Real Number Properties and Basic Word Problems

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Real Number Properties and Basic Word Problems
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Natural Numbers

• Known as Counting Numbers
• Example 1, 2, 3, 4, 5,.

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Whole Numbers

• You add the number 0 to the natural numbers.
• Example 0, 1, 2, 3, 4, 5.

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Integers

• Integers are made up of whole numbers and their
  opposites.
• Example -4,-3,-2,-1,0,1,2,3,4.

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Rational Numbers

• The set of rational numbers is made up of all of
  the following
• a. Natural Numbers
• b. Whole Numbers
• c. Integers
• d. Plus every repeating and
• terminating decimal.

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Examples of Rational Numbers

• A. ½ 0.5 (Terminating Decimal)
• B. 1.23232323 (Repeating Decimal)
• C. 0.256256256 (Repeating Decimal)
• D. 2.735 (Terminating Decimal)

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Irrational Numbers.

• Consists of Non-Terminating and Non-Repeating
  Decimals.
• Example 0.9482137507264

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Real Numbers (R)
Rational Numbers (Q)
Irrational Numbers
Integers (Z)
Decimal form is non-terminating and non-repeating
Whole Numbers
Natural Numbers (N)
1, 2, 3,
0, 1, 2, 3,
-3, -2, -1, 0, 1, 2, 3,
Decimal form either terminates or repeats
All rational and irrational numbers
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The Number Line

• A number line consists of positive numbers (right
  of 0) and negative numbers (left of 0).
• A real life example of a number line is a
  temperature thermometer.

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For example..

• -5 would represent 5 degrees below zero.
• 4 would represent 4 degrees above zero.

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Make the Comparison

• 7 degrees below 0 is (warmer/colder) than 4
  degrees above 0.
• 7 degrees below 0 is a (lower/higher) temperature
  than 4 degrees above 0.

• Colder
• Lower

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Coordinates on a Graph.

• Find the best estimate of the point.
• a. -2 b. 2 c. -1.75 d. -1.5

Answer -1.75
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Sets and Subsets

• A set is a group of numbers.
• Example Set A 1,2,3,4,5
• A subset is a group of numbers in which every
  member is in another set.
• Example Set B 1,2,3
• So, B is a subset of A.

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Which of the following would represent a subset
of integers?

• States Sales Tax Rate
• Amount of Gas in a Car
• Number of Students in Class
• A Dinner Receipt
• Strategy Eliminate those that are NOT integers.

• 7.5 - NO
• 6.5 Gallons NO
• 12 YES
• 10.31 - NO

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You TryWhich of the following would represent a
subset of integers?

• Costs of a TV
• of miles on the odometer of a car
• A persons weight
• Number of residents in South Carolina

• No
• No
• No
• Yes

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Inequalities..

• We use inequalities to compare numbers.
• The following are inequalities

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Examples.

• 4 is less than 7 -
• 9 is greater than or equal to 5 -

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You TryInsert the appropriate inequality sign.

1. -5 -2
2. -7 2
3. 4 -12

1. lt
2. lt
3. gt

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Least to Greatest

• This means to arrange numbers in the order from
  the smallest to the largest.
• HINT If there are fractions it might be easier
  to convert to decimals first.

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Which Number is Smaller?
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Which Number is Larger?.......
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You TryCompare

• -0.67 gt -0.68
• -0.8 gt -0.86

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Which Set is Ordered from Least to Greatest?

1. -3/2, -3, 0, 2/3
2. -3, -3/2, 0, 2/3
3. 0, 2/3, -3/2, -3
4. 0, -3/2, -3, 2/3

• -3/2, -3, 0, 2/3
• -3, -3/2, 0, 2/3
• 3. 0, 2/3, -3/2, -3
• 4. 0, -3/2, -3, 2/3

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• What kinds of numbers are used to represent
  numbers below zero?

• Answer
• NEGATIVE Numbers

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• Make -8 -4 a true statement.

• Answer
• lt

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Number Properties
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Quick Review
0
-400 -200
200 400

• Coordinate of A
• a) -250 b) -300 c) -325 d) -500

2) Coordinate of B a) -210 b) -350 c) -100
d) -50 3) Coordinate of C a) 350 b) 425
c) 325 d) 275
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Quick Review

• 4) Use , -8 5
• 5) Which is smaller? or
• 6) Write from smallest to largest
• -3, -3.8, -5, 5.6, -5.6

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Commutative Property-Changes Order

• For Addition
• AB BA
• Ex. 23 5
• 32 5
• 2332

• For Multiplication
• AB BA
• Ex. 4(8) 32
• 8(4) 32
• 4(8) 8(4)

THIS IS NOT TRUE FOR SUBTRACTION OR DIVISION!
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Associative Property-Changes Grouping

• For Multiplication
• A(BC) (AB)C
• Ex.
• 2 (3 5) (2 3) 5
• 2(15) (6)5
• 30 30
• 2 x (3 x 5) (2 x 3) x 5

• For Addition
• A (B C) (A B) C
• Ex.
• 5 (2 4) (5 2) 4
• 5 6 7 4
• 11 11
• 5 (2 4) (5 2) 4

This is not true for subtraction or division!
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Which Property?

1. 3x 4 4 3x
2. 6y (7 3z) (6y 7) 3z
3. (5x 7) 8y 5x (7 8y)
4. (3x)(2x 5) (2x 5)(3x)
5. 10x 4y 4y 10x
6. (2x 5)(10y) (2x)(5 10y)

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Distributive Property

• A (B C) AB AC
• (B C) A BA CA

• A (B C) AB AC
• (B C) A BA CA

Ex. -3 (4 2x) Strategy Think -3 (4 2x)
means -3 (4 -2x) -3(4)
(-3)(-2x) -12 6x
TRY THESE A) 4 (6 2a) B) -7 (-3m 5)
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Which Property?

1. -3x(y 2) 4y -3x(y) 3x(2) 4y
2. -3y 4x(y 2) -3y 4xy 4x(2)
3. 6x (3y 1) (3y 1) 6x

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What is an example of the commutative prop. of
addition?

1. 3 5m 3 (1 4)m
2. 3 5m 5m 3
3. 3 5m 5 3m
4. 3 5m 3m 5

1. 7 4m (7 4)m
2. (5 2) 4m 7 4m
3. 7 4m 4 7m
4. 7 4m 4m 7

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Word Problems
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Words that tell you to add.

• Plus
• Increased By
• Sum
• More Than

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Words that tell you to subtract

• Minus
• Decreased By
• Difference
• Less Than

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Multiply and Divide Words..

• Multiply
• Product
• times

• Divide
• Quotient
• Ratio of

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Homework

• Worksheet