Lesson 10
Using Symbols to Represent
Unknowns and Variables
TAKS
TAKS
Grade
Grade
9 Objective
# Objective
2 #
(A.3)(A)
(7.2)(#)
Mathematical problems involve relationships between known and unknown
quantities. Letters, numbers, and symbols are used to create simplified
models of mathematical problems. These models are called algebraic
expressions. An algebraic expression is a mathematical phrase that can
contain numbers, variables, and operation symbols. An equation is a
mathematical sentence that states that two expressions are equal.
New Vocabulary
• algebraic
expression
• equation
• variable
Using Variables to Represent Unknowns
EXAMPLE 1
The value of a variable
in an expression can be
changed, or “varied.” For
example, the expression
4s represents the
perimeter of a square.
The value of s can vary,
depending on the
length of the side
of a square.
Write algebraic expressions for the following problem situations.
a. Daniela’s family has x dogs and y cats. If the average of the dogs’
weights is 85 pounds, and the average of the cats’ weights is 12
pounds, what expression can give the total weight of the pets?
The average weight of x dogs is 85 pounds. Write an expression
for the total weight of the dogs: 85x.
The average weight of y cats is 12 pounds. Write an expression for the total weight of
the cats: 12y.
Now find the total weight of all the pets: 85x 12y.
b. The length of a rectangle is 3x, and its width is 2x 4. Write an expression for the area
of the rectangle.
The area of a rectangle equals its length times its width: A lw.
Substitute the given values for length and width:
A (3x)(2x 4)
Simplify.
3x(2x) 3x(4)
6x2 12x
The area of the rectangle is:
6x2 – 12x.
Quick Check 1
1a. Crane took 2 math tests. His average score
on the 2 tests was 85. Then he took a third
test and scored q points on it. Write an
expression for his average score for all
3 tests.
28
LESSON 10
■
1b. Write an algebraic expression for the area
of a triangle that has height 5a 2 and base
4a 1.
Using Symbols to Represent Unknowns and Variables
TAKS Review and Preparation Workbook
Copyright © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved.
In an algebraic expression or equation, a variable is a symbol, usually a
letter, that stands for a number. For example, the problem 5 times what
number is equal to 15? can be written 5n 15, where n represents the
unknown number. To write an algebraic expression, choose variables to
represent the unknowns. Then express the relationship between the
unknowns and the known quantities in terms of mathematical operations.
TAKS Objective 2 (A.3)(A)
LESSON 10
Writing Expressions from Problem Situations
Write mathematical expressions one piece at a time. Pay special attention
to the order in which operations must happen in order to arrive at the
correct solution. Write your expression keeping the correct order of
operations in mind.
EXAMPLE 2
Write algebraic expressions for the following problem situations.
Kyle is cutting letters out of large construction paper for decorations
at a school party. The letter L is shown below.
When you choose a
letter to represent an
unknown, use a letter
that is easily connected
to the quantity it
represents. For example,
if the unknown is a
height, use the letter h.
ƒ cm
52 cm
If more than one
number in a problem
situation is unknown,
several variables may
be needed to correctly
represent the situation.
g cm
Copyright © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved.
30 cm
Write an expression for the area of the letter L (only the shaded
region) in terms of f and g.
To find the area of a rectangle, multiply its length times its width. One way to find the
area of the letter L is to find the area of the larger rectangle and then subtract the area
of the smaller, unshaded rectangle.
Area of large rectangle:
A lw
52 30
1560
Area of small rectangle:
A lw
fg
Area of letter L:
A Area of large rectangle Area of small rectangle
1560 fg
Quick Check 2
2a. The average time Marie usually spends on a writing assignment is
2 hours. The average time she spends on a history assignment is 1.5 hours.
If this week she will do w writing assignments and h history assignments,
what expression will give her average time for all the assignments?
2b. Write an expression for the area of the shaded region below.
w ft
12 ft
x ft
TAKS Review and Preparation Workbook
LESSON 10
■
Using Symbols to Represent Unknowns and Variables
29
Name__________________________Class____________Date________
1 A class contains 12 boys and 16 girls. The
average height of boys in the class is b inches.
The average height of girls is g inches. Which
algebraic expression below correctly
represents the average height per student
for the entire class?
A
12b 1 16g
28
3 Tara purchased favors for the guests at her
party. She bought 12 hats for a total of h
dollars and 12 shirts for a total of s dollars.
She gave 1 hat and 1 shirt to each guest, and
there were 12 guests at the party. Which
expression below gives the total cost of party
favors per guest?
A 12h
s
B 12b 16g
B 12(h s)
C 28(12b 16g)
C h 12
1s
b1g
D 16 1 12
1s
D h 12
2 A garden has the dimensions shown below.
A fence is going to be installed to divide the
garden. The area of the larger portion of the
garden is 45n. How can the area of the
smaller portion of the garden be expressed
in terms of n?
4 A storage facility is divided into equally
sized units as shown below. The area of the
entire facility is y z. How can the area of
an individual unit be expressed in terms of
y and z?
fence
80 ft
z
n ft
45 ft
30
F 45n 80
F 4y 4z
G 45n
80
G 12(y z)
H 45(80 n)
H 3z
1n
J 80 45
J
LESSON 10
■
4y
(y4) (z3)
Using Symbols to Represent Unknowns and Variables
TAKS Review and Preparation Workbook
Copyright © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved.
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