Slide 1 / 156 Slide 2 / 156 8th Grade The Number System and Mathematical Operations Part 2 2015-08-31 www.njctl.org Slide 3 / 156 Slide 4 / 156 Table of Contents Click on topic to go to that section. Squares of Numbers Greater than 20 Simplifying Perfect Square Radical Expressions Approximating Square Roots Squares of Numbers Rational & Irrational Numbers Real Numbers Greater than 20 Properties of Exponents Glossary & Standards Return to Table of Contents Slide 5 / 156 Slide 6 / 156 Square Root of Large Numbers Square Root of Large Numbers It helps to know the squares of larger numbers such as the multiples of tens. Think about this... 10 2 = 100 20 2 = 400 What about larger numbers? 30 2 = 900 40 2 = 1600 How do you find ? 50 2 = 2500 60 2 = 3600 70 2 = 4900 80 2 = 6400 90 2 = 8100 100 2 = 10000 What pattern do you notice?
Slide 7 / 156 Slide 8 / 156 Square Root of Large Numbers Square Root of Large Numbers Examples: For larger numbers, determine between which two multiples of ten the number lies. Squares List of Lies between 2500 & 3600 (50 and 60) 10 2 = 100 1 2 = 1 Ends in nine so square root ends in 3 or 7 20 2 = 400 2 2 = 4 Try 53 then 57 30 2 = 900 3 2 = 9 53 2 = 2809 40 2 = 1600 4 2 = 16 50 2 = 2500 5 2 = 25 60 2 = 3600 6 2 = 36 70 2 = 4900 7 2 = 49 80 2 = 6400 8 2 = 64 Lies between 6400 and 8100 (80 and 90) 90 2 = 8100 9 2 = 81 Ends in 4 so square root ends in 2 or 8 100 2 = 10000 10 2 = 100 Try 82 then 88 82 2 = 6724 NO! Next, look at the ones digit to determine the ones digit 88 2 = 7744 of your square root. Slide 9 / 156 Slide 10 / 156 1 Find. 2 Find. Squares List of Squares List of Slide 11 / 156 Slide 12 / 156 3 Find. 4 Find. Squares List of Squares List of
Slide 13 / 156 Slide 14 / 156 5 Find. 6 Find. Squares Squares List of List of Slide 15 / 156 Slide 16 / 156 7 Find. 8 Find. Squares Squares List of List of Slide 17 / 156 Slide 18 / 156 9 Find. Simplifying Perfect Square Radical Expressions Squares List of Return to Table of Contents
Slide 19 / 156 Slide 20 / 156 Two Roots for Even Powers If we square 4, we get 16. If we take the square root of 16, we get two answers: -4 and +4. That's because, any number raised to an even power, such as 2, 4, 6, etc., becomes positive, even if it started out being negative. So, (-4) 2 = (-4)(-4) = 16 AND (4) 2 = (4)(4) = 16 This can be written as √16 = ±4, meaning positive or negative 4. This is not an issue with odd powers, just even powers. Slide 21 / 156 Slide 22 / 156 Is there a difference between... Evaluate the Expressions & ? Which expression has no real roots? Evaluate the expressions: is not real Slide 23 / 156 Slide 24 / 156 10 11 A 6 A 9 B -6 B -9 C is not real C is not real
Slide 25 / 156 Slide 26 / 156 (Problem from ) 12 13 A 20 B -20 C is not real Which student's method is not correct? A Ashley's Method B Brandon's Method On your paper, explain why the method you selected is not correct. Slide 27 / 156 Slide 28 / 156 14 15 Slide 29 / 156 Slide 30 / 156 16 17 A 3 A -3 B B C No real roots C D
Slide 31 / 156 Slide 32 / 156 Square Roots of Fractions 18 The expression equal to a is equivalent to a positive integer when b is b = b 0 A -10 B 64 C 16 D 4 16 4 = 49 = 7 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. Slide 33 / 156 Slide 34 / 156 19 C A no real solution B D Slide 35 / 156 Slide 36 / 156 20 C A B D no real solution
Slide 37 / 156 Slide 38 / 156 22 23 C C A A no real solution no real solution B D B D Slide 39 / 156 Slide 40 / 156 Square Roots of Decimals 24 Evaluate To find the square root of a decimal, convert the decimal to a fraction first. Follow your steps for square roots of fractions. A B C no real solution D = .2 = .05 = .3 Slide 41 / 156 Slide 42 / 156 25 Evaluate 26 Evaluate .06 B .6 B 11 A 0.11 A 6 1.1 C C no real solution no real solution D D
Slide 43 / 156 Slide 44 / 156 27 Evaluate 0.08 0.8 B A C D no real solution Slide 45 / 156 Slide 46 / 156 Perfect Square All of the examples so far have been from perfect squares. Approximating What does it mean to be a perfect square? Square Roots The square of an integer is a perfect square. A perfect square has a whole number square root. Return to Table of Contents Slide 47 / 156 Slide 48 / 156 Non-Perfect Squares Non-Perfect Squares Square Perfect Root Square Think about the square root of 50. You know how to find the square root of a perfect square. 1 1 2 4 Where would it be on this chart? What happens if the number is not a perfect square? 3 9 4 16 What can you say about the square Does it have a square root? 5 25 root of 50? 6 36 What would the square root look like? 7 49 50 is between the perfect squares 49 8 64 and 64 but closer to 49. 9 81 10 100 So the square root of 50 is between 7 11 121 and 8 but closer to 7. 12 144 13 169 14 196 15 225
Slide 49 / 156 Slide 50 / 156 Approximating Non-Perfect Squares Approximating Non-Perfect Squares Square Perfect Square Perfect When approximating square roots of Approximate the following: Root Square Root Square numbers, you need to determine: 1 1 1 1 2 4 2 4 3 9 · Between which two perfect squares it lies 3 9 4 16 (and therefore which 2 square roots). 4 16 5 25 5 25 6 36 · Which perfect square it is closer to (and 6 36 7 49 therefore which square root). 7 49 8 64 8 64 9 81 9 81 10 100 Example: 10 100 11 121 11 121 12 144 12 144 Lies between 100 & 121, closer to 100. 13 169 13 169 14 196 14 196 So is between 10 & 11, closer to 10. 15 225 15 225 Slide 51 / 156 Slide 52 / 156 Approximating Non-Perfect Squares Approximating Non-Perfect Squares Approximate to the nearest integer Approximate to the nearest integer < < Identify perfect squares closest to 70 < < Identify perfect squares closest to 38 < < Take square root 6 < < 7 Take square root Identify nearest integer Answer: Because 38 is closer to 36 than to 49, is closer to 6 than to 7. So, to the nearest integer, = 6 Slide 53 / 156 Slide 54 / 156 Approximating a Square Root Approximating a Square Root Another way to think about it is to use a number line. Approximate on a number line. Example: Approximate 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 10.0 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 11.0 Since 8 is closer to 9 than to 4, is closer to 3 than to 2; 115 lies between ______ and ______ so # 2.8 So, it lies between whole numbers ____ and ____. Imagine the square roots between these numbers, and picture where lies.
Slide 55 / 156 Slide 56 / 156 29 The square root of 40 falls between which two perfect 30 Which integer is closest to? squares? A 3 and 4 B 5 and 6 C 6 and 7 D 7 and 8 < < Identify perfect squares closest to 40 < < Take square root Identify nearest integer Slide 57 / 156 Slide 58 / 156 31 The square root of 110 falls between which two perfect 32 Estimate to the nearest integer. squares? A 36 and 49 B 49 and 64 C 64 and 84 D 100 and 121 Slide 59 / 156 Slide 60 / 156 33 Select the point on the number line that best 34 Estimate to the nearest integer. approximates the location of . B C E G A D F H From PARCC EOY sample test non-calculator #19
Slide 61 / 156 Slide 62 / 156 35 Estimate to the nearest integer. 36 Approximate to the nearest integer. Slide 63 / 156 Slide 64 / 156 37 Approximate to the nearest integer. 38 Approximate to the nearest integer. Slide 65 / 156 Slide 66 / 156 39 Approximate to the nearest integer. 40 Approximate to the nearest integer.
Slide 67 / 156 Slide 68 / 156 41 The expression is a number between: 42 For what integer x is closest to 6.25? A 3 and 9 B 8 and 9 C 9 and 10 D 46 and 47 From the New York State Education Department. Office of Assessment Policy, Development and Derived from Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. Slide 69 / 156 Slide 70 / 156 44 Between which two positive integers does lie? 43 For what integer y is closest to 4.5? A 1 F 6 B 2 G 7 H 8 C 3 I 9 D 4 J 10 E 5 Derived from Derived from Slide 71 / 156 Slide 72 / 156 45 Between which two positive integers does lie? 46 Between which two labeled points on the number line would be located? F 6 A 1 G 7 B 2 A B C D E F G H I J C 3 H 8 D 4 I 9 E 5 J 10 Derived from Derived from
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