commonwealth of virginia department of education - Rockingham ...

COMMONWEALTH OF VIRGINIA 

DEPARTMENT OF EDUCATION 

Superintendent of Public Instruction 

Paul D. Stapleton 

Deputy Superintendent 

M. Kenneth Magill 

Assistant Superintendent for Instruction 

Jo Lynne DeMary 

Office of Secondary Instructional Services 

Patricia I. Wright 

Director 

Maureen B. Hijar 

Mathematics Specialist 

Virginia Department of Education i Geometry Instructional Modules

Table of Contents 

Page 

Correlation of Activities to SOL iv 

Foreword vii 

Introduction viii 

I. Lines and Angles 

Exploring Vertical Angles 1 

Parallel Lines with Patty Paper 4 

Parallel Lines on the Graphing Calculator 6 

Angles and Parallel Lines 8 

The Three Parallel Lines Investigation 12 

Constructing a Line Segment Congruent 

to a Given Line Segment 14 

Constructing the Bisector of a Line 

Segment 17 

Constructing the Circumcenter of a 

Triangle 20 

Constructing the Medians and Centroids 

of a Triangle 22 

Constructing a Perpendicular to a Given 

Line from a Point not on the Line 24 

Constructing the Altitudes and Orthocenter 

of a Triangle 27 

Constructing a Perpendicular to a Given 

Line at a Point on the Line 30 

Constructing the Bisector of a Given 

Angle 33 

Constructing the Incenter of a Triangle 36 

Constructing an Angle Congruent to a 

Given Angle 38 

II. 

Triangles and Logic 

Types of Proofs 41 

Venn Diagrams 46 

Properties of Similar and Congruent 

Triangles 49 

Triangles and Midpoints 52 

Transformations and Congruence 54 

How Many Triangles? 56 

Tangram Activities 58 

Tangram Activities for Area and 

Perimeter 60 

Virginia Department of Education ii Geometry Instructional Modules

Tangram Activity for Spatial Problem 

Solving 63 

Geoboard Exploration of the 

Pythagorean Theorem 66 

Proofs of the Pythagorean Theorem 68 

Egyptian Rope Stretching 71 

III. 

Polygons and Circles 

Quadrilaterals and their Properties 72 

Quadrilateral Properties Laboratory 76 

Quadrilateral Sorting Laboratory 80 

Quadrilateral Diagonal Investigation 84 

Triangle and Quadrilateral Angle 

Laboratory 89 

The Sum of the Angles in a Polygon 94 

Interior Angles in a Polygon 99 

Circumference 102 

Cake Problem 103 

Geometer’s Sketchpad Investigation 

of Π 105 

Problem Solving with Circles 108 

Investigation of Secants and Circles 113 

IV. 

Three-Dimensional Figures 

Euler’s Formula 116 

Constructing the Soma Cube 117 

Building other Structures with the 

Soma Cube 122 

Making Two-Dimensional Drawings of 

Three-Dimensional Figures 126 

Spatial Problem Solving 129 

Architect’s Square 130 

Exploring Surface Area and Volume 134 

Gulliver’s Travels and Proportional 

Reasoning 145 

Comparing the Edge Length, Surface 

Area, and Volume of Cubes 149 

Comparing the Radius, Surface Area, 

and Volume of Spheres 151 

V. Coordinate Relations, Transformations, and Vectors 

Patty Paper Translations 154 

Patty Paper Rotations 155 

Patty Paper Reflections 156 

Mira Explorations 157 

Coordinates and Symmetry 160 

Polygons and Symmetry 162 

Virginia Department of Education iii Geometry Instructional Modules

Correlation of Activities to the Standards of Learning for 

Geometry 

Lines and Angles 

Activity Related SOL Page Number 

Exploring Vertical Angles G.3 1 

Parallel Lines with Patty Paper 

G.4 4 

Parallel Lines on the Graphing 

Calculator G.4 6 

Angles and Parallel Lines G.4 8 

Three Parallel Lines 

Investigation G.4 12 

Constructing a Line Segment 

Congruent to a Given Line 

Segment G.11 14 

Constructing the Bisector of a 

Line Segment 

G.11 17 

Constructing the Circumcenter 

of a Triangle 

G.11 20 

Constructing the Medians and 

Centroids of a Triangle 

G.11 22 

Constructing a Perpendicular 

to a Given Line From a Point 

Not on the Line 

G.11 24 

Constructing the Altitudes and 

Orthocenter of a Triangle 

G.11 27 

Constructing a Perpendicular 

to a Given Line at a Point on 

the Line 

G.11 30 

Constructing the Bisector of a 

Given Angle G.11 33 

Constructing the Incenter of a 

Triangle G.11 36 

Constructing an Angle 

Congruent to a Given Angle 

G.11 38 

Virginia Department of Education iv Geometry Instructional Modules

Triangles and Logic 


Types of Proof G.1 41 

Venn Diagrams G.1 46 

Properties of Similar and 

Congruent Triangles G.5 49 

Triangles and Midpoints G.5 52 

Transformations and 

Congruence G.5 54 

How Many Triangles? G.6 56 

Tangram Activities G.5 & G.7 58 

Tangram Activities for Area 

and Perimeter G.5 60 

Tangram Activity for Spatial 

Problem Solving G.5 63 

Geoboard Exploration of the 

Pythagorean Relationship 

G.7 66 

Proofs of the Pythagorean 

Theorem G.7 68 

Egyptian Rope Stretching 

G.7 71 

Polygons and Circles 


Quadrilaterals and their 

Properties G.8 72 

Quadrilateral Properties 

Laboratory G.8 76 

Quadrilateral Sorting 

Laboratory G.8 80 

Quadrilateral Diagonal 

Investigation G.8 84 

Triangle and Quadrilateral 

Angle Laboratory 

G.9 89 

The Sum of the Angles in a 

Polygon G.9 94 

Interior Angles in a Polygon 

G.9 99 

Circumference G.10 102 

Cake Problem G.10 103 

Geometer’s Sketchpad 

Investigation of ∏ G.10 105 

Problem Solving with Circles 

G.10 108 

Investigation of Secants and 

Circles G.10 113 

Virginia Department of Education v Geometry Instructional Modules

Three-Dimensional Figures 


Euler’s Formula G.12 116 

Constructing the Soma Cube 

G.12 117 

Building Other Structures with 

the Soma Cube 

G.12 122 

Making Two-Dimensional 

Drawings of Three- 

Dimensional Figures 

G.12 127 

Spatial Problem Solving G.12 129 

Architect’s Square G.12 & G.13 130 

Exploring Surface Area and 

Volume G.13 134 

Gulliver’s Travels and 

Proportional Reasoning G.13 & G.14 145 

Comparing the Edge Length, 

Surface Area, and Volume of 

Cubes G.14 149 

Comparing the Radius, 

Surface Area, and Volume of 

Spheres G.14 151 

Coordinate Relations, Transformations, and Vectors 


Patty Paper Translations G.2 154 

Patty Paper Rotations G.2 155 

Patty Paper Reflections G.2 156 

Mira Explorations G.2 157 

Coordinates and Symmetry 

G.2 160 

Polygons and Symmetry G.2 162 

Virginia Department of Education vi Geometry Instructional Modules

Foreword 

The Geometry Instructional Modules are intended to assist classroom teachers of 

geometry in implementing the Virginia Standards of Learning for mathematics. These modules 

are organized into the five SOL Assessment Reporting Categories for Geometry set out in the 

Virginia Standards of Learning Assessment: Test Blueprints from the Virginia Department of 

Education. Each activity in the modules is correlated to the appropriate Standards of Learning. 

The purpose of these modules is to provide additional activities for teachers of Geometry 

as they implement the use of graphing calculators, computer software, and different approaches to 

SOL instruction into their classrooms. These activities are not intended to replace complete 

instruction of any SOL. The intent is for the activities to reinforce and enhance student 

understanding of concepts that have been taught as part of the local curriculum. 

The use of technology, particularly the graphing calculator and computer software, is 

important in the instruction of the Geometry Standards of Learning. Some of the activities in 

these modules support the use of these technology tools. Basic knowledge of the technology tools 

is assumed in the activities. 

The activities were field tested in classrooms around the state in the spring of 1999 and in 

a 1998 summer workshop at the College of William and Mary. Virginia teachers are encouraged 

to modify and adapt the activities in the Geometry Instructional Modules to meet the needs of 

students in their classrooms. The activities in these modules may be duplicated as needed for use 

in Virginia. 

The Geometry Instructional Modules are provided to school divisions through an 

appropriation from the General Assembly and in accordance with the Virginia Department of 

Education’s responsibility to develop and pilot model teacher, principal, and superintendent 

training activities geared to the Standards of Learning content and assessments, and to technology 

applications. 

Acknowledgements 

The Department of Education wishes to express sincere appreciation to Dr. Marguerite 

Mason and Dr. Dana Johnson, College of William and Mary; Dr. Sara D. Moore, University of 

Kentucky; and Mr. Bruce Mason for the development of these modules. 

Virginia Department of Education vii Geometry Instructional Modules

Introduction 

The van Hiele Levels of Geometry Understanding 

Description: 

The van Hiele theory of geometric understanding describes how students learn geometry and provides a 

framework for structuring student experiences which 

should lead to conceptual growth and understanding. The sorting task included 

in this introduction is appropriate for all ages and levels of students. It can 

serve as an activity to help students advance their level of understanding as well 

as an assessment tool for teachers to use in identifying what van Hiele level the 

student is thinking at with regard to triangles. 

____________________________________________________________________________________ 

Background: 

After observing their own students, Dutch teachers P.M. van Hiele and Dina 

van Hiele-Geldof described learning as a discontinuous process with jumps 

which suggest "levels." They identified five sequential levels of geometric 

understanding or thought: 

1) Visualization 

2) Analysis 

3) Abstraction 

4) Deduction 

5) Rigor. 

Clements and Battista (1992) proposed the existence of a Level 0 which they 

call Pre-recognition. 

In Kindergarten through grade two, most students will be at Level 1. By 

grade three, students should be transitioning to Level 2. If the content in 

the Virginia Standards of Learning is mastered, students should attain 

Level 3 by the end of sixth grade. Level 4 is usually attained by students who 

can prove theorems using deductive techniques. One problem is that most 

current textbooks provide activities requiring only Level 1 thinking up through 

sixth grade and teachers must provide different types of tasks to facilitate the development of the high 

levels of thought. 

Virginia Department of Education viii Geometry Instructional Modules

The van Hiele Levels of Geometry Understanding 

____________________________________________________________________________ 

Objectives: 

Teachers will be able to describe the developmental sequence of 

geometric thinking according to the van Hiele theory of geometric 

understanding and activities suitable for each level. In addition, 

teachers will be able to assess the van Hiele levels of their students. 

____________________________________________________________________________________ 

Materials: 

• Explanation Sheets 

• Paper triangles from the Triangle Sorting Pieces Sheet, cut out and placed 

in a plastic baggy or manila envelope. 

____________________________________________________________________________________ 

Time Required: 

Approximately 1 hour 

____________________________________________________________________________________ 

Directions: 

1. Teachers should study the Explanation Sheet: The van Hiele Levels. 

2. Turn to the Explanation Sheet: Additional Points. Note for Point 1 that the 

levels are hierarchical. Students can not be expected to write a geometric 

proof successfully unless they have progressed through each level of 

thought in turn. At Point 2, college students and even some teachers have 

been found who are at Level 1 while there are middle schoolers at Level 3 

and above. (If the content in the SOL is mastered, students should attain 

Level 3 by the end of sixth grade.) As an example of an experience that can 

impede progress (Point 3), think of the illustration of the teacher who knew 

that the relationship between squares and rectangles was a difficult one for 

her fourth graders so she had them memorize "Every square is a rectangle, 

but not every rectangle is a square." When tested a couple of weeks later, 

half the students remembered that a square is a type of rectangle, while the 

other half thought that a rectangle was a type of square. It was almost impossible for these 

students to learn the true relationship between squares and rectangles because every time they 

heard the words square and rectangle together, they insisted on relying on their memorized 

sentence 

rather than on the properties of the two types of figures. 

3. Continue on to Point 4: Properties of Levels . As an example of separation, consider the meaning 

of the word "square." When someone such as a teacher who is thinking at Level 3 or above says 

"square", the word conveys the properties and relationships of a square such as having four 

congruent sides; having four congruent angles; having perpendicular diagonals; and being a 

type of polygon, quadrilateral, parallelogram, and rectangle. To a student 

thinking at Level 1, the word "square" will only evoke an image of something 

that looks like a square such as a CD case or first base. The same word 

is being used, but it has entirely different meaning to the teacher and the 

student. The teacher must keep in mind what the meaning of the word or 

Virginia Department of Education ix Geometry Instructional Modules

symbol is to the student and how the student thinks about it. For 

Attainment, it is important to note that there are five phases of learning 

that lead to understanding at the next higher level. 

4. Divide students into small groups. Distribute the sets of cut-out triangles, 

at least one set per three participants. Instruct the students to lay out the 

pieces with the letters up. Do not call them triangles. Tell the students 

that the objects can be grouped together in many different ways. For 

example, if we sorted the shapes that make up the American flag (the red 

stripes, the white stripes, the blue field, the white stars), we might sort by 

color and put the white stripes and the stars together because they are 

white, the red stripes in another group because they are red, and the blue 

field by itself because it is the only blue object. Another way the flag parts could be grouped 

would be to put all the stripes and the blue field together because they are all rectangles and all 

the stars together because they are 

not rectangles. Have students sort the shapes into groups that belong 

together, recording the letters of the pieces they put together and the 

criteria they used to sort. Have them sort two or three times, recording 

each sort. 

5. Ask the students for some of their ways of sorting. Expect answers like 

"acute, right, and obtuse triangles" or "scalene, isosceles, and equilateral". Have them compare 

their ways of sorting with those of other groups. 

6. Refer to Sample Student Sort sheets. Sample Student Sort 1 is a low 

Level 1 sort where the student is sorting strictly by size and may not even 

know that the figures are triangles. Sample Student Sort 2 is another 

Level 1 sort. Here the student thinks that triangles must have at least two sides the same length or 

possibly that triangles must be symmetric. Sample 

Student Sort 3 is another Level 1 sort. This student also believes that 

triangles must have at least two sides the same length or possibly that 

triangles must be symmetric. Additionally, this student recognized the 

figures with right angles or "corners" as a separate category. The Sample 

Student Sort 4 is at least a Level 2 or 3 sort in which the sorter focuses 

on the lengths of the sides, a criteria which separates the figures into 

categories which overlap. The student has actually sorted into groups with 

no sides the same length, two sides the same length, and all sides the 

same length. It is unclear whether the student knows that equilateral 

triangles are a type of isosceles triangle. The Sample Student Sort 5 

focuses on parts of the figures and so is a Level 2 sort, but the student does 

not have the vocabulary to adequately describe the figures. The Sample 

Student Sort 6 is similar to the Sample Student Sort 4, but the word "Perfect" 

is incorrect and indicates that the student may be thinking more of the shape as a whole rather 

than of the individual parts. This sort is probably Level 2. 

7. Compare student responses to the samples to help identify what van Hiele level describes his/her 

geometric thinking. 

Virginia Department of Education x Geometry Instructional Modules

Explanation Sheet: 

The van Hiele Levels 

Level 1: Visualization Geometric figures are recognized as entities, without any awareness of parts 

of figures or relationships between components of the figure. 

A student should recognize and name figures and distinguish a given figure from others that look 

somewhat the same. Identification is based on a visual model. 

Typical student response: "I know it's a rectangle because it looks like a door and 

I know that the door is a rectangle." 

Level 2: Analysis Properties are perceived, but are isolated and unrelated. A student should 

recognize and name properties of geometric figures. All the properties known are listed since the 

student does not perceive any relationship between the properties. 

Typical student response: "I know it's a rectangle because it is closed, it has four 

sides and four right angles, opposite sides are parallel, opposite sides are congruent, diagonals bisect 

each other, adjacent sides are perpendicular, ..." 

Level 3: Abstraction Definitions are meaningful, with relationships being 

perceived between properties and between figures. Logical implications and class inclusions are 

understood, but the role and significance of deduction is not understood. If a student is required to "know 

a definition" before attaining Level 3, 

it will be a memorized definition with little meaning to the student. 

Typical student response: "I know it's a rectangle because it's a parallelogram with right angles." 

Level 4: Deduction The student can construct proofs, understand the role of axioms and definitions, 

and know the meaning of necessary and sufficient conditions. A student should be able to supply 

reasons for steps in a proof. 

Generally speaking, most high school geometry courses are taught at this level. 

Level 5: Rigor The standards of rigor and abstraction represented by modern geometries characterize 

Level 5. Symbols without referents can be manipulated according to the laws of formal logic. A student 

should understand the role and necessity of indirect proof and proof by contrapositive. 

Virginia Department of Education xi Geometry Instructional Modules


Additional Points 

1. The learner can not achieve one level without passing through the previous levels. 

2. Progress from one level to another is more dependent on educational experience than on age or 

maturation. 

3. Certain types of experiences can facilitate or impede progress within a level or to a higher level. 

4. Properties of levels: 

• Adjacency: what was intrinsic in the preceding level is extrinsic in the current level 

• Distinction: each level has its own linguistic symbols and its own network of relationships 

connecting those symbols 

• Separation: two individuals reasoning at different levels can not understand one another 

• Attainment: the learning process leading to complete understanding at the next higher level 

has five phases: inquiry, directed orientation, explanation, free orientation and integration 

Virginia Department of Education xii Geometry Instructional Modules


Phases of Learning 

Information: Gets acquainted with the working domain (e.g., examines examples and nonexamples) 

Guided orientation: Does tasks involving different relations of the network that is to be formed 

(e.g., folding, measuring, looking for symmetry) 

Explicitation: Becomes conscious of the relations, tries to express them in words, and learns 

technical language which accompanies the subject matter (e.g., expresses ideas about properties of 

figures) 

Free orientation: Learns, by doing more complex tasks, to find his/her own way in the network of 

relations (e.g., knowing properties of one kind of shape, investigates these properties for a new shape, 

such as kites) 

Integration: Summarizes all that has been learned about the subject, then reflects on actions and 

obtains an overview of the newly formed network of relations now available (e.g., properties of a 

figure are summarized) 

Virginia Department of Education xiii Geometry Instructional Modules

Triangle Sorting Pieces 

A 

B 

C 

D 

E 

F 

G 

N 

H 

J 

S 

K 

Q 

R 

L 

M 

W 

O 

P 

V 

Z 

T 

U 

X 

Y 

Virginia Department of Education xiv Geometry Instructional Modules

Sample Student Sort 1 

F 

X 

H 

Q 

B 

N 

Z 

S 

K 

W 

R 

V 

Small 

Large 

T 

U 

Medium 

O 

A 

M 

C 

P 

E 

G 

L 

D 

J 

Virginia Department of Education xv Geometry Instructional Modules

T 

M 

A 

Z 


Q 

S 

C 

H 

K 

B 

R 

J 

O 

U 

W 

X 

Triangles 

G 

N 

NOT Triangles 

P 

L 

D 

E 

V 

Virginia Department of Education xvi Geometry Instructional Modules


H 

C 

K 

Q 

B 

U 

R 

J 

O 

X 

S 

W 

Triangles 

Z 

T 

E 

G 

N 

Look alike... 

N is smaller... 

NOT triangles 

D 

M 

V 

A 

L 

look like ramps... NOT triangles 

Virginia Department of Education xvii Geometry Instructional Modules


S 

X 

W 

K 

EQUILATERAL 

T 

E 

Z 

A 

V 

M 

D 

P 

L 

G 

SCALENE 

B 

Y 

J 

N 

U 

H 

F 

C 

O 

Q 

ISOSCELES 

Virginia Department of Education xviii Geometry Instructional Modules


B 

H 

one longest side 

C 

T 

E 

G 

P 

O 

J 

U 

irregular & very narrow 

2 sides are similar, 

one is shorter 

R 

K 

S 

Q 

Y 

W 

X 

same shape & 

small size 

F 

2 sides are similar, 

one is longer 

N 

V 

L 

M 

D 

Z 

A 

irregular sides 

3 uneven sides 

Virginia Department of Education xix Geometry Instructional Modules


N 

V 

E 

Z 

P 

M 

L 

A 

G 

T 

D 

Every side has a different size 

C 

B 

U 

J 

O 

H 

F 

Q 

Y 

2 sides =, 3rd side smaller or larger 

Perfect 

Triangles 

R 

X 

W 

S 

K 

Virginia Department of Education xx Geometry Instructional Modules

Exploring Vertical Angles 

Reporting Category: Lines and Angles 

Related SOL: G.3 

______________________________________________________________________________ 

Description: 

Students will examine a variety of vertical angles constructed with patty 

paper, geo-strips, and/or graphing software to verify that vertical 

angles are congruent. 

_____________________________________________________________________________________ 

Materials: 

• Activity Sheet 

• Patty Paper or waxed paper 

• Geo-strips and fasteners; or straws, skewers, pick-up sticks, or pencils and 

rubber bands; or paper strips and safety pins 

• Graphing calculators or graphing software, 

____________________________________________________________________________________ 


Approximately 20 minutes 

____________________________________________________________________________________ 

Virginia Department of Education 1 Geometry Instructional Modules

Activity Sheet: 

Exploring Vertical Angles 

Directions: 

Use Patty Paper or waxed paper. 

1) Fold a line on a sheet of Patty Paper. Unfold. Fold a second line intersecting 

the first line. Label the angles as shown below. 

a 

d 

b 

c 

2) What appears to be true about

4) Instead of geo-strips, this task can be done by fastening a rubber band 

around the middle of two pencils, skewers, straws, or pick-up sticks and flexing 

them back and forth. 

Use graphing software or a TI-92 calculator. 

1) Construct two lines that intersect. 

2) Measure the four angles. Verify that the vertical angles are congruent. 

Drag one of the lines to change the size of the angles. Verify that the 

vertical angles are still congruent. 





______________________________________________________________________________ 

Description: 

Students will investigate ways of making parallel lines with patty paper. 

____________________________________________________________________________________ 

Materials: 


• Patty Paper 

____________________________________________________________________________________ 



____________________________________________________________________________________ 




Question: 

How can you be sure two lines are parallel? 

Preparing for Research: 

1) Start by folding a line on your patty paper and marking a point not on the 

line. See Figure 1. 

Figure 1 

2) You can construct a line perpendicular to another line and through a 

point by folding the patty paper so that the point is on the fold and the line 

matches across the fold. 

3) How can you use this information to construct a line parallel to the line you 

made and through the point you marked? What if you construct a 

perpendicular then construct another perpendicular? 

Making Observations: 

Describe your method for constructing a parallel line. Will your method always 

Work when given a line and a point not on the line? How do you know? 

Inferring: 

1) Test your method with another line and points. 

2) Compare your method with that of a classmate. 

Drawing Conclusions: 

Can a parallel line always be constructed through a given point not on the given 

line? 

How can you use your knowledge here to construct a parallelogram using patty 

paper? 


Parallel Lines on the Graphing Calculator 



______________________________________________________________________________ 

Description: 

Students will investigate the graphs of parallel lines on a graphing calculator 

and determine how to tell if two lines are parallel by looking at their equations. 

____________________________________________________________________________________ 

Materials: 


• Graphing calculator 

____________________________________________________________________________________ 



_____________________________________________________________________________________ 



Parallel Lines on the Graphing Calculator 

Question: 

How can you use your graphing calculator to draw a line parallel to a 

given line? 


1) On the y= menu of your calculator, enter the equation Y=2X+1 into Y1. Press 

graph and adjust the viewing area so that you see the origin and the line 

(Zoom/Standard). 

2) When two lines are parallel, they share the same slope. What is the slope 

of the line you have graphed? 

3) On the Y= menu, enter a second equation with the same slope but a different 

intercept. Enter Y=2X –4 into Y2. Press graph so that you see both lines. 


1) Do the lines appear parallel? Use the Trace function to identify two pairs of 

coordinates from each line. Each pair of coordinates should have the same X 

value and your coordinates should be whole numbers to make the calculations 

easier. 

2) What is true about the distance between two lines if they are parallel? How can 

you use the distance formula to show that your two lines are the same distance 

apart? 

Remember, the distance formula is this: D = √ (x2 - x1) 2 + (y2 - y1) 2 

Inferring: 

Is the distance between each pair of points the same? What does this tell you 

about the lines? 


1) What additional information do you need to prove that the distance between 

any pair of points on two parallel lines is constant? 

2) Given the equations of two lines, can you tell if they are parallel without 

graphing them? 


Angles and Parallel Lines 



______________________________________________________________________________ 

Description: 

Students will color in corresponding angles and alternate interior angles on a 

parallelogram grid and draw conclusions about these angles. On a second 

parallelogram grid, they will prove various types of angles are congruent. 

____________________________________________________________________________________ 

Materials: 


• Colored pencils, crayons, or markers 

• Parallel line grid sheets #1 and #2. 

____________________________________________________________________________________ 



____________________________________________________________________________________ 




Questions: 

What angle relationships do you see in a grid of parallel lines which form 

parallelograms? 


1) Look at the grid on Parallelogram Grid Handout 1. What shapes do you 

see? 

2) Color in one set of corresponding angles. 

3) Using a different color, color another set of corresponding angles. 

4) Using a third and fourth color, color in two sets of alternate interior angles. 


Your drawing should have four pairs of angles marked, with each pair a 

different color. Is each pair of angles congruent? How do you know? 

Inferring: 

On Parallelogram Grid Handout 2, you see three sections of the grid, each 

with some angles marked. Use what you know about parallel lines and 

congruent angles to explain and justify the conclusions drawn beneath each 

grid. Use at least two different proof methods (flow proof, paragraph proof, 

two column proof, or another method you and your teacher agree on) to 

justify the conclusions given. 


1) When lines in a grid are parallel, are corresponding angles always congruent? 

2) When lines in a grid are parallel, are alternate interior angles always 

congruent? 



Parallelogram Grid Handout #1 



Parallelogram Grid Handout #2 

j 

k 

t 

r 

Show that angle k is congruent to angle t. 

Show that angle j is congruent to angle r. 

c 

a 

Show that angle a is congruent to angle c. 

n 

w 

p 

Show that angle n is congruent to angle p. 

Show that angle p is congruent to angle w. 


The Three Parallel Lines Investigation 



_____________________________________________________________________________ 

Description: 

Using the TI-92 or a graphing utility, the students will conjecture the 

following theorem: 

If three parallel lines cut off congruent segments on one transversal, 

then they cut off congruent segments on every transversal. 

____________________________________________________________________________________ 

Materials: 


• Graphing utility or TI-92 

____________________________________________________________________________________ 

Prerequisite Constructions: 

Parallel lines, perpendicular line, point of intersection, segments. 

____________________________________________________________________________________ 

SAMPLE SKETCH: 

D 

A 

l 

E 

B 

m 

F 

C 

n 

(adapted from lesson by Mary Staniger and Barb Koble, Cedar Falls High School, Cedar 

Falls, IA, staniger@forbin.com, found at http://www.ti.com/calc/docs/) 



The Three Parallel Lines Investigation 

In this activity you are asked to investigate the properties of segments cut on a 

transversal by three parallel lines using a graphing calculator. 

SKETCH 

Step 1: Construct 3 parallel lines. 

Step 2: Construct a perpendicular transversal to the 3 parallel lines. 

Step 3: Construct points at each intersection on the transversal. Label them A, B, and 

C. 

Step 4: Measure the lengths of segments AB and BC. 

Step 5: Position the parallel lines so that the segments are congruent. 

Step 6: Construct another transversal. 

Step 7: Construct points at each intersection on the transversal in Step 6. Label them 

D, E, and F. 

Step 8: Measure the lengths of segments DE and EF. 

INVESTIGATION 

Investigate the measures of the segments on the transversal constructed in step 6 as 

you drag it to change its slant. 

Record at least 3 observations. 

m∠D 

m∠E 

m∠F 

DE 

EF 

EB 

CONJECTURE: 



to a Given Line Segment 



____________________________________________________________________________________ 

Description: 

Students will construct the bisector of a line segment, using a 

compass and straightedge, patty paper, and a graphing utility program. 

The accuracy of each construction will be justified as well. 

____________________________________________________________________________________ 

Materials: 


• Method 1) Straightedge, Compass; Method 2) Straightedge, Patty Paper; 

Method 3) Graphing Utility 

____________________________________________________________________________________ 


Approximately 10 minutes each 

____________________________________________________________________________________ 




to a Given Line Segment 

Directions: 

Given a line segment, construct a line segment congruent to the given segment. 

__ 

Given: AB 

A 

B 

Method 1: Using a straightedge and compass. 

1. Use a straightedge to draw a line. Call it l. 

l 

2. Choose any point on l and label it X. 

l X 

3. Set your compass for radius AB. 

A 

B 

B 

4. Using X as the center, draw an arc intersecting line l. Label the point of the 

intersection Y. 

l 

X 

Y 

__ 

__ 

XY is congruent to AB . 

__ 

Justification: Since AB was used as the radius of circle X, XY 

__ 

≅ AB 

. 

Method 2: Using patty paper 

1. Use a straightedge to draw a line on a piece of patty paper. Call it l. 

l 

2. Choose any point on l and label it X. 

X 

l 


__ 

3. Put the patty paper containing l on top of AB 

superimposing point X on point A. 

l 

X 

A B 

__ 

, aligning l with AB 

and 

4. Mark off point Y on l where point B is. 

X 

Y 

l A B 

5. Remove the patty paper. 

X 

Y 

l 

__ 

__ 

XY is congruent to AB . 

__ 

__ 

Justification: Since XY was made by tracing AB 

__ 

, XY 

__ 

≅ AB 

. 

Method 3: Using a graphing utility program 

1. Click on the given line segment so that it is active. 

2. From the Edit Menu, choose “Copy.” 

3. From the Edit Menu, choose “Paste.” 

The new line segment should be a carbon copy of the original one. 

Justification: The line segment has been electronically copied. 


Constructing the Bisector of a Line Segment 



_____________________________________________________________________________ 

Description: 

Student will construct the bisector of a given line segment, using a 

compass and straightedge, patty paper, or a graphing utility program. 


____________________________________________________________________________________ 

Materials: 




____________________________________________________________________________________ 



____________________________________________________________________________________ 



Constructing the Perpendicular Bisector 

of a Line Segment 

Directions: 

Given a line segment, construct its perpendicular bisector. 

__ 

Given: AB A B 

Method 1: Using a straightedge and compass 

1. Using any radius greater than 1 AB, draw four arcs of equal radii, two with center 

2 

A and two with center B. Label the points of intersection of these arcs X and Y. 

X 

A 

B 

Y 

< __ > 

2. Draw XY . 

X 

A 

B 

Y 

< __ > __ 

XY is the perpendicular bisector of AB . ___ ___ 

Justification: Since the same radius was used to construct AX and BX, X is equidistant 

from A and B. Likewise, since the same radius was used to construct AY and BY, Y is 

< __ > __ 

equidistant from A and B. So XY is the perpendicular bisector of AB . 



__ 

1. Use a straightedge to draw AB 

on a piece of patty paper. 

2. Fold the patty paper so that point A is superimposed on point B and crease. 

__ 

3. This creased line is the perpendicular bisector of AB . 

__ 

Justification: Since the distance from point A to the intersection of AB and the crease 

and the distance from point B to this point are the same, this point is the midpoint of 

__ 

AB . 


1. Click on the given line segment so that it is active. 

2. From the Construct Menu, choose “Point at Midpoint.” 

3. Choose both the line segment and its midpoint so that they are both active. (i.e., 

hold down the space bar and click on each.) 

4. From the Construct Menu, choose “Perpendicular Line.” 

The perpendicular bisector of the given line segment will be constructed. 

Justification: The perpendicular bisector of the given line segment has been 

electronically found. 


Constructing the Circumcenter of a Triangle 



____________________________________________________________________________________ 

Description: 

Students will construct the circumcenter of various types of triangles, 

using a compass and straightedge, patty paper, or a graphing utility 

program. They will conjecture whether the circumcenter of the triangle will 

be interior, exterior, or on the triangle depending on whether the triangle is 

acute, obtuse, or right. 

____________________________________________________________________________________ 

Materials: 

• Activity sheet 

• Straightedge and Compass; or Straightedge and Patty Paper; or a Graphing 

Utility 

____________________________________________________________________________________ 



____________________________________________________________________________________ 


Constructing the Circumcenter of a Triangle 

Directions: 

1. Get into groups of 3-4. 

2. Draw several different triangles. Be sure to include obtuse, acute, and right 

triangles. 

3. The circumcenter of a triangle is the point of intersection of the three 

perpendicular bisectors of the sides of the triangle. Construct the circumcenters 

of the triangles you drew. Use any of the three methods that were investigated in 

the Constructing the Perpendicular Bisector of a Line Segment activity. 

4. Where are the circumcenters of the acute triangles located? Where are the 

circumcenters of the obtuse triangles located? Where are the circumcenters of 

the right triangles located? 

5. Compare your results with other groups and make a conjecture about the 

location of the circumcenters. 


Constructing the Medians and 

Centroid of a Triangle 



____________________________________________________________________________________ 

Description: 

Students will construct the medians and find the centroid of various 

types of triangles, using a compass and straightedge, patty paper, or a 

graphing utility program. They will conjecture the location of the centroid. 

____________________________________________________________________________________ 

Materials: 



Utility 

____________________________________________________________________________________ 



____________________________________________________________________________________ 



Constructing the Medians and Centroid of a Triangle 

Directions: 


2. Draw several different triangles, labeling the vertices A, B, and C. Be sure to 

include obtuse, acute, and right triangles. 

3. The median of a triangle is the line segment that connects the vertex to the 

midpoint of the opposite side. Use the perpendicular bisector of a line segment 

construction to locate the midpoint of each side of each triangle. Label the 

__ 

midpoint of AB 

__ 

as point D, the midpoint of BC as point E, and the midpoint of 

__ 

CA as point F. 

A 

D 

B 

F 

E 

C 

4. Join each midpoint to the opposite vertex to form the medians. 

The centroid of a triangle is the point of intersection of the three medians of the 

triangle. Label the centroids of the triangles you drew as point G. 

A 

D 

B 

G 

F 

E 

C 

5. Make a table using paper, a spreadsheet, a graphing utility program, or a table 

on a graphing calculator. Include the following categories: 

__ __ 

__ __ 

__ 

Length of AE , Length of AG , the ratio of AG t o AE , Length of BF , Length of 

__ 

__ __ __ __ 

__ 

BG the ratio of BG to BF , Length of CD , Length of CG , and the ratio of CG to 

__ 

CD . 


Location of the centroids. 


Constructing a Perpendicular to a Given Line 

From a Point Not on the Line 



______________________________________________________________________________ 

Description: 

Students will construct a perpendicular to a given line from a point not 

on the line, using a compass and straightedge, patty paper, and a graphing 

utility program. The accuracy of each construction will be justified as 

well. 

____________________________________________________________________________________ 

Materials: 




____________________________________________________________________________________ 



____________________________________________________________________________________ 



Constructing a Perpendicular to a Given Line From a Point 

Not on the Line 

Directions: 

Given a line and a point not on the line, construct a perpendicular from the point to 

the line. 

• A 

Given: line l and point A, not on l 

l 


1. Using A as center, draw two arcs of equal radii that intersect l at points X and Y. 

A 

l X Y 

2. Using X and Y as centers and a suitable radius, draw arcs that intersect at point 

Z. 

A 

l X Y 

Z 

< __ > 

3. Draw AZ . 

A 

l X Y 

Z 

< __ > 

AZ is perpendicular to l. 


__ > 

Justification: Both A and Z are equidistant from X and Y. So AZ is the 

__ < __ > 

perpendicular bisector of XY , and AZ is perpendicular to l. 


1. Fold or draw a line on a piece of patty paper. Draw a dot on your patty paper, 

not on this line, to represent the given point. 

2. Fold your patty paper so that the crease passes through the given point. The line 

folds over on itself. Use a corner of another patty paper or a protractor to check if 

the angles formed by the crease and the given line are right angles. 

Justification: This is a trial and error procedure in which you crease the patty paper so 

that the crease goes through the given point and is perpendicular to the given line. 


1. Click on the given line segment and the given point so that they are active. 

2.From the Construct Menu, choose “Perpendicular Line.” 

Justification: The perpendicular to the given line segment from a point not on the line 

has been electronically found. 


Constructing the Altitudes and 

Orthocenter of a Triangle 



____________________________________________________________________________________ 

Description: 

Students will construct the altitudes and find the orthocenter of various 

types of triangles, using a compass and straightedge, patty paper, or a 

graphing utility program. They will conjecture the location of the 

orthocenter. 

____________________________________________________________________________________ 

Materials: 



Utility 

____________________________________________________________________________________ 



____________________________________________________________________________________ 



Constructing the Altitudes and Orthocenter of a Triangle 

Directions: 


2. Draw several different triangles, labeling the vertices A, B, and C. Be sure to 

include obtuse, acute, and right triangles. 

3. The altitude of a triangle is the line segment drawn from a vertex perpendicular 

to the opposite side (or the line through the opposite side). Use the construction 

of a perpendicular to a given line from a point not on the line to construct each 

__ __ 

__ __ 

altitude. Label the altitude to side AB as CD , the altitude to side BC as AE , 

__ 

and the altitude to side AC 

__ 

as BF . 

D 

A 

F 

B 

E 

C 

Notice that not all altitudes stay inside the triangle. Sometimes the line 

containing the side needs to be extended so that the perpendicular will intersect it. 

A 

F 

E 

B 

C 

D 


4. The orthocenter of a triangle is the point of intersection of the lines containing 

the three altitudes of the triangle. Label the orthocenters of the triangles you 

drew as point G. 

A 

F 

B 

D 

A 

G 

E 

F 

C 

E 

G 

B 

D 

C 

___ 

↔ 

NOTE: CD is the altitude but CG is needed to 

find the point of intersection. 

5. Where are the orthocenters of the acute triangles located? 

Where are the orthocenters of the obtuse triangles located? 

Where are the orthocenters of the right triangles located? 


Location of the orthocenters. 


Constructing a Perpendicular to a Given Line 

At a Point on the Line 



____________________________________________________________________________________ 

Description: 

Students will construct a perpendicular to a given line at a point on the 

line, using a compass and straightedge, patty paper, or a graphing utility 

program. The accuracy of each construction will be justified as well. 

____________________________________________________________________________________ 

Materials: 




____________________________________________________________________________________ 



____________________________________________________________________________________ 



Constructing a Perpendicular to a Given Line At a Point on 

the Line 

Directions: 

Given a line and a point on the line, construct a perpendicular to that point on the line. 

Given: line l and point A on l 

l 


1. Using A as center and any radius, draw two arcs intersecting l at points X and Y. 

A 

l X Y 

2. Using X as a center and a radius greater than AX, draw an arc. Using Y as a 

center and the same radius, draw an arc intersecting the arc with center X at 

point Z. 

Z 

A 

l X Y 

A 

< __ > 

3. Draw AZ . 

Z 

A 

l X Y 

< __ > 

AZ is perpendicular to l at A. 

Justification: A is equidistant from points X and Y by the way they were constructed. 

< __ > 

Then Z was constructed equidistant from X and Y as well. So AZ is the perpendicular 

__ < __ > 

bisector of XY , and AZ is perpendicular to l at A. 



1. Fold or draw a line on a piece of patty paper. Draw a dot on your patty paper, 

on this line, to represent the given point. 

2. Fold your patty paper so that the crease passes through the given point. The line 

folds over onto itself. Use a corner of another patty paper or a protractor to 

check if the angles formed by the crease and the given line are right angles. 


that the crease goes through the given point and is perpendicular to the given line. 


1. Click on the given line segment and the given point so that they are active. 

2. From the Construct Menu, choose “Perpendicular Line.” 

Justification: The perpendicular to the given line segment at a given has been 

electronically found. 


Constructing the Bisector of a Given Angle 



____________________________________________________________________________________ 

Description: 

Students will construct the bisector of a given angle, using a compass 

and straightedge, patty paper, or a graphing utility program. The 

accuracy of each construction will be justified as well. 

____________________________________________________________________________________ 

Materials: 




____________________________________________________________________________________ 



____________________________________________________________________________________ 



Constructing the Bisector of a Given Angle 

Directions: 

Given an angle, construct its bisector. 

A 

Given: ∠ ABC 

B 

C 


__ > 

1. Using B as center and any radius, draw an arc intersecting BA at point X 

__ > 

and BC at point Y. 

A 

X 

B 

Y 

C 

2. Using X as a center and suitable radius, draw an arc. Using Y as a center and 

the same radius, draw an arc intersecting the arc with center X at point Z. 

A 

Z 

X 

B 

Y 

C 


__ > 

3. Draw BZ . 

A 

Z 

X 

B 

Y 

C 

__ > 

BZ bisects ∠ ABC. 

__ 

Justification: If you draw XZ 

__ 

and YZ 

, XBZ ≅ YBZ by Side-Side-Side Postulate. 

__ > 

Then ∠ XBZ ≅ ∠ YBZ and BZ bisects ∠ABC. 


1. Use your straightedge to draw an angle on a piece of patty paper or trace a given 

angle. 

2. Fold your patty paper so that the rays that form the angle are superimposed on 

each other. Crease the paper to form the angle bisector. 


that the crease is the angle bisector because all the points on the crease are 

equidistance from the sides of the angle. 


1. Select three points that determine the angle to be bisected. The second selected point 

will be the vertex of the angle. 

2. From the Construct Menu, choose Angle Bisector. 

Justification: The bisector of the given angle has been electronically found. 


Constructing the Incenter of a Triangle 



____________________________________________________________________________________ 

Description: 

Students will construct the incenter of various types of triangles, using 

a compass and straight edge, patty paper, or a graphing utility program. 

They will conjecture what the incenter of the triangle is equidistant from 

The vertices of the triangle. 

____________________________________________________________________________________ 

Materials: 



Utility 

____________________________________________________________________________________ 



____________________________________________________________________________________ 



Constructing the Incenter of a Triangle 

Directions: 


2. Draw several different triangles. Be sure to include obtuse, acute, and right 

triangles. 

3. The incenter of a triangle is the point of intersection of the three angle 

bisectors of the triangle. Construct the incenters of the triangles you drew. 

4. Compare the distances from the incenter to the sides of the triangles. Also, 

compare the distances from the incenters to the vertices of the triangles. 

Which distances appear to be the same? 


location of the incenters. 


Constructing an Angle Congruent to a Given Angle 



____________________________________________________________________________________ 

Description: 

Students will construct an angle congruent to a given angle, using a 

compass and straightedge, patty paper, or a graphing utility program. 


____________________________________________________________________________________ 

Materials: 




____________________________________________________________________________________ 



____________________________________________________________________________________ 



Constructing an Angle Congruent to a Given Angle 

Directions: 

Given an angle, construct an angle congruent to it. 

A 

Given: ∠ ABC 

B 

C 


__ > 

1. Draw a ray. Label it YT . 

Y 

T 

__ > 

2. Using B as center and any radius, draw an arc intersecting BA at point D 

__ > 

and BC at point E. 

A 

D 

B 

E 

C 

3. Using Y as a center and the same radius as in step 2, draw an arc 

__ > 

intersecting YT at point Z. 

Y 

Z 

T 


4. Using Y as a center and a radius equal to DE, draw an arc that intersects the 

arc containing Y at point X. 

A 

D 

X 

B 

__ > 

5. Draw YX . 

E 

C 

Y 

Z 

T 

X 

Y 

T 

Z 

∠XYZ ≅ ∠ABC. 

__ 

Justification: If you draw DE 

Then ∠XYX ≅ ∠ABC. 

__ 

and XZ 

, XYZ ≅ DBE by Side-Side-Side Postulate. 


Place a patty paper on top of the given angle. Use your straightedge to trace 

the given angle. 

Justification: Since you traced the given angle, the angle you made should be 

congruent to the given angle. 


1. Select three points that determine the angle to be bisected. The second selected 

point will be the vertex of the angle. 

2. From the Construct Menu, choose Copy. 

3. From the Construct Menu, choose Paste. 

4. These three points determine the new angle. Select the point that will be the vertex 

and one of the other points. From the Construct Menu, choose Line Segment. 

5. Select the point that will be the vertex and the third point. From the Construct 

Menu, choose Line Segment. 

Justification: The given angle has been electronically copied. 


Types of Proof 

Reporting Category: Triangles and Logic 


____________________________________________________________________________________ 

Description: 

Students will prove the theorem that the diagonals of a rectangle are 

congruent using a two column proof, a paragraph proof, a flow proof, and a 

coordinate proof. 

____________________________________________________________________________________ 

Materials: 

Example Sheets 

____________________________________________________________________________________ 



____________________________________________________________________________________ 

Directions: 

1. Discuss how one might prove that the diagonals of a rectangle are 

congruent. 

2. Depending on the expertise of the students, either have small groups 

prove the theorem, each using a different technique and then present 

the proof to the class or work as a large group. In either case, compare 

the advantages and disadvantages of each technique. 


Example Sheet: 

Two Column Proof 

The statements and reasons are organized into two separate columns. All necessary steps are 

included. 

A 

B 

Given: 

Prove: 

Rectangle ABCD 

__ __ 

AC ≅ BD 

D 

C 

Statements 

Reasons 

1. ABCD is a rectangle. 1. Given 

__ 

2. AD ≅ BC 

2. Opposite sides of a rectangle are ≅. 

3. ∠ADC and ∠BCD are right ∠s. 3. A rectangle contains 4 right ∠s. 

4. ∠ADC ≅ ∠BCD 4. All right ∠s are ≅. 

__ __ 

5. DC ≅ DC 

5. Identity 

6. ∆ADC ≅ ∆BCD 6. SAS 

__ __ 

7. AC ≅ BD 

7. Corresponding Parts of Congruent Triangles 

are Congruent (CPCTC) 



Paragraph Proof 

Statements and reasons are interwoven to make a convincing argument. They are 

written in complete sentences. 

A 

B 

Given: 

Prove: 


__ __ 

AC ≅ BD 

D 

C 

Since ABCD is a rectangle, its opposite sides are congruent and it contains 4 right 

__ 

angles. So AD ≅BC and since all right angles are ≅, ∠ADC ≅ ∠BCD. By identity, 

__ __ 

DC ≅ DC . Using the SAS Postulate, ∆ADC ≅ ∆BCD. By Corresponding Parts of 

__ __ 

Congruent Triangles are Congruent, AC ≅ BD . 



Flow Proof 

The argument is made through the use of a flowchart with justifications keyed to the 

reasoning by numbers. 

A 

B 

Given: 

Prove: 


__ __ 

AC ≅ BD 

D 

C 

1. ABCD is a rectangle. 

2. 3. 

AD ≅ BC 


Coordinate Proof 

In a coordinate proof, you place the figure on the coordinate plane. Generally, to make 

calculations easier, place as many vertices as possible on the axes. 

A 

B 

Given: 

Prove: 


__ __ 

AC ≅ BD 

D 

C 

__ 

Using the distance formula, find the length of AC 

is (0,n); B is (m,n); C is (m,0); and D is (0,0). 

__ 

AC 

__ 

BD 

____________________ 

= √ (0 – m) 2 + ( n – 0) 2 

________ 

= √ m 2 + n 2 

___________________ 

= √ ( m – 0) 2 + ( n – 0) 2 

_________ 

= √ m 2 + n 2 

__ 

and BD 

in rectangle ABCD where A 


Venn Diagrams 


Related SOL: G.1: 

____________________________________________________________________________________ 

Description: 

Students will create a variety of Venn diagrams to describe the inclusion 

relationships between various attribute pieces. They solve a puzzle 

involving attribute pieces using a Venn diagram. 

____________________________________________________________________________________ 

Materials: 


• Attribute blocks 

• Loops of string 

____________________________________________________________________________________ 



____________________________________________________________________________________ 

Directions: 

• Assess student familiarity with Venn diagrams by creating a Venn 

diagram using attribute blocks and having students identify the 

proper labels for the circles and area of overlap. 

• Have students use loops of string and attribute blocks to create Venn diagrams 

with 3 circles. 

• Once you are certain students understand how Venn diagrams work, have them 

create the diagrams specified on the activity sheet. 

• Discuss the role of Venn diagrams in organizing information about groups. 



Venn Diagrams 

Sketch a Venn diagram for each of the situations listed below. 

1. All equilateral triangles have three congruent sides. Some isosceles 

triangles have three equal sides while some have only two equal sides. 

2. Some rectangles are squares. Some rhombi are squares. No kites are 

rectangles or rhombi. 

3. Insert the appropriate name of a quadrilateral into the Venn diagram sketched 

below. 

Parallelogram 

Rectangle 

Rhombus 

Square 

Trapezoid 


Imagine the standard set of Attribute Pieces to be available and that in subset A, 

there are 6 triangles, 7 large pieces and 7 red pieces. But also you find 

• Exactly 2 pieces are large red triangles. 

• Only 2 pieces are red and triangular but not large. 

• Exactly 1 piece is triangular and large but not red. 

• Only 1 piece is large and red but not triangular. 

There are _____ pieces in subset A. Use a Venn diagram to illustrate your solution. 

Hint: The solution is not twenty. 


Properties of Similar and Congruent Triangles 



______________________________________________________________________________ 

Description: 

Students will create two congruent triangles and two similar triangles using 

geo-strips. They then compare corresponding angles and corresponding 

sides and make conclusions about the corresponding angles and 

corresponding sides in congruent and similar triangles. 

____________________________________________________________________________________ 

Materials: 


• Geo-strips 

• Protractor 

• Ruler or angle ruler 

____________________________________________________________________________________ 



____________________________________________________________________________________ 


Activity Sheet: Properties of Similar and Congruent Triangles 

Questions: 

How can you find out if two triangles are congruent or similar by examining 

properties? 


1. Using your geo-strips, construct two identical triangles with each side a 

different color. 

2. Construct a third triangle using short red, blue, and yellow strips. 

3. Construct a fourth triangle using the middle red and the longer blue and yellow strips. 


1. First, examine your two identical triangles. Measure each of the angles using your 

protractor. What do you notice about the pairs of corresponding angles? 

2. Measure the length of each side using your ruler. It is easiest to measure from one 

fastener to the next. What do you notice? 

3. Make notes of the same measurements on your third and fourth triangles. 

Inferring: 

• Compare your results with those of a classmate. What did you each find about your 

first pair of identical triangles? 

• Were your triangles the same? Even if they were not, did the measurements 

match between them? 

These triangles are congruent — all of their corresponding parts (sides and 

angles) are exactly the same. 

• Compare your second pair of triangles. Were the angles the same? 

Were the side lengths the same? 

• What is the relationship between the side lengths? 

These triangles are similar — their corresponding angles are the same and their 

corresponding side lengths are proportional. 



1. What do you need to know about two triangles in order to be sure they are 

congruent? 

By building different triangles, see if you can identify the smallest amount of 

information that will let you be sure two triangles are congruent. 

2. What do you need to know about two triangles in order to be sure they are 

similar? 

3. If you know two triangles are similar and you know the lengths of one pair of 

corresponding sides, can you find the lengths of the other sides given one of each 

pair? 


Triangles from Midpoints 



______________________________________________________________________________ 

Description: 

Students will use the Geometer’s Sketchpad or similar software or a TI-92 

graphing calculator to investigate what happens when the midpoints of the 

sides of a triangle are connected. The investigation ends with the students 

proving that the four triangles formed are congruent. 

____________________________________________________________________________________ 

Materials: 


• Geometer’s Sketchpad or similar software 

____________________________________________________________________________________ 



____________________________________________________________________________________ 



Triangles from Midpoints 

Questions: 

When you construct a triangle from the midpoints of a given triangle, what 

is true about the four triangles you create? 


1) Use your software to construct a triangle ABC. 

2) Mark the midpoints of each side and draw triangle DEF. 

C 

F 

D 

A 

E 

B 

3) Measure each of the six angles A - F. 


1. Can you show that triangles ABC and DEF are similar? 

2. Do you think this is always true? 

You can use sketchpad to try to create a counterexample. 

Remember that even if you can not find a case where this is not true, 

that does not mean it is true. 

3. Your drawing includes four small triangles. Use the measurement tools in 

Sketchpad to investigate any relationship among these triangles. 

Inferring: 

What conclusions can you draw about these four small triangles? 


How can you prove that these four small triangles are always congruent, 

no matter what the starting triangle looks like? 


Transformations and Congruence 



______________________________________________________________________________ 

Description: 

Students will use the Geometer’s Sketchpad or similar software or a TI-92 

graphing calculator to investigate using transformations to map one 

congruent triangle onto another. They also will investigate what is needed 

to show that two triangles are congruent. 

____________________________________________________________________________________ 

Materials: 


• Geometer’s Sketchpad or similar software or TI-92 calculator 

____________________________________________________________________________________ 



____________________________________________________________________________________ 



Transformations and Congruence 

Investigation One 

Questions: 

How can you use transformations to show that two triangles are congruent? 


Draw two congruent triangles anywhere on the coordinate plane. 


Use transformations (reflection, rotation, translation) to map one triangle onto 

the other. What is the smallest number of transformations needed? 

Inferring: 

What do you know about the properties of these triangles when you see them 

mapped one onto the other? 


How does this show you that they are congruent? 

Investigation Two 

Questions: 

How can you use transformations to show that two triangles are congruent? 


Construct two triangles with two sides and the included angle congruent. 

Use transformations to map one triangle onto the other. 


What do you notice about the other parts of the triangles? 

Inferring: 

Are these two triangles congruent? 


Is any pair of triangles congruent when two sides and the included angle are 

congruent? 

You have probably studied other ways of showing two triangles are 

congruent. Explore using transformations with these relationships. 


How Many Triangles? 



______________________________________________________________________________ 

Description: 

Students will use straws or strips of paper to investigate the triangle inequality 

theorem. 

____________________________________________________________________________________ 

Materials: 


• Paper straws or narrow strips of paper 12 cm long 

• Metric ruler 

• Marker 

____________________________________________________________________________________ 



____________________________________________________________________________________ 



How Many Triangles? 

Questions: 

Are there sets of three segments that will not make a triangle? What segments can 

be combined to make a triangle? How many triangles can you make from a given 

set of segments? 


Trim your straw or strip to 12 cm in length and mark it at one cm intervals. 


Folding only at your marks, make as many different triangles as you can. 

Keep a record of your triangles by sketching them and noting the side lengths. 

Inferring: 

1. Are there some combinations of side lengths that do not appear? 

2. For example, do you have a triangle with side lengths 1, 2, 9? 

3. Do you have a triangle with side lengths 3, 4, 5? 

4. Do you see a pattern in the side length combinations that appear? 

5. Think about the saying “The shortest distance between two points is a 

straight line.” What does this say about the relationship between the 

sum of two sides of a triangle and the length of the third side? 


The triangle inequality theorem states that the sum of the lengths of any two 

sides of a triangle must be greater than the length of the third side. 

Are your data consistent with this statement? 


Tangram Activities 


Related SOL: G.5 & G.7 

______________________________________________________________________________ 

Description: 

Students will construct their own tangrams and observe properties of the seven 

tangram pieces. 

____________________________________________________________________________________ 

Materials: 

• Directions for making Tangrams 

• Paper suitable for folding such as copy paper (one sheet per student) 

• Scissors 

____________________________________________________________________________________ 



____________________________________________________________________________________ 


Directions for Making Tangrams 

1 

5 

2 

6 

4 

7 

3 

1. Fold the lower right corner to the upper left corner along the diagonal. Crease sharply. 

Cut along the diagonal. 

2. Fold the upper triangle formed in half, bisecting the right angle, to form Piece 1 

and Piece 2. Crease and cut along this fold. Label these two triangles "1" and "2". 

3. Connect the midpoint of the bottom side of the original square to the midpoint of the 

right side of the original square. Crease sharply along this line and cut. Label the 

triangle cut off "3". 

4. Fold the remaining trapezoid in half, matching the short sides. Cut along this fold. 

5. Take the lower trapezoid you just made and connect the midpoint of the longest side to 

the vertex of the right angle opposite it. Fold and cut along this line. Label the small 

triangle "4" and the remaining parallelogram "7". 

6. Take the upper trapezoid you made in Step 4. Connect the midpoint of the longest side 

to the vertex of the obtuse angle opposite it. Fold and cut along this line. Label the small 

triangle "5" and the square "6". 

7. Ask students to make a square out of all seven tangram pieces. 

8. Which pieces are congruent? How do you know? 

9. Which triangles are similar? How do you know? Can you write a proportion to express 

the relationship? 


Tangram Activities for Area and Perimeter 



______________________________________________________________________________ 

Description: 

Students will consider parts of the composite tangram as fractions of the whole 

and will consider each piece relative to a given non-standard unit, namely small 

square and the small triangle. They will find the perimeter of each piece by using 

the length of the side of the small square as the unit. This will require the 

application of the Pythagorean Theorem. 

____________________________________________________________________________________ 

Materials: 


• A set of tangrams for each student 

____________________________________________________________________________________ 


Approximately 30 - 40 minutes 

____________________________________________________________________________________ 



Area and Perimeter With Tangrams 

1. If the area of the composite square (all seven pieces -- see below) is 

1 unit, find the area of each of the separate pieces. 

piece # 

area 

1 

2 

3 

4 

5 

6 

7 

2. If the smallest triangle (piece #4 or #5) is the unit for area, find the area of 

each of the separate pieces in terms of that triangle. 

piece # 

area 

1 

2 

3 

4 

5 

6 

7 

2 

7 

1 

4 

6 

3 

5 

3. If the smallest square (piece #6) is the unit for area, find the area of each 

of the separate pieces in terms of that square. Enter your findings in the 

table below. 

4. If the side of the small square (piece #6) is the unit of length, find the 

perimeter of each piece and enter your findings in the table. 


piece # area perimeter 

1 

2 

3 

4 

5 

6 

7 

5. Pieces #1, #2, #3, #4, and #5 are all similar right triangles. How can you 

verify that they are similar? 

• Compare their bases, heights, and areas. What relationships do you 

find? 

• How does change in one of the measures of the object affect the others? 

• Does this occur if the triangles are not similar? 


Tangram Activity for Spatial Problem Solving 



______________________________________________________________________________ 

Description: 

Students will develop spatial reasoning skills by using tangrams to solve 

puzzles. 

____________________________________________________________________________________ 

Materials: 

• Activity Sheets 

• A set of tangrams for each student 

____________________________________________________________________________________ 


Variable, allow 30 minutes to get started. Students may work 

independently over a period of a week or so and turn in solutions later. 

____________________________________________________________________________________ 



Spatial Problem Solving with Tangrams 

Use the number of pieces in the first column to form each of the geometric shapes that 

appear in the top of the table. Make a sketch of your solution(s). Some have more 

than one solution while some have no solution. 

Make These Polygons 

Use this 

many 

pieces ↓ 

Square Rectangle Triangle Trapezoid Trapezoid Parallel 

-ogram 

2 

3 

4 

5 

6 

7 



Tangram Puzzles 

Can you make these shapes with the seven Tangram 

pieces? Make a sketch of your solutions. 

1. 2. 

3. 4. 

5. 6. 

Design your own tangram picture. Trace the outline and give it a name. Submit the 

outline and a solution key. 


Geoboard Exploration of the Pythagorean Relationship 



_________________________________________________________________________________ 

Description: 

Students will investigate the Pythagorean relationship by constructing right 

triangles on a geoboard. 

____________________________________________________________________________________ 

Materials: 


• 11 pin geoboards or dot paper 

• overhead geoboard 

____________________________________________________________________________________ 



____________________________________________________________________________________ 

Directions: 

1. On a transparent geoboard on the overhead projector, construct a right 

triangle in which one leg is horizontal and the other is vertical. Ask a 

student to construct a square on each leg and then on the hypotenuse of 

the triangle. Ask students to find the area of each square. It may be 

difficult for some students to recognize a way to find the area of the square 

on the hypotenuse so you may need to assist them. 

2. Give students the handout “Geoboard Exploration of Right Triangles”. 

Have them fill in the data from the example done by the whole class. Then 

have small groups work to find several other examples and record them in 

the chart. 

3. Debrief the examples with the whole class. 

4. What patterns do you see? (If students have not seen it before, you may tell 

them that this is the Pythagorean relationship that will be stated later as a 

theorem.) 

5. Can you state the relationship in words? In symbols? 

6. Do you think this is always true? 

7. If you label the sides of the triangle, can you write a statement of what you 

think is true? 

8. Does this procedure provide a proof that the relationship is always true? 



Geoboard Exploration of Right Triangles 

Make a right triangle on a large geoboard or dot paper. Construct a square on each 

side of the triangle. Label the shortest side a, the middle side b, and the longest side 

(hypotenuse) c. 

Complete the Table 

Length of 

side a 


side b 


side c 

Area of 

square on 

side a 


square on 

side b 


square on 

side c 

a 2 + b 2 


Proofs of the Pythagorean Theorem 



______________________________________________________________________________ 

Description: 

Students will experience two proofs of the Pythagorean Theorem-- 

an algebraic proof; and a transformational geometry proof-- 

in which they will cut up the diagram and rearrange the pieces as 

if it were a puzzle. 

____________________________________________________________________________________ 

Materials: 


• Scissors 

• Crayons or markers (2 colors) 

____________________________________________________________________________________ 



____________________________________________________________________________________ 


An Algebraic Approach to the Pythagorean Theorem 

W b U a 

a 

T 

c 

b 

c 

b 

c 

V 

c 

a 

X a S b Y 

Fill in expressions for each of the indicated areas: 

1. Area of the large square WXYZ= (a + b) 2 = (a + b)(a + b) = ______________________ 

2. Area of the large square WXYZ = 

area of square STUV + 4(area of triangle XST) = ______________ + ______________ 

3. Set the expressions from #1 and #2 equal to each other and simplify. 

Where have you seen this before? Shade a right triangle in the drawing above for which the 

relationships hold. 


A Transformational Geometry Approach to the Pythagorean Theorem 

W b U a Z 

a 

c 

V b 

c 

b 

c 

T 

c 

X a S b Y 

1. Mark off the square that has area of b 2 in the drawing above and shade it. Mark off 

the square that has area of a 2 in the drawing above and shade it. 

2. Cut off triangle XSV and triangle SYT and rearrange them to show that the 

combined area of the shaded sections is the same as the area of the interior square 

whose area is c 2 . Label the diagram to help write an explanation or what you did. 

(Note: there are two ways to do this. One involves translating the pieces, the other 

involves rotating the pieces.) 


Egyptian Rope Stretching 



______________________________________________________________________________ 

Description: 

Students are presented with a problem that shows an application of the 

Pythagorean Theorem in ancient Egyptian culture. 

____________________________________________________________________________________ 

Materials: 

A rope that has 13 knots tied in equal intervals. 

____________________________________________________________________________________ 


Approximately 30-40 minutes 

____________________________________________________________________________________ 

Directions: 

1. Show students the rope that has 13 knots tied in equal intervals. Tell them that a 

picture of a rope like this was found on inscriptions in tombs of ancient Egyptian 

kings. Ask students to work in groups to figure out what the purpose of the rope 

might have been. 

2. Ask students to suggest ideas for the way the Egyptians used the rope. If students 

come up with the idea that it was a used to make a template for determining right 

angles, let them demonstrate. If they do not, you should have two students help 

you demonstrate. Have a student hold knot #1 and knot #13 together. Have 

another hold knot #4. You should hold knot # 8. All three of you should stretch 

the rope and have the class observe the resulting shape. Make sure they 

understand that the only shape that can be formed when you pull is a right 

triangle. 

3. Have students conjecture about how the Egyptians might have used this rope. (To 

build right angled corners on pyramids? To redraw the boundaries of the fields 

after the spring flooding of the Nile?) 

4. Ask students what other numbers of knots in a rope might be used to serve the 

same purpose of forming a right triangle. For example: 

• Could you obtain the right triangle result with a rope that has 20 

equally spaced knots? Try it. 

• Could you do it with a rope with 31 knots (30 spaces)? 

Why or why not? 


Quadrilaterals and their Properties 

Reporting Category: Polygons and Circles 


_________________________________________________________________________________ 

Description: 

Students will investigate quadrilaterals and their properties through the use of 

various manipulatives such as sorting pieces and geo-strips and the use of 

graphing utilities. The sequence of activities is designed to facilitate an increase in 

a learner's van Hiele level of thinking about quadrilaterals from Level 1 to Level 3. 

_________________________________________________________________________________________ 

Materials: 

• Quadrilateral Sorting Pieces 


_________________________________________________________________________________________ 



_________________________________________________________________________________________ 

Directions: 

1. Students sort a set of quadrilateral pieces, an activity which enables the teacher 

to assess their van Hiele levels. 

2. Then they play the game "What's My Rule?" to develop the ability to classify 

quadrilaterals by various attributes and to focus on more than one attribute at a 

time. 

Refer back to the Introduction for additional descriptions of the van Hiele levels. 

Virginia Department of Education 72 Geometry Instructional 

Modules

Activity Sheet: Quadrilateral Sort 

Directions: 

• Divide the students into small groups. Distribute the quadrilateral sorting 

sheet, have the students cut out the quadrilaterals. 

• Instruct the students to lay out the pieces with the letters up. Do not call 

them quadrilaterals. Tell the students that objects can be grouped together 

in many different ways. For example, if we sorted the shapes that make up 

the American flag (the red stripes, the white stripes, the blue field, the white 

stars), we might sort by color and put the white stripes and the stars together 

because they are white, the red stripes in another group because they are red, 

and the blue field by itself because it is the only blue object. Another way the 

flag parts could be grouped would be all the stripes and the blue field together 

because they are all rectangles and all the stars together because they are not 

rectangles. 

• Have them sort the shapes into groups that belong together, recording the 

letters of the pieces they put together and the criteria they used to sort. 

• Have them sort two or three times, recording each sort. 

• Ask the students for some of their ways of sorting. 

• Have them compare their ways with those of other groups. 

Virginia Department of Education 73 Geometry Instructional 

Modules

Quadrilateral Sorting Pieces 


Activity Sheet: What's My Rule? 

Directions: 

Divide the students into groups of 3 or 4. Pass out the sets of cut-out 

quadrilaterals, one set per group. Go over the rules with the students. 

Rules 

1. Choose one player to be the sorter. The sorter writes down a "secret rule" to 

classify the set of quadrilaterals into two or more piles and uses that rule to 

slowly sort the pieces as the other players observe. 

2. At any time, the players can call "stop" and guess the rule. The correct 

identification is worth 5 points. A correct answer, but not the written one, is 

worth 1 point. Each incorrect guess results in a 2-point penalty. 

3. After the correct rule identification, the player who figured out the rule 

becomes the sorter. 

4. The winner is the first one to accumulate 10 points. 


Quadrilateral Properties Laboratory 



______________________________________________________________________________ 

Description: 

Students will construct parallelograms, rectangles, rhombi, and squares 

using D-stix, geo-strips, or toothpicks and marshmallows and make 

observations as the figures are flexed. 

____________________________________________________________________________________ 

Materials: 


• D-stix, geo-strips, or miniature marshmallows and toothpicks cut into two different 

lengths 

• square corner (the corner of an index card or book) 

____________________________________________________________________________________ 



____________________________________________________________________________________ 

Directions: 

• Divide the students into groups of 3 or 4 and direct each group to 

experiment as you ask questions. Be sure to model constructing the 

polygons and flexing them. 

• Have the students pick two pairs of congruent segments and connect them 

as shown below. Have them flex the figure to different positions. 

. . 

. . 

• Ask 

• What stays the same? (lengths of the sides, the opposite sides are 

parallel, opposite angles are the same, sum of angles, perimeter) 

• What changes? (size of angles, area, lengths of diagonals) 

• What do you notice about the opposite sides of this quadrilateral? 

(They remain parallel and congruent.) 

A parallelogram is a quadrilateral with opposite sides parallel. 

• What is the sum of the interior angles of this quadrilateral? (360°) 

• What do you notice about the opposite angles? (congruent) 


Note to Teacher: Some students will likely turn the strips so that they 

cross forming two triangles. If no one does, you should. Ask if this figure 

is a polygon. Elicit from the group what the essential elements of a polygon 

are, i.e., 

a) composed of straight line segments connected end to end 

b) simple (the segments do not cross) 

c) closed 

d) lies in a plane (e.g., if you take a wire square and twist it so 

that it isn't flat, it is no longer a polygon) 

• Make one of the angles a right angle. (You can use the square corner 

to check your accuracy.) 

• Ask 

• What happens to the other angles? (They become right angles.) 

• Will this always be true when you make one angle of a parallelogram a right 

angle? (Yes) 

• How do you know? (The sum of the angles in a parallelogram is 360°. One 

angle is given as 90°. Its opposite angle must be the same or 90°. 

Subtracting these two angles from 360°, the remaining two angles, which 

are congruent since they are opposite angles in a parallelogram, must total 

180°. Therefore, each is 90°. Note: This is Level 3 thinking.) 

• Is it still a parallelogram? (Yes) 

• Is it still a quadrilateral? (Yes) 

• Is it still a polygon? (Yes) 

• What other name, besides polygon, quadrilateral, and parallelogram, can be 

given to it now? (rectangle) 

A rectangle is a parallelogram with four right angles. 

• Make a parallelogram that has all four sides equal in length. What 

is another name for this parallelogram? (rhombus) 

A rhombus is a parallelogram with four congruent sides. 

• Flex the figure to different positions. 

. 

. 

. 

• Ask 

• What stays the same? (lengths of the sides, the opposite sides 

are parallel, opposite angles are the same, sum of angles, perimeter) 

• What changes? (size of angles, area, lengths of diagonals) 

. 


• What is the sum of the interior angles of this quadrilateral? (360°) 

• What do you notice about the opposite angles? (congruent) 

• Is it still a quadrilateral? (Yes) 

• Is it still a polygon? (Yes) 

• Make one of the angles of this rhombus a right angle, checking with 

your square corner. 

• Ask 

• What happens to the other angles? (All right angles) 

• Is it still a parallelogram? (yes) 

• What other name, besides polygon, quadrilateral, parallelogram, 

and rhombus, can be given to this new figure? (square) 

A square is a parallelogram with four congruent sides and four right angles. 

• Is it a rectangle? (Yes) 

• How do you know? (It has four right angles.) 

• Distribute the Types of Quadrilaterals sheet and discuss the definitions 

for quadrilateral, parallelogram, rectangle, rhombus, and square. 

Discuss the examples of each, noticing their orientations and how each 

example fits the definition even though they are not necessarily the 

stereotype figure usually seen. 


TYPES OF QUADRILATERALS 

A quadrilateral is a 

four sided polygon. 

A parallelogram is a 

quadrilateral with opposite 

sides parallel . 

These sides 

are parallel. 

A square is a quadrilateral 

with 4 right angles and 

4 congruent sides . 

A trapezoid is a 

quadrilateral with exactly 

one pair of parallel sides. 

These sides 

are parallel. 

A rectangle is a quadrilateral 

with all right angles 

A rhombus is a quadrilateral 

with all sides congruent. 

A kite is a quadrilateral with 2 

pair of consecutive 

congruent sides but not all 

congruent sides. 

A dart is a concave 

quadrilateral. 


Quadrilateral Sorting Laboratory 



______________________________________________________________________________ 

Description: 

Students will record which quadrilaterals meet the various descriptions 

listed in the properties table, determine which sets are identical and are 

subsets of one another, attach labels to each category, and create a 

quadrilateral family tree. 

____________________________________________________________________________________ 

Materials: 


• Paper quadrilaterals (from Quadrilateral Sorting Pieces)cut out and placed 

in a plastic baggy or manila envelope. 

____________________________________________________________________________________ 



____________________________________________________________________________________ 

Directions: 

• Distribute the Quadrilateral Table activity sheet. Divide the students into groups of 3 or 4 

and direct each group to experiment and answer the questions on the handout. 

• After the students have filled out the table, have pairs of groups compare their answers, 

and reconcile any discrepancies. 

• Distribute the Quadrilateral Sorting Laboratory activity sheet. 

• Have the students continue with Steps 5-10. 

• For Step 11 the students can construct the family tree as small groups or as a large 

group. Pass out the Family Tree activity sheet. 


Activity Sheet: Quadrilateral Table 

___________________________________________________ 

1. has 4 right angles 

___________________________________________________ 

2. has exactly one pair of parallel sides 

___________________________________________________ 

3. has two pair of opposite sides congruent 

___________________________________________________ 

4. has 4 congruent sides 

___________________________________________________ 

5. has two pair of opposite sides parallel 

___________________________________________________ 

6. has no sides congruent 

___________________________________________________ 

7. has two pair of adjacent sides congruent, but not all sides congruent 

___________________________________________________ 

8. has perpendicular diagonals 

___________________________________________________ 

9. has opposite angles congruent 

___________________________________________________ 

10. is concave 

___________________________________________________ 

11. is convex 

___________________________________________________ 

12. its diagonals bisect one another 

___________________________________________________ 

13. has four sides 

___________________________________________________ 

14. has four congruent angles 

___________________________________________________ 

15. has four congruent sides and four congruent angles 

___________________________________________________ 


RH 

O 

MB 

I 

PA 

RA 

LL 

EL 

OG 

RA 

MS 

QU 

AD 

RIL 

AT 

ER 

AL 

S 

OT 

HE 

RS 

Activity Sheet: Family Tree 



Quadrilateral Sorting Laboratory 

Directions: 

1. Spread out your Quadrilateral Set with the letters facing up so you can see them. 

2. Find all of the quadrilaterals having 4 right angles. List them by letter alphabetically in 

the corresponding row of the Table. 

3. Consider all of the quadrilaterals again. Find all of the quadrilaterals having exactly one 

pair of parallel sides. List them by letter alphabetically in the corresponding row of the 

Table. 

4. Continue in this manner until the Table is complete. 

5. Which category is the largest? What name can be used to describe this category? 

6. Which lists are the same? What name can be used to describe quadrilaterals with these 

properties? 

7. Are there any lists which are proper subsets of another list? If so, which ones? 

8. Are there any lists which are not subsets of one another which have some but not all 

members in common? If so, which ones? 

9. Which lists have no members in common? 

10. Label each of the categories in the Table with the most specific name possible using the 

labels kite, quadrilateral, parallelogram, rectangle, rhombus, square, and trapezoid. 

For example, #1 - a quadrilateral which has 4 right angles is a rectangle. (Having 4 right 

angles is not enough to make it a square; it would need 4 congruent sides as well.) 

11. Compare your results to that of the other Lab Groups. Then fill out the family tree by 

inserting the names kites, rectangles, squares, and trapezoids into the appropriate places 

on the diagram. 


Quadrilateral Diagonal Investigation 



______________________________________________________________________________ 

Description: 

Working in small groups, students will choose all the squares from the sample 

quadrilaterals, draw in their diagonals, and make observations about 

whether they are congruent, perpendicular, or bisect one another. They 

repeat the task with rhombi, rectangles, parallelograms, trapezoids, kites, 

and darts, supplementing their quadrilateral by drawing additional ones as 

needed to test their conjectures. Finally, they fill out a table summarizing 

their findings and other observations about various types of quadrilaterals. 

____________________________________________________________________________________ 

Materials: 


• Sample quadrilateral pieces 

• Straightedge 

• Protractor or angle ruler 

____________________________________________________________________________________ 



____________________________________________________________________________________ 

Directions: 

• Divide the students into small groups. Distribute the sample quadrilateral pieces . 

• Distribute the activity sheet. 

• Have students complete the activity sheet table. 

• Summarize the characteristics found from the table. 


Sample Quadrilateral Pieces 

A 

B 

C 

D 

E 

F 


Sample Quadrilateral Pieces 

page 2 

G 

H 

J 

J 

I 

L 



Quadrilateral Properties Investigation 

Both pair opposite sides must be //. 

Both pair opposite sides must be ≅. 

All sides must be ≅. 

Both pair opposite


Proving a Quadrilateral is a Parallelogram 

Directions: 

• Divide the students into small groups. Have them examine the 

Quadrilateral Properties table completed in this activity. In particular, have them 

examine the ways that you can insure that a quadrilateral is a parallelogram; that 

a parallelogram is a rectangle; and that a parallelogram is a rhombus. 

• Summarize this information using the summary sheet. 

Summary Information 

Ways to Prove that a Quadrilateral is a Parallelogram 

Show that: 

1. both pairs of opposite sides are //. 

2. both pairs of opposite sides are ≅. 

3. both pairs of opposite angles are ≅. 

4. one pair of opposite sides is both // and ≅. 

5. the diagonals bisect each other. 

Ways to Prove that a Parallelogram is a Rectangle 

Show that: 

1. its diagonals are ≅. 

2. it contains a right angle. 

Ways to Prove that a Parallelogram is a Rhombus 

Show that: 

1. its diagonals are ⊥. 

2. two consecutive sides are ≅. 


Triangle and Quadrilateral Angle Laboratory 



______________________________________________________________________________ 

Description: 

One property of the angles of a triangle is extremely important in 

understanding which polygons tessellate: The sum of the interior angles of 

any triangle equals 180°. Students will verify that the interior angles of a 

triangle add up to 180° by cutting three random triangles out of paper, 

numbering and tearing off their vertices, and rearranging them adjacent to 

each other to form a straight angle. They will also explore the meanings of 

counterexample and conjecture. Students then verify that the sum of the 

angles in a quadrilateral is 360° in a similar manner. 

____________________________________________________________________________________ 

Materials: 



• Scissors 

• Tape (optional) 

____________________________________________________________________________________ 



____________________________________________________________________________________ 



Triangle and Quadrilateral Angles Laboratory 

Questions: 

What is the sum of the internal angles in a triangle? in a quadrilateral? 

Is it always the same? 

Preparing for research: 

1. Draw 3 large triangles using a straightedge. Cut the 3 triangles out. Label 

the vertices of one triangle as 1, 2, and 3. Label the vertices of the second 

triangles as 1*, 2*, and 3*. Label the vertices of the third triangle as 1', 2', and 

3'. 

3 

1 

2 

2. Tear off the corners. (That's right. Tear, not cut. By tearing, you can still 

determine which was the vertex. It will be the cut part.) 

1 

2 

3. Draw a dot on the page and a straight line through the dot. Place the cut 

vertex of


1. What angle do these combined corners appear to form? 

2. Repeat this procedure with the other two triangles. Does the same thing 

happen with all three triangles? 

Inferring: 

Compare your predictions to your actual findings. 

Drawing conclusions: 

Add your results to that of the other Lab Groups in your class. Then form 

conclusions focused on the following issues. 

• Do the angles from all of their triangles always equal the same amount? 

• Examine any triangle that turned out differently. Why do you think it did? 

• Based on this activity, what conjecture can you make about the sum of the 

angles in a triangle? (You have probably known about this for many years.) 

• If you and your classmates made 200 triangles, tore off the vertices, lined 

them up, and they all totaled the same, would this insure that the 201st 

triangle would come out the same? 

• What if you made 1,000 triangles this way and they all came out the same? 

Would this insure that the 1,001st triangle that you made would come out 

the same? 

• What if you made 1,000,000 triangles and they all came out the same? 

Would this insure that the 1,000,001st triangle that you made would come 

out the same? 

If you could find a single triangle that came out differently, you would disprove 

the conjecture that the sum of the measures of the interior angles in a triangle 

always is 180°. Any item that does not fit your conjecture is called a 

counterexample. 

A single counterexample is enough to disprove a conjecture. Did you or any of 

your classmates find a triangle whose angles did not add up to 180°? 

Unfortunately, not finding one does not mean that one does not exist. However, 

it does give us more confidence in our conjecture. We can prove that, in 

Euclidean Geometry: the sum of the interior angles of all triangles equals 180°. 


Now, do the same thing with quadrilaterals. 


1. Draw 3 different large quadrilaterals using a straightedge. Cut the 3 

quadrilaterals out. Label the vertices. 

2 

3 

1 

4 

2. Tear off the vertices of each figure and arrange them adjacent to one another 

around a dot you have drawn. Trace or tape them in place. 


3. Repeat this sequence of steps until all the angles from one figure are 

adjacent to one another. 


1. What angle do these combined corners appear to form? 

2. Repeat this procedure with the other two quadrilaterals. Does the same thing 

happen with all three quadrilaterals? 


Add your results to that of the other Lab Groups in your class. Then form 

conclusions focused on the following issues. 

• Do the angles from all of their quadrilaterals always equal the same 

amount? 

• Examine any quadrilateral that turned out differently. Why do you think it 

did? 

• Based on this activity, what conjecture can you make about the sum of the 

measures of the interior angles in a quadrilateral? 

• What if you made 1,000,000 quadrilaterals and they all came out the 

same? Would this insure that the 1,000,001st quadrilateral that you 

made would come out the same? 

• Does this conjecture work with concave quadrilaterals? How could you 

test it? 


The Sum of the Angles in A Polygon 



______________________________________________________________________________ 

Description: 

Students will construct various polygons on a geoboard and put in all the 

diagonals from one vertex to divide each polygon into non-overlapping 

triangles. The students will compute the sum of the degrees in the 

triangles to get the sum of degrees in the polygon. They will generalize their 

findings into the formula for the sum of the interior and exterior angles in 

any convex polygon and then the size of each interior and exterior angle in 

a regular polygon. 

____________________________________________________________________________________ 

Materials: 


• Recording Table 

• Square geoboards and rubber bands 

• Paper and pencil 

____________________________________________________________________________________ 



____________________________________________________________________________________ 



The Sum of the Angles in a Polygon 

Questions: 

What is the sum of the measures of the internal angles in any polygon? 

Is there a pattern? Is there a general formula? What is the sum of the 

measures of the external angles of any polygon? Is there a pattern? 


Make a triangle of any kind on the geoboard. You can observe that it has 3 

sides and 3 interior angles. From your previous work, you know that the 

sum of its interior angles is 180°. 

. . . . . 

. . . . . 

. . . . . 

. . . . . 

. . . . . 


1. Remove the triangle and make a quadrilateral on the geoboard. As you can 

see, it has 4 sides and 4 interior angles. Put your index finger on any vertex 

peg in your quadrilateral. Join that peg to the peg at the non-adjacent 

vertex with a rubber band (i.e., put in all the diagonals from one vertex). It 

should form 2 triangles. Does it? If the sum of the interior angles in each of 

those two triangles is 180°, the sum of the angles in both triangles is 360°. 

. . . . . 

. . . . . 

. . . . . 

. . . . . 

. . . . . 

2. Remove the quadrilateral and its diagonal. Make a pentagon on the geoboard. 

Fill in the number of sides and interior angles on the table. 

. . . . . 

. . . . . 

. . . . . 

. . . . . 

. . . . . 

3. Put your finger on one of the vertices. Put in all the diagonals from it. Fill in 

the number of diagonals and number of triangles made when you put in the 

diagonals. 


Inferring: 

• Based on the number of triangles you've made, figure out what the sum of the 

interior angles in the figure is and enter it into the table. 

• Repeat steps 2-3 for the rest of the polygons listed in the table. 


• The last entry is for an n-gon. An n-gon is a polygon with n sides. How many 

interior angles must this polygon have if it has n sides? 

• Observe the relationship between the number of sides the polygon has and the 

number of diagonals. This relationship should indicate the expression or 

formula for finding the number of diagonals you can make from one vertex in 

an n-gon. 

• Similarly, observe the relationship between the number of diagonals from one 

vertex and the number of triangles formed. Fill in your table with this 

information. 

• Now, generalize the information in your table to find an expression to find the 

sum of the interior angles of a n-gon. 


Recording Table: 

Type of # of # of # of # of s Sum of 

Polygon Sides Interior

If the sum of the interior angles of a triangle is 180°, how large is each of the 3 

congruent angles in a regular triangle? How large is any exterior angle? What 

is the sum of all the exterior angles? 

Using the table you made above to help you, fill in the following table. 

Type of # of Sum of Size of Size of Sum of 

Regular Interior Interior Each Each Exterior 

Polygon

Interior Angles in Polygon 



______________________________________________________________________________ 

Description: 

Students will compare the graph of y = 180(x-2) to the scatterplot produced 

by plotting the number of sides of a polygon vs. the sum of the interior 

angles in the polygon. 

____________________________________________________________________________________ 

Materials: 


• Graphing Calculator 

____________________________________________________________________________________ 



____________________________________________________________________________________ 



INTERIOR ANGLES OF POLYGONS 

1. Complete the chart. 

Number of Sides 

Sum of the Measures of 

the Interior Angles 

3 

4 

5 

6 

7 

8 

2. Clear any data stored in the first two lists in your calculator. 

Press STAT 2nd [L 1 ] 

, 2nd [L 2 ] ENTER 

3. Enter data from the chart into lists L1 and L2. 

Press STAT ENTER . Enter the number of sides in L1 and the 

sum of the measures of the interior angles in L2. 

4. Press WINDOW and enter the following settings: 

Xmin = 0 

Xmax = 9 

Xscl = 1 

Ymin = 0 

Ymax = 1300 

Yscl = 100 

5. Make a scatter plot. Clear all functions from the Y = list. Press Y = .and 

CLEAR as needed. Press 2nd [STAT PLOT] and select Plot1 by pressing 

ENTER . Choose the scatter plot, L1 as the Xlist, and L2 as the Ylist. Press 

GRAPH 

. Make a sketch of the results. 


6. Now graph the function y = (x-2)180. To do this, press Y = .and enter (X- 

2)180 where you see the blinking cursor. Press GRAPH . 

a. Does the graph of the function seem to go through the points on the 

scatter plot? 

b. What do the variables x and y represent in the function 

y = (x - 2)180? 

7. Change the window settings to have Xmax = 25, Ymax = 4200, and Yscl = 

200. Press GRAPH . 

8. Now find the value of y when x = 17. Press 2nd [CALC] . Enter 17 

after the prompt by pressing 17 ENTER . 

What value is displayed? 

What does this value represent? 

9. Find the sum of the measures of the interior angles for the given number of 

sides: 

12 _____ 18 _____ 22 _____ 


Circumference 



______________________________________________________________________ 

Description: 

Students will presented with a problem that will challenge their 

intuition about measurements of the circumference of a tennis ball can 

and the height of the can. 

____________________________________________________________________________________ 

Materials: 

• Can of tennis balls 

• Piece of string 

____________________________________________________________________________________ 



____________________________________________________________________________________ 

Directions: 

1) Have students bring a tennis ball can to class. Ask students which they 

think is greater: the height of the can or the circumference of the can? 

Have each participant write down his/her answer supported with 

reasons. Then they should discuss it in groups before having a whole 

class discussion. 

2) Discuss student conclusions with the whole class. 

3) Have students wrap a string around their tennis ball and compare this to 

the height of the can. Get participant reactions. 


Cake Problem 



______________________________________________________________________________ 

Description: 

Students will be presented with a problem that requires them to apply what 

they know about area of circles. 

____________________________________________________________________________________ 

Materials: 

Activity Sheet 

____________________________________________________________________________________ 



____________________________________________________________________________________ 

Directions: 

1) Present the cake problem to students. Make sure everyone understands 

the problem. Put a 10-inch diameter circle on the overhead projector to 

model the cake and ask a volunteer to make an estimate of the placement 

of the cut that solves the problem. 

2) Give out the handout and have students work in small groups to solve it. 

Have each student write up their solution. 

3) Ask for volunteers to share solutions. Discuss variations on solutions. 

4) Return to the transparency of the cake and draw the correct solution. 

How close did the estimate come? 



Cake Problem 

You have a cake that is 10 inches in diameter. You expect 12 people to share it, so 

you cut it into 12 equal pieces (see Figure A). 

Before you get a chance to serve the cake, 12 more people arrive! So you decide to cut 

a concentric circle in the cake so that you will have 24 pieces (see Figure B). 

How far from the center of the cake should the circle cut be made so that all 24 people 

get the same amount of cake? 

Figure A Figure B 


Geometer’s Sketchpad Investigation of ð 



______________________________________________________________________________ 

Description: 

Students will investigate the relationship of circumference and diameter 

using The Geometer’s Sketchpad. Using the dynamic medium of the 

Sketchpad script, they should observe that changes in the diameter of the 

circle result in a corresponding change in circumference but that the ratio 

of circumference to diameter is fixed. 

____________________________________________________________________________________ 

Materials: 


• Geometer’s Sketchpad software 

____________________________________________________________________________________ 



____________________________________________________________________________________________ 



Geometer’s Sketchpad Investigation of ð 

1) Make sure that labels will be displayed automatically. Click on DISPLAY 

from the menu bar at the top of screen. Choose PREFERENCES and then 

make sure the box “points” is checked under “Autoshow Labels.” 

2) Return to the blank screen. Click on the circle icon on the left of the 

screen. Click on the center of the screen and drag to form a circle. The 

center should be labeled A and a point on the edge of the circle should be 

labeled B. 

3) Click on the line segment icon on the left of the screen. Click on point A at 

the center of the circle, drag to point B, and release. A radius should be 

displayed. 

4) Click on the pointer arrow on the left of the screen. Click on the circle to 

highlight it. 

5) Click on MEASURE at the top of the screen on the menu bar. Slide down to 

RADIUS and release. The radius of the circle will appear on the screen. 

6) Click on MEASURE again, slide down to CIRCUMFERENCE and release. . 

The circumference of the circle will appear on the screen. 

7) To measure and display the diameter of the circle: click MEASURE, slide 

down to CALCULATOR and release. Click on the line that displays the 

measure of the radius of circle AB; it will be automatically copied onto the 

calculator window. Click ∗ and 2 from the calculator keypad to double the 

radius. Click OK and the new expression representing the diameter of the 

circle will be transferred to the main screen. 

8) Return to CALCULATOR by way of MEASURE. Click on the circumference 

expression on the script screen. Click the / symbol on the calculator 

keypad (division) and then click on the diameter expression from the main 

screen. Click OK and the quotient expression for 

(circumference)/(diameter) should appear on the main screen along with the 

value for the circle that appears on the screen. 

9) Debrief the activity with questions such as the following: 

• How did the circumference of the circle change when the radius 

increased? when the radius decreased? 

• How did the ratio of the circumference to the diameter change when you 

made the circles larger and smaller? 

• How could you verify that the constant value, 3.14, was not just a bug in 

the software -- that circumference divided by diameter of any circle is 

always a constant? 


10) Now you are ready to observe how changes in the radius affect the 

diameter and circumference of a circle. Click and drag point B to make the 

circle larger and smaller. Stop it in five different places and record the 

values for radius, diameter, circumference, and C/D in the table below. 

radius diameter circumference C/D 


Problem Solving With Circles 



______________________________________________________________________________ 

Description: 

Students will use circumference, perimeter, and the formula for volume of 

a cylinder to solve applied problems. 

____________________________________________________________________________________ 

Materials: 


• Problem Solving Sheet 

• Graphing calculator 

____________________________________________________________________________________ 



____________________________________________________________________________________ 



Teacher Guide 

Using the Activity: 

In this activity, students can use the calculator to find the lengths of three 

different ramps in-line skaters might use. Students will use the pi, square, and 

square-root keys on the calculator. The teacher may want to review the formula 

for finding the circumference (perimeter) of a circle. Discuss how to find the 

lengths of various arcs of the circle. To find the length of the third ramp, 

participants will need to use the Pythagorean Theorem. 

Another important extension to this activity is finding the steepness of the 

various ramps. Students can use this data to determine the level of difficulty of 

each ramp. 

Answers: 

First ramp: 51.41592654 ft. 

The length of the arc AB is one quarter the circumference of the circle. 

Second ramp: 40.943951 ft 

The length of the arc is one sixth the circumference of the circle. 

Third ramp: 41.540659 ft 

The length of the ramp x is found by using the Pythagorean Theorem. 

Discussion Questions: 

1. What makes one ramp better than another? 

2. Which ramp is safest? Why? 

3. Which construction is more challenging? Why? 

Thinking Cap: 

D = 2L + (2)(1/2)(Pi)d = 2L + (Pi)d 

(adapted from lesson by Ann Mele, Assistant Principal, Offsite Educational Services; NY Public 

Schools; NY, NY found at http://pegasus.cc.ucf.edu/~ucfcasio/) 




In-line skating has become a popular city sport. The parks department is 

thinking of constructing ramps in some of the local playgrounds. A "half-pipe" 

ramp is formed by two quarter circle ramps each 10 feet high with a flat space 

of 20 feet between the diameters. 

1. Find the distance a skater travels from the top of one ramp to the top of the 

other. 

∩ 

(hint: What is the length of AB?) 

2. Another launch ramp is formed by 2 arcs each with a central angle of 60 

degrees and a radius of 10 ft. Find the length from the top of one ramp to 

the top of the other. (Hint: What fractional part of the circle is each arc?) 


3. A third ramp is a straight ramp 4 ft high and 10 ft long with a flat space 

of 20 ft in point P to point R. Find the distance a skater travels from the top 

of one ramp to the top of the other. (Hint: Use the Pythagorean Theorem) 

Thinking Cap 

A school track is formed by 2 straight segments joined by 2 half circles. 

Each segment is L long and each half circle diameter is D in length. Write 

a formula for finding the distance, D, around the track. 


Problem Solving Sheet 

SODA STRAWS 

How many straws-full of pineapple juice can be taken from a 46 fl.oz. can of juice that 

is filled to the top? 

diameter of the can__________ 

height of the can____________ 

diameter of straw____________ 

length of straw______________ 

1. Guess and check off home screen 

2. Create an algebraic expression and use tables 

DUCT TAPE 

How many rolls of duct tape would it take to create a one-mile strip of tape? (you may 

not unroll the duct tape or read its length from the packaging). You may have one 6- 

inch piece of tape to experiment with. 

1. Imagine tape as concentric circles...1st method 

2. Find volume of tape...2nd method 

(adapted from lesson by Bob Garvey, Louisville Collegiate School, Louisville, KY, bgarvey@aol.com, 

found at http://www.ti.com/calc/docs/act83geom.htm) 


Investigation of Secants and Circles 



____________________________________________________________________________________ 

Description: 

Students will be given instructions on how to use the Sketchpad to investigate 

the relationship between the measure of an angle formed by two secants drawn 

from a point outside a circle and the measure of the intercepted arcs. They will 

then use what they have learned to set up a demonstration to illustrate the 

relationships of segment lengths when two secants are drawn from a point 

outside the circle. 

__________________________________________________________________________________________ 

Materials: 

Geometer’s Sketchpad 

____________________________________________________________________________________ 



____________________________________________________________________________________________ 

Directions: 

1) Introduce the following relationships: 

a) The measure of an angle formed by two secants drawn from a point 

outside a circle is equal to half the difference of the measures of the 

intercepted arcs. 

b) When two secant segments are drawn to a circle from an external 

point, the product of the lengths of one secant segment and its external 

segment equals the product of the lengths of the other secant segment 

and its external segment. 

2) Give students the worksheet and ask them to work with a partner at the 

computer to use the Geometer’s Sketchpad to investigate these two 

statements or use the software to do a whole-class demonstration. 

3) After students have a chance to try out the Sketchpad investigation, 

discuss their findings. 

4) Ask students to write, in their own words and with a diagram of their own, 

what the key ideas are from this investigation. 



Investigation of Secants and Circles 

1. Draw a circle with a radius about one inch long. The center should be labeled A 

and the point on the circle B. 

2. Draw a second point on the circle and label it C. 

3. Draw a point outside the circle and label it D. 

4. Draw segment BD and segment CD. If these segments are not secants, move point 

D and point C until BD and CD are secants. 

5. Mark the point on segment CD where the secant intersects the circle and label it 

E. 

Mark the point on segment BD where the secant intersects the circle and label it 

F. 

6. Now we want to measure arc FE. 

a) Highlight the center of the circle (A), then while holding down the shift key, 

highlight point F and then point E. 

b) Choose “Arc on Circle” from the CONSTRUCT menu. 

c) While the arc is highlighted, choose “Show Label” from the DISPLAY menu. 

(It should read a1.) 

d) While the arc is still highlighted, choose, “Arc angle” from the MEASURE 

menu. Note the measure of arc a1 is displayed on the screen. 

7. To measure arc CB: repeat the process outlined in #6. Remember that the 

Sketchpad reads the endpoints of the arc in the counterclockwise direction! 

8. Now that you have labeled and measured the two arcs that are intercepted by the 

secants, you are ready to explore the relationship of those arcs (arc FE and arc 

CB) and the angle formed by the secants (angle EDF). 

• Measure angle EDF: Hold the SHIFT key and highlight points E, D, and F. 

Then while they are still highlighted, go to the MEASURE menu and choose 

“Angle.” On the screen, you will see the measure of angle EDF stated. 

• Go to the MEASURE menu; choose “Calculate.” In the box at the top of the 

calculator, set up the expression needed to express half the difference of the 

two arcs. (Click on the expressions that appear on the Sketchpad screen to 

use them in the calculator.) 

• Compare the two quantities you have set up in part a and part b. 

• 

• 

• 

• 

• 

• 


• Keep an eye on those two quantities as you gently drag point D away from 

the circle. What do you notice? 

• Now gently drag point D closer to the circle. What do you notice? Does it 

matter how far point D is from the circle? 

9. Now that you have some experience in investigating the relationship between the 

angle formed by two secants and the intercepted arcs, use the Sketchpad to 

study the following relationship: 

When two secant segments are drawn to a circle from an external point, the 

product of the lengths of one secant segment and its external segment equals the 

product of the lengths of the other secant segment and its external segment. 

You can use the diagram you made on the Sketchpad to test this idea. 

a) For each relevant segment, highlight the endpoints, CONSTRUCT the 

“segment” and MEASURE it. 

b) Then set up and “calculate” (under the MEASURE menu) the appropriate 

products required by the statement above. 

c) Gently drag point D to different positions and see if the relationship still 

holds. 


Euler’s Formula 

Reporting Category: Three-Dimensional Figures 


______________________________________________________________________________ 

Description: 

By examining models, students will derive Euler’s formula (F+E - V =2). 

_____________________________________________________________________________________ 

Materials: 

• Spreadsheet software 

• Cubes for model building 

_____________________________________________________________________________________ 



_____________________________________________________________________________________ 

Directions: 

1) Have each student set up a spreadsheet (or provide each with a spreadsheet) 

set up as seen here: 

Shape # faces # edges # vertices F+E-V 

1 x 1 cube 

1 x 2 rectangle 




your choice #1 

your choice #2 

2) Provide each student with 10-15 unit cubes with which to build models. 

Review the terms face, edge, vertex to be sure all participants can count 

these accurately. 

3) Have each student build the first four rectangles listed in the spreadsheet 

and enter the numbers of faces, edges, and vertices into the spreadsheet. 

4) Have the students work in small groups to develop possible relationships 

among these numbers and test them on their spreadsheet. 

5) Once students have hypothesized a relationship that fits their data, have 

them use the cubes to construct two more figures and test their hypothesis 

on these. 


Introduction to the Soma Cube: 

Constructing the Soma Pieces 



______________________________________________________________________________ 

Description: 

Students will build the seven Soma Cube pieces as a spatial problem solving 

task and will find the surface area and volume of each of the seven pieces. 

_____________________________________________________________________________________ 

Vocabulary: 

Tricube: solid made of three unit cubes joined face-to-face. 

Tetracube: solid made of four unit cubes joined face-to-face. 

_____________________________________________________________________________________ 

Materials: 

• 27 wooden cubes, 27 sugar cubes, or 27 Snap cubes for each student 

• glue 

• permanent markers 

_____________________________________________________________________________________ 



_____________________________________________________________________________________ 

Directions: 

1) Give out 27 wooden 1-inch cubes or 27 Snap cubes to each student or small 

group of students if materials are limited. Present them with the following 

problem: 

• How many different ways can you join 3 cubes face-to-face? These 

are called “tricubes.” 

Have students try finding them with the cubes and discuss the 

results with the class. Tell them that if a tricube can be flipped or 

repositioned (reflected, rotated) in such a way that it is exactly like 

a tricube already made, then it is not different from the other one. 

There are only two different tricubes. However, for this activity we 

only need to save the non-rectangular one. Have students put 

aside the rectangular one (every face is a rectangle!). 

2) Have students find all possible non-rectangular tetracubes (4 unit cubes 

joined face-to-face). Discuss what they find. There should be six different 

ones (see teacher reference page that accompanies this lesson). You may 

have students glue the wooden cubes together or snap the Snap cubes 

together and number the completed pieces according to the teacher 

reference page. 


Discuss the nature of the pieces: 

• Are any of the pieces reflections of each other? (If you put a mirror 

next to one piece, will you see the other in the mirror?) 

(Pieces #5 and #6) 

• Which pieces can be placed so that they are only one unit high? 

(Pieces #1, #2, #3, #4) 

• Which pieces must occupy space that is 2 units high? 

(Pieces #5, #6, #7) 

• Which of the pieces have a line of symmetry on a given face? 

3) Review the concept of volume by identifying one of the wooden cubes or 

one of the Snap cubes as the unit. Discuss the face of the unit cube as 

the unit of area. Have students find the surface area and volume of each 

of the numbered solids #1 - #7 and complete the table in the activity sheet 

Surface Area and Volume. 

Discuss their findings: 

• Did the pieces with the same volume have the same surface area? 

• Did the pieces with the same surface area have the same volume? 

4) Give students diagrams of the top, side, and bottom view of Soma pieces 

in the Handout “Soma Views from the Top, Front, and Side” and have them 

identify the pieces from these views. Have them draw the diagrams for the 

remaining pieces. 

Extension: Ask students to build with Snap cubes, all “pentacubes” that 

can be made with five cubes each. 


Instructor Reference Sheet 

A 

1 

B 

2 

C 

4 

D 

3 

1 1 

2 2 2 

4 4 

3 

3 3 

4 

E 

7 

7 

7 

F 

5 

5 

5 

G 

6 

6 


SOMA Views from Top, Front, and Side 

top front right side 

top 

front 

right side 


Surface Area and Volume of the Soma Pieces 

surface 

piece # area 

vol. 

1 

2 

3 

4 

5 

6 

7 


Building the Soma Cube and other Structures with Soma Pieces 



______________________________________________________________________________ 

Description: 

Students will use the seven Soma pieces to build the 3x3x3 cube and 

other structures that use all seven pieces. 

_____________________________________________________________________________________ 

Materials: 


• Recording Sheet 

• Seven Soma pieces 

_____________________________________________________________________________________ 


Approximately 45 minutes; some students may want to extend the activities 

independently. 

_____________________________________________________________________________________ 

Directions: 

1. Give students some history of the Soma Cube. It was invented by Piet Hein 

in Denmark. He was listening to a lecture on quantum physics when the 

speaker talked about slicing up space into cubes. Hein then thought about 

all the irregular shapes that could be formed by combining no more than 

four cubes, all the same size and joined at their faces. In his head he 

figured out what these would be and that it would take 27 cubes to build 

them all. From there he showed that the pieces could form a 3x3x3 cube. 

2. Tell students: So now we know that the seven pieces fit together to form a 

3x3x3 cube. In fact, there are 1,105,920 different ways to assemble the 

cube. Try to find one. 

Let students work until they get a solution. 

3. Ask students how they might make a record of the solution they got before 

they take the cube apart. You may show them one way by sharing the 

solution recording sheet that accompanies this lesson. It requires that the 

numbers of the unit cubes be recorded in three layers: top, middle, and 

bottom. Have students record their solutions. (Note that some will need 

help in making the correct correspondence of the numbers to the grid). 

Then have them try to find a different solution and record it. 


Discuss: 

• How many dimensions are represented in the Soma cube? 

• How many dimensions are represented in the solution sheet? 

• Does anyone want to share any strategies that might help in 

transferring the 3-dimensional information onto the 2-dimensional 

representation? 

4. Now reverse the order of the task. Give students the activity sheet, Build 

This Cube, to do. 

Top Middle Bottom 

5 5 

2 

5 

3 4 

3 3 3 

6 

5 

2 

6 

4 4 

7 

1 1 

6 6 

2 

7 

4 

2 

7 7 

1 

5. Give students the activity sheet with pictures of structures that can be 

built with the seven Soma pieces. Once they have succeeded at 

building any of the structures, they should label the drawings 

with the numbers of the appropriate pieces. 

6. Have students complete a table in which they compare the volume of the 

structures and the surface area. Discuss their observations with the whole 

class: 

• What is the volume of the completed Soma cube? 

• What is the volume of the other structures you built? 

• How can you tell the volume of the structures by only studying the 

picture and not actually building them? 

• How can you figure out the surface area of the structures without 

actually building the figures? 



Build This Cube 


5 5 

2 

5 

3 4 

3 3 3 

6 

5 

2 

6 

4 4 

7 

1 1 

6 6 

2 

7 

4 

2 

7 7 

1 


Soma Solutions Recording Sheet 





Making 2-Dimensional Drawings 

of 3-Dimensional Figures 



______________________________________________________________________________ 

Description: 

Students will use isometric dot paper to make drawings of Soma pieces. 

_____________________________________________________________________________________ 

Materials: 

• Isometric dot paper 

• Soma pieces 

_____________________________________________________________________________________ 



_____________________________________________________________________________________ 

Directions: 

1) Distribute the isometric dot paper to students. Have them study it and 

discuss how it is different from regular graph paper. Discuss: 

• What does the prefix “iso” mean? 

• How does this apply to the way the paper is designed? 

2) Demonstrate on the overhead projector how to draw a single unit cube 

using the paper. Have students practice drawing single cubes while 

working in small groups so they can help each other. 

(Hint: the easiest way to show this is by drawing a Y in the center and 

then circumscribing a hexagon around it.) 


3) Show students how to position one of the seven Soma pieces on a 

diagonal so that they can draw it on the isometric paper. Draw one 

on the overhead while talking through the process as you go. 

4) Have students work in groups to draw all seven Soma pieces on 

isometric dot paper. Each student should complete his or her own 

drawings with the help of group members. 

5) Challenge students to design a structure using all seven Soma 

pieces and draw it on the isometric dot paper. 


Isometric Dot Paper 


Spatial Problem Solving 



______________________________________________________________________________ 

Description: 

Students will do paper folding and cutting activities and activities with 

Cuisenaire Rods to help develop their spatial problem solving skills. 

_____________________________________________________________________________________ 

Materials: 

Books by Patricia S. Davidson and Robert E. Willicutt 

1) Spatial Problem Solving with Paper Folding and Cutting available 

from Cuisenaire Co. 

ISBN 0-914040-36-7 

2) Spatial Problem Solving with Cuisenaire Rods available from 

Cuisenaire Co. ISBN 0-914040-99-5 

_____________________________________________________________________________________ 


open-ended 

_____________________________________________________________________________________ 

Directions: 

These books provide a wealth of spatial problem solving activities. Each 

book contains black line masters for duplication in the classroom. They 

can be used with the whole class or as extensions for independent work. 


Architect’s Square 



______________________________________________________________________________ 

Description: 

Students will work in small groups to create all possible four-cube figures 

and arrange them into a housing development. 

____________________________________________________________________________________ 

Materials: 

• Activ 

ity Sheets 

• Unit 

cubes (approximately 60 per group) 

• Isom 

etric dot paper 

____________________________________________________________________________________ 


Approximately 5 class sessions 

____________________________________________________________________________________ 

Directions: 

1. Have the 

students work in groups of 2-4. Instruct them to build all possible 

arrangements of four unit cubes. These will be houses, and the first 

memorandum sets the stage for the activity. Students should record their 

figures on isometric dot paper (p.129). (2 days) 

2. Onc 

e students have found all 15 possible arrangements, they should 

use the information in activity sheet #2 to calculate costs for each 

house. (1-2 days) 

3. Using their 

information, students should next plan a brochure advertising their houses. They should 

think about the features of each house for an elderly couple, a single person, and a family 

with small children. Activity Sheet #3 contains this information. 


Activity Sheet #1 

TO: All Architects 

RE: Housing Designs 

We are beginning work on a new subdivision of modular homes. Each home 

will include four units. All units are shaped as cubes and adjacent units must 

touch over a full face. It is not acceptable to have faces overlap partially, nor is 

it acceptable for cubes to touch at edges only. 

Your task is to figure out how many different houses we can design. A design is 

considered different from another if they cannot be superimposed without 

reflection. Use the cubes provided to you to design and record (on dot paper) all 

possible home designs. 



TO: All Architects 

RE: Housing Costs 

Now that you have drawn all possible houses, you must figure the cost of each house. 

There are three factors which influence the cost of our houses. Each square unit of 

land has a fixed cost, each square unit of wall has a fixed cost, and each square unit 

of roof has a fixed cost. These costs are as follows: 

Land 

$10,000 per square unit 

Wall 

$ 5,000 per square unit 

Roof 

$ 7,500 per square unit 

Using your drawings, models you build, and these figures, complete a chart like the 

one below to price each home. 

Design 

# 

Units of 

Land 

Land 

Cost 


Wall 

Walls 

Cost 




Cost 

Total 

Cost 



TO: 

RE: 

All Architects 

Advertising 

Now that your houses are designed and costs calculated, we need your help 

with a marketing plan. There are three groups we would like to target — elderly 

couples, families with small children, and single people. Look at your design 

and consider such factors as stairs to climb, space for people to spread out, and 

cost. Which home design would be best for each of these three groups? 

Plan a brochure which includes designs for at least six different homes, including the 

three you selected above. For each home, include a sketch of the home on dot paper, 

the calculations for the cost of the home, and three to five selling points for that 

design. Your brochure should be colorful and easy to read. 


Exploring Surface Area and Volume 



______________________________________________________________________________ 

Description: 

Students will derive the formulas for the surface area of a 

rectangular prism, a cylinder, and a sphere. 

_____________________________________________________________________________________ 

Materials: 


• Calculator 

• Cans 

• Scissors 

• Boxes of light cardboard (to cut into nets) 

• Oranges, waxed paper, knife to cut oranges in half 

______________________________________________________________________________ 

Teacher Note: 

Nets for shapes are commercially available from publishers such as 

ETA and Cuisinaire/Dale Seymour. 



Exploring Surface Area 

Questions: 

What are the formulas for determining surface area of solid 

figures? How can these formulas be used to solve problems? 


Working with a partner, complete the Finding Formulas activity 

sheets to derive the formulas for surface areas and volumes of 

rectangular prisms and the surface area of a sphere. 


Complete the second activity, Making Nets, to extend your 

knowledge of surface area of a variety of solid shapes. 

Inferring: 

Finally, use your learning from these first activities to complete 

the Solving Problems activity sheet. You may use your 

calculator here. 


Bring together learning from these exercises by discussing 

the relationship of surface area to two-dimensional perimeter 

and area. 



Finding Formulas 

1. After purchasing a gift for a friend, you decide to cover the box sides and 

bottom with wrapping paper. A diagram of the box appears below. How much 

wrapping paper will you need to cover the sides and bottom of the box? 

Your gift box is called an open box; it has no top surface. If this were a closed box 

— with a top surface — how much additional paper would be required to cover the 

surface? How much total paper is required? 

How can you generalize the process you used to find the surface area of the closed 

box? Let l = length, w = width, and h = height of the box. 

Compare the formula you and your partner developed to that of another group. 

Did you have the same result? You should be able to justify your formula to your 

classmates. 

2. If your gift were a can of tennis balls, the surface area would be the surface 

of the cylinder (the lateral area) and the areas of the top and bottom (the bases). 

Use a can (soup can, soda can, tennis ball can) for this activity. 

Wrap a piece of paper around the can, trim it to fit exactly, and 

spread it out flat. What shape is it? How can you find its area? 

What relationship does the length of the label have to the can? The 

height of the label? 

What shape are the bases of the can? Are the two bases identical? What is the 

area of each base? 

The surface area of the can = the lateral area + the area of the two bases. For your 

can, what is the surface area? Use your calculator to find decimal approximations 

to the nearest tenth. 


Finding Formulas (continued) 

3. The surface area of a sphere is more difficult to figure out. On a globe, a great 

circle is a circle drawn so that, when the sphere is cut along the line, the cut 

passes through the center of the sphere. The equator is a great circle on a globe. 

Take an orange from your teacher and draw a great 

circle on the orange. Cut the orange (carefully!) along 

the circle and trace five cut halves on a piece of waxed 

paper (see figure). 

Now, carefully peel both halves of the orange and fill in 

as many circles as you can with the peel. How many 

circles did your group fill? How does this compare 

with the findings of other groups? What is the class 

estimate for the number of great circles filled by the 

peel? 

Using one of your great circle tracings, find the radius of your orange and the area 

of one great circle. 

Given the area of one great circle and your estimate of the number of circles filled 

by the peel, what is the surface area of your orange? 

What is the general formula for the surface area of a sphere in terms of its radius? 



Making Nets 

A net is a flattened paper model of a solid shape. For example, the net below, when 

folded, makes a cube. 

Can you draw a different net which, when folded, will make a cube? Cut out your net 

and fold it to test your drawing. 

A net is helpful because it represents the surface area of a shape. Take a box from 

your teacher’s supply and cut it into a net. Note whether your box is open or closed. 

Sketch your box and net below. 

Use the formula you derived in Finding Formulas problem 1 to find the surface area of 

your box. Explain to a classmate how your net relates to your formula. 


Making Nets (continued) 

Now sketch a net of the can you used in Finding Formulas #2. 

How does this net relate to the surface area formula you found? 

Sketch a net of the pyramid below. Use your net to find the 

surface area of the pyramid. 

What shape does the net below make? 



Solving Problems 

1. Two cylindrical lampshades 40 cm in diameter and 40 cm high are to be covered with new fabric. 

The fabric chosen is 1m wide. If you purchase a 1.5 m length of this fabric, will you have enough to 

cover the shades? Justify your answer. 

2. An umbrella designer has created a new model for an umbrella that 

when opened has the form of a hemisphere with a diameter of 1 meter. 

If a dozen sample models are to be made using a special waterproof 

material, approximately how much fabric will be needed, allowing 0.5 m 

for seams and waste for each model? Explain your plan, strategies, and 

how you solved the problem. 



Exploring Volume 

Questions: 

How do you calculate the volume of a solid figure? How do you use 

volume to help solve problems? 

Materials: 


• Unit cubes 

• Milk cartons or other boxes of the same shape and different sizes 

• Hollow solid figures (e.g., Power Solids) 

• Sand/rice/water to pour between figures to show volume relationships 

• Rulers 

• Calculators 

• Cans 

• Graduated cylinder 


1. Distribute milk cartons (with the tops cut off) or open boxes of other sizes 

along with unit cubes. Ask students to fill the boxes with cubes as best they 

can and estimate the volume of the box based on counting the cubes used to 

fill the box. 

2. Ask students to develop a quick method (quicker than counting) for figuring 

out how many whole cubes fill the box. They should rapidly decide on 

length x width x height. 


Now ask students to generalize this finding to other prisms/cylinders 

by relating the area of the base to the height. Use a can as a model of 

a right circular cylinder. Students calculate the area of the base and 

multiply by the height to find the volume. Since 1 cm 3 = 1 ml, a 

graduated cylinder can be used to compare the estimated volume with 

the actual volume. 

Inferring: 

Ask students to use these procedures and findings to solve the 

problems on the Exploring Volume worksheet. 



Discuss the relationship of volume to surface area and the 

relationship of these measures to area and perimeter of two 

dimensional figures. Both Gulliver’s Travels and Architect’s Square 

(other activities in this toolkit) extend these concepts. 


Activity Sheet: Exploring Volume 

1. Which will carry the most water in a given length, two pipes where one has a 3 dm 

radius and the other a 4 dm radius, or one pipe with a 5 dm radius? Explain. 

2. A company delivers 36 cartons of paper to your school. Each carton measures 40 

cm x 30 cm x 25 cm. Is it possible to fit all cartons in an empty storage closet 1m 

x 1m x 2m? Justify your conclusion with a visual explanation. 

3. You have studied the pyramids and want to make a scale model of a pyramid with 

a square base and sides which are isosceles triangles. How much clay is required 

if the base of the actual pyramid is 30 m on each side and the height of the 

pyramid is 30 m? Your scale is 1 cm : 15 m. 

NOTE: You may wish to use power solids to explore the relationship between the area 

of a pyramid such as this and the area of a cube or prism with the same base 

and height. Your teacher should have a station set up with power solids and 

sand/water/rice to use in this exploration. 


Exploring Volume (continued) 

4. A movie theater decides to change the shape of its popcorn holder from a 

rectangular box to a pyramidal box. The tops of both boxes are the same and the 

height remains the same. If the rectangular box of popcorn cost $4.00, what is a 

fair price for the new box? 

5. A manufacturer of globes (approximately 1m in diameter) packs the globes in 1 

cubic meter boxes for shipping. How much packing material (e.g., styrofoam 

peanuts) is needed for a shipment of 100 globes? 

6. Take two sheets of paper the same size. Roll one 

sheet vertically and tape to form a right circular cylinder. Roll the second sheet 

horizontally and tape to form a second right circular cylinder. Tape each 

cylinder so that there is no overlap of paper; the edges should meet exactly. If 

each cylinder were filled with popcorn, would they contain the same amount? 

Explain and justify your answer. 


Gulliver’s Travels and Proportional Reasoning 



______________________________________________________________________________ 

Questions: 

How small are the Lilliputians? How can proportional reasoning help us 

understand the use of scale? Are the proportional relationships different 

for length, area, and volume? In what ways? 

_____________________________________________________________________________________ 

Materials: 

• Gulliver’s Travels excerpt (handout) 

• Calculator 

• Objects from Gulliver’s pockets (optional — handkerchief, comb, folded & 

tied paper, snuff box, pocket watch, knife and/or other objects described), 

• Measuring tools (optional — to be used with objects from Gulliver’s pockets) 

_____________________________________________________________________________________ 


1. Read the excerpt from Gulliver’s Travels, taking time to discuss the objects 

which are described there. Help students identify at least five of the objects 

described. 

2. Using the objects found in Gulliver’s pockets, estimate the size of the 

Lilliputians. If you have brought real combs, handkerchiefs, etc. to class, 

measure both the objects and the students to investigate the proportional 

relationships. 

3. At an earlier point in the story, the Lilliputians feed Gulliver and are amazed 

by the amount of food he consumes. This is related to his volume and 

overall size as much as his height. While the volume of a human is difficult 

to calculate, the volume of cubes and spheres is not. We will use these 

shapes to investigate the relationship between changes of side 

length/diameter and surface area and volume. 


1. Our investigation will begin with a cube. Using unit cubes, build at least 

three cubes with side lengths 1, 2, and 3. Use these models to review the 

formulas for surface area and volume in terms of side length. 

(V = l 3 ; SA = 6l 2 ) 


2. For 10-12 cubes of steadily increasing side length, have students enter three 

lists of data into the calculator: side length, surface area, and volume. The 

lists can be entered numerically or through formulas, depending on student 

facility with the calculations and the calculator. 

3. Use the graphing function of the calculator to plot length vs. surface area and 

length vs. volume, being careful to use different symbols for each plot. 

4. Repeat steps 1-3 for spheres, reviewing the formulas for calculating surface 

area and volume (SA = 4Πr 2 , V =4Πr 3 /3 ), entering data for 10 - 12 spheres of 

increasing diameter. Again, graph the relationships. 

Inferring: 

Discuss the data lists and graphs, noting the increasing rate of change for 

area/surface area and volume when compared to a steady rate of change for 

length. 


Students should recognize the exponential growth of area and volume 

measures when compared to the linear growth of length/diameter. They 

should be able to apply this information to such situations as unit conversion 

(if there are 100 cm in a meter, how many square centimeters are there in a 

square meter?), estimation of surface area or volume given length/diameter, 

and the interpretation of scale drawings. 


Gulliver’s Travels 

by Jonathan Swift 

Gulliver’s ship was caught in a storm. He swam to safety on a mysterious island called 

Lilliput. He was taken to the Emperor’s castle, bound and captive, and was presented 

to the Emperor and Empress. He was released from his rope bindings and the 

Lilliputians wanted to ensure that he had no weapons. This is the inventory of the 

contents of his pockets. (From part I, chapter II) 

Imprimis, In the right coat-pocket of the Great Man-Mountain [the Lilliputian’s 

name for Gulliver] after the strictest search, we found only one great piece of coarse 

cloth, large enough to be a foot-cloth for your Majesty’s chief room of state. In the left 

pocket, we saw a huge silver chest, with a cover of the same metal, which we, the 

searchers, were not able to lift. We desired it should be opened, and one of us, 

stepping into it, found himself up to the midleg in a sort of dust, some part whereof, 

flying up to our faces, set us both a sneezing for several times together. In his right 

waistcoat-pocket, we found a prodigious bundle of white thin substances, folded one 

over another, about the bigness of three men, tied with a strong cable and marked 

with black figures; which we humbly conceive to be writings, every letter almost half 

as large as the palm of our hands. In the left, there was a sort of engine [a term used 

for any mechanical device], from the back of which were extended twenty long poles, 

resembling the palisados before your Majesty’s court; wherewith we conjecture the 

Man-Mountain combs his head, for we did not always trouble him with questions, 

because we found it a great difficulty to make him understand us. In the large pocket 

on the right side of his middle cover (so I translate the word ranfu-lo, by which they 

meant my breeches) we saw a hollow pillar of iron, about the length of a man, fastened 

to a strong piece of timber, larger than the pillar; and upon one side of the pillar were 

huge pieces of iron sticking out, cut into strange figures, which we know not what to 

make of. In the left pocket, another engine of the same kind. In the smaller pocket on 

the right side, were several round flat pieces of white and red metal, of different bulk; 

some of the white, which seemed to be silver, were so large and heavy, that my 

comrade and I could hardly lift them. In the left pocket were two black pillars, 

irregularly shaped: we could not, without difficulty, reach the top of them as we stood 

at the bottom of his pocket. One of them was covered, and seemed all of a piece; but 

at the upper end of the other, there appeared a white round substance, about twice 

the bigness of our heads. Within each of these were enclosed a prodigious plate of 


steel; which, by our orders, we obliged him to show us, because we apprehended they 

might be dangerous engines. He took them out of their cases, and told us, that in his 

own country his practice was to shave his beard with one of these, and to cut his meat 

with the other. There were two pockets which we could not enter: these he called his 

fobs; they were two large slits cut into the top of his middle cover, but squeezed close 

by the pressure of his belly. Out of the right fob hung a great silver chain, with a 

wonderful kind of engine at the bottom. We directed him to draw out whatever was at 

the end of that chain; which appeared to be a globe, half silver, and half of some 

transparent metal: for on the transparent side we saw strange figures circularly 

drawn, and thought we could touch them, until we found our fingers stopped with 

that lucid substance. He put this engine to our ears, which made an incessant noise 

like that of a watermill. And we conjecture it is either some unknown animal, or the 

god that he worships: but we are more inclined to the latter opinion, because he 

assured us (if we understood him right, for he expressed himself very imperfectly), that 

he seldom did anything without consulting it. He called it his oracle, and said that it 

pointed out the time for every action of his life. From the left fob he took out a net 

almost large enough for a fisherman, but contrived to open and shut like a purse, and 

served him for the same use: we found therein several massy pieces of yellow metal, 

which, if they be of real gold, must be of immense value. 

Having thus, in obedience to your Majesty’s commands, diligently searched all 

his pockets, we observed a girdle about his waist made of the hide of some prodigious 

animal; from which, on the left side, hung a sword of the length of five men, and on 

the right, a bag or pouch divided into two cells, each cell capable of holding three of 

your Majesty’s subjects. In one of these cells were several globes or balls of a most 

ponderous metal, about the bigness of our heads, and required a strong hand to lift 

them: the other cell contained a heap of certain black grains, but of no great bulk or 

weight, for we could hold above fifty of them in the palms of our hands. 

This is an exact inventory of what we found about the body of the Man- 

Mountain, who used us with great civility, and due respect to your Majesty’s 

commission. Signed and sealed on the fourth day of the eighty-ninth moon of your 

Majesty’s auspicious reign. 

CLEFREN FRELOCK, MARSI FRELOCK. 


Comparing the Edge Length, Surface Area, 

and Volume of Cubes 



______________________________________________________________________________ 

Description: 

Students will examine the surface area and volume of a cube as a function 

of its edge length. They will graph these functions and describe the 

relationships. 

_____________________________________________________________________________________ 

Materials: 


• Graphing Calculator or Graphing Software 

• Spreadsheet (optional) 

_____________________________________________________________________________________ 



_____________________________________________________________________________________ 

Directions: 

Give students the Activity sheet and discuss their conclusions. 

_____________________________________________________________________________________ 

Background: 

To use a TI-83 for this exercise, 

• press STAT and clear the first three lists by positioning the cursor over 

the name of the list and pressing CLEAR ENTER . 

• Enter the edge lengths as L1. Position the cursor over L2 and type in the 

expression for the surface area, 6L1 2 and press ENTER . 

• Similarly, create a volume column by entering its formula for L3. 

• To graph the surface area as a function of edge length, press 2nd 

[STATPLOT] ENTER . Turn Plot1 on, choose the scatter plot picture, L1 for 

Xlist, and L2 for Ylist. 

• Press ZOOM and choose 9:ZoomStat. 

• To graph the volume as a function of edge length, repeat the procedure 

above to change Ylist to L3. 



Comparing the Edge Length, Surface Area, and Volume of Cubes 

Directions: 

Use graphing software or a graphing calculator. 

1) Sketch four cubes with different edge lengths. Record the edge length 

and find the surface area and volume of each cube. Use a spreadsheet 

or record your data in a table. 

2) Graph the surface area as a function of edge length. Describe the shape 

of the graph. 

3) Graph the surface area as a function of edge length. Describe the shape 

of the graph. 

4) What do the graphs in parts (2) and (3) tell you about how surface area 

and volume change as the edge length changes? 


Comparing the Radius, Surface Area, 

and Volume of Spheres 



______________________________________________________________________________ 

Description: 

Students will examine the surface area and volume of a sphere as a function 

of its radius. They will graph these functions and describe the relationships. 

_____________________________________________________________________________________________ 

Materials: 


• Graphing Calculator or Graphing Software 

• Spreadsheet (optional) 

_____________________________________________________________________________________ 



_____________________________________________________________________________________ 

Directions: 

Give students the activity sheet and discuss their conclusions. 

_____________________________________________________________________________________ 

Background: 

To use a TI-83 for this exercise, 

• Press STAT and clear the first four lists by positioning the cursor 

over the name of the list and pressing CLEAR ENTER . 

• Enter the radii as L1. 

• Position the cursor over L2 and type in the expression for the volume, (4/3)ΠL1 3 . 

• Similarly, create a surface area column by entering its formula for L3. 

• For L4, enter the formula L2/L3 to get the ratios of volume to surface area. and 

press ENTER . 

• To graph this ratio as a function of the radius, press 2nd [STATPLOT] 

ENTER . Turn Plot1 on, choose the scatter plot picture, L1 for Xlist, and L4 for 

Ylist. Press ZOOM and choose 9:ZoomStat. 


On the TI-89/92 

• Press APPS, 6 (Data Matrix Editor), 3 (New), ENTER . 

• Choose type as a list. Give the variable a name and press 

ENTER . 

• Enter the radii in Column 1. 

• Put your cursor over c2 and press ENTER . Put in the volume formula, 

which is 4/3Πc1 3 , ENTER . 

• Put your cursor over c3 and press ENTER . Put in the surface area 

formula, which is 4Πc1 2 , ENTER . 

• Put your cursor over c4 and press ENTER . Put in the ratio c2/c3, 

ENTER . 

• Press F2 (Plot Set-up), choose F1 (Define). Select Plot Type as scatter. 

Choose the mark you wish. X is c1; Y is c4. Use Frequency and Categories 

should be No. Press ENTER . 

• Choose GRAPH, F2 (Zoom), 9 (ZoomData), ENTER . The graph should be 

visible. 

NOTE: 

V 

In this exercise, students look at a graph of the ratio as a 

S.A. 

function of the radius. They should notice that the graph is 

linear. Challenge students to find the equation of this line. 

V 

You may also want to ask them to show the same result algebraically. ( 

S.A. 

= 1 3 

r.) For any sphere, the ratio of its volume to surface area is given by the 

formula y = 1 3 r. 

= r 3 



Comparing the Radius, Surface Area, and Volume of Spheres 

Directions: 

Use graphing software or a graphing calculator. 

1) The radii of four spheres are 1 m, 2 m, 3 m, and 4 m. Find the surface area and volume of 

each sphere. Use a spreadsheet or record your data in a table. 

2) Find the ratio of volume to surface area ( V ) for each sphere in part (1). 

S.A. 

3) Use a graphing calculator to graph V 

S.A. 

as a function of the radius. What do you notice? 


Patty Paper Translations 

Reporting Category: Coordinate Relations, Transformations, and Vectors 


______________________________________________________________________________ 

Description: 

Students will translate figures using patty paper. 

_____________________________________________________________________________________ 

Materials: 

• Patty paper 

• Pencil 

_____________________________________________________________________________________ 



_____________________________________________________________________________________ 

Directions: 

1. Have students draw a simple shape on the patty paper with a dot in the interior of the 

shape. 

2. Next, students should draw a ray from the dot to an edge of the paper and mark a 

second dot on the ray outside the simple shape. Students will translate the figure 

along this ray. This is a good time to discuss the meaning of the word translate in 

geometry. Note that the word has a different meaning around foreign language 

translating. 

3. On a second sheet of patty paper, ask the students to mark the distance between the 

two dots. This is the translation distance. 

4. Have students place a third patty paper over the first and trace the figure, ray, and 

interior dot. Now ask students to carefully slide the interior dot on the tracing along 

the ray until it is on top of the second dot on the first paper. 

5. Use the patty paper with the translation distance marked to measure the distance 

between corresponding points on the two figures. Encourage students to discuss their 

findings. Lead the discussion towards the fact that a translated figure is congruent to 

the original except that its location in the plane is different. Students should also note 

that each point moves by the same distance and in the same direction during 

translation. 

Reference: 

Serra, M. (1994). Patty Paper Geometry: Student Workbook. Berkeley, CA: 

Key Curriculum Press. 


Patty Paper Rotations 



_____________________________________________________________________________________ 

Description: 

Students will rotate a figure using patty paper. 

_____________________________________________________________________________________ 

Materials: 


• Pencil 


_____________________________________________________________________________________ 



______________________________________________________________________________ 

Directions: 

1. Ask students to draw a simple shape on the patty paper. They should place a dot 

on an edge or in the interior of the shape and place a second dot outside the shape. 

The second dot will be the center of rotation. 

2. Next, tell students to draw a ray from the second dot through the first and draw a 

second ray from the center of rotation (second dot) to create an acute angle of rotation. 

3. Students should place a second patty paper over the first and trace the figure, the dots, 

and the first ray. Using a pencil to hold the two sheets of patty paper together at the 

center of rotation, turn the second patty paper until the tracing of the first ray is 

aligned with the second ray. 

4. Ask students to place a third sheet of patty paper over this rotation and carefully trace 

the two figures and the angle of rotation. They will use this tracing for the remainder 

of the exercise. 

5. Allow students to use another patty paper or a ruler to measure the distance between 

corresponding parts of the two figures. What patterns do they notice? How is this 

finding different from a translation transformation? 

6. Use another patty paper and a straightedge to create an angle with the center of 

rotation as its vertex, a point on the original figure along one ray and the 

corresponding point on the rotated figure along its other ray. What do students notice 

about this angle and the original angle of rotation? Does this finding hold true for all 

parts of the rotated figure? 

Reference: 

Serra, M. (1994). Patty Paper Geometry: Student Workbook. 

Berkeley, CA: Key Curriculum Press. 


Patty Paper Reflections 



_____________________________________________________________________________________ 

Description: 

Students will use patty paper to reflect a figure over a line. 

_____________________________________________________________________________________ 

Materials: 


• Pencil 


_____________________________________________________________________________________ 



_____________________________________________________________________________________ 

Directions: 

1. Ask students to draw a simple shape on their patty paper and to fold the paper so 

that the fold does not intersect the shape. This line is the line of reflection. 

2. Students should fold the paper along the line of reflection and trace the figure onto the 

other side of the patty paper. 

3. Have students draw two segments, each connecting a point on the original figure with 

its corresponding point on the reflected figure. Use patty paper to measure the lengths 

of these segments. Encourage students to discuss their findings and describe what 

happens when a figure is reflected. 

4. To extend this activity, encourage students to think about the relationship between the 

line of reflection and the segments connecting corresponding parts of the two figures. 

They should notice that the line of reflection is perpendicular to the segments and that 

the line bisects each segment. 

Reference: 

Serra, M. (1994). Patty Paper Geometry: Student Workbook. 

Berkeley, CA: Key Curriculum Press. 


Mira Explorations 



_____________________________________________________________________________________ 

Description: 

Students will use a Mira to complete various constructions on shapes. The 

Mira practices reflective symmetry and students review important concepts 

about triangles, bisectors, and constructions. 

_____________________________________________________________________________________ 

Materials: 


• Mira 

_____________________________________________________________________________________ 



_____________________________________________________________________________________ 

Directions: 

1. Allow students to explore the Mira by drawing simple shapes and reflecting them 

along a line. 

2. Assist students as they work through the worksheet activities using the Mira. Be sure 

to discuss the role reflective symmetry plays in the Mira constructions they are 

completing. Encourage students to use correct vocabulary as they discuss their 

constructions. 

Reference: NCTM Addenda Series, Geometry in the Middle Grades, Activity 15. 



Mira Explorations 

1. Use the Mira to draw the perpendicular bisectors of the segments below. 

2. The circumcenter of a triangle is the point where the perpendicular bisectors of the sides 

meet. Find the circumcenter of each triangle below. 

circumcenters you located in #2 in different places? Where do you 

think the circumcenter of a right triangle is located? Use the space below and 

your Mira to test your hypothesis. 

3. 

3. Why are 

the 


4. A median of a triangle is a line from a vertex to the midpoint of the opposite side. The point 

where the three medians of a triangle meet is called the centroid. Use your Mira to draw the 

medians of the triangle below. 

5. Use your Mira to find a position where the image of ray QA maps onto ray QB. Draw the Mira 

line. What does this line do to angle AQB? 

Q 

A 

B 





__________________________________________________________________ 

Description: 

Students will generalize the changes in coordinates when points are reflected 

over the X and Y axes. 

_____________________________________________________________________________________ 

Materials: 


• Graph paper 

_____________________________________________________________________________________ 



_____________________________________________________________________________________ 

Directions: 

1. Ask students to mark the X and Y axes on their graph paper. They can construct 2 

sets of axes ranging from -10 to +10 on one sheet of graph paper. 

2. On the first set of axes, mark the X axis as the line of symmetry. On the second set 

of axes, mark the Y axis as the line of symmetry. On the worksheet, students should 

record the coordinates of each point as it is reflected across the line of symmetry 

indicated. Students then generalize the pattern for any coordinates (x,y). 

Reference: NCTM Addenda Series, Geometry in the Middle Grades 




1. Draw two sets of axes on your graph paper. Each set of axes should include the range -10 to 

+10. 

2. On the first set of axes, mark the X axis as the line of reflection. 

3. Plot the points listed below on your first set of axes. Then reflect each point across 

the line of symmetry and write in the coordinates of the reflection. 

(-6, 1) --> (__,__) (0, 4) --> (__,__) 

( 2, 3) --> (__,__) (4, 0) --> (__,__) 

(-2, -2) --> (__,__) 

What pattern do you see in the new coordinates? In general, if a point (x,y) is 

reflected across the X axis, the coordinates of the reflection are (__,__). 

4. On the second set of axes, mark the Y axis as the line of reflection. 

5. Plot the points listed below on your second set of axes. Then reflect each point across the line 

of symmetry and write in the coordinates of the reflection. 

(-6, 1) --> (__,__) (0, 4) --> (__,__) 

( 2, 3) --> (__,__) (4, 0) --> (__,__) 

(-2, -2) --> (__,__) 

What pattern do you see in the new coordinates? 

In general, if a point (x,y) is 

reflected across the Y axis, the coordinates of the reflection are (__,__). 


Polygons and Symmetry 



_____________________________________________________________________________________ 

Description: 

Students will work in pairs or small groups to construct polygons with 

varying numbers of sides and lines of symmetry. 

_____________________________________________________________________________________ 

Materials: 


• scratch paper 

• straightedge 

• Mira (optional) 

_____________________________________________________________________________________ 



_____________________________________________________________________________________ 

Directions: 

1. Review with students the line(s) of symmetry in polygons. If you wish, include 

the use of the Mira in finding lines of symmetry. 

2. Distribute the activity sheet, scratch paper, and straightedges to each small 

group. Encourage students to work collaboratively to come up with figures that meet 

the specified conditions. When students say there is no figure to meet a given 

condition, encourage them to be specific about why the condition cannot be met. 

Reference: 

NCTM Addenda Series, Geometry in the Middle Grades 



Polygons and Symmetry 

Working with your group, sketch polygons to fill in as many spaces on the grid 

below as possible. There are some that cannot be filled in. Be ready to 

explain why you think each empty space must remain empty. 

lines of 

symmetry 

1 

2 

3 

4 

5 

6 

3 sides 4 sides 5 sides 6 sides 

For a given number of sides, can you always make a polygon with no lines of symmetry? 

Can you identify patterns in your chart which would help you continue it?

commonwealth of virginia department of education - Rockingham ...

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