1. Review for the Final 302A, Fall, 2007

2. What is on the test? From book: 1.2, 1.3, 1.4, 1.7; 2.3; 3.1, 3.2, 3.3, 3.4; 4.2, 4.3; 5.2, 5.3, 5.4; 6.1, 6.2 From Explorations: 1.1, 1.4, 1.7; 2.8, 2.9; 3.1; 3.13, 3.15, 3.19, 3.20, 4.2, 4.3, 5.8, 5.9, 5.10, 5.12, 5.13, 5.14, 5.15, 5.16, 6.3, 6.4, 6.5, 6.7 From Class Notes: Describe the strategies used by the students--don’t need to know the names.

3. Chapter 1 A factory makes 3-legged stools and 4- legged tables. This month, the factory used 100 legs and built 3 more stools than tables. How many stools did the factory make? 16 stools, 13 tables

4. Chapter 1 Fred Flintstone always says “YABBADABBADO.” If he writes this phrase over and over, what will the 246th letter be? D

5. Chapter 2 Explain why 32 in base 5 is not the same as 32 in base 6. 32 in base 5 means 3 fives and 2 ones, which is 17 in base 10. 32 in base 6 means 3 sixes and 2 ones, which is 20 in base 10. So, 32 in base 5 is smaller than 32 in base 6.

6. Chapter 2 Why is it wrong to say 37 in base 5? In base 5, there are only the digits 0, 1, 2, 3, and 4. 7 in base 5 is written 12.

7. Chapter 2 What error is the student making? “Three hundred fifty seven is written 300507.” The student does not understand that the value of the digit is found in the place: 300507 is actually 3 hundred-thousands plus 5 hundreds and 7 ones. Three hundred fifty seven is written 357--3 hundreds plus 5 tens plus 7 ones.

8. Chapter 3 List some common mistakes that children make in addition. Do not line up place values. Do not regroup properly. Do not account for 0s as place holders.

9. Chapter 3 Is this student correct? Explain. “347 + 59: add one to each number and get 348 + 60 = 408.” No: 347 + 59 is the same as 346 + 59 because 346 + 1 + 60 - 1 = 346 + 60 + 1 - 1, and 1 - 1 = 0. The answer is 406.

10. Chapter 3 Is this student correct? “497 - 39 = 497 - 40 - 1 = 457 - 1 = 456.” No, the student is not correct because 497 - 39 = (497) - (40 - 1) = (497) - 40 + 1 = 458. An easier way to think about this is 499 - 39 = 460, and then subtract the 2 from 499, to get 458.

11. Chapter 3 Is this student correct? “390 - 27 is the same as 300 - 0 + 90 - 20 + 0 - 7. So, 300 + 70 + -7 = 370 + -7 = 363.” Yes, this student is correct. This is analogous to 390 = 380 + 10 = 27; 300 - 0 + 80 - 20 + 10 - 7 = 300 + 60 + 3. Note: to avoid this negative situation, we regroup.

12. Chapter 3 Multiply 39 12 using at least 5 different strategies. Lattice Multiplication Rectangular Area Egyptian Duplation 39 10 + 39 2 40 12 - 1 12 30 10 + 9 10 + 30 2 + 9 2 = (30 + 9)(10 + 2)

13. Chapter 3 Divide 259 ÷ 15 using at least 5 different strategies. Scaffold Repeated subtraction Repeated addition Use a benchmark Partition (Thomas’ strategy)

14. Chapter 3 Models for addition: Put together, increase by, missing addend Models for subtraction: Take away, compare, missing addend Models for multiplication: Area, Cartesian Product, Repeated addition, measurement, missing factor Models for division: Partition, Repeated subtraction, missing factor

15. Chapter 4 An odd number: An even number:

16. Chapter 4 Prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, … 2 factors ONE IS NOT PRIME. Composite numbers: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, … at least 3 factors Square numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, … an odd number of factors

17. Chapter 4 Prime factorization: many ways to get the factorization, but only one prime factorization for any number. Find the prime factorization of 84. 2 2 3 7, or 2 2 3 7

18. Chapter 4 Greatest Common Factor: The greatest number that can divide evenly into a set of numbers. The GCF of 50 and 75 is 25. You try: Find the GCF of 60, 80, and 200. 20: 60 = 20 3, 80 = 20 4, 200 = 20 10.

19. Chapter 4 The Least Common Multiple is the smallest number that is divisible by a set of numbers. The LCM of 50 and 75 is 150. You try: Find the LCM of 60, 80, and 200. 1200: 60 20 = 1200, 80 15 = 1200, 200 6 = 1200.

20. Chapter 4 What is the largest square that can be used to fill a 6 x 10 rectangle. 2 x 2: You can draw it to see why. (Which is involved here, GCF or LCM?)

21. Chapter 5 Fractions models: Part of a whole Ratio Operator Quotient Make up a situation for 6/10 for each of the models above.

22. Chapter 5 Name the model for each situation of 5/6. I have 5 sodas for 6 people--how much does each person get? Out of 6 grades, 5 were As. I had 36 gumballs, but I lost 6 of them. What fraction describes what is left? In a room of students, 50 wore glasses and 10 did not wear glasses.

23. Chapter 5 There are three ways to represent a fraction using a part of a whole model: part-whole discrete, number line (measurement) Represent 5/8 and 11/8 using each of the above pictorial models.

24. Chapter 5 Errors in comparing fractions: 2/6 > 1/2 Look at the numerators: 2 > 1 Look at the denominators: 6 > 2

25. Chapter 5 Appropriate ways to compare fractions: –Rewrite decimal equivalents. –Rewrite fractions with common denominators. –Place fractions on the number line. –Sketch parts of a whole, with the same size whole

26. Chapter 5 More ways to compare fractions: –Compare to a benchmark, like 1/2 or 3/4. –Same numerators: a/b > a/(b + 1) 2/3 > 2/4 –Same denominators: (a + 1)/b > a/b 5/7 > 4/7 –Look at the part that is not shaded: 5/9 < 8/12 because 4 out of 9 parts are not shaded compared with 4 out of 12 parts not shaded.

27. Chapter 5 Compare these fractions without using decimals or common denominators. 37/81 and 51/90 691/4 and 791/7 200/213 and 199/214 7/19 and 14/39

28. Chapter 5 Remember how to compute with fractions. Explain the error: 2/5 + 5/8 = 7/13 3 4/7 + 9/14 = 3 13/14 2 7/8 + 5 4/8 = 7 11/8 = 8 1/8 5 4/6 + 5/6 = 5 9/6 = 5 1/2

29. Chapter 5 Explain the error: 3 - 4/5 = 2 4/5 5 - 2 1/7 = 3 6/7 3 7/8 - 2 1/4 = 1 6/4 = 2 1/2 9 1/8 - 7 3/4 = 9 2/8 - 7 6/8 = 8 12/8 - 7 6/8 = 1 4/8 = 1 1/2

30. Chapter 5 Explain the error: 3/7 4/9 = 7/16 2 1/4 3 1/2 = 6 1/8 7/12 4/5 = 35/48 4/7 3/5 = 20/35 21/35 = 420/1225 = 84/245 = 12/35

31. Chapter 5 Explain the error: 3/5 ÷ 4/5 = 4/3 12 1/4 ÷ 6 1/2 = 2 1/2

32. Chapter 5 Decimals: Name a fraction and a decimal that is closer to 4/9 than 5/11. Explain what is wrong: 3.45 ÷.05 = 0.69

33. Chapter 5 True or false? Explain. 3.69/47 = 369/470 5.02/30.04 = 502/3004

34. Chapter 5 Order these decimals: 3.95, 4.977, 3.957, 4.697, 3.097 Round 4.976 to the nearest tenth. Explain in words, or use a picture.

35. Chapter 6 An employee making $24,000 was given a bonus of $1000. What percent of his salary was his bonus? 1000/24,000 = x/100 100,000 = 24,000x x ≈ 4.17%

36. Chapter 6 Which is faster? 11 miles in 16 minutes or 24 miles in 39 minutes? Explain.

37. Chapter 6 Ryan bought 45 cups for $3.15. “0.07! That’s a great rate!” What rate does 0.07 represent? Describe this situation with a different rate--and state what this different rate represents.

38. Chapter 6 Which ratio is not equivalent to the others? (a)42 : 49 (b)12 : 21 (c)50.4 : 58.8 (d)294 : 357

39. Chapter 6 Write each rational number as a decimal and a percent. 3 4/5 1/11 2 1/3

40. Chapter 6 Write each decimal as a fraction in simplest form and a percent. 4.9 3.005 0.073

41. Chapter 6 Write each percent as a fraction and a decimal. 48% 39.8% 2 1/2% 0.841%

42. Chapter 6 A car travels 60 mph, and a plane travels 15 miles per minute. How far does the car travel while the plane travels 600 miles? (Hint: you can set up one proportion, two proportions, or skip the proportions entirely!) Answer is the car travels 40 miles--the car travels 1 miles for each 15 miles the plane travels. 1/15 = x/600.

43. Chapter 6 DO NOT set up a proportion and solve: use estimation instead. (a) Find 9% of 360. (b) Find 5% of 297. (c) Find 400% of 35. (d) Find 45% of 784.

44. Chapter 6 DO NOT set up a proportion and solve: use estimation instead. (e) What percent of 80 is 39? (f) What percent of 120 is 31? (g) 27 is what percent of 36? (h) 87 is 20% of what number? Now, go back and set up proportions to find the exact values of (a) - (h). Were you close?

45. Chapter 6 David has 150 mg of fools’ gold. Find the new amount if: He loses 30%? He increases her amount by 90%? He decreases her amount by 40%?

46. Study Hard, and show up on time!