1. 302A final exam review

2. Final Exam Tuesday, May 12 11am – 1pm In our usual classroom Review session: Thurs, 5/7 @ 12:30pm here.

3. Some Answers to 6.1 homework 28) We need a constant ratio of men to women. 9 men=x men 4 women360 women Solve: 9 360 = 4x; x = 810 men now. If we want a ratio of 2 : 1, then we want 405 women. We have 360 women, so we need 45 more women to make the ratio 2 : 1.

4. 31a)Let’s draw a picture. Crew of 4 men In 5 days: 1 building In 10 days: 2 buildings In 15 days: 3 buildings In 20 days: 4 buildings So, a crew of 4 can build 4 buildings in 20 days. Two crews of 4 can build 8 buildings in 20 days. 10 crews of 4 can build 40 buildings in 20 days, so 40 people altogether.

5. Some Answers to 6.2 homework 1a: 4% of 450. Think 1% of 450 is 4.5, so 4% of 450 is 4 4.5, or 18. Or, 4% of 450 is close to 5% of 450. We know that 10%of 450 is 45, and so 5% is half of 10%, so half of 45 is 22.5.

6. 1h: 30 is what percent of 35? Think: part : whole = percent : 100. So, 30/35 = x/100? Well, 30/35 is close to 30/36, or 5/6, which is about 83%. Or, 30/35 is close to 35/40, or 7/8, which is 87.5%.

7. 1p: What is 100% more than 35? 100% of 35 is 35, so 100% more is 35 + 35 = 70.

8. #8. Two dresses $119 with 40% off Pay 60% 119 0.60 = $71.40 $79.99 with 20% off Pay 80% 79.99 0.80 = $63.99 Second dress is cheaper.

9. 30. To find 0.5%, we know that this means 0.5/100. If we multiply numerator and denominator by 10, we get (0.5 10)/(100 10) = 5/1000, or 1/200.

10. Exploration 6.7 Do Part 1 #1 with your group.

11. What is on the final exam? 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; 2.8, 2.9; 3.1, 3.3, 3.6, 3.13, 3.15, 3.18, 3.19; 4.2; 5.8, 5.9, 5.10, 5.12, 5.13, 5.14, 5.15; 6.3, 6.5, From Class Notes: Describe the strategies used by the students--don’t need to know the names.

12. 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

13. A test 3 problem If = 3/5, carefully draw 1/6. Two approaches:

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

15. 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.

16. 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.

17. 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.

18. Chapter 2 True or false? 57 8 + 31 8 = 100 8 2401 5 – 432 5 > 2000 5 13 9 < 22 8 < 41 7 False: 57 8 + 31 8 = 110 8 ; False: 2401 5 – 432 5 = 1414 5 ; True: 12 < 18 < 29

19. 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.

20. 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 + 60 because 346 + 1 + 60  1 = 346 + 60 + 1  1, and 1  1 = 0. The answer is 406.

21. 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.

22. Chapter 3 Is this student correct? “390  27 is the same as 300 + 90  20  7. So, 300 + 70  7 = 370  7 = 363.” Yes, this student is correct.

23. Chapter 3 Multiply 39 × 12 using at least 5 different non- traditional strategies. Lattice Multiplication Rectangular Array/Area Model Egyptian Duplation Repeated Addition, Use a Benchmark 39 × 10 + 39 × 2 40 × 12  1 × 12 30 × 10 + 9 × 10 + 30 × 2 + 9 × 2 = (30 + 9)(10 + 2)

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

25. 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/related facts Models for division: Partition, Repeated subtraction, missing factor/related facts

26. Chapter 3 Vocabulary: Addition: addend + addend = sum Subtraction: minuend  subtrahend = difference Multiplication: multiplier × factor = product Division: dividend ÷ divisor = quotient + remainder/divisor dividend = quotient × divisor + remainder

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

28. Chapter 4 Prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, … 2 factors The number 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

29. 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

30. Chapter 4 Greatest Common Factor: The greatest number that a factor of every number in 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.

31. 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.

32. 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?)

33. Chapter 5 Fractions models: Part of a whole Ratio Operator Quotient Make up a real-world problem for each model above for 6/10.

34. 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, and I lost 5/6 of them. How many are left? In a room of students, 50 wore glasses and 10 did not wear glasses. Answers: quotient (5/6 soda per person); part- whole (5/6 As); operator (5/6 of 36 gumballs, or 30 are gone--6 remain); ratio (5:1)

35. 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 pictorial models above.

36. Chapter 5 Ifrepresents 12/25, show what 2 will look like.

37. Chapter 5 If these rectangles represent 12 of something, then each rectangle represents 3 of something. One third of a rectangle represents the unit fraction. So, we need to show 50/25, which is 16 full rectangles, and 2/3 of another rectangle.

38. Chapter 5 Errors in comparing fractions: 2/6 > 1/2 Look at the numerators: 2 > 1 –Two pieces is more than one piece. Look at the denominators: 6 > 2 –We need 6 to make a whole rather than 2. There are more pieces not shaded than shaded. –If we look at what is not shaded, then there are more unshaded pieces. The pieces are smaller in sixths than in halves.

39. 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

40. 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. –Multiply by a form of 1.

41. 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 ; >; >

42. 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

43. 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

44. 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

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

46. Chapter 5 Decimals: Name a fraction and a decimal that is closer to 4/9 than 5/11. 4/9 = 0.44…; 5/11 = 0.4545… ex: 0.44445 is closer to 4/9 than 5/11 Explain what is wrong: 3.45 ÷.05 = 0.0145928… This is 0.5 ÷ 3.45. If we divide by a number less than one, than our quotient is bigger than the dividend.

47. Chapter 5 True or false? Explain. 3.69/47 = 369/470 5.02/30.04 = 502/3004 Multiply by 1, or n/n. 3.69/47 = 36.9/470, not 369/470; 5.02/30.04 = 502/3004.

48. Chapter 5 Order these decimals: 3.95, 4.977, 3.957, 4.697, 3.097 3.097, 3.95, 3.957, 4.697, 4.977 Round 4.976 to the nearest tenth. Explain in words, or use a picture. Is 4.976 closer to 4.9 or 5.0? Put on a number line, and see it is closer to 5.0.

49. Chapter 6 An employee making $24,000 was given a bonus of $1000. What percent of his new take home pay was his bonus? 1000/25,000 = x/100 100,000 = 25,000x x = 4%

50. Chapter 6 Which is faster? 11 miles in 16 minutes or 24 miles in 39 minutes? Explain. Use the rate miles/minutes. Then 11miles/16 minutes compared to 24/39. 0.6875 miles per minute > 0.6153… miles per minute. So the first rate is faster, or more miles per minute.

51. 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. $3.15/45 cups = $0.07 per cup. Another rate would be 45/$3.15 = 14.28 cups per dollar.

52. Chapter 6 Which ratio is not equivalent to the others? (a) 42 : 49(b)12 : 21 (c) 50.4 : 58.8(d)294 : 343 (b)

53. Chapter 6 Write each rational number as a decimal and a percent. 34/51/112 1/3 3: 3.0 (or 3), 300% 4/5: 0.8, 80% 1/11: 0.09, 9.09% 2 1/3: 2.3, 233.3%

54. Chapter 6 Write each decimal as a fraction in simplest form and a percent. 4.93.0050.073 4.9: 4 9/10; 490% 3.005: 3 5/1000 = 3 1/200; 300.5% 0.073: 73/1000; 7.3%

55. Chapter 6 Write each percent as a fraction and a decimal. 48%39.8%2 1/2%0.841% 48%: 48/100 = 12/25; 0.48 39.8%: 39.8/100 = 398/1000 = 199/500; 0.398 2 1/2%:2.5/100 = 25/1000 = 1/40; 0.025 0.841%: 841/100000; 0.00841

56. 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.

57. 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.

58. 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?

59. Chapter 6 Iga Tahavit has 150 mg of fools’ gold. Find the new amount if: She loses 30%? She increases her original amount by 90%? She decreases her originalamount by 40%? 105 mg; 285 mg; 90 mg

60. Percent & Proportion Questions In Giant World, a giant tube of toothpaste holds one gallon. If a normal tube of toothpaste holds 4.6 ounces and costs $2.49, how much should the giant tube cost? One gallon is 128 ounces. Ounces = 4.6 = 128 Dollars $2.49 x 4.6x = 128 2.49 About $69.29

61. Estimate In Giant World, a giant tube of toothpaste holds one gallon. If a normal tube of toothpaste holds 4.6 ounces and costs $2.49, how much should the giant tube cost? If we round, we can think: 4 ounces is about $2.50. Since we want to know how much 128 ounces is, think: 4 32 = 128, so $2.50 times 32 is $80. (or, $2.50 30 = $75)

62. Try this one The admissions department currently accepts students at a 7 : 3 male/female ratio. If they have about 1000 students in the class, how many more females would they need to reduce the ratio to 2 : 1? Currently: 7x + 3x = 1000, so x = 100; 700 males and 300 females. To keep 1000 students in class, they want 2y + 1y = 1000, so y = 333; 666 males and 333 females. They need to accept 333 - 300 = 33 more females to achieve this ratio. OR, with 700 males, they need 350 females, or 50 more.

63. Try this one Lee’s gross pay is $1840 per paycheck, but $370 is deducted. Her take-home pay is what percent of her gross pay? Part = percent = 370 = x Whole 100 1840 100 370 × 100 = 1840x; About 20% is taken out, so about 80% for take-home pay. Could also do: 1840 - 370 = 1470: 1470 = x 1840 100

64. Last one Estimate in your head: 16% of 450 10% of 450 = 45; 5% = 22.5, about 67.5 OR 10% of 450 = 45; 1% of 450 = 4.5, or about 5; 6 × 1% = 6 × 5 = 30; 30 + 45 = 75. 123 is approximately what percent of 185? Approximate: 120 is approximately what percent of 200; 120/200 = 60/100, so about 60%.

65. Good Luck! Remember to bring pencil, eraser and calculator to the exam. Study hard! Show up on time! 11:00 am – 1:00 pm Tuesday, May 12 (here)