Welcome to Math 6

1. Welcome to Math 6 Today’s lesson will be a Problem-SolvingWorkshop

2. What you can do: Make sense of the problems and persevere in solving them.

3. Persevere- to persist in spite of difficulty, opposition, or discouragement In other words: Keep going until you succeed.

4. First, seek the meaning of the problem Then look for efficient ways to solve it. Check your thinking by asking yourself: “Does this make sense?” and “Can I solve the problem in a different way?”

5. Finally, you should be able to: Check answers to problems using a different method.

6. Connector So far we have worked with prime numbers, prime factorization, and greatest common factor of two or more numbers. Today we will solve real-world problems. You will need to apply those same skills as you solve these problems.

7. Objective for this lesson: 1. Compute fluently with multi-digit numbers and find common factors.2. Find the greatest common factor of two or more numbers.

8. Key Vocabulary for this lesson Factor - A number that is multiplied by another number to get a product. When we say 60 is ‘divisible’ by 10, we can also say that 10 is a ‘factor’ of 60.

9. Prime Number A whole number greater than one that has exactly two factors, itself and 1. A whole number that is divisible by exactly two numbers: itself and 1. Key Vocabulary for this lesson

10. Key Vocabulary for this lesson Composite Number- A number, greater than one, that has more than two whole-number factors.

11. Key Vocabulary for this lesson Multiple – The product of any number and a whole number is a multiple When we say that 25 is ‘divisible’ by 5, that means that 25 is a ‘multiple’ of 5.

12. Key Vocabulary for this lesson Prime Factorization– The prime factorization of a number is the number written as the product of its prime factors. Example- The factors of 32 are 1 x 32; 2 x 16 or 4 x 8; If we continue to factor the factors of 32, we get: 2 x 2 x 2 x 2 x 2 or 25

13. Let’s jump right in!

14. Guided Practice Problem 1 Kim is making flower arrangements. She has 16 red roses and 20 pink roses. Each arrangement must have the same number of red roses and the same number of pink roses. What is the greatest number of arrangements Kim can make if every flower is used?

15. Analyze the problem Be sure you understand what is being asked. Read the problem carefully again.

16. Highlightthe significant numbers and facts. Kim is making flower arrangements. She has 16 red roses and 20 pink roses. Each arrangement must have the same number of red roses and the same number of pink roses. What is the greatest number of arrangements Kim can makeif every flower is used?

17. She has 16 red roses and 20 pink roses. Each arrangement must have the • same number of red roses • and the same number of pink roses. What is the greatest number of arrangements Kim can make • if every flower is used?

18. Okay, so Kim is dividing up two kinds of flowers and making equal groups. Each group will have an equal number of flowers. But not an equal number of pink and red roses.

19. 16 red roses and 20 pink roses. Start by finding the factors of 16 and 20. 16- 1x16, 2x8, 4x4 20- 1x20, 2x10, 4x5 The common factors are: 1,2 and 4 The Greatest Common Factor (GCF) is 4

20. The GCF of 16 and 20 is 4. Is that the correct answer? • If we divided the roses into 4 groups, we would get: • 4 red roses in each group and • 5 pink roses in each group.

21. So 4 groups is the most that we could make. But how do we know if that is correct? • I want you to evaluate your own thinking and the thinking of other students. • Pose questions like “How did you get that?”, “Why is that true?” “Does that always work?” • Explain your thinking to others and respond to others’ thinking.

22. Guided PracticeProblem 2 A school held a contest. 15 boys and 9 girls were divided into equal teams. Each team had the same number of boys and the same number of girls. What was the greatest possible number of teams if everyone was on a team.

23. Read it again. Highlight key points. A school held a contest. 15 boys and 9 girls were divided into equal teams. Each team had the same number of boys and the same number of girls. What was the greatest possible number of teams if everyone was on a team?

24. There are 15 boys and 9 girls.“…greatest number of teams possible …same number of boyson each team… andthe same number of girls on each team…”How many teams were made if each person was on a team?

25. What is being asked? What’s the greatest number of teams made if every person was on a team? Find the GCF of 15 and 9. 15: 3x5, 1x15 9: 3x3, 1x15 The GCF is 3. Does that mean there can be 3 teams?

26. 3 is the greatest common factor. If we had three teams, there could be 3 girls per team and 5 boys per team. Since 15 and 9 have no other common factors (besides 1), that is the only answer that would work. Why? Because the problem states that every person must be on a team.

27. Check your answer. Does it make sense? • It’s best to use some other method of checking than the one you used to find your answer. • In this case, you could try making a table:

28. Since 3 is the GCF of 9 and 15, we tried 3 to see if it would work. It does work. This solution has the exact total number of girls and boys.

29. Using the Greatest Common Factor was a quick way to solve the problem. There are the same number of boys (5) on each team and the same number of girls(3) on each team.

30. Guided Practice Problem 3 • Ms. Kline makes balloon arrangements. • She has 32 blue balloons, 24 yellow balloons, and 16 red balloons. • Each arrangement must have the same number of each color. • What is the greatest number of arrangements that Ms. Kline can make if every balloon is used?

31. Be sure you understand it! Ms. Kline makes balloon arrangements. She has 32 blue balloons, 24 yellow balloons, and 16 red balloons. Each arrangement must have the same number of each color. What is the greatest number of arrangements that Ms. Kline can make if every balloon is used? What are the key points?

32. Highlight the key points! Ms. Kline makes balloon arrangements. She has 32 blue balloons, 24 yellow balloons, and16 red balloons. Each arrangement must have the same number of each color. What is the greatest number of arrangements that Ms. Kline can make if every balloon is used?

33. Key points: 32 blue balloons, 24 yellow balloons, and 16 red balloons. Each arrangement must have the same number of each color. …every balloon is used… What is the greatest number of arrangements that Ms. Kline can make?

34. Key points: • Each arrangement…must have the “same number of each color… and every balloon is used…” • That is really the key phrase. • It tells us to use division

35. Find the Greatest Common Factor of 32, 24 and 16 32: 1, 32, 2, 16, 4, 8 24: 1, 24, 2, 12, 3, 8, 4, 6 16: 1, 16, 2, 8, 4 • Common Factors of all three numbers are: 1,2,4,8. • The GCF is 8

36. Let’s use 8 since it is the GCF • The quantity of each type of balloon is divisible by 8. • Ms. Kline can make 8 balloon arrangements. • Each arrangement would have 4 blue balloons, 3 yellow balloons and 2 red balloons. • There are no left over balloons.

37. Now how can we prove it? Let’s find multiples of each 8. That’s as simple as ‘skip-counting’ by 8. 8, 16, 24, 32… Notice that 16, 24 and 32 come up when we list multiples of 8.

38. Guided Practice Problem 4 Jared has 12 jars of grape jam, 16 jars of strawberry jam, and 24 jars of raspberry jam. He wants to place the jam into the greatest possible number of boxes so that each box has the same number of jars of each kind of jam. How many boxes does he need?

39. Highlight the key points! Jared has 12 jars of grape jam, 16 jars of strawberry jam, and 24 jars of raspberry jam. He wants to place the jam into the greatest possible number of boxes so that each box has the same number of jars of each kind of jam. How many boxes does he need?

40. What’s the best way to solve the problem? Look for clues… Word problems usually have a key phrase which gives a clue how to solve it. In this problem the key phrase is “each box has the same number of jars…” This tells us we can use division! Try using the greatest common factor in this type of problem.

41. Find the Greatest Common Factor of 24,16 and 12 24: 1, 24, 2, 12, 3, 8, 4, 6 16: 1, 16, 2, 8, 4 12: 1, 12, 2, 6, 3, 4 • Common Factors of all three numbers are: 1,2 and 4 • The GCF is 4

42. The GCF of 24,16 and 12 is 4 If Jared uses fourboxes, he could put in each box: 3 jars of grape, 4 jars of strawberry and 6 jars of raspberry jam.

44. Now its your turn! Next are 4 more problems that you can try on your own. Give it your best shot!

45. Independent Practice #1 The12 boys and 18 girls in Mr. Ruiz’ science class must form lab groups. Each group must have the same number of boys and the same number of girls. What is the greatest number of groups Mr. Ruiz can make if every student must be in a group?

46. Independent Practice #2 Mrs. Rodriguez is planting 4 types of flowers in her garden. She has 42 Irises, 36 daffodils, 18 tulips and 12 lilies. Each row is to have the same number of each type of flower. What is the greatest number of rows Mrs. Rodriguez can plant if every bulb is used?

47. Independent Practice #3 In a parade, one school band will march directly behind another school band. All rows must have the same number of students. The first band has 36 students, and the second band has 60 students. What is the greatest number of students who can be in each row?

48. Independent Practice #4 Pam is making fruit baskets. She has 30 apples, 24 pears, and 12 oranges. What is the greatest number of baskets she can make if each type of fruit is distributed equally among the baskets?

49. Conclusion: • When solving word problems, be sure first you UNDERSTAND what is being asked. • ALWAYS look for key words and phrases. • CHECK your answer, if possible by using some other method than the one you used to find it.

50. Assignments- • Complete “Math Level 6- Assignment: Problem-Solving Workshop- GCF” • When solving word problems, you must be able to COMMUNICATE the meaning of your answer. • Always identify numbers with words. Label what the number you come up with represents.