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CRCT 6 Mathematics
Lesson 1: Factors and Multiples
Lesson 2: Computation with Decimals
Lesson 3: Computation with Fractions and
Mixed Numbers
Lesson 4: Fractions, Decimals, and Percents
Lesson 5: Problem Solving
Unit 2
Algebra
Lesson 6: Ratio, Proportion, and Percent
Lesson 7: Patterns and Relationships
Lesson 8: Expressions and Equations
Unit 3
Geometry
Lesson 9: Plane Figures
Lesson 10: Solid Figures
Unit 4
Measurement
Lesson 11: Converting Units of Measurement
Lesson 12: Geometric Measurement
Unit 5
Data Analysis and Probability
Lesson 13: Data Analysis
Lesson 14: Probability
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READING • ELA/WRITING • MATHEMATICS
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Iowa City, Iowa 52244-2180
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www.BuckleDown.com
EMAIL: [email protected]
Catalog # 2064.GA
Georgia
6
CRCT
Mathematics
6 MATHEMATICS
Number and Operations
Georgia CRCT
The cover image depicts a protractor.
This important tool for measuring
angles is also a useful instrument for
drafting and plotting.
Unit 1
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TABLE OF CONTENTS
Introduction ..................................................................................... 1
Testwise Strategies™.......................................................... 2
Unit 1 – Number and Operations ................................................ 3
Lesson 1: Factors and Multiples ........................................ 4
GPS: M6N1.a, M6N1.b, M6N1.c
Concepts/Skills to Maintain: Multiples and factors
Lesson 2: Computation with Decimals ............................ 17
GPS: M6N1.g
Concepts/Skills to Maintain: Operations with decimal fractions
Lesson 3: Computation with Fractions and
Mixed Numbers ................................................ 26
GPS: M6N1.d, M6N1.e, M6N1.g
Concepts/Skills to Maintain: Addition and subtraction of
common fractions and mixed numbers with unlike
denominators; modeling multiplication of common fractions
Lesson 4: Fractions, Decimals, and Percents.................. 37
GPS M6N1.f, M6N1.g
Concepts/Skills to Maintain: Modeling percent
Lesson 5: Problem Solving ............................................... 43
GPS: M6P1.a, M6P1.b, M6P1.c, M6P1.d
Unit 2 – Algebra ............................................................................. 55
Lesson 6: Ratio, Proportion, and Percent........................ 56
GPS: M6A1, M6A2.b, M6A2.c, M6A2.g
Lesson 7: Patterns and Relationships ............................. 64
GPS: M6A2.a
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Lesson 8: Expressions and Equations ............................. 74
GPS: M6A2.d, M6A2.e, M6A2.f, M6A2.g, M6A3
Concepts/Skills to Maintain: Evaluating algebraic expressions
Unit 3 – Geometry ......................................................................... 91
Lesson 9: Plane Figures.................................................... 92
GPS: M6G1.a, M6G1.b, M6G1.c, M6G1.d, M6G1.e
Lesson 10: Solid Figures................................................. 102
GPS: M6G2.a, M6G2.b, M6G2.c, M6G2.d
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Table of Contents
Unit 4 – Measurement ................................................................ 111
Lesson 11: Converting Units of Measurement .............. 112
GPS: M6M1
Lesson 12: Geometric Measurement.............................. 126
GPS: M6M2.a, M6M2.b, M6M2.c, M6M3.a, M6M3.b,
M6M3.c, M6M3.d, M6M4.a, M6M4.b, M6M4.c, M6M4.d
Concepts/Skills to Maintain: Perimeter, capacity, and
area of geometric figures
Unit 5 – Data Analysis and Probability ................................. 145
Lesson 13: Data Analysis ............................................... 146
GPS: M6D1.a, M6D1.b, M6D1.c, M6D1.d, M6D1.e
Concepts/Skills to Maintain: Graphing data
Lesson 14: Probability .................................................... 166
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GPS: M6D2.a, M6D2.b, M6D2.c
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Unit 1 – Number and Operations
GPS: M6N1.a, M6N1.c
Lesson 1: Factors and Multiples
In this lesson, you will maintain your skills at finding multiples and factors. You
will find the greatest common factor and least common multiple of two whole
numbers. You will identify prime and composite numbers. Finally, you will break
numbers down into their prime factorization.
Multiples
Multiples of a number are the products that result from multiplying the number
by each of the whole numbers (0, 1, 2, 3, 4, and so on).
Example
What are the first five multiples of 6?
Multiply 6 by each of the first five whole numbers.
6•00
6•16
6 • 2 12
6 • 3 18
6 • 4 24
A number that is a multiple of two or more numbers is a common multiple
of those numbers. (Zero is not considered a common multiple.) The smallest
common multiple of two or more numbers is their least common multiple
(LCM).
Example
What is the least common multiple of 6 and 8?
multiples of 6: 0, 6, 12, 18, 24, 30, 36, 42, 48, 54, . . .
multiples of 8: 0, 8, 16, 24, 32, 40, 48, 56, 64, 72, . . .
The numbers 24 and 48 are the first two common multiples of 6 and 8.
The least common multiple of 6 and 8 is 24.
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The first five multiples of 6 are 0, 6, 12, 18, and 24.
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Lesson 1 – Factors and Multiples
GPS: M6N1.a, M6N1.c
Practice
Directions: For Numbers 1 through 5, list the first 10 multiples.
1. multiples of 4: ___________________________________________________
2. multiples of 7: ___________________________________________________
3. multiples of 9: ___________________________________________________
4. multiples of 12: ___________________________________________________
5. multiples of 16: ___________________________________________________
6. What is the least common multiple of 4 and 7? __________
7. What is the least common multiple of 7 and 9? __________
8. What is the least common multiple of 9 and 12? __________
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9. What is the least common multiple of 12 and 16? __________
10. What is the least common multiple
of 10 and 15?
11. What is the least common multiple
of 3 and 13?
A. 30
A. 13
B. 50
B. 26
C. 60
C. 39
D. 90
D. 52
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Unit 1 – Number and Operations
GPS: M6N1.a, M6N1.c
Factors
Factors of a number divide that number evenly (remainder of 0). A number
is evenly divisible by all its factors.
Example
What are the factors of 24?
Find the numbers that divide 24 evenly.
24 1 24
24 2 12
24 3 8
24 4 6
24 6 4
24 8 3
24 12 2
24 24 1
The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.
A number that is a factor of two or more numbers is a common factor of those
numbers. The largest common factor of two or more numbers is their greatest
common factor (GCF).
What is the greatest common factor of 24 and 42?
factors of 24: 1, 2, 3, 4, 6, 8, 12, and 24
factors of 42: 1, 2, 3, 6, 7, 14, 21, and 42
The numbers 1, 2, 3, and 6 are the common factors of 24 and 42.
The greatest common factor of 24 and 42 is 6.
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Example
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Lesson 1 – Factors and Multiples
GPS: M6N1.a, M6N1.c
Practice
Directions: For Numbers 1 through 5, list all the factors.
1. factors of 5: _______________________________________________________
2. factors of 10: _______________________________________________________
3. factors of 17: _______________________________________________________
4. factors of 102: _______________________________________________________
5. factors of 110: _______________________________________________________
6. What is the greatest common factor of 5 and 10? __________
7. What is the greatest common factor of 10 and 17? __________
8. What is the greatest common factor of 17 and 102? __________
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9. What is the greatest common factor of 102 and 110? __________
10. What is the greatest common factor
of 44 and 52?
11. What is the greatest common factor
of 39 and 78?
A. 1
A. 01
B. 2
B. 03
C. 4
C. 13
D. 6
D. 39
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Unit 1 – Number and Operations
GPS: M6N1.a, M6N1.c
Solving Problems
You can use common multiples and common factors to solve real-world problems.
Example
Todd and Amy volunteer at a local nursing home. Todd volunteers every
5 days and Amy volunteers every 4 days. If Todd and Amy both volunteer
today, in how many days will they volunteer together again?
Step 1: Determine whether you will use common multiples or common
factors to solve the problem.
The solution will be larger than 5 and 4, so use common multiples.
Step 2: Write the multiples of each number.
multiples of 5: 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, . . .
multiples of 4: 0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, . . .
Step 3: Find the nonzero multiples that are common.
multiples of 5: 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, . . .
multiples of 4: 0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, . . .
Step 4: Find the LCM.
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The least common multiple is 20. Todd and Amy will volunteer together
again in 20 days.
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Lesson 1 – Factors and Multiples
GPS: M6N1.a, M6N1.c
Example
The math class is making fruit baskets to donate to charity. The class has
collected 24 oranges and 36 apples. The teacher wants the same number
of oranges in every basket. The teacher also wants the same number of
apples in every basket. What is the greatest number of baskets the class
can make?
Step 1: Determine whether you will use common multiples or common
factors to solve the problem.
The solution will be smaller than 24 and 36, so use common factors.
Step 2: Write the factors of each number.
factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Step 3: Find the common factors.
factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
The class can make 1, 2, 3, 4, 6, or 12 baskets.
Step 4: Find the GCF.
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The greatest common factor is 12. The greatest number of baskets the
class can make is 12, each holding 2 oranges and 3 apples.
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Unit 1 – Number and Operations
GPS: M6N1.a, M6N1.c
Practice
Directions: For Numbers 1 through 4, first determine whether you will use
common multiples or common factors to solve the problem. Then solve the
problem.
1. One species of cicada hatches every 13 years. Another species hatches every
17 years. If both species hatch this year, how many years will it be before both
species hatch at the same time again?
Will you use common multiples or common factors to solve the problem?
How many years will it be before both species hatch at the same time again?
_______________
2. Paul is assembling books. He has 42 color pages and 77 black and white pages.
Each book must have the same number of color pages and the same number
of black and white pages. What is the greatest number of books Paul can
assemble?
What is the greatest number of books Paul can assemble? _______________
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Will you use common multiples or common factors to solve the problem?
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Lesson 1 – Factors and Multiples
GPS: M6N1.a, M6N1.c
3. Susie is making cakes at her bakery. She has 8 cups of flour and 12 eggs.
Each cake must have the same number of cups of flour and the same number
of eggs in it. What is the largest number of cakes Susie can make?
Will you use common multiples or common factors to solve the problem?
What is the largest number of cakes Susie can make? _______________
4. At Sunnydale Middle School, the sixth graders have mashed potatoes every
10 days and pudding every 6 days. If the sixth graders have mashed potatoes
and pudding today, how many days will it be before they have both on the
same day again?
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Will you use common multiples or common factors to solve the problem?
How many days will it be before the students have mashed potatoes and
pudding on the same day again?
_______________
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Unit 1 – Number and Operations
GPS: M6N1.b
Prime and Composite Numbers
A prime number has only two factors: 1 and the number. A composite number
has at least three factors. Remember, 0 and 1 are neither prime nor composite
numbers.
Examples
The number 3 has only two factors: 1 and 3. Therefore, 3 is
a prime number.
The number 4 has three factors: 1, 2, and 4. Therefore, 4 is
a composite number.
The number 6 has four factors: 1, 2, 3, and 6. Therefore, 6 is
a composite number.
Practice
1. Is 8 a prime number or a composite number? _______________________
2. Is 11 a prime number or a composite number? ______________________
3. Is 15 a prime number or a composite number? ______________________
5. List all the composite numbers between 20 and 30.
6. Which is a prime number?
12
7. Which is a composite number?
A. 37
A. 43
B. 45
B. 59
C. 51
C. 61
D. 63
D. 77
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4. List all the prime numbers between 20 and 30.
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Lesson 1 – Factors and Multiples
GPS: M6N1.b
Prime Factorization
Prime factorization is a way of expressing a composite number as the product
of prime numbers. The fundamental theorem of arithmetic states that every
counting number is either prime or can be decomposed (broken down) into its
prime factorization. You can use a factor tree to decompose a composite number
into its prime factorization.
Example
Decompose 504 into its prime factorization.
Write the number 504. Write a prime factor under the left branch and
circle it. Write the nonprime factor under the right branch. Repeat this
process under each composite number until you have two prime numbers
at the bottom of the tree. The prime factorization is the product of all the
circled numbers.
504
2
252
126
2
2
63
21
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3
7
The prime factorization of 504 is 2 • 2 • 2 • 3 • 3 • 7 or 23 • 32 • 7.
Note: There is more than one way to make a factor tree. In the first step
of this example, you could have divided by 3 or 7 instead of by 2. The
order in which you find the prime factors does not matter. However, when
you list the prime factors in your answer, list them in order from least to
greatest.
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Unit 1 – Number and Operations
GPS: M6N1.b
Practice
1. Draw a factor tree for 45.
The prime factorization of 45 is _____________________________.
The prime factorization of 120 is ______________________________.
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2. Draw a factor tree for 120.
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Lesson 1 – Factors and Multiples
GPS: M6N1.b
3. Draw a factor tree for 1,260.
The prime factorization of 1,260 is _____________________________.
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4. Draw a factor tree for 800.
The prime factorization of 800 is _____________________________.
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Unit 1 – Number and Operations
CRCT Practice
A. 16
A. 072
B. 18
B. 144
C. 12
C. 216
D. 18
D. 432
2. What is the prime factorization
of 60?
5. What is the prime factorization
of 350?
A. 5 • 6
A. 2 • 52 • 7
B. 2 • 33
B. 22 • 3 • 9
C. 3 • 4 • 5
C. 2 • 5 • 35
D. 22 • 3 • 5
D. 5 • 7 • 10
3. Tristan handed out markers and
crayons to the class. He shared
46 markers and 69 crayons
equally. If there are more than
10 but fewer than 35 children
in the class, how many children
are in Tristan’s class?
16
4. What is the least common multiple
(LCM) of 18 and 24?
6. Josie’s computer saves its files
every 6 seconds. The computer
checks for computer viruses every
10 seconds. How many times,
in 60 seconds, does the computer
save and check for viruses at the
same time?
A. 12
A. 00
B. 23
B. 01
C. 28
C. 02
D. 34
D. 30
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1. What is the greatest common
factor (GCF) of 72 and 84?