4.10 Multiplication and Division Equations
Warm Up
1.5 × 3.5 (5.25)
0.04 × 0.30 (0.012)
0.96 ÷ 0.3 (3.2)
Materials
BLM 4.10A
BLM 4.10B
1.08 ÷ 4 (0.27)
Objective
Students will simplify and solve
one-step multiplication and division
equations involving rational numbers.
Introduction
Construct Meaning
1
Lynn has traveled 9 2 miles since leaving home to go to a concert.
2
This is 5 of the distance from her house to the concert hall. How
far is the concert hall from her house?
2
1
d 9 miles
5
2
Write an equation.
2
19
d 5
2
Simplify numerical expressions.
Convert mixed numbers to improper fractions.
1
1
1
1
5 2
Pythagoras experimented with a
monochord, a simple instrument with a
single string stretched over a moveable
bridge. Plucking the string causes it to
vibrate and produce tones. Moving the
bridge changes the length of the string,
which in turn changes the pitch of the
tone. Pythagoras discovered that the
pitch of a string exactly half the
original length is one octave higher
(double the frequency).The ratio of the
new length (higher frequency) to the
original length (lower frequency) is
1/2:1, or 1:2.The ratio of a higher
frequency to the lower frequency is
2:1.The frequency ratio and the string
length ratio are reciprocals. Pythagoras’
experiments led to a method of tuning
instruments with intervals in integer
ratios.
Write the following on the board to
show students the reciprocal
relationship.
frequency ratio new frequency (higher pitch)
2
2
old frequency (lower pitch)
1
string length ratio 2 5d 2 2
Isolate the variable by multiplying both sides by the multiplicative inverse.
95
3
d 23 miles
4
4
Simplify.
5 19
Write each sentence as an equation, solve, and check.
Sentence
Equation and Solution
1
2
1x 2
3
3
One and one-third of a
quantity is the opposite
of two and two-thirds.
4
8
x 3
3
3 4
3
8
x 4 3
4
3
Check
4
8
(2) 3
3
8
8
3
3
x 2
A number divided by
negative six equals
three and five-ninths.
0.7 of a number is
equivalent to the
product of 1.4
and 1.9.
5
y
3
9
6
32
y
9
6
32
y
(6) (6) 9
6
64
1
y or 21
3
3
64
32
(6) 3
9
64
1
32
· 3
6
9
32
32
9
9
0.7z 1.4(1.9)
0.7z 2.66
0.7z
2.66
0.7
0.7
0.7(3.8) 1.4(1.9)
2.66 2.66
z 3.8
Check Understanding
Write true or false.
a. Dividing by a fraction is the same as multiplying by its multiplicative inverse. true
2k
k
1
2
3
4
k
1
b. k true
c. k true
d. k k false e. k true
3
4
4
3
4
3
5
5
9898
I t
di t C
B
Remind students that the reciprocal of a number is the same as the
1
2
multiplicative inverse of the number.The multiplicative inverse is the
1
new length
number by which a given number is multiplied to yield a product of 1.
1
old length
2
In the example above, the multiplicative inverse of 2 is 1/2.What steps
do you take to determine the multiplicative inverse of a mixed number? (Write the number as an improper
fraction, then invert the numerator and denominator.) Have students determine the multiplicative inverse
of 2 1/2. (2/5) What is the multiplicative inverse of 2 1/2? (2/5) Make sure students understand the
difference between the additive inverse and the multiplicative inverse.
Directed Instruction
1 Write the equation (4)(2/3)n 3 1/2 on the board.The first step in solving a multiplication equation is to
simplify numerical parts of the expressions on either side of the equation.What can be simplified here?
(4 · 2/3 8/3) When working algebraic equations with rational numbers, fractions are generally left as
improper fractions rather than mixed numbers. (Mixed numbers continue to be the preferred form for
expressing such quantities as units of measurement.) Since we will be multiplying and working with
116
Intermediate Course B
reciprocals, we will at the same time convert the mixed number on the right side of the equation to an
improper fraction, yielding the simplified equation: (8/3)n 7/2.To isolate the variable, multiply both sides
of the equation by the multiplicative inverse of the coefficient 8/3. Multiplying both sides by 3/8 yields
n 21/16.This solution is adequate as written. If the solution involves units, such as inches, then it would be
appropriate to convert the answer to a mixed number: 1 5/16.To check, substitute 21/16 for n into the
original equation.The equation should balance: 7/2 7/2.
2 Write the equation k/5 1 7/8.
Rewrite the equation with improper
fractions: k/5 15/8. How will we
isolate the variable k? (Since k is
being divided by 5, we multiply
both sides of the equation by 5,
yielding k 5(15/8) 75/8
or 9 3/8).) Check by substitution:
15/8 15/8.
Write the multiplicative inverse of each expression if a and b are both integers other than 0.
1
a
1
a b
2b
f. b
g. a h. i. b
b a
a
a
2b
Translate each sentence into an algebraic equation and identify the operation(s) needed to
solve.
n
10; Multiply both sides by 1.5.
j. A number divided by 1.5 is 10. 1.5
3
1
k. The product of a number and 3 4 is equal to the sum of 6 and 44. See Answer Key.
1
1
l. One-third of a number is equal to 24. 1
x 2; Multiply both sides by 3.
3
4
Did You
Practice
Know?
Write yes or no to indicate whether the value
of x shown is a solution to the given equation.
Equation
Value of x Solution?
1.
3
x 9
5
15
Yes.
2.
1
4x 12
2
1
3
8
No.
0.5
No.
1
2
Yes.
3. 35.05 7.01x
4.
Indian music also has a complex
rhythmic structure.The player
of the tabla (hand drums) often
must play a different rhythm
with each hand.Western jazz
musicians have adapted a
variety of complicated rhythms
and time signatures from the
music of other cultures.
x
1
3 1
3
8
3 Have students write in their journals
the following steps for solving
multiplication equations.
• Simplify numerical expressions on
each side of the equation if
possible; convert mixed numbers
to improper fractions.
• Isolate the variable by multiplying
both sides of the equation by the
multiplicative inverse.
• Solve, then check by substitution.
Write the multiplicative inverse of each number.
15
1
1
2
5. 2
6. 2 7. 17 2
15
32
17
See Answer Key for 8–18.
Solve and check. Remember to simplify each side of the equation before isolating the variable.
5
1
1
1
15
k
8. p 5
9. 1c 2
10. 6
6
4
7
16
2
11. 3.2x 4 0.8
1
4
1
4
1
2
14. 4d 2 3
1 1
4 2
3
4
12. t 12
f
6.3
15. 2 3
5
Write the steps for division.
• Simplify expressions if possible and
convert mixed numbers to
improper fractions.
• Isolate the variable by multiplying
both sides of the equation by the
divisor.
• Solve, then check by substitution.
3
4
13. j 3
1
2
1
2
16. m 1 1
Translate each sentence into an algebraic equation and solve.
17. Three and one-half times a number n is the opposite of seven-ninths.
18. A number n divided by three and one-half is the opposite of seven-ninths.
Apply
Write an equation for each word problem. Then find the solution.
19. One year after buying a used car, Isabel sells the car for two-thirds the price she paid
for it. If Isabel’s loss was $825, how much did she pay for the car? 1p 825; p $2475
3
20. Georgia is buying a washing machine for $540. She must make a 20% down payment,
then pay off the balance in six equal monthly installments. How much will she be
paying each month? 6p 540 0.20(540) or 6p 0.8(540); p $72
21. Riley wants to rent a van that will cost $60 plus $0.59 per mile. If he has budgeted $370
for the rental, what is the maximum number of miles he can travel?
m 525.4, or about 525 miles
Thematic Connection
Pythagoras discovered simple numerical ratios
between other musical notes. If the lengths of two
strings of the same tension are in a ratio of 2:3, the
difference in pitch is called a fifth, which
corresponds to the fifth note of an eight-note
musical scale.The fifth and the base note of the
scale produce a pleasing harmony. If the ratio of the
lengths of two strings is 3:4, the difference in pitch
is called a fourth, or the fourth note of the scale. In
an eight-note scale, these two intervals together
span one octave.When harmonies were first added
to western church music, these two intervals were
used exclusively because the harmony they produce
was considered the most perfect.
© Copyright 2004
0.59m 370 60;
99
4 Proceed to LESSON 4.10. Use BLM
4.10A Multiplication and
Division Equations for additional
practice. Use BLM 4.10B Review
of Rational Numbers to review
simplifying expressions with rational
numbers.
Math Moments
In the following square, each row, column, and
diagonal adds up to the same number. Complete
the square.
3
4
13
1
11
12
16
1
6
1
112 3
1
2
5
1
1
4
117
7.1 Constants, Variables, and Terms
Materials
• T-17
BLM 7.1A
BLM 7.1B
Warm Up
Simplify.
x · x · x · x · x (x5)
x · x · y · y · y (x2y 3)
Objective
Students will identify the components
of an algebraic expression and simplify
expressions by combining like terms.
Introduction
2 · 3 · y · y · y (6y 3)
x x y · y · y (2x � y 3)
x x x x x (5x)
x · x y · y (x2 � y 2)
Juanita purchased four T-shirts and used
a $3 coupon. Have students write an
expression for the total cost if x
represents the cost of one T-shirt.
(4x � 3 or 4x � (�3)) Display the
first model and expression on color
transparency T-17 Modeling
Expressions.The small yellow tiles
represent positive integers, and the
small red tiles represent negative
integers.What is the variable in this
expression? (x) Remind students that
variables are symbols, usually letters, that
represent different values.This
expression can be evaluated for
different values of x. Constants are
symbols or letters that represent fixed
values. List the factors of 4x. (4 and x)
Since x represents different values, it is
the variable factor.The constant factor
is 4. A coefficient, a constant factor, is the
number by which a variable is
multiplied.The coefficient in the
expression 4x is 4.
Construct Meaning
In the process of investigating, it is helpful to break down a complex issue into its
components. When you are working with algebraic expressions, it is important to identify
its parts. Identifying the parts of an expression will help you understand the expression
and combine parts correctly.
The variable x can represent
the number of people.
Variables are symbols, usually letters, that represent
different values.
Constants are symbols or letters that represent fixed values. A
constant factor is a coefficient. It is the number by which a
variable is multiplied. A number by itself in an algebraic
expression is also called a constant.
The symbol represents
a constant.
12 is the coefficient of
z in the product 12z.
Terms are numbers, variables, or products of numbers and
variables. In an expression, terms are separated by a plus or
minus sign. Like terms contain the same variables. Matching
variables are raised to the same power.
3
3x and 5 are terms in
the expression 3x 5.
6y and 2y are like terms.
3
Identify the parts of the expression 2x x 20 x .
2x 3 x 20 (x 3)
Rewrite subtraction as addition of the opposite.
2x 3 1x 20 (1)(x 3)
3
The terms are 2x , x, 20, and x 3. The like terms are 2x 3 and x 3.
The coefficients are 2, 1, and 1. The constant term is 20.
Operations with Variables and Terms
Any two variables or terms can be multiplied.
x · y xy
z2 · z3 z5
To multiply powers with the
same base, add the exponents.
Like terms can be added or subtracted. Variables do not change when combining terms.
x x 2x
5ab 12ab 7ab
Terms that do not have matching variables cannot be combined into one term.
4x 3y
x2 x3
Simplify each of the following expressions.
12x 20y x
12x 20y (x)
Rewrite subtraction as addition of the opposite.
12x (x) 20y
Apply the Commutative Property.
13x 20y
Add like terms.
4x 3 · (5x 2)
3
2
Apply the Commutative Property.
4 · (5) · x · x
Algebraic expressions that include
Multiply the coefficients and multiply the variables.
20 · x 5
subtraction and/or addition operations
20x 5
Simplify.
are made up of terms. Terms are
numbers, variables, or products of
Be diligent to present yourself approved to God, a worker who does not
numbers and variables. How many
need to be ashamed, rightly dividing the word of truth. 2 Timothy 2:15
terms are in the expression 4x 3? (2)
156
Identify each term. (4x, �3) What is
the constant term in this expression? (�3) Remind students that the negative sign is a part of the number three
since subtraction is the same as adding the opposite.
Directed Instruction
1 Display the second model on T-17, covering the expression. Have students translate the left side into an
algebraic expression. (5x � 2x) What are the terms? (5x, 2x) What are the coefficients of these terms? (5,
2) Like terms are terms that contain the same variable factors. Are these terms like terms? (Yes.) Using the
model, show students that these like terms can be combined so that the expression can be simplified to 7x.
When combining like terms, keep the variable the same and add the coefficients.Tell students that this
expression is like adding 5 oranges and 2 oranges for a total of 7 oranges.
Write 5x 2y on the board.This is like adding 5 oranges and 2 apples. It is not possible to get a total of 7
orange-apples. Review the other models on T-17.Write rs2 and 2s2r on the board. Are these like terms?
(Yes.) Although the order of the factors is different, they are like terms because matching variables are
186
Intermediate Course B
rs2? ( 1) Remind students that
raised to the same powers.What is the coefficient of
1 · rs2 and 1 s2r.
rs2 is the same as
2 Write the expression 2x/3
x/3 on the board. Remind students that x/3 is equivalent to ( 1/3)x.What is
another way to write 2x/3? ((2/3)x) This expression may also be written (2/3)x (1/3)x.What are the
terms in this expression? ((2/3)x, ( 1/3)x) Are these like terms? (Yes.) What are the coefficients of the
terms? (2/3 and 1/3) Simplify
this expression. (x/3 or (1/3)x)
Check Understanding See Answer Key for a and c.
Some students may choose to
10m2n 3y 2z 3
7z
a. Find the pairs of like terms in the chart.
5x
simplify by writing both numerators
5
3
2
3
b. In physics, g always represents acceleration due to
pq
3z y
6ba
5xy
gravity. This is an example of a constant
.
2 2
2
over
the common denominator and
10m n
3m
ab
25x
2
1
c. Are the terms 3 y 2 and 1 3 y 2 like terms? Find the
combining
like terms. Other
2m2
q5p3
9z 3y 2
7z
sum and then the product of these terms.
students may choose to combine the
d. How many terms appear in the expression 10m 2mn 8m 6n2? four terms
coefficients 2/3 and 1/3 and keep
e. Are constant terms in an expression considered to be like terms? Yes.
f. Translate “the sum of a number and fifteen more than the number” into an algebraic
the
variable the same.
expression and simplify. x
x 15 2x 15
3 Stress to students that all numbers
Practice
Copy and complete the chart by listing the items in each category.
Expression
1.
5x
7y
x5
2.
3k
6
k
2
3.
8a2b3
Terms
none
7y, x 5,
2
k
3k and 2 ,
k
3k, 6, 2 ,10
6 and 10
8a2b 3, 2b 3a2, 8a2b 33 and
2b a2
8
5x,
2
10
2b3a2
Like Terms
8
Coefficients
5,
Constant Terms
7, 1
2
3,
1
2
6, 10
8,
2
8
State whether the terms in each set are like or not. Find the product and then the sum of
each set of terms. Simplify if possible. See Answer Key.
3
1
4. 2m, m
5. 5k, k
6. 4xy, 4yx
7. 2a, a2
8. c3, c3
9. 4p2, 2p4
8
8
Simplify each expression.
10. 5a 2a 7a
2
5
2
13. 2m2
m
m2m
7
7
16. 4xy 2xy 6 6xy
6
7
11. 7x x 7 6x
3
4d f
7 m 14. 2d f 6d
1 3
4
3 3
17. p3 4
p
6 5p
5
5
12.
2c
3c
3
15. 5z
2
20
8
12
c
5
3z
5z3
3z5
2s
5r
35x
109
because the simplified expression is equal to the original expression.
4;
18.
3r
15x 20x 4.
3. 109
a. Evaluate the expression for x
b. Simplify the expression and evaluate the simplified expression for x
c. Compare your answers from a and b. Explain. They are the same
2s
2r
19. Consider the expression
3.
Apply
Translate each of the following into an algebraic expression and simplify if possible.
3
20. The sum of twice a cubed number and the opposite of the cubed number 2x
x 3; 3x 3
2
21. The sum of twice a squared number and the opposite of the cubed number 2x
x
22. Your friend has 17 more books than you have at home. Write an expression in simplest
form that represents the total number of books you have together.
23. The
constant is the characteristic that remained the same.
2b
17
Explain the use of the italicized word in each sentence.
23. “The constant in the investigation was the time of day of each crime.”
24. “The variable in the investigation was the method used to break into each home.”
The variable is the characteristic that changed.
157
and variables can be multiplied and
divided, but only like terms can be
added. Review the properties of
exponents.Write 5x · 3x 2 on the
board. Use the Commutative and
Associative Properties to show
students how to simplify this
expression (5 · 3 · x · x 2 15x 3 ).
Simplify (2/3)x · (1 1/6)x. ((2/3)x ·
(7/6)x 2/3 · 7/6 · x · x
(7/9)x 2) Remind students to
change mixed numbers to improper
fractions when multiplying. Stress to
students that the coefficients are
multiplied and the variables are
multiplied by adding their
exponents.Write (xy)2 on the board
and ask a student to simplify this
expression. ((xy) · (xy) x · x · y ·
y x 2y 2) Remind students that
exponents apply only to the adjacent
variable or expression inside the
parentheses. Distribute copies of
BLM 7.1A Variables. Have
students work in groups to complete
the chart. Students may choose to
draw models of each expression to
help them simplify.
8w 2 10w on the board. How many terms are in the expression? (3) What are the coefficients?
(3, 8, 10) Point out that the negative is part of the coefficient. Students should always read expressions as
sums of positive and negative terms (3w ( 8w 2) 10w).This is important because it allows the
interchanging of terms according to the Commutative and Associative Properties. Ask a student to circle the
like terms and then simplify the expression. ( 8w2 13w)
4 Write 3w
5 Proceed to LESSON 7.1. Use BLM 7.1B Operations with Variables for additional practice.
Math Moments
Consider the expression xy 2x 2y 3xy 3 2x 2y 6xy 6 15xy 3 x 3y.
1. How many terms are in this expression? (8)
2. Identify the pairs of like terms. (xy and 6xy, 2x 2y and 2x 2y, 3xy 3 and 15xy 3)
3. Simplify the expression. (12xy 3 x 3y 7xy 6)
© Copyright 2004
187