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Buckle Down Georgia
CRCT 7 Mathematics
Number and Operations
Lesson 1: Number Concepts and Representation
Lesson 2: Comparing and Ordering Rational Numbers
Lesson 3: Computation and Problem Solving
Unit 2
Algebra
Lesson 4: Expressions and Equations
Lesson 5: Relations and Functions
Unit 3
Geometry
Lesson 6: Geometric Constructions
Lesson 7: Geometric Figures
Lesson 8: Transformations and Symmetry
Unit 4
Data Analysis
Lesson 9: Data Analysis
Georgia
Georgia CRCT
Unit 1
Mathematics
The cover image depicts dice that
feature operational symbols. This
hands-on teaching tool helps students
understand order of operations as
well as probability.
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Iowa City, Iowa 52244-2180
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www.BuckleDown.com
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Catalog # 2066.GA
7 MATHEMATICS
Go to www.BuckleDown.com to review our complete line of CRCT materials for Grades 2–8
READING • ELA/WRITING • MATHEMATICS
7
CRCT
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TABLE OF CONTENTS
Introduction ..................................................................................... 1
Testwise StrategiesTM ......................................................... 2
Unit 1 – Number and Operations ................................................ 3
Lesson 1: Number Concepts and Representation ............. 4
GPS: M7N1.a, M7N1.d
Lesson 2: Comparing and Ordering
Rational Numbers......................................... 17
GPS: M7N1.b, M7N1. d
Lesson 3: Computation and Problem Solving ................. 31
GPS: M7N1.c, M7N1.d, M7P1.a, M7P1.b, M7P1.c, M7P1.d
Concepts/Skills to Maintain: Operations with fractions
including mixed numbers; Multiplication and division
of positive rational numbers
Unit 2 – Algebra ............................................................................. 57
Lesson 4: Expressions and Equations ............................. 58
GPS: M7A1.a, M7A1.b, M7A1.c, M7A1.d, M7A2.a, M7A2.b,
M7A2.c, M7A2.d
Lesson 5: Relations and Functions .................................. 75
GPS: M7A3.a, M7A3.b, M7A3.c, M7A3.d, M7A3.e
Unit 3 – Geometry ......................................................................... 97
Lesson 6: Geometric Constructions ................................. 98
GPS: M7G1.a
Lesson 7: Geometric Figures .......................................... 113
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GPS: M7G2.a, M7G2.b
Concepts/Skills to Maintain: Circumference of circle;
surface area and volume
Lesson 8: Transformations and Symmetry ................... 131
GPS: M7G1.b, M7G1.c
Unit 4 – Data Analysis ................................................................ 147
Lesson 9: Data Analysis ................................................. 148
GPS: M7D1.a, M7D1.b, M7D1.c, M7D1.d, M7D1.e
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Unit 1 – Number and Operations
GPS: M7N1
Lesson 1: Number Concepts and
Representation
In this lesson, you will review the different types of rational numbers. You will find
the absolute value of a number. Finally, you will maintain your skills at working
with exponents, multiples, factors, prime numbers, composite numbers, and prime
factorization.
Rational Numbers
Rational numbers can be expressed in fractional form, a, where a (the
b
numerator) and b (the denominator) are both integers and b 0. In decimal form,
the rational numbers are either terminating or repeating decimals. The following
definitions will help you understand the rational numbers.
Natural numbers are the positive numbers.
{1, 2, 3, 4, 5, 6, . . .}
Whole numbers consist of the natural numbers and zero.
{0, 1, 2, 3, 4, 5, . . .}
Integers consist of the natural numbers, their opposites, and zero.
Nonintegers consist of fractions that can be written as terminating or repeating
decimals. A terminating decimal comes to a complete stop, whereas a repeating
decimal continues the same digit or block of digits forever. Here are some examples
of nonintegers.
3
7
3
4
11
5
5.2
9.261
0.6
The following tree diagram shows the breakdown of the rational numbers.
Rational
Integers
Whole
Natural
4
Nonintegers
Negative
Integers
0
Terminating
Decimals
Repeating
Decimals
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{. . . , 3, 2, 1, 0, 1, 2, 3, . . .}
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Lesson 1 – Number Concepts and Representation
GPS: M7N1
Practice
Directions: For Numbers 1 through 8, write whether each statement
is true or false.
1. 0 is a natural number. ____________
2. 8 is a whole number. ____________
3. The decimal number 0.34398. . . neither terminates nor repeats, so it is not a
rational number.
___________
4. 57 is both a rational number and a noninteger. ____________
8
5. 0 is an integer. __________
6. Some terminating decimals are integers. ____________
7. All repeating decimals are nonintegers. ____________
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8. If you take away all of the natural numbers from the integers, only negative
numbers are left.
____________
9. Which of the following is a natural
number?
10. Which of the following is not
an integer?
A. 1
A. 1
B. 0
B. 0
C.
1
2
D. 1
C.
1
2
D. 1
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Unit 1 – Number and Operations
GPS: M7N1.a
Absolute Value
The absolute value of a number is that number’s distance from 0 on a number
line. When you write the absolute value of a number n, use the notation n.
Example
The absolute value of 7 7 7
The absolute value of 7 7 7
7 units
–10 –9 –8
7 units
–7 –6 –5 –4 –3 –2 –1
0
1
2
3
4
5
6
7
8
9 10
The absolute value of every number will be either positive or 0. Negative
signs on the outside of absolute value signs act as factors of 1. You need to
evaluate what’s inside the first, then multiply by 1.
50
1(50) 50
Practice
1. 2 __________
3. 75 __________
2. 26 __________
4. 31 __________
Directions: For Numbers 5 and 6, represent each distance or depth using absolute
value signs. Then determine each value and answer the question.
5. Tifton is 21 miles southeast of Sylvester. Albany is 19 miles in the exact
opposite direction from Sylvester. Which city is farther from Sylvester?
6. The lowest point in the United States, Death Valley, is at 282 feet. If sea level
represents 0 feet, how far is Death Valley from sea level?
6
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Directions: For Numbers 1 through 4, determine each value.
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Lesson 1 – Number Concepts and Representation
GPS: M7N1.d
Exponents
An exponent shows how many times to multiply a base number by itself.
When working with exponents, remember the following rules:
1. A base number with an exponent of 1 equals the same number.
2. Any base number (except zero) with zero as the exponent equals 1.
(00 is undefined.)
3. Any base number with a negative exponent is written as its
reciprocal with a positive exponent.
The following model shows how to find the value of 25.
exponent
‘
25 2 • 2 • 2 • 2 • 2 32
“
base
Examples
320 1
(3)2 (3) • (3) 9
7491 749
62 54 5 • 5 • 5 • 5 625
(1)3 (1) • (1) • (1) 1
1
62
1
6•6
1
36
Practice
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Directions: For Numbers 1 through 8, find the value of each expression.
1. 102 __________
5. 170 __________
2. (4)3 __________
6. (12)2 __________
3. 93 __________
7. (2)3 __________
4. 73 __________
8. 848,0001 __________
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Unit 1 – Number and Operations
GPS: M7N1.d
Multiples and Factors
Multiples of a number are the products that result from multiplying a given
number by the whole numbers (0, 1, 2, 3, 4, and so on).
Example
What are the multiples of 5?
Multiply 5 by the whole numbers: 0, 5, 10, 15, 20, . . .
A number that is a multiple of two or more numbers is a common multiple.
Example
The multiples of 2 are 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, . . .
The multiples of 5 are 0, 5, 10, 15, 20, 25, 30, 35, . . .
The numbers 10 and 20 are multiples of both 2 and 5. Therefore, 10 and
20 are common multiples of 2 and 5. (Zero is not considered.)
The smallest of the common multiples is called the least common
multiple (LCM). The least common multiple of 2 and 5 is 10.
Factors of a number will divide that number exactly (without remainder).
Examples
16 1 16
16 2 8
16 4 4
16 8 2
16 16 1
The factors of 16 are 1, 2, 4, 8,
and 16.
What numbers divide 20 exactly?
20 1 20
20 2 10
20 4 5
20 5 4
20 10 2
20 20 1
The factors of 20 are 1, 2, 4, 5, 10,
and 20.
Common factors of 16 and 20 are 1, 2, and 4. Because 4 is the largest
common factor, it is the greatest common factor (GCF) of 16 and 20.
8
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What numbers divide 16
exactly?
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Lesson 1 – Number Concepts and Representation
GPS: M7N1.d
Practice
Directions: For Numbers 1 through 4, fill in the missing multiples.
1. Multiples of 6: 6, ________, 18, ________, ________, . . .
2. Multiples of 8: 8, 16, 24, ________, 40, ________, 56, ________, 72, . . .
3. Multiples of 9: 9, 18, ________, 36, 45, ________, 63, ________, . . .
4. Multiples of 15: 15, ________, ________, 60, 75, 90, ________, 120, 135, . . .
5. What is the LCM of 6 and 8? ________
6. Dakota has art class every 4 school days and music class every 6 school days.
If he has both classes today, how many school days will pass before he takes
both classes again on the same day?
_______________
Directions: For Numbers 7 through 10, list all the factors.
7. Factors of 9: ______________________________
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8. Factors of 27: ______________________________
9. Factors of 28: ______________________________
10. Factors of 36: ______________________________
11. What is the GCF of 9 and 27? ________
12. The organizers of a school concert set up chairs in two sections. The orchestra
section has 84 chairs, and the audience section has 312 chairs. If all rows have
the same number of chairs, what is the greatest number of chairs that each
row can have?
__________
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Unit 1 – Number and Operations
GPS: M7N1.d
Prime and Composite Numbers
A prime number has only two factors: 1 and the number itself. A composite
number has at least three factors.
Example
2, 3, and 5 have only two factors. They are prime numbers.
Factors of 2: 1 and 2
3: 1 and 3
5: 1 and 5
4 has three factors. It is a composite number.
Factors of 4: 1, 2, and 4
Practice
1. Is 6 a prime number or a composite number? ___________________
2. Is 7 a prime number or a composite number? ___________________
3. Is 9 a prime number or a composite number? ___________________
4. List all the prime numbers between 10 and 20.
6. Which is a prime number?
7. Which is a composite number?
A. 23
A. 29
B. 33
B. 43
C. 49
C. 73
D. 51
D. 93
TIP: 0 and 1 are neither prime nor composite. The only even prime number is 2.
10
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5. List all the composite numbers between 10 and 20.
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Lesson 1 – Number Concepts and Representation
GPS: M7N1.d
Prime Factorization
Prime factorization is a way to express a composite number as a product
of prime numbers. Factor trees help you determine the prime factorization
of composite numbers.
Example
What is the prime factorization of 24?
Write the number 24. Put a prime factor under the left branch and circle
it. Put its nonprime factor pair under the right branch. Repeat the process
until you have circled two prime numbers at the bottom of the tree.
24
12
2
6
2
2
3
The prime factorization of 24 is 2 • 2 • 2 • 3 or 23 • 3.
Practice
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1. Draw a factor tree for 45.
The prime factorization of 45 is _____________________________.
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Unit 1 – Number and Operations
GPS: M7N1.d
Prime factorization can be used to find the LCM and the GCF of two or more
numbers.
Example
Use prime factorization to find the LCM of 75 and 90.
Step 1: Find the prime factorizations of 75 and 90.
75 3 • 52
90 2 • 32 • 5
Step 2: Circle the highest power of all the prime factors of both numbers.
75 3 • 52
90 2 • 32 • 5
Step 3: Multiply the highest power of all the prime factors from Step 2.
2 • 32 • 52 2 • 3 • 3 • 5 • 5 450
The LCM of 75 and 90 is 450.
Example
Use prime factorization to find the GCF of 96 and 180.
Step 1: Find the prime factorization of 96 and 180.
96 25 • 3
180 22 • 32 • 5
Step 2: Circle the lowest power of the common prime factors of both
numbers.
180 22 • 32 • 5
Step 3: Multiply the lowest power of the common prime factors from
Step 2.
22 • 3 2 • 2 • 3 12
The GCF of 96 and 180 is 12.
12
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96 25 • 3
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Lesson 1 – Number Concepts and Representation
GPS: M7N1.d
Practice
1. Draw a factor tree for 54.
The prime factorization of 54 is ______________________________.
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2. Draw a factor tree for 420.
The prime factorization of 420 is ______________________________.
3. What is the LCM of 54 and 420? ____________
4. What is the GCF of 54 and 420? ____________
13
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Unit 1 – Number and Operations
GPS: M7N1.d
5. Draw a factor tree for 225.
The prime factorization of 225 is ______________________________.
The prime factorization of 600 is ______________________________.
7. What is the LCM of 225 and 600? ____________
8. What is the GCF of 225 and 600? ____________
14
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6. Draw a factor tree for 600.
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Lesson 1 – CRCT Practice
CRCT Practice
1. What is the value of (2)4?
A.
16
B.
8
C. 8
D. 16
2. What is the GCF of 42 and 54?
A. 2
5. List all of the common factors of
28 and 42.
A. 1, 2, 7
B. 2, 4, 7, 14
C. 1, 2, 7, 14
D. 1, 2, 4, 6, 12
6. Which of the following is NOT a
whole number?
B. 3
A. 2
C. 6
B. 1
D. 7
C. 0
D.
1
3. What is the value of 26?
1
A. 64
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B. 64
C.
64
D.
7. Which statement is true?
A. 7 7
B. 28 28
C.
56
56
D.
4
4
1
64
4. What is the prime factorization
of 256?
A. 24
B. 26
C. 28
D. 212
15
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Unit 1 – Number and Operations
Directions: Use the following number
line to answer Numbers 8 through 10.
1
G
H
11. Which of the following is a natural
number?
A. 0
0 1
J
K
1
B. 2
8. What is the value of G?
C. 0.75
A. 6
B. 5
C.
5
D.
6
D. 1
12. What is the value of 112?
A. 22
9. What is the absolute value of H?
A. 3
B. 2
C.
2
D.
3
10. What is the absolute value of K?
A. 7
B. 111
C. 121
D. 222
13. What is the LCM of 4 and 12?
A. 4
B. 12
C. 16
D. 24
16
C.
6
D.
7
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B. 6