real number properties and basic word problems. natural numbers…… known as “counting”...
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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)
Irrational NumbersRational Numbers (ℚ)
Decimal form either terminates or repeats
Decimal form is non-terminating and non-repeating
Integers (ℤ)
Whole Numbers
Natural Numbers (ℕ)
1, 2, 3, …
0, 1, 2, 3, …
…-3, -2, -1, 0, 1, 2, 3, …
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.
Negative Positive0
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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
0 1 212
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 Try…Which of the following would represent a subset of integers?
Costs of a TV
# of miles on the odometer of a car
A person’s 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” -
74
59
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You Try……Insert the appropriate inequality sign.
1. -5 -2
2. -7 2
3. 4 -12
1. <
2. <
3. >
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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?
9
2
7
3 or
smalleris7
3
22222.09
2
42857.07
3
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Which Number is Larger?.......
32.013
5 or
.arg32.0,
32.032.0
3846.013
5
erlisSo
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You Try…Compare -0.67 > -0.68
-0.8 > -0.86
3
2 68.0
5
4
7
6
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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}
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
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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:
<
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Number Properties
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Quick Review
0-400 -200 200 400
1) 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 Review4) Use , : -8 5
5) Which is smaller? or
6) Write from smallest to largest:-3, -3.8, -5, 5.6, -5.6
7
5
8
3
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Commutative Property-Changes Order For Addition
A+B = B+A
Ex. 2+3 = 5 3+2 = 5 2+3=3+2
For Multiplication
AB = BA
Ex. 4(8) = 328(4) = 32
4(8) = 8(4)
THIS IS NOT TRUE FOR SUBTRACTION OR DIVISION!
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Associative Property-Changes Grouping
For AdditionA + (B + C) = (A + B) + C
Ex.5 + (2 + 4) (5 + 2) + 4 = 5 + 6 = 7 + 4 = 11 = 11
5 + (2 + 4) = (5 + 2) + 4
For MultiplicationA(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
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) +3z3) (5x + 7) + 8y = 5x + (7 + 8y)4) (3x)(2x + 5) = (2x + 5)(3x)5) 10x + 4y = 4y + 10x6) (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 – CAEx. -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?
A) 3 + 5m = 3 + (1 + 4)mB) 3 + 5m = 5m + 3C) 3 + 5m = 5 + 3mD) 3 + 5m = 3m + 5
A) 7 + 4m = (7 +4)mB) (5 + 2) + 4m = 7 + 4mC) 7 + 4m = 4 + 7mD) 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 –1. Product2. times
Divide –1. Quotient2. Ratio of
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Homework
Worksheet