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Grade 9 Math Unit 5 – Square Roots & Surface Area 5.1 – Square Roots of Perfect Squares To determine the area of a square or rectangle, we multiply the length by the width. In the case of a square though, the length and width will always be the same. When we multiply a number by itself, we say that we are taking the square of the number. Ex. #1 Square all numbers from 1 to 15. Answer: 12= 1 22 = 4 32 = 9 42 = 16 52 = 25 62 = 36 72 = 49 82 = 64 92 = 81 102= 100 112 = 121 122 = 144 132 = 169 142 = 196 152 = 225 Ex. #2 Find the areas of the following squares. a. Area = 8 x 8 = 64 cm2 8 cm b. 5 5 25 2 Area m 6 6 36 5 m 6 Finding the square root is the opposite of finding the square. Instead of taking a number multiplied by itself to find the square, we have the square and need to find what number multiplied by itself will equal that number. Ex. #3 Complete the following table. Area of a Square Side Length (Square Root) 49 64 121 144 16/100 25/81 4/9 36/64 7 8 11 12 4/10 5/9 2/3 6/8 = 3/4 Ex. #4 Solve the following square roots. a. 36 = 6 b. 4 = 2/3 9 Perfect Squares A number that is the square of a whole number is called a perfect square. Example: 16 is a perfect square because 16 = 42. Fractions can also be perfect squares if both the numerator and the denominator are whole numbers. Example: 25/100 is a perfect square because 25 5 1 100 10 2 When a fraction that is a perfect square is written as a decimal, then the decimal is also a perfect square. The square root will be a terminating (ex. 3.5) or repeating (ex. 0.3333…) decimal. Numbers that are terminating or repeating are called rational numbers. Example: Solve the following square roots. State whether each answer is a rational number and explain why. 100 10 a. 5 Rational because it is a whole number, therefore is terminating. 4 2 b. 25 Not rational because the decimal is not repeating or terminating. Also, 29 the denominator is not a perfect square. c. 1 1 0.3 Rational because it is a repeating decimal. 9 3 d. 25 5 0.5 Rational because it is a terminating decimal. 100 10 e. 1.69 f. 169 13 1.3 Rational because it is a terminating decimal. 100 10 1 0.5773502.... Not rational because it is not repeating or terminating. Also, 3 the denominator is not a perfect square. Learning Activity 5.1 1. Check the numbers that are rational numbers. Explain why they are or are not rational numbers. o 6 Yes - whole o 5.5 Yes - terminating o 4.123456…..No – non-terminating, non-repeating o ¾ Yes – fraction, which also equals 0.75 which is a terminating decimal o √25 Yes – equals 5 which is a whole number, and it is a perfect square 9 o Yes – it is a perfect square and equals a fraction (3/4), which also equals 0.75 16 which is a terminating decimal. 2. What number has a square root of: a. 3/8 = (3/8)2 = 9/64 b. 1.8 = (1.8)2 = 3.24 3. Is each fraction a perfect square? Explain your reasoning. a. 8/18 = 4/9 Yes, because √4/9 = 2/3 b. 16/5 No. It cannot be reduced. Although the numerator has a square root that is a whole number, the denominator does not. Therefore, 16/5 is not a perfect square. c. 2/9 No. It cannot be reduced. Although the denominator has a square root that is a whole number, the numerator does not. Therefore, 2/9 is not a perfect square. 4. Is each decimal a perfect square? Explain your reasoning. a. = Yes! 625 25 25 5 Or, reduce the fraction first: 100 4 4 2 Yes! 627 b. 0.627 = 1000 It cannot be reduced, and neither the numerator nor the denominator can be written as a product of equal factors. Therefore 0.627 is not a perfect square. Assignment Do #1-16 p. 11 Grade 9 Math Unit 5 – Square Roots & Surface Area Student Copy 5.1 – Square Roots of Perfect Squares To determine the area of a square or rectangle, we multiply the ___________ by the ____________. In the case of a square though, the length and width will always be the ____________. When we multiply a number by itself, we say that we are taking the ____________ of the number. Ex. #1 Square all numbers from 1 to 15. Ex. #2 Find the areas of the following squares. a. 8 cm b. 5 m 6 Finding the __________ _________ is the _______________ of finding the ____________. Instead of taking a number multiplied by itself to find the ____________, we have the square and need to find what number ___________________ by _______________ will equal that number. Ex. #3 Complete the following table. Area of a Square 49 64 121 144 16/100 25/81 4/9 36/64 Side Length (Square Root) Ex. #4 Solve the following square roots. a. b. 36 4 9 Perfect Squares A number that is the square of a whole number is called a ______________ ______________. Example: 16 is a perfect square because 16 = _____. Fractions can also be perfect squares if both the _________________ and the ______________ are whole numbers. Example: 25/100 is a perfect square because 25 100 When a __________________ that is a perfect square is written as a _______________, then the decimal is also a _________________ ____________. The square root will be a _______________________ (ex. 3.5) or ____________________ (ex. 0.3333…) decimal. . Numbers that are terminating or repeating are called _____________________ numbers. Example: Solve the following square roots. State whether each answer is a rational number and explain why. 100 a. 4 b. 25 29 c. 1 9 d. 25 100 e. 1.69 f. 1 3 Learning Activity 5.1 1. Check the numbers that are rational numbers. Explain why they are or are not rational numbers. o 6 o 5.5 o 4.123456….. o ¾ o √25 9 o 16 2. Calculate the number whose square root is: a. 3/8 b. 1.8 3. Is each fraction a perfect square? Explain your reasoning. a. 8/18 b. 16/5 c. 2/9 4. Is each decimal a perfect square? Explain your reasoning. a. 6.25 a. 0.627 Assignment Do #1-16 p. 11