NS8-1 Factors and Multiples Page xxxx

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1 NS8- Factors and Multiples Page xxxx Curriculum Expectations Ontario: 7m7, 7m, 8m, 8m, 8m6 WNCP: 6N, [R, C] Goals Students will learn to identify factors and multiples and learn how to find all pairs of numbers that multiply to a given number PRIOR KNOWLEDGE REQUIRED Vocabulary factor multiple Knows the times tables Can divide -digit numbers into - and -digit numbers Defining multiples and factors. See Questions -. Bonus Question 4: 7 is not a multiple of 0. The only number that is a multiple of 0 is 0 itself. Question : is not a factor of. Only is a factor of. process Expectation Organizing data Using an organized list to find factors of whole numbers. As a class, find all pairs of numbers that multiply to. Use Questions 6-8 as models. EXTRA PRACTICE: for Questions 7, 8: Find all pairs of numbers that multiply to give. Then cross out the pairs that are repeated. How to know 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 Question 8 suggests 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) process Expectation Using logical reasoning process Expectation Other ways to solve a problem process Expectation Using logical reasoning ASK: What is larger: 8 6 or 9 7? How do you know? Emphasize that 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,,, 4,, or 6. Then check each one in turn on your list. Since each possibility is ruled out, 9 must not be paired up. ASK: Can 0 be paired up? How do you know? (Students might use similar reasoning or might instead state that 48 is not a multiple of 0. Discuss both answers.) ASK: Can be paired up? Continue in this way until all students see that the only numbers larger than 6 that are paired up are paired with,,, 4,, or 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. Number Sense 8-

2 EXTRA PRACTICE: Find all pairs of numbers that multiply to a) 6 6 c) 7 7 e) 0 0 Bonus b) f) d) When the number is equal to another number multiplied by itself (i.e., a perfect square). Repeat the exercise for 00 instead of 48, this time completing the first 0 rows of the chart. Emphasize that once you reach 0 0 = 00, any number more than 0, if it s paired up, must be paired up with a number less than 0. For example, if something = 00, then the something must be less than 0. But we know which numbers are paired up with a number less than 0 because we ve done all the numbers up to 0! Once we find a number paired up with itself, we can stop. EXTRA PRACTICE: Find all pairs of numbers that multiply to a) 49 c) 8 d) 44 Bonus b) Teacher s Guide for Workbook 8

3 NS8- Factors and Multiples Page xxxx Curriculum Expectations Ontario: 7m, 7m7, 7m WNCP: 6N, [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 Listing multiples to find the lowest common multiple. See Questions -. Bonus Question : Find the LCM of,, and 4. (ANSWER: ) Question 4: Find the LCM of: ANSWERS: a) 6, 8, and 0 0 b), 4, and 6 c) 4, 6, and 9 6 Question : 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 0 are 0, 40, 60,, which are all multiples of the LCM, 0.) How can you find all the common multiples of three numbers if you know the lowest common multiple? (Answer: same as above.) Listing factors to find the greatest common factor. See Questions 6-7. Bonus Question 7: List the factors and then find the GCF of 0, 4, 60, and 96. ANSWER: 0:,,,, 6, 0,, 0 4:,,, 6, 7, 4,, 4 60:,,, 4,, 6, 0,,, 0, 0, 60 96:,,, 4, 6, 8,, 6, 4,, 96 So the GCF is 6. Investigating the GCF of two consecutive numbers. See Investigation and Question 8. process Expectation Making and investigating conjectures Bonus Investigation : Investigate the GCF of two consecutive even numbers. EXAMPLES: 6 and 8, or 8 and 0. (The GCF will always be.) Number Sense 8-

4 Question 8: Without listing the factors, find the GCF of: a) 96 and 400 (ANSWER: Since the GCF must divide 4 (= ), the only possibilities are,, and 4. Check each of these in turn, starting with 4. Since 4 is indeed a factor of both, 4 is the GCF.) b) 98 and 40 (ANSWER: Since the GCF must divide 4 (= 40-98), the only possibilities are,, and 4. Since 4 doesn t divide into them and does, is the GCF). c) 7 and 77 (GCF is or ; GCF = ) d) 9 and 9 (GCF is or ; GCF = ) e) 7 and 780 (GCF is,, or ; GCF is ) Bonus 7 64 and ANSWER: The GCF must be a factor of 0 (the difference between the two numbers), so the GCF must be,, 4,, 0 or 0. The largest of these possibilities that divides the two numbers is 4, since is not a factor of either. process Expectation Looking for a pattern See Investigation. process Expectation Looking for a similar problem for ideas Investigating the relationship between the GCF, the LCM, and the product of two numbers. Fill in the first rows of the chart as a class, then have students complete the chart on their own. When students have finished A and B, write this equation on the board: a b = GCF LCM To help students with C, tell them to look for a similar problem for ideas. Write the equation from B below the equation already on the board: a b = GCF LCM 4 = Explain that C is asking them how to get the LCM from a b and the GCF. This is like asking how to get from 4 and. Ask a volunteer to tell you how to get from 4 and when you know that 4 is, then write on the board = 4. Have a volunteer write how to get the LCM from a b and the GCF: LCM = a b GCF Bonus Find two numbers a and b where the GCF is and the LCM is. (ANSWER: 4 and 6 or and ) Have students find the LCM of these pairs by first finding the GCF: ANSWERS: a) 0 and = 70 b) and = 80 c) 4 and = 80 d) and = 4 Teacher s Guide for Workbook 8

5 Bonus and 0 00 ANSWER: = = = process Expectation Looking for a pattern To help students with D, encourage them to circle the rows in the chart where the LCM is the same as the product a b. They should notice that the GCF is always in this case. To see why, look at the formula: a b = GCF LCM If a b = LCM, then LCM = GCF LCM, so GCF =. Investigating the special case where a is a factor of b. See part E of Investigation. In this case, it is not only true that a b = GCF LCM, but in fact a = GCF and b = LCM. Extensions process Expectation Looking for a pattern. There are 6 girls and 4 boys registered for a team competition. The organizers would like the same number of girls on each team and the same number of boys on each team. a) Which of the following numbers must be a factor of both 6 and 4: the number of teams OR the number of people on each team? ANSWER: # of teams # of girls on each team = 6 and # of teams # of boys on each team = 4 Since the number of teams must divide both 6 and 4, it must be a factor of both. b) What is the greatest common factor of 6 and 4? What does this tell you about the teams? ANSWER: The GCF of 6 and 4 is 4, so 4 is the maximum number of teams the competition can have. (In fact, with 4 teams, each team would have 4 girls and boys.) process Expectation Revisiting conjectures that were true in one context. Some students may wish to look for a relationship between a b c and the GCF and LCM for a, b, and c, written as GCF(a, b, c) and LCM(a, b, c). Some numbers for a, b, and c that students can use to investigate include: 6, 0,, 0,, 4, 4, 6, 8 8,, 0, 7, 0, 4, 6, 0, 0, 4, 6, 7, 0 Students will find that there is no relationship involving just these numbers. In fact, to find a relationship, students would also need to find the GCF of each pair. The relationship is a b c GCF(a, b, c) = LCM(a, b, c) GCF(a, b) GCF(a, c) GCF(b, c). Number Sense 8-

6 You could guide motivated students to discover this by creating a chart with these headings: a b c GCF(a, b, c) GCF(a, b) GCF(a, c) GCF(b, c) LCM(a, b, c) and then a chart with these headings: a b c GCF(a, b, c) GCF(a, b) GCF(a, c) GCF(b, c) LCM(a, b, c) using the same numbers for a, b, and c as above. The reason this relationship exists is well beyond the grade 8 curriculum. However, if you are interested, here is the proof: Take any prime number, say p. If we can show that p occurs the same number of times in the prime factorization of a b c GCF(a, b, c) as it does in the prime factorization of LCM(a, b, c) GCF(a, b) GCF(a, c) GCF(b, c), then the two numbers have the same prime factorization, so they are equal. (See NS8-4 for more on prime factorization.) Suppose that the prime factorization of a includes A copies of p. Suppose that the prime factorization of b includes B copies of p. Suppose that the prime factorization of c includes C copies of p. And suppose that A < B < C. How many copies of p are in the prime factorization of a b c GCF(a, b, c)? ANSWER: A + B + C + A. How many copies of p are in the prime factorization of LCM(a, b, c) GCF(a, b) GCF(a, c) GCF(b, c)? ANSWER: C + A + A + B. So p occurs the same number of times in the prime factorization of each number. This is true for any prime number, so the prime factorizations of the two numbers are the same. Therefore, the two numbers are equal. 6 Teacher s Guide for Workbook 8

7 NS8- Prime Numbers Page xxxx Curriculum Expectations Ontario: 6m7, 7m, 7m, 7m, 7m6, 7m7, 7m WNCP: 6N, [R, C, CN, V] Goals Students will decide which numbers are prime or composite for small numbers. Students will use Eratosthenes Sieve to find all prime numbers less than 00. PRIOR KNOWLEDGE REQUIRED Can divide -digit numbers by -digit numbers Can find factors Vocabulary factor product multiple prime composite process Expectation Looking for a pattern Discovering the definition of prime and composite numbers. Display a chart with the numbers from to and their factors: Number Factors,, 4,, 4, 6,,, 6 7, 7 8,, 4, 8 9,, 9 0,,, 0,,,, 4, 6,, 4,, 7, 4,,, Tell your students that the numbers from to 0 can be classified as follows: Prime Composite Neither Prime nor Composite,,, 7 4, 6, 8, 9, 0 Ask your students to look carefully at the classification and then ASK: Do you think that is prime? What about? Add them under prime and composite respectively. Continue with, 4, and. Have students find the factors of the numbers from 6 to and then place those numbers in the correct category. Do not encourage students to say the classifying rule until all students can comfortably predict where a number belongs. Then ask students to write down what they think prime and composite numbers are. Emphasize that is neither prime nor composite: How are and its factors different from the other numbers? Summarize by saying that a prime number has exactly two factors and a composite number has more than two factors. The number is neither prime nor composite because it has exactly one factor. Since any number Number Sense 8-7

8 has and itself as factors, a number is composite if it has at least one other factor as well. NOTE: Mathematicians do not include 0 as a composite number. In order to write 0 as a product of two numbers, one of the two numbers must be 0, e.g., 0 = 0. But composite numbers can be written as the product of two smaller whole numbers. If students ask about 0, you might say that prime and composite numbers are only referring to numbers greater than. Deciding if a number greater than is prime or composite. Have students decide whether these numbers are prime by dividing to see if any number (besides and the number itself) divides evenly into them: a) 7 b) 8 c) 9 d) 0 process Expectation Reflecting on what made the problem easy or hard Reflecting on other ways to solve a problem. Looking for a similar problem for ideas. ASK: Which number was it hardest to check? (9) What made it so hard? (For the others, it was easy to see that either or divided into it, so we could stop after checking those factors, but for 9, we had to check all the numbers to 8.) ASK: Do you need to check all the numbers from to 8? Tell students that you are going to look for a pattern that will help them eliminate possible numbers as factors. We ll look at 0 instead of 9 to get ideas. List the factors of 0: Have students pair up the factors so that they multiply to give 0. ANSWER: process Expectation Using logical reasoning ASK: What is the smallest factor of 0 greater than? () What is the largest factor of 0 less than 0 itself? () Notice that these multiply to give 0. This is because if there was a larger factor of 0, it would have to multiply with something less than to give 0. But there is no whole number between and. Have students predict the largest factor (other than the number itself) of these numbers: ANSWERS: a) 0 0 b) 4 c) 6 8 d) 40 0 e) 8 4 ASK: What is the smallest factor of 4 other than? () What will the largest factor of 4 (other than 4) be? (4 = ) Why? (Because if there was a larger factor, it would have to multiply with something smaller than to give 4, but is not a factor, so there is no such number.) Since = 4, if 6 something = 4, then that something has to be less than, because 6 or more is at least 48. So is the largest factor of 4, other than 4 itself. 8 Teacher s Guide for Workbook 8

9 Have students determine the largest possible factor of these numbers (other than the number itself) by first finding the smallest factor greater than : a) b) 6 c) d) e) 48 f) 49 g) 00 h) 99 i) 7 j) 80 SAMPLE ANSWER: i) The smallest factor of 7 greater than is, so the largest factor of 7 less than 7 is 7 =. Now return to the problem of determining the factors of 9, and whether or not we really have to check all the numbers up to 8. Using what you just learned from the factors of 0 look at the smallest possible factor and predict the largest possible factor any factor of 9 must be at least and at most 4 ( 4 = 8). If a number more than 4 was a factor, what would it multiply with? It would have to multiply with something at least, but is already too big, so the factor has to be less than. We ve now reduced our work by half! But we know that is not a factor because we checked, so the next smallest possibility is. How large can the other factor be now? Well, any factor must multiply with at least to give 9, and 0 is already too large, so the factor must be at most 9! process Expectation Organizing data But isn t a factor either, which we can check by long division. The next smallest number is 4, so any factor must multiply with at least 4 to give 9, and 8 4 is already too large, so we only have to check 4,, 6, and 7. None of these are factors by long division, so 9 is prime. (Or, one could eliminate 6 and 7 by the same reasoning since 4 isn t a factor, any factor must multiply with at least to make 9, but 6 is already too large, so we only need to check, but isn t a factor either.) Bonus Question 7: Find all numbers in the 40s that are prime. Question 0: Write consecutive numbers that are also all composite. (Sample answer: 4,, 6, 7, 8) Eratosthenes Sieve for finding prime numbers up to 0 Now tell your students that you will help them discover another way to find all the prime numbers less than 0. (See also Question.) Write on the board: Start by using a very tedious method. ASK: Is prime? (no, so cross it out) Is prime? (yes, because only and are factors) Is prime? (yes, because only and are factors) Is 4 prime? (no, because is a factor, so cross 4 out) Continue in this way until you have decided that 0 is not prime. Then explain that you think this is a lot of work and you think there has got to be a faster way. Number Sense 8-9

10 Rewrite the numbers from to 0 on the board, and cross out again because it is neither prime nor composite. Explain that this time, you will eliminate all the composite numbers in a more efficient way. A number is composite if it has a factor other than and itself. So you will look for numbers with these factors in an organized way. Start by looking for numbers with the factor. Tell students that there is only one number that has as a factor and is prime. ASK: What number is that? () Emphasize that any other number with as a factor is not prime, so you can cross all such numbers out: process Expectation Reflecting on other ways to solve a problem Notice that you ve already crossed out half the numbers! This is a lot easier than checking the factors of each number separately. Now look for numbers with a factor of. Which number has as a factor but is prime? () Can any other number with as a factor be prime? (no) Why not? (because it has a factor other than and itself) Circle the and then cross out the remaining multiples of. Emphasize that some multiples of will already be crossed out (because they are also multiples of, e.g., 4) process Expectation Reflecting on what made a problem easy or hard ASK: Why were the multiples of easy to find? (they are every second number) Why are the multiples of easy to find? (they are every third number) Remind students that so far, you ve tried the numbers and as factors. What number should you try next? Students should see that they can proceed systematically by trying the natural numbers in order (,,, 4,, ) so they should try 4 next. However, you don t have to try 4 any number that has 4 as a factor also has as a factor (any multiple of 4 is also a multiple of ) so you have already crossed those numbers out. ASK: Which factor should we try next? () Is prime? (yes) Can any of the other multiples of be prime? (no) Discuss why not. Then cross out all remaining multiples of : process Expectation Using logical reasoning 0 Now explain that if a number is less than 0 and composite, it has to have a factor,, 4, or. This is because it must be the product of two of its factors, and if the factors are both more than, then they are both at least 6, in which case the number would be at least 6 6 = 6. So one of the factors must be,, 4, or. But we have already crossed out all the numbers that are multiples of,, 4, or, so we have found Teacher s Guide for Workbook 8

11 all the composite numbers less than 0 and we can circle all the remaining numbers. Try to find all prime numbers less than 00. Challenge students to explain why any composite number less than 00 will have a number less than 0 as a factor. Then make the claim that any composite number less than 00 will have either,,, or 7 as a factor, and challenge students to explain why this is true. (We know the composite number has some number less than 0 as a factor. Suppose, for example, it has 9 as a factor. Then it will have as a factor too. If it has 8 as a factor, it will have as a factor. If it has 6 as a factor, it will have both and as factors, and so on.) Encourage students to use the hundreds chart from Question to answer Questions and. Encourage students to try to find organized ways ACTIVITIES process Expectation Reflecting on what makes a problem easy or hard, Connecting to the real world Some mathematicians study codes in order to send messages in secret. For example, when you buy things online, you have to send your credit card number over the Internet, which is publicly accessible, so the information has to be encoded to be safe. To solve this problem, mathematicians like to find problems that are easy to do in one direction but hard to reverse; they can use the easy direction to encode a message, but someone would have to use the hard direction to break the code and find the message. Each number below is the product of two prime numbers. Find the prime numbers and then verify your answer by calculating the product of the two primes. a) 8 b) c) d) 9 e) 4 f) ASK: Which is easier to find: the two prime numbers that multiply to or the product of and 7? If you wanted to encode a message, would you want the public information to be and 7 (so that anyone who wants to read the message has to multiply and 7 to break the code) or would you want the public information to be the number (so that anyone who wants to read the message has to find the factors of to break the code)? Why? (It is easier to find a product, since I just have to use the standard algorithm, but to find the factors of a number, I have to try all the prime factors in order until I find one that divides the number. To make it harder to break the code, force the person to find the factors of a large number.) A mathematician wants to make codes from numbers that have exactly two prime factors. Which of the following codes would be easy to break without a computer? Which would take more work to break without a computer? (To break the code, find the two prime numbers that multiply to give the number.) Don t give students the answers written below in brackets. a) 46 ( ) b) 99 ( ) Number Sense 8-

12 c) 6 ( ) d) 94 ( 97) e) 4 ( 47) f) 9 (7 ) g) 4 ( 7) h) 47 ( 9) To build a message that is hard to break, would you use numbers that are products of two large prime numbers or numbers that are products of a small prime number and a large prime number? Why? Tell students that to ensure that people cannot break codes, even with the powerful computers available today, mathematicians use products of prime numbers that are hundreds of digits long! connection Literature process Expectation Using logical reasoning To fully understand Eratosthenes Sieve, students need to understand if-then statements such as, if,, or 4 was a factor of, then we would have crossed the out already. To familiarize your students with if-then statements, you can go through the book Anno s Hat Tricks by Akihiro Nozaki (illustrated by Mitsumasa Anno). Extensions process Expectation Looking for a pattern, Connecting. a) Explore the patterns in the ones digits of the multiples of i) ii) 8 iii) iv) 7 v) 4 vi) 6 b) How does the pattern in the multiples of compare to the pattern in the multiples of 8? Describe any other relationships you notice between the various patterns. (Here is one relationship: The pattern for is the pattern for 7 read backwards. To get the ones-digit-pattern for s, write an endless list: and continually move right places each time, starting at :, 6, 9,,, 8, To get the ones-digit-pattern for 7s, write the same endless list, but start at 7 and move right 7 places each time. Notice that this is the same as moving left places each time because the pattern repeats every 0 digits and 0-7 =. Since moving left is the opposite of moving right, the list for 7 will be the list for backwards.). Are there more prime numbers or more composite numbers less than 00? What strategy could you use to answer the question without checking every number? Solution: No even number except is prime. Half the numbers from to 00 are even. If you also consider that all multiples of are composite (and so on), clearly there are more composite numbers than primes. process Expectation Connecting. A rectangle with side lengths equal to a whole number of centimetres has area 7 cm. What is its perimeter? (The rectangle must be a 7 by rectangle since 7 is prime, so its perimeter is 6.) Teacher s Guide for Workbook 8

13 process Expectation Connecting 4. Show your students a multiplication table: ASK: Are there any prime numbers in the table? Where are they? (,, and are all in the first row and first column) Why do you think that happened? (the other numbers in the table are a product of numbers other than and the numbers themselves, and so are composite) Number Sense 8-

14 NS8-4 Prime Factorizations Page xxxx Curriculum Expectations Ontario: 6m7, 7m, 8m, 8m, 8m6 WNCP: 6N, [V, R, PS] Goals Students will make factor trees to find the prime factorization of small numbers. PRIOR KNOWLEDGE REQUIRED Vocabulary factors factor trees prime composite factorization prime factorization Can multiply Can identify prime and composite numbers Can find the factors of a number Introduce factorizations. Review the words prime and composite. ASK: Is 0 prime or composite? (composite) What is a factor of 0 other than or 0? Write on the board 0 =. Have a volunteer fill in the blanks. Explain that the student has proven that 0 is composite and has found a factorization of 0. Ask if anyone can prove that 0 is composite in a different way. Is there a different factorization of 0? (0 and 4 are both factorizations of 0) Challenge students to find factorizations of 6, 8, 9, and to prove that the numbers are composite. Can students find two different factorizations of? ( 6 and 4) ASK: Does the factorization prove that is composite? (no, because any number can be written as itself) Introduce prime factorizations. Write on the board 0 = 0 and 0 = 4 and ask if any of the numbers in these factorizations are composite. Have volunteers prove it. (0 = and 4 = ) Then write on the board: 0 = and 0 =. ASK: Are any of the numbers in these factorizations composite? (no, they are all prime) Tell your students that these are called prime factorizations of 0 because all the numbers in the factorizations are prime. Find prime factorizations using factor trees. See Questions -. process Expectation Modelling, Organizing data Extra Practice: Use a factor tree to find a prime factorization. a) b) c) d) 40 e) 4 f) 00 Bonus 04 Branching patterns. Demonstrate what is meant by a branching pattern for a factor tree (see Question 4), and have students show the branching pattern for these factor trees: a) 4 b) 4 c) Teacher s Guide for Workbook 8

15 Then have students draw a factor tree for 4 that has this branching pattern: process Expectation Organizing data Bonus How many numbers less than 0 have a factorization tree with the following branching patterns? a) b) c) ANSWERS: a) Any number with prime factors (4 =, 6 =, 9,0,4,) b) Any number with prime factors (8,, 8, 0) c) Any number with 4 prime factors (6) ACTIVITY Students will assemble small rectangles into a single larger rectangle. Students will be able to solve puzzles with the smallest rectangles by trial and error, but with larger examples, it will be helpful to mathematically figure out the size the large rectangle needs to be. This can be done by adding the total areas of the small rectangles to find the area of the large rectangle, and then determining the large rectangle s possible dimensions by finding pairs of numbers that multiply to give that area. Use the following progressively more difficult puzzles to lead students to this discovery. Students can draw rectangles with the dimensions given on cm grid paper and cut them out. To make the puzzle pieces firmer, students can glue the grid paper onto thick paper. Puzzle A: process Expectation Using logical reasoning Ask students the dimensions of the large rectangle they made. All students should say 4. Show different possible ways to fit the pieces into the rectangle, and then challenge your students to figure out why everyone came up with the same size rectangle, even though they may have come up with different ways of fitting the pieces in. Making and investigating conjectures (ANSWER: The areas of the rectangles add to = 0, so the possible lengths and widths of the large rectangle are 0, 0, and 4. Neither the 0 nor the 0 will fit the rectangle, so the large rectangle must be 4.) Puzzle B: 4 Number Sense 8-4

16 process Expectation Problem-Solving, Working backwards process Expectation Visualization Have students calculate the area of the large rectangle that will fit the pieces ( = 6) and find all the possible lengths and widths ( 6, 8, 4 4, 7 8). Then explain to students that the 6 rectangle isn t possible because it won t fit the,, and pieces. ASK: What other dimensions are not possible? (The 8 rectangle isn t possible because it can t fit the or pieces; the 7 8 rectangle isn t possible because it won t fit the piece; only the 4 4 piece is possible.) Have your students solve the puzzle knowing that the pieces must fit into a 4 4 rectangle. ASK: Did organizing your information save time? Puzzle C: These have a total area of 60, so that the possible large rectangles are 0, 4, 6 0,, and 0. The rectangles 0, 4, and 0 can all be eliminated immediately. It is a bit trickier to eliminate the, but students will see by manipulating the pieces that it is not possible to fit both the 6 and the 7 rectangles into a rectangle, so the only possibility is 6 0. Once students realize this, it is quite easy to finish the puzzle. process Expectation Organizing data Puzzle D: The total area of the pieces is = The pairs of numbers that multiply to give are, 4, 7, and 9. To see this, students can divide the whole numbers in order and stop when they reach a number already in a pair; or they can notice that is already too big (= 69), so at least one of the numbers must be less than and they only have to try numbers up to. In either case, the only possible rectangle that will fit the 6 8 rectangle is the 9 rectangle. Starting with this information makes the problem much easier to solve. Extensions process Expectation Communicating. Find all the factors of 4. Explain the strategy you used.. Give an example of a composite number and explain why it is composite.. a) Draw factor trees for various prime numbers. EXAMPLES:, 7,,, 7, 9. process Expectation Looking for a pattern b) What does the factor tree for a prime number always look like? (ANSWER: a single point) 6 Teacher s Guide for Workbook 8

17 process Expectation Looking for a pattern 4. Have students find prime factorizations for 60. Invite volunteers to write or read aloud their answers. Look for different answers. If no one found a different answer, show a different one on the board. EXAMPLES: and. ASK: What is the same about these two factorizations and what is different? ANSWER: They have the same numbers (,, and ) occurring the same number of times but written in different orders. Explain that these prime factorizations are so essentially the same they contain the same prime numbers, the same number of times that mathematicians don t distinguish between them. In fact, you might teach your students the reason why was not defined to be prime. If was defined as prime, then numbers would have many different prime factorizations. For example, different prime factorizations of 60 would be and so on. Because mathematicians wanted to talk about the unique prime factorization of a number, it was more convenient to say that is not prime.. We can use prime factorization to determine whether two products are equal without calculating the products. EXAMPLE: 8 9 = 6 since their prime factorizations are 8 9 and so 8 9 = and 6 = The products 8 9 and 6 have the same factors, they are just rearranged. Use prime factorizations to decide if the two products are the same. a) 60 0 and 4 7 b) 0 7 and 4 4 c) 4 and 0 4 d) 0 9 and 6 7 Bonus and 9 49 Number Sense 8-4 7

18 NS8- Prime Factorizations and GCFs Page xxxx Curriculum Expectations Ontario: 8m7, 8m Goals Students will learn how to find the greatest common factor (GCF) from the prime factorizations of two or more numbers. Vocabulary factor prime factorization greatest common factor (GCF) process Expectation Looking for a pattern, Making and investigating conjectures, Problem- Solving, Using logical reasoning PRIOR KNOWLEDGE REQUIRED Can find factors and prime factorizations How the prime factorizations of a number and its factors are related. Have students find the factors of 4 and the prime factorizations of all the factors. Challenge students to describe how the prime factorization of each factor is related to the prime factorization of 4. Bonus Check to see if your conjecture holds for the factors of 0. ASK: How many s are in the prime factorization of 4? () How many s? () Does any factor of 4 have more than three s? (no) Why does this make sense? (If there were more than three s in a factor, then there would be more than three s in 4, because 4 = that factor another number. Since there are only three s in 4, there cannot be more than that in the factor.) To emphasize the point, have students find the prime factorization of two numbers and their product, for example, and. ANSWER: = and = and = ( ) ( ). Notice that the number of times each factor occurs in the product is the sum of the number of times it occurs in each factor. You can t have fewer s in the product than there are in one of the factors. So if 4 has three s in its prime factorization, no factor of 4 can have more than three s any factor of 4 has at most as many s (and s) as 4. EXTRA PRACTICE: The prime factorization of 600 is. Without calculating them, decide which of the following products are factors of 600: Bonus For the products that are factors, find the number it multiplies with to make 600. EXAMPLE: multiplies with 0 (= ). Using prime factorizations to find the GCF of two numbers. See Question. Guide students through the same steps to find the GCF of 60 and Teacher s Guide for Workbook 8

19 EXTRA PRACTICE for Question : a) = 7 b) 4 = 00 = = EXTRA PRACTICE for Question 4: a) 8 and 94 b) 0 and 40 c) 68 and 96 d) 44 and e) 6 and 84 SAMPLE ANSWER: e) 6 = 7 and 84 = 7, so GCF = 7 = Using prime factorizations to find the GCF of three numbers. Revisiting conjectures that were true in one context. Now that students know how to find the GCF of two numbers from their prime factorizations, have them make a conjecture about how to find the GCF of three numbers from their prime factorizations. (ANSWER: Find all the prime factors that all three prime factorizations have in common.) Students can test their conjecture on these numbers: a) 4,, and 40 b) 4, 0, and c) 4, 8, and d) 40, 70, and 90 e) 0, 40, and 6 f) 0,, and 8 g, 40, and 4 h), 40, and 70 process Expectation Organizing data Extension Prime factorizations can make it easier to list all the factors of a number. For example, if 0 =, then we can list all the factors of 0 by listing the prime factors separately (one at a time) and then listing all the possible products of the factors (two at a time, three at a time, etc.), until we have used them all up. (Don t forget that is always a factor!) Here is the list for the factors of 0: is always a factor one at a time two at a time three at a time,,,, So,,,, 6, 0,, and 0 are all the factors of 0. List all the factors of a) 4 b) 0 c) 70 d) 770 Bonus 40 Sometimes a number has a prime factor occurring more than once, as in = or 8 =. When we list the factors for such numbers, some will occur twice. Here is the list for the factors of : is always a factor one at a time two at a time three at a time,,,, So the factors of are,,, 4, 6,. (We don t have to list and 6 twice.) List all factors by first finding a prime factorization. a) 8 b) 7 c) 7 d) 8 e) f) 90 Number Sense 8-9

20 NS8-6 Prime Factorizations and LCMs Page xxxx Curriculum Expectations Ontario: 8m Goals Students will learn how to find the lowest common multiple (LCM) from the prime factorizations of two or more numbers. Vocabulary prime factorization multiple lowest common multiple (LCM) PRIOR KNOWLEDGE REQUIRED Can find factors and prime factorizations Using prime factorizations to find the LCM of two numbers. See the Investigation and Questions -. EXTRA PRACTICE for Question 4: a) 4 = and 70 = 7 ( 7 = 60) b) = 7 and 6 = 7 ( 7 = 80) Bonus 8 = 9 and 44 = ( 9 = 86) EXTRA PRACTICE for Question : a) 84 and 08 b) 4 and 6 c) 4 and 60 SAMPLE ANSWER: a) 84 = 7 08 = So LCM = 7 = 76 Bonus Use prime factorizations to find the LCM of three numbers. a) 4, 0 and b 40, 6, and 40 c) 0,, and 6 d), 6, and 4 e) 4,, and SAMPLE ANSWER: a) 4 = 0 = = So LCM = = 480 ACTIVITY The Pool Factor (This activity adapted from work originally by Rich Cornwall.) Look at a rectangular grid as a pool table with 4 corner pockets, but no side pockets. 0 Teacher s Guide for Workbook 8

21 If you hit a pool ball at a 4 angle from the bottom left corner, the ball will travel around the grid, always bouncing off walls at a 4 angle, until it falls into a pocket. a) Count the number of times the ball hits a side wall on its trip. Count both the starting and ending points as hits. b) Count the number of squares the ball passes through on its trip. For the 6 4 grid, there are hits and the ball passes through squares. FINISH START Complete the following chart by drawing pool tables of different dimensions (base height) on grid paper and drawing the path of a pool ball hit at a 4 angle from the bottom left corner. Base Height GCF LCM Number of Hits # of Squares Passed Through Number Sense 8-6

22 Predict a rule for determining a) the number of hits b) the number of squares passed through from the base and the height. Using b for base and h for height, express your answers in terms of variables. Check your predictions with other examples of base and height. ANSWERS: a) (b + )/GCF b) LCM Extensions. Multiply two even numbers. How do you know that 4 is always a factor of the product?. What is the least number with factors 4, 0, and 8?. Ask students to look back at their work in the Activity above. Have them look at all the grids with base 7. ASK: In which corner does the ball land when the height is an odd number: top left, top right, bottom left, or bottom right? Where does the ball land when the height is even? Have students draw more grids with base 7 and GCF, and look for a pattern. Then ASK: What kind of number is 7, odd or even? Have students look at all the grids with odd bases and GCF. (They can draw more such grids too.) Does the pattern hold? Now have students look for patterns when the base is even. In general: b odd h even b odd h odd b even h even b even h odd Note that the base and height cannot both be even if their GCF if, so the ball never lands in the bottom left corner, where it started. Teacher s Guide for Workbook 8

23 NS8-7 Order of Operations Page xxxx Curriculum Expectations Ontario: 6m, 7m WNCP: 6N9, [PS] Goals Students will understand the need for brackets in expressions and for assigning an order to the operations. PRIOR KNOWLEDGE REQUIRED Vocabulary order of operations brackets NOTE: This lesson will probably take two days to do. Can add, subtract, multiply, and divide -digit and small -digit numbers The need for an order of operations. Have students solve the following problem: 8 +. Discuss how to get the answer (subtract from 8 then add ) and how to get the answer (add and first, then subtract from 8). ASK: What could we do to make it clear which operation to do first? (Students may suggest ideas other than inserting brackets if they are not familiar with brackets yet; accept all answers.) Introduce brackets. The brackets tell you to do the operations in brackets first. Writing (8 ) + means + = ; writing 8 ( + ) means 8 7 =. If there are no brackets, we do the addition and subtraction from left to right, so means the same thing as (8 - ) +. Have students evaluate: a) (8 ) + and 8 ( + ) b) (0 4) + and 0 (4 + ) c) ( + 7) 4 and + (7 4) d) (6 + ) and 6 + ( ) process Expectation Making and investigating conjectures Predict and then check whether these expressions have the same answer: a) (7 + ) and 7 + ( ) b) (7 4) + and 7 (4 + ) The order of operations for expressions involving only addition, subtraction, and brackets. Explain that when there are brackets, we evaluate the expressions in brackets first, then write the expression without brackets, and then solve from left to right. EXTRA PRACTICE: a) (4 + ) + b) (4 + ) c) (4 + ) d) (4 + + ) SAMPLE ANSWER: a) 9 + = 4 + = 7 Which pairs of expressions above have the same answer? (ANSWERS: a and c; b and d) Bonus a) (4 + + ) + 6 Students can use Question as a model for placing the brackets in Question. Some students may think of using nested brackets, such as (9 ( )). Allow them to investigate this case. Note, however, that there will be no new answer, since, for example: (9 ( )) = 9 + = (9 ). Bonus Question : Add brackets to in different ways to get as many different answers as you can. Note that this will result in only one Number Sense 8-7

24 answer, as in part i), even though both addition and subtraction are involved. The reason is that the subtraction is at the end. Expressions involving only multiplication, division, and brackets. Have students solve. Discuss how to get the answer 9 and how to get the answer. Using the same rule as for addition and subtraction (i.e., moving from left to right) what s the right answer 9 or? ( = = 9) How would students add brackets if they want to ensure someone will get the answer? ( ( )) Question 4: Add brackets to 40 0 in different ways to get as many different answers as you can. (There will be only one answer.) Discuss when changing the order the operations are done in affects the answer. (Problems with only multiplication have only one answer, no matter where you add brackets, and problems where the only division is at the end have only one answer. Changing the order for all other combinations of multiplication and division changes the answer.) Students can now complete Questions and 6. (Notice that they do not need to know that multiplication and division are done before addition and subtraction to complete the questions; they only need to know to do what s in brackets first.) Why multiplication is done before addition. Write + 4. Explain that since 4 is really a short form for adding four s, this expression can be written as Look back at the original expression. ASK: To get the same answer as the new expression, would you do + 4 first and then multiply by, or 4 first, and then add? (4 first) Have students write these expressions out using only addition: ANSWERS: a) b) c) d) Notice that 4 + = = + = 7 This is the same as doing the multiplication first: (4 ) + ( ) = + = 7. Explain that when we see multiplication in an expression with addition, we always calculate the multiplication first because multiplication is just a short form for repeated addition. Explain to students that they assume this order of operations that multiplication is done before addition when they write numbers in expanded form. EXAMPLE: 4 = For the example above, ASK: What answer would you get if you did all the operations from left to right? (ANSWER: = 4 Teacher s Guide for Workbook 8

25 = = 04) What answer would you get if you did all the addition first, and then the multiplication? (ANSWER: 04 = = 4680) These answers are both wrong! The only correct way to evaluate the expression is to do all the multiplication first: = 4. That s what we mean when we write numbers in expanded form. So in a way, this order of operations is nothing new. The order of operations for all operations and brackets. Now tell students the order hierarchy:. Evaluate all expressions in brackets.. Do multiplication and division from left to right.. Do addition and subtraction from left to right. Explain why multiplication and division are treated as a group, while addition and subtraction are treated as a separate group, as follows: Addition and subtraction are naturally grouped together because they undo each other; multiplication and division are grouped together for the same reason. We already said that multiplication is done before addition. Since multiplication is grouped with division and addition is grouped with subtraction, we do all instances of multiplication and division before all instances of addition and subtraction, unless there are brackets (in which case we do whatever is in the brackets first). Bonus Question 7: Evaluate each expression using the correct order of operations. a) ( 7) (6 6) ANSWER: ( 7) 0 = 8 0 = 40 0 = 4 b) 90 ( ) (4 + ) + ANSWER: 90 ( 0) (4 + 6) + = = = 0 + = c) (80 ( ) ( )) 6 ANSWER: (80 0 0) 6 = (8 0) 6 = (40 0) 6 = 0 6 = process Expectation Looking for a pattern Finding expressions without brackets that mean the same as given expressions with brackets. Explain that sometimes, we don t need to write any brackets if we change some of the operation symbols. Have students calculate these three expressions: 0 ( + ) Number Sense 8-7

26 process Expectation Justifying the solution ASK: Which expression without brackets has the same answer as the expression with brackets? EXTRA PRACTICE for Question : Create new expressions with the same answer only by removing brackets and, where necessary, changing operation signs. Verify your answer by calculating both expressions. a) 0 (8 ) b) 0 (8 ) c) 0 (8 ) d) 0 (8 ) e) 0 4 (0 ) f) 0 4 (0 ) Using the distributive property to write expressions without brackets. See Investigation, parts A-E. Bonus A: 6 = + + So ( + + ) = ( ) + ( )+ ( ) process Expectation Reflecting on what made the problem easy or hard Questions for B. and C.: Discuss what made it easy to calculate 7 and 4. ASK: What kinds of numbers are easiest to multiply by? Write a large number on the board, such as: ASK: What can you put in the blank so that you will know the answer easily? (0,, 0, 00, 000, and so on) Challenge students to prove their assertions can they in fact write the answer easily? Review multiplying by multiples of 0, 00, 000, and so on. Solve: a) 00 b) c) 7 80 d) e) f) process Expectation Splitting into simpler problems Challenge students to break the following products into simpler products using multiples of 0, and then solve. ANSWERS: a) (0 + ) = 0 + = 0 + = 6 b) (0 + 4) = = = 68 c) 6 6 = 6 (0 + ) = = 0 + = d) 4 4 = 4 (0 + ) = = 00 + = e) = 6 (0 + 8) = = = 8 Bonus C: a) 7 b) 4 c) 478 d) 9 67 SAMPLE ANSWER: d) 9 67 = 9 ( ) = = = Teacher s Guide for Workbook 8

27 Bonus D: 7 Bonus E: Have students write this expression without brackets: ( ) ( + 7) ANSWER: ) EXTRA PRACTICE for F: Have students break these problems into simpler problems: a) b) 4 c) 7 4 SAMPLE ANSWER: c) (0 + 7) (0 + 4) = = = 998 Bonus 4 ANSWER: 4 = ( ) ( ) = = = 7 09 Investigate types of expressions that cannot be written without brackets. See parts G and H in the Investigation. Notice that, while multiplication distributes over addition and subtraction, addition and subtraction do not distribute over multiplication. Also, in division, the dividend can be split into a sum or difference, but the divisor cannot. Extensions. Use a cube model to write (4 + ) ( + 6) ( + 7) without brackets: This expression represents the volume of the whole cube. Using the diagram at right, challenge students to find the volume of each of the 8 pieces separately, and then the volume of the whole cube. To find the volume of the 8 pieces separately, students will need to write the side lengths not shown. For example, the edge at the top right of the diagram is divided into parts of length 6 and. To find the volume of the entire cube, students add the volumes of the 8 pieces. Number Sense 8-7 7

28 process Expectation Problem-Solving Hint: You many need to use a number consisting of 4s, such as 44.. Using four 4s each time, make expressions for each value from 0 through 0. You may use brackets and any of the four operations. EXAMPLE: (4 4) (4 + 4) = 6 8 = POSSIBLE ANSWERS: 0 = (4-4) (4 + 4) 6 = (4 + 4) = = = 4 4 (4 + 4) 8 = (4 + 4) = ( ) 4 9 = = (4-4) = (44-4) 4 = ( ) 4 How many different expressions can your students come up with?. Sometimes, we can reverse the digits of both factors and get the same product! a) The digits in these factors have been reversed. Use factor trees to check which equations work (i.e., are correct). Cross out the ones that don t work. 69 = = = = = = = = 49 7 = 7 8 = = = 4 9 b) Pick one equation from part a) that works and one that doesn t work. Use the area model or the distributive law to find all 4 products from both equations as sums of smaller products. What is special about the one that works that makes it work? Make a conjecture about what makes it work in special cases and check your conjecture on all examples above. Then find another example where it works. SAMPLE ANSWER: It works for 69 = = 96, but not for 94 and 49. Using the distributive law 69 = (0 + ) (60 + 9) = = (0 + ) (90 + 6) = Since 9 = 6, it is also true that 0 60 = 0 90 The following equations are also true: 0 9 = 90 and 0 6 = 60. So what makes this work is that the product of the ones digits in the original numbers equals the product of the tens digits: 9 = 6. This isn t true in the other example: 94 = (90 + 4) (0 + ) = and 49 = (40 + 9) (0 + ) = Teacher s Guide for Workbook 8