Objectives
To review multiplying fractions and mixed numbers;
and to find reciprocals.
1
materials
Teaching the Lesson
Key Activities
Students review multiplying fractions and mixed numbers. They also practice finding
the reciprocal of a number.
Key Concepts and Skills
Math Journal 2, pp. 205 and 206
Student Reference Book, pp. 88–90
(optional)
calculator
• Use an algorithm to multiply fractions and mixed numbers.
[Operations and Computation Goal 4]
• Write a special case to illustrate a general pattern.
[Patterns, Functions, and Algebra Goal 1]
• Define and find the reciprocal of a number.
[Patterns, Functions, and Algebra Goal 4]
Key Vocabulary reciprocal
Ongoing Assessment: Recognizing Student Achievement Use journal page 206.
[Patterns, Functions, and Algebra Goal 4]
2
materials
Ongoing Learning & Practice
Students practice finding and comparing products of whole numbers and fractions
by playing Fraction/Whole Number Top-It.
Math Journal 2, p. 207
Student Reference Book, pp. 319 or 320
Students practice and maintain skills through Math Boxes and Study Link activities.
Game Master (Math Masters, p. 478)
Study Link Master (Math Masters, p. 180)
4 each of number cards 1–10 (from the
Everything Math Deck, if available)
Geometry Template
calculator (optional)
3
materials
Differentiation Options
READINESS
Students convert
mixed numbers to
improper fractions.
ENRICHMENT
Students explore
sums and products
of reciprocals.
EXTRA PRACTICE
Students find
reciprocals of
whole numbers,
fractions, and
decimals.
ELL SUPPORT
Students add
terms to their
Math Word Banks.
Teaching Masters (Math Masters,
pp. 181 and 182)
Differentiation Handbook
Technology
Assessment Management System
Journal page 206, Problems 1–10
See the iTLG.
530
Unit 6 Number Systems and Algebra Concepts
Getting Started
Mental Math and Reflexes
Math
Message
Students indicate thumbs-up if the fractions are equivalent or thumbs-down if they are
not equivalent.
Complete Problems 1–3 on
journal page 205.
Suggestions:
5
22
1.5
3
and thumbs up
5
10
7
70
and thumbs down
10
1,000
7
15
38
3
and 7 thumbs down
7
5
19
76
and thumbs up
20
80
16 and 12 thumbs up
18 and 16 thumbs down
1 Teaching the Lesson
Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
(Math Journal 2, p. 205)
Go over the answers to Problems 1–3. Draw attention to the
exception to the general pattern in Problem 2—any nonzero
number divided by itself equals 1. Zero is excluded. Division by 0
0
is undefined, so 0 is undefined.
5
3
3.14
Write the special cases 1, 16 1, and 1 on the board. Ask
students to give a general pattern in words for the three special
cases. The quotient of a number divided by 1 is that number.
ELL
Adjusting the Activity
To support English language learners, summarize students’ general
patterns in words by writing the following on the board:
A U D I T O R Y
x
x
1
x
1
x
x1x
K I N E S T H E T I C
T A C T I L E
Student Page
Date
V I S U A L
Time
LESSON
Applying Properties of Multiplication
61
Math Message
88 90
103
1. Write a general pattern in words for the group of three special cases.
Reviewing Multiplication
19 1 19
2
7
PARTNER
ACTIVITY
of Fractions
For any number n, n 1 n.
General pattern:
2. Write a general pattern in words for the group of three special cases.
(Math Journal 2, p. 205)
Multiplication of fractions and mixed numbers was covered in
Lessons 4-6 and 4-7. Pose several problems from the Student
Reference Book, pages 88–90 to check students’ understanding.
Then assign Problems 4–18 on journal page 205.
2
7
1 0.084 1 0.084
58
58
1
3
8
3
8
1
7.02
7.02
1
For any number n,
General pattern:
n
n
1.
3. Multiply. Write your answers in simplest form.
a.
19
19
4 4
b.
2
3
2
3
6
6
c.
0.5
2
2
0.5 Multiply. Write your answers in simplest form. When you and your partner have
finished solving the problems, compare your answers.
1
4
5.
6 11
8.
2 1 11.
3 14.
4
1
17.
5
1
6
3
5
4. 10
6
3
4
7. 2 4
1
1
2
10. 4
5
7
3
13. 2 5
10
7
16. 8
8
7
1
10
41
50
1
1
2
3
3
5
2
3
3
8
3
4
4
1
43
17
2 32
1
4
6
11
1
1
1
18. Write three special cases for the general pattern x 1.
x
7
1
7
1
5
9
1
1
5
9
3
7
6.
7 9.
7
3
1
3
5
6
3
7
9
2
3
12.
1 4 15.
1
100
100
1
5
89
1
Sample answers:
2.5 1
2.5
1
205
Math Journal 2, p. 205
Lesson 6 1
531
Student Page
Date
Defining the Reciprocal
Time
LESSON
Reciprocals
61
Reciprocal Property
a
b
b
a
If a and b are any numbers except 0, then 1.
a
a
b
1
a
and
b
a
93
are called reciprocals of each other.
1
a
1, so a and are reciprocals of each other.
Find the reciprocal of each number. Multiply to check your answers.
1
6
4
3
1. 6
3
3. 4
8
, or
3
2
17
3
14
3
5. 8
1
7. 8 2
2
9. 4 3
11. 0.1
13. 0.75
15. 0.375
10
–
or 1.3
–
or 2.6
4
,
3
8
,
3
Solve mentally.
5
7
5
19. 7 5 4 7
3
10
7
10
3
17
1
4. 3
13
6. 16
5
8. 3 6
1
10. 6 4
12. 0.4
14. 2.5
16. 5.6
1
32
5
47
1
1
17. 3 2 4 4
1
21. 33
2
23
2. 17
1
1
17
3
1
16
3
, or 1
13
13
6
23
4
25
5
, or 2.5
2
2
, or 0.4
5
5
, or 0.1786
28
1
2
18. 6 5 6
1
1
20. 2 8 2 2
22. 3.875 2.5 0.4
2
5
1
82
3.875
WHOLE-CLASS
DISCUSSION
of a Number
(Math Journal 2, p. 205)
Bring the class together to discuss Problems 4–18. Students may
have discovered a visual pattern: If a fraction is multiplied by the
same fraction turned upside down, then the product is 1. Pairs of
numbers whose product is 1 are called reciprocals. To support
English language learners, write reciprocal on the board and list
some examples. Ask students for additional examples to add to the
1
list. (Do not erase the board.) For example, 2 is the reciprocal of 2
1
1
and 2 is the reciprocal of 2 because 2 2 1. The numbers 1 and
–1 are their own reciprocals (1 1 1 and –1 –1 1). Zero has
no reciprocal because the product of 0 and any number is 0. Every
other number has a reciprocal that is not equal to itself. Point out
that reciprocals need not be in fraction form. For example, 0.8 and
1.25 are reciprocals because 0.8 1.25 1.
206
Math Journal 2, p. 206
Adjusting the Activity
Students can multiply 0.8 and 1.25 on a calculator to check that the
numbers are reciprocals. They could also convert the numbers to fractions
8
1
4 5
1 .
and multiply:
10
4
5 4
A U D I T O R Y
K I N E S T H E T I C
T A C T I L E
Finding Reciprocals
V I S U A L
WHOLE-CLASS
ACTIVITY
One way to test whether two numbers are reciprocals is to
find their product. If the product is 1, then the numbers are
reciprocals. Write the following number pairs on the board and
ask the class to determine which pairs are reciprocals. As you
discuss the answers, add to your list of reciprocals on the board.
1
4 and 4 Yes
7
4
4
and 7 Yes
1
33
and
3
10
Yes
2
3
2
5
1
and 3 No
and 5 No
2.5 and 0.4 Yes
Write the following numbers on the board and ask students to find
their reciprocals.
5
6
7
5
1
9
532
Unit 6 Number Systems and Algebra Concepts
6
,
5
5
7
9
1
or 15
1
8 8
5 8
21
28
8
3.125 0.32, or 25
NOTE Be sure students understand that the
Links to the Future
In Lesson 6-11 and Lesson 9-5, students will multiply both sides of an equation
by the reciprocal of the fraction or decimal coefficient of the variable term.
Recognizing and applying multiplicative inverses is a Grade 6 Goal.
Practicing Finding Reciprocals
reciprocal of a number may be represented
in many ways, not only as a fraction turned
upside down.
PARTNER
ACTIVITY
(Math Journal 2, p. 206)
Students complete the problems on journal page 206.
Ongoing Assessment:
Recognizing Student Achievement
Journal
Page 206
Problems 1–10
Use journal page 206, Problems 1–10 to assess students’ ability to name and
identify the reciprocal of a number. Students are making adequate progress if
they are able to solve Problems 1–10. Some students may be able to find
reciprocals for Problems 11–16 without a calculator and use reciprocals in
Problems 17–22 to mentally find the products of fractions and mixed numbers.
[Patterns, Functions, and Algebra Goal 4]
Adjusting the Activity
Consider having some students use the F D key on a calculator for
Problems 11–16 to convert between decimals and fractions. (Not all calculators
have a F D key.)
A U D I T O R Y
K I N E S T H E T I C
T A C T I L E
Using a Calculator to
V I S U A L
WHOLE-CLASS
DISCUSSION
Find Reciprocals
Some calculators have keys that can be used to find reciprocals.
5
For example, to find the reciprocal of 6 using a calculator that has
6
the
key, enter 5
6
. The display shows 5.
If students are using calculators that do not have the
key,
remind them that a fraction is a division problem and the
1
reciprocal of a number can be expressed as a. For example, to
3
find the reciprocal of 8, key in:
TI-15: 1 3 n 8
fx-55: 1 3
8
d
;
.
Have students use calculators to find reciprocals.
Suggestions:
1 20
20 1
3 8
78 59
1 3.14
3.14 1
1
–9 –0.1
, or – 9
NOTE Keystrokes vary for different
calculators. Check the calculator’s instruction
manual for specific keystrokes. If students are
using calculators with the x –1 key, they must
1
understand that x –1 x. Students have
1
–1
learned that 10 10 from writing
numbers in expanded notation.
Lesson 6 1
533
Student Page
Date
Time
LESSON
2 Ongoing Learning & Practice
Math Boxes
61
1. Rename each mixed number as a fraction.
7
a. 3 8
53
9
53
6
51
7
b.
c.
d.
2
e. 14 3
2. Multiply.
31
8
1
b. 2
7
6
1
2 3
5
6
1
2
c. 8 8
9
9
7
d.
8 6 44
3
14
1
, or 16
1
a. 3 4 2
8
5 9
1
Playing Fraction/Whole Number
2
9
1
5
Top-It
1
2
10
71 72
88 89
3. Circle the number sentence that describes
y
3
11
5
15
B. (2 x) 5 y
C. y 2 (5 x)
D. y 8 x
0
5
10
25
1
b. 8
3
d. 1 4
3
e. 100
1
)
10
b. (9 (7 0.976
)
1
)
100
Distribute four each of number cards 1–10 (from the Everything
Math Deck, if available) to each partnership.
(6 1
)
1,000
6. a. Use your Geometry Template to draw
a spinner with colored sectors so the
chances of landing on these colors are
as follows:
80%
12.5%
87.5%
175%
3%
7
c. 8
0
53.04
5. Write a percent for each fraction.
4
a. 5
2
a. (5 10 ) (3 10 ) (4 10
1
x
(Student Reference Book, pp. 319 or 320; Math Masters, p. 478)
4. Write each number in standard notation.
the numbers in the table.
A. y x 10
PARTNER
ACTIVITY
green
3
10
red: Students use cards to form whole numbers and fractions. They
find the products of the numbers they form and then compare
those products. Students can use Math Masters, page 478 to
record their products and comparisons for each round of play.
blue
blue: 0.33
green: 20%
red
b. On this spinner,
what is the chance
of not landing on
red, blue, or green?
Math Boxes 6 1
17%
146
59 60
207
INDEPENDENT
ACTIVITY
(Math Journal 2, p. 207)
Math Journal 2, p. 207
Mixed Practice Math Boxes in this lesson are paired with
Math Boxes in Lesson 6-3. The skills in Problems 5 and 6
preview Unit 7 content.
Writing/Reasoning Have students write a response to
the following: Explain how to mentally solve Problems 2b
1
and 2d. Sample answer: For 2b: Multiply (2 2) and
1
1
1
1
1
1
1
(2 3). Add the products: 1 6 16. For 2d: 5 2 10. Apply
1
the reciprocal property: 10 10 1.
Study Link 6 1
INDEPENDENT
ACTIVITY
(Math Masters, p. 180)
Study Link Master
Name
Date
STUDY LINK
Home Connection Students find pairs of equivalent
fractions, find the reciprocals of numbers, and multiply
fractions and mixed numbers to solve number stories.
Time
Practice with Fractions
61
Put a check mark next to each pair of equivalent fractions.
2
3
1.
✓
✓
3.
5.
5
6
and 24
30
and 56
8
and ✓
2.
73
90 93
3
4
28
16
1 and 4
5
4.
7
3
49
7
6.
2 and 3
7
and 3
8
19
4
Find the reciprocal of each number. Multiply to check your answers.
1
1
2 5
19
7. 19
8. 2
2
5
7
5
1
2
6
9. 3 10. 7
, or 2
6
6
Multiply. Write your answers in simplest form. Show your work.
3
2
1
1
7
4
11. º 1 12. 3 º 3
8
7
22
1
Solve the number stories.
13.
How much does a box containing 5 horseshoes
1
weigh if each horseshoe weighs about 2 pounds?
1212 lb
One and one-half dozen golf tees are laid in a straight
1
line, end to end. If each tee is 2 inches long, how
8
long is the line of tees?
3814 in.
2
14.
15.
1
A standard-size brick is 8 inches long and 2 inches
4
3
high and has a depth of 3 inches. What is the volume
4
of a standard-size brick?
Practice
16.
107 (82) 56 18.
85 66 (48) 81
672 in.3
17.
4 (12 18) 19.
7 (11 22) 1
Math Masters, p. 180
534
Unit 6 Number Systems and Algebra Concepts
Teaching Master
Name
3 Differentiation Options
61
1.
Converting Mixed Numbers to
5–15 Min
Improper Fractions
Products and Sums of Reciprocals
Read Statement 1. Then find each reciprocal or product to help you decide
whether the statement is true or false.
24
d.
The product of 4 and 6 is
e.
The reciprocal of the product of 4 and 6 is
f.
g.
.
1
24 .
Repeat Problems 1a–1e using a different pair of positive numbers.
Do you think Statement 1 is true or false for all positive numbers? Explain.
Sample answer: True. For any positive numbers
1
1
1
a and b, a b (a b) .
To provide experience converting mixed numbers to
improper fractions, have students use a shortcut.
2.
Read Statement 2. Then find each reciprocal or sum to help you decide
whether the statement is true or false.
Statement 2 The sum of the reciprocals of two positive numbers is equal to
the reciprocal of their sum.
1
1
5 .
10 .
a. The reciprocal of 5 is
b. The reciprocal of 10 is
3
10 .
c. The sum of the reciprocals from 2a and 2b is
Example 1:
1
–
2
3
(2 3) 1
61
7
2 2 2
∗
The sum of 5 and 10 is
e.
The reciprocal of the sum of 5 and 10 is
f.
Example 2:
15
d.
g.
.
1
15 .
Repeat Problems 2a–2e using a different pair of positive numbers.
Do you think Statement 2 is true or false for all numbers having reciprocals? Explain.
Sample answer: False. A common denominator of
a and b a b.
(16 5
)3
3 —
16
16
5
Time
Statement 1 The product of the reciprocals of two positive numbers is equal
to the reciprocal of their product.
1
1
4 .
6 .
a. The reciprocal of 4 is
b. The reciprocal of 6 is
1
24 .
c. The product of the reciprocals from 1a and 1b is
SMALL-GROUP
ACTIVITY
READINESS
Date
LESSON
80 3
83
16 16
Math Masters, p. 181
∗
Have students use the shortcut to convert the following
mixed numbers to improper fractions.
3 19
4 4 4
4 64
2 23
7 3 3
7 79
12 5 5
9 8 8
8 53
67 7
59 9
6 48
ENRICHMENT
Exploring Products and
INDEPENDENT
ACTIVITY
15–30 Min
Sums of Reciprocals
(Math Masters, p. 181)
To explore the properties of reciprocals, students use specific cases
to decide whether general statements about reciprocals are true
or false.
Lesson 6 1
535
INDEPENDENT
ACTIVITY
EXTRA PRACTICE
Finding Reciprocals
15–30 Min
(Math Masters, p. 182)
Students find reciprocals of whole numbers, fractions,
and decimals.
INDEPENDENT
ACTIVITY
ELL SUPPORT
Building a Math Word Bank
5–15 Min
(Differentiation Handbook)
To provide language support for multiplication of fractions, have
students use the Word Bank template in the Differentiation
Handbook. Ask students to write any terms with which they are
unfamiliar, draw pictures relating to each term, and write other
related words. See the Differentiation Handbook for more
information.
Teaching Master
Name
Date
LESSON
Time
Finding Reciprocals
61
Solve.
3.
1
5
1
17
5.
10
6
1.
7.
º51
2.
º 17 1
4.
2
4
º 0.6 1
6.
1
n
1
2
º 1
º 0.25 1
ºn1
Explain how you solved Problem 5.
Sample answer: I renamed the decimal
6
as a fraction (10 ) and then found
10
its reciprocal (6).
For each number, fill in the circle next to the reciprocal.
(There may be more than one correct answer.)
5
8. 6
56
1
5
1
1.2
6
5
12.
9.
2
7
7
3
7
12
7
9
1
2.7
10.
3
11.
1.25
9
3
3
9
1
3
5
4
1.3
12
5
5.21
0.8
Explain how you solved Problem 10.
Sample answer: I renamed 3 as 31 and
found its reciprocal, 13. 39 is equivalent to
1
3
3, so 9 is also a reciprocal of 3.
Math Masters, p. 182
536
Unit 6 Number Systems and Algebra Concepts