1. March 26, 2009 Who dares to teach must never cease to learn. ~John Cotton Dana

2. March 26, 2009 Review Exploration 4.2 Review prime factorization Review GCF and LCM Assign Homework

3. 4.1 & 4.2 - Review Questions: If 5 is a factor of 35, what are all of the factors of 35? If 7 is a factor of 105, what are all of the factors of 105? What is the smallest factor that 35 and 105 share? What is the largest?

4. Exploration 4.2 Read the introduction to the exploration on page 83. Fill in the table on page 85, using the information on the sieve. It will help if you think of the factors in pairs: For example, for 18: 1 & 18, 2 & 9, 3 & 6. The order does not matter.

5. Exploration 4.2 In pairs, complete 3a, 3c, 3d, 3e and 3f. (This will be turned in on Tuesday). You may turn this in as a pair, but make sure both of your names are on the paper. Be sure to include the tables from pages 85 and 87.

6. Exploration 4.3 1.List all the factors of 36. Make sure you don’t miss any! 2.List all the factors of 72. Can you use the work you did for 36 to help you? 3.True or false? 72 is a factor of 36. Explain your answer.

7. Exploration 4.3 Let’s see how the prime factorization of 36 could help us: 2 183 12 4 9 2 9 2 6 2 2 3 3 3 3 2 3 36

8. Exploration 4.3 So, 36 = 2 2 · 3 2 = 2 · 2 · 3 · 3 Use this to make an organized list of factors: 1 · 366 · ___ (6 is 2 · 3) 2 · ___ 9 · ___ (9 is 3 · 3) 3 · ___ 12 · ___ (12 is 2 2 · 3) 4 · ___ (note that 4 is 2 · 2)18 · ___ (18 is 2 · 3 2 ) Notice the repetition

9. Exploration 4.3 Prime factorization for 72 = 2 3 3 2 How can you make sure you get every factor? 1, 2, 4, 8, 3, 9, 6, 12, 24, 18, 36, 72 (Is 72 a factor of 36?)

10. Exploration 4.3 You try: Find all of the factors: 60 = 2 2 3 5 Find all of the factors: 48 = 2 4 3

11. Exploration 4.3 60 = 2 2 3 5 1, 2, 4, 3, 5, 6, 10, 15, 12, 20, 60 48 = 2 4 3 1, 2, 4, 8, 16, 3, 6, 12, 24, 48

12. 4.3 – Greatest common factor and least common multiple Since we ’ ve been discussing factors and multiples, this is a good time to review GCF and LCM. These concepts also come up frequently when working with fractions, which we will study in chapter 5.

13. 4.3 (cont’d) Let ’ s look at two numbers: 120 & 84 (find prime factorizations)

14. 4.3 (cont’d) 120 = 2 3 3 5 and 84 = 2 2 3 7 120: factors are 1, 2, 4, 8, 3, 5, 6, 12, 24, 10, 20, 40, 15, 30, 60, 120 84: factors are 1, 2, 4, 3, 7, 6, 12, 14, 28, 42, 84 Find their common factors.

15. 4.3 (cont’d) We see that they both have 1, 2, 4, 3, 6, and 12. The Greatest Common Factor is 12. From the prime factorization: 2 2 3.

16. 4.3 (cont’d) Next, look at multiples: Multiples of 120 will be 120 · (some number) or 2 · 2 · 2 · 3 · 5 · (some number) Multiples of 84 will be 84 · (some number) or 2 · 2 · 3 · 7 · (some number) Both numbers include 2 · 2 · 3; only 120 also has another 2 and a 5; only 84 also has a 7.

17. 4.3 (cont’d) If we multiply (2 · 2 · 3) · (2 · 5) · 7 = 840, this is a multiple of 120 (since it is 2 · 2 · 2 · 3 · 5 · 7) and a multiple of 84 (since it is 2 · 2 · 3 · 7 · 5 · 2) In fact, 840 is the Least Common Multiple of 120 and 84. Look at some multiples of each number: 120: 120, 240, 360, 480, 600, 720, 840, 960, … 84: 84, 168, 252, 336, 420, 504, 588, 672, 756, 840, 924, 1008

18. 4.3 (cont’d) You try: Find the LCM of 38 and 95. Find the LCM of 126 and 140.

19. 4.3 (cont’d) 38 = 19 · 2 95 = 19 · 5 LCM(38, 95) = 19 · 5 · 2 = 190 126 = 7 · 3 · 3 · 2 140 = 7 · 5 · 2 · 2 LCM(126, 140) = 7 · 5 · 3 · 3 · 2 · 2 = 1260

20. 4.3 (cont’d) Word problem example: Tara can run around the track in 5 minutes. Todd can run the same distance in 6 minutes, and Tony can do it in 8 minutes. If they start at the same time, when will they next meet at the starting line?

21. 4.3 (cont’d) (Find the LCM of 5, 6, and 8) Answer: 120 minutes or 2 hours

22. Homework Due Tuesday, 3/31 Link to online homework list: http://math.arizona.edu/~varecka/302AhomeworkS09. htm