Representing Square Numbers. Use materials to represent square numbers. A. Calculate the number of counters in this square array.


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1 1.1 Student book page 4 Representing Square Numbers You will need counters a calculator Use materials to represent square numbers. A. Calculate the number of counters in this square array number of counters in a row number of counters in a column number of counters in the array 25 is called a square number because you can arrange 25 counters into a 5by5 square. B. Use counters and the grid below to make square arrays. Complete the table. Number of counters in: Each row or column Square array Is the number of counters in each square array a square number? Yes How do you know? You can arrange those numbers of counters into a square. 8 Lesson 1.1: Representing Square Numbers NELMATANSWER L01.indd 8 9/15/08 5:06:01 PM
2 s s C. What is the area of the shaded square on the grid? Area s s 6 units 6 units 36 square units When you multiply a whole number by itself, the result is a square number. Is 6 a whole number? Yes So, is 36 a square number? Yes D. Determine whether 49 is a square number. Sketch a square with a side length of 7 units. Area 7 units 7 units 49 square units Is 49 the product of a whole number multiplied by itself? Yes So, is 49 a square number? Yes terms whole numbers the counting numbers that begin at 0 and continue forever (0, 1, 2, 3, ) square number the product of a whole number multiplied by itself Note: Answers to Part F may vary. The square of a number is that number times itself. For example, the square of 8 is can be written as 8 2 (read as eight squared ). 64 is a square number. E. Square 9 and or or Are both of these products square numbers? Yes How do you know? They are each the result of multiplying a whole number by itself. F. Identify two square numbers greater than 100. ( 11 ) ( 12 ) Lesson 1.1: Representing Square Numbers 9 NELMATANSWER L01.indd 9 9/15/08 5:06:01 PM
3 1.2 Student book pages 5 9 Recognizing Perfect Squares You will need a calculator terms perfect square (or square number) the square of a whole number prime factor a factor that is a prime number A prime number has only itself and 1 as factors. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17,. Use materials a variety of to strategies represent to square identify numbers. perfect squares. Method 1: Using diagrams The area of a square with a wholenumber side length is a perfect square. This 9by9 square has an area of 81 square units, so 81 is a perfect square. Method 2: Using factors 9 units 9 units PROBLEM A perfect square can be written as the product of 2 equal factors. Is 225 a perfect square? Draw a tree diagram to identify the prime factors of 225. Continue factoring until the end of each branch is a prime number. 225 The ones digit of 225 is 5, so 5 is a factor of 225. The factor partner is is not a prime number, because is not a prime number, because The ends of the branches are now all prime numbers: 5, 5, 3, and 3. Write 225 as the product of these prime factors. 10 Lesson 1.2: Recognizing Perfect Squares NELMATANSWER L02.indd 10 9/16/08 1:28:01 AM
4 Group the prime factors to create a pair of equal factors (5 3) ( 5 3 ) or ( 15 ) 2 Is 225 the square of a whole number? Yes So, is 225 a perfect square? Yes Whole number Perfect square PROBLEM Is 170 a perfect square? Complete the tree diagram. Write 170 as a product of prime factors Can you group the prime factors to create a pair of equal factors? No So, is 170 a perfect square? No Method 3: Look at the ones digit The table shows the first 10 perfect squares. Circle the possible ones digits for a perfect square Look at the ones digit of 187. Could 187 be a perfect square? No A number with ones digit 0, 1, 4, 5, 6, or 9 may or may not be a perfect square. Look at the table of the first 10 perfect squares. Is 6 a perfect square? No Is 36 a perfect square? Yes Reflecting Show that 400 is a perfect square without using a drawing or tree diagram. 4 (2) 2, so 400 ( 20 ) 2 Lesson 1.2: Recognizing Perfect Squares 11 NELMATANSWER L02.indd 11 9/16/08 1:28:01 AM
5 17 units 17 units Practising 3. The area of this square is 289 square units. Is the side length a whole number? Yes So, is 289 the square of a whole number? Yes So, is 289 a perfect square? Yes 4. Show that each number is a perfect square. a) 16 Sketch a square with an area of 16 square units. Side length of the square 4 units Is the side length a whole number? Yes So, is 16 a perfect square? Yes b) 1764 Represent the factors of 1764 in a tree diagram. Use divisibility rules to help you identify factors Divisibility rules If the number is even, 2 is a factor. If the sum of the digits is divisible by 3, then 3 is a factor. If the sum of the digits is divisible by 9, then 9 is a factor Write 1764 as a product of prime factors Group the factors to create a pair of equal factors ( ) ( ) or ( 42 ) 2 Is 1764 a perfect square? Yes 12 Lesson 1.2: Recognizing Perfect Squares NELMATANSWER L02.indd 12 9/16/08 1:28:01 AM
6 Hint Use to solve Maddy started to draw a tree diagram to determine whether 2025 is a perfect square. How can Maddy use what she has done so far to determine that 2025 is a perfect square? Solution: Write 2025 as the product of the factors at the ends of the branches in Maddy s tree diagram These factors are not all prime numbers, but you can rearrange them to create a pair of equal factors ( 5 9 ) ( 5 9 ) or ( 45 ) 2 Is 2025 the square of a whole number? So, is 2025 a perfect square? Yes Yes 8. Guy says: 169 is a perfect square when you read the digits forward or backward. Is Guy correct? Explain. Solution: Use the strategy of guess and test , so 169 is more than Try some squares greater than Is 169 a perfect square? Yes 169 written backward is , so 961 is more than Try Is 961 a perfect square? Yes Explain why 169 and 961 are perfect squares. Each is the square of a whole number. Lesson 1.2: Recognizing Perfect Squares 13 NELMATANSWER L02.indd 13 9/16/08 1:28:02 AM
7 1.3 Student book pages Square Roots of Perfect Squares You will need a calculator term square root ( ) one of 2 equal factors of a number For example, the square root of 25 is 5, because Using the symbol, Notice that Use a variety of strategies to identify perfect squares. A square has an area of 16 m 2. Determine the side length, s. Area of a square s 2 Solve s 2 16 m. Which whole number multiplied by itself equals 16? 4 So, s 4 m. 4 is called the square root of 16, because Using the square root symbol, Determine 144 by guess and test. PROBLEM A square has an area of 144 m 2. Determine the _ side length, s 144. Solve the related equation s Use the strategy of guess and test. A = 144 m is between 100 and 400. So, s 2 is between 10 2 and ( ) 2 s metres 20. Is 144 closer to 100 or 400? closer to 100 So, is s 2 closer to 10 2 or 20 2? closer to 10 2 Square Square s 2 _ 144, so s m. Area = 16 m 2 s metres 14 Lesson 1.3: Square Roots of Perfect Squares NELMATANSWER L03.indd 14 9/16/08 1:29:23 AM
8 Determine 225 by factoring. This factor rainbow shows all the factors of 225. Complete the table to show the factor partners. The factor with an equal partner is the square root. Factors of _ So, A perfect square is the square of a whole number. Is 225 the square of a whole number? Yes Is 225 a perfect square? Yes Determine 256 using factors. Complete the tree diagram of the factors of Then, write 256 as a product of prime numbers Group these factors to create a pair of equal factors. 256 ( ) ( ) ( 16 ) 2 _ So, Reflecting How can you check your answer when you calculate the square root of a number? Use 81 9 and to explain. Calculate 9 2. If it equals 81, then Lesson 1.3: Square Roots of Perfect Squares 15 NELMATANSWER L03.indd 15 9/16/08 1:29:24 AM
9 Checking A = 16 m 2 A = 4 m 2 A = 81 m 2 2. Calculate. a) 4 If the area of a square is 4 square units, then the side length of the square is 2 units. 4 2 b) 16 If the area of a square is 16 square units, then the side length of the square is 4 units c) 81 If the area of a square is 81 square units, then the side length of the square is 9 units Practising a) Complete the factor rainbow , so The factor partner for 7 is , so The factor partner for 9 is , so The factor partner for 21 is 21. b) Is 9 the square root of 441? No Why or why not? It is not one of 2 equal factors of 441. Which factor of 441 is the square root? 21 _ c) Square the square root to check your answer. ( 21 ) Lesson 1.3: Square Roots of Perfect Squares NELMATANSWER L03.indd 16 9/16/08 1:29:24 AM
10 Note: A number of tree diagrams are possible for _ 15. Describe 2 strategies to calculate 324. Guess and test Is 324 closer to 100 or 400? closer to 400 _ So, 324 is closer to ( 20 ) 2 than to ( 10 ) 2. Guess the number whose square is Square the number. ( 17 ) If the number you guessed is not the square _ root, continue guessing until you identify 324. ( 18 ) ( ) 2 _ So, Factoring Represent the factors of 324 in a tree diagram. Use divisibility rules to identify factors of 324. Write 324 as a product of prime numbers Group the factors to create a pair of equal factors. 324 ( ) ( ) or ( 18 ) 2 _ So, Divisibility rules If the number is even, 2 is a factor. If the sum of the digits is divisible by 3, then 3 is a factor. If the sum of the digits is divisible by 9, then 9 is a factor. Lesson 1.3: Square Roots of Perfect Squares 17 NELMATANSWER L03.indd 17 9/16/08 1:29:25 AM
11 1.4 Student book pages Estimating Square Roots You will need a calculator Estimate the square root of numbers that are not perfect squares. If a number is not a perfect square, you can estimate its square root. Estimate 10 by comparing it to roots of perfect squares. Estimate the side length of a square with an area of 10 square units. Step 1: On the grid paper, draw a 2by2 square, a 3by3 square, and a 4by4 square. Complete the table. ( 17 ) 2 Square Side length (s) Area (s 2 ) Side length ( A ) 2by by3 4by _ Lesson 1.4: Estimating Square Roots Step 2: Use the side lengths of the squares you drew to estimate is between 3 and 4, and closer to 3 than So, 10 is not a whole number Step 3: Determine 10 to 2 decimal places. Square (too low) Square (too high) The square of 3.2 is close to 10. So, 10 is approximately 3.2. NELMATANSWER L04.indd 18 9/16/08 1:30:56 AM
12 Note: Keystroke sequence circled may vary. Communication Tip The symbol means approximately equal to. When you round a number, the answer is an approximation. Use instead of when you write your answer. Determine square roots using a calculator. Calculators have a square root button, Different calculators use different key sequences. PROBLEM Calculate 10. Round the result to 3 decimal places. Try each sequence below. 10 or 10 Circle the sequence above that works with your calculator _ PROBLEM Calculate _ means the same as First, estimate Use mental math. Step 1:Use and to estimate 300. _ Step 2: _ Now, calculate Use a calculator is half of 300. Halve your estimate in step 1. _ or _ Round the result to 4 decimal places. _ Hint The symbol means is not equal to. Reflecting and are both the same distance from Why is it more likely that you chose when rounding to 4 decimal places? The convention how it s usually done is to round up , but Why is this? is an approximation of 10, so the square of is close to but not equal to 10. Lesson 1.4: Estimating Square Roots 19 NELMATANSWER L04.indd 19 9/16/08 1:30:56 AM
13 Hint Use the correct key sequence for your calculator. For example, to calculate 10, use either 10 or 10. Note: Students may choose to justify any one of 5 a), b), c) or d). Practising 4. Estimate to determine whether each answer is reasonable. Correct any unreasonable answers using the square root key on your calculator. a) The area of a square with side length 3 units is 9 square units. The area of a square with side length 4 units is 16 square units. Is 3.2 a reasonable estimate for the square root of 10? Yes Use your calculator to check b) The area of a square with side length 4 units is 16 square units. The area of a square with side length 5 units is 25 square units. Is 4.8 a reasonable estimate for the square root of 15? No Use your calculator to check Calculate each square root to 1 decimal place. Choose one of your answers and explain why it is reasonable. a) c) _ b) d) _ between 6 2 and 7 2 is reasonable because 38 is. 20 Lesson 1.4: Estimating Square Roots NELMATANSWER L04.indd 20 9/16/08 1:30:57 AM
14 8. Tiananmen Square in Beijing, China, is the largest open square in any city in the world. It is actually a rectangle of 880 m by 500 m. a) What is the approximate side length of a square with the same area as Tiananmen Square? Solution: What is the area of Tiananmen Square? Area length width 880 m 500 m m 2 What _ is the side length of a square with this area? m b) Explain how you know your answer to part a) is reasonable. _ m is reasonable because is between and Estimate the time an object takes to fall from each height using this formula: time (s) 0.45 height (m) Record each answer to 1 decimal place. a) 100 m time s b) 200 m time c) 400 m time s s Lesson 1.4: Estimating Square Roots 21 NELMATANSWER L04.indd 21 9/16/08 1:30:57 AM
15 1.5 Student book page 24 Exploring Problems Involving Squares and Square Roots You will need square tiles a calculator Create and solve problems involving a perfect square. How many tiles are in each diagram? (3) 2 9 tiles ( 2 ) tiles ( 3 ) tiles?? PROBLEM Joseph had 12 tiles. He made a square with some tiles and had 3 tiles left over. What is the side length of the square? Solve s What number added to 3 makes 12? 9 What is the square root of that number? 3 So, ( 3 ) s 3 tiles PROBLEM There are 104 tiles. What is the side length of the square? Let the variable s represent the unknown side length. s Write an equation. s Subtract 4 from each s side of the equation to isolate the variable. _ So, s Side length 10 tiles 22 Lesson 1.5: Exploring Problems Involving Squares and Square Roots NELMATANSWER L05.indd 22 9/15/08 5:27:02 PM
16 PROBLEM A game is played with a deck of 52 square cards. You deal the cards in equal rows and equal columns to form a square. Three cards are left over and not used. What is the side length of the square of cards? Solution: Draw a diagram similar to the ones on the previous page to represent the problem.? Choose a variable to represent the side length. Write an equation to represent the situation. s ( s ) Solve the equation. s s s 49 7 The side length of the square of cards is 7 cards. Hint Use one of these problemsolving strategies: Make a model Work backward PROBLEM Create a problem that uses a square number and another whole number. Answers may vary. For example: Mark has 40 tiles. He makes a square with tiles and has 4 left over. What is the length of the square? Solve the problem.? s s s 36 6 The side length of the square is 6 tiles. Lesson 1.5: Exploring Problems Involving Squares and Square Roots 23 NELMATANSWER L05.indd 23 9/15/08 5:27:02 PM
17 1.6 Student book pages The Pythagorean Theorem You will need counters Model, explain, and apply the Pythagorean theorem. cutout 1.6 terms On each right triangle label the hypotenuse c label the smallest leg a label the other leg b c a b a c b right triangle a triangle with 1 right angle (90) The hypotenuse is the longest side of a right triangle, the side opposite the right angle. The 2 shorter sides are called the legs. Hint To calculate 9 2 using a calculator: 9 x 2 Pythagorean Theorem In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the 2 legs. c 2 a 2 b 2 or a 2 b 2 c 2 Use the Pythagorean theorem to determine if the triangle below is a right triangle. length of hypotenuse: c 15 length of shortest leg: a 9 12 m 15 m length of other leg: b 12 Check if a 2 b 2 c Is a 2 b 2 c 2 true for this triangle? Yes So, is the triangle a right triangle? Yes hypotenuse c b 9 m a leg leg 24 Lesson 1.6: The Pythagorean Theorem NELMATANSWER L06.indd 24 9/15/08 5:28:49 PM
18 Is the Pythagorean theorem true for all types of triangles? Use Cutout 1.6, which shows 2 acute triangles, 1 right triangle, and 2 obtuse triangles. Each triangle has one side 60 mm long and another side 80 mm long. A. Measure the third side of each triangle to the nearest millimetre. Record the length on the cutout page. B. Write the missing side lengths (a, b, or c) in the table. Calculate the missing squares (a 2, b 2, or c 2 ). Triangle a b a 2 b 2 a 2 b 2 c c 2 Comparison A a 2 b 2 > c 2 B a 2 b 2 < c 2 C a 2 b 2 = c 2 D a 2 b 2 < c 2 E a 2 b 2 < c 2 Hint greater than less than Hint In the table above, A and B are acute, C is a right triangle, and D and E are obtuse. C. For each triangle, calculate a 2 b 2 and c 2. Compare the 2 values. Record each comparison in the table. Use <,, or >. D. Is the Pythagorean theorem true for all types of triangles? Reflecting No Explain. The theorem is only true for right triangles. Match the type of triangle with the equation or inequality. Acute triangle a 2 b 2 < c 2 Right triangle a 2 b 2 > c 2 Obtuse triangle a 2 b 2 c 2 Lesson 1.6: The Pythagorean Theorem 25 NELMATANSWER L06.indd 25 9/15/08 5:28:50 PM
19 26 Lesson 1.6: The Pythagorean Theorem Practising 3. Herman formed a triangle with gridpaper squares. How can you tell that he formed a right triangle? Solution: The side lengths of the 3 squares and the 3 side lengths of the triangle are the same. If c 2 a 2 b 2, then the triangle is a right triangle. c is the length of the longest side: 5 units a and b are the other 2 side lengths: 3 and 4 units c 2 a 2 b 2 ( 5 ) 2 ( 3 ) 2 ( 4 ) Is the triangle a right triangle? Yes 5. A Pythagorean triple is any set of 3 whole numbers, a, b, and c, for which a 2 b 2 c 2. Show that each set of numbers is a Pythagorean triple. a) a 5, b 12, and c 13 a 2 b 2 c 2 ( 5 ) 2 ( 12 ) 2 ( 13 ) b) a 7, b 24, and c 25 ( 7 ) 2 ( 24 ) 2 ( 25 ) c) a 9, b 40, and c 41 ( 9 ) 2 ( 40 ) 2 ( 41 ) NELMATANSWER L06.indd 26 9/15/08 5:28:50 PM
20 26 m path of puck 7. About how far would a hockey puck travel when shot from one corner of the rink (at the goal line) to the opposite corner (at the goal line)? Think of the rink as a rectangle divided into 2 right triangles. a c Hint Check a square root by multiplying it by itself. n n should be close to n. 54 m Label the sides of the shaded right triangle a, b, and c in the diagram above. a 26 m b 54 m Use the Pythagorean theorem to calculate the distance, c, travelled by the puck. Round your answer to the nearest whole number. Step 1: Step 2: a c 2 a 2 b cm c c 2 ( 26 ) 2 ( 54 ) m 3592 The puck would travel approximately 60 m. 9. Calculate the unknown side to 1 decimal place. a 2 b 2 c 2 c 9.0 cm b 8.0 cm b Hint The original measurements are precise to a tenth of a centimetre, so round your answer the nearest tenth of a centimetre. 8.0 cm Step 1: Step 2: a 2 b 2 c 2 a a 2 a 2 ( 8 ) 2 ( 9 ) 2 17 a cm a Lesson 1.6: The Pythagorean Theorem 27 NELMATANSWER L06.indd 27 9/15/08 5:28:50 PM
21 1.7 Student book pages Solve Problems Using Diagrams You will need a calculator Use diagrams to solve problems about squares and square roots. a ruler Joseph is building a model of the front of a Haida longhouse. He wants the model to have the measurements shown on the illustration. How can Joseph calculate length c (at the top of the model)? Solve a problem by identifying a right triangle 1. Understand the Problem Draw a diagram that includes all you know about the model. c represents the length you want to know. Complete the diagram. c 9 cm c 30 cm 30 cm 21 cm 30 cm 60 cm 28 Lesson 1.7: Solve Problems Using Diagrams NELMATANSWER L07.indd 28 9/16/08 1:33:55 AM
22 2. Make a Plan Draw a line on your diagram to connect the 2 dots at the tops of the sides of the model. This will make 2 right triangles at the top of the model. The base of each triangle is half of 60 cm, or 30 The height of the triangles is the height of the whole model minus the height of the side: 30 cm 21 cm 9 cm Write these lengths on your diagram. cm. Now you know 2 sides of each triangle. Which theorem can you use to calculate the length of the unknown side of the triangle, c? Pythagorean theorem Hint The hypotenuse (the longest side) in a right triangle is always labelled c. 3. Carry Out the Plan Write the equation that relates the sides of a right triangle. c 2 a 2 b 2 Side c in the right triangle is unknown. The lengths of the other 2 sides are known. Use one of these lengths for a and one for b. a 9 cm b 30 cm Calculate c. Step 1: Step 2: c 2 a 2 b 2 c c 2 ( 9 ) 2 ( 30 ) m 981 c is approximately 31 cm long. Reflecting How did drawing a diagram help solve the problem? It helped me see where a right triangle could be drawn and it made it easy to keep track of the lengths. Lesson 1.7: Solve Problems Using Diagrams 29 NELMATANSWER L07.indd 29 9/16/08 1:33:56 AM
23 Practising term diagonal In a 2D shape, a diagonal can join any 2 vertices that are not next to each other. 5. The diagonal of a rectangle is 25 cm. The shortest side is 15 cm. What is the length of the other side? Solution: Draw a rectangle. Write 15 cm beside the shortest side. diagonal diagonals 25 cm c 15 cm Draw a diagonal on the rectangle. Write 25 cm beside the diagonal. What does the problem ask you to determine? the length of the other side of the rectangle Is the unknown length a side of a right triangle? Yes Shade one of the right triangles formed by the diagonal. The hypotenuse, c, 25 cm. Call the shortest side of the triangle a, so a 15 cm. The unknown side is b. Use the Pythagorean theorem to calculate the other side length. Step 1: Step 2: a 2 b 2 c 2 b ( 15 ) 2 b 2 ( 25 ) b b The length of the other side is 20 cm. b Lesson 1.7: Solve Problems Using Diagrams NELMATANSWER L07.indd 30 9/16/08 1:33:56 AM
24 6. Fran cycles 6.0 km north along a straight path. She then rides 10.0 km east. Then she rides 3.0 km south. Then she turns and rides in a straight line back to her starting point. What is the total distance of her ride? Solution: The first 3 legs of Fran s ride have been drawn. Draw the path that takes Fran back to her starting point. N 10.0 km W E 3.0 km S 6.0 km START c Draw a line on the diagram to divide the shape into a rectangle and a right triangle. Hint Label the hypotenuse of the triangle c. The original measurements are precise to a tenth of a centimetre, so round the value of c to the nearest tenth of a centimetre. Let a short side of Δ b other side of Δ 6 km 3 km long side of rectangle 3 km. 10 km. Use the Pythagorean theorem to calculate c. c 2 a 2 b 2 c c 2 ( 3 ) 2 ( 10 ) The total distance of Fran s ride is 6.0 km 10.0 km 3.0 km 10.4 km 29.4 km. Lesson 1.7: Solve Problems Using Diagrams 31 NELMATANSWER L07.indd 31 9/16/08 1:33:56 AM
25 Acute triangles (all 3 angles less than 90 ) Cutout 1.6 c = 87 mm a = 60 mm b = 60 mm A b = 80 mm B a = 60 mm c = 80 mm Right triangle (1 angle is 90 ) c = 100 mm a = 60 mm C b = 80 mm Obtuse triangles (1 angle is greater than 90 ) 118 mm c = 80 mm b = 60 mm 60 mm D E 80 mm a = 32 mm NELMATANSWER L07.indd 32 9/16/08 1:33:57 AM
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