Section A
Understanding Variables and Expressions
2-1 Variables and Expressions
2-2 Translating between words and math
2-3 Translating between tables and expressions
Section A Quiz
Section B
2-4
2-5
2-6
2-7
2-8
Understanding Equations
Equations and Their Solutions
Addition Equations
Subtraction Equations
Multiplication Equations
Division Equations
Section B Quiz
Algebra Unit Test
2-1 Variables and Expressions
Vocabulary
_______________ – a symbol used to represent a quantity that can change
_______________ – a value that does not change
_______________ – a mathematical phrase that contains operations, numbers, and/or
variables
_________________________ – an expression that contains at least one variable
_______________ – to find the value of a numerical or algebraic expression
Examples of algebraic expressions:
Addition
Subtraction
Multiplication
Division
Is the following a variable (V) or a constant (C)?
1. Number of days in January _____
4. Price of a calculator _____
2. Number of students in a school _____
5. Number of inches in a foot _____
3. Number of people in a state _____
6. Number of giraffes in a herd _____
Example 1: Evaluating Algebraic Expressions
EVALUATE each expression to find the missing values in the tables.
w
55
w ÷ 11
5
(55 ÷ 11= 5)
66
77
n
1
4 x n + 62
2
3
Evaluate each expression for x = 1, 2, and 3
x
x+5
x
11 - x
Evaluate each expression for x = 2, 5, and 8
x
4x
x
Example 2: Evaluating Expressions with Two Variables
A rectangle is 2 units wide. What is the area of the rectangle if it is 4, 5, 6, or 7 units long?
l
w
4
2
5
2
6
2
7
2
lxw
Think and Discuss
1. Name a quantity that is a variable and a quantity that is a constant.
2. Explain why 45 + x is an algebraic expression.
2-2 Translating Between Words and Math
Example 1: Social Studies Applications
A. The Nile River is the world’s longest river. Let n stand for the length in miles of the Nile. The
Amazon River is 4,000 miles long. Write an expression to show how much longer the Nile is
than the Amazon.
B. Let s represent the number of senators that each of the 50 states has in the U.S. Senate.
Write an expression for the total number of senators.
Example 2: Translating Words into Math
Directions: Put the words from the box on the lines in the correct column below.
product
groups of
decreased by
difference
quotient
more than
take away
less than
increased by
of
KEY WORDS
ADDITION
SUBTRACTION
MULTIPLICATION
1. minus
1. sum
DIVISION
1. divided by
2. subtracted from
2. plus
3. ______________
3. ______________
4. ______________
4. ______________
5. ______________
1. times
2. ______________
2. multiplied by
3. ______________
4. ______________
5. ______________
6. ______________
EXAMPLES
Operation
Algebraic
Expression
Words
x + 28
28 more than x
the sum of x and 28
k – 12
k minus 12
12 less than k
8w OR 8 • w
8(w) OR (8)(w)
n ÷ 3 OR
8 times w
n divided by 3
8 groups of w the quotient of n and 3
REMEMBER:
It is helpful to put a number in place of the variable so that you can check if your
answer is REASONABLE!
Directions: Write each phrase as a numerical or algebraic expression
1. 79 minus 15
__________
2. 28 more than 37
3. 8 groups of 4
__________
__________
4. product of 20 and k
__________
5. difference of g and 6
6. the quotient of n and 3
__________
__________
10. c tripled
__________
11. 3 increased by s
__________
12. m to the fourth power
13. j squared
__________
__________
14. r cubed
__________
15. half of b
__________
7. 5 more than f
__________
16. one third of a
8. j less than 5
__________
17. Caroline made f batches of 12 cookies. How
many did she bake? __________
9. 20 less than y
__________
__________
Example 3: Translating Math into Words
Directions: Write TWO phrases for the expressions below.
A. (34) (7)
B. a – 45
1) Joe collected 200 coins. He is planning to sort them into r containers, with an equal
number of coins in each container. Circle the expression that tells how many coins will be in
each container.
a) r + 200
b) 200 ÷ r
c) 200 – r
d) 200 x r
2)Write f more than 47 as an algebraic expression: ________________________
3)Write 18 less than g as an algebraic expression: _______________________
Think and Discuss:
1. Tell how to write each of the following phrases as a numerical or algebraic expression: 75
less than 1,023; the product of 125 and z.
2. Give two examples of “a ÷ 17” expressed with words.
2-3 Translating Between Tables and Expressions
(Input / Output Tables)
Example 1: Writing an Expression
A.
Reilly’s Age
Ashley’s Age
9
11
10
12
11
13
12
14
n
When Reilly’s age is n, Ashley’s age is _____________
B.
Eggs
Dozens
12
1
24
2
36
3
48
4
e
When there are e eggs, the number of dozens is e ÷ 12, or ________
Example 2: Writing an Expression for a Sequence
Write an expression for the sequence in the table.
Position
1
2
3
4
5
Value of Term
3
5
7
9
11
n
Look for a relationship between the positions and the values of the terms in the sequence. Use
guess and check.
Practice: Write an expression for the missing value in each table
#1
#2
Go-Carts
1
2
3
4
Wheels
4
8
12
16
Position
1
2
3
4
5
Value of
Term
9
10
11
12
13
#3
Players
Soccer
Teams
22
#4
n
Weeks
Days
2
4
28
44
4
8
56
66
6
12
84
88
8
16
112
x
#5
x
n
Position
1
2
3
4
5
Value of
Term
7
12
17
22
27
n
#6 Which expression describes the sequence in the table?
Position
1
2
3
4
5
Value of
Term
6
11
16
21
26
A) n + 5
B) 5n + 1
C) 6n
n
D) 6n – 1
Think and Discuss:
1. Describe how to write an expression for a sequence given in a table.
2. Explain why it is important to check your expression for all of the data in the table.
2-4 Equations and their Solutions
Vocabulary
_______________ – a mathematical statement that two expressions are equal
_______________ – a value or values that make an expression true
Is the following an EQUATION or an EXPRESSION?
a. 15 + y = 20 __________________________
b. (m - 4) x 7 ____________________
An equation is like a scale with the equal sign in the middle
Both sides have the same value.
They must always stay in balance.
2+6 = 5+3
The SOLUTION to this equation is:
2+x =
x = ______
5
EXAMPLE 1: Determining Solutions of Equations
Determine whether the given value of the variable is a solution. YES or NO!
Circle the equation that is true for y = 3
a) 12 = 4 + y
b) 12 = 4 – y
For which value of the variable is the equation true?
a) w = 3
b) w = 9
c)
12
=4
y
d) 4 = 12 y
3w + 5 = 17
c) w = 4
d) w = 2
Example 2: Life Science Application
You can use equations to check whether measurements given in different units are equal:
One science book states that a male giraffe can grow to be 19 feet tall. According to another book, a
male giraffe may grow to 228 inches. Determine if these two measurements are equal.
What do we need to know?_______________________________________
Write it algebraically:
Substitute and solve:
________________________
_________________________
PRACTICE
Determine whether the given value of the variable is a solution. YES or NO!
Kent earns $6 per hour at his after-school job. One week, he worked 12 hours and received a
paycheck for $66. Determine if Kent was paid the correct amount of money.
Think & Discuss:
1. Tell which of the following is the solution of y ÷ 2 = 9 :
y = 14, y = 16, or y = 18. How do you know?
2. Give an example of an equation with a solution of 15.
2-5 & 2-6 Addition and Subtraction Equations
Vocabulary
_______________ _______________ – operations that undo each other: addition and subtraction,
or multiplication and division.
Taking away 14 from both sides of the scale is
the same as subtracting 14 from both sides of the
equation.
Example 1: Solving Addition Equations
Solve each equation. Check your answers.
Example 2: Social Studies Application
Johnstown, Cooperstown, and Springfield are located in that order in a straight line along a highway.
It is 12 miles from Johnstown to Cooperstown and 95 miles from Johnstown to Springfield. Find the
distance d between Cooperstown and Springfield.
distance between
Johnstown and
Springfield
________
=
distance between
Johnstown and
Cooperstown
+
distance between
Cooperstown and
Springfield
=
________
+
_________
Practice: Solve each equation. Check your answers.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10. Write “21 is 5 subtracted from y” as an equation. Then solve.
Write an equation for each statement. Then solve.
The number of eggs e increased by 3 equals 14.
The number of new photos taken p added to 20 equals 36.
When 17 is subtracted from a number, the result is 64.
NAME ______________________ J
Adding and Subtracting Equations Practice #1
Solve each equation! Check your work. Circle your answers.
1. x + 12 = 16
check
2. 23 + g = 34
check
3. r – 57 = 7
check
4. 11 = x – 25
check
5. 52 + y = 71
check
6. 87 = b + 18
check
7. a – 6 = 15
check
8. g – 71 = 72
check
9. m + 25 = 47
check
10. Write an equation for the following
statement. Then solve the equation.
The number of skittles (s)
increased by 5 equals 20
NAME ______________________ J
Adding and Subtracting Equations Practice #2
Solve each equation! Circle your answers.
1. x + 18 = 44
2. x – 12 = 6
3. 75 + x = 97
4. x + 47 = 144
5. x – 23 = 63
6. x – 72 = 2
7. x – 17 = 51
8. 66 + x = 129
9. x – 11 = 67
10. x + 52 = 72
11. 57 + x = 75
12. 82 + x = 116
13. x – 57 = 23
14. 85 + x = 127
15. 83 + x = 150
16. x – 19 = 46
17. x – 28 = 45
18. 91 + x = 177
19. Write an equation for the following statement. Then solve the equation.
The number of skittles (s) decreased by 13 equals 20
2-7 & 2-8 Multiplication & Division Equations
Example 1: Solving Multiplication & Division Equations
Solve each equation. Show ALL work. Check your answers.
Marcy spreads out a rectangular picnic blanket with an area of 24 square feet. Its width is 4 feet.
What is its length?
(Remember A = l x w à AREA = LENGTH x WIDTH)
Millipedes can have up to 752 legs! They have 4 legs per segment. How many segments could the
millipede have?
Carl has n action figures in his collection. He wants to place them in 6 bins with 12 figures in each
bin. Write and solve an equation.
Practice: Solve each equation. Check your answers.
1.
2.
3.
4.
5.
6.
7.
8.
9.
The area of a rectangle is 42 square inches. Its width is 6 inches. What is the length?
A = length x width
Taryn buys 8 identical glasses. Her total is $48 before tax. Write and solve an equation to find out
how much Taryn pays per glass.
Paula is baking peach pies for a bake sale. Each pie requires 2 pounds of peaches. She bakes 6
pies. Write and solve an equation to find how many pounds of peaches Paula had to buy.
NAME ______________________ J
Multiplying and Dividing Equations Practice #1
Solve each equation! Check your work. Circle your answers.
1. 7x = 56
check
2. 11g = 99
check
c
=4
9
check
4.
x
= 15
4
check
5. 27 = 3w
check
6. 132 = 12m
y
5
check
8.
check
10.
3.
7.
9=
9. 6y = 114
k
=1
28
j
= 10
20
check
check
check
11. The area of a rectangle is 63 square feet. Its width is 3 feet. What is its length?
NAME ______________________ J
Multiplying and Dividing Equations Practice #2
Solve each equation! Check your work. Circle your answers.
1. 7x = 56
4.
c
=4
9
7. 27 = 3w
10.
9=
y
5
13. 6y = 114
2. 11g = 99
5.
x
= 15
4
8. 132 = 12m
3. 5x = 125
6.
x
= 11
3
9. 7g = 77
11.
k
=1
28
12.
14.
j
= 10
20
15. 10k = 130
x
=9
8
16. The area of a rectangle is 84 square feet. It’s width is 4 feet. What is its length?