Unit 3 Number Sense D-1. Teacher s Guide for Workbook 7.1 COPYRIGHT 2010 JUMP MATH: NOT TO BE COPIED

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1 Unit 3 Number Sense In this unit, students will study factors and multiples, perfect squares and square roots, divisibility rules, and adding and subtracting fractions with like and unlike denominators. Meeting your curriculum Lessons NS7-15 and NS7-16, about divisibility rules, are optional for Ontario students. However, lesson NS7-14 covers essential problem-solving strategies that will be used throughout the year, concentrating on familiar material about divisibility by 2, 5, and 10. We recommend that teachers in Ontario teach this lesson to consolidate the problem-solving skills taught in NS7-13. Lesson NS7-12, on perfect squares and square roots, is optional for students following the WNCP curriculum. Students following the WNCP curriculum will learn this material in Grade 8. Teacher s Guide for Workbook 7.1 D-1

2 NS7-9 Factors and Multiples Pages Curriculum Expectations Ontario: 7m1, 7m2, 7m6, 7m7, 7m12 WNCP: 6N3; essential for 7N1, [R, C] Goals Students will learn to identify factors and multiples PRIOR KNOWLEDGE REQUIRED Knows the times tables Can divide 1-digit numbers into 2- and 3-digit numbers Vocabulary factor multiple Defining multiples and factors. Show students the two examples of factors and multiples from the top of Workbook p. 51. Then ask students how they could prove that 2 is a factor of 6. (use 2 3 = 6) ASK: What other number does this show is a factor of 6? (3) Repeat with other numbers. Examples: 5 is a factor of 10 (5 2 = 10); 3 is a factor of 12 (3 4 = 12); 8 is a factor of 80 (8 10 = 80); 10 is a factor of 100 (10 10 = 100); 1 is a factor of 2 (1 2 = 2). Write on the board: 3 is a factor of 6. Ask students to tell you a sentence that means the same thing but uses the word multiple instead of factor. (6 is a multiple of 3) Repeat with other Examples: 4 is a factor of 20 (20 is a multiple of 4); 3 is a factor of 0 (0 is a multiple of 3); 0 is a factor of 0 (0 is a multiple of 0). Include sentences that use the word not. Example: 18 is not a multiple of 4 (4 is not a factor of 18) A mnemonic for multiple and factor. Some students might confuse the meanings of the two words. Suggest the following mnemonic: there are many multiples, but few factors. Look at the factors and multiples of 6 as an example: There are many multiples, all 6 or greater (not counting 0) There are only a few factors, all 6 or less. Process ExpectatioN Organizing data Zero is a multiple of any number. Have students write a multiplication statement that proves that 0 is a multiple of 3. (3 0 = 0) ASK: Which whole numbers is 0 a multiple of? (all of them, because any whole number can be multiplied by 0 to make 0) Skip counting to find multiples. List the first five multiples of 4, including 0, on the board: 4 0 = = = = = 16 D-2 Teacher s Guide for Workbook 7.1

3 Have students continue the list by writing the next five multiples of 4. ASK: Is 26 a multiple of 4? (no) How can you tell? (it s not on the list; it is between two numbers on the list) Explain that, since each multiple of 4 is 4 more than the previous multiple, we can list all the multiples of 4 by skip counting: 0, 4, 8,. ASK: How can I use skip counting to check if 20 is a multiple of 3? (Skip count by 3s until you either reach 20 or pass it if you reach 20, then it is a multiple of 3; if you pass it, then it is not a multiple of 3.) Demonstrate doing this: 0, 3, 6, 9, 12, 15, 18, 21. We passed 20, so 20 is not a multiple of 3. ASK: What are the multiples of 0? (Only 0 is a multiple of 0; to see this, skip count by 0 from 0: 0, 0, 0,.) ASK: What are the multiples of 1? (Every number is a multiple of 1; to see this, skip count by 1 from 0: 0, 1, 2, 3, 4,.) Multiples have to be whole number multiples. Tell students that = 26. Have students verify this on a calculator. ASK: Should we call 26 a multiple of 4? What is ? (27) Should we call 27 a multiple of 4? Have students find on their calculator (24.8) ASK: Should we call 24.8 a multiple of 4? Explain that if we didn t require multiples to be whole numbers, the definition of a multiple of 4 would be useless every number would be a multiple of 4! When we talk about multiples and factors, we are only talking about whole numbers. Finding factors. ASK: How did we use skip counting to find out if 20 is a multiple of 3? (skip count by 3s if you reach 20, it is a multiple of 3; if you pass 20, it is not a multiple of 3) How can you use skip counting to find out if 3 is a factor of 20? (same method 20 is a multiple of 3 means the same thing as 3 is a factor of 20 ) Tell students that you want to find all the factors of 20. Ask in turn: Process ExpectatioN Using logical reasoning Is 0 a factor of 20? (no, skip counting by 0s gives only 0) Is 1 a factor of 20? (yes, skip count by 1s to get 0, 1, 2, 3,, 20) Is 2 a factor of 20? (yes, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20) Is 3 a factor of 20? (no, 0, 3, 6, 9, 12, 15, 18, 21) Is 4 a factor of 20? (yes, 0, 4, 8, 12, 16, 20) Is 5 a factor of 20? (yes, 0, 5, 10, 15, 20) Is 6 a factor of 20? (no, 0, 6, 12, 18, 24) Continue in this way. Encourage students to tell you if you can do any less work. Have them tell you when you can stop. Continue past 20 if necessary. ASK: Can any number greater than 20 be a factor of 20? (no) Why not? (because you jump right over 20 when skip counting by any number greater than 20) Is 20 a factor of 20? (yes, because skip counting by 20 from 0 always reaches 20 it s the first number you say) Is any number greater than 10 and less than 20 a factor of 20? (no) Why not? (As soon as you start skip counting by a number greater than 10, say 11, skip counting Number Sense 7-9 D-3

4 automatically brings you past is more than half of 20, so the double of 11 is more than 20.) Have students use this method to find all the factors of 30. Emphasize that students only need to check the numbers up to 15. Nothing greater than 15 (except for 30 itself) can be a factor, because as soon as you multiply it by 2 you get a number greater than 30. (ANSWER: The factors of 30 are 1, 2, 3, 5, 6, 10, 15, 30.) Why it is useful to know the times tables. ASK: Is 58 a multiple of 8? Encourage students to find the answer without skip counting. ASK: Do you have the 8 times table memorized? What is a multiple of 8 that is close to 58? (7 8 = 56) What is the next multiple of 8? ( = 64 or 8 8 = 64) Is 58 a multiple of 8? (no) Have students use the same method to decide the following: Is 70 a multiple of 9? (no, it s between 63 = 9 7 and 72 = 9 8) Is 36 a multiple of 4? (yes, 36 = 4 9) Is 72 a multiple of 8? (yes, 72 = 8 9) Is 28 a multiple of 3? (no, it s between 27 = 3 9 and 30 = 3 10) Process assessment 7m7, [C] Workbook Question 9 Converting between logical statements that use multiple and factor. Remind students that 6 is a multiple of 2 means 2 is a factor of 6. Then write on the board: 0 is a multiple of every number. Have students rewrite this statement using the word factor. Repeat for: 0 is the only multiple of 0 (0 is the only number that has 0 as a factor OR 0 is the only number that 0 is a factor of) Every number is a multiple of 1. (Every number has 1 as a factor OR 1 is a factor of every number.) Have students rewrite this statement using the word multiple : Only 1 is a factor of 1. (The only number that has 1 as a multiple is 1 itself.) Extension Process ExpectatioN Looking for a pattern, Connecting Explore the patterns in the ones digits of the multiples of a) 2 and 8 b) 3 and 7 c) 4 and 6 What do you notice? In particular, if you know the pattern in the ones digits for multiples of 2, how can you get the pattern in the ones digits for the multiples of 8? ANSWERS: a) After 0, which is common to both 8 and 2, the pattern for 8 is the pattern for 2 read backwards: The pattern for 2 is 0, 2, 4, 6, 8, repeat. The pattern for 8 is 0, 8, 6, 4, 2, repeat. D-4 Teacher s Guide for Workbook 7.1

5 b) The pattern for 3 is 0, 3, 6, 9, 2, 5, 8, 1, 4, 7, repeat. The pattern for 7 is 0, 7, 4, 1, 8, 5, 2, 9, 6, 3, repeat. Again, the pattern for 7 can be obtained by reading the pattern for 3 backwards. c) The pattern for 4 is 0, 4, 8, 2, 6, repeat. The pattern for 6 is 0, 6, 2, 8, 4, repeat. The pattern for 6 can be obtained by reading the pattern for 4 backwards. To understand the reason for the pattern, look at the case of 3 and 7. Notice that = 10 (just as = 10 and = 10). Adding 7 gives the same ones digit as subtracting 3 from the same number because they are ten apart. For example, if you start with 5, subtracting 3 gives ones digit 2 (5 3 = 2) but adding 7 also gives ones digit 2 (5 + 7 = 12). But we get the pattern for the 3s by adding 3, so to read it backwards, we can subtract 3. Also, we can generalize the result to the ones digits in the multiples of 1 and 9, since = 10: start with the ones digits for the multiples of 1 (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, repeat) and then obtain the ones digits for the multiples of 9 by reading backwards (0, 9, 8, 7, 6, 5, 4, 3, 2, 1, repeat) Number Sense 7-9 D-5

6 NS7-10 Organized Search Pages Curriculum Expectations Ontario: 7m1, 7m2, 7m3, 7m6, 7m7, 7m12 WNCP: 6N3; essential for problem-solving, [R, C] Vocabulary factor multiple Goals Students will use an organized method to find all pairs of numbers that multiply to a given number. PRIOR KNOWLEDGE REQUIRED Can skip count to identify multiples and factors Understands the relationship between multiples and factors Knows the times tables Can divide 1-digit numbers into 2- and 3-digit numbers Materials grid paper Process ExpectatioN Organizing data Process ExpectatioN Reflecting on other ways to solve a problem Organizing data. To illustrate the importance of organizing data, challenge students to put themselves in order according to birthday, from January to December, in 3 minutes (tell students how you will signal when time is up). Suggest that they try to do so in an organized way. When the 3 minutes have elapsed, discuss the strategies students used or should have used. For example, students could make a January to March corner, an April to June corner, and so on. Then, within each group, students could arrange themselves by month, ask each other their birthdays, and order themselves accordingly. If students did not successfully order themselves in 3 minutes the first time, have them try again using this strategy. Using an organized list to find factors of whole numbers. As a class, find all pairs of numbers that multiply to 12. Look at the finished list together and ASK: Did we do any extra work? Are there any pairs we can cross out? Why? Point out the pairs that are repeated and cross them out. Have students find all pairs of numbers that multiply to give 15, then cross out the pairs that are repeated. Process ExpectatioN Using logical reasoning Knowing when to stop listing numbers to pair up. Emphasize that you don t want to have to try all the numbers up to the number you re trying to find the factors of, and the lists you made above suggest that you shouldn t have to; it seems like half the pairs are repeated anyway. Discuss how you can know when to stop. Begin, with class participation, listing the pairs that multiply to 48. ASK: Once you know that 6 is paired up with 8, how do you know that 8 is paired up with 6? (because 6 8 = 8 6) ASK: What is larger: 8 6 or 9 7? How do you know? (when both numbers are bigger, their product is bigger too) ASK: If 9 is paired up, can it be paired up with a number larger than 6? Can 9 something greater than 6 be equal to 8 6 = 48? Explain that if 9 is paired up, it has to be paired up with either 1, 2, 3, 4, or 5. Then check each one in turn on your list. Since each possibility is ruled out, 9 must not be paired up at all. D-6 Teacher s Guide for Workbook 7.1

7 Process ExpectatioN Reflecting on other ways to solve a problem Process ExpectatioN Using logical reasoning ASK: Can 10 be paired up? How do you know? (Use similar reasoning or instead note that 48 is not a multiple of 10. Discuss both answers.) ASK: Can 11 be paired up? Continue in this way until all students see that the only numbers larger than 8 that are paired up are paired with 1, 2, 3, 4, or 5. But we know which numbers are paired up with numbers less than 6 because we ve done all numbers up to 6! Explain that once you find a number that already occurs in a pair, you know that you ve found all the pairs that multiply to 48. Extra Practice: Find all pairs of numbers that multiply to... a) 26 b) 42 c) 72 d) 63 e) 30 f) 91 Bonus 180 Answers: a) d) b) e) c) f) Bonus When the number is a perfect square. Find all the pairs that multiply to 100 instead of 48. Emphasize that once you reach = 100, any number more than 10, if it s paired up, must be paired up with a number less than 10. For example, if 11 something = 100, then the something must be less than 10. But we know which numbers are paired up with numbers less than 10 because we ve done all the numbers up to 10! Remember: Once we find a number paired up with itself, we can stop. Process assessment 7m1, 7m2, 7m7, [C, R] Extra Practice: Find all pairs of numbers that multiply to the given number. Explain how you know your list is complete. a) 49 b) 64 c) 81 d) 144 Bonus 441 ANSWERS: a) b) c) d) Bonus Number Sense 7-10 D-7

8 Process assessment 7m1, 7m7, [R, C] Workbook Questions 4, 5 Introduce factor rainbows. Show students how the pairs of numbers that multiply to give a number can be made to look like a rainbow. Show the factor rainbows for 9 and 10 from Workbook p. 54. Point out that sometimes a number is paired up with itself; for example, 3 is paired up with 3 to multiply to 9, and we draw a loop from 3 to itself to show that. Extra Practice for Question 7: Make factor rainbows for 4, 5, 14, and 15. A shortcut for making factor rainbows. See Workbook Question 8. Process ExpectatioN Using logical reasoning Why factor rainbows look like rainbows. Notice that factor rainbows look like rainbows because no arc overlaps or crosses another. For example, in the factor rainbow for 12, all the numbers paired up with a number less than 3 will be more than 4 (which is paired up with 3). ASK: Why is this the case? (if two numbers multiply to the same number as 3 4 and one number is less than 3, the other number must be more than 4 because if one was less than 3 and the other less than 4, then they would multiply together to get a number less than 3 4) Tell students that the first six factors of 90 are 1, 2, 3, 5, 6, and 9. ASK: How can you use division to finish the factor rainbow for 90? (90 9 is the next factor, then 90 6, 90 5, 90 3, 90 2 and 90 1; these are in order from smallest to largest because dividing by a smaller number makes a larger result) Finding factors by division. Have students draw a factor rainbow for 36, using the shortcut learned in Workbook Question Remember: numbers are paired by what multiplies to 36. Tell students to pretend they only knew the factors of 36 up to 6. ASK: How could you find the other factors? Write on the board: 1 = 36 2 = 36 3 = 36 4 = 36 6 = 36 ASK: How can you find the numbers to put in the blanks? What operation can you use? (division) Write on the board: 36 1 = 36 2 = 36 3 = 36 4 = 36 6 = D-8 Teacher s Guide for Workbook 7.1

9 ASK: Can 36 have any factors greater than 6 other than the ones we find this way? Have two or three volunteers try to explain why not, then summarize: If a number more than 6 multiplies with something to give 36, that something must be less than 6: 6 6 = 36 So (more than 6) (6 or more) = more than 36 But (more than 6) (? ) = 36 Note that? must be less than 6. If? was 6 or more, the product would be more than 36, which it isn t. Since? = 6 or more gives something that we know isn t true,? cannot be equal to 6 or more, so it must be less than 6. Students at this age are just beginning to use this type of logical reasoning and so may find it difficult. Tell students that this type of logical reasoning is used a lot in real life, as for example in court. When someone is on trial, jury members assume the person is innocent. If the assumption eventually forces them to make a conclusion that they know from other evidence isn t true, then they know the assumption is wrong. Now remind students that we already found all the factors that multiply with a number less than 6 to make 36, so we won t find any new factors by looking at factors more than 6 we can stop at 6. Tell students that = 400. ASK: If a number more than 20 multiplies with another number to equal 400, what can you say about the other number? Why? Have students find all the factors of 400. (1, 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, 200, 400) ASK: How do you know that you can stop when you reach 20? Have students write an explanation individually in their notebooks. Then students can pair up, compare their explanations, and look for ways to improve them (or correct them if necessary). Patterns from factor rainbows. Look at the factor rainbow of 36. ASK: What is the largest factor of 36 other than 36? (18) What is it paired with? (2) What is the smallest factor of 36 other than 1? (2) Do you think that the largest factor (other than the number itself) will always be paired with the smallest factor (other than 1)? What about the next largest and next smallest? Have students explain the basis for their prediction. (EXAMPLES: the shape of the rainbow makes it look that way the arcs don t intersect and each arc is enclosed within the other; the numbers paired up are both between the previous numbers paired up; as a number gets larger, the number it multiplies with gets smaller) Emphasize that as a number gets larger, the number you need to multiply it by gets smaller, so if you know the smallest factor other than 1, you can find the largest factor other than the number itself. Have students find the largest factor, other than the number itself, of these numbers: a) 100 b) 1000 c) 448 d) 237 e) 1325 ANSWERS: a) 50 b) 500 c) 224 d) 79 e) 265 Number Sense 7-10 D-9

10 Organized searches in a different context. 1. Find all rectangles with side lengths that are whole numbers and... a) area 9 square units b) area 10 square units c) area 25 square units d) area 30 square units e) area 9 square units and perimeter 12 units Process assessment 7m1, 7m2, 7m5, [R, PS, CN] 2. a) Find the rectangle with area 16 and whole number sides that has the smallest perimeter. b) Find the rectangle with area 25 and whole number sides that has the smallest perimeter. c) Find the rectangle with area 36 and whole number sides that has the smallest perimeter. d) Predict which rectangle with area 100 will have the smallest perimeter. Verify your prediction by finding all rectangles with area 100 and calculating their perimeters. ACTIVITY Figure 1 Figure 2 Ask students to copy the shapes from Figure 1 on cm 2 grid paper so that each square is one square centimetre. Students should cut the pieces out and try to assemble them into a single large rectangle. After students have completed the puzzle, have them tape or glue the pieces down and set the rectangle aside (they will need it later). Then ask students to do the same with the shapes from Figure 2. Process ExpectatioN Looking for a similar problem for ideas Let students work for a few minutes, and then ASK: Who is finding this puzzle more difficult than the last puzzle? Tell students that you think it would be easier if they knew the size of the rectangle they were trying to build. Have them look at the smaller, simpler rectangle they built in the first puzzle. ASK: What size is that rectangle? (4 by 5) Is there a way you could have figured that out before putting the pieces together? Can you find the area of the rectangle by just looking at the pieces? (yes) How? (by adding the areas of the smaller rectangles) Show this: = 20, so the area is 20. (Students can confirm this by counting the squares in the completed rectangle.) ASK: What are the factors of 20? What are the possible lengths and widths of a rectangle with area 20? (2 10 or 4 5) Why isn t 2 10 possible? (the 3 3 square won t fit) Tell students that the small rectangle was easy to build by trial and error, but for larger rectangles, it saves time to do the problem mathematically. Now go back to the second puzzle. Give students time to calculate the area of the larger rectangle and determine what the dimensions of that rectangle must be. Then ASK: If these rectangles all fit into one larger D-10 Teacher s Guide for Workbook 7.1

11 rectangle, what is the area of the larger rectangle? (56) What are the possible lengths and widths of the rectangle? (7 8, 14 4, 28 2, 56 1) Can we eliminate any possibilities? Why? (7 8 won t fit the 12 1 piece; 28 2 won t fit the 3 3 or 5 3 pieces; and 56 1 won t fit the 2 5, 3 3, or 5 3 pieces only the 14 4 rectangle can fit all the pieces) Have students create the larger rectangle from the smaller ones knowing that its dimensions are ASK: Did organizing the information you had save time? For another challenge, draw rectangles on the board with the following dimensions: 4 4, 1 1, 2 3, 5 6, and 1 7. These have a total area of 60, so the possible larger rectangles are 1 60, 2 30, 4 15, 6 10, 12 5, and The rectangles 1 60, 2 30, 4 15, and 20 3 can all be eliminated immediately. It is a bit trickier to eliminate the 12 5 (it is not possible to fit both the 5 6 and 1 7 rectangles in a 12 5 rectangle), but the only possibility is Once students realize this, they can quite easily finish the puzzle. Bonus Draw the following 11 rectangles on grid paper, cut them out, and assemble them into one larger rectangle: 3 5, 3 4, 3 3, 2 4, 2 3, 1 4, 1 5, 1 1, 6 8, 2 5, 2 8 Solution: The total area of the pieces is 135 (135 = ). The factor rainbow for 135 is: So the possible dimensions of the larger rectangle are 1 135, 3 45, 5 27, Only the 9 15 rectangle will fit the 6 8 rectangle. Once students determine that the larger rectangle needs to be 9 15, it is not hard to complete the puzzle. Extension a) One-headed dragons have 2 tails and four-headed dragons have 3 tails. (Students might enjoy naming these beasts.) A knight must cut off both a dragon s head(s) and tails, as dragons with a tail but no head can still breed other dragons. After a battle, a knight counts 31 heads and 37 tails. How many dragons of each type did he slay? Students will need to be very organized. To get them started, ASK: What is the greatest number of four-headed dragons slain by the knight if there are 31 heads? (7) Why can t there be 8 four-headed dragons? (8 4 = 32 > 31) Number Sense 7-10 D-11

12 Since we know there are at most 7 four-headed dragons, let s start with the possibilities for four-headed dragons. Have students fill in the chart for each possibility: 4-headed dragons Heads Tails Notice that each possibility tells us how many heads and tails the one-headed dragons must have since the heads must total 31 and the tails must total 37: 4-headed dragons Heads Tails Number of heads of 1-headed dragons to make 31 heads in total Number of tails of 1-headed dragons to make 37 tails in total = = = = But the number of heads from one-headed dragons is equal to the number of one-headed dragons, and the number of tails is equal to 2 the number of one-headed dragons, so we just have to look for numbers that fit this condition: we have to find the row for which the last column is double the second-last column. (If there are 5 four-headed dragons, they have 20 heads and 15 tails, so the one-headed dragons have = 11 heads and = 22 tails. Indeed, 2 11 = 22, so this works. So the knight must have slain 5 four-headed dragons and 11 one-headed dragons.) b) Change the question slightly, so that four-headed dragons have 5 tails and one-headed dragons still have 2 tails. This time, the knight counts 23 heads and 34 tails. From that, he knows immediately that there are 11 dragons altogether. How did he figure that out? (HINT: Each dragon has one more tail than head.) D-12 Teacher s Guide for Workbook 7.1

13 Solution: Each dragon has one more tail than head, so if there are 11 more tails than heads in the pile, and 34 is 11 more than 23, then there must have been 11 dragons.) c) Find the number of each type of dragon slain in part b). Why is solving this question less work than part a) was? Solution: Since we know the total number of dragons (11), the only possibilities are 1 four-headed and 10 one-headed, 2 four-headed and 9 one-headed, and so on until 7 four-headed and 4 one-headed. Checking all possibilities until you count 23 heads and 34 tails shows that in fact there are 7 four-headed and 4 one-headed dragons. d) Go back to the problem in part a). Using what you learned from parts b) and c), how can you tell immediately that there are 6 more oneheaded dragons than four-headed dragons? Solution: ASK: If there are 1 one-headed dragon and 1 four-headed dragon, how many heads and tails are there? How many heads and tails are there if there are 2 of each? 3 of each? If there is the same number of each dragon, then what can you say about the number of heads and tails? (they are the same) Explain the reason for this pattern. ASK: Do one-headed dragons have more heads or tails? (tails) How many more? (1) Do four-headed dragons have more heads or tails? (heads) How many more? (1) Explain that every time a one-headed dragon is slain, there is added 1 more tail than head. Every time a four-headed dragon is slain, there is added 1 more head than tail. So, if there are the same number of each type of dragon, then there are the same number of heads as tails. ASK: What if there is 1 more one-headed than four-headed dragon are there more heads or tails? How many more? Have students investigate with different numbers. (There will always be 1 more tail than heads.) Repeat for 2 more one-headed than four-headed dragons (there will always be 2 more tails than heads), and then 3 more (there will always be 3 more tails than heads). ASK: We know there are 6 more tails than heads. How many more one-headed than four-headed dragons are there? (6 more) e) Have students do part a) again, using this new information. ASK: Which way was less work? (This way is much less work. We can just try the possibilities: 1 four-headed and 7 one-headed, 2 four-headed and 8 one-headed, and so on, until we find the one that gives the right number of heads and tails.) Number Sense 7-10 D-13

14 NS7-11 LCMs and GCFs Pages Curriculum Expectations Ontario: 7m2, 7m7, 7m12 WNCP: 6N3; review, [C, PS, R] Goals Students will find the lowest common multiple (LCM) and greatest common factor (GCF) of a set of numbers by listing all the multiples or all the factors of the numbers. Students will investigate properties of LCMs and GCFs. Vocabulary lowest common multiple (LCM) greatest common factor (GCF) consecutive PRIOR KNOWLEDGE REQUIRED Can find factors and multiples Materials BLM Squares and LCMs (p D-93) BLM Squares and GCFs (p D-94) Listing multiples to find the lowest common multiple. See Workbook Questions 1 5. Extra Practice for Workbook Question 3: Find the LCM of 2, 3, and 4. (Answer: 12) Bonus for Workbook Questions 4, 5: How can you find all the common multiples of two numbers if you know the lowest common multiple? (Answer: All the common multiples are multiples of the lowest common multiple. Students can discover this by picking numbers and checking. EXAMPLE: The common multiples of 4 and 10 are 20, 40, 60,, which are all multiples of the LCM, 20.) How can you find all the common multiples of three numbers if you know the lowest common multiple? (Answer: same as above) Bonus for Workbook Question 7: a) Explain why the common multiples of 2, 3, and 5 must also be common multiples of 2 and 3. b) What is the LCM of 2 and 3? c) List the first ten common multiples of 2 and 3. d) Circle the common multiples of 2 and 3 that are also multiples of 5. What is the LCM of 2, 3, and 5? e) Find the lowest common multiple of a) 3, 4, and 6 b) 4, 6, and 9 c) 2, 5, and 6 d) 6, 8, and 9 e) 6, 8, and 10 ANSWERS: a) 12 b) 36 c) 30 d) 72 e) 120 D-14 Teacher s Guide for Workbook 7.1

15 Listing factors to find the greatest common factor. See Questions Bonus for Workbook Question 10: List the factors and then find the GCF of 30, 42, 60, and 96. Answer: 30: 1, 2, 3, 5, 6, 10, 15, 30 42: 1, 2, 3, 6, 7, 14, 21, 42 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 96: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96 So the GCF is 6. Process assessment 7m1, 7m2, [R] Workbook Investigations 1 and 2 Review the mnemonic for factor and multiple (see NS7-9): There are few factors and many multiples. This is especially important at this stage because the word lowest in lowest common multiple primes students to think of small numbers, when in fact they are thinking about numbers larger than the number they are finding multiples of. Similarly, the word greatest in greatest common factor primes students to think of large numbers, when in fact they are thinking about numbers less than the number they are finding factors of. Investigating the GCF of two consecutive numbers. See Investigation 1 on Workbook p. 57. Process ExpectatioN Making and investigating conjectures Process ExpectatioN Looking for a pattern, Making and investigating conjectures Process ExpectatioN Representing, Visualizing Process assessment 7m6, [V] BLM Squares and GCFs Bonus Investigate the GCF of two consecutive even numbers. EXAMPLES: 6 and 8, or 18 and 20. (The GCF will always be 2.) Investigating the relationship between the GCF, the LCM, and the product of two numbers. See Investigation 2 on Workbook p. 57. After students finish Investigation 2, have them check their conjecture by creating more examples. Have students complete BLM Squares and LCMs. Emphasize how constructing the squares requires counting by the side lengths of the rectangles. For example, in part a), count by 2s along one side and count by 3s along the other side until you get the same length and width this will happen precisely when you find the first number that is a multiple of both 2 and 3, i.e., the LCM of 2 and 3. Extensions 1. a) For all examples in Investigation 2, have students calculate GCF LCM. What other number is this equal to? (the product of the original two numbers) b) Use your answer to part a) to help you find two numbers, a and b, with GCF 2 and LCM 12. Solution: We know the product of the two numbers must be 2 12 = 24. The possibilities are: 1 and 24, 2 and 12, 3 and 8, or 4 and 6. The GCF of each pair is: 1, 2, 1, and 2. So both 2 and 12 Number Sense 7-11 D-15

16 and 4 and 6 have GCF 2 and product 24, which means they also have LCM 12. c) Have students find the LCM of these pairs by first finding the GCF: i) 10 and 14 ii) 35 and 40 iii) 45 and 60 iv) 32 and 44 ANSWERS: i) GCF = 2, so LCM = = 70 ii) = 280 iii) = 180 iv) = 352 Process ExpectatioN Problem-Solving 2. There are 56 girls and 42 boys registered for a team event. The organizers would like to put the same number of girls on each team and the same number of boys on each team. They figured out that they can do this and still use all the students. a) Which of the following numbers must be a factor of both 56 and 42: the number of teams OR the number of people on each team? ANSWER: # of teams # of girls on each team = 56 and # of teams # of boys on each team = 42 Since the number of teams must divide into both 56 and 42, it must be a factor of both. b) What is the greatest common factor of 56 and 42? What does this tell you about the teams? ANSWER: The GCF of 56 and 42 is 14, so 14 is the maximum number of teams the competition can have. (In fact, with 14 teams, each team would have 4 girls and 3 boys.) 3. a) Use the distributive law to explain why, if 3 is a factor of two numbers, then it is also a factor of their difference. For example, if 3 is a factor of 9 and of 15, use the distributive law to show that it is also a factor of ANSWER: If 3 is a factor of two numbers, say 3 10 = 30 and 3 8 = 24, then the difference between the two numbers is: = 3 (10 8) = 3 2 which is again a multiple of 3. So indeed 3 is a factor of the difference between the two numbers. Encourage students to prove this for various examples that they create on their own, until they are convinced it will work no matter which numbers they use. Then introduce variables. Have students fill in a chart of this form: a a Tell students that you will use the variable a to represent any whole number. ASK: What numbers does 3 a represent? (the multiples of 3) Explain that if you want to represent two different multiples of D-16 Teacher s Guide for Workbook 7.1

17 3, you can use two different variables, for example 3 a and 3 b. Then: 3 a 3 b = 3 (a b). Note that the difference between any two multiples of 3 is thus also a multiple of 3. b) Now use the distributive law to explain why any factor of two numbers (including the GCF) must also be a factor of (and thus divide evenly into) the difference between the two numbers. ANSWER: Use the same reasoning as above with 4 or 5 as a factor of two numbers instead of 3 as a factor of two numbers. Have students prove it for various numeric examples until they are convinced it will work no matter which numbers they use. Then introduce variables: If m is a factor of two numbers, say m a and m b, then the difference between the two numbers is m a m b = m (a b) and m times anything is a multiple of m, so m divides evenly into it. c) Use the result of b) to explain why two consecutive numbers have GCF equal to 1. d) What must the GCF of two consecutive even numbers be? (2) e) The GCF of any two numbers must be a factor of the difference between the two numbers. So, to find the GCF of two large numbers that are close together, follow these steps: Step 1. Find the difference of the two numbers. Step 2. List the factors of the difference. Step 3. Check each factor to see if it is a factor of the smaller number. Step 4. The largest number that is a common factor of both the difference and the smaller number is the GCF of the numbers. EXAMPLES: Find the GCF of 127 and 137. Step = Step 2. The factors of are,, and. Step 3. Which of these factors divide 127? Step 4. The GCF of 127 and 137 is Process ExpectatioN Using logical reasoning Find the GCF of and Step = 25 Step 2. The factors of 25 are,, and. Step 3. Which of these factors divide 3415? and. Step 4. The greatest common factor of and is the largest of the answers from Step 3:. In order to be a factor of both and 3 440, a number must divide into both numbers. Discuss with students how they know that their answer above does in fact divide into both numbers. (since it divides into both 3415 and 25, it must also divide into their sum, Number Sense 7-11 D-17

18 3440) ASK: How do you know that the number you found is the greatest common factor? (If a larger number divided evenly into both 3415 and 3440, that larger number would also divide evenly into their difference, 25, but there is no larger number that divides evenly into both 3415 and 25 because we found the largest one.) Find the GCF of these numbers. i) and ii) and iii) 657 and 663 iv) 590 and 610 v) 396 and 400 vi) 398 and 402 vii) 372 and 377 viii) and ANSWERS: i) 5 ii) 1 iii) 1 iv) 10 v) 4 vi) 2 vii) 1 viii) 5 Bonus and ANSWER: The GCF must be a factor of 20 (the difference between the two numbers), so the GCF must be 1, 2, 4, 5, 10, or 20. The largest of these possibilities that divides the two numbers is 4, since 5, 10, and 20 are not a factor of either. D-18 Teacher s Guide for Workbook 7.1

19 NS7-12 Perfect Squares and Square Roots Pages Curriculum Expectations Ontario: 7m1, 7m2, 7m3, 7m16, 7m17 WNCP: optional, [R, C]; 8N1 Goals Students will identify perfect squares and square roots of perfect squares. Students will also compare and order perfect squares and square roots, and will operate on square roots. PRIOR KNOWLEDGE REQUIRED Vocabulary prime number perfect square squaring a number square root greater than (>) less than (<) Can find the factors of a small number Can find the area of a rectangle Can multiply up to 9 9 Review how to find the area of a rectangle. Draw a rectangle on the board, with grid squares showing its area: Remind students that the area of a shape is the amount of space it takes up. When the shape is a rectangle, we can measure it exactly by using samesized squares as a unit and counting how many squares the rectangle takes up. ASK: What is the area of the rectangle I drew? (12 squares) Have students copy these rectangles onto grid paper and find their areas: Process ExpectatioN Organizing data Write the side lengths of the rectangles on the board and ASK: How can you find the area of a rectangle from its side lengths? (multiply the side lengths together) Emphasize that the number of squares in each row times the number of rows is the total number of squares. This means that the number of squares along the top times the number of squares along the side is the area of the rectangle. Have students find the areas of various rectangles (without grid squares showing) that you draw and label on the board. EXAMPLES: 2 by 4, 3 by 3, 5 by 6, 1 by 12, 10 by 7, 8 by 9. Find factors by drawing rectangles. Tell students that a rectangle has area 12 squares. ASK: How many squares wide and long could it be? Challenge students to find all possible rectangles that have area 12 squares. Suggest that they try a width of 1, then a width of 2, then 3, and so on. Discuss when students know they can stop. Emphasize that what they are really doing is finding pairs of numbers that multiply to 12, so they can stop when they get a number that repeats, i.e., a rectangle congruent to one they ve already drawn. Number Sense 7-12 D-19

20 Have students find all the non-congruent rectangles that have area 18. Emphasize that if they go in order of width, they can stop as soon as they find the same rectangle twice (it will be turned the second time). ASK: What are the factors of 18? (the sides of the rectangles they found: 1, 2, 3, 6, 9, 18) Process ExpectatioN Visualizing, Representing Process ExpectatioN Oganizing data, Looking for a pattern Have students do Workbook Question 1. Introduce prime numbers. Have students look at their answers to Workbook Question 1. ASK: Which numbers have only one rectangle? (5 and 7) Tell students that if a number has only one rectangle, then the only factors of the number are 1 and itself. Any number with exactly two factors 1 and itself is called a prime number. A number greater than 1 is prime if you can draw only one rectangle having that area. The number 1 itself is not prime because it has only one factor. Have students find all the prime numbers between 1 and 20. ASK: The number 2 is prime. Can any other even number be prime? (no) Why not? (because 2 will be a factor of any other even number, and 2 is neither 1 nor the number itself) Can you find two consecutive numbers that are prime? (yes, 2 and 3) Have students try to find another pair of consecutive numbers that are prime. Suggest that they look in an organized way: Pairs of consecutive numbers Are they both prime? 2, 3 Yes 3, 4 No, because 2 is a factor of 4 4, 5 No, because 2 is a factor of 4 5, 6 No, because 2 and 3 are factors of 6 6, 7 No, because 2 and 3 are factors of 6 7, 8 No, because 2 and 4 are factors of 8 Have students continue the pattern. Since 2 is always a factor of one of the numbers (the even one), the numbers cannot both be prime unless 2 is one of them. So the only pair of consecutive numbers that are both prime is 2 and 3. Process ExpectatioN Reflecting on other ways to solve a problem, Looking for a pattern Introduce perfect squares. Have students look again at their answers to Workbook Question 1. ASK: For which numbers is the rectangle a square? (4 and 9) Tell students that any number larger than 0 is called a perfect square if you can draw a square with whole number side lengths having that area. For example, a 1-by-1 rectangle is a square with area 1; 4 is a perfect square (2 by 2); 9 is a perfect square (3 by 3). Tell students that you want to find the next two perfect squares. Demonstrate trying to make a square with each area from 10 to 16 in turn. Then tell students that you think this is a lot of work. Sometimes stopping to think about another way to solve the problem is faster. SAY: We know the first four perfect squares are 1 1 = 1, 2 2 = 4, 3 3 = 9, and 4 4 = 16.What will the next one be? (5 by 5) What is the area of the 5-by-5 square? (25) So the next perfect square after 16 is 25. Have students find all the perfect squares up to 100. Tell students D-20 Teacher s Guide for Workbook 7.1

21 that they could do this by checking each number in turn to see if that number is a perfect square, but it would take a lot longer. Eliminating prime numbers as possible perfect squares. Challenge students to find all the numbers up to 10 that are not prime and also not perfect squares (6, 8, and 10) Challenge students to try to find a number less than 10 that is both a prime number and a perfect square (there aren t any). Have students complete the following Venn diagram with numbers 1 to 10: Prime Number Perfect Square Process ExpectatioN Visualizing Process ExpectatioN Organizing data Process ExpectatioN Looking for a pattern Which numbers are outside both circles? (6, 8, and 10) Which numbers are inside both circles? (none) What does that mean? (there is no number from 1 to 10 that is both a prime number and a perfect square) ASK: Can any number be both a prime number and a perfect square? (no) Why not? (because a prime number has only one rectangle, but a perfect square larger than 1 has at least two rectangles a rectangle of length itself and width 1, and a square) Finding perfect squares. Remind students that they can find perfect squares by drawing all the squares with whole number side lengths in order (as in Introduce perfect squares, above). Review doing so, then ask students to write multiplication statements for the areas of squares with whole number side lengths from 1 to 7. (1 1 = 1, 2 2 = 4, 3 3 = 9, 4 4 = 16, 5 5 = 25, 6 6 = 36, 7 7 = 49) Challenge students to find the next three perfect squares without drawing squares. Emphasize that they can just multiply any number with itself to get a perfect square. For example, 81 is a perfect square because it can be written as 9 9. On grid paper, draw the square with side length 9 to illustrate this point. 0 is a perfect square. Tell students that because 0 = 0 0, mathematicians include 0 as a perfect square even though there is no square having area 0 (although one could argue abstractly that a square with side length 0 has area 0). Process assessment 7m6, [V] Workbook Questions 6, 7 Have students find all the perfect squares between 0 and 200, using the above definition of a perfect square. (0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196) Bonus for Workbook Question 7: Find another way to show that 10 is not a perfect square. (A 3-by-3 square has area 9 and a 4-by-4 square has area 16, so any square with area 10 has side length between 3 and 4, which means it is not a whole number.) Define squaring a number. See the second teaching box on Workbook p. 59 and Question 8. Number Sense 7-12 D-21

22 Perfect squares are numbers and can be treated as such. This means that we can compare and order them, add and subtract them, multiply and divide them. Comparing and ordering perfect squares. ASK: Which is larger, the square of 5 (5 2 ) or the square of 6 (6 2 )? (6 2 ) How do you know? (because 6 is greater than 5, so 6 6 will be larger than 5 5) Point out that students do not have to evaluate either 5 2 or 6 2 to compare them. Connect this to the area of squares: A square of side length 5 fits into a square of side length 6, and so has smaller area. Have students order lists of perfect squares and explain their strategy. See Workbook Question 11. Then ask students to decide which is larger between a perfect square and another whole number. Examples: 3 2 or 8 (3 2 = 9, so 3 2 is larger), 5 2 or 32 (5 2 = 25, so 32 is larger). Finally, have students order lists of numbers, including some perfect squares written in square notation. See Workbook Question 12. Define square root. Have students find the number that fits in both boxes: a) 36 = b) 16 = c) 49 = Discuss strategies. For example, guess, check and revise is a good strategy here. Tell students that this problem is the opposite of what they ve done so far instead of squaring a number by multiplying it by itself, we are starting with a perfect square and trying to find what number it is the square of. Tell students that what they have done is find the square root of the perfect squares above. Write square root on the board. Evaluating square roots. Have students find the square root of various perfect squares. EXAMPLES: The square of 5 is 25, so the square root of 25 is. The square of 3 is 9, so the square root of 9 is. The square of is 16, so the square root of 16 is. Process ExpectatioN Guessing, checking, and revising Process ExpectatioN Looking for a pattern Tell students that you want to find the square root of 64. Explain that a good strategy is to memorize 5 2 = 25 and compare numbers to it: any number less than 25 will have square root less than 5 and any number greater than 25 will have square root greater than 5. That gives a good starting point for guessing and checking. ASK: Is 64 more than 25 or less than 25? (more) Will its square root be more than 5 or less than 5? (more) Suggest 7 as a good first guess. ASK: What is 7 7? (49) So is 7 too low or too high? (too low) Guess a higher number next, say 8. What is 8 8? (64) Using even and odd to eliminate square root possibilities. First review the words even and odd. Then ASK: When is a perfect square even? Write the numbers from 0 to 10 and beside them their squares. ASK: Which numbers have a square that is even? (0, 2, 4, 6, 8, and 10) Which numbers have a square that is odd? (1, 3, 5, 7, and 9) Point out that the even numbers have a square that is even and the odd numbers have a square that is odd. ASK: Is 64 even or odd? (even) Could 7 really have been its D-22 Teacher s Guide for Workbook 7.1

23 square root? (no, the square root of an even number is even, but 7 is odd) Tell students that this makes it even easier to find square roots. ASK: What is the square root of 81? Will it be even or odd? (odd, because the square of an even number is even, and 81 is odd) Is the square root of 81 more than 5 or less than 5? (more) How do you know? (because 81 is more than 25) Will the square root be more than 10 or less than 10? (less) How do you know? (because 81 is less than 100) What are the odd numbers between 5 and 10? (7 and 9) Try both: 7 7 = 49 and 9 9 = 81, so 9 is the square root of 81. Give students random perfect squares and have them tell you the square root. They could answer by holding up the correct number of fingers. Introduce the standard notation for square roots, as in the teaching box on Workbook p. 60. Then have students find square roots as on Workbook Question 14. Process assessment 7m7, [V] Workbook Question 20 The side length of a square is the square root of the area. Tell students that a square has side length 3 cm. ASK: What is the area of the square? (9 cm 2 ) What if I know that the area of the square is 9 cm 2 but I don t know the side length? How can I find the side length from the area? PROMPT: How did you get the 9 from the 3? (3 3 = 9) How can you get the 3 back from the 9? What number multiplied by itself is 9? (take the square root of 9) Explain that if you know the area of a square, you can find its side length by taking the square root of the area. Extra Practice for Question 17: Have students find the side lengths of squares with these areas: a) 36 m 2 b) 64 cm 2 c) 81 km 2 Process ExpectatioN Connecting Compare the notation for area units and the notation for squaring a number. Discuss the relationship between the two notations (both involve raising the number 2). It makes sense that the notations will be similar because the concepts are similar squaring a number involves finding the area of a square. Operate on square roots. Tell students that square roots are numbers too, so you can operate on them as you do on numbers. You can add, subtract, multiply, and divide them. For each expression, have students rewrite the expression by evaluating the square roots, and then evaluate the resulting expression. a) b) 25 1 c) 16 9 d) 36 9 Evaluate the first example together: = = 5 Answers: b) 4 c)12 d)2 Bonus Include problems that require doing two operations and using the order of operations: a) b) 64 ( 9 4) c) d) ( 64 9) 4 ANSWERS: a) 22 b) 8 c) 2 d) 10 Number Sense 7-12 D-23

24 Process ExpectatioN Using logical reasoning The square root of a number squared is the number itself. Tell students that they ve shown you that they know how to take the square root of a perfect square. Tell them you want them to do the same thing, but using different notation. Write on the board: Have students evaluate each of these square roots by first evaluating the square inside the square root sign. As an example, point to the third example, and ASK: What are we taking the square root of? PROMPT: What is 3 2? (9) Write: 3 2 = 9 ASK: What is the square root of 9? (3, because 3 2 is 9; that s how we found the 9) Have students do the remaining examples. ASK: What do you notice? (the square root of a number squared is the number itself) Have students solve problems with larger perfect squares, to allow them to show off. See Workbook Question 16. Comparing and ordering square roots. Since square roots are numbers, we can compare them to other numbers and ask which is larger. First compare square roots to square roots, as in Workbook Question 18 and Question 19 a). Then compare square roots to other numbers, and then to perfect squares as well. Finally, perform operations on square roots and compare the results, as in Question 19 c). Extensions 1. Where are the perfect squares on a multiplication table? (the diagonal) 2. Any number greater than 1 that is not prime is called composite (1 is neither prime nor composite). Are there more prime numbers less than 100 or more composite numbers? How could you answer the question without checking every number? Solution: Encourage students to think about the numbers in general. No even number except 2 is prime. Half the numbers from 3 to 100 are even and therefore composite. If you also consider that all multiples of prime numbers are composite (so all multiples of 5, 7, 11, and so on), clearly there are more composite numbers than primes. 3. What numbers can be the remainder of a perfect square divided by 8? Have students complete the chart below. Perfect square Remainder when divided by Perfect square Remainder when divided by 8 What pattern do you notice? What numbers can be the remainder of a perfect square divided by 8? D-24 Teacher s Guide for Workbook 7.1