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Square Roots and the Pythagoren Theorm

Square Roots and the Pythagoren Theorm. 1.1 Square Numbers and Area Models. We can prove that 36 is a square number. Draw a square with an area of 36 square units. 6 units 6 units 36 = 6 x 6 = 6 2. 6 2 = 36. We can prove that 49 is a square number.

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Square Roots and the Pythagoren Theorm

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  1. Square Roots and the PythagorenTheorm

  2. 1.1Square Numbers and Area Models

  3. We can prove that 36 is a square number. Draw a square with an area of 36 square units. 6 units 6 units 36 = 6 x 6 = 62 62 = 36

  4. We can prove that 49 is a square number. Draw a square with an area of 49 square units. 7 units 7 units 49 = 7 x 7 = 72 72 = 49

  5. A square has an area of 64 cm2 Find the perimeter. What number when multiplied by itself will give 64? 8 x 8 = 64 So the square has a side length of 8cm. Perimeter is the distance around: 8 + 8 + 8 + 8 = 32 64 cm2

  6. What is a Perfect Square? Part 1 Any rational number that is the square of another rational number. In other words, the square root of a perfect square is a whole number. Perfect Square Square Root 1 √1 = 1 4 √4 = 2 9 √9 = 3 16 √16 = 4 25 √25 = 5

  7. Perfect Squares Use a calculator to determine if the following are perfect squares Perfect Square? Square Root Per. Square? √121 = Y/N √169 = Y/N √99 = Y/N √50 = Y/N

  8. Perfect Squares - KEY Use a calculator to determine if the following are perfect squares Perfect Square? Square Root Per. Square? √121 = 11Y/N √169 = 13Y/N √99 = 9.95 Y/N √50 = 7.07 Y/N

  9. What is a Perfect Square? Part 2 Another way to look at it. If we can find a division sentence for a number so that the quotient is equal to the divisor, the number is a square number. 16 ÷ 4 = 4 Dividend divisorquotient

  10. Quiz #1 Ch 1 1) List the first 12 perfect squares. 2) If a square has a side length of 5cm, what is the area? Show your work. 3) Find the side length of a square with an area of 81 cm2. Show your work.

  11. Quiz #1 Ch 1 Key 1) List the first 12 perfect squares. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. 2) If a square has a side length of 5cm, what is the area? 5cm x 5cm = 25cm2 3) Find the side length of a square with an area of 81 cm2. 81/9 = 9cm

  12. 1.2 Squares and Roots Squaring and taking the square root are inverse operations. That is they undo each other. 42 = 16 √16 = 16/4 = 4

  13. Factors 1-30 What do you notice about all the yellow columns? • They all have an odd number of factors! • They are perfect squares! • The middle factor is the square root of the perfect square!

  14. What is a perfect square? – PART 3 A perfect square will have its factor appear twice. Ex: 36 ÷ 1 = 36 1 and 36 are factors of 36 36 ÷2 = 18 2 and 18 are factors of 36 36 ÷3 = 12 3 and 12 are factors of 36 36 ÷ 4 = 9 4 and 9are factors of 36 36 ÷6 = 66 is a factor that occurs twice Factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36 The square root of 36 is 6 because it appears twice. It is also the middle factor when they are listed in ascending order!

  15. What is a Perfect Square – Part 4 Is 136 a perfect square? Perfect squares have an odd number of factors. List the factors. 1 x 136 = 136 2 x 68 = 136 4 x 34 = 136 8 x 17 = 136 There are 8 factors in 136. Therefore, 136 is not a perfect square because perfect squares have an odd number of factors.

  16. 1.2 Quiz • Find the square root of 144. • Find 42 • List the factors of 121. Is there a square root? If so what is the square root? • Which perfect squares have square roots between 1 and 50.

  17. 1.2 Quiz • Find the square root of 144. 144 = 144/12 = 12 • Find 42 4x4=16 3. List the factors of 121. Is it a PERFECT SQUARE? If so what is the square root? Yes it is a perfect square because there is an odd number of factors. 1, 11 ,121. The square root is 11. 4. What are the perfect squares between 1and 50. 1, 4, 9, 16, 25, 36, 49

  18. 1.3 Measuring Line Segments

  19. 1.3 Measuring Line Segments

  20. 1.3 Measuring Line Segments

  21. 1.3 Measuring Line Segments

  22. 1.3 Measuring Line Segments – Outside In

  23. Practice Time Complete the 7 questions below. You will be given a hard copy (extra practice 1.3). You will need graph paper for #4. Use inside-out for # 3 and outside-in for #4.

  24. http://staff.argyll.epsb.ca/jreed/8math/unit1/03.htm

  25. 1.4 Estimating Square Roots

  26. 1.4 Estimating Square Roots Here is one way to estimate the value of the square root of a number that is not a perfect square. For example: Find √20 • Step 1: Is it a perfect square? No • Step 2: If it isn’t, sandwich it between 2 perfect squares. √16 < √20 <√25 4 < √20 < 5 (16, 17, 18, 19 20, 21, 22, 23, 24, 25: √20 is closer to 4 than 5) Now we use guess and check. • 4.6 x 4.6 = 21.16 • 4.5 x 4.5 = 20.25 • 4.4x 4.4 = 19.36 • Therefore the √20 = approximately 4.4/4.5

  27. 1.4 Estimating Square Roots Here is one way to estimate the value of the square root of a number that is not a perfect square. For example: Find √20 • Step 1: Is it a perfect square? No • Step 2: If it isn’t, sandwich it between 2 perfect squares. √16 < √20 < √25 4 < √20 < 5 (16, 17, 18, 19 20, 21, 22, 23, 24, 25: √20 is closer to 4 than 5) Now we use guess and check. • 4.5 x 4.5 = 20.25 (too large) • 4.4 x 4.4 = 19.36 (too small) • 4.45 x 4.45 = 19.80 • 4.47 x 4.47 = 19.98 • Therefore the √20 = approximately 4.47

  28. Another way to estimate √20

  29. Find √27 (to one decimal place) • Step 1: Is it a perfect square? No • Step 2: If it isn’t, sandwich it between 2 perfect squares. √25 < √27 < √36 5 < √27 < 6 - √27 is closer to 5 than 6 Now we use guess and check. • 5.2 x 5.2= 27.04 • 5.1 x 5.1 = 26.01 Therefore the √27 = approximately 5.2 Bingo, this one is closest!!!!

  30. Find √105 (to 2 decimal places) • Step 1: Is it a perfect square? No • Step 2: If it isn’t, sandwich it between 2 perfect squares. √100< √105 < √121 10 < √105 < 11 (√105 is closer to 10 than 11) Now we use guess and check. • 10.2 x 10.2 = 104.04 (not close enough 0.96) • 10.3 x 10.3 = 106.09 (not close enough 1.09) • 10.25 x 10.25 = 105.06 Therefore the √105 = approximately 10.25

  31. Place each of the following square roots on the number line below. √5, √52, and √89 √4< √5< √9 2 < √5 < 3 - √5 is closer to 2than 3 • 2.2 x 2.2 = 4.84 • 2.25x2.25 = 5.063 • 2.24 x 2.24 = 5.017 • √5= approximately 2.24 √49 < √52 < √64 - √52 is closer to 7 than 8 • 7.2 x 7.2 = 51.8 • 7.25 x 7.25 = 52.56 • 7.22 x 7.22 = 52.12 • 7.21 x 7.21 = 51.98 • √52= approximately 7.21 √81 < √89 < √100 - √83 is closer to 9 than 10 • 9.4 x 9.4 = 88.36 • 9.45 x 9.45 = 89.30 • 9.43 x 9.43 = 88.92 - • 9.44 x 9.44 = 89.12 • √89= approximately 9.43 Bingo, this one is closest!!!! Bingo, this one is closest!!!! Bingo, this one is closest!!!! √5 √89 √52

  32. Quiz 1.4 Estimate the square roots of√11 and √38 √9 < √11< √16 3 < √11 < 4 (√11 is closer to 3than 4) • 3.2 x 3.2 = 10.24 • 3.3 x 3.3 = 10.89 • 3.32 x 3.32 = 11.02 • √11= approximately 3.32 √36 < √38< √49 6 < √38 < 7 (√38 is closer to 6than 7) • 6.2 x 6.2 = 38.44 • 6.17 x 6.17 = 38.069 • 6.16 x 6.16 = 37.95 • √38= approximately 6.16

  33. 1.5 The Pythagorean Theorem

  34. Watch Brainpop:“Pythagorean Theorem” http://www.brainpop.com/math/geometryandmeasurement/pythagoreantheorem/preview.weml

  35. 1.5 The Pythagorean Theorem In any right triangle, the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares whose sides are the two legs

  36. Watch This Video! http://www.youtube.com/watch?v=0HYHG3fuzvk

  37. 1.5 The Pythagorean Theorem Side A and Side B are always the legs and they are “attached” to the right angle. Side C is always across from the right angle. It is always longer than side A or side B. If you add the squares of side A and B, it will = the square of side C

  38. 1.5 The Pythagorean TheoremThe Formula(s) a2 + b2 = c2 (most common) c2 = a2 + b2 h2 = a2 + b2 (text book) a2 + b2 = h2

  39. 1.5 The Pythagorean Theorem Find the hypotenuse. h2 = a2 + b2 h2 = 62+ 72 h2 = 36 + 49 h2 = 85 √h2 =√85 h = 9.22

  40. 1.5 The Pythagorean Theorem Find the hypotenuse. h2 = a2 + b2 h2 = 82+ 62 h2 = 64 + 36 h2 = 100 √ h2 = √100 h = 10

  41. 1.5 The Pythagorean Theorem Find the leg “x”. We will make x – a. h2 = a2 + b2 182 = a2+ 112 324 = a2+ 121 324 – 121 = a2+ 121 - 121 203 = a2 √203= √a2 14.24= a

  42. 1.5 The Pythagorean TheoremQuiz Find the missing sides 6 Steps 7 steps

  43. Key Find the missing sides a2 + b2 = h2 a2 + b2 = h2 32 + 62 = h2 a2 + 42 = 82 9 + 36 = h2 a2 + 16 = 64 45 = h2 a2 + 16 – 16 = 64- 16 √45 = √h2 a2 = 48 6.71 = h √a2 = √48 a = 6.93

  44. 1.6 Exploring the Pythagorean Theorem

  45. For each triangle below, add up the 2 areas of the squares of the legs in the 2nd column, and include the area of the square of the hypotenuse in the third column. Do you see any patterns?

  46. For each triangle below, add up the 2 areas of the squares of the legs in the 2nd column, and include the area of the square of the hypotenuse in the third column. Do you see any patterns?

  47. Use Pythagoras to determine if the triangle below is a right triangle. a2 + b2 = h2 62 + 62 = 92 ? 36 + 36 = 81 ? • ≠ 81 This triangle is not a right triangle!

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