Download The Properties of Number Systems

The Properties of
Number Systems
Objective To summarize the properties of number systems
and operations.
a
www.everydaymathonline.com
ePresentations
eToolkit
Algorithms
Practice
EM Facts
Workshop
Game™
Teaching the Lesson
Key Concepts and Skills
• Identify terminating and repeating decimals. [Number and Numeration Goal 1]
• Identify the absolute value of rational
numbers. [Number and Numeration Goal 1]
• Use the order of operations to
solve problems. Family
Letters
Assessment
Management
Common
Core State
Standards
Ongoing Learning & Practice
Reviewing a Problem-Solving
Diagram and Solving
Number Stories
Math Journal 2, p. 220
Student Reference Book, pp. 258
and 259
Students review a problem-solving
diagram and solve multistep
number stories.
[Patterns, Functions, and Algebra Goal 3]
• Apply the Identity, Commutative,
Associative, and Distributive Properties
of addition and multiplication. [Patterns, Functions, and Algebra Goal 4]
• Apply the Multiplication Property of –1. [Patterns, Functions, and Algebra Goal 4]
Key Activities
Students explore the real number system and
properties of various sets of numbers within it.
Math Boxes 6 5
Math Journal 2, p. 221
Students practice and maintain skills
through Math Box problems.
Study Link 6 5
Math Masters, p. 193
Students practice and maintain skills
through Study Link activities.
Ongoing Assessment:
Recognizing Student Achievement
Use Mental Math and Reflexes. Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Differentiation Options
ENRICHMENT
Investigating Properties
of Rational Numbers
Student Reference Book, pp. 104 –106
Math Masters, p. 194
Students explore properties of rational
numbers by determining whether
general statements are true or false.
ENRICHMENT
Renaming Repeating Decimals
as Fractions
Math Masters, p. 195
Students use a power-of-10 strategy to write
equivalent equations and rename repeating
decimals as fractions.
EXTRA PRACTICE
Plotting Numbers on a Number Line
Class Number Line
Students generate different types of numbers
in real-life contexts. They plot them on a
number line and compare them.
[Operations and Computation Goal 4]
ELL SUPPORT
Key Vocabulary
Naming and Comparing Numbers
counting numbers whole numbers integers rational numbers terminating
decimals repeating decimals irrational
numbers real numbers
Students write examples of various types
of numbers and compare them.
Materials
Math Journal 2, pp. 218–220
Student Reference Book, pp. 2–24 and 94–106
Study Link 64
calculator Real Number Line Poster
Additional Information
When you use the ready-made Everyday Mathematics posters to support English language learners, you should display the
English version only or both the English and Spanish versions simultaneously; do not display the Spanish version only.
Teacher’s Reference Manual, Grades 4–6 pp. 59–77
Lesson 6 5
553
Mathematical Practices
SMP2, SMP3, SMP4, SMP6, SMP7, SMP8
Content Standards
Getting Started
6.NS.1, 6.NS.6, 6.NS.6a, 6.NS.6c, 6.NS.7, 6.NS.7a, 6.NS.7b, 6.NS.7c,
6.NS.7d, 6.EE.2
Mental Math and Reflexes Math Message
Remind students that they have used visual models, the commondenominator method, and the invert-and-multiply method to solve fraction division
problems. Pose division problems. Students write a number model and solve.
Suggestions:
1
1
_
Geraldine is making curtains. Each curtain uses _
4 yard of fabric. She has 4 yard
1
1
÷_
= 1; 1 curtain
of fabric. How many curtains can she make? _
4
4
• How many positive numbers
are there?
• How many negative numbers are there?
• How many numbers are neither positive
nor negative?
• Name one positive and one negative
number, each of which is very close to 0.
7
1
Charlie made _
gallon of soup. He has containers that can store _
gallon.
8
3
5
7
1
21
_
_
_
How many containers does Charlie need to store the soup? _
8 ÷ 3 = 8 , or 2 8 ;
3 containers
2
1
_
Larry has 2 _
3 pints of ice cream in his freezer. One serving of ice cream is 6 pint.
2
1
_
_
How many servings of ice cream does Larry have? 2 ÷ = 16; 16 servings
3
Mental Math
Ongoing
and Reflexes Assessment:
Recognizing Student
Achievement
Study Link 6 4a
Follow-Up
Briefly go over the answers.
6
1 Teaching the Lesson
▶ Math Message Follow-Up
Use the Mental Math and Reflexes problems
to assess students’ ability to divide fractions
and mixed numbers. Students are making
adequate progress if they are able to solve
the suggested problems.
WHOLE-CLASS
DISCUSSION
Most students probably know that there is an infinite amount of
positive and negative numbers. Ask students to express this fact
in a variety of ways: Positive (negative) numbers go on without
end. It does not matter how large (small) a positive (negative)
number I think of, I can always think of one that is larger
(smaller). Zero is the only number that is neither positive nor
negative. Have students share examples of positive and negative
numbers that are close to zero. Record them on the board and
have students put them in ascending order.
[Operations and Computation Goal 4]
Student Page
Fractions
Different Types of Numbers
▶ Presenting an Overview
Counting is almost as old as the human race and has been used
in some form by every human society. Long ago, people found
that the counting numbers (1, 2, 3, and so on) did not meet
all their needs.
WHOLE-CLASS
DISCUSSION
of Our Number System
♦ Counting numbers cannot be used to express measures
1
between two consecutive whole numbers, such as 22 inches
and 1.6 kilometers.
(Student Reference Book, pp. 99 –102)
♦ With the counting numbers, division problems such as 8 / 5
and 3 / 7 do not have an answer.
Fractions were invented to meet these needs. Fractions can
also be renamed as decimals and percents. And most of the
1
1
numbers you have seen, such as 2, 56, 1.23, and 25%, are
either fractions or can be renamed as fractions. With the
invention of fractions, it became possible to express rates and
ratios, to name many more points on the number line, and to
solve any division problem involving whole numbers (except
division by 0).
However, even fractions did not meet every need. For example,
3
1
problems such as 5 7 and 24 54 have answers that are less
than 0 and cannot be named as fractions. (Fractions, by the
way they are defined, can never be less than 0). This led to the
invention of negative numbers. Negative numbers are
1
numbers that are less than 0. The numbers 4, 3.25, and
100 are negative numbers. The number 3 is read as
“negative 3.”
Negative numbers serve several purposes:
♦ to name locations such as temperatures below zero on a
thermometer and depths below sea level
Every whole number can
be renamed as a fraction.
For example, 0 can be
0
written as 1 and 8 can
8
be written as 1.
Since every whole
number can be renamed
as a fraction, every
negative whole number
can be renamed as a
negative fraction. For
7
example, 7 1.
♦ to show changes such as yards lost in a football game and
decreases in weight
♦ to extend the number line to the left of zero
♦ to calculate answers to many subtraction problems
Student Reference Book, p. 99
554
Unit 6
Number Systems and Algebra Concepts
Links to the Future
This is an introduction to sophisticated ideas that students will revisit in seventhor eighth-grade algebra courses. Identifying the properties of real-number
subsets is not a Grade-Level Goal for sixth grade.
1. Identify counting numbers.
Draw a number line on the board with 11 evenly spaced
marks on it as shown.
Write the numbers 1 through 5 on the number line. As you
discuss the following ideas, write them on the board along
with examples.
1
2
3
4
5...
These are called counting numbers. As the name implies,
they are used to count things. There are infinitely many
counting numbers. Ask students to name some counting
numbers not shown on the number line.
2. Identify whole numbers.
Add the number 0 to the number line. All of the counting
numbers and 0 make up the set of whole numbers.
0
1
2
3
4
5...
3. Identify integers.
Now write the numbers -1 through -5 to the left of 0.
...5 4 3 2 1
0
1
2
3
4
5...
All of the whole numbers and their opposites make up the set
of integers. Remind students that numbers and their
opposites are located on opposite sides of 0 on the number line.
0 is its own opposite. Ask students to name some integers not
shown on the number line.
4. Identify rational numbers.
Ask students to name some numbers that are between
consecutive numbers on the number line. Students should
1 , 2.5, -4_
1 , and _
1 . Write
name fractions and decimals, such as _
2
4
3
a few of these numbers in the appropriate places on the
number line. These are called rational numbers.
A rational number is any number that can be written as a
a , where a and b are integers and b ≠ 0.
simple fraction _
b
Rational numbers can be positive, negative, or 0. They can be
whole numbers, because any whole number can be expressed
3 ). They can also be mixed
as a fraction (for example, 3 = _
1
numbers or percents, because any mixed number or percent
can be renamed as a fraction.
Students have learned that any fraction can be renamed as a
1 = 0.5). Dividing 1 by 2 to rename _
1
decimal (for example, _
2
2
results in a remainder of 0. In other words, the division comes
to an end. These decimals are called terminating decimals.
Student Page
Date
Time
LESSON
Scavenger Hunt
65
Use Student Reference Book, pages 2–24 and 94–106 to find answers to as many of
these questions as you can. Try to get as high a score as possible.
An infinite number
Sample answers:
1.
How many rational numbers are there? (10 points)
2.
Give an example of each of the following. (5 points each)
3.
5
a.
A counting number
b.
A negative rational number
c.
A positive rational number
d.
A real number
e.
An integer
f.
An irrational number
-_12
3
_
4
_
√5
0
π
Name two examples of uses of negative rational numbers. (5 points each)
Sample answer: To express measures below sea level; and
to express temperatures below the freezing point.
4.
3
Explain why numbers such as 4, _
5 , and 3.5 are rational numbers. (10 points)
Sample answer: They can be expressed as simple fractions.
4 can be expressed as _41 , _35 is already a fraction, and 3.5 can
be expressed as _72 .
5.
Explain why numbers such as π and √2
are irrational numbers. (10 points)Sample
answer:
They are nonterminating and nonrepeating decimals.
0
0
6.
n+n=n
What is n?
7.
k = OPP(k)
What is k?
8.
j∗j=j
Which two numbers could j be?
9.
a + (-a) =
0
(15 points)
(15 points)
(15 points)
1 or 0
10.
1 =
b∗_
b
(15 points each)
1
(15 points)
Math Journal 2, p. 218
205_246_EMCS_S_G6_MJ2_U06_576442.indd 218
8/29/11 10:43 AM
Lesson 6 5
555
Student Page
Date
1 does not result in a zero
Dividing 1 by 3 to rename _
3
remainder despite how many numbers follow the decimal point
1 = 0.333333...). Not only do the digits continue, but they do
(_
3
so in a repeating pattern—in this case, 3 is repeated over and
over. Such decimals are called repeating decimals.
Time
LESSON
Scavenger Hunt
6 5
䉬
continued
Match each sentence in Column 1 with the property in Column 2 that it illustrates.
(5 points each)
11.
Column 1
Column 2
A.
a (b c) (a b) c
B.
abba
C.
a ⴱ (b c) (a ⴱ b) (a ⴱ c)
D.
a ⴱ (b c) (a ⴱ b) (a ⴱ c)
E.
aⴱbbⴱa
F.
a ⴱ (b ⴱ c) (a ⴱ b) ⴱ c
D
B
C
F
E
A
Distributive Property of Multiplication
over Subtraction
Commutative Property of Addition
Distributive Property of Multiplication
over Addition
5. Identify irrational numbers.
Nonterminating decimals whose digits do not follow a repeating
pattern are called irrational numbers. The number π is an
example of an irrational number. Pi (3.14159265389…) has
long fascinated mathematicians. Teams of researchers have
calculated the digits of pi to 1.24 trillion places and found no
pattern in the digits to the right of its decimal point. Irrational
a , where a and b are
numbers cannot be written as a fraction _
b
integers (b ≠ 0). Thus, the set of rational numbers and the set
of irrational numbers have no numbers in common.
Associative Property of Multiplication
Commutative Property of Multiplication
Associative Property of Addition
a 0. How can that be? (15 points)
12.
If a is a negative number, then its opposite will always be
greater than 0. For example, (8) 0, because (8) 8.
Complete. (2 points each, except the last problem, which is worth 25 points)
13.
OPP(1) 1
1
OPP(OPP(1)) OPP(OPP(OPP(1))) 1
1
OPP(OPP(OPP(OPP(1)))) OPP(OPP(OPP(OPP(OPP(OPP … (OPP(OPP(1))))))))) 1
100 OPPs
Explain how you found the answer to the last problem.
Sample answer: Taking
the opposite an even number of times gives a positive number,
6. Identify real numbers.
The set of numbers consisting of all rational numbers and
all irrational numbers is called the set of real numbers.
Examples of these numbers are displayed on the Real Number
Line Poster. Students have worked with all the notations
shown on the number line except the tangent of 30° (tan 30°),
the number e, and the irrational square roots, all of which are
represented by nonterminating, nonrepeating decimals.
because every pair of OPPs gives a positive number.
Math Journal 2, p. 219
Adjusting
the Activity
ELL
Summarize this section of the lesson
by drawing a diagram that shows the
relationships among the six sets of numbers.
Include examples in the diagram. Ask
students to identify the sets to which a
number belongs. Have them explain why a
particular number belongs to some sets but
not to others. (See below.)
AUDITORY
KINESTHETIC
TACTILE
▶ Reviewing Absolute Value
Ask students to find the absolute values of some of the numbers
you plotted on the number line. Then ask questions such as the
following. Tell students to use absolute value to justify their answers.
5 38
Whole
6.75 –5
1
3
–29
0
Counting
1
If Brian’s account balance is -$20, what is the size of his debt?
The size of his debt is $20 because |-20| = 20.
●
The floor of Lisa’s basement is 12 feet underground. The floor of
Jeremy’s basement is 15 feet underground. Whose basement is
deeper? Jeremy’s basement because |-15| > |-12|.
As students share their answers, be sure that the following points
are discussed:
Irrational
Integers
16
8
●
VISUAL
Real Numbers
Rational
WHOLE-CLASS
DISCUSSION
The absolute value of a number is the distance of that number
from zero on the number line.
2
π
Absolute value can be used to indicate the size, or magnitude, of
numbers representing certain real-world situations.
º 5
In some contexts, it is useful to compare the absolute value of
two numbers instead of the numbers themselves.
3
▶ Taking Part in a Mathematical
PARTNER
ACTIVITY
Scavenger Hunt
(Math Journal 2, pp. 218–220; Student Reference Book, pp. 2–24 and 94–106)
Explain that students will work in pairs to complete the
scavenger hunt on journal pages 218–220.
556
Unit 6
Number Systems and Algebra Concepts
Student Page
Encourage students to read all the questions first and then to
search the Student Reference Book for answers. Most answers
require thinking beyond the information provided. Let students
know that some questions are very challenging.
When most partnerships have finished, review the answers.
You may also want to discuss some of the following ideas:
_
_
Numbers such as √12 and √5 are irrational numbers, but not
all square roots are irrational; for example, the square root of
16 is 4, which is a counting number. Ask students to identify
other
square
_
_ roots_that are counting numbers. Sample answers:
√9 = 3, √4 = 2, √1 = 1
All square _roots have a positive and a negative value. For
example: √9 = +3 or -3, because 3 ∗ 3 = 9 and -3 ∗ -3 = 9.
_
Note that -√1 _
is shown at the -1 point on the number line,
because OPP(√1 ) = OPP(1) = -1.
1.
10–1 is a positive number because it is equivalent to _
Date
Time
LESSON
Scavenger Hunt
6 5
䉬
continued
Is 52 a positive or negative number? Explain. (15 points)
14.
Sample answer: It is positive. By the Powers of a
2
Number Property, 5 is 1 divided by 5 used as a
1
1
factor 2 times, which is 5 º 5 , or 25 .
15.
Two numbers are their own reciprocals. What are they?
16.
What number has no reciprocal?
0
1 and 1
(15 points each)
(15 points)
Number Stories
Diana wants to make a 15 ft by 20 ft section of her yard into a garden. She will plant
2
flowers in 3 of the garden and vegetables in the rest of the garden. How many
square feet of vegetable garden will she have?
2
1.
100 ft
Sample answer: I multiplied 15 by
20 to get the total square footage of 300 ft2; then I
2
multiplied 300 by 3 to find the area for the flower
garden: 200 ft2. Finally, I subtracted 200 from 300 to
find the area used for vegetables.
Explain how you got your answer.
2.
Leo is in charge of buying hot dogs for his school’s family math night. Out of 300 parents and
3
children, he expects about 5 of them to attend. Hot dogs are sold 8 in a package, and Leo
figures he will need to buy 22 packages so that each person can have 1 hot dog.
a.
How do you think he calculated to get 22 packages?
3
Sample answer: He first multiplied 5 by 300 to find
out that about 180 people will come. Then he divided
180 by 8, which is 22.5.
Will Leo have enough hot dogs? No. He did not interpret the
remainder properly. He needs to buy 23 packages.
b.
10
Math Journal 2, p. 220
2 Ongoing Learning & Practice
▶ Reviewing a Problem-Solving
INDEPENDENT
ACTIVITY
Diagram and Solving Number Stories
(Math Journal 2, p. 220; Student Reference Book, pp. 258 and 259)
With the class, review the general approach to problem solving
described on the Student Reference Book pages. Use the strategies
and diagram to discuss how students might solve the Check Your
Understanding problem at the bottom of page 259.
Over the next few days, students can estimate an answer and
write an explanation of how they arrived at their estimate.
Student Page
Date
Time
LESSON
Math Boxes
6 5
䉬
Have students solve the multistep number stories on the bottom
half of journal page 220 and explain their thinking.
▶ Math Boxes 6 5
1.
Simplify.
of 80
9
b. 8
of 2
2
c. 3
of 32
3
d. 8
INDEPENDENT
ACTIVITY
2.
3
a. 4
1
of
60
9
1
, or 2
4
4
7
,
3
1
or 23
1
6
4
9
(Math Journal 2, p. 221)
8
a. 9
3
4
7
b. 8
6
c. 9
1
3
8
d. 24
1
2
4
24
5
127
5
28
1
1 3
2
87–89
3.
Give a ballpark estimate for each quotient.
93
4.
Sample estimates:
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lessons 6-4a and 6-7. The skills in
Problems 5 and 6 preview Unit 7 content.
Writing/Reasoning Have students write a response to
3 of 80 is less than 80, while
the following: Explain why _
4
9 of 2 is greater than 2. Sample answer: Multiplying a
_
8
number by a fraction less than 1 yields a product less than the
number; multiplying a number by a fraction greater than
1 yields a product greater than the number.
Divide. Simplify if possible.
a.
643.27 5
728.09
b. 7
c.
432.67 82
d.
2,091.05 / 53
130
100
5
40
Complete each sentence using an
algebraic expression.
If Mark earns t dollars for each hour
he tutors, then he earns
1
1
3 2 t, or 32 ⴱ t dollars when he tutors
a.
1
2
for 3 hours.
b.
Madison’s dog is 3 years older than
her cat. If the dog is d years old, then
d 3 years old.
the cat is
261
5.
Write each decimal as a fraction in
simplest form.
a.
0.06 b.
0.52 c.
0.09 d.
0.64 240
6.
3
50
13
25
9
100
16
25
Roll two 6-sided dice, one red, one green.
Give the probability of rolling the
following totals.
1
6
a.
11
1
18
b.
7
c.
0
0
d.
3 or 4
e.
an even number
5
36
1
2
59–60,
74
148 149
Math Journal 2, p. 221
Lesson 6 5
557
Study Link Master
Name
Date
STUDY LINK
65
▶ Study Link 6 5
Time
Fill in the missing numbers in the tables. Look for patterns in the results.
1.
x
y
7
9
-2
12
-3
-9
_2
_5
3
6
2.7
-1.9
22
23
OPP(y)
x+y
y+x
x-y
-7
–9
16
16
-2
2
10
-12
1_12
0.8
12
-14
6
-_16
14
-6
-12
9
-_56
10
-12
1_12
1.9
0.8
-(23)
12
-2.7
-(22)
(Math Masters, p. 193)
105
OPP(x)
2
3
-_23
INDEPENDENT
ACTIVITY
Turn-Around Patterns
y-x
Home Connection Students explore the relationships
between addition and subtraction and between
multiplication and division of rational numbers.
1
_
6
-4.6
4
4.6
-4
Which patterns did you find in your completed table?
Sample answer: Addition answers are the same regardless
of order. Subtraction answers are opposite when the order
of the numbers is reversed.
2.
x
_1
y
7
9
-2
12
-3
-9
_2
_5
3
6
2.7
-1.9
22
23
_1
y
x
_1
_1
7
9
-_12
-_13
1
_
12
-_19
6
_
5
1
-_
1.9
1
_
23
3
_
2
1
_
2.7
1
_
22
xºy
yºx
x÷y
9
_
7
_
63
-24
27
-_16
3
-6
3
5
_
5
_
4
_
5
_
9
9
63
-24
27
-5.13
25
7
9
1
_
5
-5.13
25
3 Differentiation Options
y÷x
-1.42
-0.70
2
1
_
2
▶ Investigating Properties of
Which patterns did you find in your completed table?
Sample answer: Multiplication answers are the same
regardless of the order of the factors. Division answers are
reciprocals when the order of the divisor and dividend
is reversed.
Math Masters, p. 193
EM3cuG6MM_U06_180-216.indd 193
3/29/10 5:17 PM
(Student Reference Book, pp. 104–106; Math Masters, p. 194)
To further explore properties of rational numbers, students decide
whether general statements about rational numbers are true or
false. They revise false statements to make them true for all
special cases. Students should check their work by referring to the
Properties of Numbers on Student Reference Book pages 104–106.
Teaching Master
Name
Fractions
Date
6 5
Properties of Numbers
䉬
The following properties are true for all numbers. The variables
a, b, c, and d stand for any numbers (except 0 if the variable
stands for a divisor).
Time
Properties of Numbers
LESSON
For each statement below, indicate whether it is always true or can be false.
If the statement can be false, give an example.
True or false?
Properties
Examples
5 7 12
Binary Operations Property
3 8
3
523
0.5 * (4) 2
235 83 3490
7 8 8 7 15
Commutative Property
The sum or product of two numbers is the same, regardless of
the order of the numbers.
abba
a*bb*a
5 * (6) 6 * (5) 30
3
4
4
4
* (5) 5 *
3
4
a
2. b
ⴱ a (b c) (a b) c
a * (b * c) (a * b) * c
a * (b c) (a * b) (a * c)
a * (b c) (a * b) (a * c)
2 * (8 3) (2 * 8) (2 * 3)
2 * 5
16 (6)
10 10
Addition Property of Zero
5.37 0 5.37
The sum of any number and 0 is equal to the original number.
a00aa
0 (6) 6
The product of any number and 1 is equal to the original number.
a*11*aa
2
3
*1
Unit 6
2
5
5
9
9
10
4
8
5
2
Explain why giving only one example for a true statement is not enough to prove
that it is true.
Sample answer: True means true in every
case, not just true in one case.
Try This
Correct each false statement in Problems 1–4 so the statement is true for all
special cases. Give one example for each statement.
2
c
3
(a º d ) (c º b) 3
(3 º 5) (2 º 4)
; 1
1. ab d 5
20
bºd
4
20
4
32
(9 º 8) (4 º 10)
c
(a º d ) (c º b) 9
; 3. ab d 8
80
10
80
bºd
2
4
10 5
2
3
1 * 19 19
Student Reference Book, p. 104
558
ac
bd
13
5 * (8 2) (5 * 8) (5 * 2)
5 * 10 40 10
50 50
Multiplication Property of One
c
d
a
4. b
3
6.
When a number a is multiplied by the sum or difference
of two other numbers, the number a is “distributed” to each of
these numbers.
ac
bd
212 * (2 * 3) (212 * 2) * 3
212 * 6
5
*3
15 15
Distributive Property
aⴱc
bⴱd
c
d
false
true
false
true
Example
3
4
5
87
20 20
12
c
d
a
3. b
5.
The sum or product of three or more numbers is the same,
regardless of how the numbers are grouped.
ac
bd
c
d
a
1. b
(7 5) 8 7 (5 8)
Associative Property
15–30 Min
Rational Numbers
Student Page
When any two numbers are added, subtracted, multiplied, or
divided, the result is a single number.
a b, a b, a * b, and a b are equal to single numbers.
INDEPENDENT
ACTIVITY
ENRICHMENT
4
Number Systems and Algebra Concepts
Math Masters, p. 194
Teaching Master
ENRICHMENT
▶ Renaming Repeating Decimals
INDEPENDENT
ACTIVITY
15–30 Min
Name
LESSON
6 5
䉬
Date
Renaming Repeating Decimals
You can use a power-of-10 strategy when renaming a repeating decimal as a fraction.
Work through each of the examples shown below.
as Fractions
–
Example 1: Rename 0.3 as a fraction.
Let 1x 0.3333…
Because one digit repeats, multiply both
sides by 10 to eliminate the repeating
digits to the right of the decimal point.
(Math Masters, p. 195)
Subtract
To further explore repeating decimals, students use a power-of-10
strategy to write equivalent equations that they use to rename a
repeating decimal as a fraction.
Some students may notice that if the repeating decimal has (a) as
the repetend, then the fraction represented by that repeating
(a)
decimal is _
, where R is a number with the same number of
R
digits as (a), but all the digits are 9s. For example: In the decimal
45 = _
5 . Similarly,
⎯⎯, if (a) is 45, then R is 99. So, 0.45
⎯⎯ = _
0.45
99
11
153 = _
17 .
⎯⎯⎯ = _
0.153
999
111
▶ Plotting Numbers on a
SMALL-GROUP
ACTIVITY
15–30 Min
If 1x 0.333...,
then 10x 3.33... .
10x 3.333
1x 0.333
9x 3
9x
9
Divide to solve for x.
Simplify.
–
1
0.3 renamed as a fraction is .
3
9
3
9
1
3
x 3
—
Example 2: Rename 0.45 as a fraction.
Let 1x 0.4545…
Because two digits repeat, multiply both
sides by 100 to eliminate the repeating
digits to the right of the decimal point.
Subtract.
If 1x 0.454545...,
then 100x 45.45... .
100x 45.4545
1x 0.4545
99x 45
99x
99
Divide to solve for x.
Simplify.
—
5
0.45 renamed as a fraction is .
45
99
45
99
5
11
x 11
Rename each repeating decimal as a fraction.
7
–
—
9
1. 0.7 2. 0.25 3.
EXTRA PRACTICE
Time
25
99
Compare the denominators in the examples to the denominators of your
answers for Problems 1 and 2. Use any patterns you notice to mentally
–
—
rename 0.5 and 0.32. Check your answers with a calculator.
5
32
–
—
9
99
a. 0.5 b. 0.32 Math Masters, p. 195
Number Line
To provide additional practice with the number system, have
students generate real-world contexts that involve different kinds
of numbers. Encourage them to come up with a variety of contexts
that use integers, rational numbers, and irrational numbers.
Tell students to name several numbers that make sense in each
of their contexts. Have them locate each number on the Class
Number Line and compare the numbers using their locations
on the number line. Ask students to make a general statement
about the size of a number and its location on the number line.
Sample answer: The larger the number, the farther to the right it
appears on the number line.
ELL SUPPORT
▶ Naming and
SMALL-GROUP
ACTIVITY
5–15 Min
Comparing Numbers
Have students write three examples of counting numbers, whole
numbers, integers, terminating decimals, and repeating decimals
within the following ranges: -5 ≤ x ≤ 10 and 54 ≤ x ≤ 56. Ask
students to compare these types of numbers and to write about
the differences.
Lesson 6 5
559