Summer 2017 Grade Fundamentals Abbreviated Packet ... 8.4 Number Terminology ... 48, 72, 96, 120,...

Summer 2017

7th

Grade Fundamentals – Abbreviated Packet

“Mastering the Fundamentals”

Chris Millett

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Copyright © 2017

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.

Science, Math, and Technology Center of Excellence ® is a registered trademark of Chris Millett.

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Table of Content Module 8: Number Theory and Terminology ................................................................................................................................ 4

Section 8.1 Factors and Multiples ........................................................................................................................................... 5 Factor (Definition and Examples) ....................................................................................................................................... 5 Greatest Common Factor (GCF) ........................................................................................................................................ 5 Multiple (Definition and Examples) .................................................................................................................................... 5 Least Common Multiple (LCM) ......................................................................................................................................... 6 Factors and Multiples – Guided Practice ........................................................................................................................... 7

Section 8.2 Prime Numbers and Composite Numbers ........................................................................................................... 8 Prime Number (Definition and Examples) ......................................................................................................................... 8 Composite Number (Definition and Examples).................................................................................................................. 9 Key Math Factors .............................................................................................................................................................. 10 Key Multiplication Fact of Composite Numbers (1 – 100) ............................................................................................... 12 Prime Factorization (Definition and Examples) ............................................................................................................... 14 Equivalent Products .......................................................................................................................................................... 15 Prime Numbers and Composite Numbers – Guided Practice ......................................................................................... 16

Section 8.3 Principles of Numbers ........................................................................................................................................ 17 Positive Numbers ............................................................................................................................................................... 17 Negative Numbers ............................................................................................................................................................. 17 Zero .................................................................................................................................................................................... 17 Number Line Representation of Numbers ....................................................................................................................... 17 Principles of Numbers – Guided Practice ......................................................................................................................... 18

Section 8.4 Number Terminology ......................................................................................................................................... 19 Integer ................................................................................................................................................................................ 19 Consecutive Integers.......................................................................................................................................................... 19 Even Number ..................................................................................................................................................................... 19 Odd Number ...................................................................................................................................................................... 19 Rational Number ............................................................................................................................................................... 19 Irrational Number ............................................................................................................................................................. 19 Remainder ......................................................................................................................................................................... 20 Digits and Place Value ....................................................................................................................................................... 20 Number Terminology – Guided Practice .......................................................................................................................... 21

Section 8.5 Arithmetic Operations on Numbers................................................................................................................... 23 Common Arithmetic Operations on Numbers ................................................................................................................. 23 Standard Arithmetic Symbols ........................................................................................................................................... 23 Addition (Sum) .................................................................................................................................................................. 23 Subtraction (Difference) .................................................................................................................................................... 24 Multiplication (Product).................................................................................................................................................... 25 Division (Quotient) ............................................................................................................................................................ 26

Module 8 (Number Theory and Terminology) – Review Exercises ..................................................................................... 27 Factors and Multiples ........................................................................................................................................................ 27 Prime Numbers and Composite Numbers ........................................................................................................................ 29 Positive & Negative Number Arithmetic .......................................................................................................................... 32

Module 9: Principles of Fractions ................................................................................................................................................ 33 Section 9.1 Introduction to Fractions ................................................................................................................................... 34

Introduction to Fractions .................................................................................................................................................. 34 Types of Fractions ............................................................................................................................................................. 34 Simplest-Form Fraction .................................................................................................................................................... 34 Equivalent Fractions ......................................................................................................................................................... 35 Reciprocal .......................................................................................................................................................................... 35 Least Common Denominator (LCD) ................................................................................................................................ 35 Reducing (Simplifying) Fractions ..................................................................................................................................... 36 Unreducing (Unsimplifying) Fractions ............................................................................................................................. 36 Converting Improper Fractions to Mixed Fractions ........................................................................................................ 36 Converting Mixed Fractions to Improper Fractions ........................................................................................................ 36

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Comparing Fractions ........................................................................................................................................................ 37 Converting Fractions to have a Common Denominator .................................................................................................. 37 Introduction to Fractions – Guided Practice .................................................................................................................... 38

Section 9.2 Fraction Arithmetic ............................................................................................................................................ 40 Adding and Subtracting Fractions with Like Denominators ........................................................................................... 40 Adding and Subtracting Fractions with Unlike Denominators ....................................................................................... 40 Adding Mixed Fractions .................................................................................................................................................... 41 Subtracting Mixed Fractions ............................................................................................................................................ 42 Multiplying Fractions ........................................................................................................................................................ 43 Multiplying Mixed Fractions ............................................................................................................................................ 43 Dividing Fractions by Fractions ........................................................................................................................................ 43 Dividing Fractions by Whole Numbers ............................................................................................................................ 44 Dividing Whole Numbers by Fractions ............................................................................................................................ 44 Dividing Mixed Fractions .................................................................................................................................................. 45 Fraction Arithmetic – Guided Practice ............................................................................................................................ 46

Module 8: Number Theory and Terminology

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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

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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

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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?

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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)

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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)

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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 2

6 2 3

8 2 4

9 3

10 2 5

12 2 3 4 6

14 2 7

15 3 5

18 2 3 6 9

20 2 4 5 10

21 3 7

22 2 11

24 2 3 4 6 8 12

25 5

26 2 13

27 3 9

28 2 4 7 14

30 2 3 5 6 10 15

32 2 4 8 16

33 3 11

34 2 17

35 5 7

36 2 3 4 6 9 12 18

38 2 19

39 3 13

40 2 4 5 8 10 20

42 2 3 6 7 14 21

44 2 4 11 22

45 3 5 9 15

46 2 23

48 2 3 4 6 8 12 16 24

49 7

50 2 5 10 25

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Key Math Factors (continued)

Factors of Composite Numbers (continued)

51 3 17

52 2 4 13 26

54 2 3 6 9 18 27

55 5 11

56 2 4 7 8 14 28

57 3 19

58 2 29

60 2 3 4 5 6 10 12 15 20 30

62 2 31

63 3 7 9 21

64 2 4 8 16 32

65 5 13

66 2 3 6 11 22 33

68 2 4 17 34

69 3 23

70 2 5 7 10 14 35

72 2 3 4 6 8 9 12 18 24 36

74 2 37

75 3 5 15 25

76 2 4 19 38

77 7 11

78 2 3 6 13 26 39

80 2 4 8 10 20 40

81 3 9 27

82 2 41

84 2 4 6 7 12 14 21 42

85 5 17

86 2 23

87 3 29

88 2 4 8 11 22 44

90 2 3 5 6 9 10 15 18 30 45

91 7 13

92 2 4 23 46

93 3 31

94 2 47

95 5 19

96 2 3 4 6 8 12 16 24 32 48

98 2 7 14 49

99 3 9 11 33

100 2 4 5 10 20 25 50

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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 2 x 2

6 2 x 3

8 2 x 4

9 3 x 3

10 2 x 5

12 2 x 6 3 x 4

14 2 x 7

15 3 x 5

18 2 x 9 3 x 6

20 2 x 10 4 x 5

21 3 x 7

22 2 x 11

24 2 x 12 3 x 8 4 x 6

25 5 x 5

26 2 x 13 13

27 3 x 9

28 2 x 14 4 x 7

30 2 x 15 3 x 10 5 x 6

32 2 x 16 4 x 8

33 3 x 11

34 2 x 17

35 5 x 7

36 2 x 18 3 x 12 4 x 9 6 x 6

38 2 x 19

39 3 x 13

40 2 x 20 4 x 10 5 x 8

42 2 x 21 3 x 14 6 x 7

44 2 x 22 4 x 11

45 3 x 15 5 x 9

46 2 x 23

48 2 x 24 3 x 16 4 x 12 6 x 8

49 7 x 7

50 2 x 25 5 x 10

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Key Math Factors (continued)

Factors of Composite Numbers (continued)

51 3 x 17

52 2 x 26 4 x 13

54 2 x 27 3 x 18 6 x 9

55 5 x 11

56 2 x 28 4 x 14 7 x 8

57 3 x 19

58 2 x 29

60 2 x 30 3 x 20 4 x 15 5 x 12 6 x 10

62 2 x 31

63 3 x 21 7 x 9

64 2 x 32 4 x 16 8 x 8

65 5 x 13

66 2 x 33 3 x 22 6 x 11

68 2 x 34 4 x 17

69 3 x 23

70 2 x 35 5 x 14 7 x 10

72 2 x 36 3 x 24 4 x 18 6 x 12 8 x 9

74 2 x 37

75 3 x 25 5 x 15

76 2 x 38 4 x 19

77 7 x 11

78 2 x 39 3 x 26 6 x 13

80 2 x 40 4 x 20 8 x 10

81 3 x 27 9 x 9

82 2 x 41

84 2 x 42 4 x 21 6 x 14 7 x 12

85 5 x 17

86 2 x 23

87 3 x 29

88 2 x 44 4 x 22 8 x 11

90 2 x 45 3 x 30 5 x 18 6 x 15 9 x 10

91 7 x 13

92 2 x 46 4 x 23

93 3 x 31

94 2 x 47

95 5 x 19

96 2 x 48 3 x 32 4 x 24 6 x 16 8 x 12

98 2 x 49 7 x 14

99 3 x 33 9 x 11

100 2 x 50 4 x 25 5 x 10 10 x 10

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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

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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

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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?

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Section 8.3 Principles of Numbers

Positive Numbers Numbers greater than zero

Simple examples : 1 1½ 37.6 ½ 1,000 1,000,000 1,000,000,000

Negative Numbers Numbers less than zero

Simple examples : -1 -1½ -37.6 -½ -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

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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.

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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

The following number are not Integers: 8

7 -

3

10 1

2

1 3.1415926 5

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

Simple examples : -1,000,000 -1,000 -11

5 0

11

5 1,000 1,000,000

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 3 5

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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)

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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.

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Number Terminology – Guided Practice (continued)

13. Categorize the following numbers either as Rational or Irrational

__________ 7

1 __________ 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?

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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)


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)


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

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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


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


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)


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)


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

25


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


(-8) x 5 = -40 Multiplying a negative number with a negative number (-8) x (-5)


(-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

26


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


40 ÷ 5 = 8 Dividing a positive number by a negative number 40 ÷ (-5)


40 ÷ (-5) = -8 Dividing a negative number by a positive number (-40) ÷ 5


(-40) ÷ 5 = -8

Dividing a negative number with a negative number (-40) ÷ (-5)


(-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

27

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?





__________ 11. List 3 Multiples of 8.



28

Module 8 (Number Theory and Terminology) – Review Exercises (continued)

Factors and Multiples (continued)



__________ 16. What is the Least Common Multiple of 6 and 15?





29


Prime Numbers and Composite Numbers

__________ 1. List 3 Prime Numbers between 5 and 15.





__________ 6. List 3 Odd Composite Numbers between 4 and 16.



__________ 9. List 3 Odd Composite Numbers between 50 and 60

__________ 10. List 3 Odd Composite Numbers between 16 and 34

30


Prime Numbers and Composite Numbers (continued)

__________ 11. Perform Prime Factorization on 30 (fill in the blanks where appropriate)

30

/ \

_____ x _____

/ \ / \

_____ x _____ _____ x _____


36

/ \

_____ x _____

/ \ / \

_____ x _____ _____ x _____

/ \ / \ / \ / \

_____ x _____ _____ x _____ _____ x _____ _____ x _____


40

/ \

_____ x _____

/ \ / \

_____ x _____ _____ x _____

31


Prime Numbers and Composite Numbers (continued)


48

/ \

_____ x _____

/ \ / \

_____ x _____ _____ x _____

/ \ / \ / \ / \

_____ x _____ _____ x _____ _____ x _____ _____ x _____


60

/ \

_____ x _____

/ \ / \

_____ x _____ _____ x _____

/ \ / \ / \ / \

_____ x _____ _____ x _____ _____ x _____ _____ x _____

__________ 16. What is an equivalent Product of 9 x 4?





32


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)

33

Module 9: Principles of Fractions

34

Section 9.1 Introduction to Fractions

Introduction to Fractions

A comparison of the “part” (numerator) to the “whole” (denominator)

Three pieces out of eight equally cut pieces of pie 8

3

Five free throws made out of six free throws attempted 6

5

Four complete pizzas and seven pieces out of ten equally cut pieces of pizza 410

7

Types of Fractions

Proper

The numerator is smaller than the denominator

8

7 is a Proper Fraction

Improper

The numerator is larger than the denominator

7

8 is a Proper Fraction

Mixed

Contains both a whole number part and a fractional part

18

7

is a Mixed Fraction

Simplest-Form Fraction

A fraction where the greatest common factor of the numerator and the denominator is 1

8

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)

35

Section 9.1 Introduction to Fractions (continued)

Equivalent Fractions

Fractions with different numerators and denominators that represent the same value

The value 2

1 can be represented by the fractions

4

2 and

6

3

Therefore, the fractions 2

1,

4

2, and

6

3 are Equivalent Fractions

Equivalent fractions will eventually reduce to the same Simplest-Form Fraction

72

60 is an equivalent fraction to

42

35

Both fractions eventually reduce to 6

5

Reciprocal

Switching the numerator and denominator of a fraction

Reciprocal of 9

4

4

9

Reciprocal of 8 (or 1

8)

8

1

Reciprocal of 27

4 (or

7

18)

18

7

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

Example 1: what is the LCD of 3

2,

4

1, and

6

5

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

Example 2: what is the LCD of 4

3,

5

4,

8

5 and

10

7

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

36


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)

Example: Reduce 60

45

Determine GCF of 45 and 60 15

Divide numerator (45) and denominator (60) by the GCF (15)

1560

1545

4

3

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

Example: Unreduce 4

3

Determine a common value by which you multiply the numerator and denominator (i.e. 15)

154

153

x

x

60

45

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

Example: Convert 4

11 to a mixed fraction

Divide the denominator into the numerator 11 ÷ 4 = 2 (whole part with remainder 3)

Place the remainder over the denominator 4

3 (fractional part)

Put together the whole part and the fractional part 24

3

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

Example: Convert 2 4

3 to an improper fraction

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

Place this sum over the original denominator 4

11

37


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

Example: Compare 8

7 and

15

13

Multiply 7 and 15 (corresponds to first fraction) 105

Multiply 13 and 8 (corresponds to second fraction) 104

Since the first product (7 x 15) is larger than the second product (13 x 8), the first fraction (8

7) is

larger than the second fraction (15

13)

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

Example: Convert the following fractions to have a common denominator 3

2,

4

1, and

6

5

Determine the LCD for the fractions 12

Determine the number to multiply each fraction’s denominator’s to reach the LCD

o 3

2 multiply the denominator (3) by 4 to become 12

o 4


o 6


For each fraction, multiply both the numerator and denominator by the value that will make the fraction contain the LCD

o 3

2 multiply the numerator and denominator by 4:

3

2 x

4

4 =

12

8

o 4


4

1 x

3

3 =

12

3

o 6


6

5 x

2

2 =

12

10

38

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. 5

4 ________ 5.

21

12 ________ 6.

26

15 ________ 7.

51

17

Reduce the following fractions to Simplest-Form (question 8 – 11)

__________ 8. 20

12 ________ 9.

24

16 ________ 10.

35

20 ________ 11.

72

60

Create an Equivalent Fraction with the specified numerator or denominator (question 12 – 17)

_____ 12. 7

4 (numerator 20) _____ 13.

4

3 (denominator 16) _____ 14.

5

2 (numerator 12)

_____ 15. 8

7 (denominator 64) _____ 16.

6

5 (numerator 35) _____ 17.

9

7 (denominator 72)

Convert the following fractions (Improper to Mixed or Mixed to Improper) (question 18 – 17)

______ 18. 6

35 ________ 19. 4

5

3 ________ 20.

8

53 ________ 21. 8

7

5

39

Introduction to Fractions – Guided Practice (continued)

What is the reciprocal of each fraction, specified as proper or improper? (question 22 – 25)

__________ 22. 7

18 ________ 23. 1

4

3 ________ 24.

23

12 ________ 25. 5

8

7

What is the least common denominator (LCD) of each set of fractions? (question 26 – 28)

__________ 26. 5

2,

6

5,

15

7 __________ 27.

8

3,

6

1,

12

7 __________ 28.

15

7,

9

4,

5

3

Convert the following fractions to have a least common denominator (question 29 – 31)

__________ 29. 5

2,

6

5,

15

7 __________ 30.

8

3,

6

1,

12

7 __________ 31.

15

7,

9

4,

5

3

40

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

Example: 9

5 +

9

1 =

9

15 =

9

6 =

3

2

Example: 9

5 –

9

1 =

9

15 =

9

4

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


Example 9

4 +

6

1

Least Common Denominator (LCD) = 18

Unreduce 9

4 to have a denominator of 18 (LCD) fraction must be multiplied by

2

2

Unreduce 6


3

3

9

4 x

2

2 =

18

8 &

6

1 x

3

3 =

18

3

18

8 +

18

3

18

11 (already reduced)

Example 9

4 –

6

1

Least Common Denominator (LCD) = 18

Unreduce 9


2

2

Unreduce 6


3

3

9

4 x

2

2 =

18

8 &

6

1 x

3

3 =

18

3

18

8 –

18

3

18

5 (already reduced)

41

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


Example 1: 49

5 + 3

9

1 = 4 + 3 +

9

5 +

9

1 = 7 +

9

15 = 7 +

9

6 = 7 +

3

2 = 7

3

2

Example 2: 67

4 + 2

7

5 = 6 + 2 +

7

4 +

7

5 = 8 +

7

54 = 8 +

7

9 = 8 + 1

7

2 = 9

7

2

Example 3: 55

3 + 6

5

4 + 2

5

4 = 5 + 6 + 2 +

5

3 +

5

4 +

5

4= 13 +

5

443 = 13 +

5

11 = 13 + 2

5

1 = 15

5

1

Example 4: 38

3 + 2

6

1 = 3 + 2 +

8

3 +

6

1 = 5 +

24

9 +

24

4 = 5 +

24

49 = 5

24

13

Example 5: 48

7 + 5

6

5 = 4 + 5 +

8

7 +

6

5 = 9 +

24

21 +

24

20 = 9 +

24

2021 = 5

24

41 = 5 + 1

24

17= 6

24

17

Example 6: 25

4 + 3

8

5 + 6

10

7 = 2 + 3 + 6 +

5

4 +

8

5 +

10

7= 11 +

40

32 +

40

25 +

40

28 = 11 +

40

85 =

11 + 240

5 = 13

40

5 = 13

8

1

42


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


Example 1: 66

5 – 2

6

1 = 6 – 2 + (

6

5–

6

1) = 4 +

6

15 = 4 +

6

4 = 4 +

3

2 = 4

3

2

Example 2: 66

1 – 2

6

5 = 5

6

7 – 2

6

5 = 5 – 2 + (

6

7–

6

5) = 3 +

6

57 = 3 +

6

2 = 4 +

3

1 = 3

3

1

Example 3: 88

3 – 5

6

1 = 8

24

9 – 5

24

4 = 8 – 5 + (

24

9–

24

4) = 3 +

24

49 = 3 +

24

5 = 3

24

5

Example 4: 86

1 – 5

8

3 = 8

24

4 – 5

24

9 = 7

24

28 – 5

24

9 = 7 – 5 + (

24

28–

24

9) = 2 +

24

928 = 2 +

24

19 =

224

19

43


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

Example: 9

4 x

8

3 x

6

5 x

5

3 (

9

4 x

8

3) x (

6

5 x

5

3) (

3

1 x

2

1) x (

2

1 x

1

1)

3

1 x

2

1 x

2

1 x

1

1)

12

1

If you multiply first: 9

4 x

8

3 x

6

5 x

5

3 =

2160

180

12

1 (reducing takes much more effort)

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

Example: 29

4 x 1

11

3

9

22 x

11

14

9

2 x

1

14

9

28 3

9

1

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

Example: 4

3 ÷

5

2

4

3 x

2

5

8

15 (or 1

8

7)

“Keep” 4

3as is


“Flip” 5

2to

2

5

44


Dividing Fractions by Whole Numbers

Use the “keep it, change it, flip it” concept

“Keep” the first fraction as is


“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

Example 1: 4

3 ÷ 5

4

3 x

5

1

20

3

Example 2: 5

4 ÷ 8

5

4 x

8

1

5

1 x

2

1

10

1

Dividing Whole Numbers by Fractions


“Keep” the first value (whole number) as is


“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


If the answer is an improper fraction, you can convert it to a mixed fraction

Example 1: 5 ÷ 4

3 5 x

3

4

3

20 6

3

2

Example 2: 6 ÷ 11

9 6 x

9

11 2 x

3

11

3

22 7

3

1

45


Dividing Mixed Fractions

Convert each mixed fraction to the corresponding improper fraction


“Keep” the first value (the original fraction) as is


“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


If the answer is an improper fraction, you can convert it to a mixed fraction

Example 1: 25

4 ÷ 5

5

3

5

14 ÷

3

28

5

14 x

28

3

5

1 x

2

3

10

3

Example 2: 55

3 ÷ 2

5

4

5

28 ÷

5

14

5

28 x

14

5

5

2 x

1

5

1

2 2

46

Fraction Arithmetic – Guided Practice

__________ 1. What is 13

5 +

13

7?

__________ 2. What is 8

3 +

6

1?

__________ 3. What is 45

4 + 6

8

3?

__________ 4. What is 13

7 –

13

5?

__________ 5. What is 8

3 –

6

1?.

__________ 6. What is 68

3 – 4

5

4?.

__________ 7. What is 4

3 x

3

2?

__________ 8. What is 34

3 x 6

3

2?

__________ 9. What is 34

3 ÷ 6

3

2 ?

__________ 10. Which fraction has the largest value: 5

4 ,

8

7,

6

5,

4

3?

Summer 2017 Grade Fundamentals Abbreviated Packet ... 8.4 Number Terminology ... 48, 72, 96, 120,...

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