Estimating & Approximating Square Roots.

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Presentation transcript:

Estimating & Approximating Square Roots

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. A perfect square has a whole number square root.

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?

Square Perfect Root Square 1 1 2 4 3 9 4 16 5 25 6 36 7 49 8 64 9 81 10 100 11 121 12 144 13 169 14 196 15 225 Think about the square root of 50. Where would it be on this chart? What can you say about the square root of 50? 50 is between the perfect squares 49 and 64 but closer to 49. So the square root of 50 is between 7 and 8 but closer to 7.

Square Perfect Root Square 1 1 2 4 3 9 4 16 5 25 6 36 7 49 8 64 9 81 10 100 11 121 12 144 13 169 14 196 15 225 When estimating square roots of numbers, you need to determine: Between which two perfect squares it lies (and therefore which 2 square roots). Which perfect square it is closer to (and therefore which square root). Example: 110 Lies between 100 & 121, closer to 100. So 110 is between 10 & 11, closer to 10.

Estimate the following: 30 Square Perfect Root Square 1 1 2 4 3 9 4 16 5 25 6 36 7 49 8 64 9 81 10 100 11 121 12 144 13 169 14 196 15 225 Estimate the following: 30 200 215 Teacher Instructions: √30 - Lies between 25 & 36, closer to 25 but since almost right in the middle,  5.5 √200 - Lies between 196 & 225, closer to 196 so  14 √215 - Lies between 196 & 225, closer to 225 so  15

Approximating a Square Root Approximate to the nearest integer < < Identify perfect squares closest to 38 Take square root 6 < < 7 Answer: Because 38 is closer to 36 than to 49, is closer to 6 than to 7. So, to the nearest integer, = 6

Approximate to the nearest integer Identify perfect squares closest to 70 Take square root Identify nearest integer < Approximate to the nearest integer

Another way to think about it is to use a number line. Since 8 is closer to 9 than to 4, √8 is closer to 3 than to 2, so √8 ≈ 2.8 √8 2 2.2 2.4 2.6 2.8 3.0 2.1 2.3 2.5 2.7 2.9

Example: Approximate 10 10.2 10.4 10.6 10.8 11.0 10.1 10.3 10.5 10.7 10.9

The square root of 40 falls between which two perfect squares? 41 The square root of 40 falls between which two perfect squares? A 9 and 16 B 25 and 36 C 36 and 49 D 49 and 64 Answer: C

Which whole number is 40 closest to? 42 Which whole number is 40 closest to? Identify perfect squares closest to 40 Take square root Identify nearest integer < Answer: 6

The square root of 110 falls between which two perfect squares? 43 The square root of 110 falls between which two perfect squares? A 36 and 49 B 49 and 64 C 64 and 84 D 100 and 121 Answer: D

Estimate to the nearest whole number. 110 44 Estimate to the nearest whole number. 110 Answer: 10

Estimate to the nearest whole number. 219 45 Estimate to the nearest whole number. 219 Answer: 15

Estimate to the nearest whole number. 90 46 Estimate to the nearest whole number. 90 Answer: 9

What is the square root of 400? 47 What is the square root of 400? Answer: 20

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Approximate to the nearest integer. 48 Approximate to the nearest integer. Answer: 5

Approximate to the nearest integer. 96 49 Approximate to the nearest integer. 96 Answer: 10

Approximate to the nearest integer. 167 50 Approximate to the nearest integer. 167 Answer: 13

Approximate to the nearest integer. 140 51 Approximate to the nearest integer. 140 Answer: 12

Approximate to the nearest integer. 40 52 Approximate to the nearest integer. 40 Answer: 6

The expression is a number between 53 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 Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. Answer: C

Rational & Irrational Numbers

Rational & Irrational Numbers is rational because the radicand (number under the radical) is a perfect square If a radicand is not a perfect square, the root is said to be irrational. Ex:

Sort the following numbers. 24 25 32 36 40 52 64 100 200 225 625 1225 300 1681 3600 Rational Irrational

Rational or Irrational? 54 Rational or Irrational? A Rational B Irrational Answer: A

Rational or Irrational? 55 Rational or Irrational? A Rational B Irrational Answer: B

Rational or Irrational? 56 Rational or Irrational? A Rational B Irrational Answer: A

Rational or Irrational? 57 Rational or Irrational? A Rational B Irrational Answer: A

Rational or Irrational? 58 Rational or Irrational? A Rational B Irrational Answer: B

Which is a rational number? 59 Which is a rational number? A B p C D 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. Answer: C

60 Given the statement: “If x is a rational number, then is irrational.” Which value of x makes the statement false? A B 2 C 3 D 4 Answer: D 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.