Math Mammoth Canadian Grade 6-B Complete Curriculum PDF

Chapter 6: Prime Factorisation, GCF and LCM Introduction The topics of this chapter belong to a branch of mathematics known as number theory. Number theory has to do with the study of whole numbers and their special properties. In this c...

Chapter 6: Prime Factorisation, GCF and LCM Introduction The topics of this chapter belong to a branch of mathematics known as number theory. Number theory has to do with the study of whole numbers and their special properties. In this chapter, we revise prime factorisation and study the greatest common factor (GCF) and the least common multiple (LCM). The main application of factoring and the greatest common factor in arithmetic is in simplifying fractions, so that is why I have included a lesson on that topic. However, it is not absolutely necessary to use the GCF when simplifying fractions, and the lesson emphasises that fact. The concepts of factoring and the GCF are important to understand because they will be carried over into algebra, where students will factor polynomials. In this chapter, we lay the groundwork for that by using the GCF to factor simple sums, such as 27 + 45. For example, a sum like 27 + 45 factors into 9(3 + 5). Similarly, the main use for the least common multiple in arithmetic is in finding the smallest common denominator for adding fractions, and we study that topic in this chapter in connection with the LCM. Primes are fascinating “creatures,” and you can let students read more about them by accessing the Internet resources mentioned below. The really important, but far more advanced, application of prime numbers is in cryptography. Some students might be interested in reading additional material on that subject—please see the list for Internet resources. Keep in mind that the specific lessons in the chapter can take several days to finish. They are not “daily lessons.” Instead, use the general guideline that sixth graders should finish about 2 pages daily or 9-10 pages a week in order to finish the curriculum in about 40 weeks. Also, I recommend not assigning all the exercises by default, but that you use your judgement, and strive to vary the number of assigned exercises according to the student’s needs. Please see the user guide at https://www.mathmammoth.com/userguides/ for more guidance on using and pacing the curriculum. You can find some free videos for the topics of this chapter at https://www.mathmammoth.com/videos/ (choose 6th grade). The Lessons in Chapter 6 page span The Sieve of Eratosthenes and Prime Factorisation....... 13 4 pages Using Factoring When Simplifying Fractions................ 17 3 pages The Greatest Common Factor (GCF)............................. 20 3 pages Factoring Sums............................................................... 23 3 pages The Least Common Multiple (LCM)............................. 26 4 pages Chapter 6 Mixed Revision............................................. 30 2 pages Chapter 6 Revision......................................................... 32 2 pages Sample worksheet from https://www.mathmammoth.com 11  The Sieve of Eratosthenes and Prime Factorisation Remember? A number is a prime if it has no other factors besides 1 and itself. For example, 13 is a prime, since the only way to write it as a multiplication is 1 · 13. In other words, 1 and 13 are its only factors. And, 15 is not a prime, since we can write it as 3 · 5. In other words, 15 has other factors besides 1 and 15, namely 3 and 5. To find all the prime numbers less than 100 we can use the sieve of Eratosthenes. Here is an online interactive version: https://www.mathmammoth.com/practice/sieve-of-eratosthenes 1. Cross out 1, as it is not considered a prime. 2. Cross out all the even numbers except 2. 3. Cross out all the multiples of 3 except 3. 4. You do not have to check multiples of 4. Why? 5. Cross out all the multiples of 5 except 5. 6. You do not have to check multiples of 6. Why? 7. Cross out all the multiples of 7 except 7. 8. You do not have to check multiples of 8 or 9 or 10. 9. The numbers left are primes. List the primes between 0 and 100 below: 2, 3, 5, 7, ____________________________________________________________________________ Why do you not have to check numbers that are bigger than 10? Let’s think about multiples of 11. The following multiples of 11 have already been crossed out: 2 · 11, 3 · 11, 4 · 11, 5 · 11, 6 · 11, 7 · 11, 8 · 11 and 9 · 11. The multiples of 11 that have not been crossed out are 10 · 11 and onward... but they are not on our chart! Similarly, the multiples of 13 that are less than 100 are 2 · 13, 3 · 13,..., 7 · 13, and all of those have already been crossed out when you crossed out multiples of 2, 3, 5 and 7. 1. You learned this in 4th and 5th grades... find all the factors of the given numbers. Use the checklist to help you keep track of which factors you have tested. a. 54 b. 60 Check 1 2 3 4 5 6 7 8 9 10 Check 1 2 3 4 5 6 7 8 9 10 factors: ______________________________ factors: ______________________________ c. 84 d. 97 Check 1 2 3 4 5 6 7 8 9 10 Check 1 2 3 4 5 6 7 8 9 10 Sample worksheet from factors: ______________________________ factors: ______________________________ https://www.mathmammoth.com 13  For your reference, here are some of the common divisibility tests for whole numbers. A number is... A number is... divisible by 2 if it ends in 0, 2, 4, 6, or 8. divisible by 3 if the sum of its digits is divisible by 3. divisible by 5 if it ends in 0 or 5. divisible by 4 if the number formed from its last two digits is divisible by 4. divisible by 10 if it ends in 0. divisible by 6 if it is divisible by both 2 and 3. divisible by 100 if it ends in “00”. divisible by 9 if the sum of its digits is divisible by 9. Use the various divisibility tests when building a factor tree for a composite number. 135 27 135 441 49 441 / \ / \ / \ / \ 5)135 9)441 5 · ? 5 · 27 9 · ? 9 · 49 -10 / \ -36 35 3·9 81 / \ / \ -35 / \ -81 3·3·7·7 0 3·3 0 We start out by noticing that 135 is divisible by 5. Adding the digits of 441, we get 9, so it is From long division, we get 135 = 5 · 27. The divisible by 9. We divide to get 441 = 9 · 49. final factorisation is 135 = 3 · 3 · 3 · 5 or 33 · 5. The end result is 441 = 3 · 3 · 7 · 7 or 32 · 72. 2. Find the prime factorisation of these composite numbers. Use a notebook for long divisions. Give each factorisation below the factor tree. a. 124 b. 260 c. 96 / \ / \ / \ 2 · ___ 10 · ___ 3 · ___ / \ / \ / \ / \ 2 124 = 260 = 96 = d. 90 e. 165 f. 95 Sample 90 = worksheet from 165 = 95 = https://www.mathmammoth.com 14 3. Mark an “x” if the number is divisible by 2, 3, 4, 5, 6, or 9. Divisible by 2 3 4 5 6 9 Divisible by 2 3 4 5 6 9 128 209 765 6 042 4. Find the prime factorisation of a. 912 b. 528 the numbers. Use a notebook / \ for long divisions. Give each 4 · _____ factorisation below the factor tree. Note: in (a), the last two digits of 912 are “12” so it is divisible by 4. 912 = 528 = c. 76 d. 126 e. 272 76 = 126 = 272 = 164 168 5. Mia and Alex found the prime factorisation of 164 and 168, / \ / \ and were completely surprised that they got the same 4 · 42 8 · 21 factorization for both! / \ / \ / \ / \ 2·2 ·6·7 2·4 ·3·7 Investigate the situation. Is there something fishy going / \ / \ on somewhere? 2·3 2·2 164 = 23 · 3 · 7 168 = 23 · 3 · 7 6. Find all the primes between 100 and 110. How? You need to check, for each number, whether it is divisible by 2, 3, 4, 5, 6, 7, 8, 9, or 10. Sample worksheet from https://www.mathmammoth.com 15 7. Find the prime factorisation of these composite numbers. a. 196 b. 380 c. 336 196 = 380 = 336 = d. 306 e. 116 f. 720 306 = 116 = 720 = g. 675 h. 990 i. 945 675 = 990 = 945 = Find all the primes between 0 and 200. Use the sieve of Eratosthenes again (you need to make a grid in your notebook). This time, you need to cross out 1, and then every even number except 2, every multiple of 3 except 3, every multiple of 5 except 5, every multiple of 7 except 7, every multiple of 11 except 11 and every multiple of 13 except 13. Sample worksheet from https://www.mathmammoth.com 16  Using Factoring When Simplifying Fractions You have seen the process of simplifying fractions before. ÷4 In simplifying fractions, we divide both the numerator and = 12 3 the denominator by the same number. The fraction becomes = simpler, which means that the numerator and the 20 5 denominator are now smaller numbers than they were before. Every four slices have been joined together. ÷4 However, this does NOT change the actual value of the fraction. It is the “same amount of pie” as it was before. It is just cut differently. Why does this work? It is based on finding common factors and on how fraction multiplication works. In our example above, 12 4·3 the fraction can be written as. Then we can cancel out those fours:. 20 4·5 4·3 4 3 The reason this works is because is equal to the fraction multiplication ·. And in that, 4/4 is 4·5 4 5 equal to 1, which means we are only left with 3/5. Example 1. Often, the simplification is simply written or indicated this way → Notice that here, the 4’s that were cancelled out do not get indicated in any way! You only think it: “I divide 12 by 4, and get 3. I divide 20 by 4, and get 5.” Example 2. Here, 35 and 55 are both divisible by 5. This means we can cancel out those 5’s, but notice this is not shown in any way. We simply cross out 35 and 55, think of dividing them by 5, and write the division result above and below. 1. Simplify the fractions, if possible. 12 45 15 13 a. b. c. d. 36 55 23 6 15 19 17 24 e. f. g. h. 21 15 24 30 2. Leah simplified various fractions like you see below. She did not get them right though. Explain to her what she is doing wrong. 24 20 1 27 7 14 10 6 3 = = = = = = 84 80 4 60 40 16 12 8 4 Sample worksheet from https://www.mathmammoth.com 17 Sample worksheet from https://www.mathmammoth.com 34  Chapter 7: Fractions Introduction This chapter begins with a revision of fraction arithmetic from fifth grade—specifically, addition, subtraction, simplification, and multiplication of fractions. Then it focuses on division of fractions. The introductory lesson on the division of fractions presents the concept of reciprocal numbers and ties the reciprocity relationship to the idea that division is the appropriate operation to solve questions of the form, “How many times does this number fit into that number?” For example, we can write a division from the question, “How many times does 1/3 fit into 1?” The answer is, obviously, 3 times. So we can write the division 1 ÷ (1/3) = 3 and the multiplication 3 · (1/3) = 1. These two numbers, 3/1 and 1/3, are reciprocal numbers because their product is 1. Students learn to solve questions like that through using visual models and writing division sentences that match them. Thinking of fitting the divisor into the dividend (measurement division) also gives us a tool to check whether the answer to a division problem is reasonable. Naturally, the lessons also present the shortcut for fraction division—that each division can be changed into a multiplication by taking the reciprocal of the divisor, which is often called the “invert (flip)-and-multiply” rule. However, that “rule” is just a shortcut. It is necessary to memorise it, but memorising a shortcut doesn’t help students make sense conceptually out of the division of fractions—they also need to study the concept of division and use visual models to better understand the process involved. In two lessons that follow, students apply what they have learned to solve problems involving fractions or fractional parts. A lot of the problems in these lessons are revision in the sense that they involve previously learned concepts and are similar to problems students have solved earlier, but many involve the division of fractions, thus incorporating the new concept presented in this chapter. Consider mixing the lessons from this chapter (or from some other chapter) with the lessons from the geometry chapter (which is a fairly long chapter). For example, the student could study these topics and geometry on alternate days, or study a little from both each day. Such, somewhat spiral, usage of the curriculum can help prevent boredom, and also to help students retain the concepts better. Also, don’t forget to use the resources for challenging problems: https://l.mathmammoth.com/challengingproblems I recommend that you at least use the first resource listed, Math Stars Newsletters. The Lessons in Chapter 7 page span Revision: Add and Subtract Fractions and Mixed Numbers...... 37 4 pages Add and Subtract Fractions: More Practice............................... 41 3 pages Revision: Multiplying Fractions 1............................................. 44 3 pages Revision: Multiplying Fractions 2............................................. 47 3 pages Dividing Fractions: Reciprocal Numbers................................. 50 5 pages Divide Fractions....................................................................... 55 4 pages Problem Solving with Fractions 1............................................ 59 3 pages Problem Solving with Fractions 2............................................ 62 3 pages Chapter 7 Mixed Revision...................................................... 65 2 pages Sample worksheet from Fractions Revision.................................................................... 67 3 pages https://www.mathmammoth.com 35  Revision: Add and Subtract Fractions and Mixed Numbers 5 5 Example 1. Add +2. The common denominator is 24: 6 8 We need to convert unlike fractions into equivalent fractions that have 5 5 + 2 a common denominator before we can add them. The common 6 8 denominator must be a multiple of both 6 and 8 (a common multiple). ↓ ↓ 20 15 35 11 Naturally, 6 · 8 = 48 is one common multiple of 6 and 8. We could + 2 = 2 = 3 24 24 24 24 use 48. However, it is better to use 24, which is the least common multiple (LCM) of 6 and 8, because it leads to easier calculations. 1. Write the addition sentences. 3 1 7 1 a. + + b. + c. + 4 9 10 4 ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ + = + = + = + = 2. Add and subtract. Use the common denominator you found in the previous exercise. Remember, the best possible choice for the common denominator (but not the only one) is the LCM of the denominators. 5 1 1 4 5 3 a. + b. 3 + 1 c. − 16 6 12 9 6 8 5 4 11 3 45 9 d. 2 + e. 5 − 2 f. + 12 5 15 20 100 20 Sample worksheet from https://www.mathmammoth.com 37  Regroup in subtraction, We can use the same idea (regrouping) when if necessary. the fractions are written horizontally. Take one of the 7 wholes, 2 8 Here we regroup one 7 – 3 think of it as 9/9, and regroup 9 9 as 13/13. This leaves that with the fractional parts 9 wholes. There is already ↓ ↓ (with 2/9). Instead of 7 wholes, 1/13 in the column of 11 8 3 we are left with 6, and instead 6 – 3 =3 the fractional parts, so 9 9 9 of 2/9, we get 11/9. in total we get 14/13. 3. Subtract. 3 8 d. 16 – 9 9 9 3 1 1 a. 7 b. 18 c. 10 9 10 15 7 9 8 – 2 – 5 – 3 9 10 15 3 10 e. 7 – 2 14 14 4. Subtract. First write equivalent fractions with the same denominator. 3 3 9 a. 3 → 3 b. 3 → c. 8 → 4 8 11 1 5 1 – 1 → – 1 – 1 → – – 5 → – 6 12 2 5. Figure out and explain how these subtractions were done! 2 8 3 9 Emma’s way: 9 – 3 Joe’s method: 5 – 2 17 17 14 14 ↓ 2 8 6 11 3 3 6 = (9 – 3) + ( – ) = 6 – = 5 5 – 2 – 17 17 17 17 14 14 14 6 8 = 3 – = 2 14 14 Sample worksheet from https://www.mathmammoth.com 38  When adding or subtracting three ore more fractions, find a common denominator for all of them. You can always use the product of the denominators as your common denominator. However, it may be more efficient to use the LCM of the denominators if it is smaller. Example 2. Here, we could use 6 · 7 · 2 = 84 as a common denominator. 5 5 1 + − However, in this case, the LCM of 6, 7 and 2 is 42, so it is better (leads 6 7 2 to easier calculations) than using 84. ↓ ↓ ↓ 35 30 21 44 1 Another option would be to add the first two fractions (5/6 and 5/7) to + − = =1 get 65/42, and then to subtract the third fraction, 1/2, from that result. 42 42 42 42 21 6. Add or subtract the fractions. 5 1 1 2 1 1 a. + + b. + − 12 6 3 7 2 4 1 2 1 19 1 1 c. + + d. − − 10 5 3 20 3 4 7 1 2 7 3 3 e. − + f. − + 8 5 3 6 5 4 7. Joe started working at an automobile company 23 ½ years ago. However, during that time, he has taken ¼ of a year off for paternity leave, and spent another 1 ⅓ years laid off due to a recession. So, how long has he actually been working for the company? Sample worksheet from https://www.mathmammoth.com 39  While you can often compare two fractions using mental math strategies, sometimes the fractions are so close to each other that you need to rewrite both using a common denominator, then compare. 7 11 7 11 Example 3. Which is more, or ? 8 13 8 13 ↓ ↓ Let’s write both using the common denominator 104 (on the right): 91 88 We see that 7/8 is more. > 104 104 8. Compare the fractions, writing < or > between them. Use a common denominator only if you need to. 1 5 15 15 6 1 1 1 a. b. c. d. 2 9 65 34 15 2 120 75 3 8 2 8 11 3 10 2 e. f. g. h. 5 13 3 11 15 4 2 000 1 000 ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ 9. Julie is convinced that 5/6 is more than 7/8 — she even sketched a picture where it looks like it is so. How would you convince (prove to) her otherwise? 10. Order the fractions from the smallest to the biggest. 1 1 3 3 2 7 5 3 a. , , , b. , , , 4 2 8 7 3 5 4 2 ___< ___< ___ < ___ < ___ ___< ___< ___< ___ < ___ 2 8 3 3 5 5 7 5 2 6 c. , , , , d. , , , , 3 5 5 4 4 6 12 8 9 5 ___< ___< ___ < ___ < ___ ___< ___< ___< ___ < ___ Solve the equations. Hint: if the fractions confuse you, first think how the equation would be solved if it had whole numbers. Then solve the original equation the same way. 4 2 1 1 a. 8 + x = 10 b. 5 – x = 2 7 5 9 3 Sample worksheet from https://www.mathmammoth.com 40  Add and Subtract Fractions: More Practice These exercises simply give you more practice on adding and subtracting fractions and mixed numbers. Use them as directed by your teacher. 1. Add or subtract. Give your answer in lowest terms, and as a mixed number, if applicable. 17 2 11 7 13 3 a. + b. + c. + 18 9 30 12 22 4 7 3 7 1 9 31 d. 6 –1 e. 4 –1 f. 15 – 3 10 20 8 3 10 100 2. Subtract. First write equivalent fractions with the same denominator. 1 1 1 a. 5 → 5 b. 12 c. 33 2 9 3 7 2 6 – 1 → – 1 – 5 – 17 12 3 7 1 6 1 d. 8 e. 86 f. 53 9 7 6 7 1 6 – 2 – 45 – 40 12 8 7 Sample worksheet from https://www.mathmammoth.com 41 Sample worksheet from https://www.mathmammoth.com 70