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The Number System and Mathematical Operations Part 2

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The Number System and Mathematical Operations Part 2 1 8th Grade The Number System and Mathematical Operations Part 2 2015­11­20 www.njctl.org 2 Table of Contents Click on topic to go to that Squares of Numbers Greater than 20 section. Simplifying Perfect Square Radical Expressions Approximating Square Roots Vocabulary Words are bolded Rational & Irrational Numbers in the presentation. The text Real Numbers box the word is in is then Properties of Exponents linked to the page at the end Glossary & Standards of the presentation with the Teacher Notes word defined on it. 3 Squares of Numbers Greater than 20 Return to Table of Contents 4 Square Root of Large Numbers Think about this... What about larger numbers? How do you find ? 5 Square Root of Large Numbers It helps to know the squares of larger numbers such as the multiples of tens. 102 = 100 202 = 400 302 = 900 402 = 1600 502 = 2500 2 60 = 3600 Answer & 702 = 4900 Math Practice 802 = 6400 902 = 8100 1002 = 10000 What pattern do you notice? 6 Square Root of Large Numbers For larger numbers, determine between which two multiples of ten the number lies. 102 = 100 12 = 1 202 = 400 22 = 4 302 = 900 32 = 9 402 = 1600 42 = 16 502 = 2500 52 = 25 602 = 3600 62 = 36 702 = 4900 72 = 49 802 = 6400 82 = 64 902 = 8100 92 = 81 1002 = 10000 102 = 100 Next, look at the ones digit to determine the ones digit of your square root. 7 Square Root of Large Numbers Examples: 102 = 100 12 = 1 202 = 400 22 = 4 Lies between 2500 & 3600 (50 and 60) 302 = 900 32 = 9 List of 402 = 1600 42 = 16 Ends in nine so square root ends in 3 or 7 Squares Try 53 then 57 502 = 2500 52 = 25 2 2 532 = 2809 60 = 3600 6 = 36 702 = 4900 72 = 49 802 = 6400 82 = 64 902 = 8100 92 = 81 1002 = 10000 102 = 100 Lies between 6400 and 8100 (80 and 90) Ends in 4 so square root ends in 2 or 8 Try 82 then 88 822 = 6724 NO! 882 = 7744 8 1 Find. 28 Answer 102 = 100 12 = 1 202 = 400 22 = 4 302 = 900 32 = 9 List of Squares 402 = 1600 42 = 16 502 = 2500 52 = 25 602 = 3600 62 = 36 702 = 4900 72 = 49 802 = 6400 82 = 64 902 = 8100 92 = 81 1002 = 10000 102 = 100 9 2 Find. 42 Answer 102 = 100 12 = 1 202 = 400 22 = 4 302 = 900 32 = 9 List of Squares 402 = 1600 42 = 16 502 = 2500 52 = 25 602 = 3600 62 = 36 702 = 4900 72 = 49 802 = 6400 82 = 64 902 = 8100 92 = 81 1002 = 10000 102 = 100 10 3 Find. 65 Answer 102 = 100 12 = 1 202 = 400 22 = 4 302 = 900 32 = 9 List of Squares 402 = 1600 42 = 16 502 = 2500 52 = 25 602 = 3600 62 = 36 702 = 4900 72 = 49 802 = 6400 82 = 64 902 = 8100 92 = 81 1002 = 10000 102 = 100 11 4 Find. 48 Answer 102 = 100 12 = 1 202 = 400 22 = 4 302 = 900 32 = 9 List of Squares 402 = 1600 42 = 16 502 = 2500 52 = 25 602 = 3600 62 = 36 702 = 4900 72 = 49 802 = 6400 82 = 64 902 = 8100 92 = 81 1002 = 10000 102 = 100 12 5 Find. 79 Answer 102 = 100 12 = 1 202 = 400 22 = 4 302 = 900 32 = 9 List of Squares 402 = 1600 42 = 16 502 = 2500 52 = 25 602 = 3600 62 = 36 702 = 4900 72 = 49 802 = 6400 82 = 64 902 = 8100 92 = 81 1002 = 10000 102 = 100 13 6 Find. 99 Answer 102 = 100 12 = 1 202 = 400 22 = 4 302 = 900 32 = 9 List of Squares 402 = 1600 42 = 16 502 = 2500 52 = 25 602 = 3600 62 = 36 702 = 4900 72 = 49 802 = 6400 82 = 64 902 = 8100 92 = 81 1002 = 10000 102 = 100 14 7 Find. 59 Answer 102 = 100 12 = 1 202 = 400 22 = 4 302 = 900 32 = 9 List of Squares 402 = 1600 42 = 16 502 = 2500 52 = 25 602 = 3600 62 = 36 702 = 4900 72 = 49 802 = 6400 82 = 64 902 = 8100 92 = 81 1002 = 10000 102 = 100 15 8 Find. 47 Answer 102 = 100 12 = 1 202 = 400 22 = 4 302 = 900 32 = 9 List of Squares 402 = 1600 42 = 16 502 = 2500 52 = 25 602 = 3600 62 = 36 702 = 4900 72 = 49 802 = 6400 82 = 64 902 = 8100 92 = 81 1002 = 10000 102 = 100 16 9 Find. 101 Answer 102 = 100 12 = 1 202 = 400 22 = 4 302 = 900 32 = 9 List of Squares 402 = 1600 42 = 16 502 = 2500 52 = 25 602 = 3600 62 = 36 702 = 4900 72 = 49 802 = 6400 82 = 64 902 = 8100 92 = 81 1002 = 10000 102 = 100 17 Simplifying Perfect Square Radical Expressions Return to Table of Contents 18 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. 19 Square Root Of A Number Square roots are written with a radical symbol Positive square root: Negative square root: Positive & negative square roots: Negative numbers have no real square roots no real roots because there is no real number that, when squared, would equal ­16. 20 Is there a difference between... & ? Which expression has no real roots? Evaluate the expressions: Math Practice 21 Evaluate the Expressions is not real 22 10 A 6 B ­6 C is not real A Answer 23 11 A 9 B ­9 C is not real C Answer 24 12 A 20 B ­20 C is not real A Answer 25 (Problem from ) 13 Brandon's method is not correct. Brandon's method works for the Which student's method is not correct? square root of 4, but it would not A Ashley's Method work for the square root of 36. B Brandon's Method Answer Half of 36 is 18, but the square of On your paper, explain why the method you selected is 36 is 6 times 6 equals 36. Ashley not correct. describes the correct way to find the square root of a number. 26 14 5 Answer 27 15 11 Answer 28 16 A B C D B Answer 29 17 A 3 B ­3 C No real roots C Answer 30 18 The expression equal to is equivalent to a positive integer when b is A ­10 B 64 C 16 A Answer D 4 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. 31 Square Roots of Fractions a = b b 0 16 4 = = 49 7 32 Square Roots of Fractions Try These 33 19 A C B D no real solution A Answer 34 20 A C B D no real solution C Answer 35 21 A C B D no real solution B Answer 36 22 A C B D no real solution D Answer 37 23 A C B D no real solution C Answer 38 Square Roots of Decimals To find the square root of a decimal, convert the decimal to a fraction first. Follow your steps for square roots of fractions. = .2 = .05 = .3 39 24 Evaluate A B C D no real solution C Answer 40 25 Evaluate A .06 B .6 C 6 D no real solution B Answer 41 26 Evaluate A 0.11 B 11 C 1.1 D no real solution A Answer 42 27 Evaluate A 0.8 B 0.08 C D no real solution B Answer 43 28 Evaluate A B C D no real solution D Answer 44 Approximating Square Roots Return to Table of Contents 45 Perfect Square All of the examples so far have been from perfect squares. What does it mean to be a perfect square? The square of an integer is a perfect square. Math Practice A perfect square has a whole number square root. 46 Non­Perfect Squares You know how to find the square root of a perfect square. What happens if the number is not a perfect square? Does it have a square root? What would the square root look like? 47 Non­Perfect Squares Square Perfect MP1 Make sense of problems and persevere at solving them Root Square Think about the square root of 50. 1 1 and MP5 Use appropriate tools 2 4 Where would it be on this chart? 3 9 strategically are addressed on this slide 4 16 What can you say about the square 5 25 root of 50? If students get confused or stuck on 6 36 future problems, ask: 7 49 50 is between the perfect squares 49 Math Practice Which tool would be best for solving 8 64 and 64 but closer to 49. this problem? 9 81 Answer: Square root chart 10 100 So the square root of 50 is between 7 You can also ask the questions given 11 121 and 8 but closer to 7. on this slide to assist students. 12 144 13 169 14 196 15 225 48 Approximating Non­Perfect Squares Square Perfect Root Square When approximating square roots of 1 1 numbers, you need to determine: 2 4 3 9 • Between which two perfect squares it lies 4 16 (and therefore which 2 square roots).
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