4.1
Making Smaller Ballots
Goals
Explore 4.1
• Use knowledge of exponential relationships to
Distribute a sheet of quarter-inch or inch grid
paper and scissors to each pair. Students will also
need grid paper for their graphs.
Have the class cut 8-inch squares from the grid
paper. Each pair should then cut their paper
square into ballots as directed, complete the table,
and answer Questions B–E. (Students need only
actually cut the paper for the first two or three
cuts.)
Have some students put graphs on a
transparent grid for the summary.
make tables and graphs and to write equations
for exponential decay patterns
• Analyze and solve problems involving
exponents and exponential decay
Students’ experiences with exponential change
in the previous investigations all involved
exponential growth. In this problem, they revisit
the ballot-cutting activity to study the pattern of
decreasing exponential change in the area of a
ballot after each cut.
Launch 4.1
Remind students that in the ballot-cutting activity
in Investigation 1, they looked at how the number
of ballots changes with each cut. Explain that they
are going to revisit that activity, but this time, they
will look at how the area of a ballot changes with
each cut. Demonstrate, starting with an 8-inch
square of inch grid paper.
Suggested Questions Hold up your square of
grid paper.
• This sheet of paper has an area of 64 square
inches. When I make the first cut, what
happens to the area of a ballot? (It becomes
half the original area, or 32 in.2)
second cut? (16 in.2)
• What would a ballot look like if I made
10 cuts? (It would be very small.)
• Do you think it would be large enough for
you to write your name on it? (no)
although the area will become smaller and
smaller, it will theoretically never reach 0.)
• What happens to the area of a ballot with each
successive cut? (It is half the previous area, or
the previous area divided by 2.)
• Does this pattern remain consistent as you
make more cuts? (yes)
• Do the data in your table look like data from
other situations you have encountered? (In
some ways, no; the values are decreasing. In
previous situations, the values increased. In
other ways, yes; each value can be determined
by multiplying the previous value by a
constant number.)
• How can you determine the area of a
ballot from the area of the previous ballot?
1
(Multiply it by 2, or 0.5.)
• Let’s start at the beginning and generate the
1
table using the constant factor 2. If I know the
area of the original ballot is 64 square inches,
how do I get the area of a ballot after one cut?
1
(Multiply 64 by 2.)
4
• Will I ever have a ballot with an area of 0? (No;
Suggested Questions Ask students to share what
they discovered in the problem.
I N V E S T I G AT I O N
• What will be the area of each ballot after the
Summarize 4.1
Have students work in pairs.
Investigation 4
Exponential Decay
81
• What is the area after two cuts? (32 in.2)
Cuts
Area Calculation
Area (in.2)
64
0
64
1
64 ! 2
2
64 ! 2 ! 2
3
64 ! 2 ! 2 ! 2
1
32
1
1
1
1
16
1
8
• Suppose I could continue cutting. How could
I find the area of a ballot after 50 cuts?
1
[Multiply 64 by a string of 50 factors of 2; or,
1
calculate 64( 2 )50.]
• What is the area of a ballot after n cuts?
1
[64( 2 )n in.2]
• Explain how you got your equation in
Question C.
• What does the graph of this situation look
like?
• How is the graph similar to and different from
the graphs in the previous problems?
Students should realize that the graph is not a
straight line because this is not a linear
relationship. It has a curved shape similar to a
graph of exponential growth, but it is decreasing
rather than increasing.
82
Growing, Growing, Growing
Check for Understanding
Ask questions that connect the various
representations of the relationship.
• Pick a pair of values from the table and ask
students to explain what these values mean in
terms of the context, the equation, and the
graph.
• Have students explain how the variables and
numbers in the equation relate to the context of
the situation, the table, and the graph.
• Have students discuss how the pattern and
features of the graph are related to the
equation, situation, and table.
Then ask the following question:
• When will the area be about
0.01 square inches? Explain your reasoning.
At a Glance
4.1
Making Smaller Ballots
PACING 1 day
Mathematical Goals
• Use knowledge of exponential relationships to make tables and graphs
and to write equations for exponential decay patterns
• Analyze and solve problems involving exponents and exponential decay
Launch
Remind students of the ballot-cutting activity. Explain that this time they
will look at how the area of a ballot changes with each cut. Demonstrate,
starting with an 8-inch square of inch grid paper.
• This sheet of paper has an area of 64 square inches. When I make the
Materials
•
8-inch square of inch
grid paper for
demonstration
•
Inch or quarter-inch
grid paper for
students
•
Scissors (1 pair per
pair of students)
first cut, what happens to the area of a ballot?
• What will be the area of each ballot after the second cut?
• What would a ballot look like if I made 10 cuts?
• Do you think it would be large enough for you to write your name
on it?
• Will I ever have a ballot with an area of 0?
Have students work in pairs on the problem.
Explore
Distribute a sheet of quarter-inch or inch grid paper and scissors to each
pair. Students will also need grid paper for their graphs.
Have the class cut 8-inch squares from the grid paper. Each pair should
then cut their paper square into ballots as directed, complete the table, and
answer Questions B–E. (Students need only actually cut the paper for the
first two or three cuts.)
Have some students put graphs on a transparent grid for the summary.
Summarize
Materials
Ask students to share what they discovered in the problem.
• What happens to the area of a ballot with each successive cut?
• Does this pattern remain consistent as you make more cuts?
• Do the data in your table look like data from other situations you have
•
Student notebooks
encountered?
• How can you determine the area of a ballot from the area of the
previous ballot?
• Let’s start at the beginning and generate the table using the constant
1
factor 2. If I know the area of the original ballot is 64 square inches, how
do I get the area of a ballot after one cut?
• What is the area after two cuts?
continued on next page
Investigation 4
Exponential Decay
83
Summarize
continued
• What is the area of a ballot after n cuts?
• Explain how you got your equation in Question C.
• What does the graph of this situation look like?
• How is the graph similar to and different from the graphs in the
previous problems?
Students should realize that the graph is not a straight line because
this is not a linear relationship. It has a curved shape similar to a graph
of exponential growth, but it is decreasing rather than increasing.
Check for Understanding
Ask questions that connect the various representations of the
relationship. Then, ask:
• When will the area be about 0.01 square inches? Explain your
reasoning.
ACE Assignment Guide
for Problem 4.1
Area of Ballot
D.
72
Core 1, 2, 8
Other Unassigned choices from previous problems
Exercise 1 and other ACE exercises, see the
CMP Special Needs Handbook.
Answers to Problem 4.1
A.
48
40
32
24
16
8
Number of Cuts
Area (in.2)
0
64
1
32
2
16
3
8
4
4
5
2
6
1
7
0.5
8
0.25
9
0.125
10
0.0625
B. Each cut makes the area of a ballot half the
previous area.
1
C. A = 64( 2 )n
84
Area (in.2)
Adapted For suggestions about adapting
64
56
Growing, Growing, Growing
0
0 1
2 3 4 5 6 7 8 9 10
Number of Cuts
E. The pattern is different from the exponential
growth patterns in that the numbers decrease
instead of increase. It is similar in that each
value (area) can be derived from the
preceding value. Some students might notice
that each value is obtained by dividing the
previous value by 2, and some may notice
that each value is obtained by multiplying
1
the previous value by 2 .