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Unit 5 Number Sense In this unit, students will study exponents, including powers of negative integers, and the order of operations with exponents and integers. Students will also study negative fractions and decimals. COPYRIGHT © 2011 JUMP MATH: NOT TO BE COPIED Meeting Your Curriculum Ontario students should cover all topics in this unit. For WNCP students, this unit is optional. Teacher’s Guide for Workbook 8.2 Q-1 NS8-104 Introduction to Powers Pages 137–138 Ontario: 8m1, 8m2, 8m4, 8m7, 8m11 WNCP: optional, [R, T, C]; 9N1, 9N2 Vocabulary power exponent base first power, second power, and so on Process Expectation Revisiting conjectures that were true in one context. Process assessment 8m1, 8m4, [R, T] Workbook p 138 Question 11 Q-2 Goals Students will evaluate powers by using repeated multiplication and investigate properties of powers. PRIOR KNOWLEDGE REQUIRED Can multiply a sequence of more than two whole numbers (e.g., 2 × 3 × 5) Can use a calculator to multiply numbers Can add, subtract, multiply, and divide two whole numbers Knows that multiplication commutes (e.g., 2 × 4 = 4 × 2) Introduce powers as repeated multiplication. First, remind students that multiplication is a short form for repeated addition. Then point out that just as we can add the same number repeatedly, we can multiply the same number repeatedly. Introduce the concepts power, base, and exponent as on Workbook page 137. Emphasize that in math, just as in English, a base is the bottom part of something. Assign questions according to the progression on Workbook page 137 Questions 1–5. Investigate the commutativity of powers. Remind students that multiplication is commutative—the order we write the numbers in doesn’t matter. For example, 2 × 5 = 5 × 2. ASK: Do you think the same is true for powers—will it matter which number is written as the base and which number is written as the exponent? Have students explain the basis for their prediction. (Sample answers: I think it won’t matter because of the example I found in Question 5; I think it won’t matter because it doesn’t matter for both addition or multiplication and just as multiplication is repeated addition, powers are repeated multiplication; I can see in a simple example like 23 and 32 that order does matter—23 is even and 32 is odd, or 23 = 8 and 32 = 9; I think it will matter because an odd base will give an odd result and an even base will give an even result, so interchanging an even number and an odd number will make a difference.) Then have students do the Investigation on Workbook page 137 and discuss the results. Emphasize that sometimes predictions turn out not to be true even when we have good reasons for making the predictions. Introduce the terms first power, second power, and so on. See Workbook page 138 Questions 7 and 8. Add, subtract, multiply, and divide powers. Remind students of the order of operations. ASK: Which operation do you perform first, addition or multiplication? (multiplication) Why? (because multiplication is a short form for repeated addition and 3 + 4 × 5 actually means 3 + 5 + 5 + 5 + 5, so you have to add the 5 four times—that is, multiply 4 × 5—and then add the result to the 3) Emphasize that powers are repeated multiplication. When you see a power such as 23, you are really multiplying 2 three times, so Teacher’s Guide for Workbook 8.2 COPYRIGHT © 2011 JUMP MATH: NOT TO BE COPIED Curriculum Expectations 5 × 23 really means 5 × 2 × 2 × 2 = 5 × 8 = 40. This means that you have to evaluate the powers before multiplying. Tell students to evaluate powers before doing multiplication or division, just as they do multiplication or division before addition or subtraction. So the new order of operations is: 1. Evaluate powers. 2. Do all multiplication and division, in order from left to right. 3. Do all addition and subtraction, in order from left to right. (NOTE: Do not introduce brackets in the same expression as powers at this point.) Have students do Workbook page 138 Question 12. ACTIVITY The cards on BLM Magic Cards (pp Q-24–Q-25) are an application of the fact that every number is a sum of powers of 2 or one more than a sum of powers of 2. To create their own “magic cards” (BLM Question 10), students write 1, 2, 4, 8, and so on (1 and then the powers of 2) in sequence in the top left of each card, extend the table in BLM Question 4, then use the table to fill in the rest of the numbers on the cards. To extend the table in BLM Question 4, students will need to write the numbers from 32 to 63 as sums of powers of 2 (or one more than a sum of powers of 2). For example, 41 = 32 + 8 + 1, so students will know to put 41 on cards 1 (for the 1 in the sum), 4 (for the 8 in the sum), and 6 (for the 32 in the sum). Card #1 1 Card #2 2 Card #3 4 Card #4 8 Card #5 16 Card #6 32 Extensions 1. The greatest common factor of 56 and 57 is ______. (56) 2. The lowest common multiple of 56 and 57 is ______. (57) COPYRIGHT © 2011 JUMP MATH: NOT TO BE COPIED Process Expectation Modelling, Visualizing 3. Connecting algebra to geometry. This extension is good preparation for NS8-105–106 Extension 3. Have students copy the following picture of two squares onto grid paper (not in their notebooks, as students will need to cut the squares out). 12 17 Number Sense 8-104 Q-3 Have students find the area of the shaded part in two ways: a) Subtract the area of the small square from the area of the big square. (172 − 122) b) Cut out the shaded region, and cut and rearrange it into a rectangle. What is the length of the rectangle in terms of 12 and 17? (17 + 12) What is the width of the rectangle in terms of 12 and 17? (17 − 12) c) Have students write an equation using only the numbers 12 and 17 and the exponent 2, based on the two ways of finding the area of the shaded region. ANSWER: 172 − 122 = (17 + 12)(17 − 12). d) Have students make their own picture to prove that 192 − 162 = (19 + 16)(19 − 16). e) Tell students that the dimensions of the squares can be 12 and 17, or 16 and 19, or any two numbers. Because the numbers can be any numbers, we can use variables to represent them. Show students how to change the equation in c) to an equation with variables: a2 − b2 = (a + b)(a − b). Challenge students to substitute various numbers into this equation to see if it is true, and to fill in the table below (students should make up their own examples for the last two rows): Algebra a b 5 3 7 4 9 2 10 9 8 3 a + b a − b (a + b)(a − b) 8 2 16 a2 b2 a2 − b2 25 9 16 Some students might notice that even when a − b is negative, the equation still holds. This is interesting, because the picture doesn’t apply to that case, but it was the picture that helped us guess the general formula in the first place. 4. Recall that taking a number to the exponent 2 is called squaring the number. Explain that taking a number to the exponent 3 is called cubing the number; and just as 72 can be read “seven squared,” 73 can be read “seven cubed.” Why is that? (because the volume of a cube of side length 7 is 7 × 7 × 7 = 73) 5. Comparing powers without computing them directly. a) Write the smallest power of 2 that will make the statement true. Q-4 5 = 51 < 2 52 < 2 53 < 2 Teacher’s Guide for Workbook 8.2 COPYRIGHT © 2011 JUMP MATH: NOT TO BE COPIED connection ANSWERS: 3, 5, and 7. b) Continue the pattern above. Which power is bigger now? 54 ____ 2 ANSWER: 54 > 29 Challenge students to explain why the pattern doesn’t continue to hold. (Explanation: You are continually multiplying the powers of 2 by 22 = 4 and the powers of 5 by 5, so eventually the power of 5 will become larger.) c) Determine how many digits the number 212 has without calculating it, as follows. i) Use 24 > 10 to explain why 212 > 103. ii) Use 23 < 10 to explain why 212 < 104. iii) Use your answers to i) and ii) to determine how many digits 212 has. ANSWERS: i) 212 = 24 × 24 × 24 > 10 × 10 × 10 = 103 ii) 212 = 23 × 23 × 23 × 23 < 10 × 10 × 10 × 10 = 104 iii) Any number between 103 = 1000 and 104 = 10 000 has 4 digits, so 212 has 4 digits. d) i) Calculate the powers to determine the correct sign (> or <). 53 _____ 102 54 _____ 103 ii) Without calculating it, determine how many digits the number 512 has. ANSWERS: i) 53 = 125 > 100 = 102 and 54 = 625 < 1000 = 103 ii) 512 = 53 × 53 × 53 × 53 > 102 × 102 × 102 × 102 = 108 and 512 = 54 × 54 × 54 < 103 × 103 × 103 = 109. COPYRIGHT © 2011 JUMP MATH: NOT TO BE COPIED So 512 is between 108 and 109. This means that it has 9 digits. Encourage students who don’t see this immediately to use a T-table to see that if a number is between 108 and 109 (as 512 is), then it has 9 digits (some students might think it has 8 digits). The T-table could look like this: number is between… number has ___ digits 10 and 102 2 10 and 10 3 3 103 and 104 4 2 Number Sense 8-104 Q-5 rom the same pattern, a number between 108 and 109 will have 9 digits, F since the number of digits is equal to the higher power of 10. Process assessment 7m1, [R] 6. a)A rule for the sum of powers of 2. Find a rule for the sum of the first n powers of 2 by completing the following chart: n 2n sum of n powers of 2 2n + 1 1 2 2 4 2 4 2+4=6 8 3 8 2 + 4 + 8 = 14 4 16 5 6 Write a rule for the sum of the first n powers of 2. Predict the sum of the first 10 powers of 2. ANSWER: 2n + 1 − 2, so the sum of the first 10 powers of 2 is 211 − 2 = 2048 − 2 = 2046. b) How big are powers of 2? You have a choice between 2 prizes: Prize A: monthly payments of $1000 Prize B: monthly payments of powers of $2 ($2, $4, $8, $16, and so on) If you won the prize for 10 months, which prize would you prefer? (Prize A: $1000 × 10 = $10 000 but 211 − 2 = $2046) Which would you prefer if you won for 20 months? (Prize B: $1000 × 20 = $20 000 but 221 − 2 = $2 097 150) If you start receiving these prizes in January, what would be the first month where the total payout from the Prize B would be more than the total payout from Prize A? (the 13th month, which is January of the 2nd year) 7. a) Computer codes are written as sequences of 0s and 1s. SAY: There are two possible sequences of length 1 and four possible sequences of length 2. Show them on the board: 0 1 00 10 01 11 Have students write all the 0-1 sequences of length 3. ASK: How many are there? (8) Write them on the board to summarize. ASK: How many sequences of length 4 do you think there will be? Show students how you can quickly change the sequences of length 3 to get all the sequences of length 4, and how doing so demonstrates that the number of sequences of length 4 is double the number of length 3: add a 0 to the end of all the sequences of length 3 and then add a 1 to the end instead. Q-6 Teacher’s Guide for Workbook 8.2 COPYRIGHT © 2011 JUMP MATH: NOT TO BE COPIED Write on the board: 2 sequences of length 1 4 sequences of length 2 8 sequences of length 3 16 sequences of length 4 Have students continue the pattern to determine how many 0-1 sequences there are of lengths 5, 6, 7, 8, 9, and 10. ASK: Do these numbers remind you of anything? (they are powers of 2) Where have you seen any of these numbers in relation to computers? (EXAMPLE: 256 Megabytes) Process assessment 8m5, [CN] b) ASK: Which power of 2 is closest to 1000? (the 10th) Tell students that a kilobyte is really 1024 bytes, not actually 1000 bytes. c) A megabyte is about 1 000 000 bytes. Exactly how many bytes are in a megabyte? HINT: 1 000 000 = 1000 × 1000. Answer: 1024 × 1024 = 1 048 576. 8. Students will need to be very familiar with writing numbers as sums of powers of 2, or one more than a sum of powers of 2. Doing BLM Magic Cards will provide this familiarity. EXAMPLE: 217 = 128 + 64 + 16 + 8 +1 = 27 + 26 + 24 + 23 + 1. The ancient Egyptians used this observation to multiply any number by any other number! For example, to calculate 37 × 217, start with 37 and double it continually: COPYRIGHT © 2011 JUMP MATH: NOT TO BE COPIED 37 × 1 37 × 2 37 × 4 37 × 8 37 × 16 37 × 32 37 × 64 37 × 128 Since 217 = 128 + 64 + 16 + 8 +1, then 37 × 217 = 37 × (128 + 64 + 16 + 8 +1) = 37 × 128 + 37 × 64 + 37 × 16 + 37 × 8 + 37 × 1 = 4736 + 2368 + 592 + 296 + 37 = 8029 a) Have students check this answer using the standard algorithm for multiplication. b) Use the ancient method of multiplying to do the following questions and check your answer by using the standard algorithm: Number Sense 8-104 = 37 = 74 = 148 = 296 = 592 = 1184 = 2368 = 4736 15 × 19 13 × 27 37 × 29 Q-7 c) When you do these questions, you have a choice in which number to expand. For example, to multiply 15 × 19, you could either write 15 = 1 + 2 + 4 + 8 and find 19 × 1 + 19 × 2 + 19 × 4 + 19 × 8 or you could write 19 = 1 + 2 + 16 and find 15 × 1 + 15 × 2 + 15 × 16. Do each of the above questions using the other way and decide which way you like better for each question and why. d) State one thing you like about the ancient Egyptian method of multiplying. (Sample answer: As long as I can add, I only have to multiply by 2 or double numbers; I don’t have to remember the times tables.) e) State one thing you don’t like about the ancient Egyptian method of multiplying. (Sample answer: It’s a lot more work than using the standard algorithm!) COPYRIGHT © 2011 JUMP MATH: NOT TO BE COPIED Q-8 Teacher’s Guide for Workbook 8.2 NS8-105 Investigating Powers NS8-106 Powers and Expanded Form Pages 139–141 Curriculum Expectations Ontario: 8m1, 8m2, 8m7, 8m11, 8m12 WNCP: optional, [R, C]; 9N1, 9N2 Goals Students will evaluate, compare, and order simple powers and investigate properties of powers. Students will solve equations involving powers. Students will write numbers in expanded form using power notation. PRIOR KNOWLEDGE REQUIRED Vocabulary power exponent base expanded form Can evaluate a power by using repeated multiplication Can write a number in expanded form Review power, base, and exponent. Have students identify the base and exponent in several powers. Students can signal their answers by holding up the correct number of fingers. Patterns in powers of 2. See Workbook page 139 Question 1. Point out that if you know a power of 2, you can find the next power of 2 by multiplying by 2, i.e., doubling the number. Patterns in powers of 10. Have students make a chart like the one on Workbook page 139, Question 2a), for powers of 10 instead of powers of 2. Then have students do these questions: a) 103 × 102 = ______ × ______ b) 105 × 104 = ______ × ______ = ______ = ______ = 10 = 10 Continue with these problems: Process Expectation COPYRIGHT © 2011 JUMP MATH: NOT TO BE COPIED Looking for a pattern c) 104 × 103 = 10 d) 105 × 101 = 10 e) 103 × 103 = 10 f) 106 × 102 = 10 Draw students’ attention to the exponents. ASK: How can you get the exponent in the answer from the other two exponents? (add the two exponents) PROMPT: What operation can you use? ASK: Do you think this will be true for any base, or just base 10? Demonstrate why the same property holds for any base by using the teaching box on Workbook page 139. Then have students do Workbook page 139 Questions 2–3. Bonus Process Expectation Looking for a pattern, Using logical reasoning Number Sense 8-105, 106 3 2 × 33 × 34 = 3 7 × 72 × 73 × 74 × 75 × 76 × 77 × 78 = 7 Relating powers of 9 to powers of 3. Have students do Workbook page 140 Question 4. When students finish, challenge students to explain why the pattern holds. Emphasize that because each 9 in the product is being replaced by a product of two 3s, there are twice as many 3s in the product as there were 9s. Q-9 After students finish Workbook page 140 Question 4, provide the following bonus problem. Bonus Write each product as a power of 3: a) 34 × 93 = 34 × 36 = 310 b) 33 × 92 c) 92 × 27 d) 94 × 34 Comparing and ordering powers with the same base or exponent. Have students recall how easy it was to compare and order perfect squares: 432 < 472 because 43 < 47; there is no need to compute the squares to check which one is larger. Have students articulate why this was the case. (Sample answer: multiplying two smaller numbers together will get a smaller result than multiplying two larger numbers together.) Then write on the board: 433 < 473. Ask students if they think this is true and why. To help them, rewrite the powers as repeated multiplication: 43 × 43 × 43 < 47 × 47 × 47. Emphasize that multiplying three smaller numbers together will get a smaller result than multiplying three larger numbers together. ASK: Which is larger: 4319 or 4719? (4719) How do you know? (because 47 is larger than 43 and multiplying 19 larger numbers together will result in a larger number than multiplying 19 smaller numbers together) Emphasize that as long as the exponent is the same, the power with the larger base is larger. Then repeat the line of questioning for powers with the same base, but different exponents. Emphasize that, for example, multiplying more 5s together will get a larger result than multiplying fewer 5s together, so when the base is the same, the power with the greater exponent is greater. After students finish Workbook page 140 Question 6, provide the following bonus problem. Bonus 85 = 32 (Answer: 3, because both sides of the equation are equivalent to 215) Solving equations with powers. Show students how to use an organized list to solve for the variable. By trying each possibility for x—when it is either the base or the exponent—and evaluating the power, students can determine which value for x solves the equation. 8m1,8m2, [R] Workbook p 140 Question 8j) NOTE: Workbook page 140 Question 9 is preparation for finding the bases in Question 10. After students finish Workbook Question 10, provide the following bonus problem. Bonus Write each expression as a power of a prime number. a) 25 × 82 e) 25 × 55 ANSWERS: a) 211 b) 343 × 72 f) 43 × 27 b) 75 c) 3125 × 53 g) 2432 c) 58 d) 29 e) 57 d) 43 × 8 f) 213 g) 310 Powers and expanded form. Review writing numbers in expanded form and then have students write numbers in expanded form using powers of 10; see Workbook page 141 Questions 1–4. Then have students write numbers written in expanded form, using power notation, as whole numbers; see Workbook page 141 Question 5. Finally, have students compare numbers written in expanded form as in Workbook page 141 Q-10 Teacher’s Guide for Workbook 8.2 COPYRIGHT © 2011 JUMP MATH: NOT TO BE COPIED Process assessment Question 6. Encourage students who struggle with Question 6 to circle the part in the two expressions that is different. Alternatively, students could write both expressions as whole numbers in standard form. Extensions 1. How many 8s must I add together to get a sum equal to 83? (64 because 83 = 82 × 8 = 64 × 8) 2. How many 8 × 8 squares do I need to have in order to have a total area of 83? (8 because 83 = 8 × 82) 3. Have students complete the chart by filling in the answers and continuing the patterns in the last two rows: 1=1 1=1 1+2=3 13 + 23 = _____ 1+2+3=6 13 + 23 + 33 = _____ 1 + 2 + 3 + 4 = _____ 13 + 23 + 33 + 43 = _____ 1 + 2 + 3 + 4 + 5 = _____ 13 + 23 + 33 + 43 + 53 = _____ 1 + 2 + 3 + 4 + 5 + 6 = _____ 13 + 23 + 33 + 43 + 53 + 63 = _____ Have students think about and describe how you can get the numbers in the second column from the numbers in the first column. (The numbers in the second column are the square of the sums in the first column. For example: 13 + 23 + 33 + 43 + 53 = (1 + 2 + 3 + 4 + 5)2.) Process Expectation Visualizing Show students how to visualize the equations in the second column for small numbers: Since 13 + 23 = 1 × 12 + 2 × 22, draw one 1 × 1 square and two 2 × 2 squares. Then try to fit these three squares into a 3 × 3 square, since 3 = 1 + 2, and the equation says 13 + 23 = (1 + 2)2. COPYRIGHT © 2011 JUMP MATH: NOT TO BE COPIED + + = This is almost a 3 × 3 square—just the bottom right corner is missing. But the two striped squares overlap in the middle, and if we take the overlapping part of one of the striped squares and move it to the bottom right corner, we get exactly a 3 × 3 square (see next page). Number Sense 8-105, 106 Q-11 So 13 + 23 = (1 + 2)2. The next case is 13 + 23 + 33 = 1 × 12 + 2 × 22 + 3 × 32, so draw one 1 × 1 square, two 2 × 2 squares, and three 3 × 3 squares, and try to fit them into a 6 × 6 square, since 6 = 1 + 2 + 3 and the equation says 13 + 23 + 33 = (1 + 2 + 3)2. + + + + + So 13 + 23 + 33 = (1 + 2 + 3)2. COPYRIGHT © 2011 JUMP MATH: NOT TO BE COPIED Challenge students to show the next case: 13 + 23 + 33 + 43 = (1 + 2 + 3 + 4)2. Give students 1-cm grid paper (see page U-25) and scissors. Students could colour the squares of different sizes different colours, and make a poster to show their work. Q-12 Teacher’s Guide for Workbook 8.2 NS8-107 Powers of Negative Numbers NS8-108 Exponents, Integers, and Order of Operations Pages 142–143 Curriculum Expectations Ontario: 8m1, 8m2, 8m7, 8m23 WNCP: optional, [R, PS, C]; 9N1, 9N2, 9N4 Goals Students will evaluate powers of negative integers. Students will evaluate expressions that involve integers, and that include brackets and powers, using order of operations. PRIOR KNOWLEDGE REQUIRED Can add, subtract, multiply, and divide integers Can use the order of operations without powers Can write repeated multiplication using power notation + (+) = + + (−) = − − (+) = − − (−) = + Review multiplying integers. Remind students that multiplying two positive or two negative integers results in a positive integer, whereas multiplying a positive integer and a negative integer results in a negative integer. Display the chart shown in the margin. Have students practise: a) 3 × (−4) d) (+2) × (−7) b) (−2) × (−5) e) (−4) × (−5) ANSWERS: a) −12 b) 10 c) −6 c) (−3) × 2 d) −14 e) 20 Have students do examples that require multiplying many integers. Students can multiply two or more terms in each step. a) (−2) × (−3) × (4) b) (−3) × (5) × (−2) × (−4) c) (−10) × (−2) × (−3) × (−5) × (−1) × (−7) ANSWERS: a) 24 b) −120 c) 2100 COPYRIGHT © 2011 JUMP MATH: NOT TO BE COPIED Work through Workbook page 142 Question 1a) together as a class, then have students complete parts b)–e) on their own. When students are finished, discuss the pattern: which powers are positive and which are negative, and why. Emphasize that each time we multiply by a negative number, we change the sign of the product, so having an even number of negative terms makes the product positive, while an odd number of negative terms makes the product negative. Then have students complete Workbook page 142. Review the correct order of operations with brackets but no powers. 1. Do operations in brackets. 2. Do multiplication and division, from left to right. 3. Do addition and subtraction, from left to right. Emphasize, in particular, that multiplication stands for repeated addition, so it makes sense to do multiplication before addition. In the expression Number Sense 8-107, 108 Q-13 2 + 3 × 4, the 3 isn’t being added to the 2, it is telling how many 4s to add together: 2 + 4 + 4 + 4. That’s why we do multiplication before addition. Furthermore, multiplication and division are “opposite” operations that undo each other, so it makes sense to do them at the same time, and not to do one before the other. The same goes for addition and subtraction. Extra Practice: Evaluate: a) 3 − (7 + 2) b) 3 − 7 + 2 e) 8 − (5 + 1) + 3 × (−2) − 8 ÷ (−2) c) 8 × 3 − 5 d) 8 × (3 − 5) f) 15 ÷ (−3) × (−4) − 3 × (2 − 4) Solution: e)Do operations in brackets: 8 − 6 + 3 × (−2) − 8 ÷ (−2). Then do multiplication and division, from left to right: 8 − 6 + (−6) − (−4). Then do addition and subtraction, from left to right: 2 + (−6) − (−4) = −4 − (−4) = 0 f)Do operations in brackets: 15 ÷ (−3) × (−4) − 3 × (−2). Then do multiplication and division, from left to right: (−5) × (−4) − 3 × (−2) = 20 − (−6) = 26 Process assessment Review the correct order of operations with powers but no brackets. 8m2, [R] Workbook p 142 Question 4 1. Evaluate powers. 2. Do all multiplication and division, in order from left to right. 3. Do all addition and subtraction, in order from left to right. −24= −(24) (−2)4 = (−2) × (−2) × (−2) × (−2) = (+4) × (−2) × (−2) = −(2 × 2 × 2 × 2) = (−8) × (−2) = −16 = +16 So −24 is the opposite of 24. Extra Practice: Evaluate: a) 3 × 52 e) 3 + 42 ÷ (−2) b) 1000 ÷ 53 c) 32 − (−2)3 f) 1000 ÷ (−2)3 g) 7 − 32 × (−2) ANSWERS: a) 75 b) 8 c) 17 d) −16 e) −5 d) −43 ÷ (−2)2 h) 12 ÷ (−4) × 5 f) −125 g) 25 h) −15 The order of operations with brackets and powers. Emphasize that expressions inside brackets are evaluated first, even before powers. Q-14 Teacher’s Guide for Workbook 8.2 COPYRIGHT © 2011 JUMP MATH: NOT TO BE COPIED Emphasize, in particular, that a power stands for repeated multiplication, so it makes sense to evaluate the powers before doing any multiplication. For example, look at 2 × 34. This expression really means 2 × 3 × 3 × 3 × 3, because 34 means 3 × 3 × 3 × 3. Also, point out that the expressions (−2)4 and −24 mean different things: Extra Practice: Evaluate: a) (7 − 5)3 d) 52 − 82 g) 3375 ÷ 52 × (−3) b) 73 − 53 e) 5 + 34 − (2 × 5) h) (7 − 3)2 × (−2) c) (5 − 8)2 f) 3375 ÷ (52 × (−3)) ANSWERS: a) 8 b) 218 c) 9 d) −39 e) 76 f) −45 g) −405 h) −32 Include expressions in exponents. Tell students that sometimes there is an expression in an exponent that needs to be calculated before you can evaluate the power. All the numbers in the same exponent should be treated as if they were in brackets and so evaluated first. EXAMPLE: 32 + 4 = 36. Have students practise: a) 38 − 6 e) (− 2)(4 + 2) ÷ 2 b) (−2)6 ÷ 2 f) (− 2)4 + 2 ÷ 2 c) 58 − 3 × 2 d) (7 −10)3 − 2 + 1 2 4 g) (−8) ÷ (−2) × (3 + 2)20 ÷ 5 c) 25 e) −8 ANSWERS: a) 9 b) −8 d) 9 f) −32 g) 2500 Summarize the order of operations. 1. 2. 3. 4. Do operations in brackets. Calculate exponents and evaluate powers. Do multiplication and division, from left to right. Do addition and subtraction, from left to right. Have students do Workbook page 143 Question 3 for practice. Process Expectation Problem Solving Adding brackets changes the value of an expression. Write on the board and evaluate: 25 − 32 + 2 × 22 (= 24). Show students different ways of adding brackets to this expression and ask them to evaluate the new expressions using the correct order of operations: a) 25 − 32 + (2 × 2)2 d) 25 − (32 + 2 × 22) g) 25 − (32 + 2 × 2)2 COPYRIGHT © 2011 JUMP MATH: NOT TO BE COPIED ANSWERS: a) 32 b) (25 − 3)2 + 2 × 22 e) 25 − (32 + (2 × 2)2) b) 492 c) 500 d) 8 e) 0 c) (25 − 3)2 + (2 × 2) f) 25 − (32 + 2) × 22 f) −19 g) −144 Then point out that some ways of adding brackets don’t make sense. For example, it wouldn’t make sense to add an opening bracket between a base and an exponent in a power, e.g., 25 − 3(2 + 2) × 22. Write this expression on the board and ASK: Why doesn’t this make sense? (the exponent doesn’t mean anything without the base) Point out, however, that a closing bracket can be put between a base and its exponent (as in parts a, b, c, e, and g above) because the expression in brackets becomes the base for the exponent. Process assessment 8m1, 8m2, [PS] Workbook p 143 Question 6 Number Sense 8-107, 108 Have students add brackets to these expressions to get as many different answers as they can. a) 16 + 8 ÷ 2 + 6 b) (−5)6 ÷ 3 × 2 ANSWERS: a) 26, 3, 18, 17 c) (−2)5 + 2 × (−5)2 ÷ 5 b) 625 or −5 c) −22, −1500, 372.8 Q-15 NS8-109 Concepts in Powers Page 144 Goals Curriculum Expectations Students will investigate properties of powers. Ontario: 8m1, 8m7, 8m13 WNCP: optional, [R, C]; 9N1, 9N2 PRIOR KNOWLEDGE REQUIRED Can write repeated multiplication as powers Can use long division (or a calculator) to compute remainders Can evaluate small powers Vocabulary Patterns in ones digits of powers. Have students complete this chart: Process assessment 8m1, 8m7, [R, C] Workbook p 144 Question 1 Process Expectation 2n 21 22 23 24 = 2 4 8 16 ones digit 2 4 8 6 25 26 27 28 After students complete the chart, have students predict the ones digit of 29, 210, 211, and 212, and explain the basis for their predictions. Ask students to predict the ones digit for 222. (4, because every fourth term in the sequence has the same ones digit, so 222 has the same ones digit as 218, 214, 210, 26, and 22) Looking for a pattern ASK: What is the ones digit of 27051? Start by listing the powers of 2, in backwards order, that have the same ones digit: 27047, 27043, 27039. ASK: Process Expectation Is this a good method to use? Why not? Why did it work well for 222? Tell students that although the method we used for 222 doesn’t work as well for 27051, we can still look at the easier problem for ideas. Write on the board: Looking for a similar problem for ideas, Reflecting on what makes a problem easy or hard 22, 26, 210, 214, 218, 222 Then write the sequence of exponents on the board: 2, 6, 10, 14, 18, 22. ASK: How can we describe these numbers? (sample answers: they are all of the form 4n + 2; they are all 4 more than the previous term) Then ask students to think in terms of remainders: How can you describe these numbers in terms of remainders? (They all have remainder 2 when dividing by 4.) Emphasize that all these powers of 2 have the same ones digit as 22 (which is the first exponent in the sequence that has remainder 2 when dividing by 4). ASK: How does this help us with the harder problem of finding the ones digit of 27051? PROMPT: What do we have to know about 7051 to help us answer this question? (the remainder when dividing by 4) Have students use long division to find this remainder, and then use this information to determine the ones digit of 27051. (The remainder when dividing 7051 by 4 is 3, so 27051 has the same ones digit as 2 to the exponent 3. Since 23 is 8, 27051 has ones digit 8.) Bonus Q-16 W hat is the ones digit of 251, 2151, 2251, 2351, 2451, 2551, 2651, 2238651, and 2800951? (these all have remainder 3 when dividing by 4; in Teacher’s Guide for Workbook 8.2 COPYRIGHT © 2011 JUMP MATH: NOT TO BE COPIED power exponent base ones digit remainder fact the last 2 digits determine the remainder when dividing by 4; students using the WNCP curriculum might remember this fact from Grade 7) When students finish, point out that it is quite surprising that we can find the ones digit of 27051 when we know so little about the number itself. We don’t know any of its other digits, or even how many digits there are! Process assessment Extensions 8m1, 8m7, [R, C] 1. a) Write each power of 10 in standard form. 104 = _______ 103 = _______ 102 = _______ 101 = _______ b) How do we get each number on the right side of the equations in part a) from the number above it? (divide by 10) How do we get each exponent in the left side of the equation from the exponent above it? (subtract 1) c) Continue the pattern in the exponents and in the numbers: COPYRIGHT © 2011 JUMP MATH: NOT TO BE COPIED 10 = _______ d)Repeat parts a) and b) for powers of 2, powers of 3, and powers of 5. e) What is the zeroth power of any number? Explain. (Each power is obtained from the previous one by dividing by the base. But the first power of any base is the base itself, so the zeroth power is the base divided by the base, which is always 1.) f) Introduce negative powers. Have students continue the pattern: 102 = 100 101 = 10 100 = 1 10−1 = _______ 10−2 = _______ ANSWERS: 1/10 or 0.1, and 1/100 or 0.01, because you divide by 10 to get the next term in the pattern. 2. Remind students that to find two or three consecutive numbers that add to a number, we can use equations (e.g., x + x + 1 = 37). a) Use an equation to find two consecutive numbers that add to: Number Sense 8-109 i) 37 ii) 79 iii) 63 iv) 99 Q-17 b) Find half of: i) 37 ii) 79 iii) 63 iv) 99 c) How are your answers to parts a) and b) related? Explain to students that the two consecutive numbers are very close together—they are only one apart. If they were exactly the same and added to 37, they would be exactly half of 37, but because they are only close to each other and not exactly the same, they are only close to half of 37. d) Use half of 85 to find two consecutive numbers that add to 85. Tell students that you want to find three consecutive numbers that add to 60. SAY: The three numbers are very close together—they are only one apart. If the same number was added 3 times to get 60, what number would that be? (20) Tell students that they should look for numbers close to 20 to find three consecutive numbers that add to 60. (19 + 20 + 21 = 60) e) Use 1/3 of 72 to find three consecutive numbers that add to 72. (1/3 of 72 = 24, so 23 + 24 + 25 = 72) f) Which whole number, when squared, is close to 182? Calculate the square root of 182 on a calculator and use the result to find two consecutive numbers that multiply to 182. (13 × 14 = 182) g) Guess and check to find the whole number which, when cubed, is closest to 1320. (103 = 1000 and 113 = 1331, so 11) h) The product of three consecutive numbers is 1320. Find these numbers. (10 × 11 × 12 = 1320) i) The product of three consecutive numbers is 7980. Find these numbers. (19 × 20 × 21 = 7980) 2. Prime factorizations, powers, squares, and square roots. a) Write each square of a power as a power itself. EXAMPLE: Q-18 (23)2 = 23 × 23 = (2 × 2 × 2) × (2 × 2 × 2) = 26 i) (32)2 Bonus ii) (25)2 iii) (53)2 iv) (74)2 v) (85)2 (2324)2 Teacher’s Guide for Workbook 8.2 COPYRIGHT © 2011 JUMP MATH: NOT TO BE COPIED Tell students that you want to find two consecutive numbers that multiply to 156. SAY: These two numbers are close together—they are only one apart. If a number was multiplied by itself to get 156, what would the number be? (the square root of 156) What two consecutive numbers are closest to the square root of 156? (12 and 13 because 122 = 144 and 132 = 169) Tell students that if numbers that are close together multiply to 156, they should be close to the square root of 156, so a good guess for the two numbers is 12 and 13. Indeed, 12 × 13 = 156. b)Teach students to write prime factorizations using power notation. For example, 24 = 2 × 2 × 2 × 3 can be written as 24 = 23 × 3. Have students write prime factorizations for these numbers using power notation: i) 12 COPYRIGHT © 2011 JUMP MATH: NOT TO BE COPIED iv) 72 c)Why is 53 a prime factorization, but 82 is not a prime factorization? (because 5 is prime, but 8 is composite and can be factorized further) PROMPT: Draw the factor tree for each number—53 = 125 and 82 = 64. d)If the prime factorization of a number is 23 × 75 × 118, what is the prime factorization of its square? (26 × 710 × 1116) e)How can you tell immediately from the exponents in a prime factorization whether the number is a perfect square? (If all exponents are even, the number is a perfect square; if any exponent is odd, the number is not a perfect square.) f) Write each square root of a power as a power itself. EXAMPLE: 2 6 = 2 × 2 × 2 × 2 × 2 × 2 = (2 × 2 × 2)×(2 × 2 × 2) = 2 × 2 × 2 = 23 i) ANSWERS: i) 24 Algebra iii) 96 connection ii) 75 28 ii) 34 iii) 310 ii) 32 iv) iii) 35 516 iv) 58 g) If the prime factorization of a number is 28 × 310 × 516, what is the prime factorization of its square root? (24 × 35 × 58) 4. a) Mathematicians have proven that if a and b have GCF = 1, and a is a prime number, then a is a factor of ba − 1 − 1. This is called Fermat’s Little Theorem. Check this for: = 2 and b = 3 a a = 2 and b = 5 a = 3 and b = 2 a = 3 and b = 4 a = 3 and b = 5 a = 3 and b = 10 a = 5 and b = 2 a = 5 and b = 3 a = 5 and b = 4 a = 7 and b = 10 your own example: a = and b = (make sure a is prime and GCF of a and b is 1) Number Sense 8-109 b) For a = 4 and b = 7, check whether a is a factor of ba − 1 − 1. Is this a counter-example to the statement in part a)? Explain. c) For a = 5 and b = 10, check whether a is a factor of ba − 1 − 1. Is this a counter-example to the statement in part a)? Explain. Q-19 connection Literature, Measurement 5. Read the book One Grain of Rice by Demi, and then work through these questions individually, stopping after each one to check answers and discuss. a) If 1 grain of rice weighs 25 mg, how much rice (by mass) would Rani get in a week? In 2 weeks? In 3 weeks? After the whole month (30 days)? NOTE: Review the rule for finding the sum of n powers from NS8-104 Extension 6: The sum of the first n powers of 2 is 2n + 1 – 2. For example, the sum of the first 6 powers of 2 is 2 + 4 + 8 + 16 + 32 + 64 = 27 – 2 = 126. ANSWERS: 1 week: Number of grains = 1 + 2 + 4 + 8 + 16 + 32 + 64 = 127. The mass of 127 grains is 127 × 25 mg = 3175 mg = 3.175 g. 2 weeks: The number of grains after 1 week is 27 − 1. Using the same pattern, the number of grains after 2 weeks is 214 − 1 = 16 383 grains. The mass is 16 383 × 25 mg = 409 575 mg = 409.575 g. 3 weeks: 221 − 1 = 2 097 151 grains, for a total mass of 2 097 151 × 25 mg = 52 428 775 mg = 52 428.775 g = 52.428 775 kg. 30 days: There are 230 − 1 = 1 073 741 823 grains, for a total mass of 1 073 741 823 × 25 mg = 26 843 545 575 mg = 26 843 545.575 g = 26 843.545 575 kg = 26.843 545 575 tonnes. So Rani would get almost 27 tonnes of rice after 30 days. b) If the volume of a grain of rice is 33.3 mm3, what would Rani need in order to carry the rice home on the 7th day? The 14th day? The 21st day? After the whole month? ANSWERS: Notice that on day 1, she needs to carry 1 grain; on day 2, 2 grains; on day 3, 22 grains; on day 4, 23 grains; and on day n, 2n − 1 grains. 14th day: 213 grains = 8192 grains for a volume of 8192 × 33.3 mm3 = 272793.6 mm3 = 272.7936 cm3. Rani could carry the rice in a bowl. 21st day: 220 grains = 1 048 576 grains for a volume of 1 048 576 × 33.3 mm3 = 34 917 580.8 mm3 = 34 917.580 8 cm3 = 34.917 580 8 dm3. Rani needs a capacity of almost 35 L. You can demonstrate this capacity if your school has 35 thousands cubes. Four big pails should do. Q-20 Teacher’s Guide for Workbook 8.2 COPYRIGHT © 2011 JUMP MATH: NOT TO BE COPIED 7th day: 26 grains = 64 grains for a volume of 64 × 33.3 mm3 = 2131.2 mm3 = 2.1312 cm3. Rani could carry this much rice home in the palm of her hand. A whole month: 229 grains = 536 870 912 grains for a volume of 536 870 912 × 33.3 mm3 = 17 877 801 369.6 mm3 = 17 877 801.369 cm3 = 17 877.801 369 dm3 = 17.877 801 369 m3. This is almost 18 m3. Rani would need many sacks... and other villagers to help her carry them! c) After a whole month, would all the grains they have to store fit into the classroom? ANSWER: There would be 230 − 1 grains = 1 073 741 823 grains for a volume of 1 073 741 823 × 33.3 mm3 = 35 755 602 705.9 mm3 ≈ 35.8 m3 (This is likely to fit in most classrooms. Calculate the volume of your classroom, roughly, by measuring the length and width of the floor and the height of the walls.) d) The residents of the village decide they won’t start eating the rice until there is enough for everyone in the village to have 3 bowls of rice a day. If a bowl of rice has 2000 grains, and the village has 250 people, on what day can the villagers start eating the rice? ANSWER: They need to have 1 500 000 grains before they can start eating. From part a), the answer is between 2 weeks and 3 weeks. Check each day in turn: 15 days: 215 − 1 = 32 767 grains 16 days: 216 − 1 = 65 535 grains 17 days: 217 − 1 = 131 071 grains 18 days: 218 − 1 = 262 143 grains 19 days: 219 − 1 = 524 287 grains 20 days: 220 − 1 = 1 048 575 grains 21 days: 221 − 1 = 2 097 152 grains So on day 21, the villagers can each have 3 bowls of rice a day. Notice that on day 22, they would get 221 = 2 097 152 grains of rice, so they have enough to eat 3 bowls each on that day too, and so on. COPYRIGHT © 2011 JUMP MATH: NOT TO BE COPIED e) Assuming the villagers eat 3 bowls of rice a day as described in part d), what is the volume of rice the villagers have to store at the end of the month? How long can they eat from this supply of rice? ANSWER: The villagers eat for 10 days (day 21 to day 30). So we need to subtract 1 500 000 × 10 = 15 000 000 grains from the total to be stored (see part c): 1 073 741 823 − 15 000 000 = 1 058 741 823 grains of rice. To determine how many more days the villagers can eat for, divide this total by the 1 500 000 grains they eat each day: 1 058 741 823 ÷ 1 500 000 = 705.827 882. So the villagers can eat for 705 more days, or almost 2 years. Number Sense 8-109 Q-21 NS8-110 Negative Fractions and Decimals Page 145 Curriculum Expectations Ontario: 8m1, 8m6, 8m7, 8m13 WNCP: optional, [V, R, C], 9N3 Vocabulary integer opposite integer positive negative fraction decimal > and < symbols Goals Students will represent, compare, and order rational numbers (i.e., positive and negative fractions and decimals to thousandths). PRIOR KNOWLEDGE REQUIRED Can compare and order fractions Can compare and order decimals Can compare and order integers Can use long division to write fractions as decimals Review comparing and ordering positive fractions and decimals. Progress as follows: 1. Compare tenths, hundredths, and thousandths. EXAMPLE: Compare 0.132 and 0.105 2. Compare fractions with denominator 2, 4, 5, 8, or 20 to decimals by changing the fraction to a fraction with denominator 10, 100, or 1000, and then to a decimal. EXAMPLES: Compare 1 7 8 3 i) and 0.23 ii) and 0.431 iii) and 1.584 iv) and 0.36 4 20 5 8 3 3 ×125 375 = SAMPLE ANSWER: iv) = = 0.375 > 0.36. 8 8 ×125 1000 3. Compare any fraction to a decimal by changing the decimal to a fraction and comparing the fractions. EXAMPLE: Compare 2/3 to 0.65 by comparing 200/300 to 65/100 = 195/300. 4. Compare any fraction to a decimal by changing the fraction to a decimal by long division. EXAMPLE: 5. Order lists of fractions and decimals. EXAMPLE: Order the list: 5 2/3, 5 3/5, 5.712, 5.615, 5.67. Review comparing and ordering integers. First, recall that all negative numbers are smaller than all positive numbers. Second, remind students that an “opposite integer” is the number that is the same distance from 0 on a number line, but in the other direction; a number and its opposite add to 0; to obtain the opposite number, keep the number part the same, but change the sign (from + to − or from − to +). Q-22 Teacher’s Guide for Workbook 8.2 COPYRIGHT © 2011 JUMP MATH: NOT TO BE COPIED Compare 2/3 to 0.65 by writing 2/3 as the repeating decimal 0.6 , which is greater than 0.65. (NOTE: If students are not comfortable converting fractions to repeating decimals, students will need to use the method of changing the decimal to a fraction instead of changing the fraction to a decimal.) Process Expectation Using logical reasoning Process Expectation Visualizing To order negative numbers, order the opposite positive numbers and put the negative numbers in reverse order to their positive opposites. For example, since 3 < 4 < 5, we know that −5 < −4 < −3. The fact that 4 is less than 5 tells us that 5 is further from 0, but this means that −5 is further from 0 than −4, which means it is more to the left, so −5 is less than −4. Placing negative numbers on number lines. Have students mark the positive integers 3, 7, and 8 with an X on a number line from −10 to 10 that has only three points labelled: −10, 0, and 10. Then have students mark their opposites, −3, −7, and −8, on the same number line. ASK: Do you see any symmetry? (a vertical line through 0 is a mirror line) Review placing positive fractions and decimals on number lines, and then show students how to place negative fractions and decimals on number lines by using the mirror image of the opposite positive numbers. Process assessment COPYRIGHT © 2011 JUMP MATH: NOT TO BE COPIED 8m6, 8m7, [V, C] Workbook p 142 Question 2 Compare and order positive and negative fractions and decimals. Combine the concepts above. EXAMPLE: 1 5 1 Order these numbers: 2, 1.3, −1.4, − 2 , , − 1 . 4 3 2 1 5 Step 1: Order the positive numbers: 2, 1.3, , . 4 3 1 5 So, < 1.3 < < 2. 4 3 1 Step 2: Order the opposite of the negative numbers: 1.4, 2, 1 . 2 1 So, 1.4 < 1 < 2. 2 Step 3: Order the negative numbers in reverse order from Step 2: 1 −2 < − 1 < −1.4 2 Step 4: Combine the lists from Steps 1 and 3, with all negative numbers less than all positive numbers: 1 5 1 −2 < − 1 < −1.4 < < 1.3 < <2 2 4 3 Number Sense 8-110 Q-23