Representing Square 1.1 Numbers Student book page 4 You will need Use materials to represent square numbers. • counters A. Calculate the number of counters in this square array. ϭ • a calculator number of number of number of counters counters in counters in in a row a column the array 25 is called a square number because you can arrange 25 counters into a 5-by-5 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 525 4 9 4 1 Is the number of counters in each square array a square number? How do you know? 8 Lesson 1.1: Representing Square Numbers Copyright © 2009 Nelson Education Ltd. NNEL-MATANSWER-08-0702-001-L01.inddEL-MATANSWER-08-0702-001-L01.indd 8 99/15/08/15/08 55:06:27:06:27 PPMM C. What is the area of the shaded square on the grid? Area ϭ s ϫ s s units ϫ units square units s When you multiply a whole number by itself, the result is a square number. Is 6 a whole number? So, is 36 a square number? D. Determine whether 49 is a square number. Sketch a square with a side length of 7 units. Area ϭ units ϫ units square units Is 49 the product of a whole number multiplied by itself? So, is 49 a square number? The “square” of a number is that number times itself. For example, the square of 8 is 8 ϫ 8 ϭ . 8 ϫ 8 can be written as 82 (read as “eight squared”). terms 64 is a square number. whole numbers the counting numbers that begin at 0 and E. Square 9 and 10. continue forever ϫ ϭ 2 ϭ (0, 1, 2, 3, …) 9 9 or 9 10 ϫ 10 ϭ or 102 ϭ square number the product of a Are both of these products square numbers? whole number How do you know? multiplied by itself F. Identify two square numbers greater than 100. 2 ( ) ϭ 2 ( ) ϭ Copyright © 2009 Nelson Education Ltd. Lesson 1.1: Representing Square Numbers 9 NNEL-MATANSWER-08-0702-001-L01.inddEL-MATANSWER-08-0702-001-L01.indd 9 99/15/08/15/08 55:06:28:06:28 PPMM Recognizing Perfect 1.2 Squares Student book pages 5–9 You will need Use materialsa variety of to strategies represent tosquare identify numbers. perfect • a calculator squares. Method 1: Using diagrams The area of a square with a whole-number side length is a 9 units perfect square. This 9-by-9 square has an area terms 9 units of square units, so is perfect square (or a perfect square. square number) the square of a whole Method 2: Using factors number PROBLEM A perfect square can be written as the product prime factor of 2 equal factors. Is 225 a perfect square? a factor that is a prime Draw a tree diagram to identify the prime factors of 225. number Continue factoring until the end of each branch is a prime A prime number has number. only itself and 1 as factors. 225 The ones digit of 225 is , so The fi rst few prime 5 is a factor of 225. numbers are 2, 3, 5, 7, The factor partner is 225 ÷ 5 ϭ . 11, 13, 17, …. 5 ϫ 225 ϭ 5 ϫ 45 is not a prime number, because 9 ϫ ϭ 45. 9 ϫ 45 ϭ 9 ϫ 9 is not a prime number, because 9 ϭ 3 ϫ . ϫ 9 ϭ 3 ϫ 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 Copyright © 2009 Nelson Education Ltd. NNEL-MATANSWER-08-0702-001-L02.inddEL-MATANSWER-08-0702-001-L02.indd 1100 99/16/08/16/08 11:28:21:28:21 AAMM 225 ϭ 5 ϫ ϫ ϫ Group the prime factors to create a pair of equal factors. 225 ϭ 5 ϫ 5 ϫ 3 ϫ 3 ϭ (5 ϫ 3) ϫ ( ϫ ) 2 ϭ 15 ϫ or ( ) Is 225 the square of a whole number? So, is 225 a perfect square? 170 PROBLEM Is 170 a perfect square? Complete the tree diagram. Write 170 as a product of prime factors. 17 ϫ 170 ϭ 17 ϫ ϫ Can you group the prime factors to create a pair of equal factors? So, is 170 a perfect square? 2 ϫ Method 3: Look at the ones digit Whole Perfect The table shows the fi rst 10 perfect squares. number square Circle the possible ones digits for a perfect square. 00 1 1 0 1 2 3 4 5 6 7 8 9 2 4 Look at the ones digit of 187. Could 187 be a perfect 3 9 square? 4 16 5 25 A number with ones digit 0, 1, 4, 5, 6, or 9 may or may not 6 36 be a perfect square. 7 49 Look at the table of the fi rst 10 perfect squares. Is 6 a 8 64 perfect square? Is 36 a perfect square? 9 81 10 100 Refl ecting Show that 400 is a perfect square without using a drawing or tree diagram. 2 4 ϭ (2)2, so 400 ϭ ( ) Copyright © 2009 Nelson Education Ltd. Lesson 1.2: Recognizing Perfect Squares 11 NNEL-MATANSWER-08-0702-001-L02.inddEL-MATANSWER-08-0702-001-L02.indd 1111 99/16/08/16/08 11:28:21:28:21 AAMM Practising 3. The area of this square is 289 square units. Is the side length a whole number? 17 units So, is 289 the square of a whole number? So, is 289 a perfect square? 4. Show that each number is a perfect square. 17 units a) 16 Sketch a square with an area of 16 square units. Side length of the square ϭ units Is the side length a whole number? So, is 16 a perfect square? b) 1764 Represent the factors of 1764 in a tree diagram. Use divisibility rules to help you identify factors. 1764 Divisibility rules ϫ • If the number is 2 even, 2 is a factor. • If the sum of the digits is divisible by 2 ϫ 3, then 3 is a factor. • If the sum of the digits is divisible by 9 ϫ 9, then 9 is a factor. ϫ 7 ϫ Write 1764 as a product of prime factors. 1764 ϭ ϫ ϫ ϫ ϫ ϫ Group the factors to create a pair of equal factors. 1764 ϭ ( ϫ ϫ ) ϫ ( ϫ ϫ ) 2 ϭ ϫ or ( ) Is 1764 a perfect square? 12 Lesson 1.2: Recognizing Perfect Squares Copyright © 2009 Nelson Education Ltd. NNEL-MATANSWER-08-0702-001-L02.inddEL-MATANSWER-08-0702-001-L02.indd 1122 99/16/08/16/08 11:28:21:28:21 AAMM 2025 7. Maddy started to draw a tree diagram to determine whether 2025 is a perfect square. 5 405 How can Maddy use what she has done so far to determine that 2025 is a perfect square? 581 Solution: 99 Write 2025 as the product of the factors at the ends of the branches in Maddy’s tree diagram. 2025 ϭ ϫ ϫ ϫ These factors are not all prime numbers, but you can rearrange them to create a pair of equal factors. 2025 ϭ ( ϫ ) ϫ ( ϫ ) 2 ϭ ϫ or ( ) Is 2025 the square of a whole number? So, is 2025 a perfect square? 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. 102 ϭ , so 169 is than 102. Try some squares greater than 102. 112 ϭ 122 ϭ 132 ϭ Is 169 a perfect square? Hint 169 written backward is . Use 32 ϭ 9 to solve 302 ϭ , so 961 is than 302. 2 ϭ ■ 30 . Try 312. 312 ϭ Is 961 a perfect square? Explain why 169 and 961 are perfect squares. Copyright © 2009 Nelson Education Ltd. Lesson 1.2: Recognizing Perfect Squares 13 NNEL-MATANSWER-08-0702-001-L02.inddEL-MATANSWER-08-0702-001-L02.indd 1133 99/16/08/16/08 11:28:22:28:22 AAMM Square Roots of 1.3 Perfect Squares Student book pages 10–15 You will need Use a variety of strategies to identify perfect • a calculator squares. A square has an area of 16 m2. Determine the side length, s. Area of a square ϭ s2 Area = 16 m2 Solve s2 ϭ 16 m. Which whole number multiplied by s metres itself equals 16? So, s ϭ m. 4 is called the square root of 16, because 42 ϭ 16. term ___ − ϭ Ί square root (√ ) Using the square root symbol, 4 16. one of 2 equal factors − of a number Determine √ 144 by guess and test. 2 For example, the PROBLEM A square has an area of 144 m . Determine the square root of 25 is 5, ____ side length, s ϭ Ί144 . because 52 ϭ 25. Solve the related equation s2 ϭ 144. Using___ the symbol, Ί25 ϭ 5. Use the strategy of guess and test. A = 144 m2 Notice that ___ ___ 102 ϭ 202 ϭ Ί25 ϫ Ί25 ϭ 25. 144 is between 100 and 400. s metres So, s2 is between 102 and ( )2. Is 144 closer to 100 or 400? So, is s2 closer to 102 or 202? Square 11.