Number Theory and Algebraic Reasoning

1
Chapter 2

• Number Theory and Algebraic Reasoning

2
Learning Objectives

• To read exponents properly
• To evaluate positive, negative, and zero powers
• To express whole numbers as powers

3
2-1 Exponents

• Exponent tells how many times to use the base
  as a factor (little number)
• Base the number youre multiplying (the big
  number)
• Power the exponent determines the power
• 34 3 x 3 x 3 x 3 81 (three to the fourth
  power)
• 57 5 x 5 x 5 x 5 x 5 x 5 x 5 78,125
• (five to the seventh power)

4
Expressing Whole Numbers as Powers

• Write each number using an exponent and the given
  base.
• 49, base 7
• 7 x 7 49
• 72 49
• 64, base 2

5
Practice

• 26
• 111
• 105
• 73
• 44
• 100
• 81, base 3
• 343, base 7
• 625, base 5
• 64, base 2

6
Negative Exponents

• 4-1
• 5-3
• 6-4
• 2-2
• 7-3
• 8-1
• 9-4

7
Think and Discuss

• Describe the relationship between 35 and 36.
• Tell which power of 8 is equal to 26 . Explain.
• Explain why any number to the first power is
  equal to that number.
• What do you do if the exponent is negative?

8
Learning Objectives

• To convert numbers in standard form to scientific
  notation
• To convert numbers in scientific notation to
  standard form
• To multiply by powers of ten mentally
• To explain why numbers are written in scientific
  notation

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2-2 Powers of Ten and Scientific Notation

• The distance from Venus to the Sun is over
  100,000,000 kilometers. You can write this
  number as a power of ten by using a base of ten
  and an exponent.
• 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 108
• Scientists use powers of ten to write really big
  numbers in a shorter way (scientific notation)
• The powers of 10 represent all of the zeros

10
Scientific Notation vs. Standard Form

• Scientific Notation 3.5 x 109
• Decimal times 10 to an exponent
• Always only one number in front of the decimal
  point
• Standard Form 3,500,000,000
• Includes all of the zeros

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Standard Form to Scientific Notation

• The planet Neptune is about 4,500,000,000 km from
  the sun.
• Move the decimal point to get one digit in front
  of the decimal.
• 4.5
• The exponent is equal to the number of places the
  decimal point is moved.
• 4.5 x 109

12
Scientific Notation to Standard Form

• Pluto is about 3.7 x 109 miles from the Sun.
  Write this distance in standard form.
• Since the exponent is 9, move the decimal point 9
  places to the right.
• 3,700,000,000

13
Practice

• Write these numbers in scientific notation.
• 4,340,000
• 327,000,000
• 1,262,000,000

14
Practice

• Write these numbers in standard form.
• 212 x 104
• 31.6 x 103
• 43 x 106
• 56 x 107

15
Think and Discuss

• Tell whether 15 x 109 is in scientific notation.
  Explain.
• Compare 4 x 103 and 3 x 104. Explain how you
  know which is greater.
• Why do scientists use scientific notation?

16
Learning Objectives

• To memorize the order of operations (PEMDAS)
• To evaluate expressions using order of operations
  showing steps
• To solve story problems using order of operations

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2-3 Order of Operations

• Please (Parentheses)
• Excuse (Exponents)
• My (Multiplication)
• Dear (Division)
• Aunt (Addition)
• Sally (Subtraction)
• Multiplication and Division (left to right)
• Addition and Subtraction (left to right)

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Practice

• Evaluate 27 18 6
• Evaluate 36 18 2 x 3 8
• Evaluate 5 62 x 10

19
More Practice

• Evaluate 36 (2 x 6) 3
• Evaluate (4 12 4) -23
• Evaluate 44 14 2 x 4 6

20
Think and Discuss

• Apply the order of operations to determine if the
  expressions 3 42 and (3 4)2 have the same
  value.
• Determine whether grouping symbols should be
  inserted in the expression 3 9 4 x 2 so that
  its value is 13.
• Give the correct order of operations for
  evaluating
• (5 3 x 20) 13 32

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Anticipation Guide

• Please indicate True or False for each statement.
• 1. Prime numbers are those that contain only two
  factors.
• 2. Multiples are those that divide evenly into a
  number.
• 3.All numbers are either prime or composite.
• 4. Prime numbers can only be odd numbers.
• 5. The number one is considered to be a prime
  number.

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Learning Objectives

• To distinguish between a prime number and
  composite number
• To define prime and composite
• To write a number as its prime factorization
  using a factor tree
• To write a number as its prime factorization
  using the birthday cake method

23
2-4 Prime Factorization

• Prime number whole number greater than 1 that
  is divisible by only 1 and itself
• Examples
• Composite number whole number with more than 2
  factors
• Examples

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Divisibility Rules

• Divisible by 2
• Last digit is even number
• Divisible by 3
• Sum of digits is divisible by 3
• Divisible by 5
• Last digit is 5 or 0
• Divisible by 9
• Sum of digits is divisible by 9
• Divisible by 10
• Last digit is 0

25
Factor Trees

• Prime factorization a composite number can be
  written as a product of its prime factors.
• Monkeys are like prime factors!
• 36 280 252

26
More Practice

• Create a factor tree to find the prime
  factorization.
• 495 150 476

27
Think and Discuss

• Explain how to decide whether 47 is prime.
• Compare prime numbers and composite numbers.
• Tell how you know when you have found the prime
  factorization of a number.

28
Learning Objectives

• To define GCF
• To use a list to find the GCF
• To use a factor tree to find the GCF
• To use the birthday cake method to find the GCF

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2-5 Greatest Common Factor

• Greatest Common Factor (GCF) the greatest whole
  number that divides evenly into each number
• Factor the numbers that divide into a number
  evenly

30
Using a List to Find GCF

• Lets compare two of your favorite movies!
• Movie A
• Movie B

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Listing or Rainbow Method

• Find the GCF of 24, 36, 48
• 24
• 36
• 48
• List all the factors
• Circle the greatest factor that is in all of the
  lists

32
Factor Tree Method

• Make a factor tree for each number
• Circle the common prime factors
• Multiply the common prime factors
• 60 45

33
Birthday Cake Method

• Step 1 Write numbers side by side.
• Step 2 Draw a shelf under the numbers and pick
  a number to divide them both by. Put the division
  answers below the numbers.
• Step 3 Repeat step 2 with another shelf or
  layer.
• Step 4 Multiply the numbers on the left side to
  get GCF!
• 24 18

34
Practice

• Find the GCF of 12 36 54
• 40, 56
• 20, 35

35
Problem Solving

• Sasha and David are making centerpieces for the
  Fall Festival. They have 50 small pumpkins and
  30 ears of corn. What is the greatest number of
  matching centerpieces they can make using all of
  the pumpkins and corn?

36
Think and Discuss

• Tell what the letters GCF stand for and explain
  what the GCF of two numbers means in your own
  words.
• Discuss whether the GCF of two numbers could be a
  prime number.

37
Learning Objectives

• To define LCM
• To find the LCM using a list
• To find the LCM using a factor tree
• To find the LCM using the birthday cake method

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2-6 Least Common Multiple

• Multiple number that is the product of that
  number and a whole number (skip-counting)
• Ex. 5, 10, 15, 20, 25 (multiples of 5)
• Least Common Multiple (LCM) the common multiple
  of two or more numbers with the least value

39
Listing Method (can be time-consuming)

• Find the LCM of 3 and 5
• 3
• 5
• Find the LCM of 4, 6, 12
• 4
• 6
• 12

40
Birthday Cake Method

• Do the same as you did for GCF, but you multiply
  all of the numbers on the left and bottom
  together (L-shape)
• Find the LCM of 78 110
• Find the LCM of 16 128

41
Practice

• Find the LCM of
• 9
• 27
• 45
• Find the LCM of 60 130

42
Problem Solving

• Charlotte and her brother are running laps on a
  track. Charlotte runs one lap every 4 minutes,
  and her brother runs one lap every 6 minutes.
  They start together. In how many minutes will
  they be together at the starting line again?

43
Think and Discuss

• Tell what the letters LCM stand for and explain
  what the LCM of two numbers is.
• Describe a way to remember the difference between
  GCF and LCM.

44
2-7 Variables and Algebraic Expressions

• Variable letter that represents a number
• Ex. x, y, n, etc.
• Constant a number because it cannot change
• Ex. 16, 25, 1954
• Algebraic expression consists of one or more
  variables and constants and operations
• Ex. N 7
• Evaluate to substitute a number in for the
  variable
• N 5 N 7 5 7 12

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Learning Objectives

• To distinguish between a constant and a variable
• To evaluate algebraic expressions containing
  variables and constants
• To evaluate algebraic expressions using order of
  operations
• To evaluate algebraic expressions with more than
  one variable

46
Evaluating Expressions

• N 3 N 7
• x 6 x -3
• Y 12 y 4

47
Using Order of Operations

• 3x -2 for x 5
• n 2 n for n 4
• 6y2 2y for y 2

48
Evaluating with Two Variables

• 3/n 2m for n 3 and m 4
• 3x 5y for x 6 and y 2

49
Think and Discuss

• Write each expression another way.
• A. 12x B. 4/y C. 3xy/2
• Explain the difference between a variable and a
  constant.

50
Learning Objectives

• To translate words into algebraic expressions
• To translate algebraic expressions into words
• To translate real world problems into algebraic
  expressions

51
2-8 Translate Words into Math

• When solving real-world problems, you will need
  to translate words into algebraic expressions.
• Example Although they are closely related, a
  Great Dane weighs about 40 times as much as a
  Chihuahua.
• 40c or 40 x c Great Danes weight

52
Addition and Subtraction Verbal Expressions

• Addition

• Subtraction

• Add
• Plus
• Sum
• More than
• Increased by

• Subtract
• Minus
• Difference
• Less than
• Decreased by
• Take away
• Less

53
Multiplication and Division Verbal Expressions

• Multiplication

• Division

• Times
• Multiplied by
• Product

• Divided into
• Divided by
• Quotient

54
Real-World Problems

• Jed reads p pages each day of a 200-page book.
  Write an algebraic expression for how many days
  it will take Jed to read the book.
• To rent a certain car for a day costs 84 plus
  0.29 for every mile the car is driven. Write an
  algebraic expression to show how much it costs to
  rent the car for a day.

55
Practice

• Write each phrase as an algebraic expression.
• The quotient of a number and 4
• W increased by 5
• The difference of 3 times a number and 7
• The quotient of 4 and a number, increased by 10

56
More Practice

• Mr. Campbell drives at 55 mi/hr. Write an
  expression for how far he can drive in h hours.
• On a history test Marissa scored 50 points on the
  essay.. Besides the essay, each short answer
  question was worth 2 points. Write an expression
  for her total points if she answered q short
  answer questions correctly.

57
Think and Discuss

• Write three different verbal expressions that can
  be represented by 2 y.
• Explain how you would determine which operation
  to use to find the number of chairs in 6 rows of
  100 chairs each.

58
Learning Objectives

• To identify like terms in an algebraic expression
  or list
• To combine like terms given an expression
• To find the perimeter of a shape by combining
  like terms

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2-9 Combining Like Terms

• Term a number, variable, or product of numbers
  and variables
• Ex. 4a, 3k5
• Coefficient number that is multiplied by a
  variable
• Ex. 4 of 4a
• Like terms terms with the same variable raised
  to the same power
• Ex. 3x and 2x, 5 and 1.8, 2x2 and 5x2

60
Identifying Like Terms

• You cant combine apples and bananas!!
• You have to group similar objects
• Identify the like terms in the list
• 5a t/2 3y2 7t x2 4z k 4.5y2 2t
  2/3a

61
Combining Like Terms

• 7x 2x
• 5x3 3y 7x3 -2y -4x2
• 3a 4q2 2b
• 45x -37y 87

62
More Practice

• 6t -4t
• 3a2 5b 11b2 -4b 2a2 -6
• 2x 3 3x 2 x

63
Think and Discuss

• Identify the variable and the coefficient in each
  term
• A. 11t B. -3a C. 4/5n
• Explain whether 5x, 5x2, and 5x3 are like terms.
• Explain how you know which terms to combine in an
  expression.

64
Learning Objectives

• To determine whether a number is a solution of an
  equation
• To determine whether a number is a solution from
  a story problem

65
2-10 Equations and Their Solutions

• Equation a mathematical statement that two
  expressions are equal in value
• Its like a balanced scale. The left side is
  equal to the right side.
• Solution the value for the variable that makes
  the equation true
• Ex. x 3 10 7 is the solution for x

66
Determine Whether a Number is a Solution

• 18 s 7
• Is 11 a solution?
• Is 25 a solution?
• 9y 2 74
• Is 8 a solution?

67
More Practice

• 13w 2 6w 103
• Does w 15?
• 3(50 t) 10t 104
• Does t 12?

68
Practice

• Nicole has 82 CDs. This is 9 more than her
  friend Jessica has. The equation 82 j 9 can
  be used to represent the number of CDs Jessica
  has. Does Jessica have 91 CDs, 85 CDs, or 73 CDs?

69
More Practice

• Tyler wants to buy a new skateboard. He has 57,
  which is 38 less than he needs. Does the
  skateboard cost 90 or 95?

70
Think and Discuss

• Compare equations with expressions.
• Give an example of an equation whose solution is
  5.

71
Learning Objectives

• To define inverse operations
• To isolate the variable and solve bye adding or
  subtracting
• To identify inverse operations in a story problem

72
2-11 Solving Equations by Adding or Subtracting

• Solve to find the solution of an equation
• Isolate the variable get the variable alone on
  one side of the equal side
• Ex. X 3 8
• Inverse operations opposite operations that
  undo each other
• Ex. Addition and subtraction
• Ex. Multiplication and division

73
Solving an Equation with Addition

• Solve the equation x 8 17
• Solve the equation y -11 20

74
Solving an Equation with Subtraction

• Solve the equation a 5 11
• Solve the equation m 16 25

75
Practice

• Michael Jordans highest point total for a single
  game was 70. The entire team scored 117 points
  in that game. How many points did his teammates
  score?
• 70 p 117

76
More Practice

• B - 7 24
• T 14 29
• C 12 35

77
Think and Discuss

• Explain how to decide which operation to use in
  order to isolate the variable in an equation.
• Describe what would happen if a number were added
  or subtracted on one side of an equation but not
  on the other side.

78
Learning Objectives

• To solve equations by isolating the variable
  through multiplication or division
• To identify inverse operations in a story problem

79
2-12 Solving Equations by Multiplication or
Division

• Multiplication and division are inverse
  operations of each other
• They undo each other

80
Solving by Multiplication

• Solve the equation x/7 20
• Solve the equation y9 2

81
Solving by Division

• Solve the equation 240 4z
• Solve the equation 51 17x

82
Real-Life Application

• If you count your heartbeats for 10 seconds and
  multiply that number by 6, you can find your
  heart rate in beats per minute. Lance Armstrong,
  who won the Tour de France four years in a row,
  from 1999 to 2002, has a resting heart rate of 30
  beats per minute. How many times does his heart
  beat in 10 seconds?
• 6b 30

83
More Practice

• h2 13
• t5 20
• 4x 48

84
Think and Discuss

• Explain how to check your solution to an
  equation.
• Describe how to solve 13x 91.
• When you solve 5p 35, will p be greater than 35
  or less than 35?
• When you solve p5 35, will p be greater than
  35 or less than 35?