Algebraic Expressions
Objective To introduce the use of algebraic expressions to
represent situations and describe rules.
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ePresentations
eToolkit
Algorithms
Practice
EM Facts
Workshop
Game™
Teaching the Lesson
Key Concepts and Skills
• Identify and use patterns in tables to
solve problems. [Patterns, Functions, and Algebra Goal 1]
• Write algebraic expressions to model rules. [Patterns, Functions, and Algebra Goal 1]
• Use variables to write number models
that describe situations. [Patterns, Functions, and Algebra Goal 2]
Key Activities
Students complete statements in which the
variable stands for an unknown quantity. They
state the rule for “What’s My Rule?” tables in
words and with an algebraic expression.
Ongoing Assessment:
Recognizing Student Achievement
Use journal page 341. [Patterns, Functions, and Algebra Goal 2]
Family
Letters
Assessment
Management
Common
Core State
Standards
Ongoing Learning & Practice
1 2
4 3
Playing Name That Number
Student Reference Book, p. 325
per partnership: 1 complete deck of
number cards (the Everything Math
Deck, if available)
Students apply number properties,
equivalent names, arithmetic
operations, and basic facts.
Math Boxes 10 3
Math Journal 2, p. 344
compass Geometry Template
or ruler
Students practice and maintain skills
through Math Box problems.
Study Link 10 3
Math Masters, p. 299
Students practice and maintain skills
through Study Link activities.
Key Vocabulary
algebraic expression
Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Differentiation Options
READINESS
Exploring “What’s My Rule?” Tables
Math Masters, p. 300
Students use patterns in tables to
solve problems.
EXTRA PRACTICE
Writing Algebraic Expressions
Student Reference Book, p. 218
Students choose variables to write
algebraic expressions.
ELL SUPPORT
Building a Math Word Bank
Differentiation Handbook, p. 142
Students define and illustrate the term
algebraic expression.
ENRICHMENT
Analyzing Patterns and Relationships
Math Masters, p. 300A
Students analyze patterns and relationships.
Materials
Math Journal 2, pp. 341–343
Student Reference Book, p. 218
Study Link 102
Class Data Pad slate
Advance Preparation
Teacher’s Reference Manual, Grades 4–6 pp. 278–289
Lesson 10 3
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Getting Started
Mental Math
and Reflexes
Math Message
Ava, Joe, and Maria are 5th graders. Ava is
1 centimeter taller than Joe, and Joe is
2 centimeters taller than Maria. Make a table
of 4 possible heights for Ava, Joe, and Maria.
Students solve extended multiplication and division facts
problems involving powers of 10. Write the problems on the
board or Class Data Pad.
6 ∗ 102 = 600
0.56 ÷ 102 = 0.0056
0.254 ∗ 103 = 254
36.5 ÷ 103 = 0.0365
2
7.538 ∗ 10 = 753.8
8 ÷ 102 = 0.08
Heights
Ava
4.3 ∗ 103 = 4,300
Study Link 10 2 Follow-Up
7.6 ÷ 10 = 0.76
Have partners compare answers and resolve differences.
Joe
Maria
24 ∗ 107 = 240,000,000
1 Teaching the Lesson
▶ Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
Algebraic Thinking On the Class Data Pad, draw and label the
table for the Math Message. Ask students to share heights listed in
their tables. Record them on the Class Data Pad table.
Ask: How tall is Ava? Expect that students will respond with one of
the heights from the table. Ask them to explain their reasoning.
Sample answer: Ava’s height is 1 centimeter more than Joe’s. How
did you determine Joe’s height? Sample answer: Joe is 2 cm taller
than Maria. How tall is Maria? Sample answer: Maria’s height
isn’t given, so I picked a likely height for a fifth grader.
Student Page
Emphasize that Ava’s height depends on Joe’s height, and Joe’s
depends on Maria’s. We only know that Ava is 1 cm taller than Joe,
and that Joe is 2 cm taller than Maria. If Maria could be any
height, then there would be an infinite number of possibilities for
Joe’s and Ava’s heights. If Maria is 147.5 cm tall, then Joe’s height
is 149.5 cm, and Ava’s height is 150.5 cm. If Maria is 1.48 m tall,
then Joe is 1.50 m tall, and Ava is 1.51 m tall.
Algebra
Algebraic Expressions
Variables can be used to express relationships between quantities.
Claude earns $6 an hour. Use a variable to express the relationship
between Claude’s earnings and the amount of time worked.
If you use the variable H to stand for the number of hours Claude worked, you can write his
pay as H ∗ 6.
H ∗ 6 is an example of an algebraic expression. An algebraic
expression uses operation symbols (+, -, ∗, , and so on) to
combine variables and numbers.
Write the statement as an algebraic expression.
Statement
Marshall is 5 years
older than Carol.
Algebraic Expression
If Carol is C years old, then
Marshall’s age in years is C + 5.
Evaluating Expressions
Some algebraic
expressions:
2-x
m∗m
C+5
6∗H
(C + 5) (6 ∗ H)
Ask: How does Ava’s height compare with Maria’s height? Ava is
3 cm taller than Maria. If Maria is 151.5 cm tall, how tall is Ava?
154.5 cm If Ava is 150.2 cm tall, how tall is Maria? 147.2 cm
Other expressions that
are not algebraic:
7+5
6 ∗ 11
(7 + 5) (6 ∗ 11)
To evaluate something is to find out what it is worth. To
evaluate an algebraic expression, first replace each variable
with its value.
▶ Introducing Algebraic
Evaluate each algebraic expression.
6∗H
x∗x∗x
If H = 3, then 6 ∗ H is 6 ∗ 3, or 18.
If x = 3, then x ∗ x ∗ x is 3 ∗ 3 ∗ 3, or 27.
Expressions
WHOLE-CLASS
DISCUSSION
ELL
(Student Reference Book, p. 218)
Write an algebraic expression for each situation using the suggested variable.
2. Toni runs 2 miles every day. How
1. Alan is A inches tall. If Barbara is
3 inches shorter than Alan, what is
many miles will she run in D days?
Barbara’s height in inches?
Algebraic Thinking On the board or Class Data Pad, make a table
of just Maria’s and Joe’s heights. Point out that it is similar to a
“What’s My Rule?” table. Label Maria’s Height in and Joe’s Height
out. Ask: What is the rule for this table? out = in + 2
What is the value of each expression when k = 4?
3. k + 2
4. k ∗ k
5. k 2
6. k 2 + k - 2
Check your answers on page 440.
Student Reference Book, p. 218
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Ask volunteers to represent Joe’s height using an algebraic
expression. Let M represent Maria’s height in inches. Then M + 2
represents Joe’s height in inches. Add M and M + 2 to the column
headings in the table on the Class Data Pad.
Review and discuss Student Reference Book, page 218. Have
students study the examples of expressions that are algebraic and
those that are not algebraic. Ask students to explain how the
examples are similar and how they are different. Expressions use
operation symbols (+, -, ∗, ÷) to combine numbers, but algebraic
expressions combine variables and numbers.
Tell students that it is important to remember the following
points. To support English language learners, write the
statements on the board:
A situation can often be represented in several ways: in words,
in a table, or in symbols.
Algebraic expressions use variables and other symbols to
represent situations.
To evaluate an algebraic expression means to substitute values
for the variable(s) and calculate the result.
Ask students to propose algebraic expressions to fit simple
situations. To support English language learners, write the
situations and respective expressions on the board. For example:
Sue weighs 10 pounds less than Jamal. If J = Jamal’s weight,
then J - 10 represents Sue’s weight.
Isaac collected twice as many cans as Alex. If A = the number
of cans Alex collected, then 2 ∗ A, or 2A, represents the number
of cans Isaac collected.
There are half as many problems in today’s assignment as
there were in yesterday’s. If y = the number of problems in
y
1 y, _
1 ∗ y, or _
yesterday’s assignment, then there are _
problems
2 2
2
in today’s assignment.
Pose the following problems. Ask students to write an algebraic
expression for each problem on their slates.
●
Six times the sum of 9 and some number 6 (9 + n)
●
10 times the product of a number and 6 10 (n ∗ 6)
●
Triple the sum of a number and 20 3 (n + 20)
●
10 less than a number n - 10
●
7 less than the product of a number and 6 n ∗ 6 - 7
Algebraic expressions can be combined with relation symbols
(=, <, >, and so on) to make number sentences. For example,
x + 2 = 15, or 3y + 7 < 100. Ask volunteers for the name of
number sentences that contain algebraic expressions.
Algebraic equations
Links to the Future
Pictures, diagrams, and graphs are important
ways to represent situations, and they are
discussed throughout Everyday Mathematics.
Graphs in an algebra context are discussed
in Lesson 10-4.
Lesson 10 3
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Student Page
Date
Time
LESSON
Algebraic Expressions
10 3
䉬
Complete each statement below with an algebraic expression, using the suggested
variable. The first problem has been done for you.
1.
First Hometown Bank
If Leon gets a raise of $5 per week,
then his salary is
S ⫹ $5
PARTNER
ACTIVITY
PROBLEM
PRO
P
RO
R
OBL
BLE
B
L
LE
LEM
EM
SO
S
SOLVING
OL
O
LV
VIN
IIN
NG
Algebraic Thinking Go over Problem 1 on journal page 341.
Partners then complete the statements on journal pages 341
and 342.
Beth
141
Pay to the order of: Leon
Amount: S dollars
The Boss
When most students have finished, discuss students’ answers, and
point out that there are often several ways to write an algebraic
expression. The answer to Problem 7 can be written 10 + (5 ∗ D),
5D + 10, or 10 + 5D. The answer to Problem 8 can be written
1 ∗ X, _
1 X, or _
X.
_
3
3
3
Leon’s salary is
S dollars per week.
.
If Ali’s grandfather is 50 years
older than Ali, then Ali is
G ⫺ 50
4.
(Math Journal 2, pp. 341 and 342)
.
Kesia’s allowance
is D dollars.
3.
▶ Writing Algebraic Expressions
If Beth’s allowance is $2.50 more
than Kesia’s, then Beth’s allowance is
D ⫹ $2.50
2.
夹
years old.
Ali’s grandfather
is G years old.
Ali
Ask students for the algebraic equation they would write to
represent Problem 1. B = D + $2.50 Have students choose and
write the algebraic equation for two problems. They should write
their equations in the space beneath the problem answer line.
Seven baskets of potatoes weigh
7 ⴱ P, or 7P
A basket of potatoes
weighs P pounds.
pounds.
Ongoing Assessment:
Recognizing Student Achievement
Math Journal 2, p. 341
Journal
Page 341
Use journal page 341 to assess students’ ability to write algebraic expressions
that model situations. Students are making adequate progress if they correctly
identify and write the expressions for Problems 3 and 4. Some students will
correctly write the algebraic equations for the two problems of their choice.
[Patterns, Functions, and Algebra Goal 2]
▶ Expressing a Rule as an
PARTNER
ACTIVITY
Algebraic Expression
(Math Journal 2, p. 343)
Algebraic Thinking Have students complete journal page 343.
Everyday Mathematics students are very familiar with “What’s My
Rule?” tables because of their experiences with them starting in
first grade. Use the Readiness activity in Part 3 with students who
do not understand “What’s My Rule?” tables.
Student Page
Date
Time
LESSON
Algebraic Expressions
10 3
5.
If a submarine dives 150 feet,
then it will be traveling at a depth of
X + 150
continued
X ft
feet.
A submarine is traveling
at a depth of X feet.
6.
2 Ongoing Learning & Practice
The floor is divided into 5 equal-size
areas for gym classes. Each class
has a playing area of
_1 * A, _1 A, or _A
5
5
5
ft 2.
The gym floor has an
area of A square feet.
▶ Playing Name That Number
7.
(Student Reference Book, p. 325)
The charge for a book that is
D days overdue is
10 + (5 * D), 5D + 10, cents.
or 10 + 5D
Sports
Heroes
Students practice applying number properties, equivalent names,
arithmetic operations, and basic facts by playing Name That
Number. Encourage students to find number sentences that
use all 5 numbers and to use numbers as exponents or to
form fractions.
A library charges 10 cents for each
overdue book. It adds an additional
charge of 5 cents per day for each
overdue book.
8.
2
If Kevin spends _
3 of his allowance on
a book, then he has
_1 * X, _1 X, or _X
3
3
3
PARTNER
ACTIVITY
dollars left.
Kevin’s allowance
is X dollars.
Math Journal 2, p. 342
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Student Page
▶ Math Boxes 10 3
INDEPENDENT
ACTIVITY
(Math Journal 2, p. 344)
Date
Time
LESSON
“What’s My Rule?”
10 3
1. a.
State in words the rule for the “What’s My Rule?”
table at the right.
X
Y
5
1
Subtract 4 from X.
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lesson 10-1. The skill in Problem 4
previews Unit 11 content.
Writing/Reasoning Have students write a response to the
following: Explain the strategy and reasoning you would
use to solve Problem 3b with the standard multiplication
algorithm. Answers vary.
b.
INDEPENDENT
ACTIVITY
(Math Masters, p. 299)
-3
2
-2
Y=X-4
Q
Z
1
3
Y= 4 -X
3. a.
3
5
-4
-2
-3
-1
-2.5
-0.5
g
1
_
t
Circle the number sentence that describes the rule.
1
∗
Z=_
2Q 1
Z=2∗Q
State in words the rule for the “What’s My Rule?”
table at the right.
Multiply g by 4.
2
2
0
0
2.5
10
1
_
1
4
5
20
Circle the number sentence that describes the rule.
g=2∗t
1
State in words the rule for the “What’s My Rule?”
table at the right.
Z=Q+2
b.
▶ Study Link 10 3
-5
Add 2 to Q.
b.
Writing/Reasoning Have students write a response to
the following: Explain how you found the volume for
Problem 5. I knew the base (B) of the prism was 42 cm2
and the height (h) of the prism was 3 cm. I used the formula
B ∗ h to calculate the volume (V ): 42 ∗ 3 = 126 cm3.
0
-1
Circle the number sentence that describes the rule.
Y=X/5
2. a.
4
t=2∗g
t=4∗g
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Home Connection Students complete statements with
algebraic expressions. They write the rule and identify the
related number sentence for “What’s My Rule?” tables.
3 Differentiation Options
READINESS
▶ Exploring “What’s My Rule?”
PARTNER
ACTIVITY
5–15 Min
Tables
(Math Masters, p. 300)
Algebraic Thinking To provide experience with using patterns in
tables to solve problems, have students make rules and then
complete the related table. Partners work together to complete
each of their Math Masters pages.
When students have finished, discuss the rules and tables they
made. Ask partners to share what they think is important to
remember when solving “What’s My Rule?” tables. Sample
answers: If you know the in value, follow the rule to find the out
value. If you know the out value, do the opposite of the rule to find
the in value.
Student Page
Date
Time
LESSON
10 3
1.
Math Boxes
Identify the point named by each ordered number pair.
a.
(0,4)
b.
(3,3)
c.
(5,4)
d.
(4,0)
y
5
B
A
D
C
D
B
4
A
3
2
1
0
C
0
1
2
3
4
x
5
208
2.
Add or subtract.
a.
b.
c.
d.
e.
3.
10
-8 + (-17) = -25
0
-12 – (-12) =
0
-45 + 45 =
-31 - 14 = -45
Multiply. Use the algorithm of your choice.
Show your work.
20 + (-10) =
a.
43
* 78
3,354
b.
19
* 86
c.
1,634
79
* 42
3,318
92
4. a.
Draw a circle that has a
diameter of 4 centimeters.
19
5.
The rectangular prism below has a volume
153 164
of
126 cm. 3
(unit)
3 cm
Area of base
42 cm2
Write a number model for the formula.
b.
The radius of the circle is
2 cm .
42 ∗ 3 = 126 cm3
197
Math Journal 2, p. 344
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Lesson 10 3
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Study Link Master
Name
Date
STUDY LINK
Time
EXTRA PRACTICE
Writing Algebraic Expressions
10 3
Complete each statement below with an algebraic expression, using the
suggested variable.
1.
▶ Writing Algebraic Expressions
Augusto
Algebraic Thinking Students complete the Check Your
Understanding problems on Student Reference Book, page 218.
Ask students to write algebraic expressions for the first two
problems and to describe how they chose the variables to use.
Lamont and Mario
harvested
L + M carrots.
carrots.
Rhasheema and Alexis have a lemonade stand at their school fair. They
promise to donate one-fourth of the remaining money (m) after they repay the
school for lemons (l ) and sugar (s). So the girls donate
_1 ∗ (m - (l + s)), or _1 (m - (l + s))
4
4
dollars.
3. a.
b.
4. a.
b.
State in words the rule for the “What’s My Rule?” table at the right.
Multiply N by 3 and add 5.
N
Q
2
11
4
17
Circle the number sentence that describes the rule.
6
23
Q = (3 + N) ∗ 5
8
29
10
35
Q = 3 ∗ (N + 5)
Q = 3N + 5
State in words the rule for the “What’s My Rule?” table at the right.
Multiply E by 6 and add 15.
Circle the number sentence that describes the rule.
R = E ∗ 6 ∗ 15
Practice
R = (E ∗ 6) + 15
576
120 _37
384 ∗ 1.5 =
6.
50.3 ∗ 89 =
7.
843
_
=
8.
70.4 / 8 =
7
E
R
7
57
10
75
31
201
R = E ∗ 15 + 6
5.
5–15 Min
(Student Reference Book, p. 218)
Lamont, Augusto, and Mario grow carrots in three
garden plots. Augusto harvests two times as many
carrots as the total number of carrots that Lamont and
Mario harvest. So Augusto harvests
2 ∗ (L + M ), or 2(L + M)
2.
218 231
232
INDEPENDENT
ACTIVITY
3
33
108
663
ELL SUPPORT
▶ Building a Math Word Bank
5–15 Min
(Differentiation Handbook, p. 142)
To provide language support for algebra concepts, have students
use the Word Bank Template found on Differentiation Handbook,
page 142. Ask students to write the term algebraic expression,
draw pictures relating to the term, and write other related words.
See the Differentiation Handbook for more information.
4,476.7
8.8
Math Masters, p. 299
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SMALL-GROUP
ACTIVITY
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ENRICHMENT
▶ Analyzing Patterns and
INDEPENDENT
ACTIVITY
5–15 Min
Relationships
(Math Masters, p. 300A)
To extend students’ understanding of rules and patterns, students
use two sets of rules to generate two tables of values. They form
ordered pairs consisting of corresponding terms from two patterns,
and graph the ordered pairs on a coordinate plane. Students then
use the tables of values, rules, ordered pairs, or graphs to identify a
relationship between corresponding terms from each pattern.
Teaching Master
Name
Date
LESSON
Time
Patterns and Relationships
10 3
A car is traveling at a given speed over a stretch of highway. You can find the distance
the car travels by multiplying its speed by the amount of time it travels.
Car A travels at a speed of 30 miles per hour (mph). Car B travels at 60 miles per hour.
Complete the tables to find the distance each car travels for the given times.
Car A
Car B
Speed: 30 mph
Speed: 60 mph
Time (hr)
Distance (mi)
Time (hr)
Distance (mi)
0
0
0
1
30
1
2
60
90
120
2
0
60
120
180
240
4
3
4
2.
For each car, write the rule that is used to find the distance.
Car A:
Car B:
3.
Use the tables to write a set of ordered pairs in the form
(Time, Distance) for each car. Then graph the data and
connect the points for each car. Label each graph.
Multiply each hour by 30.
Car A
Car B
(0,0)
(0,0)
(1,30)
(1,60)
(2,60)
(3,90)
(2,120)
(3,180)
(4,120)
(4, 240)
Multiply each hour by 60.
270
240
210
Distance (miles)
3
180
150
120
90
60
30
0
Ca
rB
C
ar
A
1.
0
1
2
3
4
Time (hours)
4.
As the amount of time increases, explain how the distance
Car B travels compares with the distance Car A travels?
Sample answer: For each hour, Car B travels twice as far
as Car A. This is because Car B is traveling twice as fast.
Math Masters, p. 300A
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Name
Date
LESSON
Time
Patterns and Relationships
10 3
A car is traveling at a given speed over a stretch of highway. You can find the distance
the car travels by multiplying its speed by the amount of time it travels.
Car A travels at a speed of 30 miles per hour (mph). Car B travels at 60 miles per hour.
Complete the tables to find the distance each car travels for the given times.
Car A
Car B
Speed: 30 mph
Speed: 60 mph
Time (hr)
Distance (mi)
Time (hr)
0
0
0
1
30
1
2
2
3
3
4
4
Distance (mi)
2.
For each car, write the rule that is used to find the distance.
Car A:
Car B:
3.
Use the tables to write a set of ordered pairs in the form
(Time, Distance) for each car. Then graph the data and
connect the points for each car. Label each graph.
Copyright © Wright Group/McGraw-Hill
Car A
Car B
(0,0)
(1,30)
270
240
210
Distance (miles)
1.
180
150
120
90
60
30
0
0
1
2
3
4
Time (hours)
4.
As the amount of time increases, explain how the distance
Car B travels compares with the distance Car A travels?
300A
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