Academic Excellence in Mathematics
A Comprehensive Guide on Math Fundamentals
Module 7 – Module 9
Student Review Packet
Summer 2012 (Sessions 1 – 4)
L. E. A. D
“Mastering the Fundamentals”
Chris Millett
Copyright © 2011
All rights reserved. Written permission must be secured from the author to use or reproduce any part of
this book.
Academic Excellence in Mathematics ® is a registered trademark of Chris Millett.
Module 7: Number Properties and Principles
2
Section 7.1 Number Properties
Common Number Properties
 Commutative Property of Addition
 Commutative Property of Multiplication
 Associative Property of Addition
 Associative Property of Multiplication
 Distributive Property
 Identity Property of Addition
 Identity Property of Multiplication
 Property of Zero
Commutative Property of Addition
 A+B=B+A
 When you add two values, the order in which you add the values does not matter
 8 + 5 = 13
 5 + 8 = 13
 This principle also works with negative numbers
o -8 + 5 = -3
o 5 + (-8) = -3
Commutative Property of Multiplication
 AxB=BxA
 When you multiply two values, the order in which you multiply the values does not matter
 8 x 5 = 40
 5 x 8 = 40
 This principle also works with negative numbers
o (-8) x 5 = -40
o 5 x (-8) = -40
Associative Property of Addition
 (A + B) + C = A + (B + C)
 When you add three values, the order in which you add the values does not matter
 (8 + 5) + 2 = 13 + 2 = 15
 8 + (5 + 2) = 8 + 7 = 15
 This principle also works with negative numbers
o (8 + 5) + (-2) = 13 + (-2) = 11
o 8 + (5 + (-2)) = 8 + 3 = 11
3
Section 7.1 Number Properties (continued)
Associative Property of Multiplication
 (A x B) x C = A x (B x C)
 When you multiply three values, the order in which you multiply the values does not matter
 (8 x 5) x 2 = 40 x 2 = 80
 8 x (5 x 2) = 8 x 10 = 80
 This principle also works with negative numbers
o (8 x 5) x (-2) = 40 x (-2) = -80
o 8 x (5 x (-2)) = 8 x (-10) = -80
Distributive Property
 A x (B + C) = (A x B) + (A x C)
 When you multiply a value by the sum of two numbers, you can multiply the value by each number
first and then add after you multiply
 8 x (5 + 2) = 8 x 7 = 56
 (8 x 5) + (8 x 2) = 40 + 16 = 56
 This principle also works with negative numbers
o -8 x (5 + 2) = -8 x 7 = -56
o (-8 x 5) + (-8 x 2) = -40 + (-16) = -56
Identity Property of Addition
 A+0=A
 Any number plus zero is the original number
 8+0=8
 -8 + 0 = -8
Identity Property of Multiplication
 Ax1=A
 Any number times one is the original number
 8x1=8
 -8 x 1 = -8
Property of Zero
 Ax0=0
 Any number times zero is always zero
 8x0=0
 -8 x 0 = 0
4
Number Properties – Guided Practice
_______ 1. Using the Commutative Property of Addition, how can you rewrite 4 + 25?
_______ 2. Using the Commutative Property of Addition, how can you rewrite 23 + 14?
_______ 3. Using the Commutative Property of Multiplication, how can you rewrite 4 x 25?
_______ 4. Using the Commutative Property of Multiplication, how can you rewrite 12 x 9?
_______ 5. Using the Associative Property of Addition, how can you rewrite (8 + 7) + 6?
_______ 6. Using the Associative Property of Addition, how can you rewrite 9 + (7 + 6)?
_______ 7. Using the Associative Property of Multiplication, how can you rewrite (8 x 7) x 6?
_______ 8. Using the Associative Property of Multiplication, how can you rewrite 4 x (5 x 6)?
_______ 9. Using the Distributive Property, how can you rewrite (5 x 6) + (5 x 9)?
_______ 10. Using the Distributive Property, how can you rewrite 9 x (8 + 2)?
_______ 11. Using the Identity Property of Addition, how can you rewrite 45 + 0?
_______ 12. Using the Identity Property of Multiplication, how can you rewrite 45 x 1?
_______ 13. Using the Property of Zero, how can you rewrite 45 x 0?
5
Number Properties – Guided Practice (continued)
Name the Property
_________________________________ 1. 6 x (4 + 3) = 6 x 4 + 6 x 3
_________________________________ 2. 8 x 0 = 0
_________________________________ 3. 4 + 5 = 5 + 4
_________________________________ 4. 18 x 1 = 18
_________________________________ 5. (8 x 9) x 4 = 8 x (9 x 4)
_________________________________ 6. (6 + 8) + 9 = 6 + (8 + 9)
_________________________________ 7. 25 x 0 = 0
_________________________________ 8. 6 x 7 = 7 x 6
_________________________________ 9. 37 + 0 = 37
_________________________________ 10. 4 x 5 + 4 x 7 = 4 x (5 + 7)
_________________________________ 11. (4 x 9) x 10 = 4 x (9 x 10)
_________________________________ 12. (4 + 9) + 10 = 4 + (9 + 10)
6
Section 7.2 Multiplication – The Foundation to Academic Excellence in Mathematics
The Importance of Mastering Multiplication
 Most math principles require you to have solid multiplication skills
 In order to achieve “Academic Excellence in Mathematics”, you must be able to perform:
 Level 1 Multiplication (single-digit by single-digit)
 Level 2 Multiplication (single-digit by double-digit and double-digit by single-digit)
 Level 3 Multiplication (double-digit by double-digit)
Level 1 (Single-Digit by Single-Digit Multiplication)
 Before you proceed any further in mathematics, memorize these facts
1
2
3
4
5
6
7
8
9
1
1
2
3
4
5
6
7
8
9
2
2
4
6
8
10
12
14
16
18
3
3
6
9
12
15
18
21
24
27
4
4
8
12
16
20
24
28
32
36
5
5
10
15
20
25
30
35
40
45
6
6
12
18
24
30
36
42
48
54
7
7
14
21
28
35
42
49
56
63
8
8
16
24
32
40
48
56
64
72
9
9
18
27
36
45
54
63
72
81
 If you do not learn all of these multiplication facts, you are on pace to struggle in mathematics during
middle school, high school, college, and beyond
Level 2 (Single-Digit by Double-Digit and Double-Digit by Single-Digit Multiplication)
 Uses the following two math fundamentals
 Level 1 Multiplication
 The Distributive property
 Combining these two fundamentals, you can perform Level 2 Multiplication in your head (in a matter
of seconds)
 Steps to Level 2 Multiplication
 Convert the double-digit number into a sum, where one value is a tens-value
o 11  10 + 1
o 16  10 + 6
o 23  20 + 3
o 27  20 + 7
o 38  30 + 8
 Use the Distributive Property to multiply the single-digit number by the sum of the converted
double-digit number
o 12 x 8  (10 + 2) x 8  (10 x 8) + (2 x 8)  80 + 16 = 96
o 15 x 9  (10 + 5) x 9  (10 x 9) + (5 x 9)  90 + 45 = 135
o 23 x 7  (20 + 3) x 7  (20 x 7) + (3 x 7)  140 + 21 = 161
o 38 x 6  (30 + 8) x 6  (30 x 6) + (8 x 6)  180 + 48 = 228
7
Section 7.2 Multiplication – The Foundation to Academic Excellence in Mathematics
(continued)
Level 3 (Double-Digit by Double-Digit Multiplication)
 Uses the following two math fundamentals
 Level 2 Multiplication
 The Distributive property
 Combining these two fundamentals, you can perform Level 3 Multiplication in your head (in a matter
of seconds)
 This is more difficult than Level 2 Multiplication, because the numbers are much larger
 You must regularly practice Level 3 Multiplication to master it
 Steps to Level 3 Multiplication
 Convert the smaller double-digit number into a sum, where one value is a tens-value
o 11  10 + 1
o 16  10 + 6
o 23  20 + 3
o 27  20 + 7
o 38  30 + 8
 Use the Distributive Property to multiply the larger double-digit number by the sum of the
converted double-digit numbers
o 12 x 13  (10 + 2) x 13  (10 x 13) + (2 x 13)  130 + 26 = 156
o 15 x 17  (10 + 5) x 17  (10 x 17) + (5 x 17)  170 + 85 = 255
o 23 x 14  23 x (10 + 4)  (10 x 23) + (4 x 23)  230 + 92 = 322
o 24 x 23  24 x (20 + 3) (20 x 24) + (3 x 24)  480 + 72 = 552
 The “Steps to Level 3 Multiplication” work every single time, regardless of the values of the doubledigit numbers
8
Multiplication – Guided Practice
Solve the following multiplication problems without using a calculator.
__________ 1. What is 13 x 7?
(_____ + _____) x _____
(_____ x _____) + (_____ x _____)
_____ + _____
__________ 2. What is 27 x 8?
(_____ + _____) x _____
(_____ x _____) + (_____ x _____)
_____ + _____
__________ 3. What is 53 x 9?
(_____ + _____) x _____
(_____ x _____) + (_____ x _____)
_____ + _____
__________ 4. What is 86 x 8?
(_____ + _____) x _____
(_____ x _____) + (_____ x _____)
_____ + _____
9
Multiplication – Guided Reinforcement (continued)
__________ 5. What is 15 x 17?
(_____ + _____) x _____
(_____ x _____) + (_____ x _____)
_____ + _____
__________ 6. What is 24 x 13?
(_____ + _____) x _____
(_____ x _____) + (_____ x _____)
_____ + _____
__________ 7. What is 37 x 12?
(_____ + _____) x _____
(_____ x _____) + (_____ x _____)
_____ + _____
__________ 8. What is 62 x 14?
(_____ + _____) x _____
(_____ x _____) + (_____ x _____)
_____ + _____
10
Section 7.3 Multiplication Facts (1 – 100)
Importance of Multiplication Facts
 Mastering Multiplication Facts ( 1 – 100) will help you perform well in the following areas
 Algebra
 Geometry
 Trigonometry
 Statistics and Probability
 Standardized Tests
o CRCT
o ACT
o PSAT
o SAT
o Graduation Exams (Example: Georgia Graduation Exam in Mathematics)
 What Multiplication Facts should you master?
 Every product from 1 – 100 (besides 1 and the number)
o 4=2x2
o 6=2x3
o 8=2x4
o 9=3x3
o 10 = 2 x 5
o 12 = 2 x 6 and 3 x 4
o 14 = 2 x 7
o 15 = 3 x 5
o 16 = 2 x 8 and 4 x 4
o 18 = 2 x 9 and 3 x 6
o 20 = 2 x 10 and 4 x 5
o 21 = 3 x 7
o 22 = 2 x 11
o 24 = 2 x 12, 3 x 8, and 4 x 6
o 25 = 5 x 5
o 26 = 2 x 13
o 27 = 3 x 9
o 28 = 2 x 14 and 4 x 7
o 30 = 2 x 15, 3 x 10, and 5 x 6
o 32 = 2 x 16 and 4 x 8
o 33 = 3 x 11
o 34 = 2 x 17
o 35 = 5 x 7
o 36 = 2 x 18, 3 x 12, 4 x 9, and 6 x 6
o 38 = 2 x 19
o 39 = 3 x 13
o 40 = 2 x 20, 4 x 10, and 5 x 8
o 42 = 2 x 21, 3 x 14, and 6 x 7
o 44 = 2 x 22 and 4 x 11
o 45 = 3 x 15 and 5 x 9
o 46 = 2 x 23
11
Section 7.3 Multiplication Facts (1 – 100) – continued
 What Multiplication Facts should you master? (continued)
 Examples (continued)
o 48 = 2 x 24, 3 x 16, 4 x 12, and 6 x 8
o 49 = 7 x 7
o 50 = 2 x 25 and 5 x 10
o 52 = 2 x 26 and 4 x 13
o 54 = 2 x 27, 3 x 18, and 6 x 9
o 55 = 5 x 11
o 56 = 2 x 28, 4 x 14, and 7 x 8
o 57 = 3 x 19
o 58 = 2 x 29
o 60 = 2 x 30, 3 x 20, 4 x 15, 5 x 12, and 6 x 10
o 62 = 2 x 31
o 63 = 3 x 21 and 7 x 9
o 64 = 2 x 32, 4 x 16, and 8 x 8
o 65 = 5 x 13
o 66 = 2 x 33, 3 x 22, and 6 x 11
o 68 = 2 x 34 and 4 x 17
o 69 = 3 x 23
o 70 = 2 x 35, 5 x 14, and 7 x 10
o 72 = 2 x 36, 3 x 24, 4 x 18, 6 x 12, and 8 x 9
o 74 = 2 x 37
o 75 = 3 x 25 and 5 x 15
o 76 = 2 x 38 and 4 x 19
o 77 = 7 x 11
o 78 = 2 x 39 and 6 x 13
o 80 = 2 x 40, 4 x 20, 5 x 16, and 8 x 10
o 81 = 3 x 27 and 9 x 9
o 82 = 2 x 41
o 84 = 2 x 42, 3 x 28, 4 x 21, 6 x 14, and 7 x 12
o 85 = 5 x 17
o 86 = 2 x 43
o 87 = 3 x 29
o 88 = 2 x 44, 4 x 22, and 8 x 11
o 90 = 2 x 45, 3 x 30, 5 x 18, 6 x 15, and 9 x 10
o 91 = 7 x 13
o 92 = 2 x 46 and 4 x 23
o 93 = 3 x 31
o 94 = 2 x 47
o 95 = 5 x 19
o 96 = 2 x 48, 3 x 32, 4 x 24, 6 x 16, and 8 x 12
o 98 = 2 x 49, and 7 x 14
o 99 = 3 x 33 and 9 x 11
o 100 = 2 x 50, 4 x 25, 5 x 20, and 10 x 10
12
Multiplication Facts – Guided Practice
Write the Multiplication Fact Specified (Without Using the Previous Pages)
* Do not use 1 or the number specified
1) Two multiplication combinations for 12  _______________________________________
2) Two multiplication combinations for 16  _______________________________________
3) Two multiplication combinations for 18  _______________________________________
4) Two multiplication combinations for 20  _______________________________________
5) One multiplication combinations for 21  ________________________________________
6) Three multiplication combinations for 24  ______________________________________
7) Two multiplication combinations for 28  ______________________________________
8) Four multiplication combinations for 36  ______________________________________
9) Three multiplication combinations for 40  _____________________________________
10) Four multiplication combinations for 48  _____________________________________
11) Three multiplication combinations for 54  _____________________________________
12) Five multiplication combinations for 60  _____________________________________
13) Five multiplication combinations for 72  _____________________________________
14) Five multiplication combinations for 84  _____________________________________
15) Three multiplication combinations for 88  ____________________________________
16) Five multiplication combinations for 90  _____________________________________
17) Two multiplication combinations for 92  _____________________________________
18) Five multiplication combinations for 96  _____________________________________
19) Two multiplication combinations for 99  _____________________________________
20) Three multiplication combinations for 100  ____________________________________
13
Section 7.4 Multiplying With Multiples of 10
Master Multiplication with Multiples of 10
 When performing multiplication with a multiple of 10
 Temporarily ignore the zero on the number that is a multiple of 10
 Multiply the number as if the zero is not present
 After the multiplication is complete, just add the zero to the answer
 Examples
 23 x 20
o Temporarily ignore the 0 on 20 and just treat is as if it is 2
o Multiply 23 x 2  46
o Now add the zero to the answer  460
o So the final answer for 23 x 20 is 460
 14 x 30
o Temporarily ignore the 0 on 30 and just treat is as if it is 3
o Multiply 14 x 3  42
o Now add the zero to the answer  420
o So the final answer for 14 x 30 is 420
 16 x 40
o Temporarily ignore the 0 on 40 and just treat is as if it is 4
o Multiply 16 x 4  640
o Now add the zero to the answer  640
o So the final answer for 16 x 40 is 640
 15 x 80
o Temporarily ignore the 0 on 80 and just treat is as if it is 8
o Multiply 15 x 8  120
o Now add the zero to the answer  1200
o So the final answer for 15 x 80 is 1,200
 22 x 90
o Temporarily ignore the 0 on 90 and just treat is as if it is 9
o Multiply 22 x 9  198
o Now add the zero to the answer  1980
o So the final answer for 22 x 90 is 1,980
14
Multiplying With Multiples of 10 – Guided Practice
Solve the Following Multiples of 10 Multiplication Problems
1. 11 x 20 = _____
2. 12 x 30 = _____
3. 13 x 40 = _____
4. 14 x 50 = _____
5. 13 x 60 = _____
6. 12 x 70 = _____
7. 11 x 80 = _____
8. 7 x 90 = _____
9. 21 x 40 = _____
10. 23 x 50 = _____
11. 22 x 30 = _____ 12. 31 x 40 = _____
13. 35 x 40 = _____
14. 45 x 50 = _____
15. 55 x 60 = _____ 16. 12 x 80 = _____
17. 34 x 20 = _____
18. 42 x 30 = _____
19. 81 x 40 = _____ 20. 95 x 40 = _____
21. 63 x 20 = _____
22. 71 x 30 = _____
12. 82 x 40 = _____ 24. 91 x 80 = _____
15
Section 7.5 Multiplying With Multiples of 100
Master Multiplication with Multiples of 100
 When performing multiplication with a multiple of 100
 Temporarily ignore the zero on the number that is a multiple of 100
 Multiply the number as if the zeros are not present
 After the multiplication is complete, just add two zeros to the answer
 Examples
 23 x 200
o Temporarily ignore the two 0’s on 200 and just treat is as if it is 2
o Multiply 23 x 2  46
o Now add the two zeros to the answer  4600
o So the final answer for 23 x 200 is 4,600
 14 x 300
o Temporarily ignore the two 0’s on 300 and just treat is as if it is 3
o Multiply 14 x 3  42
o Now add the zero to the answer  4200
o So the final answer for 14 x 300 is 4,200
 16 x 4000
o Temporarily ignore the two 0’s on 400 and just treat is as if it is 4
o Multiply 16 x 4  640
o Now add the zero to the answer  6400
o So the final answer for 16 x 400 is 6,400
 15 x 800
o Temporarily ignore the two 0’s on 800 and just treat is as if it is 8
o Multiply 15 x 8  120
o Now add the zero to the answer  12000
o So the final answer for 15 x 800 is 12,000
 22 x 900
o Temporarily ignore the two 0’s on 900 and just treat is as if it is 9
o Multiply 22 x 9  198
o Now add the zero to the answer  19800
o So the final answer for 22 x 900 is 19,800
16
Multiplying With Multiples of 100 – Guided Practice
Solve the Following Multiples of 100 Multiplication Problems
1. 11 x 200 = _____
2. 12 x 300 = _____
3. 13 x 400 = _____ 4. 14 x 500 = _____
5. 13 x 600 = _____
6. 12 x 700 = _____
7. 11 x 800 = _____ 8. 7 x 900 = _____
9. 21 x 400 = _____
10. 23 x 500 = _____
11. 22 x 300 = _____ 12. 31 x 400 = _____
13. 35 x 400 = _____
14. 45 x 500 = _____
15. 55 x 600 = _____ 16. 12 x 800 = _____
17. 34 x 200 = _____
18. 42 x 300 = _____
19. 81 x 400 = _____ 20. 95 x 400 = _____
21. 63 x 200 = _____
22. 71 x 300 = _____
12. 82 x 400 = _____ 24. 91 x 800 = _____
17
Section 7.6 Multiplying With Multiples of 1000
Master Multiplication with Multiples of 1000
 When performing multiplication with a multiple of 1000
 Temporarily ignore the zero on the number that is a multiple of 1000
 Multiply the number as if the zeros are not present
 After the multiplication is complete, just add three zeros to the answer
 Examples
 23 x 2000
o Temporarily ignore the three 0’s on 2000 and just treat is as if it is 2
o Multiply 23 x 2  46
o Now add the two zeros to the answer  46000
o So the final answer for 23 x 2000 is 46,000
 14 x 3000
o Temporarily ignore the three 0’s on 3000 and just treat is as if it is 3
o Multiply 14 x 3  42
o Now add the zero to the answer  42000
o So the final answer for 14 x 3000 is 42,000
 16 x 4000
o Temporarily ignore the three 0’s on 4000 and just treat is as if it is 4
o Multiply 16 x 4  640
o Now add the zero to the answer  64000
o So the final answer for 16 x 4000 is 64,000
 15 x 8000
o Temporarily ignore the three 0’s on 8000 and just treat is as if it is 8
o Multiply 15 x 8  120
o Now add the zero to the answer  120000
o So the final answer for 15 x 800 is 120,000
 22 x 900,
o Temporarily ignore the three 0’s on 9000 and just treat is as if it is 9
o Multiply 22 x 9  198
o Now add the zero to the answer  198000
o So the final answer for 22 x 9000 is 198,000
18
Multiplying With Multiples of 1000 – Guided Practice
Solve the Following Multiples of 1000 Multiplication Problems
1. 11 x 2000 = _____
2. 12 x 3000 = _____
3. 13 x 4000 = _____
4. 14 x 5000 = _____
5. 13 x 6000 = _____
6. 12 x 7000 = _____
7. 11 x 8000 = _____
8. 7 x 9000 = _____
9. 21 x 4000 = _____
10. 23 x 5000 = _____
11. 22 x 3000 = _____ 12. 31 x 4000 = _____
13. 35 x 4000 = _____
14. 45 x 5000 = _____
15. 55 x 6000 = _____
16. 12 x 8000 = _____
17. 34 x 2000 = _____
18. 42 x 3000 = _____
19. 81 x 4000 = _____
20. 95 x 4000 = _____
21. 63 x 2000 = _____
22. 71 x 3000 = _____
23. 82 x 4000 = _____
24. 91 x 8000 = _____
19
Section 7.7 Division Facts (1 – 100)
Importance of Division Facts
 Similar to mastering multiplication facts ( 1 – 100), mastering division facts (1 – 100) will help you
perform well in the following areas
 Algebra
 Geometry
 Statistics and Probability
 Standardized Tests
o CRCT
o ACT
o PSAT
o SAT
o Graduation Exams (Example: Georgia Graduation Exam in Mathematics)
 What Division Facts should you master?
 Every quotient from 1 – 100 (besides 1 and the number)
 Examples
o 4÷2=2
o 6÷2=3
o 6÷3=2
o 8÷2=4
o 8÷4=2
o 9÷3=3
o 10 ÷ 2 = 5
o 10 ÷ 5 = 2
o 12 ÷ 2 = 6
o 12 ÷ 3 = 4
o 12 ÷ 4 = 3
o 12 ÷ 6 = 2
o 14 ÷ 2 = 7
o 14 ÷ 7 = 2
o 15 ÷ 3 = 5
o 15 ÷ 5 = 3
o 16 ÷ 2 = 8
o 16 ÷ 4 = 4
o 16 ÷ 8 = 2
o 18 ÷ 2 = 9
o 18 ÷ 3 = 6
o 18 ÷ 6 = 3
o 18 ÷ 9 = 2
o 20 ÷ 2 = 10
o 20 ÷ 4 = 5
o 20 ÷ 5 = 4
o 20 ÷ 10 = 2
20
Section 7.7 Division Facts (1 – 100) – continued
 What Division Facts should you master? (continued)
 Examples (continued)
o 21 ÷ 3 = 7
o 21 ÷ 7 = 3
o 22 = 2 = 11
o 22 ÷ 11 = 2
o 24 ÷ 2 = 12
o 24 ÷ 3 = 8
o 24 ÷ 4 = 6
o 24 ÷ 6 = 4
o 24 ÷ 8 = 3
o 24 ÷ 12 = 2
o 25 ÷ 5 = 5
o 26 ÷ 2 = 13
o 26 ÷ 13 = 2
o 27 ÷ 3 = 9
o 27 ÷ 9 = 3
o 28 ÷ 2 = 14
o 28 ÷ 4 = 7
o 28 ÷ 7 = 4
o 28 ÷ 14 = 2
o 30 ÷ 2 = 15
o 30 ÷ 3 = 10
o 30 ÷ 5 = 6
o 30 ÷ 6 = 5
o 30 ÷ 10 = 3
o 30 ÷ 15 = 2
o 32 ÷ 2 = 16
o 32 ÷ 4 = 8
o 32 ÷ 8 = 4
o 32 ÷ 16 = 2
o 33 ÷ 3 = 11
o 33 ÷ 11 = 3
o 34 ÷ 2 = 17
o 34 ÷ 17 = 2
o 35 ÷ 5 = 7
o 35 ÷ 7 = 5
o 36 ÷ 2 = 18
o 36 ÷ 3 = 12
o 36 ÷ 4 = 9
o 36 ÷ 6 = 6
o 36 ÷ 9 = 4
o 36 ÷ 12 = 3
o 36 ÷ 18 = 2
21
Section 7.7 Division Facts (1 – 100) – continued
 What Division Facts should you master? (continued)
 Examples (continued)
o 38 ÷ 2 = 19
o 38 ÷ 19 = 2
o 39 ÷ 3 = 13
o 39 ÷ 13 = 3
o 40 ÷ 2 = 20
o 40 ÷ 4 = 10
o 40 ÷ 5 = 8
o 40 ÷ 8 = 5
o 40 ÷ 10 = 4
o 40 ÷ 20 = 2
o 42 ÷ 2 = 21
o 42 ÷ 3 = 14
o 42 ÷ 6 = 7
o 42 ÷ 7 = 6
o 42 ÷ 14 = 3
o 42 ÷ 21 = 2
o 44 ÷ 2 = 22
o 44 ÷ 4 = 11
o 44 ÷ 11 = 4
o 44 ÷ 22 = 2
o 45 ÷ 3 = 15
o 45 ÷ 5 = 9
o 45 ÷ 9 = 5
o 45 ÷ 15 = 3
o 46 ÷ 2 = 23
o 46 ÷ 23 = 2
o 48 ÷ 2 = 24
o 48 ÷ 3 = 16
o 48 ÷ 4 = 12
o 48 ÷ 6 = 8
o 48 ÷ 8 = 6
o 48 ÷ 12 = 4
o 48 ÷ 16 = 3
o 48 ÷ 24 = 2
o 49 ÷ 7 = 7
o 50 ÷ 2 = 25
o 50 ÷ 5 = 10
o 50 ÷ 10 = 5
o 50 ÷ 25 = 2
22
Section 7.7 Division Facts (1 – 100) – continued
 What Division Facts should you master? (continued)
 Examples (continued)
o 52 ÷ 2 = 26
o 52 ÷ 4 = 13
o 52 ÷ 13 = 4
o 52 ÷ 26 = 2
o 54 ÷ 2 = 27
o 54 ÷ 3 = 18
o 54 ÷ 6 = 9
o 54 ÷ 9 = 6
o 54 ÷ 18 = 3
o 54 ÷ 27 = 2
o 55 ÷ 5 = 11
o 55 ÷ 11 = 5
o 56 ÷ 2 = 28
o 56 ÷ 4 = 14
o 56 ÷ 7 = 8
o 56 ÷ 8 = 7
o 56 ÷ 14 = 4
o 56 ÷ 28 = 2
o 57 ÷ 3 = 19
o 57 ÷ 19 = 3
o 58 ÷ 2 = 29
o 58 ÷ 29 = 2
o 60 ÷ 2 = 30
o 60 ÷ 3 = 20
o 60 ÷ 4 = 15
o 60 ÷ 5 = 12
o 60 ÷ 6 = 10
o 60 ÷ 10 = 6
o 60 ÷ 12 = 5
o 60 ÷ 15 = 4
o 60 ÷ 20 = 3
o 60 ÷ 30 = 2
o 62 ÷ 2 = 31
o 62 ÷ 31 = 2
o 63 ÷ 3 = 21
o 63 ÷ 7 = 9
o 63 ÷ 9 = 7
o 63 ÷ 21 = 3
23
Section 7.7 Division Facts (1 – 100) – continued
 What Division Facts should you master? (continued)
 Examples (continued)
o 64 ÷ 2 = 32
o 64 ÷ 4 = 16
o 64 ÷ 8 = 8
o 64 ÷ 16 = 4
o 64 ÷ 32 = 2
o 65 ÷ 5 = 13
o 65 ÷ 13 = 5
o 66 ÷ 2 = 33
o 66 ÷ 3 = 22
o 66 ÷ 6 = 11
o 66 ÷ 11 = 6
o 66 ÷ 22 = 3
o 66 ÷ 33 = 2
o 68 ÷ 2 = 34
o 68 ÷ 4 = 17
o 68 ÷ 17 = 4
o 68 ÷ 34 = 2
o 69 ÷ 3 = 23
o 69 ÷ 23 = 3
o 70 ÷ 2 = 35
o 70 ÷ 5 = 14
o 70 ÷ 7 = 10
o 70 ÷ 10 = 7
o 70 ÷ 14 = 5
o 70 ÷ 35 = 2
o 72 ÷ 2 = 36
o 72 ÷ 3 = 24
o 72 ÷ 4 = 18
o 72 ÷ 6 = 12
o 72 ÷ 8 = 9
o 72 ÷ 9 = 8
o 72 ÷ 12 = 6
o 72 ÷ 18 = 4
o 72 ÷ 24 = 3
o 72 ÷ 36 = 2
o 74 ÷ 2 = 37
o 74 ÷ 37 = 2
o 75 ÷ 3 = 25
o 75 ÷ 5 = 15
o 75 ÷ 15 = 5
o 75 ÷ 25 = 3
24
Section 7.7 Division Facts (1 – 100) – continued
 What Division Facts should you master? (continued)
 Examples (continued)
o 76 ÷ 2 = 38
o 76 ÷ 4 = 19
o 76 ÷19 = 4
o 76 ÷ 38 = 2
o 77 ÷ 7 = 11
o 77 ÷ 11 = 7
o 78 ÷ 2 = 39
o 78 ÷ 6 = 13
o 78 ÷ 13 = 6
o 78 ÷ 39 = 2
o 80 ÷ 2 = 40
o 80 ÷ 4 = 20
o 80 ÷ 5 = 16
o 80 ÷ 8 = 10
o 80 ÷ 16 = 5
o 80 ÷ 20 = 4
o 80 ÷ 40 = 2
o 81 ÷ 3 = 27
o 81 ÷ 9 = 9
o 81 ÷ 27 = 3
o 82 ÷ 2 = 41
o 82 ÷ 41 = 2
o 84 ÷ 2 = 42
o 84 ÷ 3 = 28
o 84 ÷ 4 = 21
o 84 ÷ 6 = 14
o 84 ÷ 7 = 12
o 84 ÷ 12 = 7
o 84 ÷ 14 = 6
o 84 ÷ 21 = 4
o 84 ÷ 28 = 3
o 84 ÷ 42 = 2
o 85 ÷ 5 = 17
o 85 ÷ 17 = 5
o 86 ÷ 2 = 43
o 86 ÷ 43 = 2
o 87 ÷ 3 = 29
o 87 ÷ 29 = 3
25
Section 7.7 Division Facts (1 – 100) – continued
 What Division Facts should you master? (continued)
 Examples (continued)
o 88 ÷ 2 = 44
o 88 ÷ 4 = 22
o 88 ÷ 8 = 11
o 88 ÷ 11 = 8
o 88 ÷ 22 = 4
o 88 ÷ 44 = 2
o 90 ÷ 2 = 45
o 90 ÷ 3 = 30
o 90 ÷ 5 = 18
o 90 ÷ 6 = 15
o 90 ÷ 9 = 10
o 90 ÷ 10 = 9
o 90 ÷ 15 = 6
o 90 ÷ 18 = 5
o 90 ÷ 30 = 3
o 90 ÷ 45 = 2
o 91 ÷ 7 = 13
o 91 ÷ 13 = 7
o 92 ÷ 2 = 46
o 92 ÷ 4 = 23
o 92 ÷ 23 = 4
o 92 ÷ 46 = 2
o 93 ÷ 3 = 31
o 93 ÷ 31 = 3
o 94 ÷ 2 = 47
o 94 ÷ 47 = 2
o 95 ÷ 5 = 19
o 95 ÷ 19 = 5
o 96 ÷ 2 = 48
o 96 ÷ 3 = 32
o 96 ÷ 4 = 24
o 96 ÷ 6 = 16
o 96 ÷ 8 = 12
o 96 ÷ 12 = 8
o 96 ÷ 16 = 6
o 96 ÷ 24 = 4
o 96 ÷ 32 = 3
o 96 ÷ 48 = 2
26
Section 7.7 Division Facts (1 – 100) – continued
 What Division Facts should you master? (continued)
 Examples (continued)
o 98 ÷ 2 = 49
o 98 ÷ 7 = 14
o 98 ÷ 14 = 7
o 98 ÷ 49 = 2
o 99 ÷ 3 = 33
o 99 ÷ 9 = 11
o 99 ÷ 11 = 9
o 99 ÷ 33 = 3
o 100 ÷ 2 = 50
o 100 ÷ 4 = 25
o 100 ÷ 5 = 20
o 100 ÷ 10 = 10
o 100 ÷ 20 = 5
o 100 ÷ 25 = 4
o 100 ÷ 50 = 2
27
Division Facts – Guided Practice
Write the Division Fact Specified (Without Using the Previous Pages)
* Do not use 1 or the number specified (Example: 8  8 ÷ 2 = 4 and 8 ÷ 4 = 2)
1) Two division combinations for 12  _______________________________________
2) Two division combinations for 16  _______________________________________
3) Two division combinations for 18  _______________________________________
4) Two division combinations for 20  _______________________________________
5) One division combinations for 21  ________________________________________
6) Three division combinations for 24  ______________________________________
7) Two division combinations for 28  ______________________________________
8) Four division combinations for 36  ______________________________________
9) Three division combinations for 40  _____________________________________
10) Four division combinations for 48  _____________________________________
11) Three division combinations for 54  _____________________________________
12) Five division combinations for 60  _____________________________________
13) Five division combinations for 72  _____________________________________
14) Five division combinations for 84  _____________________________________
15) Three division combinations for 88  ____________________________________
16) Five division combinations for 90  _____________________________________
17) Two division combinations for 92  _____________________________________
18) Five division combinations for 96  _____________________________________
19) Two division combinations for 99  _____________________________________
20) Three division combinations for 100  ____________________________________
28
Section 7.8 Mastering Divisibility Rules (By 2, 3, 4, 5, 6, 7, 8, 9, 10)
 Divisible By 2
 Every even number is divisible by 2
 No odd number is divisible by 2
 2,754 (an even number) is divisible by 2
 1,983 (an odd number) is not divisible by 2
 Divisible By 3
 Add the individual digits of the number
 If the sum of the digits is divisible by 3, then the original number is divisible by 3
 If the sum of the digits is not divisible by 3, then the original number is not divisible by 3
 258
o Sum the digits of 258  2 + 5 + 8 = 15
o Since the sum of the digits (15) is divisible by 3, the number 258 is divisible by 3
 718
o Sum the digits of 718 7 + 1 + 8 = 16
o Since the sum of the digits (16) is not divisible by 3, the number 718 is not divisible by 3
 Divisible By 4
 Look at the last (rightmost) 2 digits of the number
 If the rightmost 2-digit value is divisible by 4, then the original number is divisible by 4
 If the rightmost 2-digit value is not divisible by 4, then the original number is not divisible by 4
 3,476
o 76 is made from the last (rightmost) 2 digits of 3,476
o Since 76 is divisible by 4 (76 ÷ 4 = 19), then 3,476 is divisible by 4
 1,986
o 86 is made from the last (rightmost) 2 digits of 1,986
o Since 86 is not evenly divisible by 4 (86 ÷ 4 = 21½ ), then 1,986 is not divisible by 4
 Divisible By 5
 Look at the last (rightmost) digit
 If the rightmost digit is 0 or 5, the number is divisible by 5
 If the rightmost digit is not 0 or 5, the number is not divisible by 5
 3,475
o The rightmost digit of 3,475 is 5
o The number 3,475 is therefore divisible by 5
 3,480
o The rightmost digit of 3,480 is 0
o The number 3,480 is therefore divisible by 5
 1,986
o The rightmost digit of 1,986 is not 0 or 5
o The number 1,986 is therefore not divisible by 5
29
Section 7.8 Mastering Divisibility Rules (By 2, 3, 4, 5, 6, 7, 8, 9, 10) – continued
 Divisible By 6
 Every even number that is divisible by 3 is also divisible by 6
 No odd number that is divisible by 3 is divisible by 6
 If the sum of the digits is divisible by 3 and the number is even, then the original number is divisible
by 6
 If the sum of the digits is not divisible by 3 or the number is odd, then the original number is not
divisible by 3
 258
o Since the number is even, sum the digits of 258  2 + 5 + 8 = 15
o Since the sum of the digits (15) is divisible by 3 and 258 is even, the number 258 is divisible by
3
 735
o Although the sum of the digits of 735 (7 + 3 + 5 = 15) is evenly divisible by 3, the number 735
is odd and therefore cannot by divisible by 6
 Divisible By 7
 Double the rightmost digit
 Subtract this product from the value of the remaining digits
 If the difference is divisible by 7, then the original number is divisible by 7
 If the difference is not divisible by 7, then the original number is not divisible by 7
 574
o Double the rightmost digit (4)  4 x 2 = 8
o Subtract 8 from the value of the remaining digits (57)  57 – 8 = 49
o Since 49 is divisible by 7, the value 574 is divisible by 7
 734
o Double the rightmost digit (4)  4 x 2 = 8
o Subtract 8 from the value of the remaining digits (73)  73 – 8 = 65
o Since 65 is not divisible by 7, the value 734 is not divisible by 7
 Divisible By 8
 In order for a number to be divisible by 8, it must by divisible by 4
 Look at the last (rightmost) 2 digits of the number
 If the rightmost 2-digit value is not divisible by 4, then the number cannot by divisible by 8
 If the rightmost 2-digit value is divisible by 4, then perform the following steps
o If the rightmost 2-digit value is divisible by 8 and the 3rd digit from the right is even, then the
number is divisible by 8
o If the rightmost 2-digit value is not divisible by 8, but the 3rd digit from the right is odd, then the
number is divisible by 8
o For any other combination, the number is not divisible by 8
 4,832
o 32 is made from the last (rightmost) 2 digits of 4,832
o 32 is divisible by 4
o Since 32 is divisible by 8 and the 3rd digit from the right (8) is even, then 4,832 is divisible by 8
30
Section 7.8 Mastering Divisibility Rules (By 2, 3, 4, 5, 6, 7, 8, 9, 10) – continued
 Divisible By 8 (continued)
 3,476
o 76 is made from the last (rightmost) 2 digits of 3,476
o 76 is divisible by 4
o Since 76 is not divisible by 8 and the 3rd digit from the right (4) is even, then 3,476 is not
divisible by 8
 5,728
o 28 is made from the last (rightmost) 2 digits of 5,728
o 28 is divisible by 4
o Since 28 is not divisible by 8 and the 3rd digit from the right (7) is odd, then 5,728 is divisible by
8
 6,916
o 16 is made from the last (rightmost) 2 digits of 6,916
o 16 is divisible by 4
o Since 16 is divisible by 8 but the 3rd digit from the right (9) is odd, then 6,916 is not divisible by
8
 Divisible By 9
 Add the individual digits of the number
 If the sum of the digits is divisible by 9, then the original number is divisible by 9
 If the sum of the digits is not divisible by 9, then the original number is not divisible by 9
 558
o Sum the digits of 558  5 + 5 + 8 = 18
o Since the sum of the digits (18) is divisible by 9, the number 558 is divisible by 9
 258
o Sum the digits of 258 2 +5 + 8 = 15
o Since the sum of the digits (15) is not divisible by 9, the number 258 is not divisible by 9
 Divisible By 10
 Look at the last (rightmost) digit of the number
 If the rightmost digit is 0, then the original number is divisible by 10
 If the rightmost digit is not 0, then the original number is not divisible by 10
 1,340
o Since the rightmost digit of 1,340 is 0, the number 1,340 is divisible by 10
 2,458
o Since the rightmost digit of 2,458 is not 0, the number 2,458 is not divisible by 10
 General Divisibility Rule
 An odd number cannot ever be divisible by an even number
 An even number can be divisible by an odd number and/or and even number
31
Divisibility – Guided Practice
Determine which number(s), if any, will divide evenly the specified value. Circle the answer(s).
1. 67
2
3
4
5
6
7
8
9
10
none
2. 84
2
3
4
5
6
7
8
9
10
none
3. 98
2
3
4
5
6
7
8
9
10
none
4. 129
2
3
4
5
6
7
8
9
10
none
5. 240
2
3
4
5
6
7
8
9
10
none
6. 375
2
3
4
5
6
7
8
9
10
none
7. 484
2
3
4
5
6
7
8
9
10
none
8. 720
2
3
4
5
6
7
8
9
10
none
9. 840
2
3
4
5
6
7
8
9
10
none
10. 915
2
3
4
5
6
7
8
9
10
none
32
Number Principles – Guided Reinforcement
Number Properties
__________ 1. Using the Commutative Property of Addition, how can you rewrite 7 + 18?
__________ 2. Using the Commutative Property of Addition, how can you rewrite 35 + –15?
__________ 3. Using the Commutative Property of Addition, how can you rewrite –13 + 9?
__________ 4. Using the Commutative Property of Addition, how can you rewrite –17 + –23?
__________ 5. Using the Associative Property of Addition, how can you rewrite (9 + 2) + 7?
__________ 6. Using the Associative Property of Addition, how can you rewrite ((6 + (–2)) + 8?
__________ 7. Using the Associative Property of Addition, how can you rewrite (5 + 4) + (–11)?
__________ 8. Using the Associative Property of Addition, how can you rewrite –14 + (8 + (–3))?
__________ 9. Using the Commutative Property of Multiplication, how can you rewrite 23 x 4?
__________ 10. Using the Commutative Property of Multiplication, how can you rewrite 18 x (–3)?
__________ 11. Using the Commutative Property of Multiplication, how can you rewrite (–6) x 12?
__________ 12. Using the Commutative Property of Multiplication, how can you rewrite (–15) x (–5)?
33
Numbers Principles – Guided Reinforcement (continued)
Number Properties (continued)
__________ 13. Using the Associative Property of Multiplication, how can you rewrite (4 x 5) x 9?
__________ 14. Using the Associative Property of Multiplication, how can you rewrite ((3 x (–2)) x 7?
__________ 15. Using the Associative Property of Multiplication, how can you rewrite (5 x 4) x (–8)?
__________ 16. Using the Associative Property of Multiplication, how can you rewrite –6 x (2 x 3)?
__________ 17. Using the Distributive Property, how can you rewrite (8 x 3) + (8 x 5)?
__________ 18. Using the Distributive Property, how can you rewrite ((–7) x 5) + ((–7) x 8)?
__________ 19. Using the Distributive Property, how can you rewrite 6 x (4 + 7)?
__________ 20. Using the Distributive Property, how can you rewrite –3 x (9 + (–4))?
__________ 21. Using the Identity Property of Addition, how can you rewrite 18 + 0?
__________ 22. Using the Identity Property of Addition, how can you rewrite –11 + 0?
__________ 23. Using the Identity Property of Multiplication, how can you rewrite 18 x 1?
__________ 24. Using the Identity Property of Multiplication, how can you rewrite (–11) x 1?
34
Numbers Principles – Guided Reinforcement (continued)
Multiplication
Solve the following multiplication problems without using a calculator.
__________ 1. What is 16 x 7?
(_____ + _____) x 7
(_____ x 7) + (_____ x 7)
_____ + _____
__________ 2. What is 24 x 9?
(_____ + _____) x 9
(_____ x 9) + (_____ x 9)
_____ + _____
__________ 3. What is 36 x 7?
(_____ + _____) x _____
(_____ x _____) + (_____ x _____)
_____ + _____
__________ 4. What is 42 x 8?
(_____ + _____) x _____
(_____ x _____) + (_____ x _____)
_____ + _____
35
Numbers Principles – Guided Reinforcement (continued)
Multiplication (continued)
__________ 5. What is 48 x 7?
(_____ + _____) x _____
(_____ x _____) + (_____ x _____)
_____ + _____
__________ 6. What is 53 x 6?
(_____ + _____) x _____
(_____ x _____) + (_____ x _____)
_____ + _____
__________ 7. What is 66 x 5?
(_____ + _____) x _____
(_____ x _____) + (_____ x _____)
_____ + _____
__________ 8. What is 72 x 8?
(_____ + _____) x _____
(_____ x _____) + (_____ x _____)
_____ + _____
36
Numbers Principles – Guided Reinforcement (continued)
Multiplication (continued)
__________ 9. What is 83 x 6?
(_____ + _____) x _____
(_____ x _____) + (_____ x _____)
_____ + _____
__________ 10. What is 94 x 6?
(_____ + _____) x _____
(_____ x _____) + (_____ x _____)
_____ + _____
__________ 11. What is 12 x 17?
(_____ + _____) x _____
(_____ x _____) + (_____ x _____)
_____ + _____
__________ 12. What is 13 x 18?
(_____ + _____) x _____
(_____ x _____) + (_____ x _____)
_____ + _____
37
Numbers Principles – Guided Reinforcement (continued)
Multiplication (continued)
__________ 13. What is 24 x 15?
(_____ + _____) x _____
(_____ x _____) + (_____ x _____)
_____ + _____
__________ 14. What is 28 x 12?
(_____ + _____) x _____
(_____ x _____) + (_____ x _____)
_____ + _____
__________ 15. What is 33 x 15?
(_____ + _____) x _____
(_____ x _____) + (_____ x _____)
_____ + _____
__________ 16. What is 38 x 13?
(_____ + _____) x _____
(_____ x _____) + (_____ x _____)
_____ + _____
38
Numbers Principles – Guided Reinforcement (continued)
Multiplication (continued)
__________ 17. What is 43 x 14?
(_____ + _____) x _____
(_____ x _____) + (_____ x _____)
_____ + _____
__________ 18. What is 47 x 14?
(_____ + _____) x _____
(_____ x _____) + (_____ x _____)
_____ + _____
__________ 19. What is 56 x 13?
(_____ + _____) x _____
(_____ x _____) + (_____ x _____)
_____ + _____
__________ 20. What is 64 x 16?
(_____ + _____) x _____
(_____ x _____) + (_____ x _____)
_____ + _____
39
Numbers Principles – Guided Reinforcement (continued)
Multiplication with Multiples of 10, 100, and 1000
1. 17 x 40 = _____
2. 18 x 500 = _____
3. 19 x 6000 = _____
4. 21 x 4000 = _____
5. 13 x 60 = _____
6. 17 x 500 = _____
7. 18 x 300 = _____
8. 8 x 9000 = _____
9. 21 x 30 = _____
10. 23 x 40 = _____
11. 22 x 500 = _____
12. 24 x 4000 = _____
13. 34 x 40 = _____
14. 45 x 300 = _____
15. 55 x 400 = _____
16. 22 x 4000 = _____
17. 36 x 20 = _____
18. 38 x 200 = _____
19. 43 x 2000 = _____
20. 65 x 2000 = _____
21. 63 x 30 = _____
22. 71 x 200 = _____
23. 82 x 3000 = _____
24. 90 x 4000 = _____
40
Numbers Principles – Guided Reinforcement (continued)
Divisibility Rules
__________ 1. Circle the numbers by which 147 is divisible.
2
3
4
5
6
7
8
9
10
__________ 2. Circle the numbers by which 168 is divisible.
2
3
4
5
6
7
8
9
10
__________ 3. Circle the numbers by which 195 is divisible.
2
3
4
5
6
7
8
9
10
__________ 4. Circle the numbers by which 200 is divisible.
2
3
4
5
6
7
8
9
10
__________ 5. Circle the numbers by which 385 is divisible.
2
3
4
5
6
7
8
9
10
__________ 6. Circle the numbers by which 486 is divisible.
2
3
4
5
6
7
8
9
10
__________ 7. Circle the numbers by which 558 is divisible.
2
3
4
5
6
7
8
9
10
__________ 8. Circle the numbers by which 900 is divisible.
2
3
4
5
6
7
8
9
10
__________ 9. Circle the numbers by which 1,200 is divisible.
2
3
4
5
6
7
8
9
10
__________ 10. Circle the numbers by which 1,360 is divisible.
2
3
4
5
6
7
8
9
10
41
Numbers Principles – Guided Reinforcement (continued)
Divisibility Rules (continued)
__________ 11. Circle the numbers by which 1,833 is divisible.
2
3
4
5
6
7
8
9
10
__________ 12. Circle the numbers by which 2,355 is divisible.
2
3
4
5
6
7
8
9
10
__________ 13. Circle the numbers by which 3,333 is divisible.
2
3
4
5
6
7
8
9
10
__________ 14. Circle the numbers by which 5,280 is divisible.
2
3
4
5
6
7
8
9
10
__________ 15. Circle the numbers by which 5,400 is divisible.
2
3
4
5
6
7
8
9
10
__________ 16. Circle the numbers by which 5,850 is divisible.
2
3
4
5
6
7
8
9
10
__________ 17. Circle the numbers by which 6,316 is divisible.
2
3
4
5
6
7
8
9
10
__________ 18. Circle the numbers by which 7,500 is divisible.
2
3
4
5
6
7
8
9
10
__________ 19. Circle the numbers by which 7,812 is divisible.
2
3
4
5
6
7
8
9
10
__________ 20. Circle the numbers by which 8,100 is divisible.
2
3
4
5
6
7
8
9
10
42
Module 8: Number Theory and Terminology
43
Section 8.1 Factors and Multiples
Factor (Definition and Examples)
 An Integer (positive or negative) that can be multiplied by another Integer to result in an Integer
product
 Factors of 12
 1  1 x 12 = 12
 2  2 x 6 = 12
 3  3 x 4 = 12
 4  4 x 3 = 12
 6  6 x 2 = 12
 12  12 x 1 = 12
 -1  -1 x 12 = 12
 -2  -2 x 6 = 12
 -3  -3 x 4 = 12
 -4  -4 x 3 = 12
 -6  -6 x 2 = 12
 -12  -12 x 1 = 12
Greatest Common Factor (GCF)
 The largest factor common to two or more Integers
 Greatest Common Factor of 12 and 16
 Factors of 12: 1, 2, 3, 4, 6, 12
 Factors of 16: 1, 2, 4, 8, 16
 The largest factor common to both 12 and 16 (GCF)  4
 Greatest Common Factor of 16, 24, and 40
 Factors of 16: 1, 2, 4, 8, 16
 Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
 Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40
 The largest factor common to 16, 24, and 40 (GCF)  8
Multiple (Definition and Examples)
 A number which is the product of two or more of its factors
 Every value has an infinite number of multiples
 Multiples of 12
 12  1 x 12 = 12
 24  2 x 12 = 24
 36  3 x 12 = 36
 48  4 x 12 = 48
 60  5 x 12 = 60
44
Section 8.1 Factors and Multiples (continued)
Least Common Multiple (LCM)
 The smallest multiple common to two or more Integers
 Least Common Multiple of 12 and 16
 Multiples of 12: 12, 24, 36, 48, 60
 Multiples of 16: 16, 32, 48, 64, 80
 The smallest multiple common to both 12 and 16 (LCM)  48
 Least Common Multiple 16, 24, and 40
 Multiples of 16: 16, 32, 48, 64, 80, 96, 112, 128, 144, 160, 176, 192, 208, 224, 240
 Multiples of 24: 24, 48, 72, 96, 120, 144, 168, 192, 216, 240
 Multiples of 40: 40, 80, 120, 160, 200, 240, 280
 The smallest multiple common to 16, 24, and 40 (LCM)  240
45
Factors and Multiples – Guided Practice
____________________ 1. List 3 Factors of 36.
____________________ 2. What is the Greatest Common Factor of 36 and 48?
____________________ 3. List 3 Multiples of 9.
____________________ 4. What is the Least Common Multiple of 15 and 12?
46
Section 8.2 Prime Numbers and Composite Numbers
Prime Number (Definition and Examples)
 A positive integer that can be divided only by itself and by 1
 By definition 0 and 1 are not prime numbers
 2 is the smallest prime number
 2 is the only even prime number
 All other prime numbers are odd, but not all odd numbers are prime
 Memorize all Prime Numbers less than 100
 2
 3
 5
 7
 11
 13
 17
 19
 23
 29
 31
 37
 41
 43
 47
 53
 59
 61
 67
 71
 73
 79
 83
 97
 The following number are often incorrectly thought to be Prime Numbers
 21 (7 x 3)
 27 (9 x 3)
 33 (11 x 3)
 39 (13 x 3)
 49 (7 x 7)
 51 (17 x 3)
 57 (19 x 3)
 87 (29 x 3)
 91 (13 x 7)
47
Section 8.2 Prime Numbers and Composite Numbers (continued)
Composite Number (Definition and Examples)
 A positive integer that can be divided not only by itself and by 1, but also by other positive Integers
 A positive Integer that contains factors in addition to 1 and itself
 A positive Integer that is not a Prime Number
 All even numbers (besides 2) are composite
 Many odd numbers are composite
 Odd Composite Numbers less than 100
 9 (3 x 3)
 15 (5 x 3)
 21 (7 x 3)
 25 (5 x 5)
 27 (9 x 3)
 33 (11 x 3)
 35 (7 x 5)
 39 (13 x 3)
 45 (15 x 3)
 49 (7 x 7)
 51 (17 x 3)
 55 (11 x 5)
 57 (19 x 3)
 63 (21 x 3)
 65 (13 x 3)
 69 (23 x 3)
 75 (25 x 3 & 15 x 5)
 77 (11 x 7)
 81 (27 x 3 & 9 x 9)
 85 (17 x 5)
 87 (29 x 3)
 91 (13 x 7)
 93 (31 x 3)
 95 (19 x 5)
 99 (33 x 3)
48
Section 8.2 Prime Numbers and Composite Numbers (continued)
 Prime Numbers
 A prime number is a number divisible only by 1 and itself
 Recognize these numbers are prime: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47,53, 59, 61,
67, 71, 73, 79, 83, 89, 97
 Factors of Composite Numbers
 A composite number has factors besides 1 and itself
 Recognize the numbers below as composite
 Learn the factors below of all composite numbers between 1 and 100
Key Math Factors
4
6
8
9
10
12
14
15
18
20
21
22
24
25
26
27
28
30
32
33
34
35
36
38
39
40
42
44
45
46
48
49
50
2
2
2
3
2
2
2
3
2
2
3
2
2
5
2
3
2
2
2
3
2
5
2
2
3
2
2
2
3
2
2
7
2
3
4
5
3
7
5
3
4
7
11
3
13
9
4
3
4
11
17
7
3
19
13
4
3
4
5
23
3
5
4
6
6
5
9
10
4
6
8
12
7
5
8
14
6
16
10
15
4
6
9
12
5
6
11
9
8
7
22
15
10
14
20
21
4
6
8
12
10
25
49
18
16
24
Section 8.2 Prime Numbers and Composite Numbers (continued)
Key Math Factors (continued)

Factors of Composite Numbers (continued)
51
52
54
55
56
57
58
60
62
63
64
65
66
68
69
70
72
74
75
76
77
78
80
81
82
84
85
86
87
88
90
91
92
93
94
95
96
98
99
100
3
2
2
5
2
3
2
2
2
3
2
5
2
2
3
2
2
2
3
2
7
2
2
3
2
2
5
2
3
2
2
7
2
3
2
5
2
2
3
2
17
4
3
11
4
19
29
3
31
7
4
13
3
4
23
5
3
37
5
4
11
3
4
9
41
4
17
23
29
4
3
13
4
31
47
19
3
7
9
4
13
6
26
9
18
27
7
8
14
28
4
5
6
10
9
8
21
16
32
6
17
11
34
22
33
7
4
10
6
14
8
35
9
15
19
25
38
6
8
27
13
10
26
20
39
40
6
7
12
8
5
11
6
23
46
4
14
11
5
6
49
33
10
12
15
20
30
12
18
24
36
14
21
42
22
9
44
10
15
18
30
45
8
12
16
24
32
48
20
25
50
50
Section 8.2 Prime Numbers and Composite Numbers (continued)
 Multiplication Fact of Composite Numbers (1 – 100)
 Learn all of the following multiplication fact for Composite Numbers between 1 and 100
 It will tremendously help you in the following subjects
o Algebra
o Geometry
o Probability
o Statistics
o Trigonometry
o Pre-Calculus
Key Multiplication Fact of Composite Numbers (1 – 100)
4
6
8
9
10
12
14
15
18
20
21
22
24
25
26
27
28
30
32
33
34
35
36
38
39
40
42
44
45
46
48
49
50
2x2
2x3
2x4
3x3
2x5
2x6
2x7
3x5
2x9
2 x 10
3x7
2 x 11
2 x 12
5x5
2 x 13
3x9
2 x 14
2 x 15
2 x 16
3 x 11
2 x 17
5x7
2 x 18
2 x 19
3 x 13
2 x 20
2 x 21
2 x 22
3 x 15
2 x 23
2 x 24
7x7
2 x 25
3x4
3x6
4x5
3x8
4x6
13
4x7
3 x 10
4x8
5x6
3 x 12
4x9
4 x 10
3 x 14
4 x 11
5x9
5x8
6x7
3 x 16
4 x 12
5 x 10
51
6x6
6x8
Section 8.2 Prime Numbers and Composite Numbers (continued)
Key Math Factors (continued)

Factors of Composite Numbers (continued)
51
52
54
55
56
57
58
60
62
63
64
65
66
68
69
70
72
74
75
76
77
78
80
81
82
84
85
86
87
88
90
91
92
93
94
95
96
98
99
100
3 x 17
2 x 26
2 x 27
5 x 11
2 x 28
3 x 19
2 x 29
2 x 30
2 x 31
3 x 21
2 x 32
5 x 13
2 x 33
2 x 34
3 x 23
2 x 35
2 x 36
2 x 37
3 x 25
2 x 38
7 x 11
2 x 39
2 x 40
3 x 27
2 x 41
2 x 42
5 x 17
2 x 23
3 x 29
2 x 44
2 x 45
7 x 13
2 x 46
3 x 31
2 x 47
5 x 19
2 x 48
2 x 49
3 x 33
2 x 50
4 x 13
3 x 18
6x9
4 x 14
7x8
3 x 20
4 x 15
7x9
4 x 16
8x8
3 x 22
4 x 17
6 x 11
5 x 14
3 x 24
7 x 10
4 x 18
5 x 12
6 x 10
6 x 12
8x9
5 x 15
4 x 19
3 x 26
4 x 20
9x9
6 x 13
8 x 10
4 x 21
6 x 14
7 x 12
4 x 22
3 x 30
8 x 11
5 x 18
6 x 15
9 x 10
4 x 24
6 x 16
8 x 12
5 x 10
10 x 10
4 x 23
3 x 32
7 x 14
9 x 11
4 x 25
52
Section 8.2 Prime Numbers and Composite Numbers (continued)
Prime Factorization (Definition and Examples)
 A process of breaking down a Composite Number into the product of its Prime Factors
 Prime Factorization of 12
12
12
/ \
/ \
6 x 2
4 x 3
/ \
\
/ \
\
3 x 2 x 2
2 x 2 x 3
 The bottom line of Prime Factorization contains only Prime Numbers multiplied by one another
 The order of the Prime Numbers does not matter
 The order of the Prime Numbers will depend on the Factors initially chosen
 Regardless of the Factors initially chosen, the same number of Prime Numbers will also result in the
Prime Factorization (bottom) line
o The Prime Factorization of 12 will always result in the product of two 2’s and one 3
53
Section 8.2 Prime Numbers and Composite Numbers (continued)
Equivalent Products
 Equivalent Products results from multiplying combinations of factors of a number
 Equivalent Products of 72
 1 x 72
 2 x 36 (which is the double of 1 and half of 72)
 3 x 24 (which is the triple of 1 and one third of 72)
 4 x 18 (which is quadruple of 1 and one fourth of 72)
 6 x 12 (which is six times 1 and one sixth of 72)
 8 x 9 (which is eight times 1 and one eighth of 72)
 9 x 8 (which is nine times 1 and one ninth of 72)
 12 x 6 (which is twelve times 1 and one twelfth of 72)
 18 x 6 (which is eighteen times 1 and one eighteenth of 72)
 24 x 3 (which is twenty-four times 1 and one twenty-fourth of 72)
 36 x 2 (which is thirty-six times 1 and one thirty-sixth of 72)
 Results when you multiply a number by a value and divide the number by the same value
 The multiplication is offset by the division
 The division is offset by the multiplication
 Multiplying by 2 if offset by dividing by 2 (which is the same as taking one half)
 Multiplying by 3 is offset by dividing by 3 (which is the same as taking one third)
 Multiplying by 4 is offset by dividing by 4 (which is the same as taking one fourth)
 Multiplying by 5 is offset by dividing by 5 (which is the same as taking one fifth)
 Multiplying by 6 is offset by dividing by 6 (which is the same as taking one sixth)
 Multiplying by 7 is offset by dividing by 7 (which is the same as taking one seventh)
 Multiplying by 8 is offset by dividing by 8 (which is the same as taking one eighth)
 Multiplying by 9 is offset by dividing by 9 (which is the same as taking one ninth)
 Multiplying by 10 is offset by dividing by 10 (which is the same as taking one tenth)
 This pattern continues forever
54
Prime Numbers and Composite Numbers – Guided Practice
____________________ 1. List 3 Prime Numbers between 30 and 45.
____________________ 2. List 3 Odd Composite Numbers between 30 and 45.
____________________ 3. Perform Prime Factorization on 24 (fill in the blanks where appropriate)
24
/
\
_____
x
/
\
_____ x _____
/
\
/
\
_____ x _____ _____ x _____
_____
/
\
_____
x
_____
/
\
/
\
_____ x _____ _____ x _____
____________________ 4. What is an equivalent Product of 12 x 8?
55
Section 8.3 Principles of Numbers
Positive Numbers
 Numbers greater than zero
 Simple examples : 1
1½
37.6
Negative Numbers
 Numbers less than zero
 Simple examples : -1
-37.6 -½
-1½
½
1,000
1,000,000
1,000,000,000
-1,000
-1,000,000
-1,000,000,000
Zero
 A number which is neither negative nor positive
 Zero is “nothing”
Number Line Representation of Numbers
 Positive Numbers
 Positioned to the right of zero
 The further to the right the number is, the more positive (larger value) the number is
 Negative Numbers
 Positioned to the left of zero
 The further to the left the number is, the more negative (smaller value) the number is
56
Principles of Numbers – Guided Practice
__________ 1. Which of the following numbers are negative? 25, -37.8, 15.3, 45, -80 10½
__________ 2. Which of the following numbers are positive? 25, -37.8, 15.3, 45, -80 10½
__________ 3. Shade in the portion of the number line below (between -4 and 5) that is negative.
57
Section 8.4 Number Terminology
Integer
 A positive or negative number (including zero) that has neither a fractional nor decimal part
 Simple examples: -1,000,000
-25
-2
-1
0
1
2
25
1,000,000
7
10
1
 The following number are not Integers:
1
3.1415926

5
8
3
2
Consecutive Integers
 Integers listed in increasing order of size without any integers missing in between
 Can include negative and/or positive values
 Simple example: -4, -3, -2, -1, 0, 1, 2, 3, 4
 The following examples are not Consecutive Integers:
 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6  decreasing (not increasing) order of size
 -6, -4, -2, 0, 2, 4, 6  missing (odd) Integers in between
 -2.0, -1.5, -1.0, -0.5, 0, 0.5, 1.0, 1.5, 2.0  not Integers
Even Number
 An Integer that divides evenly by 2
 Must end in 0, 2, 4, 6, or 8
 Simple examples : 14
1,000
3,576
1,000,000
1,000,000,000
Odd Number
 An Integer that does not divide evenly by 2
 Must end in 1, 3, 5, 7, or 9
 Simple examples : 15
1,001
3,577
1,000,001
1,000,000,001
Rational Number
 Any number that can be written as a ratio of two numbers
 It can be written as a fraction where both the numerator and denominator are Integers
5
5
 Simple examples : -1,000,000
-1,000
0
1,000 1,000,000
11
11
Irrational Number
 Any number that cannot be written as a ratio of two numbers
 Any number that is not a Rational Number
 Simple examples : - 13
- 5
- 3
58
3

5
Section 8.4 Number Terminology (continued)
Remainder
 The “left over” value when one Integer does not evenly divide into another Integer
 Remainder of 17  3  2
 Remainder of 27  4  3
 Remainder of 40  6  4
Digits and Place Value
 In the Decimal Numbering System, the digits are
 0
 1
 2
 3
 4
 5
 6
 7
 8
 9
 The value of an Integer is determined by the value and position of the digits of the Integer
 The rightmost digit position is referred to as the “units digit”
 2,475 is
 A “four-digit” number
 Comprised of the following digits and values
o 2 (thousands position)
o 4 (hundreds position)
o 7 (tens position)
o 5 (units position)
59
Number Terminology – Guided Practice
__________ 1. Which of the following values are Integers? -50, ½ , -1,000,000, 3.87, 1,000.01
__________ 2. Which of the following values are not Integers? -50, ½ , -1,000,000, 3.87, 1,000.01
__________ 3. List three Consecutive Integers starting at 22.
__________ 4. List three Consecutive Integers starting at – 8.
__________ 5. List the four Even Numbers immediately larger than 18.
__________ 6. List the four Even Numbers immediately smaller than 18.
__________ 7. List the four Even Numbers immediately larger than – 25.
__________ 8. List the four Even Numbers immediately smaller than – 25.
__________ 9. List the four Odd Numbers immediately larger than 18.
__________ 10. List the four Odd Numbers immediately smaller than 18.
__________ 11. List the four Odd Numbers immediately larger than – 25.
__________ 12. List the four Odd Numbers immediately smaller than – 25.
60
Number Terminology – Guided Practice (continued)
13. Categorize the following numbers either as Rational or Irrational
__________
1
7
__________
7
__________ 33.3333
__________ 14. What is the remainder of 38  4?
__________ 15. In the number 12,874,935, what digit is in the millions position?
61
Section 8.5 Arithmetic Operations on Numbers
Common Arithmetic Operations on Numbers
 Addition  +
 Subtraction  –
 Multiplication  x
 Division  ÷
Standard Arithmetic Symbols
 = : is equal to
  : is not equal to
 < : is less than
 > : is greater than
  : is less than or equal to
  : is greater than or equal to
Addition (Sum)
 When a number is added to another number, the answer is call the Sum
 Adding a positive number to a positive number  8 + 5
 Start at the first number (8)
 Move to the right the number of positions of the second number (5)
 8 + 5 = 13
 Adding a positive number to a negative number  -8 + 5
 Start at the first number (-8)
 Move to the right the number of positions of the second number (5)
 -8 + 5 = -3
 Notice that the sign of the answer (negative) is the same as the sign of the larger number (-8)
 Adding a negative number to a positive number  8 + (-5)
 Start at the first number (8)
 Move to the left the number of positions of the second number (5)
o This is the same as subtracting the second number from the first number
o 8 + (-5)  8 – 5
 8 + (-5) = 3
 Notice that the sign of the answer (positive) is the same as the sign of the larger number (8)
 Adding a negative number to a negative number  -8 + (-5)
 Start at the first number (-8)
 Move to the left the number of positions of the second number (5)
o This is the same as subtracting the second number from the first number
o -8 + (-5)  -8 – 5
o Because both numbers are negative, you actually add the values and keep the negative sign
 -8 + (-5) = -13
 Notice that the sign of the answer (negative) is the same as the sign of both numbers
62
Section 8.5 Arithmetic Operations on Numbers (continued)
Subtraction (Difference)
 When a number is subtracted from another number, the answer is call the Difference
 Subtracting a positive number from a positive number  8 – 5
 Start at the first number (8)
 Move to the left the number of positions of the second number (5)
o The is the same as adding a negative number to a positive number
o 8 – 5  8 + (-5)
 8–5=3
 Notice that the sign of the answer (positive) is the same as the sign of the larger (8) value
 Subtracting a positive number from a negative number  -8 – 5
 Start at the first number (-8)
 Move to the left the number of positions of the second number (5)
o This is the same as adding a negative number to another negative number
o -8 – 5  -8 + (-5)
o Because both numbers are negative, you actually add the values and keep the negative sign
 -8 – 5 = -13
 Subtracting a negative number from a positive number  8 – (-5)
 Start at the first number (8)
 Move to the right the number of positions of the second number (5)
o The is the same as adding the second number to the first number
o 8 – (-5)  8 + 5
o Because both numbers are positive, you actually add the values and keep the positive sign
 8 – (-5)  8 + 5  13
 Subtracting a negative number from a negative number  -8 – (-5)
 Start at the first number (-8)
 Move to the right the number of positions of the second number (5)
o This is the same as adding the second number to the first number
o -8 – (-5)  -8 + 5
 -8 – (-5)  -8 + 5  -3
 Notice that the sign of the answer (negative) is the same as the sign of the larger (-8) value
63
Section 8.5 Arithmetic Operations on Numbers (continued)
Multiplication (Product)
 When a number is multiplied by another number, the answer is call the Product
 Multiplication is “repeated addition”
 8 x 5 means “repeatedly add the value 8 (5 sets of times)”  8 + 8 + 8 + 8 + 8
 5 x 8 means “repeatedly add the value 5 (8 sets of times)”  5 + 5 + 5 + 5 + 5 + 5 + 5 + 5
 Multiplication is a “short hand” way is saying “add this value over and over this many times”
 Multiplying a positive number with a positive number  8 x 5
 Results in a positive answer
 8 x 5 = 40
 Multiplying a positive number with a negative number  8 x (-5)
 Results in a negative answer
 8 x (-5) = -40
 Multiplying a negative number with a positive number  (-8) x 5
 Results in a negative answer
 (-8) x 5 = -40
 Multiplying a negative number with a negative number  (-8) x (-5)
 Results in a positive answer
 (-8) x (-5) = 40
 General Multiplication Principles
 When multiplying two values:
o When the signs are the same, the answer is positive
o When the signs are different, the answer is negative
 When multiplying more than two values:
o When there is an even number of negative values, the answer is positive
o When there is an odd number of negative values, the answer is negative
64
Section 8.5 Arithmetic Operations on Numbers (continued)
Division (Quotient)
 When a number is divided into another number, the answer is call the Quotient
 Division is “repeated subtraction”
 40 ÷ 5 means “repeatedly divide 40 in to groups containing 5 elements”
 The final answer specifies “how many groups” there are
 Repeatedly “dividing 40 into groups containing 5 elements” will result in 8 groups
o Therefore, 40 ÷ 5 = 8
 Dividing a positive number by a positive number  40 ÷ 5
 Results in a positive answer
 40 ÷ 5 = 8
 Dividing a positive number by a negative number  40 ÷ (-5)
 Results in a negative answer
 40 ÷ (-5) = -8
 Dividing a negative number by a positive number  (-40) ÷ 5
 Results in a negative answer
 (-40) ÷ 5 = -8
 Dividing a negative number with a negative number  (-40) ÷ (-5)
 Results in a positive answer
 (-40) ÷ (-5) = 8
 General Division Principles
 When dividing two values:
o When the signs are the same, the answer is positive
o When the signs are different, the answer is negative
 When dividing more than two values:
o When there is an even number of negative values, the answer is positive
o When there is an odd number of negative values, the answer is negative
65
Module 8 (Number Theory and Terminology) – Review Exercises
Factors and Multiples
For Factor questions, use factors other than 1 and the number.
__________ 1. List 3 Factors of 24.
__________ 2. List 4 Factors of 36.
__________ 3. List 4 Factors of 48.
__________ 4. List 4 Factors of 60.
__________ 5. List 4 Factors of 75.
__________ 6. What is the Greatest Common Factor of 12 and 16?
__________ 7. What is the Greatest Common Factor of 12 and 48?
__________ 8. What is the Greatest Common Factor of 39 and 52?
__________ 9. What is the Greatest Common Factor of 60 and 75?
__________ 10. What is the Greatest Common Factor of 36 and 90?
__________ 11. List 3 Multiples of 8.
__________ 12. List 3 Multiples of 12.
__________ 13. List 4 Multiples of 15.
66
Module 8 (Number Theory and Terminology) – Review Exercises (continued)
Factors and Multiples (continued)
__________ 14. List 4 Multiples of 18.
__________ 15. List 4 Multiples of 24.
__________ 16. What is the Least Common Multiple of 6 and 15?
__________ 17. What is the Least Common Multiple of 8 and 18?
__________ 18. What is the Least Common Multiple of 9 and 15?
__________ 19. What is the Least Common Multiple of 7 and 3?
__________ 20. What is the Least Common Multiple of 15 and 40?
67
Module 8 (Number Theory and Terminology) – Review Exercises (continued)
Prime Numbers and Composite Numbers
__________ 1. List 3 Prime Numbers between 5 and 15.
__________ 2. List 3 Prime Numbers between 15 and 25.
__________ 3. List 4 Prime Numbers between 20 and 40.
__________ 4. List 4 Prime Numbers between 30 and 50.
__________ 5. List 4 Prime Numbers between 40 and 60.
__________ 6. List 3 Odd Composite Numbers between 4 and 16.
__________ 7. List 4 Odd Composite Numbers between 16 and 34.
__________ 8. List 4 Odd Composite Numbers between 34 and 50.
__________ 9. List 3 Odd Composite Numbers between 50 and 60
__________ 10. List 3 Odd Composite Numbers between 16 and 34
68
Module 8 (Number Theory and Terminology) – Review Exercises (continued)
Prime Numbers and Composite Numbers (continued)
__________ 11. Perform Prime Factorization on 30 (fill in the blanks where appropriate)
30
/
_____
\
x
_____
/
\
_____ x _____
/
_____
x
\
_____
__________ 12. Perform Prime Factorization on 36 (fill in the blanks where appropriate)
36
/
\
_____
x
/
\
_____ x _____
/
\
/
\
_____ x _____ _____ x _____
_____
/
\
_____
x
_____
/
\
/
\
_____ x _____ _____ x _____
__________ 13. Perform Prime Factorization on 40 (fill in the blanks where appropriate)
40
/
_____
\
x
/
\
_____ x _____
_____
/
_____
69
x
\
_____
Module 8 (Number Theory and Terminology) – Review Exercises (continued)
Prime Numbers and Composite Numbers (continued)
__________ 14. Perform Prime Factorization on 48 (fill in the blanks where appropriate)
48
/
_____
\
x
/
\
_____ x _____
/
\
/
\
_____ x _____ _____ x _____
_____
/
\
_____
x
_____
/
\
/
\
_____ x _____ _____ x _____
__________ 15. Perform Prime Factorization on 60 (fill in the blanks where appropriate)
60
/
\
_____
x
/
\
_____ x _____
/
\
/
\
_____ x _____ _____ x _____
_____
/
\
_____
x
_____
/
\
/
\
_____ x _____ _____ x _____
__________ 16. What is an equivalent Product of 9 x 4?
__________ 17. What is an equivalent Product of 6 x 8?
__________ 18. What is an equivalent Product of 10 x 6?
__________ 19. What is an equivalent Product of 18 x 4?
__________ 20. What is an equivalent Product of 16 x 6?
70
Module 8 (Number Theory and Terminology) – Review Exercises (continued)
Positive & Negative Number Arithmetic
__________ 1. What arithmetic operation does the symbol – perform?
__________ 2. What arithmetic operation does the symbol x perform?
3. What is the value of the following arithmetic operations?
__________ 8 + 13
__________ 8 + (– 13)
__________ –8 + 13
__________ –8 + (–13)
__________ 5 – 11
__________ 5 – (–11)
__________ –5 – 11
__________ –5 – (–11)
__________ 7 x 8
__________ 7 x (– 8)
__________ (–7) x 8
__________ (–7) x (– 8)
__________ 72 ÷ 8
__________ 72 ÷ (–8)
__________ (– 72) ÷ 8
__________ (– 72) ÷ (–8)
71
Module 9: Principles of Fractions
72
Section 9.1 Introduction to Fractions
Introduction to Fractions
 A comparison of the “part” (numerator) to the “whole” (denominator)
3
 Three pieces out of eight equally cut pieces of pie 
8
5
 Five free throws made out of six free throws attempted 
6

Four complete pizzas and seven pieces out of ten equally cut pieces of pizza  4
7
10
Types of Fractions
 Proper
 The numerator is smaller than the denominator
7

is a Proper Fraction
8
 Improper
 The numerator is larger than the denominator
8

is a Proper Fraction
7
 Mixed
 Contains both a whole number part and a fractional part
7
 1 is a Mixed Fraction
8
Simplest-Form Fraction
 A fraction where the greatest common factor of the numerator and the denominator is 1
3

 the greatest common factor of 3 and 8 is 1 (this is a simplest-form fraction)
8
6

 the greatest common factor of 6 and 8 is 2 (this is not a simplest-form fraction)
8
73
Section 9.1 Introduction to Fractions (continued)
Equivalent Fractions
 Fractions with different numerators and denominators that represent the same value
1
2
3
 The value
can be represented by the fractions
and
2
4
6
1 2
3
 Therefore, the fractions , , and
are Equivalent Fractions
2 4
6
 Equivalent fractions will eventually reduce to the same Simplest-Form Fraction
60
35

is an equivalent fraction to
72
42
5
 Both fractions eventually reduce to
6
Reciprocal
 Switching the numerator and denominator of a fraction
4
9
 Reciprocal of

9
4
8
1
 Reciprocal of 8 (or ) 
1
8
4
18
7
 Reciprocal of 2 (or
)
7
7
18
Least Common Denominator (LCD)
 This is an extremely important concept when working with multiple fractions
 The LCD is the least common multiple (LCM) of the denominators of all fractions involved
2 1
5
 Example 1: what is the LCD of , , and
3 4
6
 You must determine the LCM of denominators 3, 4, and 6
 The smallest number that is a multiple of 3, 4, and 6 is 12
3 4 5
7
 Example 2: what is the LCD of , , and
4 5 8
10
 You must determine the LCM of denominators 4, 5, 8, and 10
 The smallest number that is a multiple of 4, 5, 8, and 10 is 40
74
Section 9.1 Introduction to Fractions (continued)
Reducing (Simplifying) Fractions
 Determine the greatest common (GCF) factor of the numerator and denominator
 Divide the numerator and denominator that GCF (making the resulting GCF 1)
45
 Example: Reduce
60
 Determine GCF of 45 and 60  15
 Divide numerator (45) and denominator (60) by the GCF (15)
45  15
3


60  15
4
 The GCF of 3 and 4 is 1 (therefore, the fraction has been fully reduced)
Unreducing (Unsimplifying) Fractions
 Multiply both the numerator and denominator by the a common value
3
 Example: Unreduce
4
 Determine a common value by which you multiply the numerator and denominator (i.e. 15)
3 x15
45


4 x15 60
Converting Improper Fractions to Mixed Fractions
 Divide the denominator into the numerator  becomes the whole number part
 Place the remainder over the denominator  becomes the fractional part
 Put together the whole part and the fractional part
11
 Example: Convert
to a mixed fraction
4
 Divide the denominator into the numerator  11 ÷ 4 = 2 (whole part with remainder 3)
3
 Place the remainder over the denominator 
(fractional part)
4
3
 Put together the whole part and the fractional part  2
4
Converting Mixed Fractions to Improper Fractions
 Multiply the integer by the denominator of the fraction
 Add this product to the numerator of the fraction
 Place this sum over the original denominator
3
 Example: Convert 2 to an improper fraction
4
 Multiply the integer by the denominator of the fraction  2 x 4 = 8
 Add this product to the numerator of the fraction  8 + 3  11
11
 Place this sum over the original denominator 
4
75
Section 9.1 Introduction to Fractions (continued)
Comparing Fractions
 Multiply the numerator of the first fraction by the denominator of the second fraction and the numerator
of the second fraction by the denominator of the first fraction
 Compare the two multiplied values
 The first multiplied value corresponds to the first fraction
 The second multiplied value corresponds to the second fraction
7
13
 Example: Compare
and
8
15
 Multiply 7 and 15 (corresponds to first fraction)  105
 Multiply 13 and 8 (corresponds to second fraction)  104
7
 Since the first product (7 x 15) is larger than the second product (13 x 8), the first fraction ( ) is
8
13
larger than the second fraction ( )
15
Converting Fractions to have a Common Denominator
 This is an extremely important concept when preparing to add or subtract fractions with different
denominators
 Steps to converting fractions to have a common denominator
 First, determine the least common denominator (LCD)
 Then determine for each fraction what number to multiply the current denominator to reach the
LCD
 For each fraction, multiply both the numerator and denominator by the value that will make the
fraction contain the LCD, making an equivalent fraction
2 1
5
 Example: Convert the following fractions to have a common denominator , , and
3 4
6
 Determine the LCD for the fractions  12
 Determine the number to multiply each fraction’s denominator’s to reach the LCD
2
o
 multiply the denominator (3) by 4 to become 12
3
1
o
 multiply the denominator (4) by 3 to become 12
4
5
o
 multiply the denominator (6) by 2 to become 12
6
 For each fraction, multiply both the numerator and denominator by the value that will make the
fraction contain the LCD
2
2
4
8
o
 multiply the numerator and denominator by 4:
x
=
3
3
4 12
1
1
3
3
o
 multiply the numerator and denominator by 3:
x =
3 12
4
4
5
5
2 10
o
 multiply the numerator and denominator by 2:
x
=
6
6
2 12
76
Introduction to Fractions – Guided Practice
Write the following as a fraction (questions 1 – 3)
__________ 1. 7 points received out of 10 possible points
__________ 2. 11 pieces of pie eaten out of 15 pieces of pie cut
__________ 3. 17 dogs out of 23 total animals
Write yes or no to indicate if the specified is a Simplest-Form Fraction (question 4 – 7)
__________ 4.
4
5
________ 5.
12
21
________ 6.
15
26
________ 7.
17
51
Reduce the following fractions to Simplest-Form (question 8 – 11)
__________ 8.
12
20
________ 9.
16
24
________ 10.
20
35
________ 11.
60
72
Create an Equivalent Fraction with the specified numerator or denominator (question 12 – 17)
_____ 12.
4
(numerator 20)
7
_____ 13.
3
2
(denominator 16) _____ 14.
(numerator 12)
5
4
_____ 15.
7
(denominator 64)
8
_____ 16.
5
(numerator 35)
6
_____ 17.
7
(denominator 72)
9
Convert the following fractions (Improper to Mixed or Mixed to Improper) (question 18 – 17)
______ 18.
35
6
________ 19. 4
3
5
________ 20.
77
53
8
________ 21. 8
5
7
Introduction to Fractions – Guided Practice (continued)
What is the reciprocal of each fraction, specified as proper or improper? (question 22 – 25)
__________ 22.
18
7
________ 23. 1
3
4
________ 24.
12
23
________ 25. 5
7
8
What is the least common denominator (LCD) of each set of fractions? (question 26 – 28)
__________ 26.
2 5 7
, ,
5 6 15
__________ 27.
3 1 7
, ,
8 6 12
__________ 28.
7 4 3
, ,
15 9 5
Convert the following fractions to have a least common denominator (question 29 – 31)
__________ 29.
2 5 7
, ,
5 6 15
__________ 30.
3 1 7
, ,
8 6 12
78
__________ 31.
7 4 3
, ,
15 9 5
Section 9.2 Fraction Arithmetic
Adding and Subtracting Fractions with Like Denominators
 Just add or subtract the numerators
 Keep the same denominator
 Reduce the final answer if possible
2
5
1
5 1 6
 Example: + =
= =
9
9
9
9
3
5 1
5 1 4
 Example: – =
=
9 9
9
9
Adding and Subtracting Fractions with Unlike Denominators
 Determine the “Least Common Denominator” (LCD) for the two fractions
 Now “unreduced” each fraction to have that LCD
 Since each fraction now has the same denominator, just added the numerators
 Keep the same denominator
 Reduce the final answer if possible
4
1
 Example +
9
6
 Least Common Denominator (LCD) = 18
4
 Unreduce
to have a denominator of 18 (LCD)  fraction must be multiplied by
9
1
 Unreduce
to have a denominator of 18 (LCD)  fraction must be multiplied by
6
4
2
8
1
3
3
8
3
11

x =
&
x =

+

(already reduced)
9
2 18
6
3 18
18 18
18
4
1
 Example –
9
6
 Least Common Denominator (LCD) = 18
4
 Unreduce
to have a denominator of 18 (LCD)  fraction must be multiplied by
9
1
 Unreduce
to have a denominator of 18 (LCD)  fraction must be multiplied by
6
4
2
8
1
3
3
8
3
5

x =
&
x =

–

(already reduced)
18
9
2 18
6
3 18
18 18
79
2
2
3
3
2
2
3
3
Section 9.2 Fraction Arithmetic (continued)
Adding Mixed Fractions
 Add the whole parts to one another and the fractional parts to one another
 If the fractions have the same denominators, use the steps for “Adding Fractions with Like
Denominators”
 If the fractions have different denominators, use the steps for “Adding Fractions with Unlike
Denominators”
 If the sum of the fractional parts exceeds 1, initially specify that sum as an improper fraction
 Convert the sum’s improper fraction to the corresponding mixed fraction
 Add this mixed fraction to the sum of the whole parts
 Reduce the final answer if possible
6
2
2
5
1
5
1
5 1
 Example 1: 4 + 3 = 4 + 3 + + = 7 +
=7+ =7+
=7
9
9
9
9
9
9
3
3
 Example 2: 6
4
5
4
5
45
9
2
2
+2 =6+2+ +
=8+
=8+ =8+1
=9
7
7
7
7
7
7
7
7
 Example 3: 5
3
4
4
3
4
4
3 4 4
11
1
1
+6 +2 =5+6+2+ +
+ = 13 +
= 13 +
= 13 + 2 = 15
5
5
5
5
5
5
5
5
5
5
 Example 4: 3
3
1
3
1
9
4
94
13
+2 =3+2+ +
=5+
+
=5+
=5
8
6
8
6
24
24
24
24
 Example 5: 4
7
5
7
5
21
20
21  20
41
17
17
+5 =4+5+ +
=9+
+
=9+
=5
=5+1
=6
8
6
8
6
24
24
24
24
24
24
4
5
7
4
5
7
32
25
28
85
+3 +6
=2+3+6+
+ +
= 11 +
+
+
= 11 +
=
5
8
10
5
8 10
40
40
40
40
5
5
1
11 + 2
= 13
= 13
40
40
8
 Example 6: 2
80
Section 9.2 Fraction Arithmetic (continued)
Subtracting Mixed Fractions
 Subtract the whole parts from one another and the fractional parts from one another
 If the fractions have the same denominators, use the steps for “Subtracting Fractions with Like
Denominators”
 If the fractions have different denominators, use the steps for “Subtracting Fractions with Unlike
Denominators”
 If the fraction to the right of the minus sign is great than the fraction to the left of the minus sign, you
will have to “borrow” from the whole number to the left of the minus sign
 Subtract 1 from the fraction to the left of the minus sign
 Add the denominator value on the fraction to the left of the minus sign to the numerator value to the
fraction to the left of the minus sign (creating an improper fraction)
 Subtract the whole number to the right of the minus sign from the whole number to the left of the
minus sign
 Subtract the fraction part of the number to the right of the minus sign from the improper fraction to
the left of the minus sign
 Reduce the final answer if possible
2
2
5
1
5
1
5 1
4
 Example 1: 6 – 2 = 6 – 2 + ( – ) = 4 +
=4+
=4+
=4
6
6
6
6
6
6
3
3
 Example 2: 6
1
5
7
5
7
5
2
1
1
75
–2 =5 –2 =5–2+( –
)=3+
=3+ =4+
=3
6
6
6
6
6
6
6
3
3
6
 Example 3: 8
3
1
9
4
9
4
94
5
5
–5 =8
–5
=8–5+(
–
)=3+
=3+
=3
8
6
24
24
24
24
24
24
24
 Example 4: 8
1
3
4
9
28
9
28
9
28  9
19
–5 =8
–5
=7
–5
=7–5+(
–
)=2+
=2+
=
6
8
24
24
24
24
24
24
24
24
2
19
24
81
Section 9.2 Fraction Arithmetic (continued)
Multiplying Fractions
 Reduce fractions if possible
 You can combine the numerator of one fraction with the denominator of another
 Continue the process until no further reduction is possible
 Multiply the numerators
 Multiply the denominators
 If you fully reduce all fractions before multiplying, the final answer will already be reduced
4
3 5
3
4
3
5
3
1 1
1 1
1 1
1 1
1
 Example: x x
x ( x )x( x )( x )x( x ) x
x
x )
9
8 6
5
9
8
6
5
3 2
2 1
3 2
2 1
12
4
3 5 3
180
1
 If you multiply first:
x x
x =

(reducing takes much more effort)
9
8 6 5
2160
12
 For multiplying fractions, remember this phrase  “Simplify before you multiply”
Multiplying Mixed Fractions
 Convert each mixed fraction to the corresponding improper fraction
 Reduce the fractions if possible by combining numerators and denominators (whether from the same
fraction or from different fractions)
 Multiply the numerators
 Multiply the denominators
 If the answer results in an improper fraction, convert the final answer back to a mixed fraction
4
3
22 14
2 14
28
1
 Example: 2 x 1 
x

x

3
11
9
11
9
1
9
9
9
Dividing Fractions by Fractions
 Multiply first fraction by the reciprocal of the second fraction
 Often phrased by teachers as “keep it, change it, flip it”
 “Keep” the first fraction as is
 “Change” the operation from division to multiplication
 “Flip” the numerator and denominator of the second fraction
 Do not attempt to divide fractions, but always multiply them
3
3
5
15
7
2
 Example: ÷
 x

(or 1 )
5
4
4
2
8
8
3
 “Keep”
as is
4
 “Change” the operation from division to multiplication
2
5
 “Flip”
to
2
5
82
Section 9.2 Fraction Arithmetic (continued)
Dividing Fractions by Whole Numbers
 Use the “keep it, change it, flip it” concept
 “Keep” the first fraction as is
 “Change” the operation from division to multiplication
 “Flip” the whole number to become a fraction (the reciprocal of the whole number)
 Now you multiply the first fraction by the second fraction
 See if you can reduce the fractions’ numerator and/or denominator prior to multiplying
 One all numerators and denominators have been reduced, multiply the numerators
 Then multiply the denominators
 At this point, the final answer should already be reduced
3
3
1
3
 Example 1: ÷ 5  x 
5
20
4
4
 Example 2:
4
4 1
1 1
1
÷8 x  x 
5
5
8
5 2
10
Dividing Whole Numbers by Fractions
 Use the “keep it, change it, flip it” concept
 “Keep” the first value (whole number) as is
 “Change” the operation from division to multiplication
 “Flip” the numerator and denominator of the second value (the fraction)
 Now you multiply the whole number by the “flipped” second value
 See if you can reduce the fractions’ numerator and/or denominator prior to multiplying
 One all numerators and denominators have been reduced, multiply the numerators
 Then multiply the denominators
 At this point, the final answer should already be reduced
 If the answer is an improper fraction, you can convert it to a mixed fraction
3
4
20
2
 Example 1: 5 ÷
5x

6
3
3
3
4
 Example 2: 6 ÷
9
11
11
22
1
6x
2x

7
11
9
3
3
3
83
Section 9.2 Fraction Arithmetic (continued)
Dividing Mixed Fractions
 Convert each mixed fraction to the corresponding improper fraction
 Use the “keep it, change it, flip it” concept
 “Keep” the first value (the original fraction) as is
 “Change” the operation from division to multiplication
 “Flip” the numerator and denominator of the second value (the improper fraction)
 Reduce the fractions if possible by combining numerators and denominators (whether from the same
fraction or from different fractions)
 Multiply the first value by the second
 At this point, the final answer should already be reduced
 If the answer is an improper fraction, you can convert it to a mixed fraction
4
3
14
28
14
3
1 3
3
 Example 1: 2 ÷ 5 
÷

x
 x

5
5
5
3
5
28
5 2
10
 Example 2: 5
3
4
28 14
28
5
2
5
2
÷2 
÷

x

x 
2
5
5
5
5
5
14
5
1
1
84
Fraction Arithmetic – Guided Practice
__________ 1. What is
5
7
+
?
13 13
__________ 2. What is
3
1
+ ?
8
6
__________ 3. What is 4
4
3
+6 ?
5
8
__________ 4. What is
7
5
–
?
13 13
__________ 5. What is
3 1
– ?.
8 6
__________ 6. What is 6
__________ 7. What is
3
4
– 4 ?.
8
5
3
2
x
?
4
3
__________ 8. What is 3
3
2
x 6 ?
4
3
__________ 9. What is 3
3
2
÷ 6 ?
4
3
__________ 10. Which fraction has the largest value:
85
4 7 5 3
, , , ?
5 8 6 4
Section 9.3 Advanced Fraction Principles
Fractions (Tenths, Hundredths, Thousandths, and Beyond)
 Similar to integers with place value, fractions can have place value
 Special fraction place value occurs when the numerator is 1 and the denominator contains 1 followed
by one or more zeros
 The fractions turn out to be the reciprocals of ten, hundred, thousand, ten thousand, etc.
 They have the corresponding names of tenth, hundredth, thousandth, ten thousandth, etc.
 Here are examples of these special fractions
1

 “one tenth”
10
1

 “one hundredth”
100
1

 “one thousandth”
1000
1

 “one the thousandth”
10000
 To add or subtract these types of fractions, you must convert one or more of the fractions to have a
common denominator
1
1
10
1
11

+

+

(“eleven hundredths”)
10 100
100 100
100
1
1
10
1
9

–

–

(“nine hundredths”)
10 100
100 100
100
1
1
100
1
101

+

+

(“one hundred one ten thousandths”)
100 10000
10000 10000
10000
1
1
100
1
99

–

–

(“ninety nine ten thousandths”)
100 10000
10000 10000
10000
 To multiply these types of fractions, the final answer will have the sum of the number of zeros that the
fractions involved in the multiplication have
1
1
1

x

(“one thousandth”)
10 100
1000
1
1
1

x

(“one millionth”)
100 10000
100000
 To multiply these types of fractions, multiply the first fraction by the reciprocal of the second fraction
1
1
1
1000
1000

÷

x

 100
10 1000
10
1
10
1
1
1
100
1

÷

x

(“one thousandth”)
100000 100
100000
1
1000
86
Section 9.3 Advanced Fraction Principles (continued)
Introduction to Complex Fractions
 A complex fraction contains a fraction within a fraction
 A fraction in the numerator.
 A Fraction in the denominator
 A fraction in both the numerator and the denominator.
 Examples of complex fractions



1
3
8
5
2
5
3
7
4
5
Solving Complex Fractions
 Complex fractions must be converted to “simple fractions” where there is neither a fraction in the
numerator nor a fraction in the denominator
 Treat the “division bar” of the main fraction as a “division symbol” between the main fractions
numerator and denominator
1
1
1 1
1
3

 ÷ 8 (“Dividing Fractions by Whole Numbers”)  x 
3
3 8
24
8


5
2
5
3
7
4
5
5÷

2
5
25
1
(“Dividing Whole Numbers by Fractions”)  5 x 
12
5
2
2
2
3
4
3
5
15
÷
(“Dividing Fractions by Fractions”) 
x 
7
5
7
4
28
Introduction to Word Problems with Fractions
 Word problems that cause any of the following fraction operations
 Specifying fractions
 Reducing (simplifying) fractions
 Adding fractions
 Subtracting fractions
 Multiplying fractions
 Dividing fractions
87
Section 9.3 Advanced Fraction Principles (continued)
Solving Word Problems with Fractions
 Understand what the word problem is telling you
 Understand what the word problem is asking you
 Example 1: John had 10 marbles in his pocket. He lost 3 marbles because of a hole in his pocket.
Specify the fractions of marbles lost.
 Total marbles that John had  10
 Total marbles lost by John  3
3
 Fraction of marbles lost 
10
 Example 2: John had 10 marbles in his pocket. He lost 3 marbles because of a hole in his pocket.
Specify the fractions of marbles that John still has.
 Total marbles that John had  10
 Total marbles lost by John  3
 Total marbles that John still has  10 – 3 = 7
7
 Fraction of marbles John still has 
10
88
Advanced Fraction Principles – Guided Practice
__________ 1. What is
1
1
+
?
1000 10000
__________ 2. What is
1
1
+
?
10000 1000000
__________ 3. What is
1
1
+
?
10 1000
__________ 4. What is
1
1
+
?
100000 1000000
__________ 5. What is
1
1
+
?
10 1000000
__________ 6. Simplify
__________ 7. Simplify
__________ 8. Simplify
__________ 9. Simplify
1
4
6
1
2
9
1
3
7
3
4
3
__________ 10. Simplify
4
5
8
89
Advanced Fraction Principles – Guided Practice (continued)
__________ 11. Simplify
__________ 12. Simplify
__________ 13. Simplify
__________ 14. Simplify
__________ 15. Simplify
8
3
4
10
2
5
13
2
3
16
1
5
9
9
10
__________ 16. Simplify
2
5
3
4
__________ 17. Simplify
3
4
2
5
__________ 18. Simplify
5
8
1
2
__________ 19. Simplify
5
8
1
2
__________ 20. Simplify
5
7
2
5
90
Advanced Fraction Principles – Guided Practice (continued)
__________ 21. Mary was planning to bake 12 pies for the family reunion. She ran out of time and only
baked 10 pies. Specify the pies baked as a simplified fraction of the pies planned.
__________ 22. In the state basketball tournament, Bill shot 15 free throws. He made 12 and missed 3.
Specify the free throws made as a simplified fraction of the free throws attempted.
__________ 23. In the state basketball tournament, Robert shot 18 free throws. He made 14 and missed 4.
Specify the free throws missed as a simplified fraction of the free throws attempted.
__________ 24. In the state basketball tournament, Henry shot 14 free throws. He made 9 and missed 5.
Specify the fraction of free throws missed to free throw made.
91
Module 9 (Principle of Fractions) – Review Exercises
Fraction Fundamentals
Write the following as a fraction (questions 1 – 3)
__________ 1. 20 cars sold out of 27 cars available
__________ 2. 34 book read out of 35 books assigned
__________ 3. 21 green marbles out of 25 total marbles
Write yes or no to indicate if the specified is a Simplest-Form Fraction (question 4 – 7)
__________ 4.
24
45
________ 5.
31
53
________ 6.
39
57
________ 7.
49
84
Reduce the following fractions to Simplest-Form (question 8 – 11)
__________ 8.
15
27
________ 9.
18
45
________ 10.
35
63
________ 11.
39
65
Create an Equivalent Fraction with the specified numerator or denominator (question 12 – 17)
_____ 12.
4
(numerator 28)
7
_____ 13.
3
2
(denominator 24) _____ 14.
(numerator 16)
5
4
_____ 15.
7
(denominator 72)
8
_____ 16.
5
(numerator 55)
6
92
_____ 17.
7
(denominator 108)
9
Module 9 (Principles of Fractions) – Review Exercises (continued)
Fractional Fundamentals (continued)
Convert the following fractions (Improper to Mixed or Mixed to Improper) (question 18 – 17)
______ 18.
35
8
________ 19. 7
3
5
________ 20.
53
9
________ 21. 9
4
7
What is the reciprocal of each fraction, specified as proper or improper? (question 22 – 25)
__________ 22.
26
7
________ 23. 4
3
4
________ 24.
16
27
________ 25. 8
7
9
What is the least common denominator (LCD) of each set of fractions? (question 26 – 28)
__________ 26.
2 5 3
, ,
5 6 4
__________ 27.
3 1 3
, ,
8 6 16
__________ 28.
3 1 3
, ,
8 6 16
Convert the following fractions to have a least common denominator (question 29 – 31)
__________ 29.
2 5 3
, ,
5 6 4
__________ 30.
3 1 3
, ,
8 6 16
93
__________ 31.
3 1 3
, ,
8 6 16
Module 9 (Principles of Fractions) – Review Exercises (continued)
Fraction Arithmetic
Fractional answers should be fully reduced unless otherwise specified.
__________ 1. Fully reduce the fraction
12
.
20
__________ 2. Fully reduce the fraction
36
.
48
__________ 3. Fully reduce the fraction
36
.
60
__________ 4. Fully reduce the fraction
48
.
72
__________ 5. Fully reduce the fraction
80
.
96
__________ 6. Unreduce the fraction
5
to create an Equivalent Fraction (numerator 40).
6
__________ 7. Unreduce the fraction
3
to create an Equivalent Fraction (denominator 45).
5
__________ 8. Unreduce the fraction
7
to create an Equivalent Fraction (numerator 48).
8
__________ 9. Unreduce the fraction
9
to create an Equivalent Fraction (denominator 70).
10
__________ 10. Unreduce the fraction
7
to create an Equivalent Fraction (denominator 84).
12
94
Module 9 (Principles of Fractions) – Review Exercises (continued)
Fractional Arithmetic (continued)
__________ 11. Convert the fraction
33
to Mixed.
5
__________ 12. Convert the fraction
43
to Mixed.
6
__________ 13. Convert the fraction
55
to Mixed.
6
__________ 14. Convert the fraction
55
to Mixed.
7
__________ 15. Convert the fraction
67
to Mixed.
8
__________ 16. Convert the fraction 3
5
to Improper.
8
__________ 17. Convert the fraction 5
3
to Improper.
4
__________ 18. Convert the fraction 9
7
to Improper.
8
__________ 19. Convert the fraction 12
3
to Improper.
58
__________ 20. Convert the fraction 15
1
to Improper.
4
95
Module 9 (Principles of Fractions) – Review Exercises (continued)
Fractional Arithmetic (continued)
__________ 21. What is the reciprocal of
5
?.
9
__________ 22. What is the reciprocal of
13
?.
7
__________ 23. What is the reciprocal of 2
4
?.
5
__________ 24. What is the reciprocal of 7
5
?.
6
__________ 25. What is the reciprocal of 11
__________ 26. What is
3
2
+ ?
7
7
__________ 27. What is
1
5
+ ?
8
8
__________ 28. What is 5
1
4
+4 ?
7
7
__________ 29. What is 5
4
3
+7 ?
5
5
__________ 30. What is 12
2
?.
5
8
7
+9 ?
9
9
96
Module 9 (Principles of Fractions) – Review Exercises (continued)
Fractional Arithmetic (continued)
__________ 31. What is
1
5
+ ?
5
8
__________ 32. What is
3
1
+ ?
8
6
__________ 33. What is
4
7
+ ?
5
8
__________ 34. What is 4
2
5
+7 ?
8
3
__________ 35. What is 9
7
1
+4 ?
5
8
__________ 36. What is
3
2
– ?
7
7
__________ 37. What is
1 5
– ?
8 8
__________ 38. What is 5
1
4
–4 ?
7
7
__________ 39. What is 5
4
3
–7 ?
5
5
__________ 40. What is 12
8
7
–9 ?
9
9
97
Module 9 (Principles of Fractions) – Review Exercises (continued)
Fractional Arithmetic (continued)
__________ 41. What is
1 5
– ?
5 8
__________ 42. What is
3 1
– ?
8 6
__________ 43. What is
4
7
– ?
5
8
__________ 44. What is 4
2
5
–7 ?
8
3
__________ 45. What is 9
7
1
–4 ?
5
8
__________ 46. What is
5
3
x ?
8
5
__________ 47. What is
8
15
x
?
5
16
__________ 48. What is 2
1
1
x 5 ?
3
4
__________ 49. What is 4
5
2
x 2 ?
8
3
__________ 50. What is 6
5
1
x 1 ?
8
3
98
Module 9 (Principles of Fractions) – Review Exercises (continued)
Fractional Arithmetic (continued)
__________ 51. What is
1
3
÷
?
4
4
__________ 52. What is
1
3
÷ 1 ?
4
3
__________ 53. What is 2
3
2
÷ 1 ?
5
5
__________ 54. What is 5
5
1
÷ 3 ?
6
2
__________ 55. What is 8
3
7
÷ 1 ?
8
8
__________ 56. Which fraction has the largest value:
3 1 3 2
, ,
, ?
8 3 10 5
__________ 57. Which fraction has the largest value:
3 4 1 1
,
, , ?
8 10 3 6
__________ 58. Which fraction has the largest value:
1 5 3 2
, , , ?
2 8 5 3
__________ 59. Which fraction has the largest value:
3 2 5 3
, , , ?
4 3 8 5
__________ 60. Which fraction has the largest value:
7 4 5 3
, , , ?
8 5 6 4
99
Module 9 (Principles of Fractions) – Review Exercises (continued)
Advanced Fraction Principles
__________ 61. What is
1
1
+
?
10 100
__________ 62. What is
1
1
+
?
10 1000
__________ 63. What is
1
1
+
?
10 10000
__________ 64. What is
1
1
+
?
10 100000
__________ 65. What is
1
1
+
?
10 1000000
__________ 66. What is
1
1
+
?
100 1000
__________ 67. What is
1
1
+
?
100 10000
__________ 68. What is
1
1
+
?
100 100000
__________ 69. What is
1
1
+
?
1000 100000
__________ 70. What is
1
1
+
?
1000 1000000
100
Module 9 (Principles of Fractions) – Review Exercises (continued)
Advanced Fraction Principles (continued)
__________ 71. Simplify
__________ 72. Simplify
__________ 73. Simplify
__________ 74. Simplify
__________ 75. Simplify
__________ 76. Simplify
__________ 77. Simplify
__________ 78. Simplify
__________ 79. Simplify
__________ 80. Simplify
1
5
12
1
6
12
1
8
12
4
5
12
5
6
12
3
8
12
12
1
5
12
4
5
12
3
8
12
8
3
101
Module 9 (Principles of Fractions) – Review Exercises (continued)
Advanced Fraction Principles (continued)
__________ 81. Simplify
7
8
3
4
__________ 82. Simplify
5
6
2
3
__________ 83. Simplify
5
6
3
2
__________ 84. Simplify
1
4
9
10
__________ 85. Simplify
6
5
3
5
The following scenario applies to questions 85 – 90.
During target practice with a bow and arrow, John had the following results:
1) Hit the bulls eye 3 times
2) Hit the target in an area other than the bulls eye 7 times
3) Missed the target completely 5 times
__________ 86. Write the fraction of bulls eyes made to total shots attempted.
__________ 87. Write the fraction of bulls eyes made to total missed target.
__________ 88. Write the fraction of target hit (not in the bulls eye area) to total shots attempted.
__________ 89. Write the fraction of target hit (in any area) to total shots attempted.
__________ 90. Write the fraction of target hit (in any area) to target missed completely.
102