Unit
10
Area Model
Factoring
Research-based
National Science
Foundation-funded
Learning
transforms
lives.
TTA_U10_SB_DS_r6.indd 1
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Dear Student,
When we multiply two factors, we get their
product. If we start with the product, we can
undo the multiplication to find the factors.
x
The process of discovering the factors that
x2
can be multiplied to give a number or
expression is called factoring.
x x x x
1
1
1
1
x2 + 5x + 4 = (x + ____ )(x + ____)
Unit 10 uses area models to make sense of factoring. You’ve already learned how to multiply
two algebraic expressions using an area model:
x
x
-2
x2
-14
7
(x + 7)(x – 2) = x 2 + _____ – 14
We can also use area models to un-multiply, to start with an algebraic expression and factor
it, that is, find two other expressions whose product is that original expression.
x
x
____
____
x2
3x
15
x2 + 8x + 15 = (x + ____ )(x + ____)
Unit 10 shows how multiplication, division, and factoring are all related, how division and
factoring are like opposites of multiplication, and how we can use area models to reason
about all three.
—The Authors
TTA_U10_SB_DS_r6.indd 2
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Lesson 3: Factoring
IMPORTANT STUFF
Problems 1 and 2 ask about the equation (x – 2)(x – 5) = x2 – 7x + 10 and its area model.
What calculation creates the 10 in x2 – 7x + 10?
1
x
-2
x
x2
-2x
-5
-5x
10
-2 • ____
What calculation creates the -7x in x – 7x + 10?
2
2
-2x + ____
Problems 3 and 4 ask about the equation (x + 3)(x – 8) = x2 – 5x – 24 and its area model shown below.
x
-8
x
3
x2
3x
-8x
3
What calculation creates the -24 in x2 – 5x – 24?
4
What calculation creates the -5x in x2 – 5x – 24?
-24
List all the pairs of integers (positive or negative)
whose product is 12.
5
6
Which pair has a sum of 7?
7
Which pair has a sum of 8?
8
Which pair has a sum of 13?
9
Which pair has a sum of -7?
Use an area model to factor. Complete each model and equation.
10
x2 + 7x + 12 = _______________________
x
x
x2 + 13x + 12 = _______________________
x
___
12
TTA_U10_SB_DS_r6.indd 12
x
x
12
x
x2 + 8x + 12 = _______________________
___
x2
___
12
11
x
x
12
12
x2 – 7x + 12 = _______________________
___
x2
x2
___
13
___
___
___
x2
12
Unit 10: Area Model Factoring
1/18/14 12:35 AM
Complete the model and finish Jay’s thought.
14
x2 + 3x + 2 = (
x
x
Thinking out Loud
)
Jay:
___
I can tell from the equation that I need a pair of
numbers whose product is 2 and whose sum is ___.
x2
The numbers are ___ and ___ so the factors must
2
___
15
)(
be ________ and ________.
List all the pairs of integers (positive or negative)
whose product is 30.
16
Which pair has a sum of 11?
17
Which pair has a sum of -13?
18
Which pair has a sum of 31?
19
Which pair has a sum of -17?
21
x2 – 13x + 30 = _______________________
Use an area model to factor. Complete each model and equation.
20
x2 + 11x + 30 = _______________________
x
x
x
27
Who Am I?
• t+ u = 10
• tu = 21
• t> u
Who Am I?
• t+ u = 12
• tu = 36
Lesson 3: Factoring
TTA_U10_SB_DS_r6.indd 13
23
x2 – 17x + 30 = _______________________
___
x
x2
___
30
___
x2 + 31x + 30 = _______________________
30
___
t
t
u
u
___
x2
x
30
___
x2
x
30
x
24
x
x2
___
22
___
25
Who Am I?
• t+ u = 10
• tu = 25
t
u
28
Who Am I?
• t+ u = 11
• tu = 24
• t< u
t
u
26
Who Am I?
• t+ u = 11
• tu = 30
• u > t
t
u
29
Who Am I?
• tu = 81
• t+ u = 18
t
u
13
1/18/14 12:35 AM
STUFF TO MAKE YOU THINK
30
For some expressions, factoring with the area model doesn’t give any new information.
For example, it’s not helpful to fill in the outside of this model
1
____
because b2 and 4 have no common factors besides 1.
b
_____
2
4
_____
b2
4
An expression like b2 + 4 that cannot be factored (that is, has no factors
other than
itself and 1) is called prime, just like a number whose only factors are itself
and 1!
Determine if it makes sense to factor each expression.
If so, complete the area model. If not, cross it out and label it prime.
a
____
_____
_____
2h
9
____
_____
_____
6p
60
_____
c
____
4w2
____
8w
_____
d
____
8w2
____
3
g
e
____
_____
_____
_____
10m2
30m
80
_____
f
_____
_____
c2
8c
____
____
31
b
h
_____
30a2
7w
____
24
3c
_____
____
15a
____
25b2
____
5c
Here are four ways to set up the expression n2 + 9n + 14 in a model. Three of the ways don’t help or don’t work
when you try to fill out the outside. Cross out the three that don’t help or don’t work, complete the one that does,
and write an equation to match it.
a
____
_____
_____
_____
n2
9n
14
_____
_____
____
n2
9n
____
0
14
_____
_____
b
_____
_____
____
n2
7n
____
n2
3n
____
2n
14
____
6n
14
c
14
TTA_U10_SB_DS_r6.indd 14
d
Unit 10: Area Model Factoring
1/18/14 12:35 AM
Imagine you have been asked to help design a village. The people only want roads that travel northsouth, or east-west, so that their intersections form right angles like this:
and not like this: .
All of the north-south roads must cross all the east-west roads.
32
34
Draw how you would arrange 5 roads so there are
6 intersections in the village.
Factor the expression r2 + 5r + 6.
___
___
N
W
E
33
Sketch a village map in which 7 roads
form 10 intersections.
35
Factor the expression r2 + 7r + 10.
S
___
r2
(
6
___
)(
r2
)
10
36
Suppose the people said they wanted 12 intersections but didn’t care how many roads they had. How many
different possible arrangements could you make? Draw the maps. What is the smallest number of roads that will
still give them 12 intersections?
37
With exactly 10 roads, what is the smallest number of intersections there can be? What is the greatest number of
intersections?
TOUGH STUFF
Factoring doesn’t always work out into neat and tidy factors. You may have to try a lot of different ways.
38
___
___
39
x2
___
x2 + 712x + 11 = (
Lesson 3: Factoring
TTA_U10_SB_DS_r6.indd 15
___
___
11
2x
2x 2
___
)(
)
___
2x2 + 5x + 2 = (
2
)(
)
15
1/18/14 12:35 AM
Additional Practice
Factor each expression.
A
a2 + 15a + 56 =
___
___
a
7a
___
56
___
2
___
___
C
___
x2 – 6x – 40 =
___
E
___
x2
___
-10x
D
___
I
K
x2 + 3x = ____
x (
)
___
1
___
___
___
x2
___
92x
H
___
___
J
-3x
___
30
___
y
___
x2
___
L
6x
7x2
___
k2 + 2k – 48 =
19
x2 – 14x + 33 =
___
___
1
___
x2 – 13x + 30 =
___
7x2 + 42x + 7y =
b
4q
x
42
x2 + 11 2 x + 19 =
F
q2 + 4q + bq =
___
xy
-40
72
q
There is no
squared
term here.
x
z
___
___
___
z2 + 17z + 72 =
___
G
xy + 7x + 6y + 42 =
B
33
r2 + 8r + 7 =
Draw your
own area
models.
16
TTA_U10_SB_DS_r6.indd 16
Unit 10: Area Model Factoring
1/18/14 12:35 AM
For these problems, only the inside of the area model is filled in. Find a way to complete the outside of the model and use
your work to write at least one equation (using multiplication or division) that is represented by the area model.
M
_____
_____
9y
45
____
____
_____
O
____
10a2
____
6a
P
____
_____
_____
____
p2
2p
____
10p
20
Q
S
N
R
____
_____
_____
2n2
16n
_____
_____
_____
3x2
24x
15
_____
_____
_____
ac
3a
4a2
Here are four ways to set up the expression w2 + 11w + 30 inside an area model. Three of the ways don’t help or
don’t work when you try to fill out the outside. Cross out the three that don’t help or don’t work, complete the
one that does, and write an equation to match it.
i
____
_____
_____
_____
w2
11w
30
_____
_____
____
w2
11w
____
10
20
_____
_____
ii
_____
_____
____
w2
10w
____
w2
6w
____
w
30
____
5w
30
iii
Lesson 3: Factoring
TTA_U10_SB_DS_r6.indd 17
iv
17
1/18/14 12:35 AM
Unit
TEACHING GUIDE
Area Model Factoring
Unit 10
10
Teaching Guide
June Mark
E. Paul Goldenberg
Mary Fries
Jane M. Kang
Tracy Cordner
Un
it
10
Are
Fac a Mo
tori del
ng
HEINEMANN
Portsmouth, NH
Research-based
National Science
Foundation-funded
Learning
transforms
lives.
TTA_U10_TG_DS_r6a.indd 1
3/11/14 10:26
1:48 PM
4/14/14
AM
firsthand
An imprint of Heinemann
361 Hanover Street
Portsmouth, NH 03801-3912
www.heinemann.com
Offices and agents throughout the world
Education Development Center, Inc.
43 Foundry Avenue
Waltham, MA 02453-8313
www.edc.org
© 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.
Opinions expressed are those of the authors and not necessarily those of the Foundation.
All rights reserved. No part of this book may be reproduced in any form or by any electronic or mechanical means,
including information storage and retrieval systems, without permission in writing from the publisher, except by a
reviewer, who may quote brief passages in a review, and with the exception of reproducible pages, which are identified
by the Transition to Algebra copyright line, and may be photocopied for classroom use only.
“Dedicated to Teachers” is a trademark of Greenwood Publishing Group, Inc.
Due to a printing error, the area model on the front cover of the Student Worktext may be colored incorrectly;
the shading on the lower half of the model may be missing. (Compare the Worktexts covers to the Answer Key
cover, which has the correct shading.) If your students have mis-printed copies, you may wish to discuss and
have students color the model to complete an (x + 3)(x + 2) model.
Transition to Algebra, Unit 10: Area Model Factoring Teaching Guide
ISBN-13: 978-0-325-05324-0
Transition to Algebra Teacher Resources
ISBN-13: 978-0-325-05790-3
Transition to Algebra, Unit 10 Student Worktexts 10-pack
ISBN-13: 978-0-325-05312-7
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
18
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14
RRD
1
2
3
4
5
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10
Unit
Area Model Factoring
CONTENTS
T4 Unit Introduction
T7
T10
T13
T16
T17
T20
T22
T25
T26
T29
T31
Lesson 1: Division Undoes Multiplication
Lesson 2: Area Model Inside Out
Lesson 3: Factoring
Student Reflections & Snapshot Check-in
Lesson 4: Products, Sums, and Signs
Lesson 5: Zero Product Property
Lesson 6: Solving by Factoring
Student Reflections & Unit Assessment
Exploration: Area Model Cutouts
Exploration: Signed Area Model Cutouts
Activity: Area Model Puzzles
RESOURCES
T32
T33
T34
T35
T36
Finding Pairs Cutout (Lesson 5)
Snapshot Check-in
Snapshot Check-in Answer Key
Unit Assessment
Unit Assessment Answer Key
T37 MENTAL MATHEMATICS
Factors, products & sums and percentage calculations
T38 Finding factor pairs
T39 Finding factor pairs with negatives
T40 Product and sum
T41 Finding factors using products and sums
T42 Finding factors using sums and products
T43 10% of a number
T44 20% of a number
T45 Finding 5% and 15% of a number
T46 10% off
T47 10% more
T48 20% off
T3
TTA_U10_TG_DS_r6a.indd 3
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Unit
10
Area Model
Factoring
Research-based
National Science
Foundation-funded
Learning
transforms
lives.
2/6/14 3:43 PM
Learning Goals
By the end of Unit 10, students
should be able to:
•Understand both division and factoring
as ways of undoing multiplication.
•Understand how a single area model
can represent multiplication, division,
or factoring depending on which pieces
are given and which pieces are sought.
UNIT
10
Area Model
Factoring
F
actoring can be a difficult concept for introductory algebra students to learn
and master. This unit presents factoring as a kind of “un-multiplying,” building
on the ideas of Unit 4: Area and Multiplication, which introduced the area
model as a tool for organizing multiplication of numbers and polynomials.
In this unit, students first consider division with area models, working with a
given area (the product of the multiplication being undone) and one side length of
the area model (a factor) to find the other length (the other factor of the original
multiplication). Students then connect the models to corresponding multiplication and
division equations and explore area model puzzles that provide enough information
•Factor monic quadratic trinomials by
finding the two numbers with a given
sum and product.
about the factors and product to find the missing terms. This supports students in
•Use the zero product property to solve
quadratic equations.
for factoring. The factoring problems in this unit focus primarily on expressing trinomial
understanding the roles of and relationships between the pieces of the area model
and their corresponding equations and in thinking flexibly with the model in preparation
products as two binomial factors. Students also practice finding sums for pairs of
numbers with a given product and build logic about the zero product property and its
role in solving equations by factoring.
Factoring and Area Models
As in distribution, where multiplying 2(n + 3) can appear to students to be a
different process from multiplying (x + 3)(x – 4), where the FOIL (first, outer,
inner, last) approach is often used,
Factor x2 + 4x – 77
factoring can also be seen as several
x
-7
different processes. For instance,
2
-7x
factoring x + 4x – 77 can appear
x2
x
altogether different from simplifying
-77
11x
11
a rational expression like 9hx + 6h
or
3h
x 2 + 4x – 77 = (x – 7)(x + 11)
solving an equation like x2 – 8x = 0.
T4
Transition to Algebra Unit 10: Area Model Factoring • TEACHING GUIDE
TTA_U10_TG_DS_r6a.indd 4
4/14/14 10:26 AM
The area model provides a context in which students can unify their
understanding of these seemingly different processes with a single tool.
Simplify
3h
9hx + 6h
3h
3x
2
9hx
6h
x2 – 8x = 0
x
9hx + 6h = 3h(3x + 2)
3h
x
-8
x2
-8x
x 2 – 8x = x(x – 8) = 0
Either x = 0 or x – 8 = 0
Area Model Puzzles
As Unit 4 did to some extent, this unit presents area model problems in the
form of puzzles. Parts of the product are shown, parts of the factors are
shown, and students need to treat these problems much the way they’d treat any
puzzle—by first looking to see what empty space(s)
they can fill in with certainty. Then, they can chase
3m
____
other empty spaces around the puzzle until they’ve
6mj
14j
____
filled in all the blanks. This is, of course, neither a
35
5
case of “pure” division, where the product and one
of the factors are completely known, nor a case of
pure factoring, where only the product is known and one has no knowledge
about the factors. In pure division, the puzzle is simpler: one knows exactly
where to start. In a pure factoring problem, the puzzle might take a bit of
experimentation before “chasing the empty spaces around,” but the process
of using logic together with the known structure of the area model to find the
remaining pieces is the same. Lesson 2 introduces area model puzzles to focus
attention on the roles of different parts of the area model and how these pieces
relate within a given model and to the problem at hand.
Factoring Puzzles
MysteryGrid puzzles requiring numerical factoring appear in the Stuff to Make
You Think and Tough Stuff problems in Lessons 1 and 2; then in these same
sections of Lesson 4, students confront MysteryGrid puzzles with polynomial
elements and clues that require factoring.
MysteryGrid 5, 6, 7, 8
12, +
56, •

TTA_U10_TG_DS_r6a.indd 5
2x2 + x, +
0, •
35, •
Seeking and
Using Structure
In this unit,
students use the
familiar structure of the area model
and its relationship to multiplication
to strengthen and build an intuitive
understanding of the necessary
computational skills for factoring
algebraic expressions and solving
equations by factoring. Equations such
as (x – 1)(x + 2) = 0 are viewed as
two separate “chunks” of information:
either x – 1 = 0 or x + 2 = 0 (or both).
This relies on an understanding of the
zero product property and supports
solving quadratic equations by factoring.
Using Tools Strategically
As students extend their use of area
models to the division and factoring of
polynomials, they have to be strategic
about their placement of the given
information in the model. Students
also use tables to organize their search
for pairs of numbers with a given sum
and product.
Puzzling and Persevering
All of the puzzles in Unit 10—
MysteryGrids, Who Am I? puzzles,
Mystery Number puzzles, and area
model puzzles—are designed to
engage students in mathematics
related to factoring. These puzzles
offer a natural context for perseverance
in factoring, as students must
look around, make sense of given
information, identify a reasonable
starting place, consider potential
solving strategies, identify reasonable
next steps, and determine when the
puzzle is solved.
2x, +
40, •
42, •
14, +
2, +
48, •
30, •
MysteryGrid 0, 1, x, x2
Algebraic
Habits of
Mind
x
1, +
x2 + 1, +
T5
4/14/14 10:26 AM
Who Am I? puzzles in the Additional
Practice of Lessons 2 and 4 and in the
Important Stuff of Lesson 3 require students to
consider pairs of numbers with a given product
and sum.
Mystery Number puzzles, such as the
problem with butterflies shown here, appear as
contexts for using the zero product property to
solve systems of equations in Lessons 5 and 6.
t
Who Am I?
u
• t + u = 10
• tu = 21
• t > u
•
=
+
=
= _____
Mental Mathematics: Factors, Products & Sums and
Percentage Calculations
A monic quadratic trinomial is
a polynomial with three terms
(trinomial) with a highest power of
two (quadratic) and a coefficient
of 1 on the leading term (monic).
For example, the expression
x2 – 2x – 63 is a monic quadratic
trinomial.
There are two strands of Mental Mathematics for this unit. The first strand
connects directly to the mathematical content of Unit 10; students consider
the factor pairs of a given number, calculate the product and sum for pairs of
numbers, and identify which pair of numbers will yield a given product and
sum. These activities support the factoring of monic quadratic trinomials
presented in the lessons.
The second strand of Mental Mathematics in this unit builds on students’
experiences multiplying and dividing by 10, 2, and 5 in Units 1, 4, and 5,
extending these ideas to calculations with percentages. Students calculate
5%, 10%, 15%, and 20% of given numbers and later also percent off, which
requires subtraction. These ideas are likely to be relevant contexts for mental
mathematics in students’ lives.
Explorations
The two Explorations in this unit, Area Model Cutouts and Signed Area
Model Cutouts, support greater understanding of factoring with quadratic
polynomials through a hands-on activity. Students arrange x2, x, and square
unit paper cutouts into rectangles and record their results using area models
and equations.
Related Activity
In the Related Activity, Area Model Puzzles, students build and share their
own area model puzzles, reinforcing the structure and function of the area
model and giving students an opportunity to create their own mathematics.
T6
Transition to Algebra Unit 10: Area Model Factoring • TEACHING GUIDE
TTA_U10_TG_DS_r6a.indd 6
4/14/14 10:26 AM
Lesson at a Glance
Lesson 3:
Factoring
Mental Mathematics (5 min)
PURPOSE
This lesson introduces factoring monic quadratic trinomials by identifying
which calculations result in which pieces of the product and by considering
how finding sums and products relates to factoring. By exploring equations
of the form x2 + ____ + 12 and finding that there is only one possible
factor pair for a given middle term, students prepare for the product/sum
search involved in factoring.
Mental Mathematics Begin each day with five minutes of Mental
Mathematics (pages T37–T48). In today’s lesson, students will see how to use both
products and sums in factoring quadratic trinomials.
Launch: The Roles of Sums and Products
(10 min)
•Students work through the first four
problems in the Student Worktext and
discuss the roles of sums and products in
factoring trinomials into binomial pairs.
Student Problem Solving and Discussion
(20 min)
•Allow time for students to work through
the Important Stuff and explore additional
problems.
•Students identify pairs of numbers with a
given product and then pairs with given
sums, and they use this information to
factor several expressions.
Reflection and Assessment (10 min)
Snapshot Check-in
Unit 10 Related Activity: Area Model
Puzzles (See page T31 and Student Worktext
page 37.)
Launch: The Roles of Sums and Products
Allow time for students to consider PROBLEMS 1–4 in the Student Worktext.
Problems 1 and 2 ask about the equation (x – 2)(x – 5) = x2 – 7x + 10 and its area model.
1
What calculation creates the 10 in x2 – 7x + 10?
-2 • ____
2
What calculation creates the -7x in x2 – 7x + 10?
-2x + ____
x
-2
x
x
-2x
-5
-5x
2
10
Then bring the group together to discuss student responses.
Next, draw this problem on the board.
?
What if . . .
What if students suggest something like this?
x + 9x + 18 = _____________________
2
x
x
___
___
x
___
x2
18
Discuss the form that the factors will take: (x + ____) (x + ____). Remind
students that not all trinomials are factorable, but for now, we will be looking
only at ones that are. Ask students to identify the calculations in the model that
result in the 18 and 9x terms, and work together to identify pairs of numbers
with a product of 18 and a sum of 9. Write the factors out, and verify that
these are the correct factors by completing the model.
As students offer suggestions for filling in the empty spaces, challenge them
to explain their reasoning.
x
___
x2
3
18
Draw students’ attention to the x term on
the outside left. This term requires that any
term in that row have a factor of x as well
(unless the other factor had a term with an x
in the denominator, but this is not common
in introductory algebra). Then ask students to
identify the term that must go at the top of the
rightmost column. Likely, someone will notice
that the 3 actually belongs there.
Lesson 3: Factoring
TTA_U10_TG_DS_r6a.indd 13
T13
4/14/14 10:26 AM
Student Problem Solving and Discussion
Allow time for students to work on the problems in the Student Worktext.
Listen for students describing two things:
• The coefficient of the x term is the sum of the two numbers in the
factors.
• The number (the constant term) at the end of the trinomial is the
product of these same two numbers.
If some students don’t complete
the Thinking Out Loud box on
page 13, spend some time
analyzing the model using Jay’s
thoughts: talk about the model,
how it works, why it works that
way, where there are factors,
products, sums, and terms, and
how they relate. Ask about this
model and others on this page.
Ask students to tell you which
models they’ve understood, and
then help them connect what they
understand to this model.
In PROBLEMS 5–13, students list all of the integer pairs that result in the product
12, then identify which pairs have various sums, and complete area models to
factor trinomials of the form x2 + bx + 12, where b varies among the sums.
Students then, in PROBLEM 14, complete a statement of reasoning about the
process before working through a similar set of problems with a product of 30.
Complete the model and finish Jay’s thought.
14
x2 + 3x + 2 = (
)(
Thinking Out Loud
)
Jay: I can tell from the equation that I need a pair of
___
x
numbers whose product is 2 and whose sum is ___.
x2
x
The numbers are ___ and ___, so the factors must
2
___
be ________ and ________.
PROBLEMS 24–29 feature Who Am I? puzzles in which students search for
numbers with a particular product and sum.
24
Who Am I?
t
u
t
Who Am I?
25
• t+ u = 10
• t+ u = 10
• tu = 21
• tu = 25
u
• t> u
Starting with an expression and
using the area model to find
the original factors is what has
been called “un-multiplying.”
Feel free to use this term as a
way to help students see the
interconnectedness between
multiplication, division, and
factoring.
In the Stuff to Make You Think, students consider problems in which
it is not helpful to factor, which arrangements of terms in area models are
helpful, and several perpendicular street diagrams that use similar sum and
product reasoning to that in the lesson. PROBLEMS 32–37 can be completed with
manipulatives like toothpicks or popsicle sticks.
Tough Stuff PROBLEMS 38 & 39 feature more difficult trinomial factoring,
such as an expression with a fractional coefficient and a non-monic trinomial.
___
38
___
39
x2
___
___
11
___
1
x2 + 7 2 x + 11 = (
2x
2x 2
___
)(
)
___
2
2x2 + 5x + 2 = (
Questions such as the following can support discussion:
»» If you know that the area model for x + 11x + 24
will look like this (show the model at right), how
can you determine what goes in the other two boxes?
2
T14
x
___
)(
)
x
___
x2
24
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Listen for students who use the model to explain why the two missing
pieces of the factors will have a product of 24 and a sum of 11. Students
should be able to describe that because the area of the lower right
rectangle is 24, the two side lengths will have a product of 24 and because
the remaining two pieces of area will each have one factor of x, the
coefficients of these two like terms will have to sum to 11.
»» How do the Mental Mathematics activities we’ve been doing in this unit
relate to factoring? Listen for students who identify the connection
between finding the pair of numbers with a given product and sum and the
thinking required for factoring.
»» How can we determine what size area model will best help us factor an
expression? Remind students that not all expressions are factorable and
factoring often does require some trial and error. Present expressions
like 5x2 + 35xy – 25x, x2 + 2x – 35, and x2 – 25. Whenever there is a
common factor among all the terms in the expression (such as 5x in the
first expression), it is helpful to use a 1 × something model (in this case,
1 × 3) to factor out the common factor; then additional factoring can take
place from there as needed. In the expressions x2 + 2x – 35 and x2 – 25,
however, students may identify that there is no common factor, so using
a 2 × 2 model allows for the x2 term and the constant term to be placed in
different rows and columns.
TTA_U10_TG_DS_r6a.indd 15
Lesson 3: Factoring
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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 T33. 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 10, students have:
•Translated area models into algebraic equations showing
multiplication and division.
•Used area models to divide algebraic expressions by a factor.
•Completed area model puzzles with various pieces of information
omitted.
•Considered the use of factoring in simplifying rational expressions.
•Used area models to factor monic quadratic trinomials.
Students have been developing the following Algebraic
Habits of Mind:
•Puzzling and Persevering—Students have worked through area
models inside out, using available clues to work out the missing
information. They have also solved Who Am I? number puzzles that
require thinking similar to factoring—finding two numbers with a
given product and sum.
•Using Tools Strategically—Students have extended their use of the
area models to support their thinking about division and factoring.
•Seeking and Using Structure—Students have connected the
structure of the information provided in or found using area models
to the structure of algebraic equations involving multiplicative
operations with polynomials.
T16
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Snapshot Check-in
Name:
Use the area model to write three equations: one using multiplication and two using division.
1
2x
x
-4
2x2
-8x
Draw an area model and use it to answer this division problem.
2
12w – 20 = ________________
4
Fill in the missing information to complete this area model and the equation that goes with it.
___
___
___
v2
2v
___
5v
10
3
5
4
v2 + ______ + 10 = __________________
List all the pairs of integers (positive or negative) whose product is -15.
Use an area model to factor. Complete the model and equation.
6
x2 – 2x – 15 = _______________________
x
x
___
x2
___
-15
©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 10: Area Model Factoring
Snapshot Check-in
Snapshot Check-in
T33
R5
©2014 by Heinemann and Education Development Center, Inc., from Transition to Algebra (Portsmouth, NH: Heinemann). This page may be reproduced for classroom use only.
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Snapshot Check-in
Name:
ANSWER KEY
Use the area model to write three equations: one using multiplication and two using division.
1
2x
x
-4
2x2
-8x
2x(x – 4) = 2x2 – 8x
2x2 – 8x
=x– 4
2x
2x2 – 8x
x – 4 = 2x
Draw an area model and use it to answer this division problem.
2
12w – 20 = ________________
3w – 5
4
4
3w
-5
12w
-20
Fill in the missing information to complete this area model and the equation that goes with it.
3
v
___
5
___
5
v
___
___
2
v2
2v
5v
10
4
v2 + ______
7v + 10 = __________________
(v + 5)(v + 2)
List all the pairs of integers (positive or negative) whose product is -15.
1 • -15
-1 • 15
3 • -5
-3 • 5
Use an area model to factor. Complete the model and equation.
6
(x – 5)(x + 3)
x2 – 2x – 15 = _______________________
x
3
___
x
-5
___
x
2
-5x
3x
-15
(Factors can
be written in
either order.)
©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 10: Area Model Factoring
Snapshot Check-in
T34 Transition to Algebra Unit 10: Area Model Factoring • TEACHING GUIDE
R6
©2014 by Heinemann and Education Development Center, Inc., from Transition to Algebra (Portsmouth, NH: Heinemann). This page may be reproduced for classroom use only.
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10
Unit
Mental Mathematics
Factors, products & sums and
percentage calculations
The two mental mathematics activities in this unit are not related to each
other but are both useful in high school mathematics. Students work with
number pairs corresponding to specific products and sums, which supports
the product-sum search in factoring quadratic trinomials. The structure of
these activities differs in format from the previous ones in that inputs and
outputs comprise multiple pieces of information.
Then students review common percentages. Students make mental
calculations, such as finding 20% off prices expressed in dollar amounts.
The percentage activities serve multiple purposes: they re-animate skills and
associations students have already developed, and they give the opportunity
for the lively, fast-paced back-and-forth that students may have been missing
in the first five activities.
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.
T37
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Mental Mathematics • Activity 9
$90
$81
10% off
PURPOSE
Students keep mental track of two processes: they find 10% of a given
number and then they subtract it from the original number. “10% off” is a
familiar sight for students and one worth understanding.
Introduce:
“When you see a sale that says ‘10% off,’ do you know what that means? It’s not
the same as ‘10% of.’” Let students respond. “Sure, you look at the price, figure
out 10% of it, and then subtract that from the original price. Today, you’ll take
10% off every price I name. So, if I say $200, what would you say? Start by first
saying softly to yourself what 10% of that is. Right, $20. Now subtract that
from $200. Right, $180. What if I said $400? You’d say . . . yes, $40, then $360.
•Do not pressure students
to stop verbalizing the value
for 10%. Encourage them,
always, to whisper intermediate
steps to themselves if they
find that helps. Competent
mathematicians will do the
same!
•The easiest way to calculate a
90% off sale is to take 10% of
the price!
You’ve got it. Let’s keep going!”
About this sequence:
Calculating 10% off is different from anything students have been asked to
mentally calculate before, so both steps consist mostly of inputs that are
deliberately easy calculations.
Step 1: Have students find 10% off each price you give.
Use multiples of 10 and 100, and encourage students to
verbalize (softly) their calculation of 10% before calling
out their final answer.
Step 2: Encourage students to keep whispering the
interim step (finding 10%) to themselves, but have them
say only the final result (10% off) out loud. Include some
values that are not multiples of 10.
Input
10%
Output (Input minus 10%)
$80
$72
$20
$18
$10
$1
$9
$4
$3.60
$10
$9
$300
$30
$270
$200
$180
$400
$360
$500
$50
$450
$450
$405
$70
$63
$20
$2
$18
$25
$22.50
$75
$67.50
$80
$8
$72
$300
$270
$90
$81
$2000
$200
$1800
$4000
$3600
$170
$153
$5
$4.50
$220
$198
$150
$135
$6.99
$6.29
$230
$207
$1.99
$1.79
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