G6 m4-f-lesson 22-t

Lesson 22: Writing and Evaluating Expressions—Exponents
Date: 4/15/15 220
© 2013 Common Core, Inc. Some rightsreserved. commoncore.org
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM 6•4Lesson 22
Student Outcomes
 Students evaluate and write formulas involvingexponents for given values in real-world problems.
Lesson Notes
Exponents areused in calculationsof both area and volume. Other examples of exponential applicationsinvolve
bacterial growth (powers of 2) and compound interest.
Students will need a full sizesheet of paper (8
1
2
× 11 inches) for the firstexample. Teachers should try the folding
activity ahead of time to anticipateoutcomes. If time permits at the end of the lesson,a larger sheet of paper can be
used to experiment further.
FluencyExercise (10 minutes)
Multiplication of Decimals WhiteBoard Exchange
Classwork
Example 1 (10 minutes): Foldingpaper
Ask students to predicthow many times they can fold a piece of paper in half. Allowa
shortdiscussion beforeallowingstudents to try it.
 Let’s try an experiment: How many times do you think you would be ableto fold
a piece of paper in half? The folds must be as closeto a half as possible.
Students will repeatedly fold a piece of paper until itis 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.
 2
 Fold again. Record the number of layers of paper that result.
 4
Ensure that students see that doublingthe two sheets results in four sheets. At this stage, they can be counted. During
subsequent stages, itwill be impractical to do so. Focus the count on the corner that has four loosepieces.
 Fold again. Count and record the number of layers you have now.
 8
The number of layers is doublingfromone stage to the next; so, the pattern is modeled by multiplyingby 2,not
adding 2. It is critical thatstudents find that there are eight layers here, not six.
 Continue foldingand recordingthe number of layers you make. Use a calculator if desired. Record your
answers as both numbers in standard form and usingexponents, as powers of 2.
Scaffolding:
Some students will benefit
from unfoldingand counting
rectangles on the paper
throughout Example 1. This
provides a concrete
representation of the
exponential relationship atthe
heart of this lesson.

Date: 4/15/15 221
Exercises1–3
1. Predict how many timesyou can fold apiece ofpaper inhalf.
My Prediction:
2. Before any folding (zero folds), there isonly onelayerofpaper. Thisisrecorded inthe first row ofthetable.
Fold your paper in half. Record thenumber of layers ofpaper that result. Continueaslong aspossible.
a. Are you able to continuefolding the paper indefinitely? Why or why not?
No. The stack got too thick on onecorner. It kept doubling each time.
b. How could you use acalculator tofind thenext number in theseries?
I could multiply thenumber by 𝟐to find thenext number.
c. What isthe relationshipbetween thenumber offoldsand the number oflayers?
As thenumber offolds increases by one, thenumber oflayers doubles.
d. How isthisrelationship represented inthe exponential form ofthe numerical expression?
I could use 𝟐as a baseand thenumber offolds as theexponent.
e. If you fold apaper 𝒇 times, write an expressionto show thenumber ofpaper layers.
There would be 𝟐 𝒇
layers ofpaper.
3. If the paper were tobe cutinstead offolded, the heightofthe stack would doubleat each successive stage, and it
would be possible to continue.
a. Write an expression that describeshow many layers ofpaper result from 𝟏𝟔cuts.
𝟐 𝟏𝟔
b. Evaluate thisexpression by writing it in standard form.
𝟐 𝟏𝟔
= 𝟔𝟓,𝟓𝟑𝟔
Number ofFolds
Number of Paper Layers that
Result
Number ofPaper Layers Written
as a Power of 𝟐
𝟎 𝟏 𝟐 𝟎
𝟏 𝟐 𝟐 𝟏
𝟐 𝟒 𝟐 𝟐
𝟑 𝟖 𝟐 𝟑
𝟒 𝟏𝟔 𝟐 𝟒
𝟓 𝟑𝟐 𝟐 𝟓
𝟔 𝟔𝟒 𝟐 𝟔
𝟕 𝟏𝟐𝟖 𝟐 𝟕
𝟖 𝟐𝟓𝟔 𝟐 𝟖

Date: 4/15/15 222
Example 2 (10 minutes): Bacterial Infection
 Modeling of exponents in real lifeleads to our next example of the power of doubling. Think aboutthe 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 areallowed to grow unchecked, serious illness can result.
Example 2: Bacterial Infection
Bacteriaare microscopicone-celled organismsthat reproduce in a
couple ofdifferent ways, one ofwhich iscalled binary fission. In
binary fission, abacterium increasesitssize until itislarge enough
to split into two parts that are identical. These two grow untilthey
are both large enough to splitintotwo individual bacteria. This
continuesaslong asgrowing conditionsare favorable.
a. Record the number ofbacteriathat result from each generation.
Generation Number of Bacteria
Number ofBacteriaWritten
as a Power of 𝟐
𝟏 𝟐 𝟐 𝟏
𝟐 𝟒 𝟐 𝟐
𝟑 𝟖 𝟐 𝟑
𝟒 𝟏𝟔 𝟐 𝟒
𝟓 𝟑𝟐 𝟐 𝟓
𝟔 𝟔𝟒 𝟐 𝟔
𝟕 𝟏𝟐𝟖 𝟐 𝟕
𝟖 𝟐𝟓𝟔 𝟐 𝟖
𝟗 𝟓𝟏𝟐 𝟐 𝟗
𝟏𝟎 𝟏,𝟎𝟐𝟒 𝟐 𝟏𝟎
𝟏𝟏 𝟐,𝟎𝟒𝟖 𝟐 𝟏𝟏
𝟏𝟐 𝟒,𝟎𝟗𝟔 𝟐 𝟏𝟐
𝟏𝟑 𝟖,𝟏𝟗𝟐 𝟐 𝟏𝟑
𝟏𝟒 𝟏𝟔,𝟑𝟖𝟒 𝟐 𝟏𝟒
b. How many generationswould it takeuntilthere wereover onemillion bacteriapresent?
𝟐𝟎generations will producemorethan onemillionbacteria. 𝟐 𝟐𝟎
= 𝟏, 𝟎𝟒𝟖,𝟓𝟕𝟔
c. Under the right growing conditions, many bacteriacan reproduce every 𝟏𝟓minutes. Under these conditions,
how long would it take for one bacterium to reproduceitselfinto more than one million bacteria?
It would take 𝟐𝟎fifteen-minuteperiods, or 𝟓hours.
d. Write an expression for how many bacteriawould bepresent after 𝒈generations.
There will be 𝟐 𝒈
bacteria present after 𝒈generations.

Date: 4/15/15 223
𝒘
𝒉 = 𝟐𝒘
𝒍 = 𝟑𝒘
Example 3 (10 minutes): Volume of a Rectangular Solid
 Exponents areused when we calculatethe volume of rectangular solids.
Example 3: Volume ofaRectangular Solid
Thisbox has awidth, 𝒘. The height ofthe box, 𝒉, istwicethe width. The lengthofthe box, 𝒍,isthreetimesthewidth.
That is, the width, height,and lengthofarectangular prism are in theratio of 𝟏: 𝟐: 𝟑.
For rectangular solidslikethis, thevolume iscalculated by multiplying length timeswidth timesheight.
𝑽 = 𝒍 · 𝒘 · 𝒉
𝑽 = 𝟑𝒘· 𝒘· 𝟐𝒘
𝑽 = 𝟑 · 𝟐 · 𝒘 · 𝒘· 𝒘
𝑽 = 𝟔 𝒘 𝟑
Follow the above example to calculatethe volumeofthese rectangular solids, giventhe width, 𝒘.
Width in centimeters(cm) Volume in cubic centimeters(cm3)
𝟏 𝟏 𝒄𝒎× 𝟐 𝒄𝒎 × 𝟑 𝒄𝒎 = 𝟔cm3
𝟐 𝟐 𝒄𝒎× 𝟒 𝒄𝒎 × 𝟔 𝒄𝒎 = 𝟒𝟖cm3
𝟑 𝟑 𝒄𝒎× 𝟔 𝒄𝒎 × 𝟗 𝒄𝒎 = 𝟏𝟔𝟐cm3
𝟒 𝟒 𝒄𝒎× 𝟖 𝒄𝒎 × 𝟏𝟐 𝒄𝒎 = 𝟑𝟖𝟒cm3
𝒘 𝒘 × 𝟐𝒘× 𝟑𝒘 = 𝟔 𝒘 𝟑
cm3
Closing(2 minutes)
 Why is 53
different from 5 × 3?
 53
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)
MP.3

Date: 4/15/15 224
Name Date
Exit Ticket
1. Naomi’s allowanceis $2.00 per week. If she convinces her parents to double her allowanceeach week for two
months, what will her weekly allowancebe at the end of the second month (week 8)?
Week Number Allowance
1 $2.00
2
3
4
5
6
7
8
𝑤
2. Write the expression thatdescribes Naomi’s allowanceduringweek 𝑤, in dollars.

Date: 4/15/15 225
Exit Ticket Sample Solutions
1. Naomi’sallowance is $𝟐.𝟎𝟎per week. Ifshe convincesher parentsto doubleher allowanceeachweek for two
months, what will her weekly allowance beat the end ofthe secondmonth(week 𝟖)?
Week Number Allowance
𝟏 $𝟐.𝟎𝟎
𝟐 $𝟒.𝟎𝟎
𝟑 $𝟖.𝟎𝟎
𝟒 $𝟏𝟔.𝟎𝟎
𝟓 $𝟑𝟐.𝟎𝟎
𝟔 $𝟔𝟒.𝟎𝟎
𝟕 $𝟏𝟐𝟖.𝟎𝟎
𝟖 $𝟐𝟓𝟔.𝟎𝟎
𝒘 $𝟐 𝒘
2. Write the expression that describesNaomi’sallowance during week 𝒘, indollars.
$𝟐 𝒘
Problem Set Sample Solutions
1. A checkerboardhas 𝟔𝟒squareson it.
a. If agrain of rice isput on the first square, 𝟐grainsofrice on thesecond square, 𝟒grainsofrice on thethird
square, 𝟖grainsofrice on the fourth square, etc. (doubling eachtime), how many grainsofricewould beon
the last square? Represent youranswer first in exponential form. Use the tablebelow to helpsolve the
problem.
There would be 𝟐 𝟔𝟑
= 𝟗,𝟐𝟐𝟑,𝟑𝟕𝟐,𝟎𝟑𝟔,𝟖𝟓𝟒,𝟕𝟕𝟓,𝟖𝟎𝟖grains ofrice.

Date: 4/15/15 226
Checkerboard
Square
Grainsof
Rice
Checkerboard
Square
Grains of
Rice
Checkerboard
Square
Grainsof
Rice
Checkerboard
Square
Grains of
Rice
𝟏 𝟏𝟕 𝟑𝟑 𝟒𝟗
𝟐 𝟏𝟖 𝟑𝟒 𝟓𝟎
𝟑 𝟏𝟗 𝟑𝟓 𝟓𝟏
𝟒 𝟐𝟎 𝟑𝟔 𝟓𝟐
𝟓 𝟐𝟏 𝟑𝟕 𝟓𝟑
𝟔 𝟐𝟐 𝟑𝟖 𝟓𝟒
𝟕 𝟐𝟑 𝟑𝟗 𝟓𝟓
𝟖 𝟐𝟒 𝟒𝟎 𝟓𝟔
𝟗 𝟐𝟓 𝟒𝟏 𝟓𝟕
𝟏𝟎 𝟐𝟔 𝟒𝟐 𝟓𝟖
𝟏𝟏 𝟐𝟕 𝟒𝟑 𝟓𝟗
𝟏𝟐 𝟐𝟖 𝟒𝟒 𝟔𝟎
𝟏𝟑 𝟐𝟗 𝟒𝟓 𝟔𝟏
𝟏𝟒 𝟑𝟎 𝟒𝟔 𝟔𝟐
𝟏𝟓 𝟑𝟏 𝟒𝟕 𝟔𝟑
𝟏𝟔 𝟑𝟐 𝟒𝟖 𝟔𝟒
b. Would it have been easier to writeyour answer to part (a)in exponential form or standard form?
Answers will vary. Exponential formis moreconcise: 𝟐 𝟔𝟑
. Standard form is longer andmorecomplicated to
calculate: 𝟗,𝟐𝟐𝟑,𝟑𝟕𝟐,𝟎𝟑𝟔,𝟖𝟓𝟒,𝟕𝟕𝟓,𝟖𝟎𝟖. (In word form: ninequintillion, two hundred twenty-three
quadrillion, threehundred seventy-two trillion, thirty-six billion, eighthundred fifty-four million,seven
hundred seventy-fivethousand, eighthundred eight.)
2. If an amount ofmoney isinvested at an annual interest rate of 𝟔%, it doublesevery 𝟏𝟐years. IfAlejandra invests
$𝟓𝟎𝟎, how long will it take for her investmentto exceed $𝟐,𝟎𝟎𝟎(assuming she doesn’t contributeany additional
funds)?
It will take 𝟐𝟒years. After 𝟏𝟐 years, Alejandra willhavedoubled her money andwill have $𝟏𝟎𝟎𝟎. Ifshewaits an
additional 𝟏𝟐years, shewill have $𝟐,𝟎𝟎𝟎.
3. The athleticsdirector at Peter’sschool hascreated aphonetree that is used to notify team playersin the eventthat
all games have to be canceled or rescheduled. The phonetreeisinitiated whenthe director callstwo captains.
During the second stage ofthephonetree,the captainseachcall two players. During the third stage ofthephone
tree, these playerseach call two other players. The phone treecontinuesuntil all playershave been notified. If
there are 𝟓𝟎playerson the teams, how many stageswill it taketo notify all ofthe players?
It will takefivestages. After thefirst stage, 𝒕𝒘𝒐players havebeen called, and 𝟒𝟖willnot havebeen called. After
thesecond stage, four more players willhavebeen called, for a total of six; 𝟒𝟒willremainuncalled. After thethird
stage, 𝟐 𝟑
players (eight)more will havebeen called, totaling 𝟏𝟒; 𝟑𝟔remain uncalled. After the 𝟒 𝒕𝒉
stage, 𝟐 𝟒
more
players (𝟏𝟔) will havegotten a call, for a total of 𝟑𝟎players notified. Twenty remainuncalled at this stage. The5th
round ofcalls will cover all ofthem.

Date: 4/15/15 227
White Board Exchange: Multiplication of Decimals
Progression of Exercises: Answers:
1. 0.5 × 0.5 0.25
2. 0.6 × 0.6 0.36
3. 0.7 × 0.7 0.49
4. 0.5 × 0.6 0.3
5. 1.5 × 1.5 2.25
6. 2.5 × 2.5 6.25
7. 0.25 × 0.25 0.0625
8. 0.1 × 0.1 0.01
9. 0.1 × 123.4 12.34
10. 0.01 × 123.4 1.234
Fluency work such as this exerciseshould take5–12 minutes of class.
How to Conduct a White Board Exchange:
All students will need a personal white board, white board marker, and a means of erasingtheir work. An economical
recommendation is to placecard stock insidesheet protectors to use as the personal whiteboards and to cut sheets of
felt into small squaresto use as erasers.
It is bestto prepare the problems in a way that allows you to reveal them to the class oneat a time. For example, use a
flip chartor PowerPoint presentation; write the problems on the board and cover with paper beforehand, allowingyou
to reveal one at a time; or, write only one problem on the board at a time. If the number of digits in the problem is very
low (e.g., 12 divided by 3), it may also beappropriateto verbally call outthe problem to the students.
The teacher reveals or says the firstproblem in the listand announces,“Go.” Students work the problem on their
personal white boards,holdingtheir answers up for the teacher to see as soon as they have them ready. The teacher
gives immediate feedback to each student, pointingand/or makingeye contactwith the student and respondingwith an
affirmation for correctwork such as,“Good job!”, “Yes!”, or “Correct!” For incorrectwork, respond with guidancesuch
as “Look again!”,“Try again!”, or “Check your work!”
If many students have struggled to get the answer correct, go through the solution of that problem as a class before
moving on to the next problem in the sequence. Fluency in the skill has been established when the class isableto go
through each problem in quick succession withoutpausingto go through the solution of each problem individually. If
only one or two students have not been ableto get a given problem correct when the rest of the students are finished,it
is appropriateto move the classforward to the next problem without further delay; in this case,find a time to provide
remediation to that student before the next fluency exerciseon this skill is given.

G6 m4-f-lesson 22-t

Recommended

Recommended

More Related Content

Similar to G6 m4-f-lesson 22-t

Similar to G6 m4-f-lesson 22-t (20)

More from mlabuski

More from mlabuski (20)

G6 m4-f-lesson 22-t