NYS COMMON CORE MATHEMATICS CURRICULUM
6•4
Lesson 22
Lesson 22: Writing and Evaluating Expressions—Exponents
Date:
11/19/14
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Lesson 22: Writing and Evaluating Expressions—Exponents
Student Outcomes
Students evaluate and write formulas involving exponents for given values in real-world problems.
Lesson Notes
Exponents are used in calculations of both area and volume. Other examples of exponential applications involve
bacterial growth (powers of 2) and compound interest.
Students will need a full size sheet of paper (8
1
2
× 11 inches) for the first example. Teachers should try the folding
activity ahead of time to anticipate outcomes. If time permits at the end of the lesson, a larger sheet of paper can be
used to experiment further.
Classwork
Fluency Exercise (10 minutes): Multiplication of Decimals
RWBE: Refer to the Rapid White Board Exchanges sections in the Module Overview for directions on how to administer
a RWBE.
Example 1 (5 minutes): Folding Paper
Ask students to predict how many times they can fold a piece of paper in half. Allow a
short discussion before allowing students to try it.
Predict how many times you can fold a piece of paper in half. The folds must be
as close to a half as possible. Record your prediction in Exercise 1.
Students will repeatedly fold a piece of paper until it is impossible, about seven folds.
Remind students they must fold the paper the same way each time.
Fold the paper once. Record the number of layers of paper that result in the
table in Exercise 2.
2
Fold again. Record the number of layers of paper that result.
4
Ensure that students see that doubling the two sheets results in four sheets. At this stage, the layers can easily be
counted. During subsequent stages, it will be impractical to do so. Focus the count on the corner that has four loose
pieces.
Fold again. Count and record the number of layers you have now.
8
Scaffolding:
Some students will benefit
from unfolding and counting
rectangles on the paper
throughout Example 1. This
provides a concrete
representation of the
exponential relationship at the
heart of this lesson.
NYS COMMON CORE MATHEMATICS CURRICULUM
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Lesson 22
Lesson 22: Writing and Evaluating Expressions—Exponents
Date:
11/19/14
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The number of layers is doubling from one stage to the next; so, the pattern is modeled by multiplying by 2, not
adding 2. It is critical that students find that there are eight layers here, not six.
Continue folding and recording the number of layers you make. Use a calculator if desired. Record your
answers as both numbers in standard form and exponential form, as powers of 2.
Exercises (5 minutes)
Exercises
1. Predict how many times you can fold a piece of paper in half.
My Prediction:
2. Before any folding (zero folds), there is only one layer of paper. This is recorded in the first row of the table.
Fold your paper in half. Record the number of layers of paper that result. Continue as long as possible.
Number of Folds
Number of Paper Layers that
Result
Number of Paper Layers Written
as a Power of
a. Are you able to continue folding the paper indefinitely? Why or why not?
No. The stack got too thick on one corner because it kept doubling each time.
b. How could you use a calculator to find the next number in the series?
I could multiply the number by to find the number of layers after another fold.
c. What is the relationship between the number of folds and the number of layers?
As the number of folds increases by one, the number of layers doubles.
d. How is this relationship represented in exponential form of the numerical expression?
I could use as a base and the number of folds as the exponent.
e. If you fold a paper times, write an expression to show the number of paper layers.
There would be
layers of paper.
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Lesson 22
Lesson 22: Writing and Evaluating Expressions—Exponents
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3. If the paper were to be cut instead of folded, the height of the stack would double at each successive stage, and it
would be possible to continue.
a. Write an expression that describes how many layers of paper result from cuts.
b. Evaluate this expression by writing it in standard form.
= ,
Example 2 (10 minutes): Bacterial Infection
Modeling of exponents in real life leads to our next example of the power of doubling. Think about the last
time you had a cut or a wound that became infected. What caused the infection?
Bacteria growing in the wound.
When colonies of certain types of bacteria are allowed to grow unchecked, serious illness can result.
Example 2: Bacterial Infection
Bacteria are microscopic single-celled organisms that reproduce in a
couple of different ways, one of which is called binary fission. In
binary fission, a bacterium increases its size until it is large enough
to split into two parts that are identical. These two grow until they
are both large enough to split into two individual bacteria. This
continues as long as growing conditions are favorable.
a. Record the number of bacteria that result from each generation.
Generation Number of Bacteria
Number of Bacteria Written
as a Power of
,
,
,
,
,
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Lesson 22: Writing and Evaluating Expressions—Exponents
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=
=
b. How many generations would it take until there were over one million bacteria present?
generations will produce more than one million bacteria.
= , , .
c. Under the right growing conditions, many bacteria can reproduce every minutes. Under these conditions,
how long would it take for one bacterium to reproduce itself into more than one million bacteria?
It would take fifteen-minute periods, or hours.
d. Write an expression for how many bacteria would be present after generations.
There will be
bacteria present after generations.
Example 3 (10 minutes): Volume of a Rectangular Solid
Exponents are used when we calculate the volume of rectangular solids.
Example 3: Volume of a Rectangular Solid
This box has a width, . The height of the box, , is twice the width. The length of the box, , is three times the width.
That is, the width, height, and length of a rectangular prism are in the ratio of : : .
For rectangular solids like this, the volume is calculated by multiplying length times width times height.
= · ·
= · ·
= · · · ·
=
Follow the above example to calculate the volume of these rectangular solids, given the width, .
Width in centimeters (
)
Volume in cubic centimeters (
)
× × =
× × =
× × =
× × =
× × =
MP.3
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Lesson 22
Lesson 22: Writing and Evaluating Expressions—Exponents
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Closing (2 minutes)
Why is 5
3
different from 5 × 3?
5
3
means 5 × 5 × 5. Five is the factor that will be multiplied by itself 3 times. That equals 125.
On the other hand, 5 × 3 means 5 + 5 + 5. Five is the addend that will be added to itself 3 times. This
equals 15.
Exit Ticket (3 minutes)
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Lesson 22: Writing and Evaluating Expressions—Exponents
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Name Date
Lesson 22: Writing and Evaluating Expressions—Exponents
Exit Ticket
1. Naomi’s allowance is $2.00 per week. If she convinces her parents to double her allowance each week for two
months, what will her weekly allowance be at the end of the second month (week 8)?
Week Number Allowance
1
$2.00
2
3
4
5
6
7
8
2. Write the expression that describes Naomi’s allowance during week in dollars.
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Lesson 22
Lesson 22: Writing and Evaluating Expressions—Exponents
Date:
11/19/14
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Exit Ticket Sample Solutions
1. Naomi’s allowance is $. per week. If she convinces her parents to double her allowance each week for two
months, what will her weekly allowance be at the end of the second month (week )?
Week Number Allowance
$.
$.
$.
$.
$.
$.
$.
$.
$
2. Write the expression that describes Naomi’s allowance during week in dollars.
$
NYS COMMON CORE MATHEMATICS CURRICULUM
6•4
Lesson 22
Lesson 22: Writing and Evaluating Expressions—Exponents
Date:
11/19/14
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Problem Set Sample Solutions
1. A checkerboard has squares on it.
a. If one grain of rice is put on the first square, grains of rice on the second square, grains of rice on the third
square, grains of rice on the fourth square, etc. (doubling each time), complete the table to show how many
grains of rice are on each square. Write you answers in exponential form on the table below.
Checkerboard
Square
Grains of
Rice
Checkerboard
Square
Grains of
Rice
Checkerboard
Square
Grains of
Rice
Checkerboard
Square
Grains of
Rice
b. How many grains of rice would be on the last square? Represent your answer in exponential form and
standard form. Use the table above to help solve the problem.
There would be
= , , , , , , grains of rice.
c. Would it have been easier to write your answer to part (b) in exponential form or standard form?
Answers will vary. Exponential form is more concise:
. Standard form is longer and more complicated to
calculate: , , , , , , . (In word form: nine quintillion, two hundred twenty-three
quadrillion, three hundred seventy-two trillion, thirty-six billion, eight hundred fifty-four million, seven
hundred seventy-five thousand, eight hundred eight.)
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Lesson 22: Writing and Evaluating Expressions—Exponents
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2. If an amount of money is invested at an annual interest rate of %, it doubles every years. If Alejandra
invests $, how long will it take for her investment to reach $, (assuming she does not contribute any
additional funds)?
It will take years. After years, Alejandra will have doubled her money and will have $, . If she waits an
additional years, she will have $, .
3. The athletics director at Peter’s school has created a phone tree that is used to notify team players in the event a
game has to be canceled or rescheduled. The phone tree is initiated when the director calls two captains. During
the second stage of the phone tree, the captains each call two players. During the third stage of the phone tree,
these players each call two other players. The phone tree continues until all players have been notified. If there are
players on the teams, how many stages will it take to notify all of the players?
It will take five stages. After the first stage, two players have been called, and will not have been called. After
the second stage, four more players will have been called, for a total of six; players will remain uncalled. After
the third stage,
players (eight) more will have been called, totaling ; remain uncalled. After the
stage,
more players () will have gotten a call, for a total of players notified. Twenty remain uncalled at this stage.
The fifth round of calls will cover all of them because
includes more players.
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Lesson 22
Lesson 22: Writing and Evaluating Expressions—Exponents
Date:
11/19/14
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Multiplication of Decimals
Progression of Exercises
1. 0.5 × 0.5 =
.
2. 0.6 × 0.6 =
.
3. 0.7 × 0.7 =
.
4. 0.5 × 0.6 =
.
5. 1.5 × 1.5 =
.
6. 2.5 × 2.5 =
.
7. 0.25 × 0.25 =
.
8. 0.1 × 0.1 =
.
9. 0.1 × 123.4 =
.
10. 0.01 × 123.4 =
.
