Teaching Algebra with Manipulatives

Chapter 11 Teaching Notes and Overview
Discuss the completed parts of the table in
Exercise 1. Ask students questions about the
table to check on their understanding. Complete
the missing parts of the table as a class.
Form groups of two or three to complete
Exercises 2–7. You may want to have the groups
complete two of the exercises, then discuss the
answers as a class. Then have them complete
two more and discuss the answers and so on.
Encourage students to research other fractals
and share their findings with the class.
Answers
See Teacher Wraparound Edition p. 611.
Algebra Activity
Recording Sheet
Special Sequences
(p. 285 of this booklet)
Use With the activity on page 607 in Lesson
11-6 of the Student Edition.
Objective Model, analyze, and make a
conjecture about the Towers of Hanoi game.
Materials
provide each group of students with the
following:
penny
nickel
dime
quarter
Point out to students that the object of the
Towers of Hanoi game is to move a stack of n
coins from one position to another in the fewest
number of a n
of moves. Go over the rules with
the students.
Divide the class into groups. Provide a penny,
nickel, dime, and quarter or a facsimile of these
for each group. Ask each group to keep a record
of the moves required to complete each exercise.
As soon as the groups have completed the
exercises, discuss the answers. Have students
explain how they solved the problem.
Encourage students to find other mathematical
games and to share them with the rest of the
class.
Answers
See Teacher Wraparound Edition p. 607.
Algebra Activity
Fractal Cut-Outs
(pp. 286–287 of this booklet)
Use With Lesson 11-6 as an extension.
Objective Make a visual model of a
self-similar structure.
Materials
4 sheets of 8.5-by-11 plain paper
ruler
scissors
colored pencils or markers
Have students work in pairs. Remind students
to measure and cut carefully to obtain the best
results. You may want to give each pair of
students extra paper in case they make an error
in cutting. Make sure students realize the
pattern of the placement of the cut and the
length of the cut before attempting Cards 3 and
4. The cut is always one-fourth the width of the
fold and the length of the cut is one-half of the
height of the folded section.
Students need to sharply crease their folds in
order to get the desired effect, especially on
Cards 3 and 4. You may want students to
attach their cards to poster board to make a
three-dimensional display of their work.
Answers
1. 1, 3, 7, 15
2. (Card, Area): (1, 20), (2, 25), (3, 26.25),
(4, 26.5625), (5, 26.640625), (6, 26.660156),
(7, 26.665039), (8, 26.66626),
(9, 26.666565), (10, 26.666641)
3. Sample answer: The number of boxes for
Card 1 is 1. Then for each card, you add a
power or 2. Card 2 1 2 1 or 3. Card
3 1 2 2 or 5, Card 4 1 2 3 or 7, and
so on. Card n 1 2 n1 .
Alternate answer: The number of boxes is
2 n 1, where n is the card number.
4. 1023
© Glencoe/McGraw-Hill 280 Teaching Algebra with Manipulatives