Table of Contents D
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CHAPTER3:EXPRESSIONSANDEQUATIONSPART1(4‐5WEEKS)................................................................................3
3.0AnchorProblem:TilingaGardenBorder..............................................................................................................................................6
SECTION3.1:COMMUNICATENUMERICIDEASANDCONTEXTSUSINGMATHEMATICALEXPRESSIONSANDEQUATIONS.................8
3.1aClassActivity:MatchingNumericalExpressionstoStories........................................................................................................9
3.1bClassActivity:NumericandAlgebraicExpressions.....................................................................................................................13
3.1bHomework:MatchingandWritingExpressionsforStories.....................................................................................................16
3.1bAdditionalPractice......................................................................................................................................................................................18
3.1cClassActivity:AlgebraTileExploration.............................................................................................................................................20
3.1cHomework:AlgebraTileExploration.................................................................................................................................................23
3.1dClassActivity:MoreAlgebraTileExploration................................................................................................................................26
3.1dHomework:MoreAlgebraTileExploration.....................................................................................................................................29
3.1eClassActivity:VocabularyforSimplifyingExpressions..............................................................................................................31
3.1eHomework:SolidifyingExpressions.....................................................................................................................................................34
3.1fClassActivity:IteratingGroups..............................................................................................................................................................35
3.1fHomework:IteratingGroups...................................................................................................................................................................42
3.1gClassActivity:MoreSimplifying............................................................................................................................................................43
3.1gHomework:MoreSimplifying.................................................................................................................................................................47
3.1gAdditionalPractice:IteratingGroups................................................................................................................................................48
3.1gAdditionalPractice:Simplifying..........................................................................................................................................................49
3.1hClassActivity:ModelingContextwithAlgebraicExpressions.................................................................................................51
3.1hHomework:ModelingContextwithAlgebraicExpressions......................................................................................................53
3.1iClassActivities:Properties........................................................................................................................................................................54
3.1iHomework:Properties................................................................................................................................................................................59
3.1jClassActivity:UsingPropertiestoCompareExpressions...........................................................................................................61
3.1kClasswork:ModelingBackwardsDistribution................................................................................................................................64
3.1kHomework:ModelingBackwardsDistribution..............................................................................................................................69
3.1lSelf‐Assessment:Section3.1.....................................................................................................................................................................70
SECTION3.2SOLVEMULTI‐STEPEQUATIONS................................................................................................................................................71
3.2aClassroomActivity:ModelEquations.................................................................................................................................................72
3.2aHomework:ModelandSolveEquations............................................................................................................................................77
3.2bClassActivity:MoreModelandSolveOne‐andTwo‐StepEquations..................................................................................81
3.2bHomework:MoreModelandSolveOne‐andTwo‐StepEquations.......................................................................................84
3.2cClassActivity:ModelandSolveEquations,PracticeandExtendtoDistributiveProperty........................................87
3.2cHomework:ModelandSolveEquations,PracticeandExtendtoDistributiveProperty.............................................90
3.2dClassActivity:ErrorAnalysis..................................................................................................................................................................93
3.2dHomework:PracticeSolvingEquations(selecthomeworkproblems)...............................................................................95
3.2eHomework:SolveOne‐andTwo‐StepEquations(practicewithrationalnumbers)...................................................98
3.2eExtraPractice:EquationswithFractionsandDecimals........................................................................................................100
3.2fClassActivity:CreateEquationsforWordProblemsandSolve...........................................................................................102
3.2fHomework:CreateEquationsforWordProblemsandSolve................................................................................................104
3.2g**ClassChallenge:Multi‐StepEquations........................................................................................................................................106
3.2iSelf‐Assessment:Section3.2..................................................................................................................................................................113
SECTION3.3:SOLVEMULTI‐STEPREAL‐WORLDPROBLEMSINVOLVINGEQUATIONSANDPERCENTAGES....................................114
3.3aClassroomActivity:PercentswithModelsandEquations.....................................................................................................115
3.3aHomework:PercentswithModelsandEquations.....................................................................................................................117
3.3bClassActivity:PercentProblems........................................................................................................................................................119
3.3bHomework:PercentProblems.............................................................................................................................................................121
3.3cClassActivity:MorePracticewithPercentEquations..............................................................................................................124
3.3cHomework:MorePracticewithPercentEquations..................................................................................................................126
1
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3.3dSelf‐Assessment:Section3.3.................................................................................................................................................................127
2
CHAPTER 3: Expressions and Equations Part 1
(4-5 weeks)
UTAH CORE Standard(s): Expressions and Equations
1. Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with
rational coefficients. 7.EE.1
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2. Understand that rewriting an expression in different forms in a problem context can shed light on the
problem and how the quantities in it are related. 7.EE.2
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3. Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in
any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations
to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness
of answers using mental computation and estimation strategies. 7.EE.3
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4. Use variables to represent quantities in a real-world or mathematical problem, and construct simple
equations to solve problems by reasoning about the quantities. 7.EE.4
a. Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are
specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an
arithmetic solution, identifying the sequence of the operations used in each approach. 7.EE.4a
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CHAPTER OVERVIEW:
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The goal of chapter 3 is to facilitate students’ transition from concrete representations and manipulations of
arithmetic and algebraic thinking to abstract representations. Each section supports this transition by asking
students to model problem situations, construct arguments, look for and make sense of structure, and reason
abstractly as they explore various representations of situations. In Chapter 3, students work with fairly simple
expressions and equations to build a strong intuitive understanding of structure (for example, students should
understand the difference between 2x and x2 or why 3(2x  1) is equivalent to 6x  3.) They will continue to
practice skills manipulating algebraic expressions and equations throughout Chapters 4 and 5. In Chapter 6
students will revisit ideas in this chapter to extend to more complicated contexts and manipulate with less
reliance on concrete models.
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Section 3.1 reviews and builds on students’ skills with arithmetic from previous courses to write basic
numerical and algebraic expressions in various ways. In this section students should understand the difference
between an expression and an equation. Further, they should understand how to represent an unknown in either
an expression or equation. Students will connect manipulations with numeric expressions to manipulations with
algebraic expressions. They will then come to understand that the rules of arithmetic are naturally followed
when working with algebraic expressions and equations. Lastly, students will name the properties of arithmetic.
By the end of this section students should be proficient at simplifying expressions and justifying their work with
properties of arithmetic. Section 3.2 uses the skills developed in the previous section to solve equations.
Students will need to distribute and combine like terms to solve equation. This section will rely heavily on the
use of models to solve equations. Section 3.3 ends the chapter with applications of solving equations involving
percent including ones with percent of increase and percent of decrease. Problems in this section should be
solved using models. There will be similar exercises in Chapter 4 where students will used an algebraic
equation approach.
3
VOCABULARY: expression, equation, simplify, rational number, integer, term, like terms, constant,
variable, factor, product, coefficient, unknown
CONNECTIONS TO CONTENT:
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Prior Knowledge
Students will extend the skills they learned with manipulatives in previous grades with addition/subtraction and
multiplication/division of whole numbers/integers to algebraic expressions in a variety of ways. For example, in
elementary school students modeled 4  5 as four “jumps” of five on a number line. They should connect this
thinking to the meaning of “4x”. Students also modeled multiplication of whole numbers using arrays in earlier
grades,; in this chapter they will use that logic to multiply using variables. Additionally, in previous grades,
students explored and solidified the idea that when adding/subtracting one must have “like units.” Thus, when
adding 123 + 14, we add the “ones” with the “ones,” the “tens” with the “tens” and the “hundreds” with the
“hundreds;” or we cannot add ½ and 1/3 without a common denominator because the a unit of ½ is not the same
as a unit of 1/3. Students should extend this idea to adding variables. Hence, 2x + 3x is 5x because the unit is x,
but 3x + 2y cannot be simplified further because the units are not the same (three units of x and two units of y.)
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In 6th grade students solved one step equations. Students will use those skills to solve equations with more than
one-step in this chapter. Earlier in this course, students developed skills with rational number operations. In this
chapter, students will be using those skills to solve equations that include rational numbers.
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Future Knowledge
As students move on in this course, they will continue to use their skills in working with expressions and
equations. Those same skills will be used to solve more real-life applications that use equations and well as
solving inequalities. In future courses, students will be able to solve equations of all forms by extending
properties.
4
MATHEMATICAL PRACTICE STANDARDS (emphasized):
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Students should be able to model all expression and equations throughout this
chapter. Further, they should be able to interchange models with abstract
representations.
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Model with
mathematics
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Construct
viable
arguments and
critique the
reasoning of
others
Students will, for example, note that x + x + x + x + x is the same as 5x.
Students should extend this type of understanding to 5(x + 1) meaning five
groups of (x + 1) added together, thus simplifying to 5 x + 5. For each of the
properties of arithmetic, students should connect concrete understanding to
abstract representations.
Students should be able to explain and justify any step in simplifying or
solving an expression or equation in words and/or pictures. Further, students
should be able to evaluate the work of others to determine the accuracy of
that work and then construct a logical argument for their thinking.
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Reason
abstractly and
quantitatively
Students will make sense of expressions and equations by creating models
and connecting intuitive ideas to properties of arithmetic. Properties of
arithmetic should be understood beyond memorization of rules.
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Make sense of
problems and
persevere in
solving them
Students demonstrate precision by using correct terminology and symbols
when working with expressions and equations. Students use precision in
calculation by checking the reasonableness of their answers and making
adjustments accordingly.
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Attend to
precision
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Using models students develop an understanding of algebraic structures. For
example, in section 3.2 students should understand the structure of an
equation like 3x + 4 = x + 5 as meaning the same thing as 2x = 1 or x = ½
Look for and
when it is “reduced.” Another example, in section 3.3 students will use a
make use of
model to show that a 20% increase is the original amount plus 0.2 of the
structure
original amount or 1.2 of the original amount (though they will not write
equations until Chapter 4.)
Students demonstrate their ability to select and use the most appropriate tool
Use appropriate
(paper/pencil, manipulatives, and calculators) while solving problems.
tools
Students should recognize that the most powerful tool they possess is their
strategically
ability to reason and make sense of problems.
Students will study patterns throughout this chapter and connect them to both
Look for and
their intuitive understanding and properties of arithmetic.
express
regularity in
repeated
reasoning
5
3.0 Anchor Problem: Tiling a Garden Border
Imagine that you are putting 1-foot square tiles around the edge of a square garden. Look at the answers
below—this will help you guide the discussion. Students may use a variety of methods to write each. Have
students work in groups of four and present their answers to the class. REVISIT during 3.1i.
t.
Without counting directly, how could you figure out how many tiles go
around the garden that is 4 feet by 4 feet? Write down four ways that you
could quickly “add up” the tiles.
Various methods are reasonable.
Method 1: 4(4) + 4, 4 sides of 4 tiles, then 4 corners
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4 feet across
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Method 2: 6+6+4+4, 6 on top, 6 on bottom, 4 on the left, 4 on the right
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Method 3: 62 – 42, outer square minus inner square
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Method 4: 46 − 4, 4 sides of 6, subtract the tiles counted twice
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Invest time in talking about the first representation—many possible approaches, but all simplify to the same
answer.
Without counting directly, how could you figure out how many tiles go
around the garden that is 5 feet by 5 feet? Show how you could adapt the
methods that you used above to “add up” these tiles.
Method 1: 4(5) + 4, 4 sides of 5 tiles, then 4 corners
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5 feet across
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Method 2: 7+7+5+5, 7 on top, 7 on bottom, 5 on left, 5 on right
Method 3: 72 − 52 , outer square minus inner square
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Method 4: 4  7 − 4, 4 sides of 7, subtract the tiles counted twice
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Students will begin to be fluent in the different ways of expressing the area of the border.
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Show how you could adapt the methods that you used above to
“add up” the number of tiles for a 10 foot by 10 foot square
garden.
10 feet across
Method 1: 4(10) + 4, 4 sides of 10 tiles, then 4 corners
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Method 2: 12+12+10+10, 12 on top, 12 on bottom, 10 on left,
10 on right
Method 3: 122 – 102 , outer square minus inner square
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Method 4: 412 − 4, 4 sides of 12, subtract the tiles counted
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What if the garden were 20 feet by 20 feet? Demonstrate how
each method would work now.
20 feet across
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Method 1: 4(20) + 4, 4 sides of 20 tiles, then 4 corners
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Method 2: 22+22+20+20, 22 on top, 22 on bottom, 20 on left,
20 on right
Method 3: 222 – 202 , outer square minus inner square
Method 4: 422 − 4, 4 sides of 22, subtract the tiles counted
twice
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In the discussion, help students connect the different representations and notice that all represent the same
amount/value. When they present their solutions, have them justify the logic for their representation and/or
how it is connected to other representations. Record at least one of the solutions so that you can return to it
during later lessons.
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Make sense of the problems
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Reason Abstractly
Look for and express regularity in repeated reasoning
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Section 3.1: Communicate Numeric Ideas and Contexts Using
Mathematical Expressions and Equations
Section Overview: This section contains a brief review of numerical expressions. Students will recognize
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that a variety of expressions can represent the same situation. Models are encouraged to help students connect
properties of arithmetic in working with numeric expressions to working with algebraic expressions. These
models, particularly algebra tiles, aid students in the transition to the abstract thinking and representation.
Students extend knowledge of mathematical properties (commutative property, associative property, etc.) from
purely numerical problems to expressions and equations. The distributive property is emphasized and factoring,
“backwards distribution,” is introduced. Work on naming and formally defining properties is at the end of the
section. Through the section students should be encouraged to explain their logic and critique the logic of
others.
Concepts and Skills to be Mastered (from standards)
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By the end of this section, students should be able to:
1. Use the Distributive Property to expand and factor linear expressions with rational numbers
2. Combine like terms with rational coefficients
3. Recognize and explain the meaning of a given expression and its component parts.
4. Recognize that different forms of an expression may reveal different attributes of the context.
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3.1a Class Activity: Matching Numerical Expressions to Stories
Four students, Aaron, Brianna, Chip, and Dinah, wrote a numerical expression for each story problem. Look at
each student’s expression and determine whether or not it is appropriate for the given story problem. Explain
why the expression “works” or “doesn’t work.” Teachers may want to review order of operations before
beginning. See 3.1b for where this lesson is leading.
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1. Josh made five 3-pointers and four 2-pointers at his basketball game. How many points did he score?
Expression
Evaluate Does it work?
Why or Why Not?
a. 3 + 3 + 3 + 3 + 3 + 2 + 2 +2 + 2
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Yes
Each basket is added together, giving the
total.
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No
5+3 is not an accurate interpretation of
“five 3-pointers”
c. (5 + 3)(4 + 2)
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No
5+3 is not an accurate interpretation of
“five 3-pointers”
d. 5(3) + 4(2)
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Yes
It is the same as A. 5(3) means 5 groups of
3 as above. 4(2) means 4 groups of 2.
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b. 5 + 3 + 4 + 2
$4.40
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b. 2(1.75) + 3 (0.30)
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2. I bought two apples for $0.30 each and three pounds of cherries for $1.75 a pound. How much did I spend?
Expression
Evaluate Does it work?
Why or Why Not?
a. 2(0.30) + 3 (1.75)
$5.85
Yes
The algebra describes the situation—2
groups of (0.30) etc
No
This describes two pounds of cherries and
3 apples
$5.85
Yes
The algebra describes the situation as in A
but now using repeated addition
d. (2+ 3)(0.30 + 1.75)
$10.25
No
We cannot combine groups and amounts
and then multiply.
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c. 0.30 + 0.30 + 1.75 + 1.75 + 1.75
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3. I bought two apples for $0.30 each and three oranges for $0.30 each. How much money did I spend?
Expression
Evaluate Does it work?
Why or Why Not?
a. (0.30 + 0.30 + 0.30) + (0.30 + 0.30)
$1.50
Yes
Added each purchase individually.
$1.50
Yes
This uses multiplication instead of
repeated addition
c. 5(0.30)
$1.50
Yes
This expression works because the
cost is always 0.30. **compare to 2D
d. 0.60 + 0.90
$1.50
Yes
This sums the total cost of apples and
the total cost of oranges
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b. 3(0.30) + 2(0.30)
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4. Aunt Nancy gave her favorite niece 3 dollars, 3 dimes, and 3 pennies. How much money did her niece
receive?
Evaluate Does it work?
Why or Why Not?
$3.33
Yes
This takes three of each unit of money,
then adds them together.
Yes
c. 3 + 1.00 + 0.10 + 0.01
$4.11
No
d. 3.00 + 0.30 + 0.03
$3.33
Yes
Since all the money came in groups of
three, this sums one set, then multiplies.
**compare to 2D and 3C.
This took three as its own value, rather
than operating on the other values.
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$3.33
This is like A after multiplication.
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b. 3(1.00 + 0.10 + 0.01)
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Expression
a. 3(1.00) + 3(0.10) + 3(0.01)
c. 2 (1 + 0.25 + 0.10)
$2.70
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$1.35
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b. 1 + 0.25 + 0.10
Does it work?
Why or Why Not?
Yes
This adds each piece of money in
succession.
d. 2(1) + 0.25 + 0.10
No
This is the amount given to one
nephew, not the two together.
Yes
Each nephew received the same
amount of money, thus two times the
amount one received will work.
This does not account for one of the
quarters and one of the dimes.
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Evaluate
$2.70
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Expression
a. 1 + 1 + 0.25 + 0.25 + 0.10 + 0.10
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5. Aunt Nancy gave each of her two nephews the same amount of money. Each nephew received one dollar,
one quarter, and one dime. How much did the two nephews receive altogether?
$2.35
No
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6. Aunt Nancy gave her favorite niece two dollars, 1 dime, and 3 pennies. How much money did her niece receive?
2(1) + 1(0.10) + 3(0.01)
$2.13
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7. Uncle Aaron gave 8 dimes, 2 nickels, and 20 pennies to his nephew. How much money did he give away?
8(0.10) + 2(0.05) + 20(0.01)
$1.10
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8. I bought 2 toy cars for $1 each and 3 toy trucks for $1.50 each. How much did I spend?
2(1) + 3(1.50)
$6.50
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9. The football team scored 1 touchdown, 3 field goals, and no extra points. How many points did they score in all?
Hint: a touchdown is worth 6 points, a field goal worth 3, and an extra point worth one.
1(6) + 3(3)
15 points
10. I had $12. Then I spent $2 a day for 5 days in a row. How much money do I have now?
12 – 5(2)
$2
11. I earned $6. Then I bought 4 candy bars for $0.75 each. How much money do I have left?
6 – 4(0.75)
$3
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3.1a Homework: Matching Numerical Expressions to Stories
Four students, Aaron, Brianna, Chip, and Dinah, wrote a numerical expression for each story problem. Look at
each student’s expression and determine whether or not it is appropriate for the given story problem. Explain
why the expression “works” or “doesn’t work.”
1. I bought two toy cars for $5 each and three toy trucks for $7 each. How much did I spend?
Expression
Evaluate Does it work?
Why or Why Not?
$31
Yes
This expression describes the situation
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a. 2(5) + 3(7)
$41
No
This expression pairs the wrong
quantities together.
c. (2 + 3)(5 + 7)
$60
No
Cannot combine quantities and multiply
by value if the values are different.
d. (2 + 3) + (5 + 7)
$17
No
This expression does not compensate
for repeated purchases at a price.
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b. 2(3) + 5(7)
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2. The football team scored three touchdowns, two field goals, and two extra points. How many points did
they score in all? (Hint: a touchdown is 6 points, a field goal is 3 points, and an extra point is just 1 point)
Evaluate Does it work?
Why or Why Not?
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Yes
This expression multiplies the correct
points the correct number of times.
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Expression
a. 3(6) + 2(3) + 2(1)
b. 6 + 6 + 6 + 3 + 3 + 1 + 1
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c. (6 + 6 + 6) + (3 + 3) + (1 + 1)
Yes
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Yes
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Yes
This is the same as above only using
repeated addition instead of
multiplication.
This is the same as above only
grouping the sets of addition
This is the same as A or C after the first
step of simplification.
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d. 18 + 6 + 2
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3. I earned $6. Then I bought 4 candy bars for $0.50 each. How much money do I have left?
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Expression
a. 6 – 0.50 – 0.50 – 0.50 – 0.50
Evaluate Does it work?
Why or Why Not?
$4
Yes
It gives the total, then subtracts each
expenditure.
$4
Yes
It is like A only multiplying instead of
repeated subtraction.
c. 6 – (0.50 – 0.50 – 0.50 – 0.50)
$7
No
We need to subtract the sum of 0.50
four times.
d. 6 – (0.50 + 0.50 + 0.50 + 0.50)
$4
Yes
This subtracts the sum of 0.50 four
times.
D
b. 6 – 4(0.50)
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4. I earned $5. Then I spent $1 a day for 2 days in a row. How much money do I have now?
Evaluate
$5
Does it work?
No
Why or Why Not?
This shows spending a dollar and then getting a dollar.
b. 5 – 1 – 1
$3
Yes
This expression accurately describes the situation
c. 5 – (1 – 1)
$5
No
We need to subtract the sum of the two dollars.
d. 5 – (1 + 1)
$3
Yes
We are subtracting the sum of the two dollars.
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Expression
a. 5 − 1 + 1
Does it work?
Why or Why Not?
Yes
This accounts for each of the coins.
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Evaluate
$0.37
$0.37
Yes
c. 0.20 + 0.15 + 0.02
$0.37
Yes
d. 2 + 0.10 + 3 + 0.05 + 2 + 0.01
$7.16
This grouped the dimes and pennies,
which is accurate.
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b. 2(0.10 + 0.01) + 3(0.05)
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Expression
a. 2(0.10) + 3(0.05) + 2(0.01)
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5. Uncle Aaron gave 2 dimes, 3 nickels, and 2 pennies to his nephew. How much money did he give away?
No
This is like A after multiplication
This misrepresents the idea of 2 dimes
as 2 + 0.10
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Write an expression of your own for each problem. Then evaluate the expression to solve the problem.
There are various accurate expressions for these problems. They should each result in the value given.
6. Josh made ten 3-pointers and a 2-pointer at his basketball game. How many points did he score?
10(3) + 1(2)
32 points
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7. I bought three apples for $0.25 each and 3 pounds of cherries for $1.75 a pound. How much money did I
spend?
3(0.25) + 3(1.75)
$6.00
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8. I bought five apples for $0.30 each and 5 oranges for $0.30 each. How much money did I spend?
5(0.30) + 5(0.30)
$3.00
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9. I bought two t-shirts at $12 each and 3 sweaters for $20 each. How much did I spend altogether?
2(12) + 3(20)
$84
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3.1b Class Activity: Numeric and Algebraic Expressions
Read each story problem. Determine if you think the expression is correct. Evaluating the expression for the
given value. Explain why the expression did or didn’t work for the given problem. Through this work, discuss
how ideas are related to the previous day’s work. Note changes in order and grouping and issues with “-“.
1. Ryan bought 3 CDs for x dollars each and a DVD for $15. How much money did he spend?
Evaluate
x=7
$25
Did it work?
Why or why not?
No
“3 + x” does not translate “3 CDs for x
dollars”.
b. 15x + 3
---
$108
No
“15x” does not represent
“a DVD for $15”
c. 15 + x + x + x
---
$36
Yes
There is 15 for the DVD, then three CDs,
each represented with an x.
d. 3x + 15
---
$36
Yes
ep
rin
R
N
ot
D
o
This uses multiplication to make
expression C more concise.
O
nl
y.
a. 3 + x + 15
t.
Correct
expression?
---
Expression
Correct
expression?
---
Evaluate
y=6
3
12 – 3 – y
b.
12 – (3 – y)
---
c.
12 – (3 + y)
---
d.
9–y
---
Did it work?
Why or why not?
Yes
This accurately represents losing 3 and
then y.
15
No
I lost the sum of 3 and y.
3
Yes
3
Yes
This expression groups all the jellybeans
that were eaten and subtracts them from
the original amount.
Yes, this expression begins with the
amount I had after Sam ate Jellybeans.
D
ra
f
t:
fo
rI
ns
tru
ct
a.
io
na
lP
ur
p
Expression
os
es
2. I started with 12 jellybeans. Sam ate 3 jellybeans and then Cyle ate y jellybeans. How many jellybeans
were left?
13
3. Kim bought a binder for $5, colored pencils for $2, and 4 notebooks for n dollars each. How much did she
spend?
Do you think
it will work?
---
Evaluate
(use n = 3)
$14
Did it
work?
No
b. 4n + 5 + 2
---
$19
Yes
This sums 4 times the cost of a notebook
with the other two costs.
c. 7 + 4n
---
$19
Yes
This expression has already summed the
other two costs.
d. n + n + n + n + 7
---
$19
Yes
This expression uses repeated addition for
the notebooks.
Expression
“4 + n” does not accurately represent
“4 notebooks for n dollars each”.
N
ot
R
ep
rin
t.
a. 5 + 2 + 4 + n
Why or why not?
y.
D
o
For each context below, draw a model for the situation, label all parts and then write an expression that answers
the question. The first exercise is done for you.
io
na
lP
ur
p
os
es
O
nl
4. Jill bought 12 apples.
a. Jan bought x more apples than Jill. How many apples did Jan buy?
Jan bought 12 + x apples. Help students connect the model with the expression. Ask students if they
could draw a model of the context differently.
Jill’s 12 apples
Jill’s 12 apples
Jan bought x more
x
ns
tru
ct
Jill bought 12 apples, Jan bought x more
D
ra
f
t:
fo
rI
5. Josh won 12 tickets. Evan won p tickets less than Josh. How many tickets did Evan win?
12 – p
14
N
ot
R
ep
rin
t.
6. Tim is 3 years younger than his brother. If his brother is y years old, how old is Tim?
y–3
io
na
lP
ur
p
os
es
O
nl
y.
D
o
7. I washed w windows less than Carol, who washed 8 windows. If I get paid $2 for each window I wash, how
much did I earn?
$2(8 − w)
D
ra
f
t:
fo
rI
ns
tru
ct
8. Jan bought a more apples than Jill. Jill bought 4 apples. Each apple costs $0.10. How much money did Jan
spend on apples?
$0.10(4 + a)
15
3.1b Homework: Matching and Writing Expressions for Stories
Read each story problem. Determine which expressions will work for the story problem given. Try evaluating
the expression for a given value. Explain why the expression did or didn’t work for the given problem.
1. Bob bought 5 books for x dollars each and a DVD for $12. How much money did he spend?
Do you think
it will work?
---
Evaluate
(use x = 5)
$22
Did it
work?
No
b. 5(x)12
---
$300
No
The DVD is in addition to the books.
c. x + x + x + x + x + 12
---
$37
Yes
This expression uses repeated addition to
account for the cost of the books.
d. 5x + 12
---
$37
Yes
This expression uses multiplication to
account for the cost of the books.
t.
“5 + x” does not accurately represent
“5 books for x dollars each.”
O
nl
y.
D
o
N
ot
R
a. 5 + x + 12
Why or why not?
ep
rin
Expression
2. Jim won 30 tickets. Evan won y tickets less than Jim did. How many tickets did Evan win?
b. y  30
c. y + 30
Why or why not?
This is an accurate representation of the
expression
---
24 tickets
No
Jim won more tickets than Evan, thus we
must subtract from Jim’s amount.
---
36 tickets
No
Evan won less tickets than Jim, thus we
need to subtract.
---
5 tickets
No
The difference between their amounts is
absolute; we must subtract.
rI
ns
tru
ct
d. 30 ÷ y
Did it
work?
Yes
es
io
na
lP
ur
p
a. 30 – y
Evaluate
(use y = 6)
24 tickets
os
Do you think
it will work?
---
Expression
fo
Draw a model and then write an expression for each problem.
D
ra
f
t:
3. I did 4 more problems than Minnie. If I did p problems, how many did Minnie do?
p–4
4. I bought x pairs of shoes for $25 each and 2 pairs of socks for $3 each. How much did I spend?
$25x + 2($3)
16
ep
rin
N
ot
R
6. Paul bought s sodas for $1.25 each and chips for $1.75. How much did he spend?
s($1.25) + $1.75
t.
5. I bought m gallons of milk for $2.59 each and a carton of eggs for $1.24. How much did I spend?
m($2.59) + $1.24
D
ra
f
t:
fo
rI
ns
tru
ct
io
na
lP
ur
p
os
es
O
nl
y.
D
o
7. Bob and Fred went to the basketball game. Each bought a drink for d dollars and nachos for n dollars. How
much did they spend on two drinks and two orders of nachos?
2d + 2n
17
3.1b Additional Practice
Draw a model and write an expression for each problem. :
ep
rin
t.
1. Marina has $12 more than Brandon. Represent how much money Marina has.
b is the amount of money Brandon has. Then Marina has b + 12 dollars.
D
o
N
ot
R
2. Conner is three times as old as Jackson. Represent Conner’s age.
j is Jackson’s age. Then Conner is 3j or 3j years old.
io
na
lP
ur
p
os
es
O
nl
y.
3. Diane earned $23 less than Chris. Represent how much Diane earned.
c is amount Chris earned. Diane earned $23 less than Chris or c – 23.
ct
4. Juan worked 8 hours for a certain amount of money per hour. Represent how much Juan earned.
x is amount of money earned per hour. Juan worked at this rate for 8 hours so 8x.
ra
f
t:
fo
rI
ns
tru
5. Martin spent 2/3 of the money in his savings account on a new car. Represent the amount of money Martin
spent on a new car.
x is amount in Martin’s account. He spent two-thirds of this amount: (2/3)  x or (2/3)x.
D
6. Brianne had $47 dollars. She spent $15 on a new necklace and some money on a bracelet. Represent the
amount of money Brianne has now.
y is amount of money spent on bracelet. Brianne now has $47  $15  y or $32  y.
18
7. For 5 days Lydia studied math for a certain amount of time and read for 15 minutes each day. Represent the
total amount of time Lydia studied and read over the 5 day period.
m is amount of time she studied math each day. This was repeated for 5 days so 5( m + 15 ).
y.
D
o
N
ot
9. Nalini has $26 dollars less than Hugo. Represent the amount of money Nalini has.
h is the amount that Hugo has. Nalini has $26 less than him or h – 26 dollars.
R
ep
rin
t.
8. Carlos spent $8 on lunch, some money on a drink and $4 on ice cream. Represent how much money Carlos
spent.
d is amount Carlos spent on a drink. He spent $8 + d + $4 or $12 + d.
os
es
O
nl
10. Bruno ran four times as far as Milo. Represent the distance Bruno ran.
m is the distance Milo ran. Bruno ran a distance 4 times as long: 4m.
ct
io
na
lP
ur
p
11. Christina earned $420. She spent some of her earnings on her phone bill and spent $100 on new clothes.
Represent the amount of money Christina now has.
p is the amount spent on the phone bill. $420 – p – $100 or $320 – p.
fo
rI
ns
tru
12. Camille has 4 bags of candy. Each bag has 3 snicker bars and some hard candy. Represent the amount of
candy Camille has.
h is the amount of hard candy in each bag. Camille has 4(3 + h) pieces of candy.
D
ra
f
t:
13. Heather spent ¼ of the money in her savings account on a new cell phone. Represent the amount of money
Heather spent on the new cell phone.
a is the amount in her account. She spent ¼ of this or (¼)  a or (1/4)a or a/4 on the cell phone. Discuss
these various representations with the class.
14. Miguel is 8 years older than Cristo. Represent Miguel’s age.
c is Cristo’s age. Miguel is 8 years older or c + 8.
19
3.1c Class Activity: Algebra Tile Exploration
In using Algebra tiles, every variable is represented by a rectangle, positive or negative and every integer is
represented by a square, positive or negative.
Key for Tiles:
x
=x
x
= –x
=1
Model,
Structure
1
1
1
1
1
1
1
x
1
1
x
1
1
io
na
lP
ur
p
os
es
x+4
O
nl
x
-1
5.
x
x
x
x
tru
rI
fo
x
x
x
1
1
1
x
1
ra
f
D
2x
1
x
1
20
x
x
1
x
1
x + (2) or –x – 2
x
x – 2 or x + (2) talk about these
two different ways to write the
expression
6.
x
x
x
1
t:
x
ct
x
ns
4.
D
o
1
y.
1
N
ot
Write an expression for what you see and then write the expression in simplest form.
1.
2.
3.
R
1 = –1
ep
rin
t.
1
Algebra tiles can be easily made with card stock. It is best to
start with manipulatives and then move to drawing only. Have
students draw all their models. Attention will have to be paid to:
a – (–b) = a + b and a – b = a + (–b). For example, #4 can be
expressed as –4x – 1 or –4x + –1; take time to discuss this. For
problems 10 through 18 have students create the model in more
than one way (e.g. 2x+3 can be modeled with 5x + –3x +6 + –3).
x + (1) or x – 1
x
Use the algebra tile key above to model each expression on your desk. Sketch a picture in the space below.
8.
–3 x + –2
9.
4x–1
R
ep
rin
t.
7.
x+6
D
o
N
ot
Is there more than one way to model each of the expressions above? Justify. Discuss other possible
representations (zero pairs in the representations.)
O
nl
y.
Draw a model for each of the expressions below using the key from previous page. Simplify if you can.
11.
12.
10.
3x +4 + (–2) 3x + 2
2x + x – 2 + 3 3x +1
2x + 1 + x
es
x
io
na
lP
ur
p
os
x
1
x
tru
ct
3x + 1
13.
–3x + 1 + –x
3
D
ra
f
t:
fo
rI
ns
4x + 1
14.
2x + –3 + –2x
21
15.
–x + 3 + 4x
3x + 3
16.
–2x + 4 + x  7
(3)
17.
4x – 3 + 2 – 2x
x – 3 or –x +
18.
–4x – 1 + 3x + 2 – x
2x – 1
4x
21.
–x – x – x – x 4x
y.
20.
x – x – x – x -2x
Look at 20 and 21 together
tru
ct
io
na
lP
ur
p
os
es
O
nl
19.
x+x+x+x
D
o
N
ot
R
ep
rin
t.
2x + 1
23.
4 – 2 – 4x – 2
-4x
D
ra
f
t:
fo
rI
ns
22.
3 – 2x + x  5 –x – 2 or –x + (–2)
22
24.
2x – x – 3 + 5 x + 2
3.1c Homework: Algebra Tile Exploration
Use the key below to interpret or draw the algebraic expressions in your homework.
Algebra tiles can be easily made with card stock. It is best to
start with manipulatives and then move to drawing only. Have
students draw all their models throughout this chapter.
Attention will have to be paid to: a – (–b) = a + b and a – b =
a + (–b). For example, #4 can be expressed as –4x – 1 or –4x
+ –1; take time to discuss this. For problems 10 through 18
have students create the model in more than one way (e.g.
2x+3 can be modeled with 5x + –3x +6 + –3).
=x
1 = –1
x
= x
Write an expression for each model below.
1.
2.
x
x
x
ct
tru
ns
x
rI
fo
x
x
x
x
1
1
1
1
1
1
x
x
x
x
x
x
23
1
1
6.
x
x
1
x – 2 or
x + (–2)
x
x
t:
x
x
1
1
x
N
ot
1
x
D
o
1
x
3x – 3 or
3x + ( 3)
ra
f
D
x
x
-2
5.
x
1
1
x
x
1
1
x
x – 1 or x + (–1)
4.
1
x
y.
1
1
O
nl
x
1
1
es
1
x
os
x
1
3.
io
na
lP
ur
p
1
ep
rin
x
=1
R
1
t.
Key for Tiles:
1
–2x + 3
1
12.
–2x + 4 – 3
D
o
11.
2x + 1 + 3 – 5
tru
ct
io
na
lP
ur
p
os
es
O
nl
y.
10.
3x + 2 – 2x
N
ot
R
ep
rin
t.
Model each integer or expression on your desk. Sketch a model in the space below.
7.
8.
9.
2x + 4
x–5
2x – 3 + 5
14.
5x – 3 – 4 + x
15.
–3x + 1 + 2x – 3
D
ra
f
t:
fo
rI
ns
13.
–2x + 3 + 5x – 2
24
17.
–x – 3 + 2x – 2
18.
4x – 3 – 7x + 4
D
ra
f
t:
fo
rI
ns
tru
ct
io
na
lP
ur
p
os
es
O
nl
y.
D
o
N
ot
R
ep
rin
t.
16.
x + 4 + –3x – 7
25
3.1d Class Activity: More Algebra Tile Exploration
Miguel saw the following two expressions:
17 + 4 + 3 + 16
43 – 8 – 3 + 28
He immediately knew the sum of the first group is 40 and the sum of the second set is 50. How do you think he
quickly simplified the expressions in his head?
R
ep
rin
t.
In this lesson you will focus on commuting. You are not teaching the property explicitly, rather you should
highlight that you are changing the order of addition to simplify. Students will have done this in previous
grades. In this section you are adding the idea that a – b = a + (–b) allows us to change order when dealing with
“ – “ . Further, you are extending the notion to algebraic expressions.
N
ot
Reason Abstractly
y.
D
o
Structure
=x
x
= –x
io
na
lP
ur
p
1 = –1
x
es
=1
os
1
O
nl
Key for Tiles:
For 1-16 model each expression using Algebra Tiles. Then simplify each expression.
10x + 7
2. 3 + 2x + x
3 + 3x
ra
f
t:
fo
rI
ns
tru
ct
1. 8x + 2x + 7
D
3. 5x – 9x
–4x
4. –6 + 4x + 9 – 2x
26
3 + 2x
–2x + 8
6. 9x – 12 + 12
7. 1 + x + 5x – 2
6x − 1
8. –4x – 5x
9x
N
ot
R
ep
rin
t.
5. –3 + 3x + 11 – 5x
io
na
lP
ur
p
os
es
O
nl
y.
D
o
–9x
fo
rI
ns
tru
ct
Your friend is struggling to understand what it means when the directions say “simplify the expression.” What
can you tell your friend to help him? Answers will vary. Discuss “simplify” v “evaluate” v “solve” and
“expression” v “equation”. Also why we simplify—when does it help and when is it easier to not simplify?
D
ra
f
t:
Your friend is also having trouble with expressions like problems #5 and #8. He’s unsure what to do about the
“ – “. What might you say to help him? Discuss a – b = a + (–b); changing all subtract to “add the opposite.”
27
For 9 – 16 use the key below.
=x
= –1
x
= –x
x+x+y+x+2
y
=y
y
10.
3x + y + 2
= y
2x + y + x + 3 + 3y + 2
3x + 4y + 5
t.
9.
x
ep
rin
1
=1
O
nl
y.
D
o
N
ot
R
1
12. x + y – x – y + 2
es
11. 3x + –x + y + –y + 2
2
io
na
lP
ur
p
os
2x + 2
14. –2 + 3x – 4 + 2x – y + 2y
tru
ct
13. –2x + 2y – y – y – x
5x + y – 6
D
ra
f
t:
fo
rI
ns
–3x
15. 5x – 2y + 4 – 3x + y + x – 2
16. –5 + x – y – 2y + 3x – 7
4x – 3y –12
3x – y + 2
28
3.1d Homework: More Algebra Tile Exploration
Key for Tiles:
1
x
=x
x
= –x
=1
ep
rin
t.
1 = –1
For 1-16 model each expression using Algebra Tiles. Then simplify each expression, combining like terms.
.
8x + 5
1 + 4x
R
2. 1 + 3x + x
O
nl
y.
D
o
N
ot
1. 5x + 3x + 5
4. 7 – 2x – 9 + 4x
es
–2x
–2 + 2x
ct
io
na
lP
ur
p
os
3. 3x – 5x
6. 5 – 4x + 5x
–2x – 5 or –2x + (–5)
x+5
D
ra
f
t:
fo
rI
ns
tru
5. –4x – 5 + 2x
7. –4x – 5 + x + 7x
4x − 5 or 4x + (–5)
8. x – 6x
29
–5x
For 9 – 16 use the key below.
9.
x
=x
y
= –1
x
= –x
=y
y
2x + x + 2y + x + 1
= y
10. x + 3y + x + 2 + y + 1
R
2x + 4y + 3
t.
1
=1
ep
rin
1
O
nl
y.
D
o
N
ot
4x + 2y + 1
12. x + 3y – 1x + 2y + 1
es
11. x + –2x + 3y + y + 3
5y + 1
io
na
lP
ur
p
os
–x + 4y + 3
14. 5 + x – 4 + x – 2y + y
tru
ct
13. –2x + y – 3y + y + x
ns
2x – y + 1
D
ra
f
t:
fo
rI
–x – y or –x + (-y)
15. 5x – y + 2 – 4x + 2y + x + 2
16. –2 + 2x – 2y + x – 3
3x – 2y – 5 or 3x + (–2y) + (–5)
2x + y + 4
30
3.1e Class Activity: Vocabulary for Simplifying Expressions
In groups of 2 or 3 students, consider the following expressions: a) 2x + 5 + 3y, b) 2x + 5 + 3x, and
c) 2x + 5x + 3x. How are these expressions similar? How are they different?
Students will note that a) cannot be simplified, b) can be simplified to two terms and c) can be simplified to one term. They may also
note that the three are algebraic expressions (none are strictly numeric.) Have them explain why some of the expressions can be
simplified but others not. Help student note that terms can be combined if they are “the same.” Note, too, that they likely do not have
the vocabulary to attend to precision when talking about the expression. Introduce the need for terminology.
R
ep
rin
t.
Parts of an Algebraic Expression: Use the diagrams to create definitions for the following vocabulary words.
Be prepared to discuss your definitions with the class.
Terms
Constants
N
ot
x42y35y
x42y35y
The constants are –4 and 3. (Recall that subtracting is
like adding a negative number.)
y.
D
o
There are five terms in this expression.
The terms are x, –4, 2y, 3, and –5y
O
nl
Coefficients
os
es
x42y35y
Like Terms
x42y35y
io
na
lP
ur
p
The coefficients are 1, 2, and –5
2y and –5y are like terms. –4 and 3 are also like terms.
Students will create various acceptable definitions. Here are some referenced definitions, however, it is
important to attend to precision.
Terms: a part of an algebraic expression, either a product of numbers and variable(s) or simply a number
ct
Constant: a fixed value, not a variable
tru
Coefficient: a factor of the term
rI
ns
Like Terms: terms with the same variable(s)
D
ra
f
t:
fo
2. Identify the terms, constants, coefficients, and like terms in each algebraic expression.
Expression
Terms
Constants
Coefficients
Like Terms
4x, –x, 2y, –3
–3
4, –1, 2
4x, –x
3z, 2z, 4z, –1
–1
3, 2, 4
3z, 2z, 4z
2, 3b, –5a, –b
2
3, –5, –1
3b, –b
a, b, –c, d
none
1, 1, –1, 1
None
31
3. Simplifying Algebraic Expressions. Use the vocabulary words “constant,” “coefficient,” and “like terms”
to explain in writing how to simplify each algebraic expression. Exercises 9-28 are review.
ep
rin
t.
3x + 4x
Various correct answers. i.e: "3x and 4x are like terms. The coefficient “3” for the first term tells us there are 3 x
terms, the coefficient “4” for the second term tells us there are 4 x terms. Because they are like terms we can
combine them to say there are 7 x terms or 7x.
8n + 4 + 4n
N
ot
R
6x + 4 – 5x + 7
10. 5y + x + y
7x  y
O
nl
6y + x
12. 10b + 2 – 2b
es
11. 2y + 8x + 5y – 1
8b + 2
io
na
lP
ur
p
os
7y + 8x – 1
13. 8y + x – 5x – y
15. –2x – 6 + 3y + 2x – 3y
14. 9x + 2 – 2
9x
16. 6m + 2n + 10m
16m + 2n
18. 2a – 3 + 5a + 2
fo
rI
17. 7b – 5 + 2b – 3
ns
tru
–6
ct
–4x + 7y
7a – 1
ra
f
t:
9b – 8
y.
9. 2x – y + 5x
D
o
Simplify each expression.
D
19. 8x + 5 – 7y + 2x
20. 4y + 3 – 5y – 7
10x – 7y + 5
21. 6x + 4 – 7x
–y – 4
22. 2x + 3y – 3x – 9y + 2
–x + 4
–x – 6y + 2
32
24. m – 5 + 2 – 3n
23. –2b + a + 3b
m – 3 – 3n
b+a
4 + 2r + q
4h + 3k – 3
27. 5t – 3 – t + 2
28. c + 2d – 10c + 4
–9c + 2d + 4
D
ra
f
t:
fo
rI
ns
tru
ct
io
na
lP
ur
p
os
es
O
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y.
D
o
N
ot
R
4t – 1
t.
26. 5h – 3 + 2k – h + k
ep
rin
25. 4 + 2r + q
33
3.1e Homework: Solidifying Expressions
b) 2x – 6
3. _ j___ 3x + 5x
c) –x + y
4. _i___ 16a + a
d) 9x – 4y
5. _f___ 2x – 2y + y
e) –2x
6. _b___ 2x – 2 – 4
f) 2x – y
7. _h___ x – y + 2x
g) x + y
8. _c___ –y + 2y – x
h) 3x – y
9. _g___ 5x + 4y – 3x – x + 3y – 6y
i) 17a
10. _d___ 4x + 3y + 5x – 7y
j) 8x
D
o
y.
O
nl
Simplify each expression by combining like terms.
11.
4b
es
21.
41x
13.
33y
14.
10a – 2b
15.
12b + 2
23.
z + 2
24.
4y − 20
25.
10x + 1
–21.4x – 3.4y + 5.6
27.
ra
f
or
–2u + 11
28.
D
19.
6m − 18
26.
t:
fo
rI
ns
16.
18.
22.
tru
ct
io
na
lP
ur
p
os
12.
17.
ep
rin
2. _a___ 4a – 12a
R
a) –8a
N
ot
1. _e___ 3x – 5x
t.
Matching: Write the letter of the equivalent expression on the line.
-22.4y + 92.54z + 26.3
29.
2.5x + 1.75y + 9.8
20.
4b  1
30.
–2w + 24
5.4
34
3.1f Class Activity: Iterating Groups
N
ot
R
ep
rin
t.
Review:
Show different ways to expand 5(10).
Have students come up with as many ways as possible to expand 5(10). Record student responses, have then
justify that their expression is equivalent to 5(10). Possible expansions:
10+10+10+10+10
5(0+10), 5(1+9), 5(2+8),…
5(0)+5(10), 5(1)+5(9), 5(2)+5(8),…
0+50, 5+45, 10+40,…
Students explored the distributive property in 6th grade, so these are not new concepts.
ct
io
na
lP
ur
p
os
es
O
nl
y.
D
o
Draw different array and number line models to show 5(10) is 50; use your answers above to come up with
different representations. Students should have done a similar exploration in elementary school as early as 3rd
grade. Two ideas should be highlighted:
1) Area representation and its relationship to the distributive property: Help students note that all models for 50
are “rectangles” with areas of 50—some are single rectangles (as in 1x50 or 5x10) while others are the sum of
rectangles (as in 5(3+7) is a larger rectangle of two smaller rectangles of area 15 and 35.) Sample models are
shown below—student pages do not include the illustrations below (an example of 1x50 did not fit on the page.)
Look specifically at all the rectangles when one side is 5. For all of those, the other side is always 10, even if the
10 is split into two parts. Later in this section, time will be spent on the distinction between 5(2) and 52, it would
be useful to note the representation of 5(5+5) below.
2) Iteration and its relationship to the distributive property: For example, one can see two iterations of twentyfive for 2 x 25 or five iterations of ten for 5 x 10. Another way to see the iterations is on a number line model
(the last model example shown.)
25
50
D
ra
f
t:
fo
rI
ns
tru
2
10
5
50
35
7
15
35
25
25
D
o
5
io
na
lP
ur
p
os
es
O
nl
y.
5
5
N
ot
R
ep
rin
t.
5
3
D
ra
f
t:
fo
rI
ns
tru
ct
5(3+7) = 15 + 35
If no students draw this model, suggest it to students and then ask them if they can reorder the “jumps.” i.e. help
them see they can pair a jump of 3 and 7 to make jumps of 10.
36
1
1
=1
x
=x
= –1
x
= –x
y
=y
y
= y
t.
Use the key above for the following:
ep
rin
Can you write 2x + 2y in a different way? How?
x+x+y+y
1. Model: 2x + 2y
N
ot
R
2(x + y)
D
o
Student should note a) there are TWO groups of one x
and one y , i.e. two groups of (x + y). b) They are
reordering terms.
io
na
lP
ur
p
os
es
O
nl
y.
Also, have students look carefully at the rectangle they
create with each. Note that the rectangle for 2(x + y)
is the most “square.” This idea will be tied to the
greatest common factor later in this section.
D
ra
f
t:
fo
rI
ns
tru
ct
2. Suppose for the expression 2x + 2y that y = 5. In the space below, create a new model for the expression and
write the expression in different ways. In prior grades, students worked with expressions like 5(3 + 7) and
learned that this could be written as either 5(10) or 5(3) + 5(7). This is the first time they will be extending the
concept to algebraic expressions—this is often a very difficult transition for students. Help students see that it
doesn’t matter whether they are working with constants or unknowns, everything “works” the same. Help them
see that numeric expressions like 5(3 + 7) can be written as EITHER 5(10) or 5(3) + 5(7), but an expression like
5 (x + 7) can only also be written as 5x + 35—distributing is the only way to eliminate the grouping. You are
developing intuitive understanding in this discussion. Properties will be formalized later in this section.
37
Can you write 4x + 12 in different ways? How?
3. Model: 4x + 12
x+x+x+x+1+1+1+1+1+1+1+1+1+1+
1+1
2x + 2x + 2 + 2 + 2 + 2 + 2 + 2
2(2x + 6)
ep
rin
t.
4(x + 3)
y.
D
o
N
ot
R
There are FOUR groups of (x + 3). There are also
TWO groups of 2x + 6. Have students look at the two
models. They should note that 4(x + 3) is the most
“square” representation. This will not matter in this
section. But later it will matter, so it’s good to note it
now.
Can you write 6x + 12 in different ways?
O
nl
4. Model 6x + 12
os
es
x+x+x+x+x+x+1+1+1+1+1+1+1+1+
1+1+1+1
3(2x + 4) THREE groups of (2x + 4)
6(x + 2) SIX groups of (x + 2)
Note: 6(x + 2) is the most “square” representation.
ra
f
t:
fo
rI
ns
tru
ct
io
na
lP
ur
p
2x + 2x + 2x + 2 + 2 + 2 + 2 + 2 + 2
D
How are problems # 3 and # 4 related? How are they related to # 1? Answers will vary. Note that in both 3 and
4 there are a couple choices for writing the expression in written form (later n the section this will be discussed
as “reverse distribution.”) They should note that all are equivalent forms of the same expression.
38
5. Model 5(2x + 1) and then simplify.
10x + 5
Have students draw their models and then present how they arrived at their simplified versions. Ask
students to critique the arguments presented.
D
o
N
ot
R
ep
rin
t.
Students should explain saying something like: There are FIVE groups of (2x + 1). Then you reorder the
terms so that you are combining all the x terms and all the constant terms.
os
io
na
lP
ur
p
6. Model 4(3x – 2) and simplify.
12x – 8
es
O
nl
y.
Write 5(2x +1) in two different ways.
10x + 5; 5(2x) + 5(1)
ns
tru
ct
Take time to discuss the different ways that this expression can be written (below.) On the next page you will
push to formalize a(b – c) = a(b + (–c)) = ab + –ac
D
ra
f
t:
fo
rI
Write 4(3x – 2) in three different ways. 12x – 8; 12x + (–8); 4(3x) – 4(2); 4(3x) + (–4(2))
What does the number in front of the parentheses tell you about the grouping? The number tells you how many
groups of the amount in the “( )” there are. Discuss equivalence here.
39
N
ot
R
ep
rin
t.
7. Model 3(2x – 5) and simplify.
D
o
Write 3(2x – 5) in three different ways. e.g. 3(2x) – 3(5); 6x – 15; 6x + (–15)
ct
io
na
lP
ur
p
os
es
O
nl
y.
In 6th grade you learned that expressions like 6(2 + 3) could be written as 6(5) or 6(2) + 6(3). We extended that
thinking to expressions like 5(2x + 1) and found it could be written as 5(2x) + 5(1) or 10x + 5. In exercises 6 and
7 we saw that expressions like 4(3x – 2) can be written as 12x – 8; 12 + (–8); 4(3x) – 4(2); 4(3x) + (–4(2)).
How can you use what we’ve learned about integers and what we know about writing expressions with
parentheses to re-write expressions that have “ – “ in the groupings?
Have students work in pairs to discuss this and present their thinking.
a(b – c) = a(b + (–c)) = ab + –ac
Explanations will vary: There are a groups of (b + –c) which gives you a groups of b and a groups of –c.
ns
tru
Use what you have learned to rewrite each numeric expression. The first one is done for you:
9. 3(1 + 6)
10. 5 (4 – 1)
8. 4(2 + 3)
D
ra
f
t:
fo
rI
= 4(2) + 4(3) =8 + 12 = 4(5) = 20
11. 2(7 – 2)
12. 4(3 – 5)
13. 5(2 + 3)
40
Draw a model for each expression, then rewrite the expression in an equivalent form.
15. 2(3x + 5) 6x + 10
16. 3(x + 1) 3x + 3
17. 4(3x – 1) 12x + (–4) or 12x – 4
18. 2(3 + x) 6 + 2x
19. 3(3x – 2) 9x + (–6) or 9x – 6
io
na
lP
ur
p
os
es
O
nl
y.
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o
N
ot
R
ep
rin
t.
14. 3(x + 2) 3x + 6
tru
ct
In sixth grade you talked about order of operations, what is the order of operations and how is it related to
what’s you’ve been doing above?
D
ra
f
t:
fo
rI
ns
PEMDAS—two things should be discussed here: 1) we simplify groupings first if we can, as in problems # 8
through # 12. When we cannot, we use our understanding of the expressions to simplify. In other words, for
expressions like 3(x + 2), we understand that to mean there are 3 groups of (x + 2). This allows us to reorder our
terms and combine like terms.
2) Using a – b = a + (–b) allows us to change order when there is subtraction.
41
3.1f Homework: Iterating Groups
Simplify each of the following. Draw a model to justify your answer.
2. 2(3x + 2) 6x + 4
4. 2(3x – 1) 6x – 2 or 6x + (–2)
5. 3(2x – 3) 6x + (–9)
3. 4(x + 3) 4x + 12
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o
N
ot
R
ep
rin
t.
1. 3(x + 1) 3x + 3
ns
tru
ct
io
na
lP
ur
p
os
es
O
nl
y.
6. 5(x – 1) 5x + (–5)
fo
rI
The expressions 2(5x – 3) can be written and 10x – 6 OR 10x + (–6).
Write the following expressions in two different ways as the example shows:
D
ra
f
t:
7. 4(3x – 5)
8. 2(7x – 3)
42
3.1g Class Activity: More Simplifying
Review: Discuss the following questions in groups of 2 to 3:
What is the opposite of “forward 3 steps”?
ep
rin
t.
What is the opposite of “turn right”
N
ot
R
What is the opposite of “forward three steps then turn right”?
y.
D
o
You are moving students to thinking very ABSTRACTLY in this section. They will need
to MAKE SENSE of the STRUCTURE to interpret and USE A MODEL.
O
nl
Using that logic above, what do you think each of the following means?
–(x) the opposite of x—note that these first two mean
exactly the same thing.
io
na
lP
ur
p
os
es
–x
the opposite of x
(1  x) the opposite of the entire quantity 1 – x;
–1 + x or x + (–1) or x – 1. All should be discussed.
ra
f
t:
fo
rI
ns
tru
ct
(x+1) the opposite of x + 1. Talk about taking the
opposite of ALL of x + 1. Relate to above. Thus it
means –x – 1 or –x + (–1). If students build this, they
may note that it is also the same as –1 – x or –1 + (–x)
D
What does “ – “ in front of a set of parentheses tell us? Take the opposite of the “stuff” on the inside the
grouping. It would be a good idea to have student generate their own examples here.
43
Review:
t.
Draw a model of 3(x + 1) then simplify:
O
nl
y.
D
o
N
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R
ep
rin
What do you think –3(x + 1) means?
Students need to make sense of the expression. This means the OPPOSITE of 3 groups of (x + 1)
– [3(x + 1)]. You want students to reason that they can take 3 groups of (x + 1) and then take the OPPOSITE of
that expression. Take time to explore this expression.
io
na
lP
ur
p
os
x  2
4x + 4
es
In groups of 2 or 3 students, simplify each of the following. Be ready to justify your answer.
2. – (x + 2)
3. – (3x + 2)
1. 4(x + 1)
4. – 2(x + 3)
5. – 3(x – 2)
6. 2(3 – x)
3x + 6
6  2x
rI
t:
8. – (4 – 3x)
9. – 4(2x + 3)
4 + 12x
8x  12
D
ra
f
10 + 6x
fo
7. – 2(5 – 3x)
ns
tru
ct
2x  6
3x  2
44
Combining ideas:
Review and combine ideas:
Simplify the following expression:
3x + 5 – x 2x + 5
ep
rin
t.
3(x + 2) 3x + 6
N
ot
R
– 3(x + 2) + x – 4 –2x + (–10)
D
o
In groups of 2 or 3 students, model and simplifying the following exercises. Be prepared to justify your answer.
10. 3x + 5 – x + 3(x + 2)
11. 3x + 5 – x – 3(x + 2)
x  1 Compare problems 10 and 11
io
na
lP
ur
p
os
es
O
nl
y.
5x + 11
12. 2(x – 1) + 4x – 6 + 2x
– 2(x – 1) + 4x – 6 + 2x
4x  4 Compare problems 12 and 13
ra
f
t:
fo
rI
ns
tru
ct
8x  8
13.
D
Explain your strategy for simplifying problems 10 through13.
45
Practice:
14. 7x – 2(3x +1)
2x – 1
x–2
15. 6x – 3 + 2x – 2(3x + 5)
16. –9x + 3(2x  5) + 10
2x – 13
3x – 5
17.
18. –(4x – 3) – 5x + 2
R
N
ot
D
o
y.
O
nl
es
(5 – 3x) – 7x + 4
9x + 5
io
na
lP
ur
p
os
10x + 9
tru
ct
19. 9 – 8x – (x + 2)
20. 15 – 2x – (7 – x)
x + 8
D
ra
f
t:
fo
rI
ns
9x + 7
46
ep
rin
t.
13. 5 + 2(x – 3)
3.1g Homework: More Simplifying
2. – 3(2x + 1)
3. – 3(2x – 1)
6x – 3
4. – (x + 4)
5. – (x – 4)
6. – (4 – x)
x + 4
4 + x or x – 4
8. – 5(3x + 2)
9. – 7(2x – 5)
7. – 2(4x – 3)
14x + 35
y.
15x – 10
11. 5x + 2(x – 3)
7x – 6
io
na
lP
ur
p
7x + 6
os
10. 5x + 2(x + 3)
es
O
nl
8x + 6
D
o
N
ot
R
x – 4
6x + 3
ep
rin
6x + 3
t.
Simplify the following expressions:
1. 3(2x + 1)
14. 3x + 2 – 4x + 2(3x + 1)
ct
13. 5x – 2(x – 3)
3x – 6
15. –7x + 3 + 2x – 3(x +2)
5x + 4
8x – 3
17. 4x – 5(2x  5) – 3x + 4
18. x – 7 – 2(5x – 3) + 4x
9x + 29
5x – 1
rI
ns
tru
3x + 6
12. 5x – 2(x + 3)
D
ra
f
5x + 8
t:
fo
16. 10x – 4 – 7x – 4(2x  3)
47
3.1g Additional Practice: Iterating Groups
Matching: Write the letter of the equivalent expression on the line
1. __b__
a)
2. __i__
b)
3. __j__
c)
t.
d)
4. __a__
ep
rin
e)
5. __g__
f)
6. __h__
h)
8. __f__
i)
D
o
9. __c__
N
ot
7. __e__
R
g)
j)
O
nl
y.
10. __d__
12.
3y − 9
20.
8x − 64
29.
4 – 24n
30.
os
7x + 21
io
na
lP
ur
p
11.
es
Practice: Simplify each expression.
21.
–6x + 6
9.8p + 50.96
31.
23.
–6 – 12n
32.
0.25n  1.75
24.
–t − 3
33.
36 – 63m
rs + rt
25.
–4 − k
17.
6x + 4
26.
–3x + 6
35.
18.
3.2v − 3.2
27.
15x + 6
36.
21n − 14
28.
14j − 168
37.
–24 + 4t
3k + 12
14.
2b − 4
ns
tru
13.
7y −7z
34.
D
ra
f
t:
fo
rI
15.
16.
ct
22.
0.4 + 0.56x
–24 + 24p
19.
–c + 3
48
–4k + 4
3.1 g Additional Practice: Simplifying
3.
5x  10y  2x  4y  3x
14
10
2x  5y  2x  4y  2z
4.
7w  3q  5  8q  6  10w
2
D
o
O
nl
10.
io
na
lP
ur
p
rI
t:
16.
D
(7h  2k)
3
8
6(4  2y)
(3  5h)(3)
1(x  2y)
–x + 2y
(3k  5)
18.
–3x − 6
19.
3y  2x  5y  5x  10x
15h – 9
ns
tru
14.
4
(3x  6)
17.
3
12y – 24
ra
f
3
8
fo
6
15.
ct
8x – 20
12v + 16
10y  10y  3
12.
(8  6v)(2)
14
20
4(2x  5)
13.
8.
5
9. 31y  5x  4  12  13x  23y
8
8
8
11.
14
es
2
17v  2  12v 12 15v
10
3c  6c  5c  2d  4d  3d
11
y.
2
6.
3
R
3p  2q  4 p  4q  6  4
7
7.
5
os
5.
6
3k + 5
20.
–7h  2k
7(x  5q)
–7x + 35q
49
t.
4
2.
ep
rin
5x  10y  2x  4y  3x
N
ot
1.
3  4(2x  5)
12  6(4  2y)
22.
12y − 12
8x − 17
5  (8  6v)
4  (3  5h)
24.
5y  1(x  2y)  6
26.
–x + 7y + 6
5x  (3x  6)  6
27.
D
o
12h − 6
5k  (3k  5)  8
28.
8k + 13
O
nl
2x
7x  7(x  5q)
es
4h  (7h  2k)  5
30.
io
na
lP
ur
p
os
29.
D
ra
f
t:
fo
rI
ns
tru
ct
–3h – 2k – 5
50
14x − 35q
ep
rin
2(6h  8)  10
25.
t.
–5h + 7
R
–6v − 3
N
ot
23.
y.
21.
3.1h Class Activity: Modeling Context with Algebraic Expressions
t.
Look back at the anchor problem. In particular, look back at your work for the two situations below. Recall you
were putting 1-foot square tiles around the edge of a square garden and you were trying to figure out how many
tiles you’d need for different gardens. Your task was to express the number of tiles you’d need in four different
ways. This should be a class discussion. Ask students how their understanding has evolved since doing this
problem at the beginning of the section. Review the work students did originally. Focus on equivalence of
expressions.
ep
rin
What if the garden were 100 feet by 100 feet? Demonstrate how
each method would work now.
R
Method 1: 4(100) + 4, 4 sides of 100 tiles, then 4 corners
N
ot
100 feet across
D
o
Method 2: 102+102+100+100, 102 on top, 102 on bottom,
100 on left, 100 on right
y.
Method 3: 1022 – 1002 , outer square minus inner square
os
es
O
nl
Method 4: 4102 − 4, 4 sides of 102, subtract the tiles counted
twice
io
na
lP
ur
p
What if the garden were n feet by n feet? Demonstrate how each
method would work now. Simplify each method.
Method 1: 4(n) + 4, 4 sides of n tiles, then 4 corners
n feet across
Method 3: (n+2)2 – n2 , outer square minus inner square
Method 4: 4(n+2) − 4, 4 sides of (n+2), subtract the tiles
counted twice
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f
t:
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tru
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Method 2: (n+2) + (n+2)+ n + n, n+2 on top, n+2 on bottom, n
on left, n on right
51
In the context above you wrote several expressions for each situation; often there is more than one equivalent
way to algebraically model a context. Below are contexts, write two equivalent expressions for each situation.
It may be helpful to draw a model.
1. Marty and Mac went to the hockey game. Each boy bought a program for 3 dollars and nachos for n
dollars. Write two different expressions that could be used to represent how much money the boys spent
altogether.
Expression 2:
t.
2(3 + n)
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R
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Expression 1:
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p
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2. The cooking club would like to learn how to make peach ice cream. There are 14 people in the club. Each
member will need to buy 3 peaches and 1 pint of cream to make the ice cream. Peaches cost x cents each,
and a pint of cream costs 45 cents. Write two different expressions that could be used to represent the total
cost of ingredients for all 14 members of the club. Simplify each expression.
14(3x + 1·45)
14·3·x + 14·1·45
630 + 42x cents
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ns
tru
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3. Leo and Kyle are training for a marathon. Kyle runs 10 mile per week less than Leo. Write two expressions
to represent the distance Kyle ran over 12 weeks if L equals the distance Leo ran every week.
12(L − 10)
12·L + 12(–10)
12L – 120 miles
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4. Harry is five years younger than Sue. Bridger is half as old as Harry. Write two different expressions that
could be used to represent Bridger’s age in terms of Sue’s age. Simplify each expression. (Hint: use a
variable to represent Sue’s age.)
s 5
s 5
( s  5)
1

( s  5)
2
2
2 2
2
52
3.1h Homework: Modeling Context with Algebraic Expressions
Below are contexts. Write two different expressions for situations 1 and 2 and then simplify. For situations 3
and 4, write an algebraic expression. Draw a model for each.
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t.
1. Marie would like to buy lunch for her three nieces. She would like each lunch to include a sandwich, a
piece of fruit, and a cookie. A sandwich costs $3, a piece of fruit costs $0.50, and a cookie costs $1. Write
two different expressions that could be used to represent the total price of all three lunches. Then simplify
each expression that you wrote.
3(3 + 0.50 + 1)
3·3 + 3·0.50 + 3·1
9 + 1.50 + 3
$13.50
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2. Boris is setting up an exercise schedule. For five days each week, he would like to play a sport for 30
minutes, stretch for 5 minutes, and lift weights for 10 minutes. Write two different expressions that could
be used to represent the total number of minutes he will exercise in five days. Then simplify each
expression that you wrote.
5(30 + 5 + 10)
5(45)
5·30 + 5·5 + 5·10
225 minutes
3. Five girls on the tennis team want to wear matching uniforms. They know skirts will costs $24 but are not
sure about the price of the top. Write two different expressions that could be used to represent the total cost
of all five skirts and tops if x represents the price of one top. Simplify each expression.
524 + 5x
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tru
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5(24 + x)
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t:
4. Drake, Mike, and Vinnie are making plans to go to a concert. The tickets will cost $30 each, and each boy
plans to buy a t-shirt for t dollars. Write two different expressions that could be used to represent the total
cost for all three boys. Simplify each expression.
3(30 + t)
30 30 30 t t t
90 3tdollars
53
3.1i Class Activities: Properties.
In mathematics, there are things called “properties;” you may think of them as “rules.” Properties are the rules
for a set of numbers. In today’s lesson, we are going to formally define the properties of arithmetic that you’ve
used all along in math. There is nothing new in the properties discussed in this section. Everything you expect
to work will work. We’re just giving vocabulary to what you’ve been doing so that when you construct an
argument for an answer, you’ll be able to use language with precision. By the end of this section, you should be
able to define and explain the properties in pictures, words, and symbols.
ep
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t.
Commutative Property
R
The word “commute” means “to travel” or “change.” It’s most often used in association with a location. For
example, we say people commute to work.
D
o
N
ot
For each of the following pairs of expressions, the operation is the same, but the constants have been
commuted. Determine if the pairs are equivalent, be able to justify your answer. From these pairs, we are going
to try to define the Commutative Property.
2. not equivalent
12 + 4
9.8 – 3.4
4 + 12
3.4 – 9.8
4. equivalent
os
3. not equivalent; however 12 + (4) = (4) + 12
es
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y.
1. equivalent
io
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12 – 4
5·4
4·5
6. not equivalent
3 · 0.9
18 ÷ 6
tru
ct
4 – 12 Ask students if there is a way to commute with
subtraction.
5. equivalent
6 ÷ 18
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ns
0.9 · 3
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t:
fo
What pattern are you noticing?
Commutative Property:
a + b = b + a; ab = ba; Addition and multiplication are commutative (you can change the order of addition or
multiplication.) Also discuss that subtraction expressions can be changed to addition expressions with integers
and how doing this allows order to change. After you’ve talked about multiplicative inverse (below), come back
here and discuss how division expression could be changed to multiplication expressions to change order.
54
Associative Property
The word “associate” means “partner” or “connect.” Most often we use the word to describe groups. For
example, if a person goes to Eastmont Middle School and not Indian Hills Middle School, we would say that
person is associated with Eastmont Middle School.
ep
rin
t.
For each of the following pairs of expressions, the operations are the same, but the constants have been
associated in different ways. Determine if the pairs are equivalent; be able to justify your answer. From these
pairs, we’re going to try to define the Associative Property.
(12 – 4) – 3 not equivalent
12 + (4 + 6)
12 – (4 – 3) Ask how both could be written to make
them equivalent 12 + (–4 + –3) = (12 + –4) + –3
(3 + 5) + 7.4 equivalent
(20.9 – 8) – 2 not equivalent
3 + (5 + 7.4)
20.9 – (8 – 2)
(5 · 4) ·
(18 ÷ 6) ÷ 3 not equivalent
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y.
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es
equivalent
)
io
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p
5 · (4 ·
R
(12 + 4) + 6 equivalent
(6 · 2) · 5 equivalent
(24 ÷ 12) ÷ 3 not equivalent
24 ÷ (12 ÷ 3)
tru
ct
6 · (2 · 5)
18 ÷ (6 ÷ 3) Ask how both could be written to make
them equivalent(18 · 1/6) · (1/3) = 18 · (6 · (1/3))
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t:
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What patterns do you notice about the problems that were given?
Associative Property: (a + b) + c = a + (b + c); (ab)c = a(bc); Addition and multiplication are associative (you
can change the grouping), but subtraction and division are not. As above, discuss how one might work with
subtraction expressions to change grouping. Discussion of division should happen in conjunction with
multiplicative inverse.
55
Identity Property
The word “identity” has to do with “sameness.” We use this word when we recognize the sameness between
things. For example, you might say that a Halloween costume cannot really hide a person’s true identity.
Above we defined the Associative and Commutative Properties for both addition and multiplication. We need to
do the same thing for the Identity Property.
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t.
What do you think the Identity Property for Addition should mean? Have students brainstorm. Answers will
vary. Look for something like “doesn’t change the identity of the expression.”
N
ot
R
Give examples of what you mean:
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y.
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Identity Property of Addition: a + 0 = a; You can add “0” to anything and it won’t change the expression.
Discuss “zero pairs” from Chapter 2 here.
io
na
lP
ur
p
os
es
What do you think the Identity Property for Multiplication should mean? Have students brainstorm. Answers
will vary. Look for something like “doesn’t change the identity of the expression.”
Give examples of what you mean:
tru
ct
Identity Property of Multiplication: a(1) = a; You can multiply anything by 1 and it won’t change the
expression.
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ns
For advanced students, a discussion of fields or
groups would be good in this section. For
example you may want to talk about properties
with different Isometries
fo
Multiplicative Property of Zero
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t:
What do you think this property tells us?
Multiplicative Property of Zero a(0) = 0; 0 times anything results in 0.
56
Distributive Property of addition over multiplication
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t.
Like all the other properties above, we’ve used this property throughout section 3.1. Below, first show the
property to show 2(3 + 4) and then show it for a(b + c).
Inverse Properties
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The word “inverse” means “opposite” or “reverse.” You might say, forward is the inverse of backward. There is
an inverse for both addition and multiplication.
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es
What do you think should be the additive inverse of 3?
io
na
lP
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p
What do you think would be the additive inverse of –3?
ct
What do you think would be the multiplicative inverse of 3?
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What do you think would be the multiplicative inverse of 1/3?
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Inverse Property of Addition: a + (–a) = 0; discuss how this property is related to the additive identity
D
Inverse Property of Multiplication: a(1/a) = 1 for a  0; discuss how this property is related to the multiplicative
identity.
57
1.63
2.68
2
3
Adding zero to a number
does not change the number.
∙0
Multiplying a number by zero
results in zero.
0
Commutative
Property of
Addition
O
nl
∙
∙
∙
Changing the grouping of
multiplication does not
change the result.
∙
∙
∙
“a” groups of (b + c)
1
2
3
12
3.92
2 1
3 2
1
2
3
2
1
2
3
1
6
4∙ 2∙3
2 1 1
∙ ∙
3 2 6
3.92 1.5
1 2
2 3
3.2 ∙ 4.75
1 2
2 3
1
2
3
2
1
2
4∙2 ∙3
2 1 1
∙ ∙
3 2 6
3(2 + 5) = 3(2) + 3(5)
3(2 – 5) = 3(2) + 3(–5)
fo
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∙
ct
Associative
Property of
Multiplication
io
na
lP
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p
Changing the grouping of
addition does not change the
result.
1.63
2.68
2
3
1∙0 0
9.52 ∙ 0 0
2
∙0 0
3
4.75 ∙ 3.2
2 1
3 2
Reversing the order of
multiplication
does not change the result.
os
∙
Associative
Property of
Addition
Distributive
Property of
Addition over
Multiplication
1.5
es
Commutative
Property of
Multiplication
0
1∙1
12 ∙ 1
2
∙1
3
y.
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Reversing the order of
addition
does not change the result.
0
0
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∙1
Multiplying a number by one
does not change the number.
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Multiplicative
Property of Zero
Examples
R
Identity Property
of Multiplication
Meaning
t.
Properties of Mathematics:
Name Property Algebraic Statement
Identity Property
of Addition
0
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t:
Additive Inverse
D
0
Multiplicative
Inverse
∙
1 for a  0
A number added to its
opposite will result in zero.
Multiplying a number by its
inverse will result in one.
58
1
6.1
2
3
1
0
6.1 0
2
0
3
9.8 ∙
1
9.8
2 3
∙
3 2
1
1
3
1
6
3.1i Homework: Properties
t.
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rin
3. Multiplicative Property of Zero:
∙0 0
Show the Identity Property
of Addition with 3
3 0
3
Show the Identity Property
of Multiplication with 3b
3 ∙1 3b
Show the Multiplicative
Property of Zero with 4xy
4 ∙0 0
R
2. Identity Property of Multiplication:
∙ 1
Show the Identity Property
of Addition with 2
2.17 + 0 = _______
Show the Identity Property
of Multiplication with 23
23 ∙ 1 23
Show Multiplicative
Property of Zero with
43.581 43.581 ∙ 0 0
N
ot
Complete the table below:
1. Identity Property of Addition:
0
4. Commutative Property of Addition:
is the same as:
is the same as:
∙ 6
os
6k
(1.8 + 3.2) + 9.5 is the same as: 1.8 + (3.2 + 9.5 )
io
na
lP
ur
p
6. Associative Property of Addition:
(a + b) + c = a + (b + c)
∙
O
nl
∙
es
5. Commutative Property of Multiplication:
ab = ba
y.
x + z is the same as:
D
o
4.38 + 2.01 is the same as: 2.01 + 4.38
(x + 1) + 9 is the same as: x + (1 + 9)
7. Associative Property of Multiplication:
(2.6 · 5.4) · 3.7 is the same as: 2.6· (5.4· 3.7)
tru
ct
(wh)l is the same as:
Use the listed property to fill in the blank.
ns
8. Multiplicative Inverse:
1
3
=1
5
=0
9
+
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9. Additive Inverse: a + (–a) = 0
59
¼( 4)=1
x + –x = 0
Name the property demonstrated by each statement.
9∙7
10.
3 6
Commutative Property of Multiplication
3∙6
Associative Property of Multiplication
1
5
1
Multiplicative Inverse
N
ot
5
13.
Additive Inverse Property
R
25 + (–25) = 0
12.
ep
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t.
11.
7∙9
(x + 3) + y = x + (3 + y)
Associative Property of Addition
15.
1mp = mp
Identity Property of Multiplication
16.
9 + (5+35) = (9+5) + 35
Associative Property of Addition
17.
0 + 6b = 6b
18.
7x  0 = 0
19.
4(3z)=(43)z
20.
x+4=4+x
21.
3(x + 2) = 3x + 6
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es
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14.
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Identity Property of Addition
Multiplicative Property of Zero
Commutative Property of Addition
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Associative Property of Multiplication
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Distributive Property
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3.1j Class Activity: Using Properties to Compare Expressions
es
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Evaluate the following pairs of expressions. Write whether or not the two expressions are equivalent. If the
expressions are not equivalent, correct expression 2 to make it equivalent. Use properties to explain how you
can know that the expressions are equivalent or not without evaluating them.
Expression 1
Equivalent?
Expression 2
Explanation
 or ≠
Multiplication is Associative, I can group
=
multiplication in different ways and still get the
3 25 ∙ 4
3 ∙ 25 4
same answer
Subtraction is not Commutative, I cannot change
the order of subtraction. If I change all subtractions
5
47 63
47 5
63
to the addition of the opposite term, then I can make
changes. Correct: –47 + 53 + (–63)
Addition is Commutative, I can change the order of
=
88 + 133(2) + 14
88 14 133 2
addition and still get the same answer.
Addition is Commutative, I can change the order of
25 + 4(3 + 1)
=
4(3 + 1) + 25
addition and still get the same answer.
os
Using Properties to Justify Steps for Simplify Expressions
Step
3 + 12 + 17 + 28
No change, this is where she started.
This expression was given.
3 + 17 + 12 + 28
The 17 and the 12 traded places.
Jane chose to add the numbers in pairs
first, which is like inserting parentheses.
Jane found the sums in the parentheses
first.
Commutative Property of Addition
fo
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f
And so . . . 3 + 12 + 17 + 28 = 60
D
60
t:
20 + 40
rI
(3 + 17) + (12 + 28)
ns
ct
Statement
tru
io
na
lP
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p
Example: Jane wants to find the sum: 3 + 12 + 17 + 28. She uses the following logic, “3 and 17 are 20, and 12
and 28 are 40. The sum of 20 and 40 is 60.” Why is this okay? The table below shows how to justify her
thinking using properties to justify each step.
61
Justification
Associative Property of Addition
Jane is now following the Order of
Operations.
1) The expression 3(x – 4) – 12 has been written in three different ways. State the property that allows each
change.
Expression
Step
Justification
This expression was given, only rewritten
3(x + (–4)) + (–12)
No change
using the idea that a – b = a + (–b)
Commutative Property of
(–12) + 3(x + (–4))
Addition order changed
Addition/Addition is Commutative
Three groups of (x + (–4)) written out
Distributive Property
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t.
(–12) + 3x + 3(–4)
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2) The expression 2(3x + 1) + –6 x + –2 has been written in four different ways. State the property that allows
each change.
Expression
Step
Justification
No change
Given expression
6x + 2 + –6x + –2
Multiplied 2 by both 3x and 1
6x + –6x + 2 + –2
Changed the order of the terms
Distributive Property
Commutative Property of
Addition/Addition is Commutative
0+0
6x + (–6x) and 2 + (–2) both sum to 0
y.
D
o
2(3x + 1) + –6x + –2
es
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Additive Inverse
1
1
1
1
1
1
1
1
ns
fo
Step 1: this is a representation of the expression
1
Step 2: 1 + (–1) = 0, so each of these pairs are zero: –1
is the additive inverse of 1.
1
1
Step 3: this is a representation of what remains when
the zero pairs are removed.
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1
rI
1
Justify
1
ct
1
tru
1
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p
3) Model 3 + (–5) to find the sum
3 + (–5)
Step 1
Step 2
os
Review: Look back at Chapter 2 and review addition/subtraction with the chip model.
D
Step 3
1
1
62
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4) In chapter 2 you learned that a negative times a negative produces a positive product. We used models to
discover why this is true. In groups of 2-3, write a more formal proof. Answers will vary. Look for logic and
the use of properties to justify steps. Students may use models. Have students present their proof to the class.
t:
–1(–1) + –1(1) = 0
D
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f
–1(–1) + 1(–1) = 0
–1(–1) + –1 = 0
Justification
Multiplicative Property of Zero
0 was replaced with (–1 + 1)
Additive Inverse Property
–1 was multiplied by each term in
parentheses
The –1(1) got switched to 1(–1)—changed
the order of multiplication
(–1)1 was replaced with –1
Distributive Property
rI
fo
–1(–1 + 1) = 0
Step
Given
ns
Statement
–1(0) = 0
tru
ct
Here is a proof: we start with –1(0) = 0
–1(–1) must equal 1 because if we get 0 when we
add it to –1, it must be the additive inverse of –1
Commutative Property of multiplication
Identity Property of Multiplication
Additive Inverse Property
63
3.1k Classwork: Modeling Backwards Distribution
Review: below is a review of modeling multiplication with an array.
23
33
N
ot
y.
Use the Key below to practice using a multiplication model.
x
44
xx
t:
fo
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p
22
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x2
1
Factors: 3, 3
There are three groups of 3
Product/Area: 9
R
Factors: 2, 3
There are two groups of 3
Product/Area: 6
D
o
Factors: 1, 3
There is one group of 3
Product/Area: 3
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t.
13
Factors: 4, 4
Factors: x, x
Product/Area: 4
Product/Area: 16
Product/Area: x2
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f
Factors: 2, 2
Look at the three models above. Why do you think 22 is called “two squared”? 32 is called “3 squared”? and
42 is called “four squared”? The difference between 4 × 4 (42) and 4 + 4 (2(4)) is difficult for many students.
Discuss the geometric representations.
64
1. Build the factors for 3(x + 2) on your desk. Then build the area model. Draw and label each block below.
What are the factors of the multiplication problem? 3, x + 2
What is the area ? 3x + 6
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t.
What is the product of the multiplication problem? 3x + 6
y.
What are the factors of the multiplication problem? 3, 2x +1
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2. Build the factors for 3(2x +1) on your desk. Then build the area model. Draw and label each block below.
O
nl
What is the area? 6x + 3
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What is the product of the multiplication problem? 6x + 3
3. Build the factors for 2(x + 4). Build the area and draw.
ct
What is the area or product? 2x + 8
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What are the factors? 2 and x + 4.
4. Build the factors for x(x + 3). Build the area and draw.
What is the area or product? x2 + 3x
What are the factors? x and x + 3
65
5. Build the factors for x(2x + 5). Build the area and draw.
What is the area or product? 2x2 + 5x
N
ot
R
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t.
What are dimensions or factors? x and 2x + 5
D
o
Review concepts:
os
es
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nl
y.
6. Write each as the sum of two whole number and the product of two integers. Model your expression:
b) 9
c) 35
a) 15 answers will vary:
0 + 15
1 + 14
2 + 13 etc.
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na
lP
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p
1 · 15
3·5
tru
ct
Discuss: sometimes we break
numbers apart using addition,
other times we break them apart
with multiplication. Why?
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7. Simplify each; use a model and words to explain the difference between the two expressions:
a) x + x 2x
b) x  x x2
D
Discuss the geometric representation of each.
66
Example: Model the expression 2x + 4 on your desk. Find the factors and write 2x + 4 factored form.
What are the dimensions (factors) of your rectangle? Draw them.
Length: x + 2 Width: 2
t.
What is the area (product) of the rectangle? 2x + 4
y.
9) 3x + 12
10) 5x + 10
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8) 6x + 3
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R
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Write 2x + 4 in factored form: 2(x + 2)
9) 3x +12
ct
8) 6x + 3
Factors: 3 and x + 4
Factors: 5 and x + 2
3x + 12 in factored form:
3(x + 4)
5x +10 in factored form:
5(x + 2)
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t:
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6x + 3 in factored form:
3(2x + 1)
tru
Factors: 3 and 2x + 1
10) 5x + 10
67
13. x2 + 3x
12. x + 4
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rin
t.
11. 6x + 2
x+4
x2 + 3 x
Factors: 2 and 3x + 1
Factors: 1 and x + 4
Factors: x and x + 3
6x + 2 in reverse distributed form:
2(3x + 1)
x + 4 in reverse distributed form:
x + 4 or 1(x + 4)
2x 2 + 4x in reverse distributed
form:
x(x + 3)
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nl
y.
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N
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R
6x + 2
3(m – 5)
18.
ct
2(2n – 1)
25b – 5
5(5b – 1)
19. 4x – 8 4(x – 2)
D
ra
f
t:
fo
rI
ns
tru
17. 4n – 2
io
na
lP
ur
p
os
es
Practice: Write each in reverse distributed form. Use a model if you’d like.
14. 30x + 6 6(5x + 1)
15. 4b + 28 4(b + 7)
16. 3m – 15
Look at problems 14, 15, and 19. How else might these expressions be factored? e.g. 14 could be written and
2(15x + 3). Discuss “most square” representation of each—this should not be a lesson on factoring.
68
3.1k Homework: Modeling Backwards Distribution
Write each in reverse distributed from. Use a model to justify your answer.
2. 3x + 12
1. 2x + 4
3. 2x + 10
3(x + 4)
2(x + 5)
N
ot
R
ep
rin
t.
2(x + 2)
x(x + 5)
y.
x(x + 2)
io
na
lP
ur
p
os
es
O
nl
3(x + 6)
6. x2 + 5x
D
o
5. x2 + 2x
4. 3x + 18
Simplify each expression. Draw a model to justify your answer.
8. (2x)(3x)
6x2
D
ra
f
t:
fo
rI
ns
tru
ct
7. 2x + 3x
5x
69
3.1l Self-Assessment: Section 3.1
Consider the following skills/concepts. Rate your comfort level with each skill/concept by checking the box
that best describes your progress in mastering each skill/concept.
Deep
Understanding,
Skill Mastery
N
ot
R
1. Use the Distributive Property to expand and
factor linear expressions with rational
numbers.
2. Combine like terms with rational
coefficients.
D
o
3. Recognize and explain the meaning of a
given expression and its component parts.
D
ra
f
t:
fo
rI
ns
tru
ct
io
na
lP
ur
p
os
es
O
nl
y.
4. Recognize that different forms of an
expression may reveal different attributes of
the context.
70
t.
Developing
Skill and
Understanding
ep
rin
Skill/Concept
Beginning
Understanding
Section 3.2 Solve Multi-Step Equations
Section Overview:
ep
rin
t.
This section begins by reviewing and modeling one- and two-step equations with integers. Students then
learn to apply these skills of modeling and solving to equations that involve the distributive property and
combining like terms. Students learn to extend these skills to solve equations with rational numbers. Next,
students will write and solve word problems that lead to equations like those that they have learned to solve.
Students will continue to practice all the skills that they have learned, including critiquing another’s work to
find the error.
Concepts and Skills to be Mastered (from standards )
D
ra
f
t:
fo
rI
ns
tru
ct
io
na
lP
ur
p
os
es
O
nl
y.
D
o
N
ot
R
By the end of this section, students should be able to:
1. Solve multi-step equations fluently including ones involving calculations with positive and negative
rational numbers in a variety of forms.
2. Convert between forms of a rational number to solve equations.
3. Use variables to create equations that model word problems.
4. Solve word problems leading to linear equations.
5. Connect arithmetic solution processes that do not use variables to algebraic solution processes that use
equations.
6. Critique the reasoning of others.
71
3.2a Classroom Activity: Model Equations
ep
rin
t.
Use any method you’d like to solve each of the following. Draw a model to justify your answer: student
workbook does not have models as shown in teacher edition below.
2. 8 = k – 2
3. 3n = 18
1. m + 3 = 7
N
ot
6. y + 3 = –5
To find y, we must “take away 3
from both the y +3 and the –5”
io
na
lP
ur
p
os
es
O
nl
y.
D
o
4. 17 = 2j + 1
R
To find k, we must “take away –2
from both k – 2 and 8”
5. j/2 = 6
It may have been easy to solve some (or all) of the above in your head, that’s good; that means you’re
making sense of the problem. In this section, we are going to focus on the structure of equations and how
properties of arithmetic allow us to manipulate equations. So, even though the “answer” is important, more
important right now is that you understand the underpinnings of algebraic thinking.
tru
ct
Evaluate the expressions 2x + 1 for each of the given values:
7. Evaluate 2x + 1 for x = 3
8. Evaluate 2x + 1 for x = –2
-3
Discuss how/why 7-8 simplify to
different values.
-5
ra
f
t:
fo
rI
ns
7
Discuss how the unknown in an
expression can represent any
number. Also, discuss “evaluate”
v. “solve.”
9. Evaluate 2x + 1 for x = –3
10. 2x + 1 = –5
x=2
x = -3
D
Solve each equation in any way you want:
8. 2x + 1 = 5
9. 2x + 1 = 9
x=4
72
What is the difference between an equation and an expression?
1=
es
O
nl
y.
–1 =
N
ot
–x =
D
o
x =
R
ep
rin
t.
For this activity use the following Key to represent variables and integers. Note: “x” or “–x” can be any
variable.
io
na
lP
ur
p
os
11. Use a model to solve x – 1 = 6. Write the algebraic procedure you followed to solve.
=
ct
Throughout these exercises, students
should discuss properties of arithmetic.
x–1=6
rI
ns
tru
x–1=6
+ 1 +1
ra
f
t:
fo
additive inverse
D
x=7
73
12. Use a model to solve x – 3 = 5. Write the algebraic procedure you followed to solve.
t.
ep
rin
N
ot
R
x–3=5
+ 3 +3
D
o
Talk about “equality;” adding “3” to
both sides of the equation maintains
equality. 3 and (-3) are additive
inverses.
=
=
x–3=5
x=8
io
na
lP
ur
p
os
es
O
nl
y.
=
additive inverse
13. Use a model to solve 8 = 7 + m. Write the algebraic procedure you followed to solve.
8=7+m
8=7+m
+(–7) +(–7)
=
additive inverse
D
ra
f
t:
fo
rI
ns
tru
ct
=
=
1=m
74
14. Use a model to solve 6 = 3x. Write the algebraic procedure you followed to solve.
6 = 3x
ep
rin
t.
=
O
nl
es
io
na
lP
ur
p
os
=
15. Use a model to solve 8 = –2m. Write the algebraic procedure you followed to solve.
Students might take the
“opposite” of both sides:
-8 = 2m. Allow students time
to Make Sense of the Problem
=
ra
f
t:
fo
rI
ns
tru
ct
=
D
multiplicative
inverse
y.
D
o
N
ot
R
=
(1/3)6 = (1/3)3x
or
6/3 = 3/3 x
=
75
2=x
8 = –2m
16. Use a model to solve – 5 + 3n = 7. Write the algebraic procedure you followed to solve.
=
N
ot
R
ep
rin
t.
=
D
ra
f
t:
fo
rI
ns
tru
ct
io
na
lP
ur
p
os
es
O
nl
y.
D
o
=
76
3.2a Homework: Model and Solve Equations
Model and solve each equation below. Draw algebra tiles to model. Use the Key below to model your
equations.
–x =
1=
ep
rin
t.
x =
N
ot
R
–1 =
D
o
1. x – 6 = –9
io
na
lP
ur
p
os
es
O
nl
y.
=
D
ra
f
t:
fo
rI
ns
tru
ct
=
=
77
x  6 = 9
2. –15 = x – 14
15 small negative squares
–15 = x – 14
ep
rin
t.
=
14 small negative squares and a
positive x (rectangle)
D
o
N
ot
R
=
es
O
nl
y.
=
io
na
lP
ur
p
11 negative small squares
=
D
ra
f
t:
fo
rI
ns
tru
ct
1 rectangle, 2 small squares
os
3. m + 2 = –11
78
m + 2 = –11
4. 4n = –12
ep
rin
t.
=
D
o
N
ot
R
=
os
es
O
nl
y.
=
io
na
lP
ur
p
5. –15 = –3m
D
ra
f
t:
fo
rI
ns
tru
ct
students may take the opposite of both
sides to model--. 15=3m
=
=
=
79
6. 3t + 5 = 2
2 small squares
t.
=
3 rectangles, 5 small squares
N
ot
R
ep
rin
=
os
es
O
nl
y.
D
o
=
io
na
lP
ur
p
7. 8 = 2p – 4
=
D
ra
f
t:
fo
rI
ns
tru
ct
=
=
80
3.2b Class Activity: More Model and Solve One- and Two-Step Equations
Now let’s formalize the solving equation process. The answers are obvious in these first few equations. We
use basic equations to think about solving more complicated ones. Example 1 is done for you.
What are the solving action?
Record the steps using Algebra
Model/Draw the Equation
1. x + 5 = 8
Add –5 to both sides.
x58
x58
(3)  5  8
t.
True, so the
solution is
correct.
O
nl
y.
D
o
N
ot
R
ep
rin
- 5 -5
x
=3
=
Check solution in
the equation.
2. 5 = x + 8
Show work
es
x = –3
tru
ct
io
na
lP
ur
p
os
=
fo
rI
ns
Look at # 1 and # 2, why are the answers different?
D
ra
f
t:
Discuss structure.
81
Show check
3. 3x = –6
Show check
Show work
x = –2
R
ep
rin
t.
=
D
o
N
ot
Explain the logic above in #3.
O
nl
y.
How might you use related logic to model x/3 = –6?
4. x/3 = –6
Show check
io
na
lP
ur
p
=
os
x = –18
Show work
ns
(1/2)x = 3
x=6
=
D
ra
f
t:
fo
rI
5.
tru
ct
1/3 of an x is -6, so if
we triple both sides, a
whole x is -18.
es
Show work
In problems # 3 and # 4, what happened to the terms on both sides of the equation?
Both sides of the equation were multiplied by the multiplicative inverse of the coefficient of x.
82
Show check
6. 2x – 5 = –9
Show check
Show work
x = –2
7. 7 = 3x – 2
t.
=
Show check
ep
rin
Show work
x=3
D
o
N
ot
R
=
8. –5 = –3 + 2x
Show check
O
nl
y.
Show work
x = –1
7 + x/2 = –3
Show work
Show check
x = -20
ct
9.
io
na
lP
ur
p
os
es
=
fo
rI
ns
tru
=
10.
–3 = x/2 + 2
D
ra
f
t:
Show work
x = –10
=
83
Show check
3.2b Homework: More Model and Solve One- and Two-Step Equations
What are the solving action?
Record the steps using Algebra
Model/Draw the Equation
1. 2 = x + 5
Check solution in
the equation.
Show check
Show work
x = –3
2. –12 = 3x
x = –4
–(x/4) = –8
Show work
Show check
x = 32
rI
Show work
–2 = (1/3) x
fo
4.
ns
tru
ct
=
io
na
lP
ur
p
3.
os
es
O
nl
y.
=
Show check
D
o
Show work
N
ot
R
ep
rin
t.
=
=
D
ra
f
t:
x = –6
84
Show check
5. –9 = x/2 – 5
Show check
Show work
x = –8
R
ep
rin
t.
=
7. –11 = –4x – 3
D
ra
f
t:
fo
rI
ns
tru
ct
=
io
na
lP
ur
p
os
es
O
nl
y.
=
D
o
x=5
N
ot
6. –3x + 2 = –13
85
x=2
8. –4 + n/3 = –2
Show check
Show work
x=6
Show work
x/3 – 5 = –2
10. 2 + 5x = –8
Show work
Show check
x = –2
Show work
(1/2) x – 5 = – 3
t:
11.
fo
rI
ns
tru
ct
=
io
na
lP
ur
p
os
es
O
nl
y.
D
o
=
Show check
N
ot
x=9
R
9.
ep
rin
t.
=
D
ra
f
x=4
=
86
Show check
3.2c Class Activity: Model and Solve Equations, Practice and Extend to Distributive Property
Practice: Use a model to solve each. Show your algebraic manipulations on the right.
Additional practice is available at 3.2h
1. –16 = 6a – 4
–2
D
o
7 = 6 – n/7
7
O
nl
y.
2.
N
ot
R
ep
rin
t.
=
io
na
lP
ur
p
os
es
=
=
ns
tru
ct
3. –10 = –10 – 3x
**This will cause questions:
Students will be unsure what to
do with 0 = –3x. Discuss
fo
rI
x=0
D
ra
f
t:
4. Review: Create a model and then use the model to simplify each of the following expressions
a) 3(2x + 1)
b) –2(3x + 2)
c) –4(2x – 3)
87
Use a model to solve each. Show your algebraic manipulations on the right.
5. 2(x + 1) = –8
x = –5
=
D
o
N
ot
R
ep
rin
t.
Students may distribute first OR
may chose to divide first.
Discuss both approaches.
O
nl
y.
6. 6 = –3(x – 4)
x=2
tru
=
fo
rI
ns
7. –12 = –3(5x – 1)
ct
io
na
lP
ur
p
os
es
=
D
ra
f
t:
x=1
88
8. –4(3 – 2m) = –12
**Again, they may be unsure
what to do when nothing is left
on side of the equation. i.e.
they will get 8m = 0. Look
back at Review #3.
=
N
ot
R
ep
rin
t.
m=0
D
o
9. –(1/2)(4x + 2) = –5
x=2
io
na
lP
ur
p
os
es
O
nl
y.
=
**Look for different strategies:
doubling both sides OR
distributing the –½. Talk about
both.
=
x=3
D
ra
f
t:
fo
rI
ns
tru
ct
10. 3(2x – 4) + 6 = 12
89
3.2c Homework: Model and Solve Equations, Practice and Extend to Distributive Property
Use a model to solve each. Show your algebraic manipulations on the right.
1. 9 = 15 + 2p
–3
N
ot
R
ep
rin
t.
=
y.
D
o
2. –7 = 2h – 3
–5x – 12 = 13
=
x = –5
D
ra
f
t:
fo
rI
ns
tru
ct
3.
io
na
lP
ur
p
os
es
O
nl
=
h = –2
90
4. 6 = 1 – 2n + 5
n=0
8x – 2 – 7x = –9
D
o
5.
N
ot
R
ep
rin
t.
=
x = –7
io
na
lP
ur
p
os
es
O
nl
y.
=
n=3
=
D
ra
f
t:
fo
rI
ns
tru
ct
6. 2(n – 5) = –4
91
7. –3(g – 3) = 6
=
D
o
N
ot
R
ep
rin
t.
g=1
O
nl
y.
8. –12 = 3(4c + 5)
D
ra
f
t:
fo
rI
ns
tru
ct
io
na
lP
ur
p
os
es
=
92
**This is the first equation that
has a fraction answer.
c = –9/4
3.2d Class Activity: Error Analysis
Students in Mrs. Jones’ class were making frequent errors in solving equations. Help analyze their errors.
Examine the problems below. When you find the mistake, circle it, explain the mistake and solve the
equation correctly. Be prepared to present your thinking.
Student Work
Explanation of Mistake
Correct Solution Process
1. 6t  30
ep
rin
t.
The student forgot the negative.
Multiplicative inverse is (–1/6)
D
o
The Multiplicative invers is 4/3
N
ot
R
2.
y.
8  5c  37
O
nl
3.
io
na
lP
ur
p
os
es
On the second line the student
forgot to keep the negative.
a – b = a + (–b)
4x  3  17
D
ra
f
5.
t:
fo
rI
ns
tru
ct
4.
+3
The student subtracted when
division was necessary. Student
applied additive inverse rather
than multiplicative inverse.
93
+3
6. 3(2x  4)  8
t.
The student did not distribute
accurately. There are 3 groups, of
(2x – 4). So there will be 6x and
– 8.
O
nl
y.
8.
io
na
lP
ur
p
os
es
The student failed to note a – (–
b) = a + b. Taking away –2x is
the same as adding 2x.
The student did not distribute
correctly. Likely not accounting
for the negatives.
fo
rI
ns
tru
ct
9. 2(x  2)  14
ra
f
t:
10. 3(2x  1)  4  10
D
R
N
ot
D
o
The student subtracted 2x twice
from the same side of the
equation. The equality was not
maintained.
ep
rin
7. 3x  2x  6  24
The student added terms that
were not like terms.
94
3.2d Homework: Practice Solving Equations (select homework problems)
Solve each equation, use Algebra Tiles if that will help you. Assign a portion of the problems: divide the
problems among groups of students. Each student should do 6 - 10 problems (for example, a column if in
groups of three), then student pairs check each other’s homework the next day.
1. 8  t  25
3. 3 y  13
2. 2n  5  21
13 = n
–10 = y
5. 5  b  8
6. 5  6a  5
8. 8 
11. 9 
os
io
na
lP
ur
p
13. 8  6   p  8
14. 7  8x  4x  9
4=x
–6 = p
tru
ct
16. 6(m  2)  12
rI
ns
4=m
t:
fo
19. 3(x 1)  21
D
ra
f
6=x
22. 4 14  8m  2m
–1 = m
25. 2 p  4  3p  9
-1 = p
24 = y
y.
n
6
8
–120 = n
–6 = t
y
 2  10
3
9.
21 = x
t
42
3
10.
x
5
7
O
nl
–91 = n
N
ot
n
5
7
0=a
es
7. 8 
–13 = b
D
o
4=k
R
4. 12  5k  8
ep
rin
t.
33 = t
17. 5(2c  7)  80
4.5 = c
20. 7(2c  5)  7
3=c
23. 1  5 p  3p  8  p
1=p
26. 8  x  5  1
12 = x
95
y
 4  12
5
12.
–80 = y
15. 8x  6  8  2x  4
3=x
18. 5(2d  4)  35
1.5 = d
21. 6(3d  5)  75
2.5 = d
24. 5 p  8 p  4  14
–6 = p
27. 12  20x  3  4x
x=15/24=5/8
3.2e Class Activity: Solve One- and Two-Step Equations with Rational Numbers (use algebra to find
solutions)
Before we begin… Additional practice is available at 3.2h
…how can we find the solution for this problem? …do you expect the value for x to be larger or smaller
than 4 for these problems? Explain.
Throughout this assignment, students should
make sense of the equation. Drawing models is
important throughout. As students become more
proficient, discuss multiplicative inverse.
t.
ep
rin
…how can you figure out the solutions in your head?
R
Model
N
ot
Structure
.25x  4
y.
D
o
Review equivalent forms of rational numbers from Chp
1
os
es
O
nl
Solve the equations for the variable. Show all solving steps. Check the solution in the equation (example #1
check: –13(3) = 39, true.). Be prepared to explain your work.
1.  13m  39
3. y  25  34
4. 2y  24
2.
m
=
–32
m=3
y = 59
y = –12
tru
ct
io
na
lP
ur
p
Compare 4 and 7. In 4)
two “y’s” are -24, in 7
half an x is 6. In both
we want to know what
one unknown is.
rI
fo
x = –¼
6. 13  25  y
12 = y
7.
D
ra
f
t:
5.
ns
Check:
96
x = 12
8.
x=8
Check:
10.
a = 0.246
11.
d = –1.624
12.
ep
rin
t.
9.
14.
15.
a = –2.5
es
O
nl
y.
D
o
13.
N
ot
R
Check:
17.
x = 40
18. 8.38v 10.71  131.382
v = –14.4
ns
tru
ct
16. 9.2r  5.514  158.234
r = 16.6
io
na
lP
ur
p
os
Check:
fo
t:
ra
f
19.
rI
Check:
21.
x=2
D
n = –3.094
20.
Check:
97
3.2e Homework: Solve One- andTwo- Step Equations (practice with rational numbers)
Solve the equations for the variable in the following problems. Use models if desired. Show all solving
steps. Check the solution in the equation.
1. 22  11k
3. x 15  21
2.
x = –49
x = –36
x = –5
6. 54  16  y
–70 = y
Check:
4. 3y  36
5.
Check:
io
na
lP
ur
p
8.
os
es
O
nl
y.
D
o
y = 12
N
ot
R
ep
rin
t.
–2 = k
7.
9.
x = 12
x = –8
tru
ct
x = 12
rI
ns
Check:
fo
10.
12.
j = -20
D
ra
f
t:
m = 98.992
11.
Check:
98
13. 5b  0.2
14.
15.
16. 3.8  13.4 p  460.606
17. 0.4 x  3.9  5.78
18.
x = 4.7
N
ot
m = –6.188
Check:
19.
io
na
lP
ur
p
20.
os
es
O
nl
y.
D
o
p = 34.09
R
Check:
ep
rin
t.
b = 0.04
x = 20
fo
D
ra
f
t:
Check:
rI
ns
tru
ct
x=2
99
21.
x = 25
3.2e Extra Practice: Equations with Fractions and Decimals
1.
2.
3.
N
ot
R
ep
rin
t.
Discuss “clearing fractions”
conceptually first.
5.
6.
io
na
lP
ur
p
os
es
O
nl
y.
4.
D
o
Check:
ct
Check:
9.
8.
ns
tru
7.
x = 3.5
D
ra
f
t:
fo
rI
n = 0.78125
Check:
100
10.
11.
12.
a = 3.75
ep
rin
t.
x=6
14.
k = 79.2
io
na
lP
ur
p
os
es
O
nl
y.
q = 15.5
15.
D
o
13.
N
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R
Check:
Check:
17.
D
ra
f
t:
fo
rI
ns
w = 120
tru
ct
16.
Check:
101
18.
h = –36
3.2f Class Activity: Create Equations for Word Problems and Solve
For each, draw a model to represent the context and then determine which of the equations will work to answer
the question. Explain your reasoning.
R
ep
rin
t.
1. Brielle has 5 more cats than Annie. If Brielle has 8 cats, how many does Annie have?
Will it work?
Why or Why Not?
D
o
Equation
N
ot
What is the unknown/variable? a = the number of cats that Annie has
Yes
This accurately describes the situation.
b. 5 = 8 + a
No
This has swapped the value that Brielle has with its increase over Annie's.
c. 8 = 5a
No
This says Brielle has 5 times as many cats as Annie.
d. 5 = 8a
No
This is a few steps off: wrong operator, wrong place.
os
es
O
nl
y.
a. 8 = 5 + a
ns
tru
ct
io
na
lP
ur
p
2. Three pounds of fruit snacks cost $4.25. How much does one pound of fruit snacks cost?
fo
Equation
rI
What is the unknown? (What does the variable represent?) y = the cost of one pound of fruit snacks
t:
a. y + 3 = 4.25
Will it work?
Why or Why Not?
This says one pound costs three dollars less than $4.25.
Yes
This shows three pounds by multiplying one pound by three.
c. y + y + y = 4.25
Yes
This shows three pounds by repeated addition of one pound.
d. 4.25 = 3 y
Yes
This is the same as b, but order has changed.
ra
f
No
D
b. 3y = 4.25
102
ep
rin
t.
3. Jim bought a tie for x dollars and a jacket for $37.50. Jack bought the same items as Jim. Together,
they spent $80. How much did each tie cost?
What is the unknown? (What does the variable represent?) x = the cost of a tie
Will it work?
Why or Why Not?
R
Equation
Yes
This equation lists all items and sums them.
B 2x + 2(37.50) = 80
Yes
This equation counts each item twice, then sums them.
C 2(x + 37.50) = 80
Yes
This equation sums each boy's cost, then multiplies by two.
D 2x + 80 = 37.50
No
This equation has mixed the total and the jacket cost.
O
nl
y.
D
o
N
ot
A x + x + 37.50 + 37.50 = 80
io
na
lP
ur
p
os
es
4. Bo bought some songs for $0.79 each, an album for $5.98, total price $8.35. How many songs did he
buy?
What is the unknown? (What does the variable represent?) x = the number of songs that Bo bought
Equation
Will it work?
A 0.79x + 5.98 = 8.35
Why or Why Not?
This equation multiplies the song number by the song price.
No
This equation is multiplying the album cost by the song number.
C 8.35 + 5.98 + 0.79 = x
No
This equation shows x as a sum of prices.
D 5.98 + 0.79x = 8.35
Yes
This is the same as equation via the commutative property.
tru
D
ra
f
t:
fo
rI
ns
B 5.98x + 0.79 = 8.35
ct
Yes
103
3.2f Homework: Create Equations for Word Problems and Solve
Draw a model for each context and then write the sentence as an algebraic equation. An example is given
below.
N
ot
R
ep
rin
t.
My mom’s height (h) is 8 inches more than my height (60 inches).
or
60 + 8 = h
y.
Equation: h – 8 = 60
D
o
What is unknown? My mom’s height.
es
O
nl
Solution: My mom’s height = 68 inches
io
na
lP
ur
p
os
Write out what each unknown stands for, write an equation to model the problem, then (if possible) find the
solution. Note: A solution can only occur if enough information is given. Make sure your equation matches
your model.
1. The blue jar has 27 more coins than the red jar.
2. My age is twice my cousin’s age.
tru
ct
b is the number of coins in blue jar
r is the number of coins in red jar
m is the my age
fo
rI
ns
c is the cousin’s age
m = 2c
4. Art’s jump of 18 inches was 3 inches higher than
Bill’s.
D
ra
f
t:
b = r +27
3. A car has two more wheels than a bicycle.
c is the number of wheels on a car
b is the height of Bill’s jump
b is the number of wheels on a bicycle
18 is the b + 3
c = b +2
b = 15
104
5. The sum of a number and its double is eighteen.
6. The $5 bill was $3 more than the cost of the
notebook
n is the cost of the notebook
n + 2n = 18
5=n+3
ep
rin
n = 2 The notebook costs $2.
8. The number of minutes divided by sixty gives us 3
hours.
N
ot
R
n = 6 The unknown number was 6.
7. The visiting team’s score was five points less than
our score (50 points).
t.
n is the unknown number
m is the the number of minutes
D
o
v is the visiting team’s score
3
Solution: The number of minutes (m) is 180.
10. A large popcorn and a drink together cost the same
as the movie ticket. I spent $10 on all three.
os
es
9. Bill has twice as much money as I do. Our money
together is $9.
O
nl
y.
50 – 5 = v
The visiting team's score (v) was 45 points.
io
na
lP
ur
p
m is the my money
D
ra
f
t:
fo
rI
ns
tru
ct
m + 2m = 9
My money (m) is $3
105
t is the cost of a movie ticket
t + t = 10
The cost of a movie ticket (t) is $5.
3.2g **Class Challenge: Multi-Step Equations
1. The following problems will involve all 5 steps below. Discuss these steps as a class—make sure everyone
agrees and understands the five steps.
Distribute
Collect like terms
Variables on one side, constants on other side
Divide (or multiply by fraction) to get a variable coefficient of 1
Check solution.
ep
rin
t.
1.
2.
3.
4.
5.
2. Using the 8 problems below, do one of the following.
R
N
ot
x = –1
c. 3(x – 6) − 4(x + 2) = – 21
x = –5
d. 7(5x – 2) − 6(6x – 1)= – 4
x = –4
e. 2a + (5a – 13) + 2a – 3 = 47
a =7
io
na
lP
ur
p
f. 3a + 5(a – 2) – 6a = 24
D
o
b. 3 = 2(x + 3) + x + 2 x + 2
y.
x =1
os
a. 2(4x + 1) − 11 x = –1
O
nl

Put one at a time on the chalkboard and have groups “relay race” to complete. Then compare and correct
the steps and solutions
Have groups be responsible for one problem to present to the class.
es

a = 17
c =1
h. 3(y + 7) – y = 18
y = 1.5
ct
g. 13 = 2(c + 2) + 3c + 2c + 2
ns
tru
3. Can you write and solve an equation for this problem?
t:
fo
rI
You are playing a board game. You land on a railroad and lose half your money. Then you must pay
$1000 in taxes. Finally you pay half the money you have left to get out of jail. If you now have $100,
how much money did you start with?
D
ra
f
Various equation may work. This is one example
You started with $2,400.
106
100
3.2h Extra Practice: Solve Multi-Step Equation Review
1. 6.2d  124
d = –20
k = –32.5
3.
a = 100
N
ot
R
ep
rin
t.
2.
4. g 12.23  10.6
D
o
Check:
io
na
lP
ur
p
os
es
h = –15
O
nl
g = 22.83
6.
y.
5.
7.
8.
9.
D
ra
f
t:
fo
rI
ns
w = 2.9736
tru
ct
Check:
Check:
107
10. 28  8x  4
11.
x=4
12.
x = –4
R
ep
rin
t.
s = 26
14.
w = –21
io
na
lP
ur
p
os
es
O
nl
y.
x=1
15.
D
o
13.
N
ot
Check:
ct
Check:
17.
v=9
D
ra
f
t:
fo
rI
ns
tru
16.
Check:
108
18. 3(x  1)  2(x  3)  0
x=9
21.
20.
19.
w = 1.2
Solve Multi-Step Equations (distribution, rational numbers)
22. 4(x  2x)  24
23. 5(5  x)  65
24. 6(4  6x)  24
ep
rin
t.
v = 46
N
ot
D
o
x = 2
R
Check:
x=0
io
na
lP
ur
p
os
es
O
nl
y.
x=8
Check:
26. 2(4x  1)  42
tru
x = 5.5
D
ra
f
t:
fo
rI
ns
x = –8
ct
25. 3(2x  3)  57
Check:
109
27. 7(2x  7)  105
x = 11
29. 5(7x  5)  305
30. 5(1 7x)  320
x=6
x=8
x=9
N
ot
R
ep
rin
t.
28. 3(7x  8)  150
Check:
32. 5(2x  3)  96
33.
y.
x = 8.1
a = 22.5479
io
na
lP
ur
p
os
es
O
nl
x=4
D
o
31. 3(5x  6)  78
tru
ct
Check:
ns
34.
x = –1
D
ra
f
t:
fo
rI
x = 10
36.
35.
Check:
110
x = –10
39.
38.
37.
n = 10.6003
b = 7.6032
R
ep
rin
t.
m = 21.29436
40. 2(4x  8)  32
D
o
N
ot
Check:
42. 2(4x  2)  76
x = 10
io
na
lP
ur
p
os
es
x=1
O
nl
x = –6
y.
41. 7(5x  8)  91
44. 4(9  x)  12
x=6
D
ra
f
t:
fo
rI
ns
43. 7(3x  7)  175
tru
ct
Check:
Check:
111
45. 5(7  6x)  175
x=7
46. 2(1x  4)  18
47. 3(7  4x)  33
x=1
x=3
49. 5(1x  7)  40
50. 8  3(5  2x) 1
N
ot
x=1
r = 22.0992
Check:
53.
io
na
lP
ur
p
52.
os
es
O
nl
y.
D
o
x=1
51.
R
Check:
ep
rin
t.
x=5
48. 3(10  6x)  84
x=2
ns
tru
ct
x=3
54.
fo
rI
Check:
56.
57.
ra
f
t:
55.
k = 153.395184
D
k = 0.4264
Check:
112
r = 0.5012
3.2i Self-Assessment: Section 3.2
Consider the following skills/concepts. Rate your comfort level with each skill/concept by checking the box that
best describes your progress in mastering each skill/concept.
Skill/Concept
Beginning
Understanding
Developing Skill
and
Understanding
N
ot
R
ep
rin
t.
1. Solve multi-step equations fluently
including ones involving calculations with
positive and negative rational numbers in a
variety of forms.
2. Convert between forms of a rational number
to solve equations.
D
o
3. Use variables to create equations that model
word problems.
O
nl
es
os
D
ra
f
t:
fo
rI
ns
tru
ct
io
na
lP
ur
p
6. Critique the reasoning of others.
y.
4. Solve word problems leading to linear
equations.
5. Connect arithmetic solution processes that
do not use variables to algebraic solution
processes that use equations.
Deep
Understanding,
Skill Mastery
113
Section 3.3: Solve Multi-Step Real-World Problems Involving
Equations and Percentages
Section Overview:
t.
Students will learn how to solve percent problems using equations. They will begin by modeling percent
problems using a drawn model. Then they will translate that model into an equation which they will then solve.
Students will use similar reasoning to move to problems of percent of increase and percent of decrease. Finally,
students will put all of their knowledge together to solve percent problems of all types.
ep
rin
Concepts and Skills to be Mastered (from standards )
O
nl
y.
D
o
N
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R
By the end of this section, students should be able to:
1. Recognize and explain the meaning of a given expression and its component parts when using percents.
2. Solve multi-step real-life percent problems involving calculations with positive and negative rational
numbers in a variety of forms.
3. Convert between forms of a rational number to simplify calculations or communicate solutions
meaningfully.
D
ra
f
t:
fo
rI
ns
tru
ct
io
na
lP
ur
p
os
es
Teachers note: In this section, students will solve percent problems using equations. Percent problems will be
revisited in Chapter 4 and solved using proportional thinking. You will want to relate the concepts in that
chapter. In this chapter, students may still prefer models to solve problems involving percent, that’s fine. Help
students connect models to equations throughout.
114
3.3a Classroom Activity: Percents with Models and Equations
Draw a model to help you solve the problems below. Then choose the algebraic equation(s) that
represents the situation. Solve that equation.
To pass a test you need to get at least 70% correct. There are 50 questions on the test. How many do you
need to answer correctly in order to pass?
a. Model:
es
O
nl
y.
D
o
N
ot
R
1.
ep
rin
t.
Access background knowledge regarding drawing models for percentage problems (chapter 1) and fluency
between fractions, decimals and percent. Students have not experienced equations and percentages
simultaneously. Observe their reasoning about which equation works to find the solution. Help them reason
through what each equation says/means, then practice on their own. There are several ways students may write
equations.
io
na
lP
ur
p
os
b. Choose the appropriate equation(s). Justify your choice.
c = 35 questions correct
D
ra
f
t:
fo
rI
ns
tru
ct
2. 25% of a box of cookies is oatmeal. If there are 50 oatmeal cookies in a box, how many cookies are there
total?
a. Model:
b. Choose the appropriate equation(s). Justify your choice.
(1/4)(50) = c
c(1/4) = 50
c = 50(4)
c = 200 total cookies
115
1/4 = c/50
Draw a model for each and then write an algebraic equation to solve each percent problem.
t.
3. 30% of my books are science fiction. If I have 60 science fiction books, how many books do I have?
ep
rin
0.30x= 60 so x = 200 books
y.
D
o
N
ot
R
5. 15% of the day was spent cleaning the house. If there are 24 hours in the day, how many of them were
spent cleaning the house?
h = 3.6 hours
O
nl
0.15(24) = h
io
na
lP
ur
p
os
es
6. I got 52 out of 60 questions right on the last History test. What percent correct did I get?
ct
60x = 52 so x = 86 2/3%
D
ra
f
t:
fo
rI
ns
tru
7. A pair of jeans are on sale for $35. Originally they were $45. What percent of the original price is the
sale price?
45x = 35 77.77…%
116
3.3a Homework: Percents with Models and Equations
Draw a model to help you solve the problems below. Then choose the algebraic equation(s) that would
represent the problem. Solve that equation.
ep
rin
t.
1. To get an A in math class, I need to get a 90% on the test. If the test has 40 questions, how many do I
need to get right in order to get an A?
a. Model:
y.
r = 36 questions correct
D
o
N
ot
R
b. Choose the appropriate equation(s). Justify your choice.
io
na
lP
ur
p
os
es
O
nl
2. 25% of the club came to the meeting. 3 people were at the meeting. How many people are in the club?
a. Model:
p = 12 total members
tru
ct
b. Choose the appropriate equation(s). Justify your choice.
D
ra
f
t:
fo
rI
ns
3. 32 of the 48 people at the gym are wearing blue. What percent are wearing blue?
a. Model:
b. Choose the appropriate equation(s). Justify your choice.
b = 0.66… so 66 2/3% are wearing blue
117
Draw a model for each and then write an algebraic equation to solve each percent problem.
4. 65% of the population needs to vote for the new law in order for it to pass. There are 800 voters. How
many need to vote for the new law in order for it to pass?
p = 520 voters
t.
800(0.65) = p
R
ep
rin
5. 5% of the apples have worms in them. 10 apples had worms in them. How many apples are there total?
w = 200 total apples
N
ot
w(0.05) = 10
O
nl
y.
D
o
6. 250 students dressed up for Spirit Day. There are 800 students. What percent dressed up for Spirit
Day?
es
p = 31.25% dressed for Spirit Day
io
na
lP
ur
p
os
7. 55% of my shirts are purple. If I have 20 shirts, how many of them are purple?
20(0.55) = p
p = 11 purple shirts
tru
ct
8. 5 out of the 7 ducklings have yellow feathers. What percent of the ducklings have yellow feathers?
ns
y = 71.4…% have yellow feathers
s(0.30) = 24
s = 80 total sodas
ra
f
t:
fo
rI
9. 30% of the sodas are grape. There are 24 grape sodas. How many sodas are there total?
D
10. I made 36 cupcakes, that’s 80% of what I need. How many cupcakes will I be making all together?
0.8x = 36. I need 45 cupcakes total
118
3.3b Class Activity: Percent Problems
For each context below: a) draw a model, b) select the expression that represents the situation, and then c)
justify your answer:
y.
D
o
N
ot
R
ep
rin
t.
1. Last week, Dirk jumped y inches in the long jump. This week, he increased the length of his jump by
10%.
a. Draw a model to represent this situation.
0.90y
1.10y
y + 0.10y
io
na
lP
ur
p
os
0.10y
es
O
nl
b. Which of the following algebraic expressions represents the length of his jump now? (Be
prepared to explain your answers and how you know they are correct!)
c. I need all of y and then 0.1 of y. Discuss why the other two are common mistakes.
fo
rI
ns
tru
ct
2. Hallie wants to buy a pair of jeans for h dollars. She knows she will have to pay 6% sales tax along with
the price of the jeans.
a. Draw a model to represent this situation.
D
ra
f
t:
b. Which of the following algebraic expressions represents the price of the jeans with tax?
0.06h
0.94h
0.6h
1.6h
1.06h
c. If the original price of the jeans was $38, what is the price with tax? Show at least three different
ways to get your answer.
$40.28
1.06(38) = p
38+38(.06) = p
38(1+.06) = p
119
3. Jamie has a box with x chocolates in it. Her little brother ate 25% of the chocolates.
a. Draw a model to represent this situation.
x
0.75x
x – 0.25x
(1 – 0.25)x
ep
rin
0.25x
t.
b. Which of the following algebraic expressions represents the amount of chocolate in her box
now? (Be prepared to explain your answers and how you know they are correct!)
O
nl
y.
D
o
N
ot
R
4. Drake wants to buy a new skateboard with original price of s dollars. The skateboard is on sale for 20% off
the regular price.
a. Draw a model to represent this situation.
s
b. Which of the following algebraic expressions represents the sale price of the skateboard?
0.8s
es
0.2s
s – 0.20
1s – 0.20 s
os
s – 0.80
io
na
lP
ur
p
c. If the original price of the skateboard was $64, what is the sale price? Show at least three
different ways to get your answer. $51.20
64(0.8) = p
64 − 64(0.20) = p
64(1 − 0.20) = p
fo
rI
ns
tru
ct
5. Alayna makes delicious cupcakes. She estimates that one-dozen cupcakes cost $7.50 to make. She wants a
50% mark up on her cupcakes. How much should she sell one-dozen cupcakes for?
a. Draw a model to represent this situation:
ra
f
t:
b. Write an expression to represent what she should charge.
D
1.5($7.50)
c. How much should she charge? $11.25
6. There were 850 students at Fort Herriman Middle School last year. The student population is expected to
increase by 20% next year. What will the new population be?
120
Draw a model to represent this situation.
Write an expression to represent the new population.
t.
1.20(850)
ep
rin
What will the new population be? 1020
N
ot
R
7. A refrigerator at Canyon View Appliances costs $2200. This price is a 25% mark up from the whole sale
price. What was the wholesale price?
Write an expression to represent the whole sale price
y.
D
o
Draw a model to represent this situation.
io
na
lP
ur
p
os
es
O
nl
$2200/1.25
What was the whole sale price? $1760
ns
tru
ct
8. Carlos goes the ski shop to buy a $450 snowboard that’s on sale for 30% off. When he gets to the store, he
gets a coupon for an addition 20% the sale price. What will he pay for the snow board?
Draw a model to represent this situation.
Write an expression to represent the problem
situation.
D
ra
f
t:
fo
rI
(0.80)(0.70)($450)
What will Carlos pay for the snowboard? $252
3.3b Homework: Percent Problems
1. Dean took his friend to lunch last week. His total bill was b dollars. He wants to tip the waitress 20%.
How much will Dean pay, including the 20% tip? (Don’t worry about tax in this problem.)
121
b. Draw a model to represent this situation.
c. Which of the following expressions represent the amount that Dean will pay?
0.8b
b + 0.2b
1.2b
1.8b
1 + 0.2b
$27
22.50 + 2.250(0.2) = p
22.50(1+.2) = p
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22.50(1.2) = p
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ways to get your answer.
t.
d. If Dean’s bill was $22.50, how much will Dean pay including the 20% tip? Show at least three different
y.
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5. Philip took a vocabulary test and missed 38% of the problems. There were q problems on the test.
a. Draw a model to represent this situation.
q – 0.38
q – 0.38q
0.62q
(1 – 0.38)q
es
0.38q
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b. Which of the following expressions represent the number of problems that Philip got correct?
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p
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c. Which of the following expressions represent the number of problems that Philip missed?
0.38q
1q – 0.62q
q – 0.38q
0.62q
(1 – 0.38)q
d. If there were 150 problems on the test, how many did Philip get correct? Show at least three
different ways to get your answer.
93 correct
0.62(150) = c
150(1−0.38) = c
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150 – 0.38(150) = c
6. Chris would like to buy a picture frame for her brother’s birthday. She has a lot of coupons but is not sure
which one to use. Her first coupon is for 50% off of the original price of one item. Normally, she would
use this coupon. However, there is a promotion this week and the frame is selling for 30% off, and she has
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a coupon for an additional 20% any frame at regular or sale price. Which coupon will get her the lower
price? She is not allowed to combine the 50% off coupon with the 20% off coupon.
a. Draw a model to show the two different options.
50% off coupon:
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t.
30% off sale with additional 20% off coupon:
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b. Let x represent the original price of the picture frame. Write two different expressions for each
option.
x – x(0.5)
y.
x(0.5)
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50% off coupon:
(x − 0.3x)0.8
es
(0.7x)0.8
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30% off sale with additional 20% off coupon:
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c. Which coupon will get her the lowest price? Explain how you know your answer is correct.
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The 50% off coupon will get her the lowest price.
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3.3c Class Activity: More Practice with Percent Equations
ep
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t.
1. Selina gave the waiter a $2.25 tip at the restaurant. If her meal cost $12.50, what percent tip did she give?
a) Let t represent the percent of the tip. Which of the following equations are true?
b) What is percent tip did she give?
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0.18 = t so the tip was 18%
es
x = 50
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0.06x = 3
y.
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Write an equation with a variable for each problem below, then solve the equation. Justify your work.
Have students do these in groups of 2-3 and then present their answers to the class
2. Beatrice bought a new sweater. She paid $3 in sales tax. If sales tax is 6%, what was the original price of the
sweater?
os
The original price of the sweater was $50 dollars.
io
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lP
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p
3. Jill bought a bracelet that cost $5. Sales tax came out to be $0.24. What is the sales tax rate?
5t = 0.24
t = 0.048
ct
Sales tax is 4.8%.
d = (1.25)*(1.25)(12)
On Saturday, she will run 18.75 miles.
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t:
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4. Kaylee is training for a marathon. Her training regiment is to run 12 miles on Monday, increase that distance by
25% on Wednesday, and then on Saturday increase the Wednesday distance by 25%. How far will she run on
Saturday?
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R
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t.
5. Juan is trying to fit a screen shot into a report. It’s too big, so he reduces is first by 30%. It still doesn’t fit,
so he reduces that image by 20%. What percent of the original image did he paste into his report? If the
original image was 8 inches wide, how wide is the twice reduced image?
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(0.8)(0.7)(8 inches) The image is now 4.48 inches wide.
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p
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y.
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6. The size of Mrs. Garcia’s class increased 20% from the beginning of the year. If there are 36 students in her
class now, how many students were in her class at the beginning of the year?
ct
x = the number of students at the beginning of the year. x + 0.2 x = 36, 30 students at the beginning of the
year.
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t:
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7. Mika and her friend Anna want to give 20% of the money they make at a craft fair to charity. If Mika makes
$500 and they want to give a total of $150. How much will Anna have to make?
Let A be the amount Anna will need to make; 0.2(500 + A) = 150 A=$250
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3.3c Homework: More Practice with Percent Equations
Write an equation with a variable for each problem. Then solve the equation. Justify your answer.
1. John paid $3.45 in sales tax on his last purchase. What was the original price if the tax rate is 3%?
p(0.03) = 3.45
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t.
p = 115
John spent $115 on his last purchase.
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2. Newt paid $45 in sales tax for his new television. If tax is 6%, what was the original price of the television?
p(0.06) = 45
D
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p = 750
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y.
The original price of the television was $750.
os
es
3. Carter gave the waitress a tip of $8.75. If the original price of his meal was $24.95, what percent of the price
was the tip?
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lP
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p
24.95t = 8.75
t = 0.3507
Carter gave a 35% tip.
t = 0.05021
The tax rate is 5%.
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tru
ct
4. Louie paid a total of $12.55, with tax, for his new frying pan. If the original price of the pan was $11.95,
what was the tax rate?
11.95 + 11.95t = 12.55
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5. Robert paid $14.41, with tax, for his model airplane kit. If tax was 6%, what was the original price of the kit?
(1.06)p = 14.41
p = 13.59
The original price of the kit was $13.59.
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3.3d Self-Assessment: Section 3.3
Consider the following skills/concepts. Rate your comfort level with each skill/concept by checking the box that
best describes your progress in mastering each skill/concept.
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1. Recognize and explain the meaning of a
given expression and its component parts
when solving problems with percents.
Deep
Understanding,
Skill Mastery
t.
Developing
Skill and
Understanding
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Skill/Concept
Beginning
Understanding
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y.
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2. Solve multi-step real-life percent problems
involving calculations with positive and
negative rational numbers in a variety of
forms.
3. Convert between forms of a rational
number to simplify calculations or
communicate solutions meaningfully.
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