read aloud as “two 1’s” or “21.” Then the “21” is read as “one 2, one 1” or “1211.” Then the “1211” is read 

as “one 1, one 2, two 1’s” or “111221.” Notice that whenever there are more than one of a number next to each 

other, they are counted in a group. For example, if there was a number with “111” in it, it is read as “three 1’s” 

or “31.” Ask the students to figure out the next few numbers in the sequence without your writing them down. 


• http://en.wikipedia.org/wiki/Sequence 

• http://en.wikipedia.org/wiki/Fibonacci_number 

185


Notes: 

186

Lesson #35: Fun with the Fibonacci Sequence 





• Identify occurrences of the Fibonacci sequence in nature and art 

Materials: 


• Whiteboards and dry-erase marker 

• Copies Supplements 1-3 

Part 1: Bee Lineage 


The Fibonacci sequence shows up a lot in nature and can be used to model many mathematical relationships. 

First we will look at how we can use it to model the lineage of a male honeybee. Begin by asking 

somebody to remind the group of what the Fibonacci sequence is. Then have everybody write down 

the first ten or so numbers in the sequence. (If they need a reminder, tell them that it starts with 1, 1.) 

Now read this aloud to the students (or explain it in your own words): If a female bee lays an egg and 

a male does not fertilize it, it will hatch a male. If a female bee lays an egg and it is fertilized, it will hatch 

a female. So, all female bees have two parents (a female and a male) and all male bees have one parent (a 

female). The illustration below may be helpful for visualizing this. 

Parents: Female Male Female 

\ / | 

Offspring: Female Male 

The most important thing here is to make sure everybody understands that female bees have two parents 

and male bees have one parent. Now consider a male bee and his ancestors. There is an important 

distinction to make here. When I taught this lesson, there was some confusion about what we were mapping. 

Some students looked at the offspring of a male bee, and then their offspring and then their offspring, 

etc. This is not what we want. Rather, we want to look at a male bee’s parent (which we know 

must be one female), and then her parents and then their parents, etc. For this reason, I suggest starting 

with the male bee at the bottom of the page and drawing a tree which expands upward. (I found that starting 

at the top and working your way down the page gives the feeling of mapping offspring rather than 

parents.) 

Show the students how to draw the first couple of generations of this family tree, as illustrated below 

(M stands for male, F stands for female). Then have them draw as many generations as they can. (They’ll 

187


find that they run out of room on the paper quite quickly.) You might suggest that they start with their paper 

in the landscape position (sideways). 

188 

Total F M 

Before continuing, have everybody stop and take a look at their trees. Encourage the students to look for 

anything that seems interesting. Does anybody see any pattern(s)? 

Let’s look at the number of bees in each generation. I’ve drawn seven generations above, and the numbers 

in the left-most column of each row are the total number of bees in each generation. What do you 

notice about these numbers? (They’re the Fibonacci sequence!) 

Why does this happen? Open up a discussion about answers to this question, keeping in mind that the 

answer is somewhat complicated. It might be a good idea to give the students five minutes of silent thinking 

time before starting this discussion. They probably won’t be able to explain why the Fibonacci sequence 

occurs, but they should be able to come up with some good starting thoughts. 

Now look at only the number of female bees in each generation. What do you notice? (It’s the Fibonacci 

sequence again!) Is the same true for the number of male bees in each generation? Keep track of these 

numbers in columns next to the total number of bees in each generation, as in the diagram above. Discuss 

the patterns you find. Does this help anyone understand why the total number of bees in each generation is a 

Fibonacci sequence?

Part 2: The Golden Rectangle 


Pose this problem: Suppose I have a new outdoor patio. Right now it’s just a plain, boring, concrete rectangle. 

I want to tile it with square tiles. The catch is that I don’t want any of the square tiles to be the same 

size. Is this possible? This part of the lesson will lead you through an interesting approach to ansewring the 

question, but you should be aware ahead of time that the method described below does not actually follow 

the rule of using all differently-sized squares. 

Have each student draw a big rectangle on his or her whiteboard and experiment with different ways of 

covering it in squares of different sizes. This can also be done with a pencil and graph paper. Whiteboards 

have the advantage of being easily erased for multiple trials, and graph paper has the advantage of making it 

easier to draw exact squares. After everybody has had the chance to try a few different ideas or techniques, 

have everybody put their whiteboards (and/or pieces of paper) in the middle of the table. Discuss the different 

strategies people tried and the different problems that they ran into. Was anybody successful? Do 

students think this is possible? Discuss. 

When I taught this lesson, one student came up with the following strategy. Hopefully one or more 

students in your group will discover this method as well. If this happens, have him/her/them explain what 

they did. If nobody used this method, you should suggest it now. (And even if one or more students tried 

this method on their own, lead all of the students through a thorough explanation of this method.) Do this 

by first giving each student one of the pre-drawn Golden Rectangles, but do not say that there is anything 

special about this rectangle. Don’t even call it a Golden Rectangle. The hope is that one or more students 

will eventually figure out that there is something special about this particular rectangle. Here is the method, 

along with some questions you should ask the group while you go along: 

• Suggest drawing the largest square possible inside the rectangle. 

* What will the length of the sides of the square be? (The length of the shorter side of the rect- 

angle.) 

* Where would be the best place to put the square? The best thing to do would be to use one of 

the shorter sides of the rectangle as one of the sides of the square as shown below. Have on or 

more of the students explain why this is in his or her own words. For example, why wouldn’t 

we want to put it in the middle of the rectangle? 

• Now, color in the part of the rectangle that is covered by the big square tile and focus on the remaining 

white part. 

* What shape remains? (Another rectangle!) Ask the group what they think should be done 

next. Hopefully somebody will suggest doing the same thing again. That is, draw the largest 

square possible in the new (smaller) rectangle. If nobody suggests this, try giving the group 

some sort of hint or guidance until they figure it out. 

• Continue doing this. What eventually happens? (We end up with two squares of the same size instead 

of one square and one rectangle!) 

189


Above is a picture of what the rectangle should look like after each new square is drawn in. I’ve written 

in the lengths of the sides of the squares. You might notice that they are numbers in the Fibonacci sequence. 

See if you can help the group figure this out without explicitly telling them. You might suggest exploring the 

lengths of the sides of the squares. Or you could be more broad and ask them to look for patterns or anything 

interesting. 

One thing that’s just kind of fun to do at this point is to create a spiral through the squares. Below is an 

example. Note that it requires drawing the squares at the appropriate end of each new rectangle—otherwise 

you’ll find that you are unable to draw the spiral. For example, in the picture below on the right, the highlighted 

square was drawn at the wrong end, and thus the spiral cannot be continued. 

190


• http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html 


Now ask some closing questions: 

• Did we answer the question? (No.) We found a way that almost allows us to tile a rect- 

angle with squares that are all different sizes. Does this necessarily mean that it’s impos- 

sible to do so? (No.) So is it possible? (Maybe!) 

• But this is a special rectangle. Why is this rectangle unlike other rectangles? (Because 

the dimensions are numbers in the Fibonacci sequence!) 

• What happens when you try this method on a different rectangle, without the Fibonacci 

dimensions? 

• Have you changed your opinion about whether or not it is possible to tile a rectangle with 

different sized squares? Why or why not? 

Part 3: Occurrences in Art, Architecture and Nature 

Hand out the Supplements with photos on them and let the students explore and discuss what they see. 

If the group shows a lot of interest in the Fibonacci sequence and the Golden Ratio and want to find more 

examples of where they are found in nature, architecture and art, you might bring the group to the media 

center and spend part of a day (or a whole day) doing research on the computer. Wikipedia and Google are 

both great places to start. 

The images in the Supplement were taken from the following websites: 

• Parthenon: 

• Hand and flower petal information: 

• 

Body/art: < http://britton.disted.camosun.bc.ca/goldslide/jbgoldslide.htm> 

191

Fun with Numbers Supplement 1: Blank Graph Paper 

192

Supplement 2: Golden Rectangle 


193


Did you ever wonder why there aren’t many flowers with 2 petals? Most 

flowers have a number of petals that is in the Fibonacci sequence! 

3 petals: Lily 

5 petals: Buttercup 

8 petals: Delphiniums 

13 petals: Black-Eyed Susan 

21 petals: Aster 

34 petals: Plantain, Pyrethrum 

55, 89 petals: Michaelmas daisies 

194 

Supplement 3: Occurrences in Art, Architecture and Nature

Notes: 


195


196 

Lesson #36: Introduction to Phi 





• Approximate Phi (also called the Golden Ratio) using the Fibonacci sequence 

• Find instances of Phi in the human body 

Materials: 

• Paper and pencils 

• Yardstick(s) or tape measure(s) 

• Calculators 

• One copy per student of Data Sheet and Da Vinci’s Vitruvian Man (Supplements 1 and 2) 

Part 1: Finding Phi in the Fibonacci Sequence 

We will begin this lesson by reminding ourselves of the Fibonacci sequence. Ask one or more students 

to explain the construction of the Fibonacci sequence. Then have each student write out the first 

twelve numbers of this sequence: 

1 1 2 3 5 8 13 21 34 55 89 144 . . . 

The next part may be a little tricky for some and therefore may require slow, careful explanation and 

repetition. What we’re going to do is the following: 

• Take two numbers that are next to each other in the sequence. You can start anywhere you’d like, 

but a good place to start is with 1 & 2. 

• Divide the larger of the two numbers by the smaller one. In this case, we do 2 ÷ 1 = 2. 

• Do this for all pairs of consecutive numbers in the sequence. That is, for each set of two numbers, 

divide the larger by the smaller as shown below: 

2 ÷ 1 = 2 

3 ÷ 2 = 1.5 

5 ÷ 3 = 1.66667 

8 ÷ 5 = 1.6 

13 ÷ 8 = 1.625 

21 ÷ 13 = 1.61538 

34 ÷ 21 = 1.61905 

55 ÷ 34 = 1.61765 

89 ÷ 55 = 1.61818 

144 ÷ 89 = 1.61798


What do you notice? (The quotients seem to approach a number around 1.6 – cool!) If we kept 

doing this for many numbers in the Fibonacci sequence, we would find that the quotient approaches 

1.618033988749895. . . This number is called the Golden Ratio or Phi (pronounced like‘fly’ with no ‘l’ or 

alternatively like ‘fee’)! 

Part 2: Finding Phi in the Human Body 

Each student will take four measurements: their total height (floor to top of head), the distance between 

the floor and their belly-button, the distance between their elbow crea se and their longest fingertip, and the 

distance between their elbow crease and their wrist crease. 

To do this, have the students pair off and help take each other’s measurements. Each student can decide 

whether they want to take measurements in centimeters or inches, but they must use the same unit for all 

four measurements. Standing against a wall is a good idea while taking the total height and belly-button 

height. Similarly, you might suggest resting one’s arm flat and straight on a table while taking the arm and 

hand measurements. Once everybody has all four measurements recorded on their Data Sheet (Supplement 

1), complete these calculations: 

Total height Total arm length (to fingertip) 

Belly-button height Forearm length (to wrist crease) 

What do you notice? The values that students compute should all be approximately equal to the Golden 

Ratio. Whose measurements gave the calculation that is the closest? It might be interesting to take the average 

of all everybody’s calculations and see how close that number is to Phi. Ask the group if they can think 

of other places on the human body that we might find the Golden Ratio (for examples, see Further Reading). 

Give each student a copy of the Vitruvian Man (Supplement 2) and ask them to test some of their hypotheses. 

Did anybody find a good example? Have students share their findings. 


• http://en.wikipedia.org/wiki/Golden_ratio 

• http://goldennumber.net/body.htm (A great site with lots of examples of Phi in the human body.) 

197

Fun with Numbers Supplement 1: Data Sheet 

Total height: _____ 

198 

Data Sheet 

Belly-button height: _____ 

Total height divided by belly-button height: 

Arm length: _____ 

(elbow crease to fingertip) 

Forearm length: _____ 

(elbow crease to wrist crease) 

Arm length divided by forearm length: 


= 

= 

Data Sheet 








= 

= 


Data Sheet 









= 

= 

Data Sheet 








= 

=

Supplement 2: Da Vinci’s Vitruvian Man 


199


Notes: 

200

Lesson #37: Prime Numbers and the Ulam Spiral 

Approximate lesson length:1 day 


• Find all prime numbers between 1 and 225 

• Construct part of the Ulam Spiral and discuss its features 

Materials: 

• 2 pieces of butcher paper (ask the classroom teacher to show you where it’s kept) 

• 2 yardsticks (at least) 

• Colored pencils or markers (erasable colored pencils are best) 


Begin the lesson by asking the group to explain what it means to be a prime number. The answer: a 

number that is divisible only by 1 and by itself, nothing else. The goal here is to develop a sound understanding 

of primes and then develop a method for finding primes. Once everybody is comfortable with the 

definition of a prime number, ask this question: How could we identify all prime numbers between 1 and 

225? Discuss students’ answer(s) to this question. When I did this lesson, the group immediately figured 

that the easiest thing to do would be the following. (If no student in your group suggests this method, you 

should do so.) Here is the method: 

• Write down all numbers between 1 and 200. 

• Cross out all multiples of 2. 



* Why did we skip crossing out multiples of 4? (Because all multiples of 4 are also mul- 

tiples of 2, which we have already crossed off.) 

• Continue in this way until you’ve crossed out all multiples of 2 through 14. 

* If during this process a student asks: Why can we stop crossing out multiples after 14? 

You can tell them it’s because 14 is the biggest number with a square less than 200. (14 × 14 = 

196, while 15 × 15 = 225.) 

It might also be helpful to note that all multiples of 14 will already be crossed out because they are also 

multiples of 7. So we really only need to cross out all multiples through 13. Once all multiples through 

13 have been crossed out, go back and circle all the numbers that are not crossed out. These are all the 

primes! 

This method for finding primes will be the basis for the rest of the lesson. Once everybody understands 

this method and is convinced that it will work, divide the students into two groups. Each group 

gets one piece of butcher paper and a yardstick. Both groups will be writing down all numbers from 1 to 

200 in a grid, but they will be doing so in different ways. Group A will create a grid of numbers by writ- 

201


ing the numbers in rows from left to right. This group may choose how many columns to use. There are 15 

columns in my example below. Group B will create a grid of numbers by putting 1 in the center and then 

spiraling the rest of the numbers counter-clockwise around it. Pictures of both grids are shown below. 

202 

Group A’s grid 

Group B’s grid


Once each group has created its grid (this will take the bulk of the time for the day and require good 

teamwork), instruct them to cross out all the multiples of 2, 3, 5, 7, . . . , 13. Once everybody has finished, 

lead a discussion about the different grids. Here are some questions you can ask: 

• 

• 

What are the strengths and weaknesses of each method? 

* It is easier to cross out multiples on Group A’s grid because many of them lie in straight 

or diagonal lines (for example, all the multiples of 5 lie in one column). 

* Group B does not have this luxury; they must follow the spiral around to cross out the 

multiples by crossing out every 2nd number, every 3rd number, etc. 

Are there any interesting patterns that you see with regard to where the prime numbers lie? 

* Group B’s grid should have some interesting characteristics. The prime numbers seem 

to generally line up along diagonal lines! Nobody has been able to give a complete mathematical 

explanation for this, but it’s quite interesting. This method of mapping the prime 

numbers is called the Ulam Spiral. 

* Depending on how many columns Group A’s grid has, some interesting patterns may or 

may not show up. In my example below (which has 15 columns), you can see some 

checker board-like patterns showing up in some places. Cool! 

Below are the completed grids for both Group A and Group B. The gray boxes represent those that 

students should have crossed out (multiples of 2, 3, 5, etc.) and the white boxes represent those numbers that 

are left (primes). You may want to use these to check the groups’ final results. Did they find all of the prime 

numbers correctly? 

Group A 

203


204 

Group B 

As one last note, you may want to make a copy of the 399 x 399 Ulam Spiral (Supplement 1) on the 

next page and show it to the group at the end of the lesson. On this 399 x 399 grid the prime numbers are 

represented by black dots. One of the reasons this picture is so amazing is that you can see how the primes 

still tend to line up along diagonal lines, even when they are quite large. I’ve added a box in the center that 

contains Group B’s grid to give you a sense of just how large this new grid is. Can anyone spot the prime 

numbers 3, 2, and 11? (They make an “L” in the center.) 


• http://mathworld.wolfram.com/PrimeSpiral.html 

• http://en.wikipedia.org/wiki/Ulam_spiral

Supplement 1: 399 x 399 Ulam Spiral 

The Ulam Spiral on a 399 x 399 Grid 


205


Notes: 

206

Different Bases: Beyond Base 10 

207


208 

Lesson #38: Binary in Computer Science 




• Have a general understanding of the basic working of a computer 

• Explain why binary is important and useful for computers 

Materials: 

• Pencil and paper for yourself (students don’t need anything) 

Background 

The basic idea behind this lesson is to teach the students about the importance of binary as it relates to 

the internal workings of a computer. Most people, when they think of a computer, think of the screen and 

possibly the mouse and keyboard because those are the parts of it that they interact most directly with. In 

reality, those parts are just that: peripherals that enable a human user to interact with a computer. The real 

center of operations is housed inside the big ugly box (for a laptop it’s somewhere under the keyboard and 

for an iMac it’s somewhere behind the screen). In particular, the real work is being done by the processor. 

The processor is a small chip, roughly a 2 inch square, to which all other parts of the computer are connected. 

The processor has the ability to do basic manipulation of numbers very very very quickly. Everything 

you see on the screen is actually just the product of a bunch of numerical calculations being done by the 

processor at mind-boggling speed. An over-simplified example that can help explain this concept (and one 

that the students can understand well) is the idea of letters. Computers store an internal chart that matches 

each possible letter or character to a number. When the underlying program is told to display a letter, it 

goes to this chart, finds the corresponding number, and then knows which pixels on the screen to turn on. 

Ultimately, it has no idea that it has just displayed a letter, only that it processed a number in the way the 

chart told it to. It is up to the user to interpret it as an actual letter. 

Taking this explanation one step further, the processor takes electrical signals as input and sends electrical 

signals as output (it is an electric machine after all). This means that there needs to be some way to 

interpret electrical signals as numbers. The most straightforward way to do this is to measure how intense 

a signal is and assign a numerical value to it. This is where the idea of binary, or base 2, numbers comes 

in. Since most human arithmetic is done in base 10 (the kind we all learned in early elementary school), 

the natural idea would be to interpret these electrical signals on a scale of 0 through 9, perhaps using 0 to 

indicate no signal and 9 to indicate the strongest signal. This turns out to be pretty hard since it’s very difficult 

to regulate the intensity of electricity so precisely. It is much easier to tell if electricity is present or 

not (on or off). To indicate this we need only two digits, zero and one, and we use this new base to perform 

a new arithmatic which involves only zero and one. Using only two digits when we’re used to having 

ten to work with may seem like a limitation, but as you can see in the following lesson all numbers that we 

are used to in base 10 can also be represented in binary. Now that you are hopefully convinced of binary’s 

importance to computing, it’s time to convince the students.

Part 1: Draw a computer 


Ask the students to start naming off the pieces of a computer (a desktop system not a laptop). As they 

name off parts, draw a rough sketch and label them as you go. This doesn’t need to be a work of art, just 

clear enough so it can be recognized. A sample sketch might look something like this: 

Once they have named all the parts they can think of, ask the students which part they think is most 

important (or does the most work). With luck, someone will point to the box, but if not, you can just explain 

that that’s where the real work is done. Next, ask them if they have any idea what’s inside the box. They 

probably won’t mention the processor, but you might be lucky and have a student that does. Either way, 

the next step is to explain to them that the processor is a tiny piece of electrical circuitry that can process 

numbers really really fast. It looks something like this: 

Once they’ve accepted that this processor is what does all the work for a computer, tell them that all it 

can do is take in electrical signals interpreted as numbers, perform math on them, and spit out the results. 

The logical question then is how these electrical signals can be interpreted as numbers. 

Part 2: How loud am I talking? 

The first step in answering this question is to suggest the idea that you measure how much electricity 

is coming in to the processor. Ask the students if they’ve ever had to rate something on a scale. What sort 

of scale do they think you could measure the electricity on given that you are trying to make it look like 

a number? The logical answer is 10. As an example, tell them that you are going to say a word (doesn’t 

matter what) and that they should each rate how loud you are talking on a scale of 0 to 9 (or 1 to 10). Say 

something that should fall somewhere in the middle range of volume (you don’t have to have an exact value 

to it). Unless they collaborate or have already guessed your motive, you should get a pretty wide range of 

answers. This demonstrates how hard it is to accurately interpret electricity as a number if we are using base 

10. Next, tell them to guess on a scale of 0 to 1 where 0 is not talking and 1 is talking. Say a word and have 

them guess again. They should all guess 1. You can repeat this demo as needed to communicate the idea 

that it’s much easier to simply have two modes (sound or no sound) than to differentiate based on a scale of 

no sound, a little bit of sound, a little bit more, etc... . 

209


Part 3: We’ve only got 2 digits... Oh no! 

So at this point we’ve decided that we’d like to use only 0 and 1 as digits when talking about numbers 

for computers. The question is, can we still do all the fun stuff we do using 0 through 9 as digits? The answer 

is yes if we use a base 2 numerical system. If there’s time left, move on to working on the next lesson, 

but if not, tell them that for the next lesson they will learn how computers actually do math. 


• http://en.wikipedia.org/wiki/Binary_numeral_system 

210

Notes: 


211


212 

Lesson #39: Binary Number System 





• Give the students an understanding of how numbers can be represented in binary (base 2) 

• Write numbers in base 2 (count from 0 to 16 in base 2) 

• Convert numbers from base 2 to base 10 

• Convert numbers from base 10 to base 2 

• Add numbers in base 2 

Materials: 


Base 2, or binary, is a system of numbers that uses only two digits (0 and 1) instead of the ten digits 

we are used to (0, 1, 2, 3, 4, 5, 6, 7, 8, 9). In base 10 we have the 1’s column, the 10’s column, the 100’s 

column, the 1000’s column, etc... . The important thing to note here is that each column’s value is a power 

of 10. A number in base 10 can be decomposed into the sum of its column values like this: 

Chart for 324 (base 10) 

column 10 2 = 100 10 1 = 10 10 0 = 1 

number 3 2 4 

column value 300 20 4 

Binary works the exact same way except that the columns are powers of 2 instead of 10. A binary 

number can also be decomposed in a similar manner: 

Chart for 101 (base 2) 

column 2 2 = 4 2 1 = 2 2 0 = 1 

number 1 0 1 

column value (base 10) 4 0 1 

total (base 10) = 5 

Part 1: The Basics and Counting 

Break 324 down into its columns 

3 in the 100 column, 2 in the 10 column, and 4 in the 1 column 

Add the column values to get 324 back! 

Break 101 down into its columns 

1 in the 4 column, 0 in the 2 column, and 1 in the 1 column 

Add the column values to get 5 in base 10 

Show these charts to the students to help them understand how binary works. Have them write out 

their own similar charts with the base 2 columns at the top up to 2 4 . Starting with 0, have them write out 1 

through 16 (base 10) in binary. The results should look like this:


base 10 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 

base 2 0 1 10 11 100 101 110 111 1000 1001 1010 1011 1100 1101 1110 1111 10000 

Alternately, 0 can be written as 00000, 1 can be written as 00001, 10 can be written as 00010, etc... so 

that they are filling in all of the columns in the chart for each number. If the students are struggling, have 

them keep in mind the values of each column (16, 8, 4, 2, 1) and ask them appropriate questions to help 

them figure it out. For example, if a student is trying to write 6 (base 10) in base 2, you could ask them what 

the highest valued column is that 6 is greater than or equal to. The answer here is the 2 2 = 4 column. Have 

them put a 1 in the 2 2 = 4 column and then subtract 4 from 6 (since they’re trying to figure out what other 

columns need to be filled to make 6). Ask them the same question for the remainder of 2. It should fit into 

the 2 1 = 2 column, so have the student put a 1 in the 2 1 = 2 column. Subtract 2 from 2, resulting in 0. Since 

you’ve reached a remainder of 0, you should just put 0s in all the other columns. The result should look like 

00110 or 110. 

Part 2: Playing with Binary 

Now that the students have a grasp on how to write binary numbers, it’s time to show them that binary is 

just as powerful as base 10. To start, give each student two big numbers in base 2 (101101 for example) and 

have them convert them to base 10 using the chart strategy from step 1. Have them add the base 10 results 

together and write down the answer somewhere it won’t get lost. Next teach them how to add in binary. 

Addition in binary is the same as it is in base 10 except that you carry a 1 when the sum reaches 2 instead of 

carrying a 10. Here’s an example: 

1011 

+ 1010 -> 

1 

carry the 1 

1 

1011 

+ 1010 -> 

01 

1 

1011 

+ 1010 -> 

101 

carry the 1 

1 1 

1011 

+ 1010 -> 

1101 

1 1 

1011 

+ 1010 

11101 

Once the students have added their binary numbers, have them convert the result back to base 10. Hopefully 

this result will match the sum they got earlier. If there’s extra time, give the students more example 

problems to work on. For an extra challenge, see if they can figure out subtraction. It works the same way 

as subtraction in base 10. The only tricky part is borrowing, but just remember that you’re still borrowing in 

base 2 and it works just fine. 



• http://www.math.grin.edu/~rebelsky/Courses/152/97F/Readings/student-binary (An excellent tuto- 

rial on the Binary system, take a look at the Binary Addition section.) 

213


Notes: 

214

Lesson #40: Exploding Dots 


Following lesson #39 



• Become familiar with the idea of other bases (not 10 or 2) 

• Understand addition in different bases 


Materials: 

• Instructor’s white board / chalkboard (if not available, try taping blank sheets of paper to a wall) 


Ask the group if they think there are number systems in bases other than 10 (which we’ve been taught 

all along) and 2 (which we just learned last lesson). They will likely be eager to explore the possibility of 

base 3, base 4, and more! Luckily, the concepts from the binary number system can easily be extrapolated 

to any base you want, and this lesson is designed to illustrate how in a fun and interactive manner. To 

write a number in some base x, just use the same chart idea with columns that are powers of x that you used 

in the previous lesson. Here’s the generalized chart: 

column x 2 

number 

column value (base 10) 

Addition still works the same way too, although it can be a bit tougher to understand for larger bases. 

This lesson uses an abstraction to take the numbers and the symbols out of the way and get to the heart of 

how representation and addition works for any base you want. The fascinating and somewhat unfathomable 

theme of the lesson is the fact that numbers exist in and of themselves and writing them down as 0, 1, 

2, 3, 4, etc. is just our common way of representing them. To see just how many ways we could communicate 

a quantity, check out Supplement 1. Cool! 

Part 1: Loading up the containers 

x 1 

x 0 = 1 

Start by drawing a series of containers on the board that look like this (the number of containers isn’t 

important, just draw a bunch of them): 

These containers are going to represent the place columns for your number (from right to left: x 0 , x 1 , x 2 , 

x 3 ...). You want to start off with a low base number, so have the students give you some numbers between 

215


0 and 9 and choose a low one to be the base. This base will be referred to as the magic number from now 

on. For this example we’ll use base 3. Label the containers with their appropriate column labels like this: 

216 

3 6 = 279 

3 5 = 243 

3 4 = 81 

3 3 = 27 

Next, ask for a number from the students that isn’t too large. For this example, lets use 15. Make a pile 

of 15 dots to the right of your containers. 

3 6 = 279 

3 5 = 243 

3 4 = 81 

3 3 = 27 

Now, one by one add them to the right most container. 

3 5 3 = 243 

6 = 27 9 

3 4 = 81 

3 3 = 27 

When the right most container has the magic number of dots in it (in this case 3), it explodes and all the 

dots disappear except for one that flies into the next container to the left. 

3 6 = 279 

3 6 = 279 

3 5 = 243 

3 5 = 243 

3 4 = 81 

3 4 = 81 

3 3 = 27 

3 3 = 27 

Continue this process until you have used up all the dots in the pile. Every time a container has the 

magic number of dots in it, it explodes and one dot flies off to the next container to the left. This can sometimes 

cause a chain reaction. If there are two dots each in the first two containers and you add one more, the 

first container will explode making the second one have 3 which causes it to explode. The end result would 

be one dot in the third container and none in either the first or second. After walking through one example, 

the students should have a pretty good idea how it works to convert a number from base 10 (the number you 

started with) to whatever your new base is. If you think you’ll have time, or the students still seem confused, 

it can’t hurt to do another example with a different base. 

Part 2: Addition in Base x 

Decide as a group on any two numbers you’d like to add together (36 + 7, for example). Write them on 

the board and convert them using the exploding dots method into the new base you’ve been working with. 

Let’s keep using base 3 for the example here, and lets use 21201 for our example number.* Fill the appropriate 

containers up with the right number of dots, as seen on the next page. 

3 2 = 9 

3 2 = 9 

3 2 = 9 

3 2 = 9 

3 2 = 9 

3 1 = 3 

3 1 = 3 

3 1 = 3 

3 1 = 3 

3 1 = 3 

3 0 = 1 

3 0 = 1 

3 0 = 1 

3 0 = 1 

3 0 = 1

3 6 = 279 

3 5 = 243 

3 4 = 81 

3 3 = 27 

3 2 = 9 

3 1 = 3 


3 0 = 1 

Next, get a second number, lets use 1221, and place its dots above the appropriate containers. 

3 6 = 279 

3 5 = 243 

3 4 = 81 

3 3 = 27 

Start adding at the rightmost container. Following the same rules as before, whenever a container has 

the magic number of dots in it it explodes and one dot flies to the left. The only catch here is that even once 

a container explodes, if there are still more dots to add to that container, you have to add the rest of them before 

you start adding any to the next one. The dot that flies to the left gets added to the pile of dots waiting 

to get added but it doesn’t actually get added until after all the dots are finished being added to the current 

container. 

3 6 = 279 

3 6 = 279 

3 5 = 243 

3 5 = 243 

3 4 = 81 

3 4 = 81 

3 3 = 27 

3 3 = 27 

To finish the addition, first finish adding all of the dots that are hanging above the containers. The result 

of this example looks like this: 

3 6 = 279 

3 5 = 243 

3 4 = 81 

3 3 = 27 

Do a few examples of addition and try using different bases. Don’t worry if the students don’t understand 

it for all bases right away. The next part gives you time to talk to them individually. 

*Remember that a number in base x can have any digit whose value is less than x. For example if you 

are using base 6, a number can only have 0, 1, 2, 3, 4, 5 for digits. Similarly, if you are using base 16, a 

number can have 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 for digits and then you can make up symbols to represent digits 

equal to 10, 11, 12, 13, 14, 15. A common practice here is to assign letters for the higher digits (a, b, c, 

etc...). 

Part 3: Challenge Problems 

This step switches back to just using numbers. Give each student a few addition problems in different 

3 2 = 9 

3 2 = 9 

3 2 = 9 

3 2 = 9 

3 1 = 3 

3 1 = 3 

3 1 = 3 

3 1 = 3 

3 0 = 1 

3 0 = 1 

3 0 = 1 

3 0 = 1 

217


bases (start out with the same bases that you’ve done examples with). Let each student go to work trying to 

solve the problems. Walk around and watch for trouble spots. Take this opportunity to help clarify the ideas 

to the student on an individual basis. This can prove to be the most useful step for getting the students to fully 

understand the idea behind addition in different bases. 



• http://www.math.grin.edu/~rebelsky/Courses/152/97F/Readings/student-binary (An excellent tuto- 

rial on the Binary system, take a look at the Binary Addition section.) 

• http://mathworld.wolfram.com/Base.html (This provides a good outline of how to do math with dif- 

ferent bases as well as a list of other commonly used bases) 

Note: This lesson was adapted from one taught by Robert and Ellen Kaplan, founders of the 

Math Circle. Visit http://www.themathcircle.org/ 

218

Supplement 1 

How can you count the apples? 

Base 

Base 10 

Base 2 

Base 3 

Base 4 

Base 5 

Base 6 

Number of Apples 

5 

101 

12 

11 

10 

5 


219


Notes: 

220

Lesson #41: Jeopardy 




• Provide review and practice for the material covered over the last few lessons 

• A fun last lesson before switching to a new topic 

Materials: 

• White board (or other writing surface for the questions) 

• Copy of Jeopardy Questions (Supplement 1) 

• Scratch paper and pencils for the students 

• (optional) candy for prizes 


This lesson uses the familiar game of jeopardy to provide a fun and mildly competitive setting to review 

and practice working with numbers in different bases. The concepts covered are primarily addition 

in different bases and conversion between a non-standard base and base 10. The challenge problems at 

the end involve subtraction in different bases which hasn’t yet been covered in the lessons. It should be 

a challenge for the students to apply what they already know about subtraction in base 10 and addition in 

different bases to figure out subtraction in different bases. 

Part 1: Setup 

Split the group into two teams. The turns alternate between teams. When a team’s turn starts, one 

player on the team selects a question. The team then gets to work on the problem together. In order to receive 

points, however, the person that chose the question has to be able to explain the answer that they got. 

The person that chooses the question rotates from turn to turn so that every group member gets to choose a 

question and explain the answer. If a team gets a question wrong, the other team has a chance to answer it 

to steal the points. 

Part 2: Questions 

Depending on whether your group understands addition or base conversion better, you may want to 

rearrange which questions get which point values. There are three regular categories (each with a specific 

base to work in), and one challenge category where each question is worth extra points. 


• http://en.wikipedia.org/wiki/Jeopardy 

221


222 

Base 2: 

100pts: 110101 + 100110 

answer: 1101011 

200pts: convert 11010 from base 2 to base 10 

answer: 26 

300pts: write 0 through 7 (base 10) in base 2 

answer: 000, 001, 010, 011, 100, 101, 110, 111 


answer: 10111 

Base 3: 

100pts: 220 + 102 

answer: 1022 

200pts: 22222 + 2222 

answer: 1022221 


answer: 194 


answer: 122 

Base 7: 

100pts: 265 + 346 

answer: 644 


answer: 309 


answer: 122 

400pts: convert 36 form base 7 to base 3 

answer: 1000 

Challenge: 

200pts: (base 2) 110 – 11 

answer: 11 

400pts: (base3) 111 – 22 

answer: 12 

1000pts: (base 11) 1A9 – 8A 

answer: 11A 

(note: digits in base 11 are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A so A = 10 in base 10) 

Supplement 1: Jeopardy Questions

Notes: 


223

224 

Connecting Dots and Coloring Graphs

Lesson #42: The Four Color Theorem 



• Practice making a hypothesis 

• Work together to discover the Four Color Theorem 

Materials: 

• Pencil and blank white paper 

• Packet of markers or colored pencils 

• Copies of Supplements 1-4 

Connecting Dots and Coloring Graphs 

At first this lesson will appear to be about coloring; rest assured that math (of the Graph Theory kind!) 

will emerge. Let each student pick one of the uncolored graphs to color in, and instruct them to put it 

down in front of them but DON’T start coloring. Pose this question: What’s the FEWEST number of 

colors you’ll need to color each shape on the graph a different color and make sure that none of the adjacent 

regions are the same color? Students will likely have questions about what “adjacent” means, which 

is good. Ask each student to share their best guess of what “adjacent” means; there’s a good chance they’ll 

be able to figure it out together. When you think it’s time, jump in and confirm to them that adjacent means 

touching each other, even if only at a corner. In the southwest of the United States, for example, Utah, Arizona, 

New Mexico and Colorado come together at the four corners, thus each is adjacent to the other three. 

So, what is the FEWEST number of colors you’ll need to color all the regions of your graph if none of 

the adjacent regions can be the same color? When a student gives a reply (I think it’s at least 8!) be sure to 

ask them why. Talk about the students’ different hypotheses for as long as you can, constantly challenging 

them to come up with why they think it’ll take that number of colors. After trying to talk it through for a 

minute the conversation will reach a point where the students are referencing the coloring pages in front of 

them. 

Here is the incorrect conclusion you will probably see students jumping to: A student points to a state 

(Oklahoma, for example) on the U.S. map and declares that all its bordering states (Texas, New Mexico, 

Colorado, Kansas, Missouri, and Arkansas) have to be different colors, so the minimum number of colors 

is one plus the number of bordering states, which in this case is seven. 

225


This is a predictable way for students to start reasoning, and it’s a great start. But you’ll notice there’s a 

problem with the approach: While none of the bordering states can be the same color as Oklahoma, do they 

each have to be different colors from each other? Or is it possible for Texas and Colorado to be the same 

color? Texas and Colorado can, in fact, be the same color without breaking our rule. For that matter, so 

can Missouri. Remember, we want to use the fewest different colors possible, so there’s no need to bring in 

more colors unless it’s absolutely necessary. Give the students this Oklahoma example, and then tell them 

the best way to find out if their hypotheses are right is to go ahead and start coloring. Emphasize again that 

they don’t want to add a new color unless it’s absolutely necessary. 

As they color, look for students that may be taking slightly different approaches, comment on what differences 

you see, and ask them to put into words the way they’re doing it and why. 

Example: If you notice a student start with red then add some green, but then go back and add more 

red, there must be some reasoning behind that. Ask them why they went back to red instead of using a 

third color. 

Example: If you notice a student starting in a certain part of the graph (the middle, the top, the bot 

tom) ask them if they decided to start there for a reason. Ask also if they think it would be different 

to start in a different place on the graph. 

The answer to the question is four colors; that’s all it takes! Mathematicians call this discovery the 

Four Color Theorem because it holds for any graph, not just the ones you’re given here as supplements. 

Don’t give this away to the group as they’re coloring. Instead, wait for them to uncover the theorem for 

themselves. You may have a student who colors their whole graph and announces that it only took them five 

colors. If this is the case, you can be confident that there is a place on their graph where they introduced a 

5th color when it wasn’t necessary; remember, it has been mathematically proven that each and every graph 

can be colored with just four colors, given a few restraints not discussed here that don’t take away from the 

coolness of the theorem. Luckily, you don’t need to study their map and find this place in order to urge them 

to keep working. The students will feel convinced that they have used the least number of colors possible, 

so it is useless to “argue” with them. Instead act like a mathematician would and try for a more involved 

proof of their answer... 

Tell them: “Okay, I’m convinced that it can be done with five colors because I can see it on your graph. 

You still have to convince me that it CAN’T be done with one, two, three, or four colors.” Start with this 

one-color disproof using the following graph: It is impossible to color a graph with only one color if we 

226


follow the rule that no two adjacent regions can have the same color. The students should all be able to 

convince each other that this case is impossible. Next, ask the students to prove with words and a demonstration 

that it can’t be done with two colors. To complete this disproof, follow the steps outlined below. 

Then do it again with three colors, and again with four. Except when your students follow the following 

steps with four colors, they’ll see that it’s possible to color any graph with only four colors! 

1. Pick a region in the middle of the graph and color it (for the supplemental graphs which are larger 

than this example, ‘in the middle’ means one of the regions close to the center of the country, such as the 

states Missouri, Kansas, Colorado, Kentucky, or Tennessee): 

2. Now look at the group of regions adjacent to your middle region. Think of this as the first ring. You 

must use a second color for one of these first ring regions, because you can’t use the first: 

3. Use that second color for as many more first ring regions as possible: 

227


228 

4. Now go back and use the first color everywhere you can in the second ring (the set of regions which 

lay outside of but adjacent to the first ring regions): 

5. Still in the second ring, use your second color to fill in all possible white regions as pictured below. 

For larger graphs, and depending on which middle region you start with, there will be several rings. Continue 

this process of switching between your two colors as you move from the middle through each subsequent 

ring until you get to the edge of the graph. 

At this point you have no more options for using your first two colors, so it must be that at least three 

colors are needed. Look closely at the white regions remaining above. If any of them are adjacent to each 

other, then isn’t it the case that at least two more colors are required to complete the graph coloring? Indeed, 

we know that if two white regions are adjacent, we’ll need two different colors for them. 

The three white boxes at the bottom in the middle of the graph above can’t all be green because they 

touch, and we already know that they can’t be red or blue because we worked with these colors until there 

was no more space to use them. Thus, after a student uses a first and second color as explained in steps 1 

through 5, their graph will have the following in common with the illustration in step 5: Of the remaining 

white regions, each one will be adjacent to at MOST two other white regions. 

6. Pick one of those that’s adjacent to two others and color it a third color:

7. Use that third color to fill in as many other white regions as possible: 


8. Notice that after exhausting our options with the three colors none of the remaining white 

regions are adjacent, so it requires just one more color to complete the graph! 

The really amazing thing about graphs is that no matter what your graph looks like when you start, 

you will always be able to follow the steps above and get to this point. Once students see that they can 

use only three colors to reduce the graph to a bunch of colors interspersed with white regions that are not 

adjacent to other white regions, they will have proved the Four Color Theorem because they can color each 

of these remaining white regions with the same fourth color. 

229


230 


• http://en.wikipedia.org/wiki/Four_color_theorem 

• http://www.mathpages.com/home/kmath266/kmath266.htm

Supplement 1: U.S. Map 


231

Connecting Dots and Coloring Graphs Supplement 2: Geometric Map 

232

Supplement 3: More Maps 


233

Connecting Dots and Coloring Graphs Supplement 4: Examples of Four-colorings 

234

Notes: 


235


236 

Lesson #43: The Seven Bridges Problem 



• Understand the Seven Bridges of Konigsberg problem: What is it asking? 

• Learn what a graph is 

• Be able to draw the seven bridges as a graph 

• Define the degree of a vertex 

• Explore Eulerian paths and Eulerian circuits 

Materials: 

• Whiteboard and dry-erase markers 


Part 1: Introduction to the Seven Bridges of Konigsberg Problem 

Above are two representations of the seven bridges of Konigsberg, adapted from images taken from the 

Wikipedia article “Seven Bridges of Konigsberg” (see Further Reading for the link). Pose this question: 

Is it possible to walk in such a way that you cross each bridge exactly once? Then introduce the following 

definition by telling your students that what they are searching for is called an Eulerian path. An Eulerian 

path is a route that uses each edge exactly once. Thus in this case, each bridge represents an edge. 

Have the students replicate the map on their own whiteboards or notebooks. As students begin exploring 

the answer, ask for some predictions. Do they think it’s possible? Why or why not? Can anyone find 

an Eulerian path? The answer should be no! (If a student claims to have found an Eulerian path, they have 

probably added an extra bridge or gone over one of the bridges twice without realizing.) 

Here are a few more questions to ask your students as they explore this problem: 

• Why, in your own words, can’t you cross each bridge exactly once?


• If you could change one thing in the problem to make it possible, what would it be? Why is that help- 

ful? 

• Is there more than one way to change something in the problem to make it possible? (Adding or sub 

tracting one bridge? Which bridge?) 

Part 2: Redrawing the Bridges as a Graph 

The Seven Bridges Problem can be drawn with vertices and edges. Show this diagram to the students, 

explaining that the vertices (dots) represent land masses and the edges (lines) represent bridges: 

Make sure the students have a good understanding of what is going on before proceeding. If it helps, draw 

each of the vertexes and label them as “North Bank,” “South Bank,” “Island,” and “East Bank.” Then pick 

a bridge in the picture, find its corresponding vertexes, and draw in the link. Again, ask them to find a route 

that uses each edge exactly once to make sure they believe that this new figure, called a graph, represents the 

same problem. 

Now define the term degree of a vertex as the number of edges touching a vertex. For example, the vertex 

that represents “Island” has degree 5 and the “North Bank” has degree 3. What is the degree of the “South 

Bank?” The “East Bank?” Ask the students to write the degrees next to their corresponding vertices. 

Next, ask each student to construct their own unique graph with vertices and edges. Then as a group look 

at each student’s graph and decide whether or not it is possible to find an Eulerian path for that graph. (Recall 

that an Eulerian path is a route that uses each edge exactly once.) Make two groups of graphs: ones that have 

Eulerian paths and ones that don’t. Then have the students write down the degree of each vertex for all the 

graphs and look for patterns. This is the crux of this part of the lesson. 

237


238 

Encourage the students to gather the following information and write it down: 

• Does this graph have a path? (This is equivalent to asking: Can you find a route that crosses each 

edge exactly once?) 

• If a graph has a path, where does it start and stop? 

• If it has multiple paths, where does each one start and stop? 

• What do the graphs that have paths have in common? 

• What do the graphs that don’t have paths have in common? 

• What makes it so that a graph has a path? 

After working on this for a while, if the students need a hint, tell them to think about whether the degrees 

of the vertices are even or odd. Once the students have determined as much as they can from the graphs that 

they have drawn, ask them to generate a solution based on their findings. Then, present them with the following 

solution, both parts of which are explained below. 

PART 1 OF THE SOLUTION: A graph has an Eulerian path if it has exactly 0 or exactly 2 vertices of 

odd degree. 

PART 2 OF THE SOLUTION: If a graph has exactly 2 vertices of odd degree, then one of those vertices 

must be the starting point and the other the ending point of the Eulerian path. 

Discuss the students’ findings compared to this solution. What parts of the students’ solution matched the 

above solution, and what parts were different? As a final exercise for this part of the lesson, ask the students 

to explain why the Seven Bridges of Konigsberg problem has no Eulerian path. (It has no Eulerian path because 

it has 4 vertices of an odd degree, and only graphs with 0 or 2 vertices have an Eulerian path.) In their 

answers, encourage them to use their new vocabulary words, like “vertex” and “degree.” At this point in the 

lesson, if you feel the group could benefit from some more examples, ask each student to draw a graph that 

has an Eulerian path according to the solution above, and then to find that path. 

Part 3: Circuits 

This part is a small extension of Part 2. We have defined a path as a route that traverses each edge exactly 

once. We will now define an Eulerian circuit as an Eulerian path that has the same starting and ending 

point. (If you need to burn some extra time, you may want to open up a discussion about other contexts in 

which the word circuit is used. Does our definition of a circuit coincide with other uses of the word?) Then, 

related to this lesson, pose the following questions: 

• What must be true about a graph if it has an Eulerian circuit? (First, the graph must be an Eulerian 

path, so it must have exactly 0 or 2 vertices of odd degree. If it has 2 vertices of odd degree, then we 

know by the second part of the solution that the graph starts and ends at different points. Thus a graph 

must have exactly 0 vertices of odd degree in order to be an Eulerian path.) 

• Are all circuits also paths? (Yes!) Are all paths also circuits? (No.) 

* The answers to these questions are similar to the answers when asking: “Are all people in North- 

field in Minnesota? Are all people in Minnesota in Northfield?” 


• http://en.wikipedia.org/wiki/Seven_Bridges_of_Königsberg 

• http://mathworld.wolfram.com/KoenigsbergBridgeProblem.html

Notes: 


239


240 

Lesson #44: The Three Utilities Problem 




• Understand the Three Utilities Problem 

• Explore planar graphs and their characteristics 

• Apply Euler’s forumla to the Three Utilities Problem 

Materials: 

• Whiteboard and dry-erase markers 


Part 1: Introducing the Problem 

The problem: There are three houses (A, B, and C) and three utilities (E for electricity, G for gas, and 

W for water). Is it possible to connect each house to each utility without having the lines cross? Begin by 

having the students draw the problem: 

A B C 

E G W 

This lesson is highly self-directed. Students will soon realize that the problem does not have a simple 

solution. How can something so simple be so difficult? When I taught this lesson, the students came up 

with some great questions, most of which you can answer easily by saying: I don’t know – try it! If the 

students do not offer these or other questions after ten minutes or so, pose them yourself: 

• Can we move the houses and utilities around? (Sure! Try it. Arrange the houses and utilities in 

different configurations. What happens?) 

• What if there were only two houses? Or only two utilities? (Try it and see! Does it make any dif- 

ference whether you remove a house or a utility? Why or why not?) 

• What if there were four houses and two utilities? 

• What if we had three dimensions? 

• What if the houses and utilities weren’t on a flat piece of paper, but instead on a sphere (imagine the 

houses and utilities on Earth)? 

When there are ten to fifteen minutes left, have everyone erase what they’ve drwan and put their materials 

aside. What does the group think? Is the Three Utilities Problem possible? Why or why not? Talk 

about the problems students encountered and their intuitions about the solution. The answer, perhaps not


surprisingly, is no. It is not possible to connect each house to each utility without the lines crossing. 

Part 2: Exploring Planar Graphs 

On your own whiteboard, write the following (an equation attributed to Leonard Euler, pronounced like 

“oiler”): 

V – E + F = “Euler’s number” 

In this equation, 

• V stands for the number of vertices (the dots you start with) 

• E stands for the number of edges (lines connecting two vertices – remember that the places where 

edges intersect are not vertices) 

• F stands for the number of faces (regions bounded by edges, including the outer region) 

For example, consider the following manner of connecting each of the houses to each of the utilities. 

This example has 6 vertices because there are 6 dots. 

Though it may be tempting to count the intersection of two 

lines as a vertex (there are three places where this occurs in 

this example), doing so will not only give you the incorrect 

number of vertices, but may also cause you to count the 

number of edges incorrectly as well. 

The bold line is an example of one edge. 

There are 9 edges total: 

• 3 vertical ones (WA, GB, EC) 

• 4 diagonal ones (WB, GA, 

GC, EB) 

• 2 that go around (AE, WC) 

This bold line does NOT count as an edge 

because it doesn’t connect two vertices. 

The most important thing to remember when 

counting faces is that the outer region counts 

as a face. This graph has 8 faces. 

241


242 

Therefore, in our first example, we compute “Euler’s number” to be 5: 

v = 6 

e = 9 

f = 8 

v - e + f = 

6 - 9 + 8 = 5 

For other examples, consider these configurations, or any configurations that the students produced themselves: 

v = 6 

e = 9 

f = 8 

v - e + f = 

6 - 9 + 8 = 5 

Now look at some graphs whose edges do not cross. These are called planar graphs. Note that these 

graphs do not represent the Three Utilities Problem.


Have each student create their own planar and non-planar graphs (graphs whose edges never cross and 

graphs whose edges cross at least once, respectively). Then ask them to compute the Euler number for each 

graph using the equation V – E + F. 

Ask the group what they notice about the values that they find for the Euler number. (It’s always 2 when 

the graph is planar, and it’s never 2 when the graph is non-planar.) Recalling the Three Utilities Problem, ask 

the students to connect the graph(s) of the Three Utilities Problem to what we’ve just learned. What conclusion 

can we draw? 

The major points of this part of the lesson are the following: 

• Graphs whose edges (lines) never cross always have an Euler number of 2. 

• The graph of the Three Utilities Problem always has an Eulers number not equal to 2. 

• Since we know that in the Three Utilities Problem V = 6 (because each house and each utility repre 

sents a vertex, so 3 houses + 3 utilities = 6 vertices) and E = 9 (because there must be 9 lines since 

3 houses × 3 utilities each = 9 lines), in order to make the Euler number equal to 2, first we have the 

equation: 

V - E + F = 2 

243


244 

Then, since we know that V = 6 and E = 9, we get the equation: 

6 – 9 + F = 2 

What must F equal for this equation to be correct? (F must be equal to 5.) 

Keeping this in mind, try to draw the Three Utilities Graph in such a way as to make 5 faces. Now are 

you convinced that it’s impossible? 


Many of the examples of graphs in this lesson come from these websites, which also give detailed explanations 

of the problem. 

• http://en.wikipedia.org/wiki/Water,_gas,_and_electricity 

• http://www.cut-the-knot.org/do_you_know/3Utilities.shtml 

• http://mathworld.wolfram.com/UtilityGraph.html

Notes: 


245

246 

Topology: Coffee Cups and Doughnuts

Lesson #45: Playdough and Mobius Strips 



• learn how two different objects can have the same topological “shape” 

• discover the wonders of the Mobius strip 

Materials: 


• Container of playdough for each student 

• Scissors for each student 

Part 1: Playdough 

Topology: Coffee Cups and Doughnuts 

Topology, in the broadest possible sense, is the mathematical study of space. In topology, two figures are 

equivalent if one can be deformed into the other without changing the “shape” of the object. So what exactly 

constitutes “shape?” Let’s look at some activities involving playdough to find out. 

Begin by handing out a container of playdough to each student. Ask each student to make a doughnut out of 

their playdough. Each student’s doughnut should have a hole in it. Now, ask each student to turn their doughnut 

into a coffee cup, and observe how the students choose to do this. One way (the topologically “correct” way) is 

to turn the hole of the doughnut into the handle of the coffee cup and to make a large indent in one side of the 

doughnut to make the cup part of the coffee cup. For a great visual, visit the first link in the Further Reading 

section. 

In topology, two shapes are the same if one can be transformed into the other just like a doughnut can be 

transformed into a coffee cup. Ask the students what they think the rules of transformation are. The answer: 

If there is a hole in the playdough, you cannot close it up, and if there is no hole in the playdough, you cannot 

make one. Thus objects always have the same number of holes no matter how they are deformed. 

Ask the students to each explain in their own words why a doughnut and a coffee cup are the same. They 

may explain this by saying that they can turn a doughnut into a coffee cup without poking any new holes 

through the playdough. 

Now ask the students to make, for example, the number 3 from the number 2. Can they do this? Yes, because 

neither number has a “hole” in it. What about the numbers 9 and 0 – can you make a 9 into a 0? Yes, because 

both numbers have a hole in them. Now what about the numbers 8 and 7? These numbers cannot be transformed 

into each other because the 8 has two holes and the 7 has no holes. Ask the students to see how many 

different shapes the numbers 0 through 9 have. The answer is 3: the numbers with 0 holes make up one shape, 

the numbers with one hole make up another, and the lone number 8 makes up the third shape with 3 holes. 

247


Now do the same with the letters of the alphabet. Which letters are the same shape? What about the letters 

“i” and ”j” that have dots? It turns out, that if a shape is in two (or more) parts, you cannot put the parts together, 

and thus the shape of an “i” will always have two parts no matter how you transform it. Ask the students to 

determine how many shapes make up the letters of their names. 

Part 2: Mobius Strips 

In Part 1 we looked at how many holes and how many parts shapes have. If two shapes had the same number 

of holes and the same number of parts, they are topologically equivalent. Now, instead of looking at holes 

and parts we are going to look at the number of sides an object has. 

Begin by asking each student to cut a long strip of paper and make a loop without any twists in it. Tape the 

two ends of the paper together to form the loop. How many sides does the loop have? (It has two.) An easy way 

to be sure of this is to take a pencil and draw a line parallel to the edges of the strip on one side of the strip. 

This line will loop back to itself, and then you can draw a second line on the other side of the paper that will 

again loop back to itself. Since it took two lines to trace out the sides of the loop, the loop has two sides. Now, 

with the same strip, ask the students to cut the loop in half the long way. What do they think will happen? Will 

another loop be formed? How many sides will that loop have? (Another loop will be formed, and that loop will 

also have two sides.) 

Now, ask the students to cut another long strip of paper. This time, they will twist it half a time and then tape 

the ends together. This is called a Mobius strip. How many sides does a Mobius strip have? Like before, the 

students can start drawing a line with a pencil along the paper. When their line loops back on itself, ask them to 

look at the whole paper. They will see that the line they drew covered all the “sides” of the strip. Thus, since it 

only took one line to trace out the entire Mobius strip, the Mobius strip only has one side! Now, what do they 

think will happen if they cut the Mobius strip in half lengthwise like along the dotted line in the image below? 

(It will form one large loop!) Now what about if they cut it into thirds lengthwise? (It will form two separate 

loops linked together, one big and one small!) Note that it only takes one long cut to cut a Mobius strip into 

thrids. Make sure the students predict what will happen before they cut their loops. Each time they cut their 

loop, ask them to look at the new loop(s) that form and see how many sides those new loops have. 

248 

Mobius strip cut in half Mobius strip cut in thirds 

What about a loop with a full twist in it? Or a twist and a half? Or two full twists? How many sides do those 

loops have? And what happens if you cut those loops lengthwise into two, three, or even four pieces? Ask your 

students to make predictions, and then to see for themselves. From here, the students can lead the direction of 

the lesson, exploring in more detail whichever sorts of loops they like. 

It may be helpful to suggest that the students write down the number of twists, sides, and loops so they can 

look for patterns and differences. One possible way to do this is to make a chart.


• http://en.wikipedia.org/wiki/Topology 

• http://en.wikipedia.org/wiki/Mobius_strip 


249


Notes: 

250

Lesson #46: Euler Number 




• Determine the Euler number for solids with zero and one hole 

Materials: 


• Scissors for each student 

• Tape 

• Toothpicks 

• Gumdrops or playdough 



You can begin by asking your students if they know what a polyhedron is. They may have never heard this 

word before. If they need a clue, you can tell them that a cube is a polyhedron, but a square and a sphere are not. 

So how does a cube differ from a square? A cube has three dimensions, while a square only has two. How does 

a cube differ from a sphere? A cube has straight edges and flat faces, while a sphere has no edges and a curved 

face. Thus the definition of polyhedron is that it is a three-dimensional object with flat faces and straight edges. 

At this point, your students may not know the terminology “face” and “edge,” and instead may use different 

words to describe them. So how exactly do we define “face” and “edge”? Have your students construct the cube 

in the supplement to find out! They should cut out the shape along the outside boundary, make mountain folds 

along the inner lines, and then tape the edges together. A mountain fold looks like this: 

Once each student has constructed a cube, you can explain that the six flat sides of the cube are called “faces” 

and that edges are formed where two sides meet. 

Now before we talk about Euler numbers, ask your students how they would define a vertex (pl. vertices) 

on their cube. The vertex is the spot at which two or more edges meet to form a single point. 

251


With their cubes in hand, ask the students to count the number of faces, edges, and vertices, and to record 

their results. Next, each student can also construct the tetrahedron, octahedron, and triangular prism, and can 

then count and record the number of faces, edges, and vertices for each shape. You can now tell your students 

that, for each polyhedron, these numbers have something in common. See what patterns they find! One examples 

of is that the number of edges of each polyhdron are always greater than the number of vertices and the 

number of faces. Yet sometimes there are more faces than vertices, and other times more vertices than faces. 

Another characteristic of all polyhedrons is that the number of vertices (V) plus the number of faces (F) minus 

the number of edges (E) always equals two! So, to write that out another way: V - E + F = 2. This is true not 

only of the four polyhedra your students have created, but of all polyhedra! The formula V - E + F calculates the 

Euler number of the object, and thus we can say that all polyhedra have an Euler number of two. 

It turns out, though, that not all three-dimensional objects have an Euler number equal to two. There is a 

type of shape called a torus (pl. tori) that has a different Euler number. So what is this number? You can begin 

by asking your students to construct the two tori outlined in the Supplement 1. Again, they should cut along the 

perimeter of the shape and make mountain folds on all the inner lines except for the thicker gray inner lines on 

Torus 2; those lines should be valley-folded like so: 

Once your students have created their two tori, ask them to find the Euler number of each torus. They will 

find that the Euler number of both tori is 0! So how does a torus differ from a polyhedron? The answer: The key 

distinction between them is that a torus has a hole in it, while a polyhedron doesn’t have any holes. By definition, 

polyhedra have no holes in them, and by definition a torus has one hole. Thus, to recap, we have now 

learned that the Euler number for a polyhedron (no holes) is always two, and the Euler number for a torus (one 

hole) is always zero. 

To confirm this is true, ask your students to each make a Mobius strip. To make a Mobius strip, cut out a 

long strip of paper, twist it half a turn, and bring the ends together to form a loop. What is the Euler number of 

a Mobius strip? Ask the students to think about what they’ve learned, and make an informed prediction. Before 

they saw that all shapes with a hole have an Euler number of zero, so they could predict that the Mobius strip 

also has an Euler number of zero. Now the students can calculate the Euler number of the Mobius strip. They 

will find that it has one face, one edge, and zero vertices, and thus it has an Euler number of 0. 

Now, to further reinforce the lesson, you can use gumdrops or playdough and toothpicks to make more polyhedra 

and tori. To do this, each student can stick the toothpicks into the gumdrops and make a three dimensional 

shape. Each gumdrop represents a vertex, each toothpick represents an edge, and the faces, which are harder to 

count, are the flat faces formed by the toothpicks. Two warnings about constructing shapes in this manner: 1) 

make sure that the faces the students create are actually flat. To test this, they should be able to lay each face on 

the tabletop and have it not rock back and forth, and 2) the students could use a gumdrop to connect two toothpicks 

that both actually make up one edge, so in this case the gumdrop in the middle would not be a vertex. 

252


Encourage the students to make three-dimensional objects that they have not made yet. They could even try 

and construct an object with two holes, and see what the Euler number of such a shape is! Also, if a student calculates 

an “incorrect” Euler number for their object, for example calculates an Euler number of three for a polyhedron, 

you can suggest that the student trades objects with a partner and that they each calculate each other’s 

Euler numbers. Also, it may be possible that a shape constructed has a curved side for example, so the students 

could work together to determine what makes their object not a polyhedron. 


• http://en.wikipedia.org/wiki/Polyhedron 

• http://en.wikipedia.org/wiki/Euler_characteristic 

• http://www.davidparker.com/janine/mathpage/topology.html (The source of the supplement graphics.) 

253

Topology: Coffee Cups and Doughnuts Supplement 1: Cut-outs 

254

Supplement 1 (cont’d): Cut-outs 


255


Notes: 

256

Thinking Puzzles 

257


258 

Lesson #47: A Pico Fermi Bagel Exploration 




• Discover game-playing strategies 

• Try to find the solution in the fewest number of guesses 

Materials: 


Part 1: Learning and Playing 

When played without focusing on the logic behind the game, Pico Fermi Bagel can seem like just another 

fun game with numbers. (If you have a few extra minutes at the end of the lesson, this game could be great to 

play then.) This lesson, though, will explore the game in more depth. So how do you play the game? 

To play the game, begin by thinking of a three-digit number with no repeating digits. For example, the numbers 

459 and 120 are okay, but the number 494 is not because it has two 4’s. This number is the secret number, 

and it’s the students’ goal to use logic to figure out exactly what that number is. 

To determine your number, the students take turns guessing three-digit numbers, and you respond to their 

guesses in a certain way. If none of the three digits in the student’s guess are in your number, you say, “Bagel.” 

If one of the digits the student guesses is in your number but is not in the correct place, you say, “Pico.” (If two 

or three of the digits the student guesses are in your number but none are in the correct places, you say, “Pico 

Pico” or “Pico Pico Pico.”) Lastly, if one (or two, or three) of the digits is in your number and is in the correct 

place, you say, “Fermi” (or “Fermi Fermi,” or “Fermi Fermi Fermi.”) Note that, if you respond with three “Fermi’s” 

this means that the student has guessed the secret number! It’s also possible that, for example, one digit of 

the student’s guess is correct but in the wrong place, another digit is correct and in the right place, and the third 

digit is not in your number at all. Then you would respond “Pico Fermi.” Note that you only say “Bagel” when 

none of the digits of the student’s guess are in your secret number. 

Now for an example. Let’s say that you thought of the secret number 489. 

Guess #1: 362 Bagel – no digit is correct 

Guess #2: 820 Pico – the 8 is in the wrong place 

Guess #3: 418 Pico Fermi – the 8 is in the wrong place and the 4 is in the correct place 

Guess #4: 518 Pico – the 8 is in the wrong place 

Guess #5: 487 Fermi Fermi – the 4 and 8 are in the correct place 

Guess #6: 489 Fermi Fermi Fermi – all digits are in the correct place 

Once the students begin to understand the rules of the game, they can take turns thinking up a secret num-


ber, and you can include yourself in the guessing. Sometimes, once you begin to play, the students will understand 

the rules better. You can also play with four or five digit numbers, as long as you don’t repeat digits. 

After you’ve played a couple of rounds with the students, it’s a good time to start thinking about guessing 

strategies. To begin with, if the students have not already begun to write down each guess along with the 

response (“Pico Pico,” for example), you could ask them how they could better remember what numbers have 

been guessed before. 

Now, each time a students makes a guess at the secret number, ask him or her to explain to the group his or 

her logic for making that guess. Let’s look at the example from before to determine the logic behind the guesses. 

In quotes off to the right are possible student responses to their guesses. 

Guess #1: 362 Bagel – “The initial guess must always be random 

Guess #2: 820 Pico – “Since nothing was correct in #1, I tried three new digits.” 

Guess #3: 418 Pico Fermi – “Only one digit from #2 was correct, so I kept the 8 from #2 in 

my guess and put it in a new spot, and then tried two more digits 

we hadn’t tried before.” 

Guess #4: 518 Pico – “One digit from before was in the correct place, so in my guess I 

kept two digits the same in the same places (the 1 and the 8) and 

then changed the third one.” 

Guess #5: 487 Fermi Fermi – “The 4 must be in the correct place since there were no Fermi’s 

in #4 and there was one in #3. And either the 8 goes in the 

second place or the 1 goes in the third place, but not both since 

there was only one “Pico” in #4.” 

Guess #6: 489 Fermi Fermi Fermi – “Either the 7 or the 8 is in the correct place, but not both. So if 

the 8 is in the correct place, then the third place must be a number 

we’ve never guessed before.” 

As each student explains their reasoning behind their guess, encourage the other students to ask questions 

and even to disagree, as long as they do it respectfully. Also, once one or two students understand the logic behind 

a certain guess, the students that understand it can explain it to the students that don’t. 

Part 2: How Many Guesses? 

In this section, the students will discuss the minimum number of guesses necessary to figure out the secret 

number. At first, the students may say that only one guess is necessary, because they could guess the secret number 

on the first guess. This is true. But then you could ask them, is it always possible to guess the secret number 

in one guess? The answer is no, because usually they won’t be right the first time. You could then ask them if 

they could always figure out the secret number in fewer than, say, eight guesses? 

To help them figure this out, you could begin by imaginging that the secret number has only one digit. Thus, 

there would be no such thing as a “Pico” since if the number is correct, it is also in the correct place. To guess a 

number with one digit, it is always possible to guess the number in ten or fewer guesses since there are ten digits, 

including zero. 

Now you could look at two-digit numbers. What is the minimum number of guesses necessary to always 

guess the secret number? Note that it is possible to guess the number in a fewer number of guesses, but sometimes 

it will take that number of guesses to figure it out. 

So let’s start to work on this problem for a two-digit number. One way you could approach this is by asking 

259


the students what the “worst case scenario” is for guessing the secret number. They can even experiement individually 

with pencil and paper to see how many guesses it takes if they, already knowing the answer, guess as 

“poorly” as possible while still following the logical rules of the game. 

Let’s say that the secret number is 89. The first three guesses could be 12, 34, and 56, which would all be 

“Bagel.” If the next guess is 70, then that would be a “Bagel” too, which means that the remaining digits are an 

8 and a 9. If the student then guesses 98, that would be a “Pico Pico,” so then the student would guess 89 and 

would get the answer correct. This took six guesses. 

What if the student first guesses again 12, 34, and 56, which are all “Bagel,” and then then guesses 78 and 

90 for their last two guesses? The 78 and the 90 are each “Pico.” With this information, only two more guesses 

are necessary to guess the correct number. If the student assumes that the 0 is the Pico in the guess 90, then the 

0 must be in the first spot in the answer. So the answer would be 0 _. The 9 cannot fill in the blank, because 

then the response to 90 would have been “Pico Pico,” so i tmust be a 7 or an 8. If the 8 was in the second place, 

there would have been a Fermi in the response to 78. Thus the 7 must be in the correct place, and the guess is 

07. Here, 07 is either the secrect number, and thus a “Fermi Fermi” or is the wrong number and is a “Bagel.” In 

this case it is the wrong answer, and thus we know that the correct answer must be 89, which in fact it is! This 

process took seven guesses. 

Now if the first three guesses are “Bagel” and the subsequet two are a “Pico” for one and a “Fermi” for another, 

the same logic as the previous example works to show that at most it will take seven guesses to solve the 

puzzle. Is there ever a scenario where it will take more than seven guesses? The answer is no! Thus it is always 

possible to solve the puzzle for a two digit number in seven guesses or fewer. Now you and the students can 

play the game for real with a two-digit number, always trying to solve it in seven or fewer guesses? 

What about for a three-digit number? How many guesses does it take to solve then? This answer is more 

complicated, and it would take a lot more time to figure out. If your students are really into it, then you could 

continue on with three-digit numbers, usuing the same sorts of arguments that you made with one and two-digit 

numbers. 


• http://en.wikipedia.org/wiki/Bagel_%28game%29 

• http://en.wikipedia.org/wiki/Deductive_reasoning 

• http://en.wikipedia.org/wiki/Mastermind_%28board_game%29 

260

Notes: 


261


In order to solve this challenge, your students must already be familiar with the game Pico Fermi Bagel, 

which is explained in Lesson #47. Following is a description of the problem and its solution. This description 

is provided so that you can fully understand the answer in order to best be able to help your students find it for 

themselves. Although your students may want to work individually on the problem, you could also set up a team 

whiteboard on which all the students add their individual contributions. This problem will seem even more complicated 

if your students try to remember which numbers can and can’t go where, so you could keep a master 

chart on the whiteboard of what the students have figured out. Also, when solving this problem, the students 

will frequently come up with incorrect solutions. Instead of telling them that their solutions are wrong, simply 

ask leading questions that will help them figure out on their own where they made a mistake. 

Here is the problem: Given the following four guesses, it is possible, without any further guesses, to determine 

the secret number. The four guesses are: 

262 

Guess #1: 6152 Pico Fermi 

Guess #2: 4182 Pico Pico 

Guess #3: 5314 Pico Pico 

Guess #4: 5789 Fermi 

Lesson #48: A Pico Fermi Bagel Challenge 

The arguments that help you figure out the secret number are a bit complex, so if you work through the solution 

on your own first, you might better be able to ask the group helpful prompting questions. The solution is as 

follows: 

Since the 1 and 2 are in the same places in Guess #1 and Guess #2, neither of those digits can be a Fermi. 

Thus, from #1, either the 6 or the 5 but not both is a Fermi. The 5 is in the same place in #3 and #4, so the 5 cannot 

be a Pico or a Fermi. Thus the 6 is in the correct place in #1. 

So far we know the answer is: 6 _ _ _ . 




• Discover game-playing strategies 

• Try to find the solution with the fewest number of guesses 

Materials: 


• One whiteboard and dry-erase marker 

Now either the 1 or the 2 in #1 but not both is a Pico. Thus, either the 4 or the 8 in #2 is a Pico but not both.


Yet the 8 is in the same place in #2 and #4, so it cannot be a Pico or a Fermi. Therefore the 4 in #2 is a Pico. 

Since we know there are no 5’s or 8’s in the secret number, the Fermi in #4 must be either the 7 or the 9. So now 

we must only find the values of the remaining two digits. 

Since we know there is no 5 in the secret number, from #3 we see that two of the three digits 3, 1, and 4 

must be Pico’s. Well, we already know that 4 is a Pico, so now either the 3 or the 1, but not both, is also a Pico. 

From #1 we know that either the 1 or the 2 is a Pico, so the only way to satisfy both these is if the 1 is a Pico. So 

now we know that we must use the 1 and the 2 in our final answer. Since we tried the 1 in the second and thrid 

places without getting a Fermi, we know that the 1 must go in the fourth place. 

So now our answer is: 6 _ _ 1. 

Thus the 9 in #4 cannot be a Fermi, so the 7 in #4 is the Fermi. Now our answer is: 67 _ 1. And to fill in the 

last hole we already determined that 4 is a Pico, so the 4 must fill in that last hole. 

Thus the secret number is: 6741. 

Here are a couple questions that you can ask your students when they need some prompts: 

• In any two or more of the four guesses, does the same number ever appear in the same spot? What does 

this tell us? 

• By process of elimination, what numbers can’t go in which spots? Why is this? 


• http://en.wikipedia.org/wiki/Bagel_%28game%29 

• http://en.wikipedia.org/wiki/Deductive_reasoning 

• http://en.wikipedia.org/wiki/Mastermind_%28board_game%29 

263


Notes: 

264

Lesson #49: Secret Code 



• Articulate in their own words how learning math involves observation, practice, and 

trial and error 

• Work together as a team and reflect on their experience 

Materials: 


• Permanent marker 

• Roll of masking tape 


This activity involves some set-up before you see the students. Use the masking tape on a carpeted 

floor to create a 5x5 grid like the one pictured here. Label start and finish with a permanent marker, and 

put enough X’s on the floor next to the grid for all but one student in your group to stand on. Label the first 

X with “up next” and the second X with “in the hole.” 

X (“up next”) X (“in the hole”) X X X 

The Secret Code is the path, determined by you, from start to finish using steps up (U), to the right 

(R), to the left (L), and down (D). Note that the directions remain the same regardless of which way 

you’re facing while staning in the grid. A sample code for the grid above is: URRDRRULULULD- 

LUURDRURDRU. You can check for yourself that this path leads a student form the start box through the 

grid, not necessarily in the most direct path, and ends in the finish box. The goal of the activity is for the 

265


students to discover the correct path from start to finish using trial and error. 

Each student has a designated place to stand at all times during the activity. One student will be in the 

grid, and the remaining students must stand on one of the X’s to the side. The student on the grid begins 

by standing in the start box and then steps to one of the other boxes available. In this case, standing on the 

start box, the student may step up or to the right. If the student chooses the correct move, you will make a 

‘ding’ noise to indicate that they can go on. The student proceeds making choices to move up, down, left or 

right, until they make a move that is not a part of the secret code. When a mistake is made, you will make a 

‘buzzer’ noise indicating the end of their turn. Each time the student in the grid gets buzzed, they go to the 

last X and everyone in line moves up one, with the player ‘up next’ entering the grid. 

266 

The following are rules you should communicate to the students before starting: 

1. There is NO TALKING during this activity. 

2. Boxes on the grid may be visited more than once in the secret code. 

3. The secret code may lead you away from the finish box at times, circling back or scooping down; the 

longer (and harder) the code is, the less direct the route will be. 

4. Only moves up, down, left or right are allowed in the grid; diagonal steps are not allowed. 

It will likely take around ten minutes for your group to crack the first secret code. When they do, give 

them a harder one with more moves involved. After doing the activity twice, pause and get the group together 

to discuss and answer the following questions: 

What is hard about the activity? 

Look for students to comment on how hard it is to remember the path the more and more moves are discovered 

and the longer it gets. 

What strategy did you use individually? 

Look for students to comment on how they observed and memorized the correct moves. Did they pay 

attention only when they were on the grid? Did they learn parts of the code from watching others in the 

group? Would it be easier or harder to remember the code if you didn’t ever get to go in the grid and just 

had to watch your group members do the activity? Hopefully students will agree that physically walking 

through the grid is most helpful because they can remember it with their feet AND their mind. Tell the students 

that this is just like all learning in math: you have to use trial and error, you have to build on what you 

learned before, and you have to practice a lot to get it right in the end. 

What strategy did you use as a group? Are there strategies you wish you had tried? 

If the group developed an alternative mode of communicating (since they weren’t allowed to talk) during 

the activity, talk about it now. Congratulate them on thinking outside the box and finding ways help each 

other out. 

What would make the secret code even harder? 

Students will likely agree that adding moves (for example: diagonal steps or stepping two boxes in any 

direction) would make the secret code more difficult. Let the group decide on one or two moves to add. 

Then you will write down a new secret code and conduct the activity again.


Depending on how fast your group works through the lesson, you may have time for a student volunteer 

to write their own secret code and lead the activity themselves, switching places with you so you’re 

actually participating in the grid-walking. This is a nice way to end the lesson, developing youth leadership, 

but doesn’t have to be included in the lesson. 


• http://en.wikipedia.org/wiki/Teamwork 

• http://en.wikipedia.org/wiki/Trial_and_error 

267


Notes: 

268


Lesson #50: Playing Games: A Lesson in Strategy and Logic 



• Define logic in their own words 

• Identify why and how logic and math are related 

• Play and experiment with two different logic puzzles 

• Develop and describe their own strategies for solving two different logic puzzles 

Materials: 


• Computers with internet access, preferably one for each student 

• Dictionary 

At a table separate from computers, pass out a copy of Supplement 1 to each student. Then ask the 

students to write on their sheet the answers to the first three questions. Once everyone is done writing, ask 

each student to share one way they use logic in their daily lives. Now discuss as a group what a good definition 

of the word “logic” would be. Get a number of examples, then ask for one student to get a dictionary 

and help by looking it up. Read the definition aloud and discuss whether the students learned anything 

new from the dictionary definition. 

Split the group in two and assign half the students to Marbles and the other half to Turn Out the Lights. 

Move as a group to the computers, and help each student get to the appropriate web page to play their assigned 

game. The sites for each are listed in the Further Reading section. Let them play for 5 to 10 minutes 

while you roam purposefully among them. (You’ll need 15 more minutes out of the lesson for group 

time, so if you have a half hour left total that means 15 minutes for computer time. Divide this 15 minutes 

between the two puzzles for a total of about 7 minutes with each puzzle.) If you observe a student clicking 

wildly or seeming to play the puzzle randomly, make it clear that this is not what challenge math time is 

for. A good way to redirect this kind of behavior is to tell the student they must think three moves ahead; 

in other words, don’t let them use the mouse for a move until they’ve decided what their next three moves 

will be. They can do all three moves at once, then they must pause again to plan their next three. When a 

student thinks three moves ahead, they will almost automatically switch gears and start to think strategically. 

After 5-10 minutes (depending on the length of your challenge math period) direct each student to answer 

the strategy question on the worksheet. Have the two groups switch puzzles and experiment with the 

new one for another 5-10 minutes. Stop and move away from the computers to write the strategy reflection 

for the second round, and then use the remainder of your time for students’ to share their strategies aloud 

with the group. Highlight the differences and similarities in your groups’ strategies and ask students what 

strategy they’d try if they were to go back to the puzzle tomorrow. 

269


270 


• http://en.wikipedia.org/wiki/Logic_puzzle 

• http://www.woodlands-junior.kent.sch.uk/Games/lengame/alllights.html (Turn Out the Lights) 

• http://www.woodlands-junior.kent.sch.uk/Games/lengame/iqgame.html (Marbles)

Supplement 1: Student Worksheet 

Logic Games in Challenge Math 

What is logic? 

Is logic a part of math? Why? 

How do you use logic in your daily life? List at least three ways. 

Describe your strategy for Turn Out the Lights: 

Describe your strategy for Marbles: 


271


Notes: 

272

Space Fillers and Mathematical Games 

273


The Wolf, The Goat, and The Cabbage 

Your goal is to transport the wolf, the goat, and the cabbage across the river in your boat. You can only 

take one passenger with you in the boat at a time, and you must be in the boat in order to row it across. If you 

leave the wolf alone with the goat on one side of the river, the wolf will eat the goat. And if you leave the 

goat alone with the cabbage, the goat will eat the cabbage. How do you transport all three safely across the 

river? To explain the solution, let’s call the original riverbank “side A,” and your destination “side B.” Here is 

the solution: 

274 

- Bring the goat from side A to side B and leave it there 

- Row back to side A, pick up the cabbage, and bring it to side B 

- Leave the cabbage on side B, and bring the goat from side B back to side A 

- Leave the goat on side A, and bring the wolf to side B 

- Leave the wolf with the cabbage on side B 

- Row back to side A to pick up the goat 

Note that if you pick up the wolf instead of the cabbage in the second step, you still solve the puzzle. 

Encourage your students to find both solutions. 

What Color is the Bear? 

If you walk south for one mile, east for one mile, and then west for one mile, you end up where you 

started. When you return to your starting place, you see a bear. What color is the bear? 

The easier answer: The bear is white; it’s a polar bear! Your starting and ending location is the North 

Pole. If you look at a globe with your students, you can trace out a south-east-north route following the longitude 

and latitude lines to see how you end up where you began. 

The harder answer: Begin your journey 1 + 1 / (2 × π) miles north of the South Pole. Next, walk south 

for one mile so you end up 1 / (2 × π) miles from the South Pole. Now walk east. When you walk due east 

around the globe, you walk in a circle, and in this case the circle you’re walking in has a radius of 1 / (2 × π) 

miles. The circumference of a circle is 2 × π multiplied by the radius of the circle, so the circle you’re walking 

has a circumference of one mile, since [2 × π] × [1 / (2 × π)] = 1. Thus, when you walk east for one mile 

around this circle, you’ll end up where you started walking east. Now if you walk north for another mile 

you’ll end up where you started in the very beginning! So what color is the bear? Well, there are no bears at 

the South Pole…so maybe it was actually a penguin. 

Divide a Line in Half 

Ask your students to draw a line of any length. Next ask them to divide it in half. (They can do this with 

or without a ruler; precision is not important.) Now, ask them to divide the right half of that line in half. And 

then again, ask them to divide the right half of the previous line segment in half. Continue asking them to divide 

the rightmost line segment in half. Will they ever reach the end of the line? The answer: No, they won’t. 

If they divide the line in half forever, there will always be space between their last tic mark and the endpoint 

of the line. Your students may answer yes to this question, because the width of their pencil lead will make it 

look like they have reached the end, but encourage them to draw a close-up picture of their line so they can 

see that they will never reach the end.

3-D Tic-Tac-Toe 


Begin by asking your students if they know how to play Tic-Tac-Toe. If they know the rules, ask them to 

pair up and play a game or two to refresh their memories. If they don’t know, then you can teach them. For a 

refresher of the rules, see . 

Now ask your students how they could 3-D Tic-Tac-Toe. In this version, the object of the game is still 

to get three in a row, but now you play the game in a cube with 27 “squares” instead of 9. To represent this 

cube on paper, you could suggest that your students draw three normal Tic-Tac-Toe grids, where the first grid 

represents the top layer of the cube, the second grid the middle layer, and the third grid the bottom layer. 

Also, if your students have an interest in Tic-Tac-Toe, you could create a lesson where you discover winning 

strategies for the traditional 2-dimensional version. 

Connect the Points 

Ask each student to draw a 3 x 3 grid of evenly spaced dots on their paper. Then ask them to connect all 

the points with four straight lines. Here is the solution: 

Pico Fermi Bagel 

To play the game, begin by thinking of a three-digit number with no repeating digits. For example, the 

numbers 459 and 120 are okay numbers, but the number 494 is not because it has two 4’s. This number is the 

secret number, and it’s the students’ goal to use logic to figure out exactly what that number is. 

To figure out your number, the students take turns guessing three-digit numbers, and you respond to their 

guesses in a certain way. If none of the three digits in the student’s guess are in your number, you say, “Bagel.” 

If one of the digits the student guesses is also in your number but is not in the correct place, you say, 

“Pico.” (And if two or three of the digits the student guesses are in your number but none are in the correct 

places, you say, “Pico Pico” or “Pico Pico Pico,”) And lastly, if one (or two, or three) of the digits is in your 

number and is in the correct place, you say, “Fermi” (or “Fermi Fermi,” or “Fermi Fermi Fermi.” (Note that, 

if you respond with three “Fermi’s” this means that the student has guessed the secret number!) It’s also 

possible that, for example, one digit of the student’s guess is correct but in the wrong place, another digit is 

275


correct and in the right place, and the third digit is not in your number at all. Then you would respond “Pico 

Fermi.” Note that you only say “Bagel” when none of the digits of the student’s guess are in your secret number. 

276 

Now for an example. Let’s say that you thought of the secret number 489. 

Guess #1: 362 Bagel – no digit is correct 

Guess #2: 820 Pico – the 8 is in the wrong place 

Guess #3: 418 Pico Fermi – the 8 is in the wrong place and the 4 is in the correct place 

Guess #4: 518 Pico – the 8 is in the wrong place 

Guess #5: 487 Fermi Fermi – the 4 and 8 are in the correct place 

Guess #6: 489 Fermi Fermi Fermi – all digits are in the correct place 

Multiplication War 

Remove all the queens and kings from a deck of playing cards, and then split the deck into two even parts 

between you and your opponent. You and your opponent turn over the top card on your part of the deck at 

the same time, placing them together on the table or floor in front of you. The first person to correctly state 

the product of the two cards is the winner and takes both those cards. In the event of a tie, each player puts 

out three additional cards face down and then turns up their fourth card. The first person to say the product 

of these two new cards takes all ten cards. The winner is the person who has the most cards at the time the 

game ends. Aces are 1, twos are 2,…Nines are 9, Tens are 10 and Jacks are 11. For example, the product of 

an eight and a jack would be 8 x 11 = 88 so if these two cards are turned up the first player to say “eightyeight” 

will take them. 

Sudoku 

A familiar favorite. In a Sudoku puzzle, each box must be filled with the nine digits 1-9 and you can 

only use each digit once. In addition, each row must use each of the 9 numbers exactly once, and each 

column follows this rule as well. Here is a website with Sudoku puzzles of varying degrees of difficulty: 

< http://www.sudoweb.com/> 

Thumper 

This game is ideal for those unanticipated moments of down time. With everyone in a circle (sitting on 

the floor or at a table) have players put both hands flat down in front of them and then move their right hand 

under the hand of the person sitting to their right. If done properly, everyone should now have two hands 

(one from each of the people sitting next to them) in between their own two hands. The goal of the game is to 

‘pass the tap’ around the circle in the order the hands are in now. If you start the circle by tapping your right 

hand on the table and directing it to go clockwise, there will be two hands which each need to tap in order 

before the tap reaches your left hand and you tap that one to keep passing it. Practice going around the entire 

circle clockwise and counterclockwise, then add this challenging rule to make the game more fun: tapping 

twice on the table (two quick taps one right after the other; you can’t leave room between the two because it 

gets too confusing) reverses the direction from clockwise to counterclockwise or vice versa. Any time a hand 

so much as flinches out of turn that hand is out and must be removed from the table or floor surface.

Crypto 


This fun game uses a deck of playing cards with all the face cards removed. Each player gets four cards 

dealt to them and the remainder of the deck is placed in the middle of the group with the top card turned up. 

The number on this card is the goal number. The object of the game is to use a series of mathematical operations 

(addition, subtraction, multiplication and division) on the four cards in your hand—using each card 

only once—and end up with the goal number. The first player to find such a way to produce the goal number 

says crypto and shows the group their hand and their solution. A new hand and a new goal number can be 

dealt as often as you want, though it’s nice to wait for each player to find a ‘crypto’ even after the round has 

already technically been won. 

Example: Given a hand of 10,8,7,3 and a goal number of 5, one of many possible solutions is to subtract 

the 3 from the 7 to get 4, then divide the 8 by 4 to get 2, then divide the 10 by 2 to get 5. In correct order of 

operations, this looks like: 10 / (8 / (7 - 3)). 

Buzz 

Let the group pick a buzz number between 3 and 9. The object of the game is to count from 1 to 100 

going around in a circle. Someone begins with 1, the person next to them counts 2, then the person next to 

them counts 3, and so on and so forth. The catch: when the count gets to the buzz number the player must 

say ‘buss’ in place of the number or they are out. Players must also say buzz in place of any multiple of the 

buzz number, any number whose digits add up to the buzz number, and any number that contains the buzz 

number. 

If the buzz number is 3, for example, the correct count would go like this: 

1, 2, buzz (3 is not allowed), 4, 5, buzz (6 is a multiple of 3), 7, 8, buzz (9 is a multiple of 3), 10, 11, 

buzz (as is a multiple of 3 AND its digits add up to 3), buzz (13 has a 3 in it), 14, buzz (15 is a multiple 

of 3), 16, 17, buzz (18 is a multiple of 3), 19, 20, buzz (21 is a multiple of 3), 22, buzz (23 has a 

3 in it), buzz (24 is a multiple of 3), 25, 26, buzz (27 is a multiple of 3), 28, 29, buzz (30 is a multiple 

of 3 AND has a 3 in it), buzz (31 has a 3 in it), buzz (32 has a 3 in it), buzz (33 is definitely a buzz), 

buzz (34 has a 3 in it), and so on. 

277

278

Actue 

Used to describe an angle less than 90 degrees. 

Glossary 

Glossary 

Addition in different bases 

It is conducted in exactly the same way that it is in base 10. The only difference is that carrying occurs at 

a different point. For example, in base 3, when adding a 2 and a 2, the result is 11 because you carry a 1 

when you hit 3. Note that 11 in base 3 is actually equal to 4 in base 10 (just like we’d expect from adding 

2 and 2) since there is a 1 in the 3’s column and a 1 in the 1’s column. 

Adjacent 

Nearest in space or position; immediately adjoining without intervening space; having a common boundary 

or edge; touching. 

Algebra 

Algebra is a branch of mathematics that studies the relationships of numbers and variables. Some example 

applications of algebra are solving for a single variable in an equation and solving for multiple variables in 

a system of equations. 

Angle 

The space between two lines or planes that intersect; the inclination of one line to another; measured in 

degrees. 

Angle-Angle-Angle 

Abbreviated A-A-A, this phrase is used in the context of constructing similar triangles. Given any triangle, 

knowing the sizes of all three of its angles is sufficient information for creating a triangle similar to it. It 

should be noted that this phrase is NOT used in the context of constructing congruent triangles. 

Angle-Side-Angle 

Abbreviated A-S-A, this phrase is used in the context of constructing congruent triangles. It is denotes the 

fact that, given any triangle, knowing the sizes of two of its angles and the length of the side between those 

two angles is sufficient information for constructing another triangle congruent to it. 

Angle Sum 

The result of adding the measurements of a triangle’s three angles. 

Area 

The size of a two-dimensional surface, measured in units-squared. 

Area of a triangle 

The length of the base multiplied by the height, all divided by two. Often written as (base × height) / 2. 

Artificial life 

The field of study of simulations that mirror certain life processes. 

Average 

Around the middle of a scale of physical measures. 

279

Glossary 

Axes 

The x-axis and the y-axis are the pair of perpendicular lines that intersect at the point (0, 0) and are used for reference 

when drawing a graph. To set up a pair of axes that includes all four quadrants, draw a pair of lines that 

intersect at a right angle and form a large + sign. Starting with (0,0) at the intersection, mark off units along the 

lines, making sure to keep them evenly spaced. Positive numbers go to the right and up from the intersection 

while negatives go to the left and down. 

Bar Graph 

A chart with rectangular bars whose lengths are proportional to the values that they represent. 

Base (of a triangle) 

Whichever side of the triangle is parallel to the bottom of the page. 

Base (in number systems) 

The base for a number system is the number of unique digits used. The standard number system that most 

people are familiar with is base 10 which uses the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Another common number 

system is base 2 or binary which uses only the digits 0, and 1. 

Base 10 

The standard number system that we use. Base 10 uses 10 distinct digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. 

Binary (also called Base 2) 

A number system that uses only 0 and 1 for digits. Any number in base 10 (the kind we’re used to) can be represented 

in binary, and all mathematical operations can also be performed in binary. 

Cartesian Coordinate Plane 

A two-dimensional graph with two axes that meet at right angles. The horizontal axis is traditionally the x-axis, 

and the vertical axis is the y-axis. Each axis is a number line, and the axes cross where they each equal zero. 

Points are plotted in the form (x, y) 

Cartesian Coordinate System 

A way of representing data points on the Cartesian Coordinate Plane. 

Catalan 

A Belgian mathematician born in 1814. 

Catalan number 

Any number that appears in the Catalan sequence: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 

208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, … 

Center (of a circle) 

The point inside a circle that is equidistant from all the points on the circle. 

Chord 

Any line that cuts through two points of a circle. 

Circuit (see Eulerian circuit) 

280

Circle 

A two-dimensional figure created such that each point on the figure is equidistant from the point its center. 

Circumference 

The length of the perimeter of a circle. 

Glossary 

Column (place value) 

The columns, or place values, in a number system are the values assigned to the location of each digit, starting 

from the right and moving left. In base 10, the columns are all powers of 10, meaning that we have the 1’s column, 

the 10’s column, the 100’s column, etc… This means that every digit placed in the 1’s column gets multiplied 

by 1, every digit in the 10’s column gets multiplied by 10, and so forth. This system holds for other bases 

too. For base 2, the columns are powers of 2 (the 1’s column, the 2’s column, the 4’s column, the 8’s column, 

etc…) 

Conditional probability 

The conditional probability of an event happening is the probability that it will happen given the fact that you 

already know something else about the situation. For example, the conditional probability of rolling a sum of 

12 with two dice, given that you have already rolled a 6 on the first die, is 1/6 since you only have one more die 

to roll and there is only one roll (out of six possibilities) that will give you a sum of 12. 

Congruent triangles 

Congruent triangles have the same shape and size. All three sets of corresponding angles and corresponding 

sides are equal. 

Constant 

A constant is an unknown number in an equation. Unlike a standard variable, however, this number doesn’t 

change. For example, in the equation y = x + a, a is a constant since it acts just like a real number. 

Converge 

To get closer and closer to a certain number. 

Conversion between bases 

Any number can be written in any base. To convert from one base to another, it’s may be easier to convert to 

base 10 first and then convert to the desired base. 

Counting numbers 

The counting numbers, also called the natural numbers, are the set of numbers that start with 0, 1, 2, 3, and continue 

on infinitely by adding 1 each time. 

Data 

A collection of numbers that describe how and/or when an event occurred. 

Deform (in topology) 

To change how an object looks without adding or removing any holes in it, and without making it into more or 

fewer pieces. 

Degree (of a vertex) 

The number of edges connected to the vertex. 

281

Glossary 

Dependent events 

In probability, when the outcome of previous trials does influence the outcome of the current trial. 

Diameter (of a circle) 

The distance across the circle, measured through the center. 

Digit 

A digit is a symbol used in mathematics to represent a number. Depending on the base of the number system 

being used, there will be a different number of digits. In base 10 (the base everyone uses), there are 10 distinct 

digits (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9). In base 2 (the base that computers use), there are only 2 distinct digits (0 

and 1). For bases higher than 10, new symbols must be created. For example, in base 16 (another base used in 

computer science), it is common to use letters to represent bigger digits, giving you the set 0, 1, 2, 3, 4, 5, 6, 7, 

8, 9, a, b, c, d, e, and f. 

Diverge 

To move farther and farther away from all number, thus always moving toward positive or negative infinity. 

Edge 

A one-dimensional line, defined at the intersection of two planes. 

Edge (of a graph) 

A line that that connects two vertexes in a graph. 

Electrical circuitry 

An electrical circuit is a group of metal wires that carry electrical signals. Computers use special types of 

electrical circuits, called logic circuits, to perform basic mathematical operations on the signals. An example of 

such a logical circuit is the AND circuit. The AND circuit has two input wires, a single output wire, and a series 

of wires connecting them. The intermediate wires are set up so that the output wire gets a signal (binary value 

of 1) only if both input wires have signals as well. If either of the input wires is turned off, the output wire will 

also be turned off. 

Electrical signals 

An electrical signal, in the context of computing, is a pulse of electricity on a metal wire. Computers use a 

single electrical pulse to represent the binary digit 1 and a lack of such a pulse as the binary digit 0. 

Equidistant 

When the distance between two points is the same as the distance between another two points. 

Estimate 

To guess at an actual calculation without using completely accurate measurements. 

Estimation 

An approximate calculation of quantity or degree or worth. 

Eulerian circuit 

A graph has an Eulerian circuit if you can trace a route that uses each edge exactly once and begins and ends at 

the same vertex. Note that all Eulerian circuits are Eulerian paths, but not all Eulerian paths are Eulerian circuits. 

282

Eulerian path 

A graph has an Eulerian path if you can trace a route that uses each edge exactly once. 

Glossary 

Euler number 

The equation for Euler’s number is V – E + F, where V stands for the number of vertices, E stands for the number 

of edges, and F stands for the number of faces. 

Event 

In probability, the desired outcome. 

Face 

A two-dimensional surface that is the side of a polyhedron, for example. 

Face (of a graph) – A face in a graph is a region bounded by edges. It is important to note that the outer region 

(i.e.: the area around the graph) counts as a face. 

Factor (multiplication): A factor in multiplication is a number by which you multiply something. If you want 

to make an expression twice as big, you expand it by a factor of 2 meaning that you multiply the whole expression 

by 2. 

Factorial 

A mathematical operation written as an exclamation point after a number. In general, n! = n x (n-1) x (n-2) x 

(n-3)… or, for example, 7! = 7 x 6 x 5 x 4 x 3 x 2 x 1. 

Fibonacci 

An Italian mathematician, considered by some the most talented mathematician of the Middle Ages. 

Fibonacci sequence 

This sequence begins with 1, 1. Thereafter, each number in the sequence is obtained by summing the previous 

two numbers. So, the third number in the sequence is the sum of the first and second: 1 + 1 = 2. The fourth 

number in the sequence is the sum of the second and third: 1 + 2 = 3. The first ten numbers in the sequence are: 

1, 1, 2, 3, 5, 8, 13, 21, 34, 55 

Finite 

Not infinite. 

Finite sequence 

A finite sequence does not continue forever. In other words, it has a certain number of objects. 

Formula 

A way to calculate a number that always works. For example, to find the area of a triangle, you can always use 

the formula (base × height) / 2. 

Four Color Theorem 

A famous conjecture about the coloring of graphs, first made in 1852 but not proven until 1976. 

Fractal 

A self-similar shape created by infinite iterations. 

283

Glossary 

Function 

A function is simply a vehicle that takes some sort of input, manipulates it in some way and returns output. In 

mathematics, this takes the form of an equation with two variables where one is the input variable and the other 

is the output. An example of such an equation is f(x) = x + 1. In this case f is the output variable and x is the 

input. The notation f(x) indicates that f is dependent on the x. 

Function families 

A family of functions is one that has some base function and various related functions that are simply transformed 

versions of the original. An example of such as family is the f(x) = x2 family. Other functions in the 

family take the form of f(x) = a(x + b)2 + c, where a, b, and c are constants. 

Golden Rectangle 

A Golden Rectangle is a rectangle whose sides are the lengths of two consecutive numbers in the Fibonacci 

sequence. Example: If the long side of a certain rectangle is 8” and the short side is 5”, then it is a Golden Rectangle. 

Graph (in Graph Theory) 

A set of vertices connected by edges. 

Graph scale 

The scale on a graph is the distance between each mark on the axes. The scale should always be uniform on any 

individual axis, but can sometimes be different for different axes in order to improve visual representation (if 

the y values go really high while the x values stay small it can be advantageous to have a smaller scale on the 

y-axis so that it will all fit). 

Grid 

A pattern of lines on a chart or map, such as those representing latitude and longitude, which helps determine 

absolute location. 

Height (of a triangle) 

The length of the line constructed from the point opposite the base of triangle perpendicular to the base itself. 

Hole (in topology) 

An empty space in a three dimensional object, like the hole in a doughnut. 

Hypotenuse 

The side opposite the right angle in a right triangle. 

Hypothesis 

A guess: a message expressing an opinion based on incomplete evidence. 

Independent events 

In probability, when the outcomes of previous trials do not influence the outcome of the current trial. 

Infinite set 

An infinite set is one that has an infinite number of elements in it. A common example is the counting numbers 

which start with 0, 1, 2, 3, and continue on infinitely. 

284

Glossary 

Infinite sequence 

An infinite sequence continues forever. In other words, it does not have a certain number of objects. Rather, for 

any given object in the sequence, it is guaranteed that there will be another object that follows it. 

Input 

The input for a function can be anything so long as it conforms to the type of function. A mathematical function 

expects a number for input since adding 1 to an apple doesn’t make any sense. 

Integers 

The integers are all positive and negative numbers that contain no fractional or decimal components. The set 

extends in both directions from 0, looking something like {..., -2, -1, 0, 1, 2, ...}. 

Intersection 

The intersection of two sets is defined as the set of all elements that appear in both sets. 

Isolate the variable 

The process of isolating the variable is one by which an equation with one variable is manipulated so that the 

variable is left by itself on one side with its value left on the other side. This process uses the Golden Rule of 

Algebra, doing the same thing to both sides of the equation to maintain equality, to remove all numbers from the 

side of the equation that the variable is on. 

Iteration 

The process of repeating a process or a set of rules over and over again, applying the same rules to each new 

outcome. 

Legs 

The two sides of a right triangle that meet at a right angle. 

Maximum 

The largest possible quantity. 

Minimum 

The smallest possible quantity. 

Mobius strip 

The loop created by twisting a strip of paper one-half a turn and connecting the ends of the strip together. A 

Mobius strip has one edge and one side. 

Negative numbers 

Every positive number has a negative counterpart and vice versa. The negative version of a positive number 

is the same distance from 0 as the positive version, only in the negative direction (left on a number line) rather 

than the positive direction. Adding a negative number is the same as subtracting the positive version of it, and 

multiplying by a negative number reverses the sign (so a positive times a negative is a negative and a negative 

times a negative is a positive). 

Number line 

1. A straight line that represents the real numbers. Typically, a number line has a hatch mark in the center representing 

0, and then has positive numbered hatch marks to the right of 0 and negative numbered hatch marks to 

the left. The marks must be evenly spaced. 

2. A line that extends forever in both directions with positive and negative numbers marked off at intervals on 

285

Glossary 

the line, with negative numbers to the left of positive numbers. 

Object (in a sequence) 

For our purposes, an object in a sequence is one of the numbers in a sequence. More generally, objects can be 

variables, symbols or numbers that make up a sequence. 

Obtuse 

Used to describe an angle greater than 90 degrees. 

One-to-one matching 

A one-to-one matching between sets A and B is a system by which every element in set A gets paired with one 

and only one element of B and every element in B is part of such a pair. If a one-to-one matching can be found 

between two sets, they must have the same size. 

Operations/Mathematical operators 

Mathematical operators are the standard means by which we manipulate numbers. Addition, subtraction, multiplication, 

division, square roots, and powers are all examples of mathematical operators. 

Ordered sequence 

For a list of objects to be ordered, it simply means that the order of the objects matters. This is an important 

characteristic of sequences because it makes sequences that are comprised of the exact same set of objects 

distinct from one another. For example, the sequence 1, 2, 3, 4, 5 is distinct from 5, 4, 3, 2, 1, even though both 

sequences contain the exact same set of integers. If sequences did not have this characteristic, these two lists of 

numbers would be equivalent. 

Order of Operations 

The order in which operations are applied, commonly remembered by the acronym PEMDAS, which stands for 

Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction. The operations are then applied in 

that order. 

Outcome 

In probability, any of the possible results of the experiment. 

Output 

The output of a function is whatever result is created based on the given output. For the function f(x) = x + 1, 

an input of x = 2 results in an output of 3. 

Parallel 

Describes lines that never intersect when extended to infinity in both directions. 

Parallelogram 

A polygon with four sides, with opposite sides parallel. 

Path (see Eulerian path) 

Percent 

A way of expressing a number as a fraction of 100. 

Perpendicular 

Intersecting in or forming 90 degree angles. 

286

Glossary 

Phi 

(Pronounced ‘fee’ or ‘fye’) Phi is the Greek symbol that represents the Golden Ratio, which is an irrational 

number approximately equal to 1.618. It can be approximated using the Fibonacci sequence by taking two 

consecutive Fibonacci Numbers and dividing the larger one by the smaller one. The further into the sequence 

these numbers are, the better the approximation is. For example, dividing the 50th number in the sequence 

by the 49th number in the sequence will give a closer approximation of the true value of Phi than dividing the 

8th number in the sequence by the 7th. 

Pi (π) 

The circumference of a circle divided by the diameter of a circle. It has an approximate value of 3.14159. 

Pie Chart 

A circle divided into “pie pieces” with each pie piece representing a certain percent of the circle. 

Planar graph 

A planar graph has no edges that intersect. It’s Euler number is always 2. 

Plane 

A two-dimensional surface that extends forever in all sideways directions. 

Polygon 

A two-dimensional closed figure made up of a finite number of line segments. 

Polyhedron (pl. Polyhedra) 

A three-dimensional object with flat sides, straight edges, and no holes. 

Polyhedra 

See Polyhedron. 

Power (of a number) 

see exponent. 

Prime number 

A number is prime if its only divisors are 1 and itself. Example: 5 is prime because its only divisors are 1 and 

5. Example: 6 is not prime because its divisors are 1, 2, 3, and 6. 

Probability 

1. The study of the chances of a certain outcome occurring. 

2. A measure of how likely it is that some event will occur; a number expressing the ratio of favorable cases 

to the whole number of cases possible. 

Processor 

The processor in a computer is a collection of electrical circuits that are arranged in such a way that they can 

do mathematical operations at incredible speed. The processor is responsible for all of the computer’s work 

(which is all performed as mathematical operations). All other parts of the computer are connected to the 

processor and are either systems for giving the processor input or for interpreting and displaying the processor’s 

output. 

287

Glossary 

Proof 

In Mathematics, a thorough and detailed explanation backing up a conclusion; used to ‘convince strangers’ of 

mathematical discoveries and truths. 

Protractor 

Semicircle-shaped tool for measuring angles. 

Pythagorean Theorem 

The lengths squared of the two legs of a right triangle equal the length squared of the hypotenuse. Often written 

as a2 + b2 = c2, where a and b represent the lengths of the legs of the triangle, and c represents the length of the 

hypotenuse. 

Pythagorean triple 

A Pythagorean triple is a set of three integers a, b, c such that a2 + b2 = c2. Example: The integers 3, 4, 5 compose 

a Pythagorean triple since 32 + 42 = 52 (9 + 16 = 25). 

Quadrants 

The four quadrants refer to the four sections of the Cartesian coordinate system. In quadrant 1, both x and y are 

positive. In quadrant 2, x is negative and y is positive. In quadrant 3, both x and y are negative. And in quadrant 

4, x is positive and y is negative. 

Radius (of a circle) (pl. radii) 

The distance from the center of a circle to the side of the circle. 

Rational numbers 

The rational numbers are all numbers that can be written as fractions (ratios). This set is not the same as the real 

numbers since there exist numbers that cannot be written as fractions such as pi. 

Real numbers 

The real numbers are all numbers that do not involve the square root of -1. This includes all counting numbers, 

fractions, and decimals (even those that cannot be written as fractions). 

Region 

Each area of a graph that is completely enclosed. 

Right Angle 

An angle with a measure of 90 degrees. 

Right Triangle 

A triangle with one right angle. 

Seed 

The initial configuration of objects or states of objects. 

Self-similar 

When an object looks the same at different levels of magnification. 

Sequence 

A sequence is an ordered list of objects. For our purposes, it is a list of numbers in a specific order. Example: 

The integers 2, 4, 6, 8, 10 form a sequence, and the integers 10, 8, 6, 4, 2 form a different sequence (even 

288

though both contain the exact same set of objects). 

Set element 

An element in a set is simply an item that appears in that set. 

Glossary 

Set size 

The size of a set is the number of elements that it contains. In the case of finite sets, the size can be counted. In 

the case of infinite sets, the size becomes harder to determine. The most knowledge that can be gained about the 

size of an infinite set is how it compares to other infinite sets. In the case of infinite sets, size is more technically 

referred to as cardinality. 

Set theory 

A structured approach to looking at sets of thing. The things, or elements, of the sets need not have any special 

properties, but in mathematics, sets are often composed of numbers or variables. 

Side Length 

The length measurement for any of the sides of a shape. 

Side-Angle-Side 

Abbreviated S-A-S, this phrase is used in the context of constructing congruent triangles. It is denotes the fact 

that, given any triangle, knowing the lengths of two of its sides and the size of the angle between those two 

sides is sufficient information for constructing another triangle congruent to it. 

Side-Side-Side 

Abbreviated S-S-S, this phrase is used in the context of constructing congruent triangles. It denotes the fact 

that, given any triangle, knowing the lengths of all three of its sides is sufficient information for constructing 

another triangle congruent to it. 

Similar triangles 

Similar triangles have the same shape but are different sizes. Their corresponding angles are congruent, and 

their corresponding sides are all in the same proportion. Example: Triangle A has angles that measure 37°, 53°, 

and 90° and Triangle B’s respective angles measure 37°, 53°, and 90°. Triangle A has sides that measure 3”, 4”, 

and 5” and Triangle B’s respective sides measure 6”, 8”, and 10”. 

Square (of a number) 

The square of a number is the number times itself and is written as x 2 . 

Square (shape) 

A four sided shape where each side is the same length, and the sides intersect at right angles. 

Strategy 

A plan of action characterized by thinking ahead and designed to achieve a particular goal. 

Subtraction in different bases 

Much like addition in different bases, subtraction in a different base works the same way that it does in base 10. 

The trickiest part is borrowing when doing subtraction by hand. Just keep in mind that when you borrow a 1 

from a column to the left it is equal to a multiple of your base and not a multiple of 10. 

Tangent 

Any line that touches a curve at only one point. 

289

Glossary 

Topology 

The study of the space objects inhabit. 

Tori 

See Torus. 

Torus (pl. Tori) 

A three-dimensional object with one hole. 

Triangle 

A polygon with three sides and three vertices. 

Trial and Error 

Experimenting until a solution is found. 

Triangle Inequality 

The sum of the lengths of two sides of a triangle is always greater than or equal to the length of the third side of 

that same triangle. Often written as a + b ≥ c, where a, b, and c each represent the length of a side. 

Ulam Spiral 

The Ulam Spiral is a method of mapping the prime numbers. It is created by making a grid of the positive integers 

starting with 1 and spiraling out in a counter-clockwise direction, then finally crossing out all of the nonprime 

numbers. This method of representing the prime numbers is interesting to mathematicians because of 

the curious patterns of diagonal lines that are visible, even when thousands of numbers are mapped. The Ulam 

Spiral was created by a man named Stanislaw Ulam in 1963 while doodling during a meeting. 

Union 

The union of two sets is defined as the set of all elements that appear in either set. Elements that appear in both 

sets are counted only once. 

Units (such as feet or inches) 

Lables which distinguish one type of measurable quantity from other types. 

Variable 

A letter or non-numerical symbol used to represent a number when the value of that number is not known. 

Vertex (pl. Vertices) 

A zero-dimensional point at which two or more lines meet. In a graph it is most often a dot or node in a graph, 

usually connected to other nodes by edges. 

Vertices 

See vertex. 

Volume 

The amount of 3-dimensional space occupied by an object, measured in cubed units. 

Whole numbers 

(see integers) 

290

With replacement 

In probability, when you replace an object after choosing it. 

Glossary 

Without replacement 

In probability, when you do not replace and object after choosing it and go on to calculate a new probability 

without including previously drawn objects. 

291

Lesson #2 - Augsburg College

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?