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```Chapter Focus
1
1
Using the Interactive
Student Guide
The Interactive Student Guide (ISG)
can be used in conjunction with
Algebra 1.
ISG
Lesson
Expressions, Equations, and Functions
CHAPTER FOCUS Learn about some of the Common Core State Standards that
you will explore in this chapter. Answer the preview questions. As you complete each
What You Will Learn
N.RN.3 Explain why the sum or product of two
rational numbers is rational; that the sum of a rational
number and an irrational number is irrational; and
that the product of a nonzero rational number and an
irrational number is irrational.
Extend 10–2
1.2
Lesson 1–1
1.3
Lesson 1–2
1.4
Lessons 1–3, 1–4
1.5
Lesson 1–5
1.6
Lesson 1–6
1.7
Lesson 1–7
SMP 3 When you multiply two rational numbers,
is the result always another rational number? Why or
why not?
The result is always another rational number
because the result can be expressed as a
quotient of two integers.
Algebra 1
1.1
Preview Question
Lesson 1.1: Rational and Irrational Numbers
SMP 7 Give an example of a product of two
irrational numbers that is also irrational and a product
of two irrational numbers that is rational.
_
_
_
Sample answer: √2 · √3 = √6 is irrational.
_
_
√2 · √2 = 2 is rational.
Lesson 1.2: Variables and Expressions
A.SSE.1a Interpret parts of an expression,
such as terms, factors, and coefficients.
A.SSE.2 Use the structure of an expression
to identify ways to rewrite it.
SMP 2 The expression 14x + 5 represents the
cost of ordering x copies of a book online, including
a flat fee for shipping. Which term in the expression
represents the shipping fee? How do you know?
The 5 in the expression is the shipping fee
because this term of the expression does not
depend on the number of books ordered.
Lesson 1.3: Order of Operations
A.SSE.1b Interpret complicated expressions by
viewing one or more of their parts as a single entity.
A.SSE.2 Use the structure of an expression to
identify ways to rewrite it.
SMP 6 How do you evaluate the expression
52 - 18 ÷ 6 + 3? Do you get a different result if the
expression is written as 52 - 18 ÷ (6 + 3)? Why or
why not?
Per order of operations, exponents are simplified
first followed by multiplication and division from
left to right and then addition and subtraction
from left to right. With parentheses, the value is
23 rather than 25 because (6 + 3) is found first.
2 CHAPTER 1 Expressions, Equations, and Functions
002_003_ALG1_C1FP_ISG_672788.indd 2
2 CHAPTER 1 Expressions, Equations, and Functions
2/20/15 5:21 PM
SMP 2
Teaching Tip
The preview question for Lesson 1.2
gives students a taste of SMP 2
(Reason abstractly and
quantitatively). Point out to students
that the expression 14x + 5, by itself,
is an “abstract” mathematical
expression. In other words, it can be
manipulated and understood using
mathematical rules, but it does not
necessarily carry a real-world
meaning. Explain that the scenario of
ordering books online gives the
expression a “context.” Students will
see many additional examples of this
throughout the course and have many
opportunities to move back and forth
between abstract mathematics and
contextualized mathematics.
What You Will Learn
Preview Question
Lesson 1.4: Properties of Numbers
A.SSE.2 Use the structure of an expression to
identify ways to rewrite it.
A.SSE.1b Interpret complicated expressions by
viewing one or more of their parts as a single entity.
SMP 7 How can you rewrite the expression
8 · 89 so it can be simplified using mental math?
8 · 89 = 8(90 - 1) = 8 · 90 − 8 · 1 =
720 - 8 = 712
Lesson 1.5: Equations
A.CED.1 Create equations and inequalities in
one variable and use them to solve problems.
A.REI.3 Solve linear equations and inequalities
in one variable, including equations with coefficients
represented by letters.
A.REI.1 Explain each step in solving a simple
equation as following from the equality of numbers
asserted at the previous step, starting from the
assumption that the original equation has a solution.
Construct a viable argument to justify a solution
method.
SMP 4 Jonas spent \$7.75, excluding tax, at the
store when he bought a notebook that cost \$2.50 and
3 pens. Write and solve an equation to find the price
of each pen.
Let x = the cost of each pen, 3x + 2.50 = 7.75;
x = 1.75; each pen costs \$1.75
Lesson 1.6: Relations
A.REI.10 Understand that the graph of an
equation in two variables is the set of all its solutions
plotted in the coordinate plane, often forming a curve
(which could be a line).
SMP 1 Use set notation to write the domain and
range of the relation {(-2, 3), (4, 4), (4, 5), (5,-8)}.
Domain: {-2, 4, 5}; Range: {3, 4, 5, -8}
Lesson 1.7: Functions
F.IF.1 Understand that a function from one set
(called the domain) to another set (called the range)
assigns to each element of the domain exactly one
element of the range. If f is a function and x is an
element of its domain, then f(x) denotes the output
of f corresponding to the input x. The graph of f is the
graph of the equation y = f(x).
F.IF.2 Use function notation, evaluate functions
for inputs in their domains, and interpret statements
that use function notation in terms of a context.
F.IF.5 Relate the domain of a function to its
graph and, where applicable, to the quantitative
relationship it describes.
SMP 3 A student said that the relation {(-2, 3),
(4, 4), (4, 5), (5,-8)} is a function. Do you agree? Why
or why not?
No; the domain value 4 is paired with two
SMP 4
Teaching Tip
The preview question for Lesson 1.5
can serve as a jumping-off point for a
brief discussion of SMP 4 (Model
with mathematics). Explain to
students that mathematical modeling
is the process of using mathematics
to describe a real-world situation
and/or solve a real-world problem.
Ask students to state the real-world
situation in this question in their own
words. Then ask them how they think
mathematics could be useful in
solving the problem. Tell students
that equations are one way to model a
real-world situation and explain that
they will have many opportunities to
develop and use this type of model
throughout this course.
different range values, so the relation cannot be
a function.
SMP 2 The amount of money in Trina’s bank
account, in dollars, n months after she opens the
account is given by the function b(n) = 50n + 100.
What are the meanings of the values 50 and 100?
What is the balance in Trina’s account after 10 months?
The 50 means that Trina adds \$50 to her
account each month. The 100 means that Trina
opened her account with a deposit of \$100. Trina
will have \$600 in her account after 10 months.
CHAPTER 1 Chapter Focus 3
2/24/15 11:53 PM
002_003_ALG1_C1FP_ISG_672788.indd 3
CHAPTER 1 Chapter Focus 3
1.1
Rational and
Irrational Numbers
1.1 Rational and Irrational Numbers
STANDARDS
Objectives
STANDARDS
Content: N.RN.3
Practices: 2, 3, 4, 5, 6, 7
Use with Extend 10–2
• Explain why the sum or product of two rational numbers is rational.
N.RN.3 Explain why the sum or
product of two rational numbers is
rational; that the sum of a rational
number and an irrational number is
irrational; and that the product of
a nonzero rational number and an
irrational number is irrational.
• Explain why the sum of a rational number and an irrational number
is irrational.
• Explain why the product of a nonzero rational number and an
irrational number is irrational.
a
A rational number is a number that can be written in the form __b , where a and b are
integers and b ≠ 0. The decimal form of a rational number is a repeating decimal or a
terminating decimal.
a
An irrational number is a number that cannot be expressed in the form __b , where a and b
are integers and b ≠ 0. The decimal form of an irrational number is neither repeating
nor terminating.
EXAMPLE 1
Standards for Mathematical
Practice: 2, 3, 4, 5, 6, 7
Sums of Rational and Irrational Numbers
N.RN.3
EXPLORE The table shows examples of rational and irrational numbers.
a. CALCULATE ACCURATELY Choose two of the rational numbers from the
table and find their sum. Is the sum rational or irrational? Explain how
you know. SMP 6
1
1
2
11
1
__
__
__
__
3 + 0.4 = 3 + 5 = 15 . The sum of 3 and 0.4 is
Rational Numbers
__1
__3
3 0.4- 7- 5
Irrational Number
__
__
√2 π-√5
a
rational because the sum can be written as __
b , where a and b
PREREQUISITES
are integers b ≠ 0.
• Identify rational and irrational
numbers
• Use basic properties of square roots
Teaching Tip
The sum of two rational numbers is another rational number.
c. USE TOOLS Choose a rational number and an irrational number from the table
and find their sum using a calculator. Does the sum appear to be rational or
SMP 5
irrational? Why? SMP 5
Part c offers an opportunity to
tools strategically). In particular, you
may want to discuss the limitations of
using a calculator to decide whether a
number is rational or irrational.
Remind students that calculators can
only show a limited number of digits,
but that they can look for terminating
or repeating decimals to make an
number is rational.
neither repeating nor terminating.
d. MAKE A CONJECTURE Compare your result for part c with those of other students.
Then make a conjecture about the sum of a rational number and an irrational number. SMP 3
The sum of a rational number and an irrational number is an irrational number.
4 CHAPTER 1 Expressions, Equations, and Functions
Math Background
004_007_ALG1_C1L1_ISG_672788.indd 4
22/08/14 9:18 PM
Every point on the number line represents a real number. Every real
number is either rational or irrational. In this lesson, students state
and justify generalizations about certain combinations of rational and
irrational numbers, such as the sum of two rational numbers.
Students may ask about other combinations, such as the sum of two
irrational numbers. In this case, the results are more complicated. In
fact, the sum of_two irrational
numbers may be rational or irrational.
_
For example, √2 + √_3 is irrational.
_ However, the sum of the
irrational numbers √2 and (1 - √ 2 ) is 1, which is rational. Students
will explore these combinations, and other examples of irrational
numbers (such as e), in later mathematics courses.
• In part c, how can a calculator help
you decide if the sum is rational or
irrational? Look at the decimal
form of the sum to see if it
repeats or terminates.
4 CHAPTER 1 Expressions, Equations, and Functions
Scaffolding Questions
• What should you do to decide if the
sum of the two rational numbers
you chose is rational or irrational?
Check to see if the sum can be
written as a ratio of two integers.
1
3 + π ≈ 3.474925987. The sum appears to be irrational because the decimal is
EXAMPLE 1
b. MAKE A CONJECTURE Compare your result for part a with those of other students.
SMP 3
Then make a conjecture about the sum of two rational numbers. EXAMPLE 2
You can use reasoning and the definitions of rational and irrational numbers to construct
logical arguments to show that the conjectures you made are true.
EXAMPLE 2
Justify a Conjecture
Teaching Tip
N.RN.3
Explain to students that they based
their conjecture on several specific
examples, but a general proof of a
conjecture must rely on definitions
and properties. Therefore, variables
are used to represent the numbers in
this argument. This ensures that the
argument will hold for all rational
numbers.
Complete the following argument to explain why the sum of two rational numbers
is rational.
a. USE STRUCTURE Suppose x and y are both rational numbers. You must show that
x + y is also a rational number. Since x and y are rational numbers, you can write
a
c
x = __b and y = __d . What must be true about a, b, c, and d? SMP 7
a, b, c, and d are integers, with b ≠ 0 and d ≠ 0.
b. USE STRUCTURE Write the sum of x and y as a single fraction in terms of a, b, c, and d. ad + bc
a
c
__
______
x + y = __
b+d=
bd
c. CONSTRUCT ARGUMENTS Explain why this shows that x + y is a rational number. SMP 7
SMP 3
ad + bc is a sum of products of integers, so it is an integer; bd is a product of nonzero integers,
so it is a nonzero integer; therefore, x + y is a rational number because it can be written as a
quotient of integers.
EXAMPLE 3
Justify a Conjecture Using a Contradiction
Scaffolding Questions
• What do you think we will do in
order to show that x + y is a rational
number? Show that it can be
written as a quotient of integers.
N.RN.3
Complete the following argument to explain why the sum of a rational number and
an irrational number is irrational.
a. USE STRUCTURE Suppose x is a rational number and y is an irrational number. You
must show that x + y is irrational. In this argument, you will assume that the sum is a
rational number and show that this leads to a contradiction. Suppose x + y = z and z is
a
c
a rational number. Then you can write x = __b and z = __d . What must be true about
SMP 7
a, b, c, and d? a
b. USE STRUCTURE Solving for y shows that y = z - x. Use this to write y as a single
fraction in terms of a, b, c, and d. SMP 7
a
cb - da
__
______
y = z - x = __
d-b=
db
c. USE STRUCTURE What does this say about y? Explain why this is a contradiction. c
+ ___ as a single
• How do you write ___
b d
fraction? Find a common
a, b, c, and d are integers, with b ≠ 0 and d ≠ 0.
c
SMP 3
• How do you know that bd ≠ 0?
Since b ≠ 0 and d ≠ 0, the
product cannot be 0.
SMP 7
This says that y is a rational number because it is written as a quotient of two integers. This is a
contradiction because y is an irrational number.
d. CONSTRUCT ARGUMENTS What does the contradiction allow you to conclude? Why? SMP 3
The contradiction shows that the assumption that x + y is a rational number must be false.
EXAMPLE 3
Therefore, x + y is an irrational number, which is what was to be proved.
1.1 Rational and Irrational Numbers 5
Emphasizing the Standards for Mathematical Practice
Example 3 is an essential connection to SMP 3 (Construct viable
arguments and critique the reasoning of others) because it
introduces students to a form of reasoning that they will use in future
also known as indirect proof, can be
_
used to show that √2 is irrational and that there are infinitely many
prime numbers. In geometry, students will use indirect proof to
explain why a triangle cannot have two right angles.
004_007_ALG1_C1L1_ISG_672788.indd 5
22/08/14 9:18 PM
Be sure to highlight the main steps of the process for students: (1)
Assume the opposite of the conjecture that you are trying to prove;
(2) Show that this leads to a contradiction of a known fact; (3)
Conclude that since the assumption is false, the original conjecture
must be true.
Teaching Tip
SMP 3
Discuss why the different methods of
proof are used for Examples 2 and 3.
Focus the discussion on the idea that
there is a form in which all rational
numbers can be expressed, but there is
no form in which to easily express
irrational numbers. This is why the
indirect proof is used for Example 3.
Scaffolding Questions
• What are the known facts at the
beginning of this argument? x is
rational and y is irrational.
• What are you trying to show? x + y
is irrational.
• How do you solve x + y = z for y?
Subtract x from both sides of the
equation.
1.1 Rational and Irrational Numbers 5
PRACTICE
PRACTICE
Connecting Exercises
to Standards
requirements of N.RN.3. Exercises
1–4, 9, and 10 require students to
irrational numbers or to provide a
counterexample to a given statement.
1. CONSTRUCT ARGUMENTS Explain why the product of two rational numbers
N.RN.3, SMP 3
is rational. a
c
__
Suppose x and y are both rational numbers. Then x = __
b and y = d , where a, b, c, and d are
a __
c
ac
__
integers, with b ≠ 0 and d ≠ 0. So, xy = __
b · d = bd . Both ac and bd are integers since they are
the product of integers. Therefore, xy is a rational number because it can be written as the
quotient of two integers.
2. The product of a nonzero rational number and an irrational number is irrational.
a. CONSTRUCT ARGUMENTS Explain why this statement is true. Use an argument
Suppose x is a nonzero rational number and y is an irrational number. Assume that xy = z and
a
c
__
that z is a rational number. Then x = __
b and z = d , where a, b, c, and d are integers, with a ≠ 0,
c
Exercises 5–7 require students to
determine whether the means of
given sets of numbers can be an
irrational number.
a
c
b
cb
__
__ __
__
b ≠ 0, and d ≠ 0. Solving for y shows that y = z ÷ x = __
d ÷ b = d · a = da . This shows that y is a
rational number since it is written as the quotient of two integers. This is a contradiction, so z
must be an irrational number.
b. COMMUNICATE PRECISELY Why is the word nonzero important in the statement?
How do you use this fact in your argument? N.RN.3, SMP 6
The statement is not true when the rational number is 0. For example, the product of 0 and
Exercise 8 requires students to find
two irrational numbers that satisfy
given conditions.
π is 0, which is rational. You use the fact that the rational number x is nonzero in the argument
when you divide both sides of xy = z by x.
3. CRITIQUE REASONING Alyssa said the square of an irrational number must also be
irrational. Do you agree or disagree? Justify your answer. N.RN.3, SMP 3
__
Dual Coding
CCSS
SMP
1
N.RN.3
3
2
N.RN.3
3, 6
3–4
N.RN.3
3
5–7
N.RN.3
6
8
N.RN.3
4
9
N.RN.3
2
10
N.RN.3
3
4. CONSTRUCT ARGUMENTS A set is closed under an operation if for any numbers in
the set, the result of the operation is also in the set. For example, the set of integers is
closed under addition because the sum of two integers is another integer. Is the set of
irrational numbers closed under multiplication? If so, explain why. If not, give a
N.RN.3, SMP 3
counterexample. __
__
No; the product of two irrational numbers is not necessarily rational. For example, √2 and √8 are
__
__
Exercise
__
Disagree; √2 is irrational, but (√2 )2 = 2, which is rational.
__
irrational, but √2 · √8 = √16 = 4, which is rational.
6 CHAPTER 1 Expressions, Equations, and Functions
Common Errors
Students may not understand that a conjecture must be proven to be
true for all cases, but that it can be shown to be false with a single
well-chosen counterexample. In Exercise 3, some students might use
their calculator to evaluate π2, notice that this value appears to be
irrational, and conclude that Alyssa’s statement is true. Point out to
students that they would need to give a general argument, similar to
those in Exercises 1 and 2, in order to show that Alyssa’s statement is
true. On the other hand, a single example of an irrational number
whose square is rational is enough to conclude that Alyssa’s
statement is false.
004_007_ALG1_C1L1_ISG_672788.indd 6
6 CHAPTER 1 Expressions, Equations, and Functions
22/08/14 9:18 PM
COMMUNICATE PRECISELY The table shows how Ms. Rodriguez assigns
scores in her biology class. Determine whether each statement is always,
sometimes, or never true. Explain. N.RN.3, SMP 6
5. The average of a student’s midterm exam score and ﬁnal
exam score is a whole number.
Sometimes; the average of two whole numbers is either a
whole number or halfway between two whole numbers.
Category
Common Errors
In Exercise 4, students may recognize
that the set of irrational numbers is
not closed under multiplication, but
they may have trouble giving a valid
counter example. For instance,
_
4
students
may
offer
the
numbers
√
_
and √9 as a counter example, since
_
the product of these numbers, √36 , is
rational. Help students understand
that the first step in finding a counter
example consists of choosing two _
irrational
_ numbers. The numbers √ 4
and √9 do not work as a
counterexample since these values
are rational. A brief class discussion
about what works as a valid
counterexample is an excellent way to
arguments and critique the
reasoning of others).
Type of Score
Midterm Exam
Whole number
from 0 to 100
Final Exam
Whole number
from 0 to 100
Quizzes
Rational number
from 0 to 10
Homework
0, 4 , 2 , 4 , 1
3
__
1 __
1 __
6. The average of a student’s quiz scores is a rational number.
Always; the sum of rational numbers is rational, and dividing this sum by a whole number results
in a rational number.
7. The average of a student’s homework scores is an irrational number.
Never; the scores are rational numbers and the sum of rational numbers divided by a whole
number cannot be an irrational number.
y ft
8. USE A MODEL Determine irrational values for x and y so that the area
in square feet of the rectangular carpet is a rational number greater
than 100 but less than 200. Justify your answer. N.RN.3, SMP 4
____
____
Sample answer: x = √125 , y = √180 ; the area of the carpet
____
is √125 ·
____
x ft
_______
√180 = √22,500 = 150 ft2.
9. REASON QUANTITATIVELY Without performing any calculations, determine if the
2
represents a rational number. Justify your answer. expression 5.323232 . . . + 6 ___
3
N.RN.3, SMP 2
Yes; a repeating decimal is a rational number, and so is a mixed fraction with integers. The sum of
two rational numbers is a rational number.
10. CRITIQUE
___
___ REASONING Amanda claims that the product
√3 · (√3 - 7) is irrational. Her argument is shown at the right.
Do you agree with Amanda’s argument? What about her
conclusion? Explain. N.RN.3, SMP 3
Amanda’s reasoning is flawed. In her final statement,
she should not have concluded that the product of
The difference √3 – 7 can be written as sum of
an irrational and (nonzero) rational number,
as √3 + (–7). This represents an irrational
number, because the sum of an irrational and
rational number is irrational. The number √3 is
also irrational, so the product of two irrational
numbers is also irrational.
two irrational numbers is irrational. Her conclusion is
__
__
correct, because the product can be written as 3 + (-7 √3 ). -7 √3 is the product of a rational
number and an irrational number, and is therefore irrational, and the sum of a rational number
__
and an irrational number is irrational, so 3 + (-7 √3 ) is irrational.
1.1 Rational and Irrational Numbers 7
Emphasizing the Standards for Mathematical Practice
Exercises 5–7 provide an opportunity for students to work with SMP 6
(Attend to precision), because they will need to “make explicit use of
definitions” of different types of numbers. For example, students may
decide to explore Exercise 5 by finding the average of some pairs of
whole numbers that represent test scores, such as 80 and 82. In this
case, the average is 81, which is another whole number. Students who
think carefully about their calculations might search for a pair of whole
numbers that results in a different outcome. The whole numbers 90
and 91 have an average of 90.5, which is not a whole number. Stepping
through this thought process requires an understanding of the
definition of a whole number. Remind students to consult the glossary
in their text if they are unsure about any definitions.
004_007_ALG1_C1L1_ISG_672788.indd 7
8/26/14 11:37 AM
1.1 Rational and Irrational Numbers 7
1.2
Variables and
Expressions
1.2 Variables and Expressions
STANDARDS
Objectives
STANDARDS
A.SSE.1a Interpret parts of an
expression, such as terms, factors,
and coefficients.
A.SSE.2 Use the structure of an
expression to identify ways to
rewrite it.
Content: A.SSE.1a, A.SSE.2
Practices: 2, 3, 4, 7, 8
Use with Lesson 1–1
• Write algebraic expressions.
• Use the structure of an expression to identify ways to rewrite it.
• Interpret parts of an expression.
An algebraic expression consists of sums and/or products of numbers and variables.
A term of an expression may be a number, a variable, or a product or quotient of numbers
and variables. For example, in the expression 24m + 5n + 0.1, there are three terms:
24m, 5n, and 0.1.
EXAMPLE 1
Write an Algebraic Expression
A.SSE.2
EXPLORE The Watkins family is designing a new path for their garden. They use
black and white square tiles to make a pattern that they will use to build the path.
Standards for Mathematical
Practice: 2, 3, 4, 7, 8
Stage 1
PREREQUISITES
• Perform operations with rational
numbers
EXAMPLE 1
Scaffolding Questions
• How many white tiles would you
need for Stage 10? How many black
tiles? 20; 13
• How can you check that the
expressions you wrote are correct?
Check to see if the expressions
give the correct number of tiles
when n = 1, 2, and 3.
Stage
1
2
3
4
5
Number of
White Tiles
2
4
6
8
10
Number of
Black Tiles
4
5
6
7
8
Total Tiles
6
9
12
15
18
b. FIND A PATTERN Suppose the Watkins family wants to make stage n of the pattern.
Write an expression for the number of white tiles they will need. Explain how you wrote
SMP 8
the expression. 2n; the number of white tiles is 2 times the stage number; this is 2 × n or 2n.
c. FIND A PATTERN Write an expression for the number of black tiles they will need to
make stage n of the pattern. Explain how you wrote the expression. SMP 8
n + 3; the number of black tiles is 3 more than the stage number; this is n + 3 or 3 + n.
d. USE STRUCTURE Write an expression for the total number of tiles they will need to
make stage n of the pattern. Is there more than one way to write the expression?
Explain. SMP 7
Sample answer: 2n + n + 3; you can also write this as n + 3 + 2n if you start with the number
of black tiles and then add the number of white tiles.
8 CHAPTER 1 Expressions, Equations, and Functions
Math Background
008_011_ALG1_C1L2_ISG_672788.indd 8
An algebraic expression is a way to record a series of one or more
computations with numbers and/or variables. In this lesson, the
emphasis is on interpreting the parts of an expression. Students who
are fluent mathematical thinkers are able to see an expression such
as 3y + 7 as “3 times a number plus 7,” but they are also able to think
of it as two distinct part: 3y, which might represent the cost of y
sandwiches that cost \$3 each, and 7, which might represent the cost
of a \$7 vegetable platter. This lesson develops this idea using simple
expressions, but students will gain more experience, with more
complicated expressions, as they progress through the course.
Note that students sometimes confuse expressions and equations.
An equation is a mathematical statement that two expressions
are equal.
8 CHAPTER 1 Expressions, Equations, and Functions
22/08/14 9:18 PM
• How does each stage of the pattern
compare to the one before? A
vertical row consisting of one
black tile and two white tiles is
added to the right side of the
previous stage
Stage 3
SMP 8
SMP 5
Teaching Tip
You may wish to connect this to
SMP 5 (Use appropriate tools
strategically). Kinesthetic learners in
particular may benefit from using
concrete manipulatives to build the
stages of the pattern themselves.
Using tiles or other objects to make
Stages 4 and 5 of the pattern can also
help students check that they have
filled in the table correctly.
Stage 2
a. FIND A PATTERN Complete the table. EXAMPLE 2
e. USE STRUCTURE Evaluate your expression from part d for n = 1, 2, 3, 4, 5 to verify
that it produces the same values as in the bottom row of the table in part a. SMP 7
2(1) + 1 + 3 = 6; 2(2) + 2 + 3 = 9; 2(3) + 3 + 3 = 12; 2(4) + 4 + 3 = 15; 2(5) + 5 + 3 = 18
Teaching Tip
Some students may have trouble
interpreting the coefficients of the
expression as dollar amounts. It may
be helpful to rewrite the coefficient
12.5 as 12.50 to make it clear that
this coefficient represents a cost of
\$12.50 per cap.
A verbal expression like “3 more than the number of white tiles” may be written as the
algebraic expression n + 3 because the words “more than” correspond to addition.
KEY CONCEPT
Complete the table by writing the operation that corresponds to each set of verbal phrases.
Verbal Phrases
more than, sum, plus, increased by, added to
less than, subtracted from, difference, decreased by, minus
product of, multiplied by, times, of
quotient of, divided by
EXAMPLE 2
Interpret Parts of an Expression
Operation
Subtraction
Multiplication
Division
Scaffolding Questions
• How many terms are in the given
expression? 3
• What makes some of the terms
different from the others? The
first two terms, 8x and 12.5y,
contain a variable. The last
term, 6, is a constant.
• How would you use the expression
to find the total cost of ordering
4 T-shirts and 6 caps? Substitute
x = 4 and y = 6, then simplify
the expression.
A.SSE.1a, SMP 7
The expression 8x + 12.5y + 6 gives the total cost in dollars of ordering x T-shirts and
y caps from a Web site. The cost includes a fee for shipping that is the same no matter
how many shirts or caps you order.
a. USE STRUCTURE What does the coefficient 8 represent in the expression? What
does the term 8x represent? Explain.
The 8 represents the cost of each shirt; 8x represents the total cost of the shirts; you multiply
the number of shirts x by the cost per shirt to get the total cost.
b. USE STRUCTURE What does the coefficient 12.5 represent in the expression?
What does the term 12.5y represent? Explain.
The 12.5 represents the cost of each cap; 12.5y represents the total cost of the caps; you
multiply the number of caps y by the cost per cap to get the total cost.
c. USE STRUCTURE What is the fee for shipping? Explain how you know.
SMP 7
\$6; this part of the expression does not depend on the number of shirts or caps you order.
d. USE STRUCTURE How would the expression for the total cost be different if the
company decides to increase the price of each shirt and cap by \$1.50? Explain.
9.5x + 14y + 6; the price of each shirt is \$9.50, the price of each cap is \$14, and the shipping
fee does not change.
1.2 Variables and Expressions 9
Emphasizing the Standards for Mathematical Practice
Example 2 provides an opportunity to address SMP 7 (Look for and
make use of structure) . The standard states that students should be
able to “see complicated things, such as some algebraic expressions,
as single objects or as composed of several objects.” Help students see
the expression in the example as composed of two parts: a variable
that depends on the number of shirts and caps ordered, and a fixed
part that does not depend on the quantities ordered. In addition,
students should be able to go one level deeper to analyze the individual
terms, 8x and 12.5y. In each case, the coefficient represents the unit
cost of a shirt or a cap.
008_011_ALG1_C1L2_ISG_672788.indd 9
2/20/15 5:21 PM
1.2 Variables and Expressions 9
PRACTICE
PRACTICE
Connecting Exercises
to Standards
In Exercises 1, 4, 5, and 7 students
must write an expression for a given
situation in two different ways,
satisfying A.SSE.2.
In Exercises 2, 3, 6, and 8 students
interpret the terms of given
expressions in terms of their context
or write expressions to match a given
situation, satisfying A.SSE.1a.
1. Diego used gray and white counters to make the pattern shown here. Stage 1
Stage 2
A.SSE.2
Stage 3
a. FIND A PATTERN Write an expression for the number of gray counters at stage n
and an expression for the number of white counters at stage n. What do the
SMP 7
expressions tell you about the number of counters of each color at stage n? gray: n2; white: 2n + 1; the number of gray counters is the stage number times itself; the
number of white counters is 1 more than 2 times the stage number.
b. USE STRUCTURE Write two different expressions for the total number of
counters at stage n. Explain how you know the two expressions are both correct. SMP 7
n2 + 2n + 1 or 2n + 1 + n2; you can start with the number of gray counters and add the number
of white counters, or you can start with the number of white counters and add the number of
gray counters.
Exercise
Dual Coding
CCSS
SMP
1
A.SSE.2
7, 8
2
A.SSE.1a
7
A.SSE.1a
2
4
A.SSE.2
7
5
A.SSE.2
3
6
A.SSE.1a
7
7
A.SSE.2
4
8
A.SSE.1a
7
a. What is the coefficient of d? What does it represent?
13.25; it represents the cost per day of renting the bike without the helmet (\$13.25).
b. How would the expression be different if the cost of the helmet were doubled?
The expression would be 13.25d + 13
3. REASON ABSTRACTLY Gabrielle makes a pattern using pennies. The pattern grows in
a predictable way at each stage of the pattern. The expression 3n + 1 gives the total
number of pennies at each stage of the pattern. Make a sketch to show what
Gabrielle’s pattern might look like. Draw stages 1, 2, and 3. A.SSE.1a, SMP2
Stage 1
Stage 2
3
2. USE STRUCTURE The expression 13.25d + 6.5 gives the total cost in dollars of
renting a bicycle and helmet for d days. The fee for the helmet does not depend upon
A.SSE.1a, SMP 7
the number of days. Stage 3
10 CHAPTER 1 Expressions, Equations, and Functions
Common Errors
In Exercise 1, students may write an expression that works for some
stages of the pattern, but not all stages of the pattern. For instance, a
student might notice that there are 3 white counters in stage 1, and
write 3n or n + 2 for the number of white counters. Both of these
expressions are incorrect, since they do not work for the other stages
of the pattern.
008_011_ALG1_C1L2_ISG_672788.indd 10
10 CHAPTER 1 Expressions, Equations, and Functions
If students have difficulty writing correct expressions, suggest that
they perform the intermediate step of making a table and looking for
a pattern, as they did in Example 1. Then have them check that their
expressions work for all stages in the table.
22/08/14 9:18 PM
The table shows the prices of several items at an office supply store.
Use the table for Exercises 4–6. Item
4. USE STRUCTURE Jemma buys s staplers and g gel pens. She has
a coupon for \$2 off the total cost of her purchase. Write two
different expressions that can be used to find her final cost
before tax. A.SSE.2, SMP 7
Common Errors
In Exercise 7, students may write the
expression nb + c for the area of the
apartment. This is incorrect because
the expression is missing
parentheses. The expression should
be n(b + c). To convince students that
these two ways of writing the
expression are different, have them
find the area of the apartment using
specific numerical values for n, b,
and c. Note that students will gain
parentheses in the next two lessons.
Price
Stapler
\$5.99
Tape dispenser
\$3.50
Sticky notes
\$3.25
Gel pen
\$1.75
Sample answer: 5.99s + 1.75g - 2; 1.75g + 5.99s - 2
5. CRITIQUE REASONING DeMarco buys x tape dispensers and x packs of sticky notes.
He says he can use the expression 6.75x to find the total cost of the items before tax.
A.SSE.2, SMP 3
Do you agree? Why or why not? Agree; a tape dispenser and a pack of sticky notes costs \$3.50 + \$3.25 = \$6.75 and he buys x
pairs of tape dispensers and packs of sticky notes, which costs 6.75x.
6. USE STRUCTURE Tyler buys s packs of sticky notes and one gel pen. He uses the
expression 1.08(3.25s + 1.75) to find the final cost. What do you think the 1.08 in the
expression represents? Explain.
A.SSE.1a, SMP 7
There is a sales tax of 8%; multiplying by 1.08 gives the total cost including sales tax.
b ft
7. USE A MODEL The figure shows a floor plan for a two-room apartment.
Write an expression for the area of the apartment, in square feet, by first
finding the area of each room and then adding. Then describe how you
can write the expression in a different way. A.SSE.2, SMP 4
bn + cn; you can also find the length of each side of the
c ft
n ft
apartment, n and b + c, and then multiply: n(b + c)
8. USE STRUCTURE Describe a situation that could be represented by
A.SSE.1a, SMP 7
each expression. a. 9.95 + 0.75b
Sample answer: The total cost of a pizza, if a cheese pizza costs \$9.95 plus \$0.75 for each
b. 15x - 5x
Sample answer: The cost of purchasing x hats, if the regular price of each hat is \$15 and they
are on sale for \$5 off.
c. 59c - 25c - 30
Sample answer: a store’s profit on tablet cases, if the store pays a \$30 shipping charge plus
\$25 for each tablet case and sells them for \$59 each
1.2 Variables and Expressions 11
Emphasizing the Standards for Mathematical Practice
Use Exercise 3 to address SMP 2 (Reason abstractly and
quantitatively). This exercise begins in the abstract, by providing an
algebraic expression that represents the number of pennies needed to
build each stage of a pattern. Help students move to the concrete by
discussing the fact that the expression 3n + 1 has a “growing part” (3n)
and a “constant part” (1). This provides an important clue to the
pattern. Students may wish to begin developing a suitable pattern by
placing one penny at the edge of the pattern; this represent the part
that does not change. Then they can add rows of pennies that grow
from stage to stage according to the rule 3n.
008_011_ALG1_C1L2_ISG_672788.indd 11
2/20/15 5:21 PM
1.2 Variables and Expressions 11
1.3
Order of Operations
1.3 Order of Operations
STANDARDS
Objectives
STANDARDS
A.SSE.1b Interpret complicated
expressions by viewing one or more
of their parts as a single entity.
A.SSE.2 Use the structure of an
expression to identify ways to
rewrite it.
• Evaluate expressions by using the order of operations.
• Use the structure of an expression to identify ways to rewrite it.
• Interpret parts of an expression.
To evaluate an expression means to find its value. When you evaluate a complicated
expression containing multiple operations, you must perform the operations in the correct
order to get the correct value for the expression.
EXAMPLE 1
• Perform operations with rational
numbers
• Find areas and perimeters of simple
figures
• Calculate percentages
Evaluate Expressions
A.SSE.2
EXPLORE Rima created a game. She wrote expressions on slips of paper that represent
different sums of money. The player has 10 seconds to match each expression to
its sum.
Standards for Mathematical
Practice: 1, 2, 3, 4, 5, 6, 7
PREREQUISITES
Content: A.SSE.1b, A.SSE.2
Practices: 1, 2, 3, 4, 5, 6, 7
Use with Lesson 1–2
26 - 15 + 4 - 1 + 14
48 ÷ 2 + 4 × 2
5[10 - (2 + 5)]
(1 + 6)2 - (3 + 7)
\$15
\$32
\$39
\$28
a. PLAN A SOLUTION Evaluate the expressions. Draw a line from each expression to the
SMP 1
correct value. b. CRITIQUE REASONING Ben said that to evaluate 26 - 15 + 4 - 1 + 14 you perform
SMP 3
the two additions and then the two subtractions. Do you agree? Explain. No; this gives 26 - 19 - 15 = -8, which is incorrect. To get the correct value, add and subtract
from left to right.
c. USE TOOLS Use your calculator to evaluate 48 ÷ 2 + 4 × 2. Does it give the correct
value? If so, in what order does it perform the operations? If not, in what order do you
SMP 5
think it performs the operations? Answer depends upon calculator; calculators that give the correct value multiply and divide
EXAMPLE 1
SMP 1
Teaching Tip
This Explore offers a connection to
SMP 1 (Make sense of problems and
persevere in solving them). If
students are not sure how to proceed,
encourage them to try evaluating the
expressions in different ways. When
they evaluate an expression correctly,
they will find the value among the
given choices. Letting students
grapple with the expressions in this
way gives them a chance to find their
own path for navigating the problem.
• For 26 − 15 + 4 − 1 + 14, do
you get the same value if you add
26 + 4 + 14 and then subtract
the remaining quantities versus
performing the operations left to
right? Yes; both values are 28.
• How do you know which of those
methods is correct? The second
method is correct because \$28
is one of the answer choices.
Evaluate the expression inside the innermost set of grouping symbols first; then evaluate
exponents.
12 CHAPTER 1 Expressions, Equations, and Functions
Math Background
012_015_ALG1_C1L3_ISG_672788.indd 12
3/5/15 7:17 AM
The order of operations is a standard procedure for evaluating
numerical expressions. By using these agreed-upon steps, everyone
who evaluates an expression will get the same result.
Some students may have seen the mnemonic PEMDAS (or Please
Excuse My Dear Aunt Sally) for remembering the order of operations:
parentheses, exponents, multiply/divide, add/subtract. This may
cause students to think that multiplication is performed before
division and that addition is performed before subtraction. If students
use the mnemonic, be sure they understand that all multiplications
and divisions should be performed, in the order they occur, from left
to right. The same is true for all additions and subtractions.
12 CHAPTER 1 Expressions, Equations, and Functions
Scaffolding Questions
d. COMMUNICATE PRECISELY What rules can you state about grouping symbols, such
as parentheses, and exponents in order to get the correct values for 5[10 - (2 + 5)]
SMP 6
and (1 + 6)2 - (3 + 7)? Copyright © McGraw-Hill Education
before adding; calculators that give incorrect values perform operations from left to right.
EXAMPLE 2
The rule that describes the sequence in which you should perform operations is called the
order of operations.
Teaching Tip
KEY CONCEPT
Complete the table by writing the order of operations.
Operations
Order of Operations
Multiply and/or divide from left to right.
Step 1: Evaluate expressions inside grouping symbols.
Add and/or subtract from left to right.
Step 2: Evaluate all powers.
Evaluate expressions inside grouping symbols.
Step 3: Multiply and/or divide from left to right.
Evaluate all powers.
Step 4: Add and/or subtract from left to right.
EXAMPLE 2
Write and Evaluate an Expression
Jared is buying carpet for a square room with sides that are
s feet long. The table shows the price of the carpet and the
price of the metal strip that holds down the edge of the carpet.
Item
Price
Carpet
\$2.65 per square foot
Metal Strip
\$0.20 per foot
a. USE A MODEL The metal strip holds down the carpet around the entire perimeter of
the room, except at the doorway, which is 3 feet wide. Write an expression for the total
A.SSE.1b, SMP 4
length of the strip that Jared will need. Explain. 4s - 3; the perimeter of the room is 4 times the length of a side; then subtract 3 feet for
the doorway.
b. USE A MODEL Write an expression that Jared can use to calculate the total cost of
the carpet and the metal strip for a room with sides s feet long. Explain what each term
A.SSE.1b, SMP 4
of your expression represents. 2.65s2 + 0.2(4s - 3); the total cost is the cost of the carpet (2.65 times the area of the carpet, s2)
plus the cost of the metal strip (0.2 times the length of the metal strip).
c. USE STRUCTURE Explain how you can write the expression in a different way. Copyright © McGraw-Hill Education
0.2(4s - 3) + 2.65s2.
A.SSE.2, SMP 6
Use the order of operations to evaluate the expression for s = 16;
2.65 · 162 + 0.2(4 · 16 - 3) = \$690.60.
e. USE STRUCTURE How would the expression from part b be different if Jared had a
coupon for 10% off the total cost? (Hint: 10% off the total cost means Jared pays
90% of the original cost). How much would Jared pay in this case? A.SSE.1b, SMP 7
0.9[2.65s2 + 0.2(4s - 3)]; \$621.54
1.3 Order of Operations 13
Emphasizing the Standards for Mathematical Practice
Use Example 2 to address SMP 4 (Model with mathematics). A key
element of the standard states that students “are able to identify
important quantities in a practical situation.” You may want to have
part to identify important quantities. Students can use a highlighter
to mark any numerical information that they think will be important in
the solution of the problem. For example, the fact that the doorway is
3 feet wide is an essential piece of information that is embedded
within the statement of part a. Highlighting this will help students
remember to take it into account when they write an expression for
the length of the metal strip.
012_015_ALG1_C1L3_ISG_672788.indd 13
Part e is an excellent opportunity to
standards A.SSE.1b and A.SSE.2.
Students may want to apply the 10%
discount by writing the original
expression for the cost minus
an expression for 10% of the cost.
This results in the expression
2.65s2 + 0.2(4s − 3) − 0.1[2.65s2 +
0.2(4s - 3)], which is complicated but
correct. Help students see that an
equivalent expression, based on
taking 90% of the original cost,
results in an expression that is much
easier to work with.
Scaffolding Questions
• What expression represents the
perimeter of the room including the
doorway? 4s
A.SSE.2, SMP 7
Sample answer: Find the cost of the metal strip first; then add the cost of the carpet:
d. COMMUNICATE PRECISELY Explain how you can use the expression from part b to
find the total cost of the carpet and metal strip for a room with sides 16 feet long. SMP 7
2/20/15 5:21 PM
• What should you do to the
expression to account for the
doorway? Why? Subtract 3 from
the expression 4s, since the
metal edge does not
cross the doorway.
• What expression gives the total
cost of the metal strip? Do you
need to use parentheses? Explain.
0.2(4s − 3); you need
parentheses because you
multiply the length of the strip,
4s − 3, by 0.2, and 0.2(4s − 3) is
different from 0.2 · 4s − 3.
1.3 Order of Operations 13
PRACTICE
PRACTICE
Connecting Exercises
to Standards
CALCULATE ACCURATELY Evaluate each expression. Exercises 1−6, 8, 14, and 15 give
evaluating a numerical expression and
evaluating an algebraic expression
after substituting values for the
variables, satisfying A.SSE.2.
CALCULATE ACCURATELY Evaluate each expression if x = 2, y = 7, and z = −1. Exercises 7, 9, and 13 address
A.SSE.1b, requiring students to
evaluate numerical expressions for
real-world situations.
1. 102 − 20 + 43
2. 32 − (7 − 1)2 -4
144
4. z(2x − y)
A.SSE.2, SMP 6
3. 64 ÷ (2 + 6) − 14
5. y2 − 4y ÷ 2
3
-6
A.SSE.2, SMP 6
6. x(z + 5y) − 2x2 35
60
7. The table shows how scores are calculated at diving
competitions. Each of the five judges scores each dive
from 1 to 10 in 0.5-point increments.
a. CALCULATE ACCURATELY Roberto performs a dive with a degree
of difficulty of 2.5. His scores from the judges are 8.0, 7.5, 6.5, 7.5,
and 7.0. Write and evaluate an expression to find his score for
the dive. A.SSE.2, SMP 6
Calculating a Diving Score
Step 1
Drop the highest and lowest
of the five judges’ scores.
Step 2
find the raw score.
Step 3
Multiply the raw score by the
degree of difficulty.
2.5(7.5 + 7.5 + 7.0); 55
b. CONSTRUCT ARGUMENTS Jennifer performs a dive and uses the expression
(7.5 + 8.5 + 8.0) × 3.2 to find her score. What is her score for the dive?
What can you conclude about the highest score she received from the
five judges? Explain. A.SSE.1b, SMP 3
76.8; the highest score was 8.5 or greater since the highest score is dropped and does not
appear in the expression.
Exercises 10–12 require students to
evaluate an algebraic expression and
determine whether given statements
satisfying A.SSE.1b.
Exercise
Dual Coding
CCSS
SMP
1−6
A.SSE.2
6
7
A.SSE.1b,
A.SSE.2
2, 3, 6
8
A.SSE.2
3
9
A.SSE.1b
5
10−12
A.SSE.1b
2
13
A.SSE.1b
4
14−15
A.SSE.2
2
Sample answer: 7, 7, 6, 8, 5; her score is 3.5(7 + 7 + 6 ) = 70
d. REASON QUANTITATIVELY Eli also performs a dive with a degree of difficulty of
3.5. His score is 75.25. What scores could Eli have received from the five judges?
Explain. A.SSE.1b, SMP 2
Sample answer: 7, 7, 7, 7.5, 8; his score is 3.5(7.5 + 7 + 7) = 75.25
e. REASON QUANTITATIVELY Skylar does a dive with a degree of difficulty of
3.3. Four of his scores are 7.5, 7.0, 6.5, and 6.5. Toby does a dive with a degree of
difficulty of 3.1 and receives scores of 7.0, 7.5, 7.5, 6.0, and 8.0. If Skylar’s final
score was greater than Toby’s, what can you say about Skylar’s fifth score?
Explain. A.SSE.1b, SMP 2
Skylar’s fifth score must be 7.5 or greater; Toby’s score is 3.1(7.5 + 7.5 + 7.0) = 68.2; If Skylar’s
c. REASON QUANTITATIVELY Mai performs a dive with a degree of difficulty of 3.5.
Her score for the dive is 70. What scores could Mai have received from the five
A.SSE.1b, SMP 3
judges? Explain. fifth score is 7.0 or lower, then the 7.5 score would be dropped, and the highest his score
could be is 3.3 (7.0 + 7.0 + 6.5) = 67.65. If his fifth score is 7.5 or higher, then his score is
3.3(7.5 + 7.0 + 6.5) = 69.3.
14 CHAPTER 1 Expressions, Equations, and Functions
Emphasizing the Standards for Mathematical Practice
Exercise 7c addresses SMP 3 (Construct viable arguments and
critique the reasoning of other). Be sure students understand that
this is an open-ended problem with many possible answers. A key part
of the problem is having students justify their answers to show that
the judges’ scores they choose result in a score of 70 for the dive.
Remind students that they must provide five scores, even though only
three are used in calculating the score for the dive. You may want to
ask some follow-up “what if” questions. For example, “What if one of
your judges’ scores were lower? Would that change the score for the
dive? Why or why not?”
012_015_ALG1_C1L3_ISG_672788.indd 14
14 CHAPTER 1 Expressions, Equations, and Functions
2/20/15 5:21 PM
8. CRITIQUE REASONING A student was asked to evaluate an
expression. The student’s work is shown at right. Critique the
student’s work. If there are any errors, describe them and ﬁnd the
A.SSE.2, SMP 3
correct value of the expression. Common Errors
In Exercise 10, students may
correctly rewrite the expression as
100 − 5(b + 2) and then conclude that
the value of the expression is always
less than 100. Students who make
this error may be thinking only of
positive values of b, causing them to
see (b + 2) as a positive quantity.
Suggest that students try evaluating
the expression with a variety of
values for b. Once students try a
negative value, such as b = −10, they
should recognize that −5(b + 2) is
positive and that the value of the
expression is greater than 100.
62 - 5 x 2 + 2(9 - 7)
= 62 - 5 x 2 + 2(2)
= 36 - 5 x 2 + 2(2)
The next to last line is incorrect. The student should have
= 36 - 10 + 4
added/subtracted from left to right. The correct value is 30.
= 36 - 14
= 22
9. USE TOOLS Kelly buys 3 video games that cost \$18.95 each.
She also buys 2 pairs of earbuds that cost \$11.50 each. She has a
coupon for \$2 oﬀ the price of each video game. Kelly uses a
calculator, as shown, to ﬁnd that the total cost of the items is
\$77.85. The cashier tells her that the total cost is \$73.85. Who is
A.SSE.1b, SMP 5
correct? Explain. The cashier is correct. Kelly should have entered the expression
into her calculator as 3(18.95 − 2) + 2(11.50).
REASON QUANTITATIVELY Determine whether each statement about the expression
A.SSE.1b, SMP 2
a2 - 5(b +2) is always, sometimes, or never true. Explain. 10. If a = 10, then the value of the expression is less than 100.
Sometimes; when b + 2 is negative, the value of the expression is greater than 100.
11. If b = 1, then the expression is equivalent to a2 - 15.
Always; a2 - 5(1 + 2) = a2 - 5(3) = a2 - 15.
12. The value of the expression is 0.
Sometimes; if a = 0 and b = -2, then the value of the expression is 02 - 5(-2 + 2) = 0.
b1
13. USE A MODEL The side panel of a skateboard ramp is a trapezoid, as
h
(b + b2) can be used to ﬁnd the area
shown in the ﬁgure. The expression ___
2 1
of a trapezoid. Write and evaluate an expression to ﬁnd the amount of
wood needed to build the two side panels of a skateboard ramp where
A.SSE.1b, SMP 1
h = 24 inches, b1 = 30 inches, and b2 = 50 inches. 24
2
2 • __
2 (30 + 50 ); 1920 in
h
b2
14. REASON QUANTITATIVELY Write an expression that includes the numbers 2, 4, and
5, and has a value of 50. Your expression should include one set of parentheses. A.SSE.2, SMP 2
15. REASON QUANTITATIVELY Isabel wrote the expression 6 + 3 × 5 - 6 + 8 ÷ 2 and
asked Tamara to evaluate it. When Tamara evaluated it, she got a value of 19. Isabel
told Tamara that her value was incorrect and said that the value should have been 38.
With whom do you agree? Explain. A.SSE.2, SMP 2
Tamara is correct. When evaluating this expression, first perform the multiplication and division,
and then the addition and subtraction.
1.3 Order of Operations 15
Emphasizing the Standards for Mathematical Practice
Exercise 13 is an opportunity for students to work with SMP 1
(Make sense of problems and persevere in solving them). Some
students may miss the fact that two side panels are being built, and
this may cause them to arrive at the wrong answer. Encourage
students to read the problem carefully, mark the important given
information, and persevere in the process of writing and evaluating an
appropriate expression.
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1.3 Order of Operations 15
1.4
Properties of
Numbers
1.4 Properties of Numbers
STANDARDS
Objectives
STANDARDS
A.SSE.1b Interpret complicated
expressions by viewing one or more
of their parts as a single entity.
A.SSE.2 Use the structure of an
expression to identify ways to
rewrite it.
Standards for Mathematical
Practice: 2, 3, 4, 6, 7, 8
PREREQUISITES
• Perform operations with rational
numbers
EXAMPLE 1
• Based on the results of parts c
and d, what two numerical
expressions must be equal? 3.4(3.6) + 3.4(2.1) = 3.4(3.6 + 2.1)
• Use the structure of an expression to identify ways to rewrite it.
• Interpret parts of an expression.
EXAMPLE 1
Explore Properties of Numbers
SMP 7
EXPLORE Rectangle ABCD represents Arletta’s garden.
She plants part of the garden with vegetables and part of
the garden with flowers, as shown.
a. USE STRUCTURE Arletta wants to put a straight path along
the garden from D to C. Write two different expressions she can
use to find the length of the path. Explain why it makes sense
A.SSE.2
that the two expressions give the same length. A
3.6 m
2.1 m
vegetables
ﬂowers
D
B
3.4 m
C
3.6 + 2.1 or 2.1 + 3.6; you get the same length regardless of whether you add the length of the
flower plot to the length of the vegetable plot, or vice versa.
b. USE STRUCTURE Arletta also wants to put a fence along the border of the garden
from A to B to C. Write an expression she can use to find the length of the fence. Then
show two different ways she can group a pair of numbers in the expression. Does she
A.SSE.2
get a different result depending on the grouping? Explain. 3.6 + 2.1 + 3.4; (3.6 + 2.1) + 3.4 or 3.6 + (2.1 + 3.4); with either grouping the length of the fence is
the same, 9.1 m.
c. USE STRUCTURE Arletta wants to find the area of the garden. She finds the area of
the vegetable plot and the area of the flower plot then the sum of these areas. Write
A.SSE.1b
and evaluate an expression to show how she finds the area. 3.4(3.6) + 3.4(2.1) = 12.24 + 7.14 = 19.38 m2
d. USE STRUCTURE Arletta’s friend, Troy, finds the area of the garden by adding to find
the distance from A to B and then multiplying by the distance from B to C. Write and
evaluate an expression to show how Troy finds the area. Does he get the same result as
A.SSE.1b
Arletta? Explain why this makes sense. 3.4(3.6 + 2.1) = 3.4(5.7) = 19.38 m2; he gets the same result as Arletta; it makes sense that the
results are the same because both expressions represent the same total area.
16 CHAPTER 1 Expressions, Equations, and Functions
Math Background
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The properties of numbers in this lesson play an important role in
students’ growth as mathematical thinkers. Students sometimes see
these properties as little more than complicated names for “obvious”
facts. However, these properties serve as the logical underpinnings of
numerical and algebraic computations.
As students progress through high school mathematics courses, they
will be called upon to give increasingly sophisticated logical
arguments. Students will be expected to justify every statement with
a valid reason. Having students justify algebraic steps by citing a
property of real numbers can help develop a habit of mind that will
serve them well when they are asked to justify more complicated
16 CHAPTER 1 Expressions, Equations, and Functions
Scaffolding Questions
• In part c, how do you find the area of
a rectangle? Find the product of
the length and width of the
rectangle.
• Evaluate expressions by using properties of numbers.
SMP 7
Teaching Tip
One aspect of SMP 7 (Look for and
make use of structure) is recognizing
equivalent numerical expressions by
considering properties of equality
and the Distributive Property.
A geometric model, such as the
garden shown in the example, is a
good way to illustrate why the
expressions are equivalent. Have
students that are having difficulties
finding different ways to write the
expressions label the lengths of each
side on the diagram.
Content: A.SSE.1b, A.SSE.2
Practices: 2, 3, 4, 6, 7, 8
Use with Lessons 1–3, 1–4
EXAMPLE 2
In the previous exploration, you may have discovered the following properties of numbers.
KEY CONCEPT
Teaching Tip
Complete the table by using numbers to write examples for each property.
Property
Commutative Property
For any numbers a and b, a + b = b + c and
a b = b a.
Associative Property
For any numbers a, b, and c, (a + b) + c = a + (b + c) and
(ab)c = a(bc).
Distributive Property
For any numbers a, b, and c, a(b + c ) = ab + ac and
a(b - c) = ab - ac.
EXAMPLE 2
Examples
5+9=9+5
59=95
3 + (4 + 10) = (3 + 4 ) + 10
(3 4)10 = 3(4 10)
8(5 + 2) = 8 5 + 8 2
8(5 - 2) = 8 5 - 8 2
Use Mental Math
Giovanni is buying equipment for his soccer team. The table shows
the price of some of the items he is buying.
Item
Price
Soccer Balls
\$22 each
Portable Goal
\$93 per pair
a. USE STRUCTURE Giovanni is buying 7 soccer balls. Write an expression for the total
cost of the soccer balls. Then explain how he can use the Distributive Property to
rewrite the expression to find the cost using mental math. A.SSE.1b, SMP 7
Sample answer: 7(22); rewrite the expression as 7(20 + 2); by the Distributive Property, this equals
7(20) + 7(2) = 140 + 14 = 154.
b. USE STRUCTURE Giovanni is buying 6 pairs of portable goals. Show two different
ways he can find the cost of the goals using the Distributive Property. A.SSE.2, SMP 7
Sample answer: 6(93) = 6 (90 + 3) = 6(90) + 6(3) = 540 + 18 = 558; 6(93) = 6(100 - 7) = 6(100)
- 6(7) = 600 - 42 = 558.
c. USE STRUCTURE Fred’s business donated 3 soccer balls and 3 portable goals to
Giovanni’s team. Write an expression using the Distributive Property for the cost of the
donated items. SMP 8
Sample answer: 3(22 + 93) = 3(20 + 2 + 90 + 3) = 3(110 + 5) = 330 + 15 = 345.
d. DESCRIBE A METHOD Describe a general method for using the Distributive Property
A.SSE.1b, SMP 8
to find a product by mental math. Write the greater of the two factors as a sum or difference where one term is a multiple of 10 and
the other is a single digit. Then apply the Distributive Property.
1.4 Properties of Numbers 17
Emphasizing the Standards for Mathematical Practice
Example 2 part c connects to SMP 8 (Look for and express regularity
in repeated reasoning). Once students have completed parts a and b,
they should begin to recognize a general procedure. If students have
trouble recognizing this general method, suggest that they try a few
more specific examples, such as 5(72) and 4(89). Then help students
break the process into steps. Ask: “What is the first thing you do?” If
necessary, help students see that a key step is writing the larger
factor as a sum in which one of the numbers is a multiple of 10. If
students have trouble seeing this, ask whether it would be helpful to
rewrite 5(72) as 5(59 + 13), and why or why not.
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SMP 7
This example requires students to
work fluently with whole numbers by
breaking them into parts that are
convenient for calculations. You may
want to spend a moment working with
students to express a number, such as
74, as the sum of a multiple of 10 and
a single digit. Be sure students
understand that they can use addition
(70 + 4) or subtraction (80 − 6).
Scaffolding Questions
• In general, how do you find the total
cost when you buy several of the
same item? Multiply the price of
one item by the number of items
• In part a, why is it convenient to
write 7(22) as 7(20 + 2)? It is
difficult to calculate 7(22) directly
using only mental math, but it is
easier to calculate 7(20 + 2)
because you can use the
Distributive Property to write it as
7(20) + 7(2). Each of the
products in this sum is easier to
calculate than the original one.
• How can you check each of the
mental math calculations? Sample
in a different way or use a
calculator.
1.4 Properties of Numbers 17
EXAMPLE 3
Teaching Tip
SMP 6
In this example, students use and
identify properties of numbers. If
students have trouble remembering
the names of the properties, work
with them to develop mnemonics. For
example, when you commute, you
travel back and forth to work or
school; in the Commutative Property,
numbers or variables travel back and
forth around the operation symbol.
Like terms are terms that contain the same variables, with corresponding variables having
the same exponent. In the expression 5x2 + 4x + 7x2 + 2y2, the terms 5x2 and 7x2 are like
terms. You can use properties of numbers to simplify expressions by combining like terms.
EXAMPLE 3
Combine Like Terms
Follow these steps to simplify the expression -4m3 + 4m + 6n3 + 2m.
a. USE STRUCTURE What are the like terms in the expression? Explain how
A.SSE.2, SMP 7
you know. 4m and 2m; they have the same variable raised to the same power.
b. COMMUNICATE PRECISELY Which property allows you to rewrite the expression as
A.SSE.2, SMP 6
-4m3 + 4m + 2m + 6n3? Why? Commutative Property; the property says that you can add in any order.
c. USE STRUCTURE Show how to use the Distributive Property to rewrite the middle
A.SSE.1b, SMP 7
two terms (4m and 2m) as a single term. What is the simplified expression? 4m + 2m = (4 + 2)m = 6m; -4m3 + 6m + 6n3
d. DESCRIBE A METHOD Describe how you can simplify an expression by combining like
A.SSE.2, SMP 8
terms. For each group of like terms, add the coefficients to get the coefficient of the corresponding term
in the simplified expression.
Scaffolding Questions
• Why is it useful to rewrite the
original expression as −4m3 + 4m +
2m + 6n3? Once you put the
terms 4m and 2m next to each
other, you can combine them
using the Distributive Property.
USE STRUCTURE Which property is illustrated by each equation? 1. 9y + 3x + 6y = 3x + 9y + 6y
Commutative Property
2. c(5 + d) = 5c + cd
A.SSE.2, SMP 7
3. 4(5m) = (4 5)m
Associative Property
Distributive Property
USE STRUCTURE The table shows the prices of tickets to a
theme park. Use the table for Exercises 4–6. For each situation,
explain how to use the Distributive Property and mental math to
A.SSE.1b, SMP 7
find the total cost of the tickets. Type of Ticket
Price
\$29
Students
\$21
Seniors
\$18
Sample answer: 7(29) = 7(30 - 1) = 7(30) - 7(1) = 210 - 7 = \$203
5. 9 tickets for students
Sample answer: 9(21) = 9(20 + 1) = 9(20) + 9(1) = 180 + 9 = \$189
6. 12 tickets for seniors
Sample answer: 12(18) = 12(20 - 2) = 12(20) - 12(2) = 240 - 24 = \$216
USE STRUCTURE Simplify each expression. 7. -2k + 2k2 - 3k + k3
-5k + 2k2 + k3
• How could you check that the final
expression is equivalent to the
Substitute the same values for
the variables in both expressions
and check that you get the same
result.
PRACTICE
A.SSE.2, SMP 7
8. 2n2 + 5n4 - 6n2 - 3n4
-4n2 + 2n4
9. -b + 3a + b2 - 2b + 6a - 9a
-3b + b2
18 CHAPTER 1 Expressions, Equations, and Functions
18 CHAPTER 1 Expressions, Equations, and Functions
2/20/15 5:21 PM
Common Errors
In Exercise 1, some students may see that the expression on each
side of the equal sign has three terms and quickly decide that the
relevant property must be the Associative Property. Encourage
students to look closely at the expressions and ask them to describe
how they are different. Once students see that two terms have
“traded places,” they should be able to connect the equation more
easily to the Commutative Property.
016_019_ALG1_C1L4_ISG_672788.indd 18
10. CRITIQUE REASONING Angela is shipping 8 bags of granola to a customer.
Each bag weighs 22 ounces and the maximum weight she can ship in one box
is 10 pounds 5 ounces. She makes the calculation at right and decides that
she can ship the bags in one box. Do you agree? Explain. A.SSE.1b, SMP 3
PRACTICE
= 8(20 + 2)
8(22)
= 8(20) + 2
Connecting Exercises
to Standards
= 160 + 2
= 162 ounces
No; 10 pounds 5 ounces is 10(16) + 5 = 165 ounces, but Angela used
the Distributive Property incorrectly. She should have written
Exercises 1–3, 7–9, 12, and 13
require students to recognize how
properties of numbers allow
expressions to be written in different
ways, per A.SSE.2.
8(20 + 2) = 8(20) + 8(2) = 160 + 16 = 176 ounces.
m seats
11. A theater has m seats per row on the left side of the aisle
and n seats per row on the right side of the aisle. There are
A.SSE.1b
r rows of seats. a. USE A MODEL Explain how you can use the
Distributive Property to write two diﬀerent
expressions that represent the total number of seats
in the theater. SMP 4
Aisle
n seats
r rows
The total number of seats is the number of seats per row, m + n, times the number of rows, r.
This is r(m + n). By the Distributive Property, this is also rm + rn.
b. CONSTRUCT ARGUMENTS Suppose you double the number of seats in each row
on the left side of the aisle. Does this double the number of seats in the theater?
Use one of the expressions you wrote in part a to justify your answer. SMP 3
No; the original number of seats in the theater is rm + rn. If you double m, you double only the
Exercises 4–6, 10, and 11 require
students to rewrite an expression
using the Distributive Property
before evaluating, satisfying
A.SSE.1b.
first term of the expression, but not the second term, so you do not double the total number of
seats in the theater.
12. CONSTRUCT ARGUMENTS Is there a Commutative Property or Associative
Property for subtraction? Explain why or why not for each property. A.SSE.2, SMP 3
Exercise
Neither property exists; a counterexample shows that there is no Commutative Property for
CCSS
SMP
1−3
A.SSE.2
7
4−6
A.SSE.1b
7
7−9
A.SSE.2
7
10
A.SSE.1b
3
11
A.SSE.1b
3, 4
12
A.SSE.2
3
13
A.SSE.2
2
subtraction: 7 - 4 ≠ 4 - 7; a counterexample shows that there is no Associative Property for
subtraction: 10 - (2 - 1) ≠ (10 - 2) - 1.
13. REASON QUANTITATIVELY Provide a counterexample to show that there is no
Commutative Property or Associative Property for division. What is the relationship
A.SSE.2, SMP 2
between the results when the order of division of two numbers is switched? 4 ÷ 8 ≠ 8 ÷ 4, so there is no Commutative Property for division 16 ÷ (8 ÷ 4) ≠ (16 ÷ 8) ÷ 4,
so there is no Associative Property for division. As long as neither number is 0, when the order
of division of two numbers is switched, the results are multiplicative inverses of each other.
Dual Coding
1.4 Properties of Numbers 19
Emphasizing the Standards for Mathematical Practice
Exercise 12 offers a chance to address SMP 3 (Construct viable
arguments and critique the reasoning of others). In particular, you
may want to use the exercise as the jumping-off point for a brief
discussion about counterexamples. Be sure students understand that
it takes only one counterexample to show that a conjecture is false.
Therefore, a single example in which the Commutative Property does
not work for subtraction is enough to conclude that this property does
not exist. Similarly, students only need to find a single set of numbers
where the Associative Property does not work with subtraction.
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Common Errors
Exercise 10 highlights one of the
most common errors in algebra
classes. In the calculation shown, the
8 is not completely “distributed” to
all of the terms in the second factor.
If students make this error, remind
them that the Distributive Property
states that the factor outside the
parentheses must be fully distributed
to each of the terms inside the
parentheses. It may help to show
students an example with several
terms in the parentheses. For
instance, 5(4 + 2 + 8) may be
rewritten as 5(4) + 5(2) + 5(8).
1.4 Properties of Numbers 19
1.5
Equations
1.5 Equations
STANDARDS
Objectives
STANDARDS
Content: A.CED.1, A.REI.1, A.REI.3
Practices: 1, 2, 3, 4, 7, 8
Use with Lesson 1–5
• Write equations in one variable and use them to solve problems.
A.CED.1 Create equations and
inequalities in one variable and use
them to solve problems.
A.REI.1 Explain each step in
solving a simple equation as
following from the equality of
numbers asserted at the previous
step, starting from the assumption
that the original equation has a
solution. Construct a viable
argument to justify a solution
method.
• Solve linear equations in one variable and explain the steps of
the solution.
A mathematical sentence that contains an equals sign (=) is an equation. An equation
states that two expressions are equal.
EXAMPLE 1
Investigate Equations
EXPLORE A group of friends rented bicycles from
different bike shops. The table shows the expression that
each shop uses to calculate the cost of renting one of
their bikes for h hours.
Shop
Cost for h Hours (\$)
Real Wheels
Easy Bike
a. USE A MODEL Kaden rented his bike from Easy Bike and Pedal Power
he paid \$26 for the rental. Write an equation that relates
the expression the bike shop uses to calculate the cost and the amount
Kaden paid. Do you think Kaden rented his bike for 6 hours? Justify your
A.CED.1, SMP 4
answer using the equation you wrote. 5(h + 1)
3.5h + 8.5
2(3h - 1)
3.5h + 8.5 = 26; no; if he rented the bike for 6 hours, then the expression on the left side of the
equation would be 3.5(6) + 8.5 = 29.5, which does not equal 26.
b. CONSTRUCT ARGUMENTS Do you think Kaden rented his bike for 5 hours? Justify
A.REI.3, SMP 3
Standards for Mathematical
Practice: 1, 2, 3, 4, 7, 8
Yes; when h = 5, the left side of the equation is 3.5(5) + 8.5 = 26, so the two sides of the equation
are the same.
c. CRITIQUE REASONING Megan rented her bike from Real Wheels and she paid \$25.
She claims that she rented the bike for 5 hours. Do you agree? Use an equation to
A.CED.1, SMP 3
explain why or why not. which does not equal 25; so the number of hours cannot be 5.
• Perform operations with rational
numbers
• Use order of operations
• Use properties of numbers
d. USE REASONING Kim-Ly rented her bike from Pedal Power and she paid \$22. Did she
rent the bike for 3, 4, or 5 hours? Use an equation to explain your answer. A.CED.1, SMP 2
No; the equation is 5(h + 1) = 25; when h = 5, the left side of the equation is 5(5 + 1) = 5(6) = 30,
PREREQUISITES
4 hours; the equation is 2(3h - 1) = 22; only h = 4 makes the left side of the equation equal to the
right side.
20 CHAPTER 1 Expressions, Equations, and Functions
EXAMPLE 1
Teaching Tip
SMP 4
An equation is one type of model that can
be used to describe a real-world situation.
Help students write an appropriate model
by giving them an “equation frame.” For
example, [expression Easy Bike uses] =
• In part a, what should you do to
decide if Kaden rented his bike for
6 hours? Substitute h = 6 in the
equation and check to see if both
sides are equal.
22/08/14 9:19 PM
A solution of an equation is a value of the variable that makes the
equation true. An equation may have no solution, one solution, several
solutions, or infinitely many solutions. In this lesson, students will see
equations with one solution, no solution, or infinitely many solutions
(i.e., all real numbers). When students study quadratic equations, they
will work with many equations that have two solutions. For example,
the equation x2 − 1 = 8 has the solutions x = 3 and −3.
Students can recognize equations with no solutions by simplifying
both sides of the equation until they recognize a contradiction. For
example, the equation m + 7 = m + 9 has no solution. This makes
sense since adding 7 to a number cannot give the same result as
adding 9 to the number. The solution set in this case is the empty set,
which may be written as { } or Ø.
20 CHAPTER 1 Expressions, Equations, and Functions
Scaffolding Questions
• Is there another correct way to write
the equation in part a? Yes; for
example, you could write it as
26 = 3.5h + 8.5.
Math Background
020_025_ALG1_C1L5_ISG_672788.indd 20
EXAMPLE 2
A solution of an equation is a value of the variable that makes the equation true. A set of
numbers from which replacements for a variable may be chosen is called a replacement
set. A solution set is the set of all solutions in the replacement set.
EXAMPLE 2
Teaching Tip
Solve an Equation
Students will simplify the given
equation to get the equivalent
equation 3p = 12. Encourage
students to solve this equation by
inspection. Later, students will be
expected to solve the equation by
dividing both sides by 3 (Division
Property of Equality), but for now
“3 times what number equals 12?”
Complete these steps to solve the equation (82 ÷ 4 - 11)p - 2p = 12.
a. USE STRUCTURE Use the order of operations to simplify the expression in
parentheses and write the resulting equivalent equation. Explain your steps. A.REI.1, SMP 7
First evaluate the exponent: 82 ÷ 4 - 11 = 64 ÷ 4 - 11
Next, perform the division: 64 ÷ 4 - 11 = 16 - 11.
Finally, subtract: 16 - 11 = 5. The resulting equation is 5p - 2p = 12.
b. USE STRUCTURE Explain how to simplify the resulting equation. What property
A.REI.1, SMP 7
justifies this step of the process? 5p - 2p = 12 can be simplified by combining like terms: 3p = 12. This is justified by the
Substitution Property.
c. USE STRUCTURE What is the solution of the equation? How do you know? A.REI.3, SMP 7
p = 4; this is the only value for p that can be multiplied by 3 to get a product of 12.
d. INTERPRET PROBLEMS Explain how you can check that your solution is correct. A.REI.3, SMP 1
Check that substituting p = 4 in the original equation results in a true statement.
Scaffolding Questions
• What is the first step in simplifying
the expression in parentheses?
Why? Evaluate the exponent;
this is the first step in the order
of operations.
(82 ÷ 4 - 11)4 - 2(4) = (5)4 - 2(4) = 20 - 8 = 12, so the solution checks.
e. CRITIQUE REASONING The set {3 < p < 5} is given as the replacement set for the
equation (82 ÷ 4 - 11)p - 2p = 15. This set is shown on the number line. Martina says
that this changes the solution of the equation. Do you agree with Martina? Explain your
0
1
2
3
4
5
6
7
8
9
10
Sample answer: Yes; the equation simplifies to 3p = 15, so the solution is p = 5. Since 5 is not a
value in the replacement set, the equation has no solution.
f. DESCRIBE A METHOD If no replacement set is given for an equation, describe how you
could go about solving the equation. How does having a replacement set given change
A.REI.1, SMP 8
your approach? Sample answer: If there is no replacement set given and if the equation is simple enough, I can try
values and check them in the equation, refining my guess each time. If the equation is not simple,
I can simplify the equation using properties of real numbers until the equation is simple enough
that I can tell what the solution is. If a replacement set is given, I may be able to check the values
from the set in the equation instead of simplifying the equation first.
1.5 Equations 21
Emphasizing the Standards for Mathematical Practice
An important element of SMP 1 (Make sense of problems and
persevere in solving them) is checking that answers make sense.
Explain to students that a solution of an equation makes the equation
“balance.” That is, a value of the variable is a solution if it makes the
two sides of the equation equal. In order to check a solution,
substitute the value of the variable in the original equation. Ask
students why they think it is important that they substitute the value
in the original equation, and help students understand that this
ensures that no errors were made in the intermediate steps of the
solution.
020_025_ALG1_C1L5_ISG_672788.indd 21
SMP 7
22/08/14 9:19 PM
• When you have the equation
5p − 2p = 12, are there any like
terms? Explain. Yes; 5p and −2p
are like terms since they have
the same variable raised to the
same power.
• In part e, what are some examples
of numbers that are included in the
shaded portion of the number line?
1
2,
4.11111, etc.
1.5 Equations 21
EXAMPLE 3
Teaching Tip
SMP 7
It may be beneficial for visual learners
to use a highlighter to mark each step
of the solution to show which part of
the equation has changed from one
line to the next.
Scaffolding Questions
• Why does 9x + 3 = 9x + 2 have no
solution? Adding 3 to a quantity
cannot give the same result as
• If the constant terms in the
simplified equation were different,
would the equation have a
solution? As long as the constants
are different from each other, the
equation will have no solution.
Teaching Tip
SMP 7
Students may find it strange that an
equation can have any real number as
a solution. In order to help students
become familiar with this idea, you
may want to ask them to write their
own examples of such equations.
Students might start by writing
identical expressions on either side of
the equation, such as x + 4 = x + 4,
and then modify one or both sides to
make them look different from each
other. For example, x + 4 = 2 + x + 2.
Solve an Equation
a. USE STRUCTURE Use one or more properties to justify each step of the solution
process shown below. A.REI.1, SMP 7
5x + 4x + 3 = 2x + 2 + 7x
Original equation
9x + 3 = 2x + 2 + 7x
Substitution Property
9x + 3 = 2x + 7x + 2
Commutative Property
9x + 3 = 9x + 2
Substitution Property
b. CONSTRUCT ARGUMENTS What is the solution to the original equation?
A.REI.3, SMP 3
Justify your answer. There is no solution; there is no value of x for which 9x + 3 can have the same value as 9x + 2.
c. REASON QUANTITATIVELY Change one term in the original equation so that the
equation has exactly one solution, when x = 1. How do you know that you are correct? A.REI.1, SMP 2
Sample answer: 5x + 4x + 3 = 3x + 2 + 7x; the equation simplifies to 9x + 3 = 10x + 2;
substitute 1 for x: 9 + 3 = 10 + 2.
EXAMPLE 4
Solve an Equation
Complete these steps to solve the equation -3b + 9b + 17 = 5b + 15 + b + 2.
a. USE STRUCTURE Use one or more properties to justify each step of the solution
process shown below. A.REI.1, SMP 7
-3b + 9b + 17 = 5b + 15 + b + 2
Original equation
-3b + 9b + 17 = 5b + b + 15 + 2
Commutative Property
-3b + 9b + 17 = 6b + 17
Substitution Property
6b + 17 = 6b + 17
Substitution Property
b. CONSTRUCT ARGUMENTS What is the solution to the original equation?
The solution is any real number; for any value of b, the left side of the equation is equal to the
right side.
c. REASON QUANTITATIVELY Suppose the 17 in the original equation was an 18.
Without solving another equation, what would the solution to this equation be?
Explain how you know. A.REI.1, SMP 2
There would be no solution; the final equation would be 6b + 18 = 6b + 17, which is never true for
any value of b.
22 CHAPTER 1 Expressions, Equations, and Functions
Differentiating Instruction
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If students have difficulty identifying the properties that justify each
step in these solutions, ask them to first explain how each line of the
solution is different from the previous line. Once students have
verbalized what has changed, they may have an easier time connecting
this to one of the properties of numbers they have seen. You may want
to post a list of the properties of numbers on a wall of the classroom.
This will give students a “menu” of properties to choose from.
• Why is it helpful to use the
Commutative Property first when
you simplify the expression
5b + 15 + b + 2? This allows
you to put like terms next to
each other; then you can
combine like terms.
22 CHAPTER 1 Expressions, Equations, and Functions
Scaffolding Questions
• When you solve an equation, does it
matter which side of the equation
you simplify first? No
EXAMPLE 3
Complete these steps to solve the equation 5x + 4x + 3 = 2x + 2 + 7x.
EXAMPLE 4
Some equations have no solution. Other equations have more than one solution. An
equation that is true for every value of the variable is called an identity. For example,
x + 3 = 3 + x is an identity.
EXAMPLE 5
Write and Solve an Equation
EXAMPLE 5
The table shows the fees charged for overdue items at the Cedarville Library.
a. USE A MODEL Let d be the number of days that a book is
overdue. Let F be the total late fee that is charged for the
book. Write an equation of the form F = [expression] that
shows how to calculate the late fee. Explain how you wrote
the equation. A.CED.1, SMP 4
Teaching Tip
Item
Late Fees
Book
\$1.10 plus \$0.25 per day
CD
\$1.50 plus \$0.75 per day
DVD
\$3.00 plus \$1.25 per day
F = 0.25d + 1.10; the late fee is 0.25 times the number of days, d, plus the fee of \$1.10, which does
not depend on the number of days.
b. USE A MODEL Jamar has a book that is 5 days overdue. Show how to solve an
equation to find the late fee for the book. Use the spaces provided to show the steps of
A.REI.3, SMP 4
your solution and write an explanation of each step in the spaces at the right. F
= 0.25d + 1.10
Original equation from part a
F
= 0.25(5) + 1.10
Substitute 5 for d.
F
= 1.25 + 1.10
Multiply.
F
= 2.35
The total late fee for Jamar’s book is \$2.35.
d. USE A MODEL Write an equation that shows how to calculate the total late fee F for a
CD that is d days overdue. Then show how to use the equation to find the total fee for a
A.CED.1, SMP 4
CD that is 7 days overdue. F = 0.75d + 1.50; F = 0.75(7) + 1.50 = 5.25 + 1.50 = 6.75; the late fee is \$6.75.
Encourage students to express their
equation, F = 0.25d + 1.10, in words.
(”The fee is \$0.25 times the number
of days, plus \$1.10.”) Working back
and forth between different
representations—in this case, an
equation and a verbal description—is
an essential skill for success in
algebra.
Scaffolding Questions
• In part b, do you substitute 5 for F
or for d in the equation? Why? d; d
represents the number of days,
and Jamar’s book is 5 days
overdue.
c. REASON QUANTITATIVELY Write a sentence explaining what the last line of your
A.REI.3, SMP 2
solution tells you. e. USE A MODEL Write an equation that shows how to calculate the total late fee F for a
DVD that is d days overdue. Hailey has a DVD with a late fee of \$10.50. Use your
equation to determine whether Hailey’s DVD is 9 days overdue. Explain your answer. SMP 4
• How do you simplify 0.25(5) + 1.10?
This is required by the order of
operations.
A.CED.1, SMP 4
F = 1.25d + 3.00; d = 9 is not a solution of the equation 10.50 = 1.25d + 3.00, so the DVD is
not 9 days overdue.
f. REASON QUANTITATIVELY Liza returns a CD to the library and his charged a late fee
of \$6.00. How many days overdue was Liza’s CD? Explain your reasoning. A.REI.1, SMP 2
Sample answer: 6 days; An equation for the situation is 6 = 1.5 + 0.75d. I can substitute whole
number values for d until I find one that makes the equation true.
1.5 Equations 23
ELL Strategies
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Example 5 includes some vocabulary that may be unfamiliar to ELL
students. You may want to begin with a brief discussion in which you
ask the class if anyone can provide synonyms for these words. This
will help students understand that a fee is an amount that is charged.
Also, be sure students realize that overdue is a synonym for late that
is often used for items that have been borrowed.
1.5 Equations 23
PRACTICE
PRACTICE
Connecting Exercises
to Standards
Three Web sites sell food for dogs that require a special diet. The
table shows expressions that give the total cost of ordering the
dog food from each Web site. Use the table for Exercises 1–3.
In Exercises 1–3 and 8 students must
write an equation to represent a given
real world situation and solve a
problem, satisfying A.CED.1.
1. USE A MODEL Natasha bought dog food from Super Chow and
Super Chow
paid a total of \$16.50. Write an equation that relates the
expression the Web site uses to calculate the cost and the amount
Natasha paid. Then use the equation to explain whether you think Natasha
A.CED.1, SMP 4
bought 7 pounds of dog food. In Exercises 4, 6, and 7 students must
justify each step in the solution of a
linear equation, satisfying A.REI.1.
Web Site
Cost for p pounds (\$)
Canine Kitchen
1.75(p- 1)
Pet Zone
2.50(p + 2)
1.50p + 6
1.50p + 6 = 16.50; Yes, Natasha bought 7 pounds of dog food because when p = 7, the left side of
the equation is 1.50(7) + 6 = 16.50, so the two sides of the equation are the same.
2. REASON QUANTITATIVELY Isaac bought dog food from Pet Zone and paid a total of
\$32.50. Did he buy 10, 11, or 12 pounds of dog food? Use an equation to justify your
A.CED.1, SMP 2
answer. 11 pounds; the equation is 2.50(p + 2) = 32.50; only p = 11 makes the left side of the equation
equal to the right side.
In Exercise 5 students solve an
equation using values from a provided
replacement set, satisfying A.REI.3.
3. CONSTRUCT ARGUMENTS The equation 1.75(p - 1) = 42 can be used to find the
amount of dog food the Cheng family ordered from Canine Kitchen. Did the Chengs
order more or less than 20 pounds of dog food? Justify your response. A.CED.1, SMP 3
More; when p = 20, the left side of the equation is 1.75(20 - 1) = 33.25, since this is less than
\$42, the Chengs must have ordered more than 20 pounds of dog food.
Dual Coding
CCSS
SMP
1
A.CED.1
4
2
A.CED.1
2
3
A.CED.1
3
4
A.REI.1
7
5
A.REI.3
7
6–7
A.REI.1
7
8
A.CED.1
2, 3, 4
Evaluate the exponent: 16 = x + 4 + (16 - 6); evaluate the expression in parentheses:
16 = x + 4 + 10; add: 16 = x + 14; the solution is x = 2; the solution is correct because
substituting x = 2 in the original equation results in a true statement.
5. USE STRUCTURE The values on the number line are the
replacement set for the equation m2 + 1 = 5. Which of
the values, if any, are solutions of the equation?
Explain. A.REI.3, SMP 7
-5 -4 -3 -2 -1
0
1
2
3
4
5
2 and -2; substituting m = 2 or m = -2 in the equation makes the left side of the equation equal
Exercise
4. USE STRUCTURE Describe the steps you use to solve the equation 16 = x + 4 + (24 - 6).
Then explain how you know your solution is correct. A.REI.1, SMP 7
to 5, so 2 and -2 are both solutions.
24 CHAPTER 1 Expressions, Equations, and Functions
Common Errors
In Exercise 5, students might check the given values on the number
line, working from left to right. Once they determine that x = -2 is a
solution of the equation, they may stop, assuming that they have
found “the solution.” Remind students that an equation may have more
than one solution and that they should continue to check the other
values.
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24 CHAPTER 1 Expressions, Equations, and Functions
USE STRUCTURE Solve each equation. First simplify the equation and use a property to
A.REI.1, SMP 7
justify each step. Then write the solution and explain how you found it.
6. c + 12 + 6c = 10c - 3c + 11 + 1
Original equation
c + 12 + 6c = 7c + 12
Substitution Property
c + 6c + 12 = 7c + 12
Commutative Property
7c + 12 = 7c + 12
Substitution Property
Solution: The solution is any real number; for any value of c, the left side of the equation
is equal to the right side.
7. 2n + 2 + 2n = 6n - 2n - 2
Original equation
2n + 2 + 2n = 4n - 2
Substitution Property
2n + 2n + 2 = 4n - 2
Commutative Property
4n + 2 = 4n - 2
Substitution Property
Solution: There is no solution; there is no value of n for which 4n + 2 can have the same value as
4n - 2.
8. A Web site offers its members a special rate for online
number of movies you watch and let C be the total cost
to watch the movies. A.CED.1
a. USE A MODEL Write an equation that relates
the total cost to the number of movies you
watch. SMP 4
Movie Mania
One-time sign-up fee: \$6.85
Then watch as many movies as you like
for just \$2.99 per movie!
C = 2.99m + 6.85
b. USE A MODEL Jeffrey watches 16 movies this month. Explain how to use the
SMP 4
equation to find his total cost. Common Errors
Students who are familiar with solving
an equation by performing the same
operation on each side may make an
error in Exercise 6. Once they simplify
the equation to 7c + 12 = 7c + 12,
they may subtract 7c from both sides
and subtract 12 from both sides,
leaving them with 0 = 0. Or they may
subtract 12 from each side and then
divide each side by 7c to get 1 = 1.
Students might conclude that the
original equation therefore has no
solution. Encourage students to stop
at an earlier stage of the process
when they can recognize that any
value of the variable makes the two
sides equal.
Substitute m = 16 in the equation; C = 2.99(16) + 6.85 = \$54.69
c. CRITIQUE REASONING Madison said she joined the site and paid exactly \$19 to
watch some movies. Her sister said this is impossible. Who is correct? Explain. SMP 3
Her sister is correct. The cost of 4 movies is 2.99(4) + 6.85 = \$18.81 and the cost of 5 movies
is 2.99(5) + 6.85 = \$21.80, so no number of movies costs exactly \$19.
d. REASON QUANTITATIVELY SuperFlix has no sign-up fee, just a flat rate per movie.
If renting 13 movies at MovieMania costs the same as renting 9 movies at
SuperFlix, what does SuperFlix charge per movie? SMP 2
2.99(13) + 6.85 = 45.72; 45.72 = 9p; p = 5.08; SuperFlix charges \$5.08 per movie.
1.5 Equations 25
Emphasizing the Standards for Mathematical Practice
You may want to use Exercise 8 to discuss aspects of SMP 4 (Model
with mathematics). One part of the standard is reflecting on whether
results make sense. Once students have used their model (equation)
to determine the cost of watching 16 movies, ask them whether their
result is reasonable, and why. To answer this, students might decide to
determine the cost without using the model. For instance, they could
use the text of the advertisement to reason that watching 16 movies
costs \$2.99 per movie, which is \$47.84, plus the sign-up fee of \$6.95,
which is a total of \$54.69. This should match the result students got
by solving an equation, which shows that the answer makes sense.
020_025_ALG1_C1L5_ISG_672788.indd 25
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1.5 Equations 25
1.6
Relations
1.6 Relations
STANDARDS
Objectives
A.REI.10 Understand that the
graph of an equation in two
variables is the set of all its
solutions plotted in the coordinate
plane, often forming a curve (which
could be a line).
Standards for Mathematical
Practice: 1, 2, 3, 4, 6
• Represent a relation in multiple ways.
• Identify the domain and range of a relation.
• Interpret the graph of a relation.
A relation is a set of ordered pairs. The set of the first numbers in the ordered pairs is the
domain. The set of the second numbers in the ordered pairs is the range.
EXAMPLE 1
Represent a Relation
they babysat. Each teenager gave the number of hours he or she babysat
and the amount he or she earned. The graph shows the results.
a. USE A MODEL Write the set of ordered pairs shown in the graph. What
do the numbers in each ordered pair represent? A.REI.10, SMP 4
{(1, 10), (2, 20), (3, 30), (3, 35), (6, 40)}; the first number is the
Amount Earned Babysitting
50 y
Amount Earned (\$)
STANDARDS
Content: A.REI.10, F.IF.1
Practices: 1, 2, 3, 4, 6
Use with Lesson 1–6
40
Lee
Chantel
Reynaldo
30
20
Eliza
10
Jordan
x
number of hours, the second number is the amount earned.
0
2
4
6
8
10
Time (hours)
PREREQUISITES
• Plot points on the coordinate plane
EXAMPLE 1
• What would the ordered pair
(5.5, 52) represent? A teenager
babysat for 5.5 hours and was
paid \$52.
y
1
10
2
20
3
30
3
35
6
40
Domain
Range
1
2
3
6
10
20
30
35
40
c. INTERPRET PROBLEMS Write the domain and range for the relation. Write each as a
F.IF.1, SMP 1
set within brackets, { }. Domain: {1, 2, 3, 6}; range: {10, 20, 30, 35, 40}
d. USE A MODEL How can you tell from the graph which of the teenagers babysat for the
A.REI.10, SMP 4
same number of hours? Chantel and Reynaldo babysat for the same number of hours because the points for their ordered
pairs lie on the same vertical line.
26 CHAPTER 1 Expressions, Equations, and Functions
Math Background
026_029_ALG1_C1L6_ISG_672788.indd 26
2/25/15 12:02 AM
Lessons 1.6 and 1.7 introduce students to relations and functions.
A relation is simply a set of ordered pairs. The ordered pairs can be
specified by a list, in which case the list is usually written using set
notation, such as {(-2, 4), (3, -5), (0, 4)}. The ordered pairs may also
be given in a table, a graph, or an equation. For example, the equation
y = x + 3 describes the relation with an infinite set of ordered pairs
that includes (-4, -1), (8, -1), and (π, π + 3).
The set of the x-values of the ordered pairs is the domain; the set
of the y-values in the ordered pairs is the range. Note that elements of
the domain and range are usually written in ascending order and
repeated elements are only listed once. For the first relation given
above, the domain is {-2, 0, 3} and the range is {-5, 4}.
• Why are there only 4 domain
values? Two of the teenagers
babysat for 3 hours each.
26 CHAPTER 1 Expressions, Equations, and Functions
Scaffolding Questions
• How do you find the ordered pair
for Jordan? Start at the origin and
count units to the right to find the
x-value; then count units moving
upward to find the y-value.
x
Teaching Tip
SMP 4
As students work through this
exploration, you may want to have a
brief discussion about the pros and
cons of each representation of the
given relation. For example, the graph
may make it easy to see any patterns
in the ordered pairs, while the
mapping makes explicit the
connection between domain values
and their corresponding range values.
b. USE A MODEL You can use a table or a mapping to represent a relation. A mapping is
a diagram that shows how each element in the domain is paired with an element in the
F.IF.1, SMP 4
range. Complete the table and mapping shown below.
EXAMPLE 2
e. REASON QUANTITATIVELY Which teenager was paid the highest hourly rate? Justify
Teaching Tip
Chantel; she made \$35 for 3 hours, which is \$11.67 per hour. Jordan, Eliza, and Reynaldo each
In a relation, the value that determines the output is the independent variable. The
variable with a value that is dependent on the value of the independent variable is the
dependent variable.
Interpret a Graph
Michelle’s Walk
a. USE A MODEL What are the independent variable and dependent
variable? Explain. F.IF.1, SMP 4
Independent variable: time; dependent variable: distance; the
distance depends on the amount of time Michelle has been walking
Distance from Home
EXAMPLE 2
Michelle started at her house and went for a long walk. The graph represents
her distance from home since the walk began.
Time
b. COMMUNICATE PRECISELY Describe what happens in the graph. A.REI.10, SMP 6
As time increases, distance increases, this shows Michelle walking further from home. The graph
becomes a horizontal line, so the distance from home is not changing but time increases. This may
mean she stops for a rest. Then she continues to walk further from home until the peak of the
graph. At this point, she begins to walk back to her house until she arrives at home.
house. Then she reaches her house and starts to walk away from
her house, stops, walks further away from her house, and stops
again. In the first graph, Michelle started and ended at her house.
In the second graph, Michelle started and ended somewhere
Michelle’s Walk
Distance from Home
There would be a horizontal segment on the downward-slanting part of the graph.
Sample answer: As time increases, Michelle walks toward her
• What is happening to Michelle’s
distance from home during the
horizontal part of the graph? Her
distance from home is not
changing (it remains constant).
Time
besides her house.
1.6 Relations 27
Emphasizing the Standards for Mathematical Practice
Example 2 can foster an interesting discussion related to SMP 4
(Model with mathematics). In particular, the example requires
students to think carefully about one representation of a relation
(a graph) so that they can translate it into another representation
(words). You might challenge students by asking them if the horizontal
portion of the graph must mean that Michelle stopped walking.
Although this is one possibility, the horizontal portion of the graph
only shows that her distance from home is not changing. It is possible
that she is walking in a circle, at a fixed distance around her home.
While this second option may not seem likely within the real-world
context, recognizing this possibility requires students to have a deep
understanding of what the graph represents.
026_029_ALG1_C1L6_ISG_672788.indd 27
Some students may look at the graph
of Michelle’s walk, notice its shape,
and assume that she was walking on a
steep mountain. Be sure students
understand that the graph is not a
picture or map of Michelle’s route. It
only tells her relative distance from
home at various times. To help
students understand this, have them
slowly trace the graph from left to
right with their finger. As they do so,
ask them to describe how Michelle’s
distance from home is changing.
Scaffolding Questions
• Why does it make sense that the
graph starts at the origin? At
time 0, the distance from home
is 0 because Michelle starts at
her house.
c. USE A MODEL How would the graph be different if Michelle decided to stop at her
aunt’s house on the way home and spend the night there? A.REI.10, SMP 4
d. COMMUNICATE PRECISELY The graph shows another walk that Michelle
took. What does this graph show? Compare the starting and ending points
to those in the first graph. A.REI.10, SMP 6
SMP 4
22/08/14 9:20 PM
• What point along the graph
represents the moment when
Michelle is farthest from home?
How do you know? At the peak of
the graph she is farthest from
home because that is when the
distance from home is the
greatest.
1.6 Relations 27
PRACTICE
PRACTICE
Exercises 1 and 3–6 provide students
with an opportunity to practice
interpreting a graph that represents a
relation between two quantities,
satisfying A.REI.10.
Exercise
Dual Coding
SMP
1
A.REI.10,
F.IF.1
1, 4
2
F.IF.1
1, 3, 4
3–4
A.REI.10
6
5
A.REI.10
3
6
A.REI.10
4
7
F.IF.1
3
8
F.IF.1
4
F.IF.1, SMP 1
Domain: {2, 3, 5.5, 8}; range: {0.5, 2, 4.5}
b. USE A MODEL What are the independent and dependent variables? F.IF.1, SMP 4
Independent variable: length; dependent variable: weight
c. USE A MODEL Complete the table, the mapping, and the graph below. x
y
2
0.5
3
0.5
3
2
5.5
4.5
8
4.5
Domain
Range
2
3
5.5
8
0.5
2
4.5
F.IF.1, SMP 4
Snake Sizes
5 y
4
3
2
1
x
0
2
4
6
8
10
Length (ft)
d. USE A MODEL How can you tell from the graph which snakes have the same
weight? A.REI.10, SMP 4
If the points lie on the same horizontal line, they have the same weight.
e. USE A MODEL What does it mean if there are two points on the graph that lie on a
vertical line? A.REI.10, SMP 4
If there are two points on any vertical line on the graph, then two snakes of the same length
have different weights.
2. The graph shows the number of items that eight customers bought at a
supermarket and the total cost of the items. F.IF.1
a. INTERPRET PROBLEMS Write the domain and range for the
relation. SMP 1
Domain: {2, 3, 5, 6, 8, 9}; range: {2, 6, 8, 10, 12, 14, 16, 18}
b. USE A MODEL Explain how you can tell from the graph how many of
the customers spent more than \$12. SMP 4
3 customers; find the number of points that lie above the
horizontal line y = 12.
Supermarket Costs
20 y
16
12
8
4
x
0
2
4
6
8
Number of Items
10
CCSS
a. INTERPRET PROBLEMS Write the domain and range for the relation. Total Cost (\$)
Exercises 1, 2, 7, and 8 provide
students with an opportunity to
practice recognizing the domain and
range of a relation, satisfying F.IF.1.
1. The following ordered pairs give the length in feet and the weight in pounds of
five snakes at the reptile house of a zoo: {(5.5, 4.5), (3, 0.5), (3, 2), (8, 4.5), (2, 0.5)}
Weight (lb)
Connecting Exercises
to Standards
c. CRITIQUE REASONING A student said you can add the values in the domain to find
the total number of items these customers bought. Do you agree? Explain. SMP 3
No; two customers bought 5 items and 8 items, so you cannot add domain values.
28 CHAPTER 1 Expressions, Equations, and Functions
Common Errors
In Exercise 1a, some students may write the domain as {2, 3, 3, 5.5, 8}.
Remind students that in a set, repeated elements are only listed once.
Thus, the set {2, 3, 3, 5.5, 8} is actually the same as the set {2, 3, 5.5, 8}.
In addition, students should be aware that changing the order of the
elements does not change a set, so the domain could be written as
{8, 2, 5.5, 3}. However, it is customary to list set elements in ascending
order.
026_029_ALG1_C1L6_ISG_672788.indd 28
28 CHAPTER 1 Expressions, Equations, and Functions
2/25/15 12:10 AM
3. COMMUNICATE PRECISELY Describe what happens in Tim’s
graph. SMP 6
Tim drives away from the pizzeria, stops to make a delivery,
continues to drive away from the pizzeria, stops to make another
Pizza Deliveries
Distance from Pizzeria
Tim and Lauren use their cars to deliver pizzas. The graph represents
their distance from the pizzeria starting at 6 pm. Use the graph for
Exercises 3–6.
A.REI.10
Lauren
Tim
Time
delivery, and then returns to the pizzeria.
4. COMMUNICATE PRECISELY Describe what happens in Lauren’s graph. SMP 6
At 6 PM, Lauren is not at the pizzeria; she drives directly there, without stopping.
5. CRITIQUE REASONING A student said that Tim’s and Lauren’s graphs intersect, so
their cars must have crashed at some time after 6 pm. Do you agree or disagree?
SMP 3
Explain. Disagree; the intersection point represents a time when Tim and Lauren were both at the same
distance from the pizzeria.
6. USE A MODEL After 6 pm, which delivery person was the first to return to the
SMP 4
pizzeria? How do you know? Lauren; Her graph intersects the x-axis before Tim’s graph. This means her distance from the
Common Errors
In Exercise 5, some students may see
the graph as a ”road map” that shows
the paths of the cars. In this case,
students may conclude that the point
of intersection represents a point of
collision. Help students understand
that the vertical axis represents the
drivers’ distances from the pizzeria.
The point of intersection shows that
there is a moment when both drivers
are at the same distance from the
pizzeria. However, they could be on
separate roads or even in different
towns.
pizzeria was 0 before Tim’s distance from the pizzeria was 0.
7. CRITIQUE REASONING Cameron said that for any relation, the number of elements in
the domain must be greater than or equal to the number of elements in the range. Do
you agree? If so, explain why. If not, give a counterexample. F.IF.1, SMP 3
Disagree; sample counterexample: in the relation {(1, 2), (1, 3)}, the domain is {1}, so it has one
8. USE A MODEL The graph shows the height of an elevator above the
ground. Describe the domain and range for this relation in words
and by using inequalities. Then give three ordered pairs in the
relation. F.IF.1, SMP 2
Domain: all real numbers from 0 to 4, {0 ≤ x ≤ 4};
Range: all real numbers from 0 to 80, {0 ≤ y ≤ 80};
Sample answer: (0, 0), (1, 20), (2, 40)
Height of an Elevator
100 y
80
Height (ft)
element, while the range is {2, 3}, which has two elements.
60
40
20
x
0
2
4
6
8
10
Time (s)
1.6 Relations 29
Emphasizing the Standards for Mathematical Practice
You can use Exercise 8 to connect to SMP 2 (Reason abstractly and
quantitatively). Students may be able to look at the graph and
describe the domain as ”all real numbers from 0 to 4,” but not be able
to translate this into a meaningful statement about the real-world
situation. In this case, the domain shows that the elevator traveled for
4 seconds. Similarly, the range shows that the elevator ascended from
a height of 0 feet (ground level) to a height of 80 feet. The fact that all
real numbers from 0 to 80 are included in the range shows that the
elevator passed through all the possible heights from 0 feet to 80 feet.
026_029_ALG1_C1L6_ISG_672788.indd 29
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1.6 Relations 29
1.7
Functions
1.7 Functions
F.IF.1 Understand that a function
from one set (called the domain)
to another set (called the range)
assigns to each element of the
domain exactly one element of
the range. If f is a function and x is
an element of its domain, then f(x)
denotes the output of f
corresponding to the input x. The
graph of f is the graph of the
equation y = f(x).
Standards for Mathematical
Practice: 1, 2, 3, 4, 6
Content: F.IF.1, F.IF.2, F.IF.5
Practices: 1, 2, 3, 4, 6
Use with Lesson 1–7
• Understand the definition of a function and identify relations that
are functions.
• Use and interpret function notation.
• Relate the domain of a function to its graph.
A function is a relation in which there is exactly one output for each input. In other words,
each element of the domain is assigned to exactly one element of the range.
EXAMPLE 1
Identify Functions
F.IF.1
EXPLORE Tristan surveyed students at some local high schools. At each school, he asked
six students how long they studied for their last exam and the score they received on the
exam. His data is shown in the table, mapping, and graph below.
Central High Score
Time (h)
Score
0.5
81
1
81
3
92
1.5
75
2
90
1.5
94
Westlake High School
Time (h)
Score
1
1.5
2
2.5
3
4
77
Miller High School
100 y
78
80
85
60
93
Score
STANDARDS
STANDARDS
Objectives
40
20
x
0
1
2
3
4
5
Time (h)
a. USE A MODEL For each school, is the relation a function? Why or why not? SMP 4
Central: No; the input value 1.5 is assigned to two output values, 75 and 94.
Westlake: Yes; each input value is assigned to exactly one output value.
• Plot points on the coordinate plane
EXAMPLE 1
SMP 4
Teaching Tip
Help students make sense of each
model by first asking them to identify
the input (the amount of time a
student studies) and the output (the
student’s score).
• In the graph, how can you tell that
an input is assigned to two
outputs? Two points lie on the
same vertical line.
Yes; the relation is not a function because the input value 3 is assigned to two output values,
81 and 87. To make the relation a function, Tristan could change the ordered pair (3, 81) to (3.5, 81).
30 CHAPTER 1 Expressions, Equations, and Functions
Math Background
Functions are a key tool for describing real-world phenomena. In many
real-world situations, it makes sense that each “input” value has
exactly one “output” value. For example, a function might provide the
temperature in downtown Houston at any given time. In this case, the
input is a specific time of day, and it makes sense that there can only
be one temperature (output) corresponding to that time of day.
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Students may wonder why they need to use function notation. Point
out that it provides a shorthand for naming functions and
distinguishing among multiple functions. Students will also see the
utility of function notation in later courses when they work with
f(x2) - f(x1)
complex expressions such as _______________
x -x .
30 CHAPTER 1 Expressions, Equations, and Functions
2
1
Scaffolding Questions
• In the mapping, how can the arrows
mapping is a function? If no input
value has more than one arrow,
the mapping is a function.
Miller: No; the input value 3.5 is assigned to two output values, 80 and 90.
b. CRITIQUE REASONING Tristan surveyed six students at Chavez High School, and
wrote the data as this set of ordered pairs: {(3, 87), (4, 98), (2.5, 70), (1.5, 70), (0.5, 67),
(3, 81)}. He claimed that the relation is not a function, but he said that he could change
just one input value or one output value and make it a function. Do you agree with
Tristan? Explain. SMP 3
PREREQUISITES
EXAMPLE 2
A graph that consists of points that are not connected is a discrete function. A function
with a graph that is a line or a smooth curve is a continuous function.
EXAMPLE 2
Teaching Tip
Graph a Function
Tickets to the county fair cost \$5 each.
F.IF.1, SMP 4
Yes; each input value is assigned to exactly one output value.
25 y
20
Total Cost (\$)
b. USE A MODEL Is the relation a function? Explain. If students have trouble going directly
from the verbal description to the
graph, suggest that they first make a
table of values and then plot the
ordered pairs from their table.
County Fair Tickets
a. USE A MODEL Make a graph that shows the relationship
between the number of tickets you buy and the total cost
F.IF.1, SMP 4
of the tickets. 15
10
5
c. COMMUNICATE PRECISELY Is the function a discrete or continuous
function? Why? Use the real-world context to explain why your
F.IF.1, SMP 6
0
2
4
6
8
10
Number of Tickets
Discrete; the graph consists of points that are not connected.
This makes sense because you can only buy a whole number of tickets to the fair.
d. REASON QUANTITATIVELY What is the domain of the function? How is the domain
related to the real-world context? F.IF.5, SMP 2
The domain is the set of whole numbers, {0, 1, 2, ...}. You can only buy a whole number of tickets.
e. REASON QUANTITATIVELY What is the range of the function? How is the range
related to the real-world context? F.IF.1, SMP 2
The range is the multiples of 5, {0, 5, 10, ...}. Since the total cost is \$5 times the number of tickets,
the total cost must be a multiple of \$5.
f. CRITIQUE REASONING A student said that if you write the function as a set of
ordered pairs, the ordered pair (25, 100) will be an element of the set. Do you agree or
F.IF.1, SMP 3
disagree? Explain. Scaffolding Questions
• For each point on the graph, how is
the y-value related to the x-value?
The y-value is 5 times the
x-value.
• Would it make sense to plot the
point (1.5, 7.5)? Why or why not?
No; you cannot buy 1.5 tickets to
the fair.
• Does it make sense to include the
point (0, 0) in the graph? Why or
why not? Yes; if you buy 0 tickets,
you pay 0 dollars.
Disagree; the first number in the ordered pair is the number of tickets you buy; if you buy 25
tickets, then the cost is 5(25) = \$125, not \$100, so (25, 100) is not in the set.
SMP 4
g. COMMUNICATE PRECISELY Describe a situation that could be modeled by a function
that includes the same ordered pairs as the function for the cost of county fair tickets,
but is of the type that you did not select as your answer to part c. Explain why this
situation leads to a function of the other type. F.IF.1, SMP 6
Sample answer: Malia rides her bike at a speed of 5 miles per hour away from her home. The distance
she has traveled is 5 times the number of hours she has been biking. This situation would be
represented by a continuous function. It includes ordered pairs such as (1, 5) and (2, 10), like the ticket
cost situation, but the function also makes sense for all values of time between whole numbers of hours.
• What do you notice about all the
points on your graph? They all lie
along a straight line that passes
through the origin.
1.7 Functions 31
ELL Strategies
ELL students may be unfamiliar with the word context. Explain that a
context is a real-world setting for a problem. In Example 2, the
context is a county fair where tickets cost \$5 each. You may want to
check for understanding by asking students to describe the context of
Example 1 in their own words. If students have difficulty, suggest that
they reread the given information and underline any words that
provide information about the real-world setting of the problem.
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1.7 Functions 31
EXAMPLE 3
Teaching Tip
SMP 1
If students need help getting started
with their table, suggest that they
first fill in the row for the values of x
and use equally-spaced values, such
as x-values that increase by 1. Then
have students find the corresponding
y-values. By choosing x-values that
are equally-spaced, students may be
able to identify patterns that can help
them fill in the y-values.
The vertical line test is a way to check whether a graph represents a function. If there is a
vertical line that intersects the graph in more than one point, then the graph is not a
function. Otherwise, the relation is a function.
EXAMPLE 3
Graph a Function
Follow these steps to graph the equation y - 2x = 4 and determine whether the
equation represents a function.
a. INTERPRET PROBLEMS Complete the table below by finding five ordered pairs that
satisfy the equation. F.IF.1, SMP 1
Sample ordered pairs shown.
x
-2
-1
0
1
2
y
0
2
4
6
8
b. INTERPRET PROBLEMS Use the table to help you graph the equation on the
F.IF.1, SMP 1
coordinate plane provided. c. CONSTRUCT ARGUMENTS Is the relation a function? Use your graph to justify
Yes; there is no vertical line that intersects the graph in more than one
point, so the relation is a function by the vertical line test.
Scaffolding Questions
• For this relation, what happens to
the y-value as the x-value increases
by 1? The y-value increases by 2.
• Why does it make sense to draw the
graph as a continuous function?
The domain is all real numbers.
−8 −6 −4
O
y
2 4 6 8x
−4
−6
−8
d. REASON ABSTRACTLY What are the domain and range of the function? Explain how
F.IF.5, SMP 2
these are related to the graph of the function. The domain and range are all real numbers. The line continues infinitely to the left and to the right,
showing that the domain is all real numbers. The line continues infinitely up and down, showing
that the range is all real numbers.
e. COMMUNICATE PRECISELY Describe how you can use the graph of the function to
find the output value that corresponds to the input value -6. F.IF.1, SMP 6
Find the value -6 on the x-axis. Then move down vertically to the graph of the function. Then
move horizontally to the y-axis and read the corresponding value. The output value is -8.
f. COMMUNICATE PRECISELY How did you know to draw the graph as a continuous
function? Use the given equation in your justification. How can you tell if a relation
is discrete? F.IF.5, SMP 6
Sample answer: The graph of y − 2x = 4 is continuous because the domain and range are all real
numbers. A relation would be discrete if the equation or the context of the situation resulted in
• How can you check that you drew
the graph correctly? Choose a
point on the graph and check
that the x- and y-coordinates of
the point satisfy the given
equation.
8
6
4
2
outputs that are not connected when graphed.
32 CHAPTER 1 Expressions, Equations, and Functions
Differentiating Instruction
Kinesthetic learners may benefit from using a physical object to help
them perform the vertical line test. For example, students might use a
pen or pencil or an uncooked strand of spaghetti to represent the
vertical line. Have them hold it vertically and slowly pass it across the
graph from left to right. This may make it easier for students to
determine if there is ever an x-value for which the line intersects the
graph in more than one point.
030_035_ALG1_C1L7_ISG_672788.indd 32
32 CHAPTER 1 Expressions, Equations, and Functions
22/08/14 9:20 PM
EXAMPLE 4
Function notation is a way to use an equation to write a rule for a function. For example, in
function notation, the equation y = x + 7 is written as f(x) = x + 7. The graph of the
function f(x) is the graph of the equation y = f(x).
Teaching Tip
In function notation, f(x) denotes the element of the range corresponding to the element x
of the domain. For example, for the function f(x) = x + 7, f(9) represents the output value
that corresponds to the input value x = 9. Therefore, f(9) = 9 + 7 = 16.
EXAMPLE 4
SMP 2
It might be helpful for students to
think of a function as a “machine” that
takes an input, performs one or more
operations on it, and produces an
“output.” A simple sketch based on the
function in this example is shown
below.
17.5
Use and Interpret Function Notation
Candace runs a company that installs fences. She calculates the total cost C of installing
a fence using the function rule C(x) = 5x + 25, where x is the length of the fence in feet.
a. REASON QUANTITATIVELY What is the value of C(17.5)? What does C(17.5)
F.IF.2, SMP 2
represent? C(17.5) = 112.5; C(17.5) represents the cost of installing a fence that is 17.5 feet long. So it costs
\$112.50 to install a fence 17.5 feet long.
b. USE A MODEL Explain how Candace can use the function rule to find the cost of the
F.IF.2, SMP 4
installing a fence that is 11 yards long. 11 yards is 33 feet, so calculate C(33) by substituting x = 33 in the equation;
5x + 25
C(33) = 5(33) + 25 = 190; the cost of installing the fence is \$190.
c. USE A MODEL Graph C(x) on the coordinate plane at the right. F.IF.1, SMP 4
d. REASON ABSTRACTLY What are the domain and range of
the function? Explain how these are related to the graph of
F.IF.5, SMP 2
the function. 40
The range is all real numbers greater than or equal to 25; the
line starts at y = 25 and continues upward infinitely.
e. COMMUNICATE PRECISELY Did you draw the graph as a
discrete function or as a continuous function? Justify
F.IF.1, SMP 6
The domain is all real numbers greater than or equal to 0; the
line starts at the y-axis and continues infinitely to the right.
112.5
Cost of Installing a Fence
50 y
Scaffolding Questions
30
20
10
x
0
2
4
6
8
10
Length of Fence (ft)
• How do you use the order of
operations to evaluate C(17.5)?
First multiply 17.5 by 5; then,
Continuous; you can order a fence with a length that is any
• In this problem, what units are
associated with the inputs? What
units are associated with the
outputs? feet; dollars
real number greater than or equal to 0, so it makes sense to draw a continuous graph.
f. COMMUNICATE PRECISELY Would your answer to part e change if the fencing
material were only sold in one-foot increments? If someone needed to fence a length
that was not a whole number, what should they do? Explain. F.IF.5, SMP 6
Yes, I would draw a discrete function if the material could only be purchased in one-foot
increments. The domain would only consist of whole numbers. If someone needed to fence a
length that was not a whole number, they should buy a length of fence that is the next whole
number greater than the length they need.
1.7 Functions 33
Emphasizing the Standards for Mathematical Practice
(Construct viable arguments and critique the reasoning of others).
For example, you might ask students to discuss the connection
between relations and functions: Is every relation a function? Is every
function a relation? Why? Students might draw a simple Venn
diagram, as shown below, to help them respond to these questions
030_035_ALG1_C1L7_ISG_672788.indd 33
22/08/14 9:20 PM
• In the graph, why does it make
sense that the line slopes upward
as you move from left to right? This shows that the cost of the
fence increases as the length of
the fence increases.
relations
functions
1.7 Functions 33
PRACTICE
PRACTICE
Exercise 1 requires students to work
with multiple representations to
identify relations that are functions,
per F.IF.1.
In Exercise 2, students graph a
function based on a verbal description
of a real-world situation, satisfying
F.IF.5.
1. USE A MODEL Mario collected data about some of the players on a women’s
basketball team. The data is shown in the table, mapping, and graph. Is each relation
a function? Why or why not? F.IF.1, SMP 4
Age and Height
Team History
Years on
Team
Games
Played
1
24
2
45
3
82
3
88
5
120
Age
Height (in.)
22
70
71
72
73
74
23
25
Stats from Last Game
20 y
Total Points Scored
Connecting Exercises
to Standards
16
12
8
4
x
0
2
4
6
8
10
Team history: No; the input value 3 is assigned to two output values, 82 and 88. Age and height:
No; the input values 22 and 25 are both assigned to two output values. Stats from last game: Yes;
each input value is assigned to exactly one output value.
a. USE A MODEL Make a graph that shows the relationship
between the number of servings and the number
of eggs. F.IF.1, SMP 4
b. USE A MODEL Is the relation a function?
Explain. F.IF.1, SMP 4
10 y
8
6
4
2
Yes; each input value is assigned to exactly one output
value.
x
0
2
4
6
8
10
Number of Servings
c. REASON QUANTITATIVELY What is the domain of the function?
How is the domain related to the real-world context? F.IF.5, SMP 2
The domain is all whole numbers greater than or equal to 1. You can only make 1 or more
servings, and the number of servings must be a whole number since you can only use a whole
Exercise 6 is a reasoning exercise
that requires students to think
carefully about the definition of a
Exercise
Dual Coding
SMP
1
F.IF.1
4
2
F.IF.1,
F.IF.5
2, 4
3–4
F.IF.1
1
5
F.IF.2,
F.IF.5
2
6
F.IF.1
3
number of eggs.
34 CHAPTER 1 Expressions, Equations, and Functions
Common Errors
In Exercise 2 part c, students may state that the domain is all real
numbers greater than 0. Students might argue that is possible to use
proportional reasoning to multiply or divide quantities in a recipe and
make any number of servings, such as 0.7 servings or 8.5 servings.
While this is theoretically true, the recipe depends on using a whole
number of eggs, which restricts the number of servings that can be
made to whole numbers greater than or equal to 1.
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34 CHAPTER 1 Expressions, Equations, and Functions
22/08/14 9:20 PM
CCSS
In Exercise 5, students interpret
function notation, perform
calculations using function notation,
and interpret the domain of the
function in the context of the situation,
satisfying both F.IF.2 and F.IF.5.
2. A recipe for homemade pasta dough says that the number of
eggs you need is always one more than the number of servings
you are making. Number of Eggs
Exercises 3 and 4 give students
practice in graphing a function from
INTERPRET PROBLEMS Graph each equation. Then explain whether or not the equation
represents a function.
F.IF.1, SMP 1
3. 2x + y = 6
8
6
4
2
−8 −6 −4
O
4. y = x2
y
8
6
4
2
2 4 6 8x
−8 −6 −4
−4
−6
−8
O
y
2 4 6 8x
−4
−6
−8
Function; there is no vertical line that
Function; there is no vertical line that
intersects the graph in more than one point.
intersects the graph in more than one point.
5. REASON QUANTITATIVELY The height h of a balloon, in feet, t seconds after it is
released is given by the function h(t) = 2t + 6. SMP 5
a. What is the value of h(20), and what does it tell you? F.IF.2
h(20) = 46; the height of the balloon 20 seconds after it is released is 46 feet.
b. Explain how to use the function to find the height of the balloon 2 minutes after it
F.IF.2
is released.
2 minutes is 2(60) = 120 seconds, so calculate h(120) by substituting t = 120 in the equation;
h(120) = 2(120) + 6 = 246; the height of the balloon is 246 feet.
c. What is the height of the balloon just before it is released? How do you know?
F.IF.2
6 feet; t = 0 before the balloon is released, and h(0) = 6.
d. Are there any restrictions on the values of t that can be used as inputs for the
function? If so, how would this affect the graph of the function? Explain. F.IF.5
Common Errors
In Exercise 6, students may state that
the missing value cannot be 3 or 5.
Point out that if the missing value
were 3 or 5, one of the ordered pairs
would be repeated within the set.
Although elements of a set are usually
listed only once, these repeated
elements would not mean that the
relation is not a function. Suggest
that students reread the definition of
a function and ask them to think
about whether there is a way to
replace the question mark with a
value so that a single input is assigned
to more than one output. This may
help students realize that the missing
value cannot be −4 or −3.
Sample answer: The values of t must be greater than or equal to zero because a negative value
for the time does not make sense for the given situation. The graph would start at the vertical
axis and go only to the right.
6. CONSTRUCT ARGUMENTS The following set of ordered pairs represents a function,
but one of the values is missing and has been replaced by a question mark: {(-4, -1),
(-3, -1), (3, 2), (5, 2), (?, 2)}. What conclusions can you make about the missing
F.IF.1, SMP 3
value? Explain. The missing value cannot be -4 or -3. If the missing value were either of these, then the set of
ordered pairs would no longer be a function since there would be an input value assigned to two
output values.
1.7 Functions 35
Emphasizing the Standards for Mathematical Practice
You may want to use Exercise 5 to address SMP 5 (Use appropriate
tools strategically). For example, students might enter the function
h(t) = 2t + 6 as function Y1 in their graphing calculator. Students can
use the calculator to display a table for the function and look for
patterns in the y-values. Then students can display a graph and use it
(or the table) to check their answers to the questions in this exercise.
2/20/15 5:22 PM
030_035_ALG1_C1L7_ISG_672788.indd 35
1.7 Functions 35
1
Finding a Sale Price
Finding a Sale Price
Students will use a graph to write
functions, find domain and range, and
calculate sale prices.
Provide a clear solution to the problem. Be sure to show all of your work,
Sander’s Market is having a special on cherries. For every pound of cherries
purchased beyond 3 pounds and up to 6 pounds, the price per pound is discounted
by \$1. Sander’s Market limits customers to 6 pounds of cherries. The graph shows
the cost y in dollars for purchasing x pounds of cherries.
y
STANDARDS
A.CED.1, A.REI.10, F.IF.1, F.IF.2,
F.IF.5
Cost (\$)
10
8
6
4
2
O
Standards for Mathematical
Practice: The Chapter 1
Mathematical Practices SMP 1,
SMP 2, SMP 3, SMP 4, and SMP 6.
2
4
6
x
Weight of Cherries (lb)
Part A
Find the domain and range. Describe their meaning in the context of this situation.
• Name some ordered pairs that lie
(0, 0), (2, 5), (4, 9)
Jump Start
If students have trouble getting
started with the task, suggest that
they first focus on the given graph.
Use some or all of the following
questions to help students
understand the graph.
36 CHAPTER 1 Expressions, Equations, and Functions
Emphasizing the Standards for Mathematical Practice
This Performance Task is closely aligned to SMP 2 (Reason abstractly
and quantitatively). Throughout the task, students will need to move
back and forth between the abstract mathematics and the contextual,
real-world situation. For example, to highlight the abstract nature of
the task, ask students to identify the point on the graph with an x-value
of 2. Students should find that the point has coordinates (2, 5). They
might check that this point is a solution of the equation y = 2.5x, which
describes the portion of the graph for 0 ≤ x ≤ 3. To highlight the
contextual side of the problem, ask students what the point (2, 5)
represents. Be sure students understand that this ordered pair
carries information about the real-world situation; namely,
2 pounds of cherries cost \$5.
036_037_ALG1_C1PT1_ISG_672788.indd 36
• Why does it make sense that (0, 0)
lies on the graph? 0 pounds of
cherries cost 0 dollars.
Next, have students read Part A.
If they difficulty understanding the
question, have them use their
glossary, as needed, to look up any
unfamiliar terms, such as domain,
range, rational, or irrational.
36 CHAPTER 1 Expressions, Equations, and Functions
• How can you tell that the price per
pound decreases for more than
3 pounds? The graph becomes
less steep for more than
3 pounds.
22/08/14 9:10 PM
SMP 1
Teaching Tip
Part C provides an opportunity to
connect to SMP 1 (Make sense of
problems and persevere in solving
them). Ask students why it makes
sense that this real-world relation is
a function. If necessary, help students
see that for any weight of cherries,
there is only one corresponding cost.
In other words, each input (weight) is
paired with exactly one output (cost).
It would not make sense if a single
weight had more than one cost
associated with it.
Part B
Use the graph to determine the cost per pound for buying 3 or fewer pounds of cherries.
Write and solve an equation to find the total cost for purchasing 2.5 pounds of cherries.
Part C
Explain whether the graph represents a relation and/or a function.
Part D
Write a function f(x) that gives the cost for purchasing x pounds of cherries, where
3 < x ≤ 6. Explain how you arrived at your answer.
Scoring Rubric
036_037_ALG1_C1PT1_ISG_672788.indd 37
Common Errors
Some students may have difficulty
writing a function rule in Part D. In
particular, students might write
f(x) = 3x + 1.5 instead of f(x) =
1.5x + 3. Encourage students to
check their rule by seeing if it works
for some of the ordered pairs on the
graph. For instance, the point (4, 9)
lies on the graph, so students should
be able to evaluate their function for
x = 4 and find that f(4) = 9. If they get
a different result, have students
revisit the steps they used to develop
the function rule.
22/08/14 9:10 PM
Part
Max
Points
A
2
The domain is 0 ≤ x ≤ 6 because it is impossible to purchase a negative amount of cherries and a customer
cannot purchase more than 6 pounds. The range is 0 ≤ y ≤ 12 because it is impossible to spend a negative
amount of money and if a customer purchases the maximum 6 pounds of cherries he or she will spend \$12.
B
2
The line y = 2.5x represents the cost of buying 3 or fewer pounds of cherries. y = 2.5(2.5) or y = 6.25
represents the cost of buying 2.5 pounds of cherries. So 2.5 pounds of cherries cost \$6.25. You can check your
answer by making sure the graph passing through the point (2.5, 6.25).
C
2
The graph represents a relation, because it is a set of ordered pairs. The graph also represents a function,
because each x-value (number of pounds) has a unique corresponding y-value (cost).
D
2
f(x) = 3 + 1.5x; Find two points on the graph when 3 < x ≤ 6. Use these two points to find the slope of the
line passing through them. Use the slope and one point to find the y-intercept. Then use the slope and
y-intercept to write the function.
Total
8
Full Credit Response
1
At the Box Office
At the Box Office
Students will write expressions and
write and solve an equation to
determine a cost-efficient way to buy
tickets.
Provide a clear solution to the problem. Be sure to show all of your work,
Toby and his friends want to go see a play in the new theater downtown. He has the
following options to save money on a purchase of several tickets. Each ticket agent
sells the tickets at the same full-price before any discounts are applied.
Ticket-Time
SUPERSTUB
Bill’s Box Office
For every two full-price
one free ticket!
20% off any order of
4 or more tickets!
Get \$10 off your
total order price.
STANDARDS
A.CED.1, F.IF.2
Part A
Standards for Mathematical
Practice: The Chapter 1
Mathematical Practices SMP 1,
SMP 2, SMP 3, SMP 4, and SMP 7.
Define a variable and write expressions to represent the cost of 10 tickets if purchased
at Ticket-Time, Superstub, or Bill’s Box Office. Simplify each expression. Based on your
expressions, is it possible to determine whether the total price for 10 tickets is less at
Jump Start
• What is the given information in
Part A? the number of tickets
purchased
• Aside from the discount
information, what other information
is missing? the cost of a full-price
ticket
Next, have students read Part A. If
students have difficulty determining
the Ticket-Time expression, use a
table to consider how many tickets
are actually free for 2, 3, 4, 5, 6, 7,
and 8 tickets.
38 CHAPTER 1 Expressions, Equations, and Functions
Emphasizing the Standards for Mathematical Practice
This Performance Task is closely aligned to SMP 1 (Make sense of
problems and persevere in solving them). The piecewise behavior of
the cost for tickets purchased at Ticket-Time may present a challenge
to some students. Students will write and graph such functions later
in the course. Part C of the Performance Task is a multi-part question
that requires several steps before arriving at the required answer. In
Part C, students must make the connection that purchasing tickets in
multiples of 3 means that for each grouping of 3 tickets, 2 are paid for
and 1 is free. Hence, dividing n by 3 actually gives the number of free
tickets.
038_039_ALG1_C1PT2_ISG_672788.indd 38
38 CHAPTER 1 Expressions, Equations, and Functions
2/20/15 5:22 PM
SMP 2
Teaching Tip
Part B provides an opportunity to
connect to SMP 2 (Reason abstractly
and quantitatively). In this case,
students must reason both abstractly
and quantitatively. Students must
reason abstractly to make the
connection between the expressions
and the cost, and that 8b > 7b for
b > 0. Quantitatively, since b is a
positive whole number and the
expressions represent a dollar
amount, Ticket-Time will always cost
less than Superstub for 10 tickets.
Students may make comparisons
between other ticket agents, but only
this comparison is independent of the
value of b.
Part B
Suppose a school buys 34 tickets for \$500 using Bill’s Box Office. Could the school have
saved more money if it bought the tickets at Superstub? Show your work.
Part C
Suppose a full-price ticket costs \$15. A customer buys n tickets at Ticket-Time, where
n is a multiple of 3. Write a function C(n) that gives the cost of the tickets with the
discount. Evaluate C(6) and describe its meaning.
Part D
Explain which ticket agent gives the worst or the best deal, as the number of tickets
purchased increases.
Scoring Rubric
038_039_ALG1_C1PT2_ISG_672788.indd 39
Part
A
B
2/20/15 5:22 PM
Max
Points
Full Credit Response
2
Let b = the original price of a ticket; Ticket-Time: 7b,
Superstub: 8b, Bill’s Box Office: 10b - 10. You can only tell
that the price for Ticket-Time will be less than Superstub
because for any positive whole number value of b, 8b > 7b.
2
Yes, they could have saved \$92; 500 + 10 = 510,
510 ÷ 34 = 15, so the original price of a ticket is \$15.
0.8(15)(34) = 408, so the cost after the 20% discount at
Superstub is \$408. \$500 - \$408 = \$92
Common Errors
Make sure students are correctly
interpreting each discount. For
less than 4 tickets involves no
results in a 20% discount. Students
should know that multiplying by 0.8
(or taking 80% of a number) is
equivalent to finding 20% of the
original price and then subtracting to
find the cost after discount.
C(n) = 10n; For every 3 tickets purchased, one will be free,
2
C
2
D
2
Total
8
so ___
n is the number of tickets that will be paid for if
3
ticketes are purchased in multiples of 3. C(6) = 10(6) or 60,
so 6 tickets will cost \$60.
As the number of tickets increases, the worst deal is Bill’s
Box Office, because the discount is \$10 regardless of the
number of tickets purchased. The best deal is Ticket-Time,
if tickets are purchased in multiples of 3, because the free
tickets equate to about a 33% discount, and 33% > 20%.
Diagnosing Errors
incorrectly may be confusing x-values
and y-values when locating points on
the graph. Remind them that the input
values of the function are the
x-values, and the outputs are the
y-values.
1. Six expressions are shown. Select all the
expressions that are equivalent to 3x - 12y.
4. The graph shows part of the function y = f(x).
A.SSE.2
y
4
2
4(x - 3y)
−2
2x - 6y - (x + 6y)
2
O
x
−2
3(x - 4y)
x + 4y + 2(x − 8y)
6x - 6y - (3x - 6y)
4(x − 3y) − x
last year than Hector.
Write an equation that represents the number of
A.CED.1
y=x+5
3. Theo is solving the equation 3x - 2 = -4.
He adds 2 to both sides of the equation and then
he divides both sides of the equation by 3.
Select all the properties shown below that allow
Theo to justify these steps. A.REI.1
Distributive Property
Complete the following. F.IF.1
f(2) = 3 .
f(1) = 2 .
f( -1 ) = 0
f( 0 ) = 1
5. Consider the following expression.
2(x − 1) + 4x2 + 2(x2 − 1)
When the expression is completely simplified, the
coefficient of the x2-term is 6 . A.SSE.1a
6. In the graph, each ordered pair gives the low
temperature and the high temperature for each
day of a recent winter week in Pinewood. Draw a
vertical line on the graph that can be used to show
why this relation is not a function. F.IF.1
Temperatures in Pinewood
y
8
6
4
2
x
Commutative Property
Multiplication Property of Equality
Division Property of Equality
0
2
4
6
8
Low Temperature (°F)
x(x + 1)
Students who have difficulty with
Item 6 may not understand the
various ways to identify a function.
This is an opportunity to discuss
multiple representations. For
example, students may understand
that in a function each input value
must be paired with exactly one
output value, and they may be able to
recognize a function when it is
presented in a mapping diagram or
table of values, but they may not
understand how the definition of a
function can be displayed as a graph.
Be sure students recognize that any
vertical line can only pass through one
point of a function. If there is a
vertical line that passes through more
than one point, the relation cannot be
a function because the input value
(x-value) corresponding to the line is
paired with more than one output
value (y-value).
Standardized Test Practice
High Temperature (°F)
Standardized Test
Practice
7. For the function f(x) = ___________ with domain
2
{1, 2, 3, 4, 5}, what is the range? F.IF.1
Range: {1, 3, 6, 10, 15}
40 CHAPTER 1 Expressions, Equations, and Functions
Test-Taking Strategy
040_041_ALG1_C1STP_ISG_672788.indd 40
2/20/15 5:22 PM
40 CHAPTER 1 Expressions, Equations, and Functions
Some students may have trouble with Item 2 because they are not
write the values of a and b. Point out to students that they should be
sure to write down all the intermediate steps in their solution process.
Once students have written the final equation y = x + 5 or y = 1x + 5,
they can draw boxes around the values of a and b in the equation. This
will help ensure that they write these values correctly in the answer
boxes.
Diagnosing Errors
8. Consider each product or sum. Select Rational or Irrational for each row. Then explain why you selected
Rational or Irrational for each product or sum. N.RN.3
Product or Sum
Rational
Irrational
___
___
The product of √2 and √7 .
___
___
The product of √2 and √18 .
___
3
The sum of √17 and ___.
5
__
√2 · √18 = 6
The sum of π and 19.
___
The sum of √16 and 0.9.
__
√14 is irrational.
__
In Item 8, students who incorrectly
___
_____
identify the product of √2 and √18
as irrational may not be using
properties of square roots to multiply
the given values. Be
_ sure
___ students
_____
understand that √a √b = √ab .
Explanation
The sum of a rational number
and an irrational number is irrational.
The sum of a rational number
and an irrational number is irrational.
__
√16 + 0.9 = 4.9
9. Consider each relation. Is each relation a function? Select Yes or No in each row. For any row in which you
select No, explain why the relation is not a function. F.IF.1
Is the relation a function?
Yes
last part of Item 9 may not
understand that the given relation
maps every positive integer greater
than 1 to at least two outputs. It may
be helpful for students to write down
some sample input/output
combinations. For instance, the
integer 8 maps to 1, 2, 4, and 8.
Seeing a specific example like this
may help students recognize that the
relation is not a function.
No
{(-3, 2), (3, 2), (2, 2)}
{(-2, -5), (- 1, 0), (0, -3), (1, 4), (1, 6)}
The relation that maps each real number to 2 times the number
The relation that maps every real number to 0
The relation that maps each positive integer to its factors
For the relation in the second row the input value 1 is assigned to two output values, 4 and 6.
For the relation in the fifth row, each positive integer other than one has at least two factors,
so for each input other than 1 there would be multiple outputs.
10. Solve the equation 0.2(4t + 10) = 6.8. Show your work and justify each step of the solution process. A.REI.1
4t + 10 = 34 (Division Prop. of =); 4t = 24 (Subtraction Prop. of =); t = 6 (Division Prop. of =)
11. Aya plans on buying a new television with a retail price of x dollars. The store is having a sale, and there is
also a rebate. The function p(x) = 0.8x - 150 gives the price p that Aya will end up paying for the television.
a. Interpret the meaning of the terms 0.8x and 150 in the function, in terms of the sale and the rebate.
Rubrics
A.SSE.1a
0.8x represents the price after a 20% discount; 150 represents a \$150 rebate.
b. Evaluate p(1500) and describe what it means. Item 10
[2] Correct answer of t = 6 and at
least two correct properties for
the solution
[1] Correct answer of t = 6 only
[0] no response OR incorrect answer
and reasoning
F.IF.1, F.IF.2
p(1500) = 0.8(1500) - 150 = 1050; Aya will pay \$1050 for a television that retails for \$1500.
c. Aya plans to end up paying between \$1400 and \$1800 for the television. Find the retail prices that
she can shop for when at the store. How do these values relate to the domain and range of the function?
F.IF.1, F.IF.5
\$1937.50 to \$2437.50; domain: 1937.5 ≤ x ≤ 2437.5, range: 1400 ≤ p ≤ 1800
CHAPTER 1 Standardized Test Practice 41
Test-Taking Strategy
040_041_ALG1_C1STP_ISG_672788.indd 41
22/08/14 9:12 PM
Some students may find the tables in Items 8 and 9 overwhelming.
Suggest that students focus on a single row at a time. In order to help
them do this, students may want to use a sheet of blank paper to
cover all the rows below the one they are working on. This will also
help ensure that students place a check mark in the appropriate cell
in the correct row.
Item 11
[4] All 3 parts answered correctly and
completely
[3] 2 parts answered correctly and
completely
[2] 2 parts answered correctly, but
incompletely