TEACHING GUIDE
Exponents
11
Unit
Unit 11
Teaching Guide
June Mark
E. Paul Goldenberg
Mary Fries
Jane M. Kang
Tracy Cordner
Un
it
11
Exp
one
nts
Research-based
National Science
Foundation-funded
Learning
transforms
lives.
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AM
firsthand
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© 2014 by Education Development Center, Inc.
Co-Principal Investigators and Project Directors: E. Paul Goldenberg and June Mark
Development and Research Team: Tracy Cordner, Mary Fries, Mari Halladay, Jane M. Kang, and Josephine Louie
Contributors: Cindy Carter, Susan Creighton, Jeff Downin, Doreen Kilday, Deborah Spencer, and Yu Yan Xu
This material is based on work supported by the National Science Foundation under Grant No. ESI-0917958.
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Transition to Algebra, Unit 11: Exponents Teaching Guide
ISBN-13: 978-0-325-05325-7
Transition to Algebra Teacher Resources
ISBN-13: 978-0-325-05790-3
Transition to Algebra, Unit 11 Student Worktexts 10-pack
ISBN-13: 978-0-325-05313-4
Transition to Algebra Student Worktexts, 10 Sets of All 12 Units
ISBN-13: 978-0-325-05791-0
Printed in the United States of America on acid-free paper
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11
Unit
Exponents
CONTENTS
T4 Unit Introduction
T7
T10
T13
T16
T18
T19
T22
T24
T26
T27
T30
T32
Lesson 1: Multiplication World
Lesson 2: Exponents
Lesson 3: Extending Exponents
Lesson 4: Equivalent Expressions
Student Reflections & Snapshot Check-in
Lesson 5: Area Models
Lesson 6: Simplifying Expressions
Lesson 7: Fractions in Exponents
Student Reflections & Unit Assessment
Exploration: Cubes
Exploration: Towers
Activity: Algebra MysteryGrids
RESOURCES
T33
T34
T35
T36
T37
T38
T39
T41
Blank MysteryGrids (Activity: Algebra MysteryGrids)
Matching Expressions Cards Cutouts (Lesson 4)
Cube Visualization Display (Exploration: Cubes) Cube Visualization Handouts (Exploration: Cubes)
Snapshot Check-in
Snapshot Check-in Answer Key
Unit Assessment
Unit Assessment Answer Key
T43 MENTAL MATHEMATICS
Making Sense of Exponents
Powers of 2
Powers of 3, 4, and 5
Square roots on the number line
Estimating square roots
Estimating values of positive integers raised to powers
Estimating values of negative integers raised to powers
Powers of 2 and -2
Choose a review
T44
T46
T48
T49
T50
T52
T54
T55
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Unit
11
Exponents
Research-based
National Science
Foundation-funded
UNIT
11
Exponents
Learning
transforms
lives.
Learning Goals
By the end of Unit 11, students
should be able to:
•Describe how growth by multiplication
differs from growth by addition.
•Understand positive integer exponents
as recording repeated multiplication.
•Extend the logic of positive exponents
to make sense of exponents that are
zero, negative, and rational.
I
f students learn only “rules” for working with exponents, the mathematics can
seem arbitrary and even counterintuitive. Beginners often trip on notations like
a0, expecting it to be 0 rather than 1, and it’s common to expect that negative
exponents, like 2-3, produce negative numbers. Even a3 • a5 = a8 can feel arbitrary
until it’s understood: the operation is multiplication, so why are we adding? Unit 11
helps students use their own logic to make sense of exponents. Students record and
analyze the patterns in decreasing exponents, for instance, working backwards from
212 and dividing by 2 until they reach the unfamiliar 21, 20, 2-1, and so on in order to
see why, for the mathematics to be consistent, it must be that 21 = 2, and 20 = 1, and
2-1 = 12 . Students practice using exponents in area models and rational expressions
and conclude the unit examining rational exponents. Students consider why rational
•Understand how to multiply and divide
exponential expressions that have a
common base.
103, we might choose a notation like 102 to name them.
•Use multiplication and division to
generate equivalent expressions with
exponential expressions.
Extending a Pattern
•Use area models to multiply
polynomials.
T4
exponents would exist; for example, in order to talk about numbers between 102 and
1
2
Students have a sensible way of understanding integer exponents greater
than or equal to 2. For example, 72 is understood to be “multiplying two 7’s
together,” 25 is “multiplying five 2’s together,” and (-3)4 is “multiplying four -3’s
together.”
In this context, what does 51 mean? Or 90? Or 4-2? With the above
thinking, they don’t yet mean anything. How does one “multiply a single 5
by itself ”? (Multiplying 5 “by itself ” generally means squaring it.) Even more
problematically, how does one “multiply zero 9’s” or “multiply -2 copies of 4”?
While the understanding that an exponent indicates “the number of copies
of a base number to multiply together” is correct for positive integer exponent
(and is what the notation was designed to indicate), this understanding makes
clear sense only for integer exponents greater than or equal to 2. In order
Transition to Algebra Unit 11: Exponents • TEACHING GUIDE
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to extend our understanding of exponents, we look for a way to extend the
existing pattern. Students first examine the sequence of powers of 2:
22
23
24
25
26
27
28
29
210
211
212
4
8
16
32
64
128
256
512
1024
2048
4096
Then to extend their understanding, students examine the sequence backwards
and observe that as the exponent decreases by 1, the quantity halves; we are
repeatedly dividing by 2. In this way, students find one way to logically fill in
the rest of the table.
26
25
24
23
22
21
20
2-1
2-2
2-3
2-4
64
32
16
8
4
2
1
1
2
1
4
1
8
1
16
Students can, in this way, extend the pattern to give sensible meanings for
all integer exponents. Moreover, students can see how exponent “rules” are
consistent with this extended pattern. For example, students can show that
23 • 24 = 27 by showing 8 • 16 = 128. And when the exponents a and b are
greater than or equal to 2, students can directly observe that 2a • 2b = 2a + b by
understanding exponents as repeated multiplication. This rule also applies
beyond these positive integer exponents because exponents that are zero or
negative were defined in a way consistent with the mathematics behind this rule.
Furthermore, students will see that this rule applies to all rational exponents as
1
1
1
well, by making sure that, for example, 2 2 • 2 2 = 2 or, in other words, 2 2 = √2.
Mental Mathematics: Making Sense of Exponents
The Mental Mathematics activities support the work of Unit 11 by giving
students experience with the effect of exponents on magnitude. Students
become especially comfortable with the powers of 2 because this sequence of
numbers shows up in a variety of contexts in mathematics and because it builds
on students’ prior experience with doubling and halving. Students raise 2 to the
12th power by repeated multiplication and then work backwards from 212,
dividing by 2 until they reach the unfamiliar 21 = 2, 20 = 1, and 2-1 = 12 . They
then generalize to a1 = a, a0 = 1 and a -1 = 1a , etc. Students also estimate square
roots and classify numbers with exponents into categories by magnitude (e.g.
between 0 and 10, between 10 and 100, and greater than 100).
In these positive integer cases,
the exponent counts the number
of times that 2 was used as the
only factor in this multiplication.
It is natural for students, at first,
to confuse “how many 2’s?” with
the division question they have
already spent so much time on:
“how many 2’s are there in 8?”
This used to be completely clear
when all students used were the
four basic operations, but the
question is now ambiguous! The
answer is four 2’s if we’re adding
(multiply 2 by 4), three 2’s if we’re
multiplying (raise 2 to the 3rd
power).
Algebraic
Habits of
Mind
Seeking and
Using Structure
Students make
sense of the “rules
of exponents” by seeing that they
encode observable patterns for positive
integer exponents and are extended
to apply to negative and rational
exponents. For example, students
establish the rule an • am = an + m with
examples such as a3 • a5 = a8 (because
3 copies of a and 5 copies of a are
all multiplied to give 8 copies of a).
Then they see that since according
to the rule, 58 • 50 = 58, a consistent
way to interpret 50 must be that
50 = 1. Students also reexamine the
relationship between dividing by n and
1
multiplying by n as they make sense
of numbers with negative exponents
like n-1.
Explorations
The first Exploration, Cubes, gives students a sense of how quickly the
sequence of cubic numbers grows. In the second Exploration, Towers,
students write an algebraic expression for a linear pattern, and in the Further

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Algebraic
Habits of
Mind
Communicating
with Precision
Because Unit 11
focuses on
using logic to
understand exponents, it frequently
asks students to explain their thinking,
identify errors, and compare different
ways of thinking about a quantity.
Part of clear communication is using
precise notation, and students must
get comfortable with the different
meanings of expressions such as
a3 • a3 and a3 + a3, and also with
seeing equivalences such as those
between 1 ÷ c4, c-4, and c-1 • c -3.
T6
Exploration, students revisit the expressions for triangular numbers that they
discovered in Staircase Patterns (Unit 7) and Triangular Numbers (Unit 8).
They use this pattern to write an algebraic expression for the quadratic growth
in the total number of blocks used to build a tower.
Related Activity
Students create and share their own MysteryGrid puzzles involving algebraic
expressions. This encourages students to deeply about the structure of these
puzzles, the nature of the clues, what makes a solution unique, and the
strategies they can use in solving them.
Transition to Algebra Unit 11: Exponents • TEACHING GUIDE
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Lesson at a Glance
Lesson 1:
Multiplication World
Preparation Students will experience
multiplicative growth by folding a sheet of
paper. If possible, provide each student with
a sheet of 8.5 × 11 scrap paper from the
recycle bin.
Mental Mathematics (5 min)
PURPOSE
Lesson 1 contrasts growth by addition and growth by multiplication to
provide a context for the study of exponents. Instead of learning the
rules for working with exponents right away, students first experience
how repeated multiplication exhibits a different kind of growth from the
linear growth exhibited by repeated addition. Students will later see how
exponents help to describe growth by multiplication.
Mental Mathematics Begin each day with five minutes of Mental
Mathematics (pages T43–T55). The first activity familiarizes students with powers
of 2 and introduces the connection between positive, zero, and negative exponents.
Students will examine these patterns more thoroughly in Lessons 2 and 3.
Launch: Addition versus Multiplication
Have students work on PROBLEMS 1 & 2 in small groups before discussing them
together as a class.
1
Growth by addition:
For every hour you work,
you earn $8. Fill in the table.
Hour
1
2
3
Money earned
$8
$16
2
Growth by multiplication:
You invite three friends to a
social networking site and
every friend invites three
friends of their own.
a
Fill in the table.
b
Draw some more
stages of the diagram.
Stage
Friends Invited
0
1 (Just you)
1
3
2
9
Launch: Addition versus Multiplication?
(15 min)
•Students solve and discuss problems
1 and 2 contrasting additive and
multiplicative growth. Then students fold a
sheet of paper as many times as they can
and predict the number of sections the
folds have made.
Student Problem Solving and Discussion
(25 min)
•Allow students to work through the rest of
the Important Stuff and explore additional
problems.
•Discuss student responses to Discuss &
Write problems 7 & 8 about the graphs
showing additive versus multiplicative
growth.
Unit 11 Related Activity: Algebra
MysteryGrids (See page T32 and Student
Worktext page 36.)
3
4
4
5
5
6
6
7
8
9
You
10
Discuss the difference in growth by addition and growth by multiplication.
Students may observe that growth by multiplication occurs much more quickly
than addition in this example. Ask students questions to encourage them to
examine this situation more carefully.
»» In these problems, we see that growth by multiplication by 3 is faster than
growth by addition of 8. What if we compared constantly multiplying 3 to
constantly adding 100? Or adding 1000? Let students observe that growth
by multiplication would still out-pace growth by addition in the long run.
Lesson 1: Multiplication World
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In problem 2, we already see that by stage 6, constantly multiplying 3
results in 726, which is larger than 600, the result from constantly adding
100. Encourage students to estimate values of later stages of multiplying
by 3 and compare them to constantly adding 1000. For example, since the
value at stage 6 is around 700, the value at stage 7 would be around 2100,
at stage 8 would be around 6300, and at stage 9 would be around 18000.
We see that constantly multiplying by 3 out-paces constantly adding 1000
by around stage 9.
Algebraic
Habits of
Mind
Seeking and
Using Structure
Help students
understand that
both growth by addition and growth
by multiplication have a predictable
structure. Growth by multiplication is
likely newer to them and so is less
familiar at this point, but it is still a
consistent kind of growth, although it is
now growth by a constant percentage
(constant multiplier), rather than growth
by constant addition.
Some students may appreciate
the connection to the NCAA
basketball championship
tournament, where 64 teams are
narrowed down to one champion
in just six rounds.
»» We see that constantly multiplying by 3 makes numbers grow rapidly. Does
all growth by multiplication make numbers grow larger no matter what
number we use to multiply? The purpose of this question is to keep students
from thinking that repeated multiplication always results in “faster
growth.” Consider constantly multiplying by a fraction between 0 and 1
to show that constant multiplication can result in diminishing values. Also
consider repeated multiplication with a negative number and discuss how
strange this pattern is, constantly jumping between positive and negative
values (when the exponent is an integer). Students may also consider the
case of repeated multiplication by 1: in this case, the graph would show no
change over time.
For both problems 1 and 2, connect the regularity of the growth to the
language used in the problem. The consistent additive growth of 8 is connected
to earning “$8 every hour” and the consistent multiplicative growth by 3 is
connected to “every friend inviting three friends.” Compare and contrast the
language used to indicate addition and multiplication.
Finally, have students each fold a piece of paper according to the directions
on the sticky note beneath problems 1 and 2. Unless students are using really
thin sheets of paper, they probably won’t be able to make more than 6 folds.
When they open up their pieces of paper, they may see why: by the time they
are trying to make their seventh fold, they are essentially folding 64 pieces of
paper in half. Students will probably experience surprise at the large number of
sections created by just six folds. Have students share the number of sections
they predicted, their reactions to the large number of sections actually created,
and their explanations for how six folds could produce 64 sections.
Student Problem Solving and Discussion:
In PROBLEMS 3-5, students contrast graphs showing additive and multiplicative
growth.
T8
Growth by addition and growth by multiplication look different. Match each story with the appropriate graph.
The line to buy movie tickets starts with 5 people,
and 3 more people join the line every minute.
b
Only 10 people saw the movie when it opened,
but each day, twice as many people come to see
the movie as the day before.
I
II
number of
people
a
number of
people
3
time
time
Transition to Algebra Unit 11: Exponents • TEACHING GUIDE
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This graph shows
These graphs are only sketches to demonstrate the
doubling.
general shape of the graph. Use these problems
and PROBLEMS 7 & 8 in the Discuss & Write box to
talk about the difference in shape between a graph
showing growth by addition and one showing
growth by multiplication. To support student
understanding of curved graphs showing growth
time
by multiplication, mark units of time on the time
axis (as shown on the graph to the right) to demonstrate how the curved graph
shows doubling, tripling, or another multiplicative pattern. Listen for students
who recognize that growth by addition has a constant slope (which is the
number by which the pattern grows) and that graphs showing multiplicative
growth do not have a constant slope. Also listen for students who describe the
graphs of growth by multiplication (by a number greater than 1) by explaining
that the graph grows slowly at first, then faster for larger values of x (or time).
Consider using these additional prompts to discuss growth by addition versus
multiplication:
»» What are some real-life situations of growth by addition? Growth by
multiplication? Draw connections between the language that students use
in their descriptions and the mathematical behavior in the scenario.
»» You have a choice between two deals: Either someone will pay you $20 every
day, or someone will only pay you $1 but will double your money every day.
Explain why the second deal is better in the long run. This problem uses
similar reasoning to that in problems 1 and 2, comparing addition and
multiplication. This problem draws attention to the fact that growth by
multiplication can be slower than addition in the short-term.
1 question
2 questions
3 questions
»» If a true/false quiz has just one
TTT
question, you have a 50/50 chance
TT
of guessing the right answer
TTF
True (T)
because there are only two ways
TFT
TF
you could answer: true or false. If
TFF
a true/false quiz has two questions,
FTT
how many possible ways are there
FT
FTF
to answer? There are four: both
False (F)
FFT
true, both false, true then false, or
FF
false then true. What if there are
FFF
three quiz questions? Now there
are eight possibilities. This pattern grows by multiplication, and can be
illustrated with a diagram like the diagram in problem 2.
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Lesson 1: Multiplication World
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CHECK
IN
Student
Reflections &
Snapshot Check-in
Ask students to reflect on their learning:
What are some things you’ve learned so far in this unit?
What questions do you still have?
Assess student understanding of the ideas presented so far in the unit
with the Snapshot Check-in on page T37. Use student performance
on this assessment to guide students to select targeted Additional
Practice problems from this or prior lessons as necessary.
So far in Unit 11, students have:
•Contrasted growth by addition and growth by multiplication
numerically and graphically.
•Established reasoning for and used the rule an • am = an + m for
multiplying terms with exponents.
•Extended what they know of positive exponents to make sense of
exponents that are zero or negative.
•Recognized equivalent expressions involving exponents.
Students have also focused on the following Algebraic
Habit of Mind:
•Seeking and Using Structure—In order to develop rules for working
with exponents, students have paid attention to structure. They first
understood positive exponents in terms of repeated multiplication,
which led to the observation that an • am = an + m. Then they used
the structure of this statement about multiplication to come up
with a consistent way to understand exponents that are zero or
negative. Students also used structure when they wrote equivalent
expressions involving exponents by rearranging the form of the
expression.
18
Transition to Algebra Unit 11: Exponents • TEACHING GUIDE
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SSESSMENT
Student
Reflections &
Unit Assessment
Before the Unit Assessment, have students reflect on their learning:
What are some things you learned in this unit?
What questions do you still have?
Reflections can be done orally, on paper, or some combination of
both. Use feedback from students to help them identify the big ideas
from the unit and to select Additional Practice problems to help them
prepare for the Unit Assessment included on pages T39 & T40. Before
giving this assessment, consider spending a class period working
through the Unit Additional Practice problems.
Since the Snapshot Check-in, students have:
•Used area models to multiply, divide, and factor polynomials.
•Simplified rational expressions involving exponents and variables.
•Made sense of rational exponents by extending their understanding
of integer exponents.
Throughout Unit 11, students have focused on developing
the following Algebraic Habit of Mind:
•Seeking and Using Structure—Students began by understanding
positive integer exponents as indicating the number of copies of a
number to multiply (e.g., 23 = 8 because multiplying 3 copies of 2
gives 8). They used this understanding to establish the pattern that
an • am = an + m, by observation. Students then examined the effect
of decreasing exponents (e.g. from 23 to 22 to 21) and extended this
pattern of division to exponents that are zero or negative (e.g. to 20
and 2-1 and 2-2). The multiplication rule an • am = an + m was found to
be consistent with exponents that are zero or negative and students
confirmed that, for example, 30 = 1 was a sensible interpretation
of 30 because this value is the only one that is consistent with the
statement 38 • 30 = 38. Finally, students extended this rule to rational
exponents. By considering that n • n = n1, students identified
that n = √n. Students developed this habit of mind by relying on a
structure (in this case, the rule an • am = an + m) to help them make
sense of quantities that were harder to calculate (such as 10 ).
1
2
1
2
1
2
1
2
26
Transition to Algebra Unit 11: Exponents • TEACHING GUIDE
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
EXPLORATION
Cubes
PURPOSE
Students build cubes to get a physical sense of the magnitude of cubic
numbers and experience the surprise at how quickly they grow. This
Exploration helps develop students’ visualization skills as they generalize
from experiments they can perform to experiments with numbers too large
for calculation.
Mental Mathematics Begin each day with five minutes of Mental
Mathematics (pages T43–T55). Just as many of these activities help students get a
sense of the magnitudes of numbers with exponents, today’s Exploration will also
support students’ understanding of and feel for cubic numbers.
Student Exploration and Discussion
Since part of the purpose of the Exploration is for students to experience
surprise at the magnitude of cubic numbers, students are asked to make a lot
of predictions. It’s okay if students’ predictions are wildly off—in fact, the
Exploration is more powerful if students come up with estimates that are too
low. Even for people who understand mathematically that a cube with a side
length of 4 units has a volume of 64 cubic units, there is a moment of secondguessing that occurs when faced with having to actually gather 64 blocks
to make a cube only four blocks wide. This is the central experience of this
Exploration.
Communicate to students that many people are surprised by the magnitude
of cubic numbers. Unless they work out the number exactly, it is no surprise
if their predictions are low. The writing prompts are provided not to bring
attention to inaccurate predictions but to record the (legitimate) surprise of
having to use so many blocks to construct their cube. And this exploration only
examines an exponent of 3! You might have students ponder the magnitudes
involved with even larger exponents.
Problems 9, 10, and the Further Exploration encourage students to describe
the growth of cubes. Problem 9 starts with a cube they’ve built (the 3 × 3 × 3
cube) and extend it to a 4 × 4 × 4 cube, Problem 10 starts with a 10 × 10 × 10
cube and extends to a 11 × 11 × 11 cube, and the Further Exploration asks
students to describe the growth algebraically. Discuss student responses as a
class, using the images on the “Cube Visualization” handout and/or display
page to facilitate the discussion if desired.
Use these prompts to discuss how building cubes is related to exponents.
Exploration at a Glance
Preparation
•Prepare enough cube blocks for every
student to have 10 blocks. Do not hand
out the blocks at the beginning of class;
students will come to you for the cubes,
as needed.
•Prepare to display the “Cube Visualization
Display” (available in the Resources PDF;
a reference copy is provided on page T35).
•(optional) To help students discuss
problems 9–11, you may find it useful to
photocopy and cut “Cube Visualization”
(available in the Resources PDF; a
reference copy is provided on page T36).
There are four copies of the images per
page.
Mental Mathematics (5 min)
Student Exploration and Discussion
(40 min)
•Students explore problems 1–8 as a
class, with different numbers of students
gathering together to see if they have
enough blocks to make a cube of
each given size. Problems 9–10 move
students towards thinking of the problem
algebraically.
•Invite students to share their responses
to problems 9–10 and/or discuss how
the images on the “Cube Visualization”
handout can be used to describe the
growth between cubes with consecutive
integer lengths.
Unit 11 Related Activity: Algebra
MysteryGrids (See page T32 and Student
Worktext page 36.)
The Student Worktext makes a
subtle distinction between its use
of the terms “cube” and “block.”
A cube is a three-dimensional
geometric figure whose length,
width, and height must be equal.
The term “cube” is used in this
Exploration to refer to the figure
students build together. To reduce
confusion, the term “block” is
used in this Exploration to refer
to the unit cubes students use to
build the larger cubes.
EXPLORATION Cubes
TTA_U11_TG_DS_r4b.indd 27
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Algebraic
Habits of
Mind
Seeking and
Using Structure
This Exploration
brings focus to
the connection between exponents
and geometry. Students are given
the dimensions of a cube, predict its
volume, and physically use unit blocks
to build it in order to experience the
dramatic growth between cubes of
consecutive integers. Students make
sense of and describe this growth
by focusing on the structure of the
physical cube. For example, they
consider a cube with side length 3
and find a way to describe how many
more blocks are needed to build a
cube with side length 4 without actually
building the cube. One way students
might use structure is by visualizing
the problem in a way consistent with
the diagram below. Students may also
use this image to generalize the growth
between cubes of consecutive integers.
In the Further Exploration, students can
use this image to express (x + 1)3 as
x3 + 3x2 + 3x + 1 to show that a cube
with a side length of x + 1 is made up
of a cube with a side length of x, three
squares with area x2, three straight
pieces of length x, plus 1.
T28
»» The numbers of identical cubes that can be arranged to completely fill a cube
are called “cubic numbers.” Which cubic numbers did you find? This is an
opportunity to summarize what students found and make a connection to
exponential notation. Students had the opportunity to make (or at least
predict) 13 = 1, 23 = 8, 33 = 27, 43 = 64, 53 = 125, and 63 = 216. As students
look at this sequence, they are not likely to observe any pattern except
that they get larger fast, and even the rate at which they increase goes up.
If students have done the Further Exploration, this is an opportunity to
make the connection that, by using algebra, they have figured out a way
to express how this pattern grows. The growth is based on x2 and x, so is
complicated to see by inspection, but possible to express with algebra.
»» How can we use blocks to illustrate the square numbers? Use this
opportunity to connect exponents with geometry. Cubic numbers are the
product of three identical numbers. With blocks, we built cubes, threedimensional figures with equal side lengths whose volume is the product
of its length, width, and height. The cubic number s3 is the volume of the
cube whose side length is s. Likewise, we can use square tiles to illustrate
square numbers by building squares in two dimensions with equal side
lengths. In this case, the square number s2 is the area of the square whose
side length is s. We can even go down to one dimension by building a
single row of toothpicks. In that case, the number s1 represents the length
of the row.
»» How can we visualize numbers to the fourth power? In this case, we’re
physically limited by our three-dimensional world. As mathematicians,
though, we can imagine that, in the same way that square numbers can be
pictured as the amount of space in two dimensions and cubic numbers can
be pictured as the amount of space in three dimensions, numbers to the
fourth power can be pictured as the amount of space in four dimensions.
Using numbers, we can actually calculate these values: 14 = 1, 24 = 16,
34 = 81, etc. And using algebra, we can even describe how this pattern
grows from one value to the next: (x + 1)4 = x4 + 4x3 + 6x2 + 4x + 1.
Transition to Algebra Unit 11: Exponents • TEACHING GUIDE
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Further Exploration
The Further Exploration asks students to find a way to express the volume
of a cube with side length x + 1. Students are generalizing from their work
in problems 9 and 10 to describe an expression for (x + 1)3. They can
write (x + 1)3 = x3 + 3x2 + 3x + 1, or (x + 1)3 = x3 + (x + 1)2 + 2x2 + x, or
(x + 1)3 = (x + 1) (x + 1)2 or various other expressions. Students can use area
models to verify that they can also find these expressions using an algebraic
approach. To multiply (x + 1)3, students must use two area models, one to find
(x + 1)2 and then one to find (x + 1)(x + 1)2.
x
1
x
x2
x
1
x
1
x2
2x
1
x
x3
2x 2
x
1
x2
2x
1
To multiply (x + 1)3 using a single model, students might even visualize a
“volume model” such as the one below that shows what it would look like to
multiply in three dimensions.
x
1
1
x
x3
x
1
1
x
x
x2
1
x
x
x
1
x
x2
x
x
x2
This Exploration introduces
students to cubic numbers. As
a matter of vocabulary, note
that students are not exploring
“exponential growth.” Exponential
growth is used to describe a
pattern described by an equation
like y = 3x. When x = 0, 1, 2,
3, 4, and 5, the values of the
exponential function y = 3x are
y = 1, 3, 9, 27, 81, and 243.
In this Exploration, students
examine cubic numbers,
generated by the equation y = x3
(an example of a polynomial). The
first few values of the polynomial
y = x3 are y = 0, 1, 8, 27, 64, and
125. Students encounter both
kinds of functions during Unit 11,
but it is not important that they
focus on this distinction in their
vocabulary as they are not asked
to contrast these functions.
Instead, students’ attention
should remain on examining the
effects of exponents on numbers.
x
1
EXPLORATION Cubes
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Snapshot Check-in
1
A factory produces 7 cars every day for a
whole week.
a Fill in the table.
b
3
Name:
2
A company starts with 7 people and doubles
every year for the first 7 years.
a Fill in the table.
Day
Total cars produced
Year
People in company
1
7
1
7
2
14
2
14
3
3
4
4
5
5
6
6
7
7
Does this table show growth by addition or
growth by multiplication?
a3 • a4 = a12
b
Does this table show growth by addition or
growth by multiplication?
Uh-oh! This equation is wrong.
Correct the mistake and explain one way to think correctly about this problem.
Fill in the blanks.
4
35 • 37 = _______
7
28 • 2n = 220
10
n = _______
5
511 • 5-2 = _______
8
7m • 72 = 718
What is one equivalent way to write 3-2?
m = _______
11
6
105 • 10-3 • 108 = _______
9
810 • 8w = 89
w = _______
What is one equivalent way to write 40?
©2014 by Heinemann and Education Development Center, Inc., from Transition to Algebra (Portsmouth, NH: Heinemann). This page may be reproduced for classroom use only.
Unit 11: Exponents
Snapshot Check-in
Snapshot Check-in
37
R6
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Unit Assessment
Name:
1
Write a situation that shows growth by addition and sketch a graph of your story, labeling the axes.
2
Write a situation that shows growth by multiplication and sketch a graph of your story, labeling the axes.
3
Figure out the pattern in each table and describe the growth. Is the pattern based on constant addition or constant
multiplication? By how much?
a
x
y
x
y
1
3
1
4
2
6
2
12
3
12
3
20
4
24
4
28
5
48
5
36
6
96
6
44
b
Fill in the blanks.
4
a5 • a5 = _____
5
a5 + a5 = _____
6
4a3 • a3 = _____
4a3 + a3 = _____
Explain how you think about problems 4 and 5 to help you get the right answers.
©2014 by Heinemann and Education Development Center, Inc., from Transition to Algebra (Portsmouth, NH: Heinemann). This page may be reproduced for classroom use only.
Unit 11: Exponents
Unit Assessment
Unit Assessment
T39
R8
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Unit Assessment
7
9
(continued)
Fill in the blanks.
8
a
225 • 22 = _______
b
n15 • n-1 = _______
c
64 • 6 • 6-2 = _______
d
q-9 • q5 = _______
36 • 3k = 316
f
106 • 10x = 105
x = _______
g
4c • 4-17 = 40
c = _______
Write three equivalent expressions for each of
the following.
h9
b
c0
c
The equivalent expressions all equal _________.
x7
x9
x5x-7
x0x -2
x -6x3
x8 ÷ x10
x2
x4
k = _______
e
a
Cross out the one expression that isn’t equivalent
to all the others.
Simplify these expressions.
10
6a9 =
10a5
11
32c3 =
8c5
12
7n8 =
14n7
13
8x 5y4 =
12x 7y
v3
v8
Fill in the area models and use them to complete each equation. Multiple correct answers may be possible. Find one.
n6
14
2n4
-5n
15
_____
_____
5a5
35a3
n3
____
n3(n 6 + 2n4 – 5n) =
5a 5 + 35a 3 =
c3
16
8
17
_____
4
4x6
c3
____
x12
-6
____
9x6
(c3 + 8)(c3 – 6) =
x12 + 13x 6 + _____ =
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Unit 11: Exponents
Unit Assessment
T40 Transition to Algebra Unit 11: Exponents • TEACHING GUIDE
R9
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Unit
11
Mental Mathematics
Making sense of exponents
In Unit 11, students work with exponents. The corresponding mental
mathematics activities use repeated calculations to generate expectations
about how exponents should behave. Students use the structure they see to
extend exponents into new territories—negative and fractional exponents—
that have no initial intuitive meaning (What does it mean to multiply x by
itself -3 times?) and need to be assigned a sensible meaning that maintains
the patterns found in calculations with integer exponents. Students also
approximate square roots and categorize exponents in broad categories,
estimating (by using “good-enough” calculations) to classify ab between 0
and 10, 10 and 100, greater than 100.
In nearly all of these mental mathematics activities, students “enact a function”: an
input-output rule is established at the outset, and students give the output for each
input they hear. Each function rule focuses on a key mathematical idea or property
(e.g. complements or the distributive property) that students begin to feel intuitively.
After introducing the day’s task, the teacher deliberately does not reiterate the task but
says only the input numbers for students to transform. Minimizing words lets students
focus on the numerical pattern of the activity, helping them perceive the structure
behind the mathematics. A lively pace maximizes practice and keeps students engaged.
43
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Mental Mathematics • Activity 1
Powers of 2
PURPOSE
By using repeated multiplication and then division, students calculate
powers of 2. Over the course of the unit, students will come to see why it
makes sense that any number raised to the 0th power will equal 1, instead
of just trying to remember it as another arbitrary fact. More importantly,
they see how they can figure such things out any time they forget.
Instructions:
Display or draw the table on the next page.
Beginning with 22, have students double repeatedly,
out loud, until they find 210 (1024). Write each
response in the table as students call them out.
Before going on, stop to write 24 as 2 • 2 • 2 • 2 to call
explicit attention to the “four twos” involved in this
multiplication. Then, erase all but 1024 and have
students halve until they reach 21. Write in their
responses as they say them. Following the pattern
of halving, students will see that 21 must be 2. In
another step, they see why it must be true that 20
is 1. Continue to halve until students complete the
1 1 1
table, and then discuss how the 2 , 4 , 8 , and so on
that result from the negative exponents mirror the
2, 4, 8, and so on that are generated by the positive
exponents.
About this sequence:
The activity only spans 2-6 to 210 so as to fit
reasonably when displayed on the board, but
students should be well able to calculate from 2-12
all the way to 212.
A completed table is shown at right.
44
210
1024
29
512
28
256
27
128
26
64
25
32
24
16
23
8
22
4
21
2
20
1
2-1
1
2
2-2
1
4
2-3
1
8
2-4
1
16
2-5
1
32
2-6
1
64
Transition to Algebra Unit 11: Exponents • TEACHING GUIDE
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Powers
Powers
ofof
2 2 (Mental
(Mental
Mathematics
Mathematics
Activity
Activity
1) 1)
210
29
28
27
26
25
24
23
22
21
20
2-1
2-2
2-3
2-4
2-5
2-6
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Unit 11: Exponents
Mental Mathematics • Activity 1
R12
45
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Mental Mathematics • Activity 2
Powers of 3, 4, and 5
PURPOSE
Students generalize what they learned about positive and negative powers
of 2 by seeing that all of the behaviors apply equally well to powers of 3,
4, and 5.
•Remind students of their
strategies for multiplying
by 4 (doubling twice) and 5
(multiplying by 10 and dividing
by 2 in either order).
•You may choose to go farther
than these tables do, for
example, by having students
calculate 45 or 5-4. This is fine,
but the focus is on the pattern
displayed by exponents less
than 2.
46
Instructions:
“Last time we multiplied and divided by 2 to find the powers of 2. Today we’ll
do the same with 3, 4, and 5. Let’s start with something we know, like 32 (9). So
to find 33, we have to . . . right, multiply by 3 again.” Have students fill out the
powers of 3 table on the next page, and then walk them backward through their
responses until they reach 31, which they will deduce must be 3. Carry on to the
negative powers. Do the same for 4 and 5, beginning with the squares, working
to the fourth power, then working backward into the negative exponents.
The completed tables are shown below.
35
243
34
81
44
256
54
625
33
27
43
64
53
125
32
9
42
16
52
25
31
3
41
4
51
5
30
1
40
1
50
1
3-1
1
3
4-1
1
4
5-1
1
5
3-2
1
9
4-2
1
16
5-2
1
25
3-3
1
27
4-3
1
64
5-3
1
125
Transition to Algebra Unit 11: Exponents • TEACHING GUIDE
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Powers
Powers
ofof
3,3,
4,4,
and
and
5 5 (Mental
(Mental
Mathematics
Mathematics
Activity
Activity
2) 2)
35
34
44
54
33
43
53
32
42
52
31
41
51
30
40
50
3-1
4- 1
5-1
3-2
4- 2
5-2
3-3
4- 3
5-3
©2014 by Heinemann and Education Development Center, Inc., from Transition to Algebra (Portsmouth, NH: Heinemann). This page may be reproduced for classroom use only.
Unit 11: Exponents
Mental Mathematics • Activity 2
R13
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