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1 Section 1.1 Square Numbers and Area Models Quadrilateral = a 4 sided figure Rectangle Square *Has 4 right (90o) angles *Opposite Sides are parallel * Has 4 right (90o) angles *Opposite Sides are parallel *All side lengths are the same Area is the # of square units needed to cover a region. Area of a rectangle or square can be found by multiplying Length x Width Base x Height The units for area are cm2, m2, etc... Investigation: Rectangles and Squares Using grid paper, draw and label as many different rectangles as you can with each area. A) 4 square units B) 6 square units C) 8 square units D) 9 square units E) 12 square units F) 15 square units G) 16 square units H) 20 square units Questions (Complete in your exercise) 1. For how many areas were you able to make a square? Which ones? 2. What is the side length of each square you made? 3. How is the side length of each square related to its area? 4. Find two areas greater than 20 square units, for which you can create a square. How did you know you could make a square for these areas? 5. A) Copy and complete the table below listing all the factors for each area, in order from least to greatest. The first two are done for you. Factor – numbers that multiply to give a product. It is a number that divides exactly into another number. 2 4 = 1 x 4, 2 x 2 6 = 1 x 6, 2 x 3 Factors of 4 are: 1, 2, 4 (we only write 2 once, even though it showed up twice) Factors of 6 are: 1, 2, 3, 6 8= Factors of 8 are: 9= Factors of 9 are: 12 = Factors of 12 are: 15 = Factors of 15 are: 16 = Factors of 16 are: 20 = Factors of 20 are: 5. B) What do you notice about the factors for the areas that created squares? Numbers like 1, 4, 9, etc... are known as perfect square numbers. Perfect square numbers are formed when we multiply a number by itself, or "square" the number. For example: 2x2=4 22 = 4 The square of 2 is 4 What is 5 squared? ________________ What is 7 to the power of 2? ________________ 52 5 is the BASE 2 is the EXPONENT The entire 52 is called a POWER 72 ____ is the Base ____ is the Exponent ____ is the Power On grid paper we are going to make a list of Perfect Squares from 1 to 12, including a diagram. 3 Example Questions #1: A square has a side length of 3cm. What is the area of the square? Solution: #2: A square has an area of 100cm2. What is the side length of the square? Solution: What is the perimeter of the square? (distance around a figure- add up all the sides) Solution: Complete Pages 8 - 9 #'s 4, 9, 10, 11, 14, 16, 17 4 Section 1.2 Squares and Square Roots Finding the square root of a number is doing the opposite. It's finding the number that multiplied by itself to give the result. The symbol for square root is In your exercise book make a list of all the perfect square roots. The dimensions or side length of the square can be viewed as the square root What # multiplies by itself to give 49? Notice in the diagram there are 49 little squares (perfect square #) and the side length is 7 (square root). 49 7 7 7 Example Questions #1: List all the factors of 30 and arrange them least to greatest. There are an EVEN number of factors. No factor occurred twice, therefore 30 is NOT a perfect square #2: List all the factors of 64 to show it is a perfect square. Complete Pages 15 - 16 #'s 5, 6, 7abc, 11, 13abcdef, 14, 15 There is an ODD number of factors. Since 8 occurred twice, 64 is a perfect square and the square root of 64 is 8 (the middle factor when factors are arranged from least to greatest). 5 Section 1.1 and 1.2 Practice (Complete in your exercise book) 1: Explain using a diagram and words why 10 is not a square number. 2: Explain using a diagram and words why 16 is a square number. 3: Calculate the perimeter for each of the squares below. a) 36cm2 b) c) 121cm2 49cm2 d) 9cm2 4: A square patio has area 100 m2. a) Find the dimensions (side lengths) of the patio. b) The owner wants to put lights around the patio. How many metres of lights are needed? c) If each string of lights is 5 m long. How many strings of lights are needed? 5: Find. a) 52 = h) 144 b) 102 = i) 81 c) 72 = j) 9 d) 42 = k) 1 e) 112 = l) 36 f) 22 = m) 64 g) 82 = n) 25 6 Finding the Square Roots of Bigger Numbers Remember 9 _____ because _____________ 100 _____ because _____________ What if 900 ? What # multiplied by itself gives 900? Think Factor Tree 900 9 100 *Hint: When the number ends with two zeros divide by 100 first *keep dividing until you notice two equal groups of numbers 3 3 10 10 (3 10)(3 10) (30)(30) So *To get the square root, multiply one group of numbers 900 30 Try these in your exercise book....don’t forget to write today’s date! A) 400 Lets’ find B) 14400 C) 6400 576 We must use another method to find this square root, called prime factorization When using prime factorization just keep dividing the number by a prime number until you notice two equal groups. Prime number: a number whose only factors are one and itself. For example: 2, 3, 5, 7, etc.... HINTS If the # is EVEN begin dividing by 2 If the # is ODD, begin dividing by 3 If the # ends in 5, begin dividing by 5 7 576 2 288 576 is even so start dividing by 2 2 2 144 2 2 12 12 we can make two equal groups (2 12)(2 12) (24)(24) So 576 24 Try these in your exercise book..... D) E) 484 441 Prime Factorization can also be used to show that a number is NOT a perfect square. 280 2 140 2 2 70 2 2 2 35 22267 These factors cannot be grouped equally so it’s NOT a perfect square number Practice #1: #2: Determine the square root of each of the following. Show all your workings in your exercise. A) 2500 B) 8100 C) 12100 D) 4900 E) 324 F) 256 G) 625 H) 1156 Use prime factorization to determine if the numbers below are perfect squares. If they are perfect squares, state their square root. A) 225 B) 400 C) 360 D) 484 E) 396 8 Section 1.4 Estimating Square Roots Using a Number Line Without a Number Line 9 Pick a perfect square # just above and just below the number you are looking for. ~ 3.5 12 16 36 ~ 6.1 38 49 Complete the following in your exercise book....remember to write the section number and date. #1. #2. 30 #5 #6 17 #9. 20 #10. 5 50 90 #3. #7. #11. 15 62 45 Complete Page 20 #3, 4, 5, 6 #4. #8. 90 103 #12. 142 9 Estimating Square Roots Practice Complete the following in your exercise....remember to date your work. 1. Between which two consecutive numbers (#'s in a row) will you find: A) B) C) 70 120 27 D) E) F) 20 55 2 2. Find a number with a square root between each of the following: A) 6 and 7 B) 3 and 4 C) 11 and 12 3. Estimate the length of one side of each square. 54m54 A) B) 54m1 54m2 4. Place the letter of the question on the number line below. 1 2 A) 5. 109m2 135 3 4 B) 70 5 6 C) 108 7 D) 8 58 9 E) 90 10 11 F) 22 Which statements are true, and which are false? A) C) 20 is between 19 and 21. 20 is closer to 4 than 5 B) D) 20 is between 4 and 5. 20 is between 19 and 21 6. Find the approximate side length (one decimal) of the square with each area. A) Area = 50cm2 B) Area = 125cm2 C) Area = 18cm2 7. What length of fencing is required to surround a square field with area 110m2? Side Length = Perimeter = _______ + _______ + _______ + _______ = _______ 12 10 Pythagorean Theorem Investigation Instructions Working with a partner, you will receive a baggie containing perfect squares. Arrange the perfect squares with side lengths 3, 4 and 5 to form a triangle. Is the triangle formed a right triangle? Look at the areas of the squares. Is there a relationship? Record the information in the table below. Repeat for the other side lengths indicated in the table. Side Lengths Is the triangle formed a right triangle? YES/NO Areas of the Squares Is there a relationship with the Areas of the Squares? 3–4–5 5 – 12 – 13 7 – 9 – 10 4 – 7 – 10 8 – 6 - 10 What conclusion(s) can you make from your table above? _____________________________________________________________ _____________________________________________________________ _____________________________________________________________ 11 Section 1.5 The Pythagorean Theorem A right triangle is a triangle that contains one right (90o) angle. The side opposite the right angle is called the hypotenuse. The hypotenuse is always the longest side of a triangle. The other two sides are called legs. hypotenuse leg leg The Pythagorean Theorem In a right triangle, the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the other two sides. 9 + 16 = 25 32 + 42 = 52 This is ALWAYS true for right triangles c In General a a2 + bb = c2 (leg)2 + (leg)2 = (hypotenuse)2 b 12 Complete the following examples in your exercise book.....remember to write the section and date. #1: Determine the areas of the indicated squares below. A) B) When given any two sides we can find the length of the missing side of a right triangle using the Pythagorean Theorem. Complete the following examples in your exercise book #2: Find the missing hypotenuse in each diagram below. A) B) Complete Page 34 # 3,4, 5 and Page 35 #8 13 Complete the following examples in your exercise book Determine the length and area of each square. A) B) C) Calculate the Side Length and Area of the 3 squares on page 20 #7, using the Pythagorean Theorem. Finding Missing Leg Length If sides "a" or "b" are missing... you will need to subtract the answer must be smaller than the hypotenuse (c) #3: Complete the following examples in your exercise book Find the missing leg indicated in each diagram below. A) B) C) 14 Complete Pages 34-35 #’s 6 and 7 Section 1.6 Exploring the Pythagorean Theorem Given 3 side lengths we can determine if the triangle is right or not. Remember that the hypotenuse (c) is always the longest side. #1: Determine whether each triangle is a right triangle. A) 6 – 7 – 9 longest side (hypotenuse) So 6 ⨉ 6 = 36, 7 ⨉ 7 = 49 and 9 ⨉ 9 = 81 Since 36 + 49 = 75 and NOT 81, this does NOT form a right triangle. B) 7-24-25 So 7 ⨉ 7 = 49, 24 ⨉ 24 = 576 and 25 ⨉ 25 = 625 Since 49 + 576 = 625, this does form a right triangle. A set of whole numbers that satisfies the Pythagorean Theorem is called a Pythagorean Triple. 7-24-25 is a Pythagorean Triple. Complete Pages 43-44 #’s 3, 4, 7abc Section 1.7 Applying the Pythagorean Theorem 1: 2: Complete the following examples in your exercise...remember to include the section and date. The foot of a 6m long ladder is placed 2m from the base of the building. How far up the building does the ladder reach, to the nearest tenth of a meter (one decimal place)? A ship travels for 14km toward the south. It then changes direction and travels for 9km toward the east. How far does the 15 ship have to travel to return directly to its starting point? (Answer to 2 decimal places). Complete Pages 49-50 #’s 9, 10, 11, 13, 16 Unit Review 1. Simplify each of the following: A) 121 B) 14400 C) 15 2 D) 6 2 2. What is the area of a square that has a side length of .... A) 9cm B) 5 3. What is the side length of a square that has an area of .... A) 49 cm2 B) 70 cm2 4. A) What is the square of 4? B) What is the square root of 4? 5. What is the perimeter of a square that has an area of 25 cm2? 6. Between what two consecutive numbers is 68 found? 7. TRUE/FALSE A) 8.5 is a good estimate for 85 . B) 115 is between 10 and 11. C) 7cm is a good estimate for the missing side in the right triangle below. 11 cm 4 cm D) The following three side lengths will produce a right triangle. 8cm – 5cm – 12cm 8. Use prime factorization to determine 676 . 9. Find the missing value. A) B) 16 ? ? 9m 6m 12 m 4m C) D) ? 32 m 680 m 2 2 ? 53 m 2 250 m 2 10. A) List all the factors of 36. How can the list of the factors determine whether or not 36 is a perfect square number? B) The factors of 372 are listed below: 372: 1, 2, 3, 4, 6, 12, 31, 62, 93, 124, 186, 372 Is 372 a perfect square? Explain your answer. 11. A) Find the area of the tilted square. B) Find the side of the tilted square. 17 12. The foot of a 6m ladder is placed 2m from the base of a building. How far up the building does the ladder reach? Give your answer to one decimal place. Draw and label a diagram and show all your workings. 13. Cory has two pet hamsters in a cage. Hammy walked from A to C and then from C to D. Tammy walked straight from A to D. Which hamster walked the shorter distance? By how much? A B 10 cm C 5 cm D