Test bank for Reasoning About Numbers and Quantities and Reasoning About Algebra and Change Below we have listed assessment items for most sections of Part I: Reasoning About Numbers and Quantities and Part II: Reasoning About Algebra and Change. The items were selected from those that instructors used while using the materials over several years at San Diego State University. For some sections, there are very few items. The number of items is related both to the number of times that the particular section of the module was piloted and to the emphasis given to the material. Space here, of course, is reduced from that provided on actual tests or quizzes. The test-bank is a Word document rather than a PDF document so that you can select items for tests without having to re-enter them. Some test items are similar to previous test items. They provide the opportunity to use slightly different items on different versions of tests. Also, some test items are more difficult than others, and they are marked with an asterisk. However, you may not agree, and thus it is important that you check each item to be sure that it is of the level of difficulty that you wish to have in your examination. Request: The items are different for Parts I and II. Part I has text items for sections whereas Part II items are listed for each chapter. Also, Part I answers are embedded in the set of test items, whereas answers for Part II are at the end of the set of test items. We would appreciate information from users concerning which format is more useful. Please note: We often use the following directions for true/false items on exams, and it should be included in the directions of any exam given that contains such items– For each of the statements below indicate whether the statement is True or False by CIRCLING the proper word. IF THE STATEMENT IS FALSE, THEN BRIEFLY EXPLAIN WHY IT IS FALSE OR RESTATE IT SO THAT IT IS TRUE. Chapter 1 Reasoning about Quantities 1.1 What Is a Quantity? and 1.2 Quantitative Analysis 1. What is a quantity? Give an example. What is a possible value for your example? E.g., The length of this room is a quantity. A possible value is 20 feet. 2. Name 5 quantities that you have dealt with so far today. Reasoning About Numbers and Quantities Test-Bank Items with Answers page 1 E.g., distance from home to class, time spent traveling from home to class, amount of gasoline purchased, amount of milk drunk at breakfast, amount of money spent on a Starbucks coffee, etc. 3. Name 3 quantities that relate to you, and tell how they are measured. E.g., weight (in pounds or kilograms), height (in inches), arm span (inches), shoe size (standard sizes for shoes), waist size (in inches), etc. 4. Would student motivation be difficult or easy to quantify? Explain. Tell how you might go about quantifying student motivation in this class. Probably difficult. Factors influencing motivation might include need for a passing grade, desire to understand content, parental pressure, peer pressure, etc. A scale (such as 1 low to 10 high) could be designed to measure these factors. 5. List at least five relevant quantities that are involved with this problem situation. For each quantity, if the value is given write it next to the quantity. If the value is not given, write the unit you would use to measure it. Pat and Li left the starting line at the same time running in opposite directions on a 400 meter oval-shaped race track. Pat was running at a constant rate of 175 meters per minute. They met each other for the first time after they had been running for 1.5 minutes. How far had Pat run when Li completely finished one lap? Sample answers (quantity, value or unit if value unknown; other units possible—e.g., seconds instead of minutes): Length of track, 400 meters Pat’s speed, 175 meters per minute Time until they meet for first time, 1.5 minutes Distance Pat has traveled when they meet for first time, meters Distance Li has traveled when they meet for the first time, meters Li’s speed, meters per minute Time for Li to run one lap, minutes Time for Pat to run one lap, minutes Distance Pat has run when Li finished one lap, meters (The above are relevant to one solution, but the following are quantities in the situation as well.) Difference in times for one lap for Pat and Li, seconds or minutes Difference in speeds, Pat and Li, meters per minute … 6. Carry out a quantitative analysis of the following problem situation by answering each of the questions that follow. Jennie got on the freeway at 2:00 PM, using the entrance closest to her home, and traveled at 55 mph to the College Avenue exit, where she turned off at 2:12 PM. Her Reasoning About Numbers and Quantities Test-Bank Items with Answers page 2 roommate Cassie had finished her morning classes and was headed home about the same time. Cassie entered the freeway from the College Avenue entrance at 2:08 PM, and traveled to the home exit at 60 mph. What time did Cassie arrive at the exit ramp to go home? a. What quantities here are critical? b. What quantities here are related? c. What quantities do I know the value of? d. What quantities do I need to know the value of? a. Jennie’s starting time, Jennie’s exit time, time Jennie traveled, speed Jennie traveled, distance Jennie traveled, Cassie’s starting time, distance Cassie traveled, speed Cassie traveled, time Cassie traveled. b. All in part a are related, but in different ways. c. Jennie’s starting time, Jennie’s exit time, speed Jennie traveled, Cassie’s starting time, speed Cassie traveled d. Time Jennie traveled, distance Jennie traveled (= distance Cassie traveled), time Cassie traveled, to get Cassie’s exit ramp time. 7. Consider this problem situation: The school cafeteria is ready to serve two kinds of sandwiches, tuna and ham, and two kinds of pizza, pepperoni and vegetarian. There are 48 servings of pizza prepared. There are 8 more tuna sandwiches prepared than there are servings of pepperoni pizza. There are 4 fewer ham sandwiches prepared than there are servings of vegetarian pizza. Altogether, how many sandwiches are prepared? a. List 8 quantities involved in this problem. b. Sketch a diagram to show the relevant sums and differences in this situation. c. Solve the problem. It will be difficult for your students to avoid algebra or trial-and-error on this problem; decide whether you wish to prohibit the use of algebra. You might also consider omitting part c. a. E.g., number of kinds of sandwiches, number of kinds of pizza, number of servings of pizza prepared, difference in number of tuna sandwiches prepared vs number of servings of pepperoni pizza, difference in number of ham sandwiches prepared vs number of servings of vegetarian pizza, total number of sandwiches prepared, number of tuna sandwiches prepared, number of ham sandwiches prepared, number of servings of pepperoni pizza, number of servings of vegetarian pizza, difference in number of tuna sandwiches and number of ham sandwiches… Reasoning About Numbers and Quantities Test-Bank Items with Answers page 3 b. There are other possible praiseworthy drawings possible, but the following suggests the solution (for the total number of sandwiches) pretty easily. 48 #VP #PP 4 8 #HS #TS c. There are 52 sandwiches prepared in all. (8 + 48 – 4) 8. Consider the following problem situation: Two trains leave from different stations and travel toward each other on parallel tracks. They leave at the same time. The stations are 217 miles apart. One train travels at 65 mph and the other travels at 72 mph. How long after they leave their stations do they meet each other? List six quantities in the problem (note that you are not asked to solve this problem). If a value is given, write it next to the quantity. If no value is given, write an appropriate unit of measure. Samples (quantity, value or unit if value unknown)… Distance between stations, 217 miles Speed of one train, 65 miles per hour Speed of other train, 72 miles per hour Total speed of the two trains, miles per hour Time until trains meet, hours (or minutes) Distance first train has traveled when they meet, miles Distance second train has traveled when they meet, miles 9. Carry out a quantitative analysis of the following problem situation by answering each of the questions that follow, and then solve the problem: 1 A butcher had two pieces of bologna, A and B, with A weighing 3 and 3 times as much as B. After the butcher cut 1.8 pounds off A, A was still 2 13 times as heavy as B. How many pounds does piece B weigh? a. What quantities here are critical? b. What quantities here are related? c. What quantities do I know the value of? d. What quantities do I need to know the value of? e. What is the weight of B, in pounds? Reasoning About Numbers and Quantities Test-Bank Items with Answers page 4 a. Weights of pieces A and B before cut; comparison (ratio) of pieces before cut, comparison (ratio) of pieces after cut, weight of piece cut from A. b. Same as a, along with weight of B. c. Weight of A, weight cut from A, ratio of A to B before cut, ratio of A to B after cut. d. Weight of B. e. Before cut: Piece A 1.8 pounds Piece B After cut: Piece A Piece B Piece B must weigh 1.8 pounds. 10. Consider the following problem situation: Two boats simultaneously left a pier and traveled in opposite directions. One traveled at a speed of 18 nautical miles per hour and the other at 22 nautical miles per hour. How far apart were they after 2.5 hours? List five relevant quantities that are involved in this problem. For each quantity, if a value is given, write it next to the quantity. If the value is not given, write the unit you would use to measure it, and its value if possible. Speed of first boat; 18 nautical miles per hour Speed of second boat, 22 nautical miles per hour Distance traveled by first boat in 2.5 hours, nautical miles; 2.5 18 = 45 n.m. Distance traveled by second boat in 2.5 hours, nautical miles 2.5 22 = 55 n. m. Total distance between boats at 2.5 hours nautical miles: 45 n.m. + 55 n.m. = 100 n. m. After 2.5 hours they are 100 nautical miles apart. 11. My brother and I go to the same school. My brother takes 50 minutes to walk to school, and I take 40 minutes. If he gets a 49-minute head start one day, can I catch him before he gets to school? Explain, without referring to any short-cut in your explanation. (Hint: Do not do a lot of calculation.) No. Brother needs only 1 more minute to get to school, and in 1 minute, I can travel only 1/40 of the distance to school. 12. My sister can walk from school to home in 40 minutes. I can walk from school to home in 30 minutes. But today I stayed for some extra help, and my sister was already 25 of the way home when I started. If I walk at my usual speed, can I catch my sister before she gets home? Reasoning About Numbers and Quantities Test-Bank Items with Answers page 5 If "Yes," exactly what fraction of the trip have they covered when I catch her? If "No," exactly what fraction of the trip have I covered when my sister gets home? In either case, write enough (words, numbers, drawings) to make your thinking clear. _____ (yes/no) Explanation, including fraction of the trip: No, I cannot catch up with my sister. My sister has 3/5 of the way to go, which should take 3/5 of 40 minutes, or 24 minutes before arriving at home. But in 24 minutes, I can cover only 24/30 or 5/6 of the way to home. On the diagram, my sister is at the second colored dot (16 minutes) when I begin, and has 24 minutes of walking before arrival. In those 24 minutes, I can walk only to X. Location after traveling so many minutes My sister 8 16 24 32 40 Home 6 12 18 24 30 School Me (Comment: The Brother and I exercise in Section 1.2 is usually much more difficult than either #11 or #12, so either of these is reasonable for a timed test.) 13. The big dog weighs 5 times as much as the little dog. The little dog weighs 2/3 as much as the medium sized dog. The medium sized dog weighs 9 pounds more than the little dog. How much does the big dog weigh? a. List 3 quantities associated with this problem. If possible, give the associated value. b. Draw a diagram to represent the quantities in this problem. c. This diagram was provided by a 5th grader. Tell why it is not helpful. Large dog Small dog Medium dog d. Solve the problem and explain your solution process. a. E.g., weight of large dog, weight of medium dog, weight of small dog, large dog's weight in terms of small dog's weight; small dog's weight in terms of medium dog's weight. b. Large dog: 5S Reasoning About Numbers and Quantities Test-Bank Items with Answers page 6 Small dog: 1S (which is 2/3 M) Medium dog: (M is 3/2 of S ) c. The diagram does not tell anything about their sizes other than whichwas larger and smaller than the medium dog. d. If the medium dog is 9 pounds more than the small dog, then the medium dog weighs 27 pounds, and the small dog weights 18 pounds. The large dog weighs five times as much as the small dog, so is 5 x 18 = 90 pounds. 14. Give two quantities that one could have in mind when he/she says, "That's a big athlete!" Height, weight, popularity, … 15. Give two quantities that one could have in mind when he/she says, “This has been a good day.” Outside temperature, amount of work accomplished, amount of time spent playing ball and/or picnicking, ..... 1.3 Values of Quantities ON TRUE/FALSE ITEMS, ASK FOR AN EXPLANATION IF FALSE. 1. The label on a can of chicken broth claims that its weight is 1.4 kg. Use your metric knowledge to tell how many milligrams this would be. 1,400,000 mg 2. The larger the unit of measure used to express the value of a quantity, the larger its numeric value will be. True False False. The larger the unit, the smaller the numerical value will be, for describing the same measurement. 3. Using benchmarks, find an estimate of the following and explain how you did it. The length of this line in metric units: (NOTE to instructor: measure this line as it is printed out before using it in a test.) 4. Complete the following: a) 2.3 km = ___ m 2300 b) 2 cm = ___ km 0.00002 c) 2.14 g = ___ kg 0.00214 Reasoning About Numbers and Quantities Test-Bank Items with Answers page 7 5. What metric prefix means one-hundredth? ____________ 6. If a Pascal is some unit of measure, use your knowledge of metric prefixes to complete: 4 kiloPascals = ____Pascals. centi 4000 7. Name a metric unit that is analogous to a quart. Which is larger? A liter is slightly larger. 8. Name a metric unit that is analogous to a yard. Which is larger? A meter is slightly larger. 9. Size 1 Pampers fit babies who weigh 4 to 6 kg. Maggie weighs 11 pounds. Will Size 1 Pampers fit her? Justify your answer. (Recall that a kilogram is about 2.2 pounds.) Maggie weighs 5 kilograms and Size 1 Pampers will fit her. (Alternatively, use the 2.2 pounds for 1 kilogram to change the 4 kg-6 kg range to 8.8 pounds-13.2 pounds.) 10. What are some advantages to using the metric system of measurement? The metric system allows easy conversion of units because units differ by powers of ten. It is used in science for this reason, in all countries. Most countries use it for all measures. 1.4 Issues for Learning 1. Some children, when asked to solve a story problem, try different operations on the numbers, and then decide which one seems to give the best answer. What is the danger of solving problems in this way? Students will not know what the answer means. They do not understand the problem, thus they try to find an acceptable answer for the teacher, even though they cannot explain it. 2. Many teachers teach “key words” for solving word problems. What are the limitations of this strategy? Same answer as above. Also, key words can be misleading—they only work part of the time. 3. Use diagrams to solve the following problems. (Hint: Use strip diagrams such as in the exercises for 1.4.) a. Jesse collects stamps. He now has 444 stamps. He has three times a many stamps from European countries as he does from Asian countries. How many of his stamps are from European countries? b. Silvia and Juan are buying a new table and new chairs for their dining area. Chairs with arm rests are $45; those with no arms are $8.50 cheaper The table is 4 times as much as a chair without arm rests. If they buy a table and six chairs, two with arms and four without, what is the total price they pay? Reasoning About Numbers and Quantities Test-Bank Items with Answers page 8 c. Joe lives 8 miles from campus. Jim lives 2 miles further away from campus that Joe does. If each drives a car to campus, how many miles altogether do Joe and Jim drive to and from campus? d. A Grade 3-4 elementary school classroom has 29 students. There are 7 more third graders than there are fourth graders. How many students are there in each grade? a. European 444 in all, so 111 Asian and 333 European Asian $45 b. Chairs with arm rests $8.50 Chairs without arm rests Table 2 x $45 + 4 x ($45 – $8.50) + 4 x ($45 – $8.50) = $382 c. Joe 8 miles Jim 2 miles Jim drives 10 miles one way, so 20 miles both ways, and Joe drives 16 miles both ways, so together they drive 36 miles per day. d. 18 in third grade, 11 in fourth grade 3rd Grad 4th 7 st 29 students in all Grade Grade 3 has 18 students, Grade 4 has 11 students. Reasoning About Numbers and Quantities Test-Bank Items with Answers page 9 Chapter 2 Numeration Systems 2.1 Ways of Expressing Values of Quantities, and 2.2 Place Value ON TRUE/FALSE ITEMS, ASK FOR AN EXPLANATION IF FALSE. 1. Is the old Greek numeration system ( = 1, = 2, = 3, etc.) a place-value system? Explain. No. A place-value system requires that the placement of the symbol have meaning, which is not true of the Greek system. 2. Most present-day societies use the Hindu-Arabic numeration system. True False True 3. How many tens are in 7654? How many whole tens are in 7654? 765.4, 765 4. How many hundreds are in 23? How many whole hundreds? 0.23, 0 5. How many tenths are in 1.03? How many whole tenths? 10.3, 10 6. How many ones are in 4352.678? How many whole ones? 4352.678, 4352 7. In base ten, 3421 is exactly __________ ones, is exactly __________ tens, is exactly ___________ hundreds, is exactly ___________ thousands; also, 3421 is exactly ___________ tenths, is exactly ___________hundredths. In base ten, 3421 is exactly 3421 ones, is exactly 342.1 tens, is exactly 34.21 hundreds, is exactly 3.421 thousands; also, 3421 is exactly 34210 tenths, is exactly 342100 hundredths. 8. In base ten, 215.687 is exactly __________ ones, is exactly ____________ tens, is exactly ____________ hundreds, is exactly _____________ thousands; also, 3421 is exactly ___________ tenths, is exactly ___________hundredths. In base ten, 215.687 is exactly 215.687 ones, is exactly 21.5687 tens, is exactly 2.15687 hundreds, is exactly 0 .215687 thousands; also, 215.687 is exactly 2156.87 tenths, is exactly 21568.7 hundredths. 9. (Roman numerals) IX = ____________ten and XI = ____________ten 9, 11 10. 34,597 has 345 whole thousands in it. True False False. 34,597 has 34 thousands in it. (Or, if you have emphasized describing the exact number, 34,597 has 34.597 thousands in it.) 11. 34.597 has 345 whole tenths in it. Reasoning About Numbers and Quantities True Test-Bank Items with Answers False True page 10 12. 56 has 560 tenths in it. True False 13. 23 has 230 hundredths in it. True False True False. It has 2300 hundredths in it. 14. 45 has 4500 hundredths in it. True False True 15. 632.1 has 632.1 ones in it. True False True 16. A soap factory packs 100 bars of soap in each box for shipment. If the factory makes 15,287 bars of soap, how many full boxes will they have for shipment? Explain. 152, because there are 152 hundreds in 15,287. 17. How many $10 bills could one get for $10 million? A. 1,000,000 B. 100,000 C. 10,000 D. 1000 E. None of A-D A 18. How many $100 bills could one get for a billion dollars? A. 100,000,000 B. 10,000,000 C. 1,000,000 D. 100,000 E. None of A-D 19. How many $100 bills would make $45 billion? B 450,000,000 20. Judy says, "Well, hundredths are smaller than tenths. So 0.36 is smaller than 0.4." Comment on Judy’s reasoning. Although Judy does choose the smaller number correctly, her reasoning is risky. If the numbers were 0.56 and 0.4, using just her reasoning would give an incorrect choice for the smaller number. 21. Grady thinks that 0.36 is bigger than 0.4 because 36 is bigger than 4. Comment on Grady’s reasoning. Grady is reasoning as though the numbers were whole numbers. Grady does not recognize that 4 tenths will be bigger than 3 tenths and only 6 hundredths 22. A teacher gave her class the challenge to find how many ways the number 423.1 could be thought about. Following are four children’s answers. For each answer, mark whether it is correct or incorrect. If it is incorrect, please explain. Dale’s answer: 423.1 could be thought about as 42,310 hundredths a) Is Dale's answer correct or incorrect? Correct __ Incorrect __ b) If Dale’s answer is incorrect, please explain the error. Pat’s answer: 423.1 could be thought about as 400 ones and 23.1 tenths Reasoning About Numbers and Quantities Test-Bank Items with Answers page 11 a) Is Pat's answer correct or incorrect? Correct __ Incorrect __ b) If Pat’s answer is incorrect, please explain the error. Lesley’s answer: 423.1 could be thought about as 41 tens, 12 ones, and 11 tenths a) Is Lesley's answer correct or incorrect? Correct __ Incorrect __ b) If Lesley's answer is incorrect, please explain the error. Jan’s answer: 423.1 could be thought about as 420 tens and 31 tenths a) Is Jan's answer correct or incorrect? Correct __ Incorrect __ b) If Jan’s answer is incorrect, please explain the error. Dale’s answer: a) Correct Pat’s answer: a) Incorrect b) 400 ones and 231 tenths, or… Lesley’s answer: a) Correct Jan’s answer: a) Incorrect b) 420 ones and 31 tenths, or… 2.3 Bases Other Than Ten ON TRUE/FALSE ITEMS, ASK FOR AN EXPLANATION IF FALSE. 1. For whole numbers, any two-digit numeral in base five represents a smaller number than the same two-digit numeral in base twenty. True False True 2. In base b there are b – 1 different digits. True False False. There are b digits: 0, 1, 2, 3, ... b–1. 3. These are the digits that are needed for a base seven place-value system: 0, 1, 2, 3, 4, 5, 6, 7. True False False; 7 is not a digit used in a base seven place-value system. 4. In base b, 3 + 2b3 + b would be written ________________. 2013b 5. A place-value, base-twenty system would require _____ digits. 20 6. 524 eight = __________ ten 340 ten 7. 287ten = __________ four 10133 four 8. 1012five = ______________ in base ten. Reasoning About Numbers and Quantities Test-Bank Items with Answers 132 ten page 12 9. 32ten = _______________ in base four. 200four 10. 2.31four = ________________ as a mixed number in base ten. 11. 6 23 in base ten = __________ in base three. 13 2 16 ten 20 102 three , or 20.2 three 12. 1ten = ____twelve 1 twelve 13. 214.3five = ___ in base ten 59.6 ten or 59 53 ten 1002three 14. 29ten = ___ in base three 15. 7 ten = ___ in base nine 7 nine 16. 203.6 ten = ____________ five 1303.3five 17. 2003five = _____________ten 253 ten 18. 200.3five = _____________ten 50.6 ten, or 50 53 ten 19. Write 49ten in base seven. 100 seven 20. Do the "translations" in parts A-D. Show your work. A. 3102five = _____________ten B. 310.2five = _____________ten C. 203.6ten = ____________five D. (base six pieces with small block as the unit) 402 80.4 or 80 25 1303.3 = ______________ten 336 21. You are living and working on a planet that uses only base five. How many five-dollar bills can you get for $1234.20five? Write your answer in base five since you are living on the planet. Write enough (numbers, words,...) to make your thinking clear. 123. As in base ten, 1230five = 123x10five = 123 fives. More symbolically, Reasoning About Numbers and Quantities Test-Bank Items with Answers page 13 1234.20 (1 five3 ) (2 five2 ) (3 five) something less than five ((1 five2 ) five) ((2 five) five) (3 five) (small) or = ((1 five2 ) (2 five) 3) five (small) =123 five five (small) So: in base 5, $1234.20five is: 123 fives, + something small) 22. In base five, the two whole numbers immediately before 2001five are _________five and ____________ five. 1444 and 2000 (either order) 23. If you are counting in base five, what would be the next six numerals after 2314five? 2320, 2321, 2322, 2323, 2324, and 2330 24. If you have been counting in base five, what would the five numerals before 2314five have been? 2304, 2310, 2311, 2312, 2313 25. Write how many fingers you have, in base five. In base two. In base ten. In base… 20five. 1010two. 10ten. … 26 . Which is larger? 21four or 21 five? Explain. When a numeral has more than one digit, it will vary in value if written in different bases because the place values will differ. 21four = 9 ten and 21 five = 11 ten 27. Consider: x = 81765fifteen and y = 81765thirteen. Which of x and y is greater? Explain. x because each digit other than the one’s place represents more in base fifteen than in base thirteen. 28. Consider: x = 74213sixteen and y = 74213fourteen. Which is greater, x or y (or are they equal)? Explain. x because each digit other than the one’s place represents more. 29. Consider x = 0.3147eight and y = 0.3147nine. Which of x and y is greater? Explain. (Be careful.) x is larger. 0.3nine = 3 9 Consider only x = 0.3eight and y = 0.3nine. 0.3eight = in base 10. 3 8 is larger than Reasoning About Numbers and Quantities 3 9. 3 8 in base 10 and Extending this reasoning, x is larger. Test-Bank Items with Answers page 14 Answer: 2b2 4 (or, 2b2 0b 4) 30. Write an algebraic expression for 204b. 31. If a base-eight flat = 1, the numeral ______________ would give the numerical value of the small cube. (You may give your answer either in base eight or in base ten--just make clear which.) 0.01eight, or 1 64 in base ten 32. Base eight pieces, with the small cube (a dot here) is asunit the unit. = ______________ten 1220ten 33. Sketch the wooden pieces that show 1203seven, and give the English words for the base ten value of each different sized piece of wood. Answer: with large dot representing a small cube, as the unit. large cube = 7 3 343 , or three hundred forty-three flat = 72 49 , or forty-nine small “cube” = 1, or one 33. (small cube = 1; no longs) 34. Write the base b numeral for 2b4 + b2 + 3b + 1. 20131b 35. Write out 32004m in the algebraic form of the last item. Answer: 3m4 2m3 4, or 3m4 2m3 0m2 0m 4 36. The best coins to use in thinking about the first three whole-number place values in base five would be the penny, the nickel, and the quarter. True False True 37. The best coins to use in thinking about the first three whole-number place values in base ten would be the penny, the dime, and the half-dollar. True False False. Best would be the penny, the dime, and the silver dollar *38. If 10000ten + 10b = 10023ten, what is base b? Reasoning About Numbers and Quantities Test-Bank Items with Answers page 15 b = 23. (The given equation gives 10b = 10023ten – 10000 ten, or 10 b = 23 ten —i.e., b = 23.) 39. Define your unit and sketch base blocks to represent 32.67eight. Using the flat = 1, 3 large cubes, 2 flats, 6 longs, 7 small cubes. 40. Sketch the wooden pieces that show 1203nine, and give the English words for the base ten value of each piece of wood. Answer to 40: large cube = seven hundred twenty-nine; flat = eighty-one small cube = one 61. 41. 53six names the same number as which of these base ten numerals? A. 186 B. 183 C. 12 D. 85 E. 33 E E. None of A-D C 42. In base ten, 111five would be written... A. 421 B. 155 C. 31 D. 21 43. The base b numeral 321b means... A. 3.b2 + 2.b1 + 1 B. 3.b3 + 2.b2 + 1.b1 D. 3.b + 2.b + 1 E. None of A-D C. 6b A 44. In base five, 32ten would be written... A. 152five B. 112five B C. 62five D. 17five E. None of A-D 45. The base two numeral 100two equals the base ten numeral... A. 1100100 B. 1011100 C. 8 D. 4 D E. None of A-D 46. In base ten, 32four would be written... A. 400 B. 200 C. 122 D D. 14 E. 8 47. The base four numeral 11.1four could be written in base ten as... A. 33 14 B. 33 101 C. 11 14 Reasoning About Numbers and Quantities D. 5 14 D E. None of A-D Test-Bank Items with Answers page 16 48. The base ten decimal 18.5 could be written in base six as ... A. 10.5six B. 20.3six 49. The base ten fraction A. 0.2eight 1 4 C. 30.3six D. 128.5six C E. None of A-D equals which base eight numeral? B. 0.14eight C. 0.02eight A D. 1.4eight E. None of A-D *50. If 31b = 28ten, then b = ... A. 4 B. 5 C. 7 D D. 9 E. This is impossible for any whole number b. 51. What base does the following counting work in: 1, 2, 3, 4, 10, 11, 12, 13, 14, 20, 21, … Base two Base four Base five Base six Base five 52. Which of the following is the base ten fraction representation for 1.21 four? A. 1 9 16 B. 1 3 4 C. 1 21 100 D. 1 3 5 E. None of these. A 2.4 Operations in Different Bases 1. Write an addition equation for (# fingers) + (# toes) = (answer) in some base other than base ten. (Samples) Base five: 20 + 20 = 40 Base three: 101 + 101 = 202 Base eight: 12 + 12 = 24 2. 3five 2five = _______five 11 five 3. What is 34 five ÷ 23 five ? 6 1 11 23 five or 1 13 ten 4. Show 3four x 21 four using drawings of base four materials (cubes, flats, longs, singles). Show all the steps involved, including the intermediate steps. Make clear what your choice of one is. Using the small cube as the unit: First show three groups, each with 2 longs and 1 small cube. Combine the three small cubes, and then the six longs, finally trading four longs for 1 flat, leaving 2 longs and the 3 small cubes. 123four 5. A. Add 24five + 33five in base five. (The numbers are already written in base five, so there should be no conversions done.) Reasoning About Numbers and Quantities Test-Bank Items with Answers page 17 B. How would you illustrate this with the base five blocks using drawings and showing the intermediate steps? A. 112five B. With the small cube = 1, first drawing shows 2 longs, 4 small cubes and 3 longs, 3 small cubes. Next, five of the seven small cubes are traded for a long, giving six longs and 2 remaining small cubes. Next, five longs are traded for a flat, giving 1 flat, 1 remaining long, and the 2 remaining small cubes, or 112 five. 6. A. 0.5ten = ________ eight. C. 84 ten = ________ three B. 312.2four + 22.3four= ________ four D. 2five x 43five = ________ five E. 33.3six = ________ ten A. 0.4eight B. 1001.1four C. 10010three *7. Determine the possible value(s) for base b: D. 141five E. 21.5 ten 321b – 234 b 43 b b = six (11 b – 4 b = 3 b, or 3 b + 4 b = 11 b ) 1 8. To the right is a partially completed addition, written in connection 2 1 4five with wooden pieces. At the time of the work to the right, what pieces of wood would be displayed, if the small block is the unit? + 3 3five (Drawings or word descriptions are okay.) 2 Finish the numerical calculation. (You do not have to draw the wooden pieces for the rest of the work.) Two flats, five longs, and 2 small cubes, at the time of the work. (The trade of the five longs for a flat is not reflected in the work yet.) Final sum: 302 9. 241six + 135six 420six 10. 127nine 58nine 58nine 11. 4.4five + 3.3five 13.2 five 12. 0.24seven 0.06 seven Reasoning About Numbers and Quantities Test-Bank Items with Answers page 18 – 0.15seven 13. 21six + 35.2six 100.2 six 13. Use drawings of multibase blocks to illustrate 231ten + 87ten Answer using a small square/block as the unit: . ....... Place together then trade ten longs for a flat: ........ Answer: Using small squares (dots here) as the unit: put the ones together to form 8 ones; put the tens (longs) together to form 11 tens; trade 10 tens for a 100 (flat). One would now have three hundreds (flats), one ten (long), and eight ones. The answer is 318. 14. Use drawings of multibase blocks to illustrate 32five + 23five Answer: the small square is being used as the unit .. ... ..... The five small squares can be traded for a five (long) leaving 0 ones. There are now six longs. Five would be traded for a flat of twenty-five, leaving one five (long). The answer is therefore 110 five 15. What base does the following addition NOT work in: A. Base six B. Base seven C. Base eight 13 +13=26 D. Base ten A E. It works in all of these bases. Reasoning About Numbers and Quantities Test-Bank Items with Answers page 19 16. What step is wrong in the following (base eight): 13Eight x12Eight 6 20 30 100 156 A. 6 B. 20 C. 30 D. 100 E. None E 17. A. Subtract the following in base five. Show all your work: 2 2 1 five – 4 2 five 11 2 2 11 – 4 2 1 2 4 B. Use your work in part A to explain how the way we regroup in base five subtraction is similar to the way that we regroup in base ten subtraction. We first consider the ones. Regrouping may be necessary to subtract, as in A, where we regrouped to make six ones, and again when we regrouped to make seven fives. 18. Use drawings of base ten blocks to show that 3 x 15 = 45 Using the long as the unit (although another unit could be selected here) Group ten longs to get a fourth hundred, and an answer of 45. Reasoning About Numbers and Quantities Test-Bank Items with Answers page 20 Chapter 3 Understanding Whole Number Operations 3.1 Additive Combinations and Comparisons 1. Two basketball coaches, A and B, are talking. A says to B: "Your tallest player is 6 inches taller than my tallest player!" B says to A: "Yes, but your second-tallest player is 8 inches taller than my second-tallest player." A says to B: "Hmm. My second-tallest player is 4 inches shorter than my tallest player." Make a drawing, and tell the difference in heights of Coach B's two tallest players. Drawing: Diff. in heights, Coach B's _____ There are several possible arrangements. Below is one that helps to see that the difference asked for (BT vs B2T) is 18 inches. Students may assign an arbitrary number to the height of B’s tallest player rather than rely on their drawing. Point out that they have unnecessarily (probably) ignored their drawing in arriving at their answer. 6" 4" ? 8" BT AT A2T B2T 2. To determine how much older your father is than you, you need to make an additive comparison of his and your ages. True False True 3. Marge bought several types of candy for Halloween: Milky Ways, Tootsie Rolls, Reese's Cups, and Hershey Bars. Milky Ways and Tootsie Rolls together were 6 more than the Reese's Cups. There were 4 fewer Reese's Cups than Hershey Bars. There were 12 Milky Ways and 28 Hershey Bars. How many Tootsie Rolls did Marge buy? List 5 quantities involved in this problem. Sketch a diagram to show the relevant sums and differences in this situation. Solve the problem. Reasoning About Numbers and Quantities Test-Bank Items with Answers page 21 The five quantities are usually easy: E.g., the number for each type of candy, and some of the explicit comparisons mentioned. Here is a diagram, with the deduced numbers of bars in parentheses, giving 18 TRs (start with the HBs, then determine the RCs, then the TR+MW total and finally the TRs). (30) 6 4 #TR (18) #MW 12 #RC (24) #HB 28 4. The school cafeteria is ready to serve two kinds of sandwiches, roast beef and peanut butter, and two kinds of pizza, cheese and vegetarian. There are 60 servings of pizza prepared. There are 8 fewer roast beef sandwiches prepared than there are servings of cheese pizza. There are 6 more peanut butter sandwiches prepared than there are servings of vegetarian pizza. All together, how many servings of sandwiches are prepared? a. List 8 quantities involved in this problem. b. Sketch a diagram to show the relevant sums and differences in this situation. c. Solve the problem. Again, depending on whether you have used the earlier, similar problem, many of your students will use algebra or trial-and-error on this problem; we suggest, for now, prohibiting algebra. You might also consider omitting part c. But the problem can be solved with the use of a drawing, as seen below. a. Number of kinds of sandwiches, number of kinds of pizza, number of servings of pizza prepared, difference in number of roast beef sandwiches prepared vs number of servings of cheese pizza, difference in number of peanut butter sandwiches prepared vs number of servings of vegetarian pizza, total number of sandwiches prepared, number of roast beef sandwiches prepared, number of peanut butter sandwiches prepared, number of servings of cheese pizza, number of servings of vegetarian pizza, difference in number of roast beef sandwiches and number of peanut butter sandwiches, total number of servings of pizza and sandwiches,… b. There are other possible praiseworthy drawings possible, but the following suggests the solution (for the total number of pizza servings and sandwiches) pretty easily. c. The number of sandwiches is (60 + 6) – 8 = 58. Reasoning About Numbers and Quantities Test-Bank Items with Answers page 22 60 #CP #VP 8 6 #RB #PB 5. A local community college has two sections of Math 210 (Sections A and B), and two sections of Math 211 (Sections C and D). Together, Sections C and D have 46 students. Section A has 6 more students than Section D. Section B has 2 fewer students than Section C. How many students are there in Section A and Section B all together? a. For each given value write the quantity next to it. b. Sketch a diagram to show the relevant sums and differences in this situation. c. Solve the problem. Show all your work here. a. 46 students, total number of students in C and D 6 students, difference in numbers of students in A and D 2 students, difference in numbers of students in B and C b. (sample drawing) 46 C D 2 6 B A c. (46 – 2) + 6 = 50 students for Sections A and B together 3.2 Ways of Thinking About Addition and Subtraction 1. Here are two word problems. How do they differ conceptually? Silvia had 14 books, and then received 4 more books. How many books does she have now? Silvia has 14 books on one shelf and 4 books on another. How many books are on the two shelves? Reasoning About Numbers and Quantities Test-Bank Items with Answers page 23 In the first, there is an action implied. In the second there is not. Because it is harder to “act out” the second problem, in may be more difficult for some young children. 2. A first grade teacher always reads subtraction statements such as “7 – 5 = 2” to his class as “seven take away five is two.” That is, he always reads the minus sign as “take away.” Comment on why this might not be a good practice. Reading “–“ only as “take away” ignores the fact that other situations—comparison and missing addend--might also involve subtraction. 3. Write a missing-addend problem using $35.95 and $19.50. Various possibilities. Each should involve an addition situation describable by 19.50 + n = 35.95 (or n + 19.50 = 35.95). 4. Suppose you are using toothpicks to act out the following story problem: Jack had 8 candy bars. Bill had 4. a. How many more candy bars did Jack have than Bill? b. How many toothpicks would you need to act the problem out? Explain your answer. What type of subtraction is this? a. 4 b. 12, because there are the two separate amounts; this situation involves an additive comparison. 5. For a, b, and c below, state: 1) the operation you would use to answer the question, 2) the situation in which the problem fits, and 3) an expression which yields the answer, with the answer circled. a. Susan has $175. She wants to go on a skiing trip that costs $250. How much more money does she need? b. John is 6 ft 1 in. tall and Steve is 5 ft 9 in. tall. How much taller than Steve is John? c. Karen has four fish in her aquarium. She puts three more in. How many fish are in the aquarium now? a. 1) subtraction b. 1) subtraction c. 1) addition 2) missing addend 2) comparison 2) join Reasoning About Numbers and Quantities 3) 250 – 175 = 75 (75 circled) 3) 6’1” – 5’9” = 4” (4” circled) 3) 4 + 3 = 7 (7 circled) Test-Bank Items with Answers page 24 6. Rita is given this problem: Zetta has $39, but she needs $78 to buy a jacket she wants. How much more does she need? Rita's reply: "79 minus 40 is 39, so she needs $39." Explain Rita's reasoning. What is your reaction to this method of doing the problem? Rita has increased the minuend and subtrahend by the same amount, so the difference stays the same. (Think of a comparison subtraction drawing, even though this is a missing-addend setting.) 7. Filene is asked “What is 79 minus 32?" She responded: “8 to 40, 30 to 70, and 9 more, so the answer is 47.” Explain Filene’s reasoning. What is your reaction to this method of doing the problem? Filene is using what is sometimes called “shopkeeper math" (see the next problem). She counts up from 32 to 40 (8), 40 to 70 (30), then 9 more to 79, and adds up the numbers 8, 30, and 9. 8. A bill for school supplies was $87.35. Josh paid with two $50 bills. Rikki, at the cash register (one which did not tell the change to be given to the buyer), counted Josh’s change. “40, 50, $1, and $10 makes $100. How much change did Josh receive? In what currency? Is that what he should have received? Josh received a nickel (to 40 cents), a dime (to 50 cents) then probably two quarters to make $1, then a ten dollar bill, which would add up to $11.65. This was not correct; he should have received $12.65. After the coins, Josh should have been given another dollar. “40, 50, 88, 89, 90, and 100” is probably what the cashier said, distinguishing coins from bills as she handed them out. 9. a. Make drawings of circular "pizzas" to illustrate 6 – 2, take-away view. The drawing should show 6 circles, with 2 being removed by arrows or otherwise marked out in some way. b. A child is shown 9 apples and 6 oranges, and asked “How many more apples than oranges?" She says that apples and oranges are different things, and so she doesn’t understand the question. What might you do to help her? (One possible way....) Line up the apples then the oranges below, and ask how many apples don’t have an orange partner, then ask whether there are more apples than oranges, and how many more. 10. Finish the story so that your question could be answered by the given calculation, and so that your story involves the view given. a. 6 – 2.5, missing addend. "The two joggers decided to run at the beach... Reasoning About Numbers and Quantities Test-Bank Items with Answers page 25 b. 6 – 2.5, comparison. "The two joggers decided to run at the beach... a. They usually run 6 miles. How much farther do they have to run, if they have already run 2.5 miles? b. One runs 6 miles and the other runs 2.5 miles. How much farther does the first jogger run than the second jogger does? 11. In each of the following, which way of thinking about subtraction is involved? a. This story problem: "Basketball score: Aztecs 82, Opponents 69. By how many points did the Aztecs win?" b. The following thinking/drawing strategy (sometimes used with children having trouble with their basic subtraction facts): For 15 – 7, think of going "up the hill," going to 10 along the way… +5 +3 15 ( add 3 to get to 10, then 5 more to get to 15) So, 15 – 7 = 8 10 ________________________ 7 a. comparison b. missing addend 12. Give our label (e.g., take-away, etc.) for the situation in each story problem, and write the equation you would write for the problem. Hint: How would you act it out? a. University X wants to enroll 5000 new freshmen. It currently has enrolled 4275 new freshmen. How many more freshmen does University X need to enroll? b. This year's budget is $1.6 million. Last year's budget was $1.135 million. How much larger is this year's budget than last year's? a. missing addend. 4275 + n = 5000 (or 5000 – 4275 = n) b. comparison. 1.6M – 1.135 M = n 13. a. Finish this story problem so that it involves a comparison subtraction that could be solved by 5 – 3 12 . You made 5 gallons of lemonade for a school party... b. For the same problem, finish the problem to so that it involves a take-away subtraction. c. . For the same problem, finish the problem to so that it involves a missing-addend subtraction. Samples a. .....and 3 12 gallons of Kool-Aid. How much more lemonade than Kool-Aid did you make? b. At the party, the people drank 3 12 gallons of the lemonade. How much of the lemonade was left after the party? Reasoning About Numbers and Quantities Test-Bank Items with Answers page 26 c. 3 12 gallons were made from frozen lemonade and the rest from fresh lemons. How much of the lemonade was made from fresh lemons? 14. a. Finish this story problem so that it involves a comparison subtraction that could be solved by 26–12. Laresa had $26 when she went into the store.... b. For the same problem, finish the problem to so that it involves a take-away subtraction. c. For the same problem, finish the problem to so that it involves a missing-addend subtraction. Samples a. .....and her friend Tisha had $12. How much more did Laresa have than Tisha? b. She bought a wallet for $12. How much did she have left. c. She had $12 and then received cash for baby-sitting. How much did she earn babysitting? 15. Give the rest of the "family of facts" for k – 3 = p. Any order: 3+p=k p+3=k k–p=3 3.3 Children’s Ways of Adding and Subtracting 1. Following is only the start of a child's work (in base ten). What seems lacking in this child's understanding? 402 – 39 506 –149 …7 …3 The child seems to be unaware of what subtraction means, and is just working with the subtraction of the digits in the column without regard to the order. She does not have place-value understanding. 2. Perform the following using the "equal additions" method, as used by one student in the samples in your textbook. 432 –287 4 1 3 12 –3 2 98 7 1 4 5 Reasoning About Numbers and Quantities Test-Bank Items with Answers page 27 OR: Adding a ten to the 2 to make 12, and a ten to the 80 to make 90, then ten tens to the 30 to make 13 tens, and 1 hundred to the 200 to make 300, allows all the subtractions (12 – 7, 13 – 9, 4 – 3), giving 145. 3. The work of two students is shown below. Each student "invented" the method used, that is, it was not taught to the student. For each student figure out what the student was thinking while doing the problem. Then (i) work the second problem using the same method as the student, and (ii) comment on the student's method in terms of the "number sense" exhibited. a. 732 -2 4 5 513 (i) 8 3 4 1 -4567 (ii) b. 19 x 35. Well, 20 x 35 is like 10 x 35 two times, so that's 350 two times, which is 700. But that's 20 35s and I only want 19 of them. So 700 minus 30 is 670 minus 5 is 665. (i) 21 x 43 (ii) a. (i) 4226 (ii) This student is calculating larger – smaller in each place value, ignoring what is being subtracted from what. The student is showing no number sense, or awareness of what subtraction means. b. (i) 903: 20 43 = 860: 860 + 43. 860 + 40 = 900. 900 + 3 = 903. (ii) This student is showing excellent number sense (and operation sense, in that he/she knows that 21 forty-threes can be obtained by adding 20 forty-threes and another 43), in working with place values independently, and in a fashion that shows awareness of “easy numbers.” 4 Is this child's thinking all right? If it is, complete the second calculation using the child's method. If the thinking is not all right, explain why not. (given) 675 – 198 (child does) second calculation (or explanation if not ok) 677 453 – 200 – 295 477 Okay. Second calculation (in columns): 458 – 300 = 158 5. A visitor to a first-grade classroom saw a teacher ask a child to solve this problem: Jaime gets $5 a week for keeping the yard in good shape. He is saving his money for the country fair. After 4 weeks, how much has he saved? She thinks to herself: This is a multiplication problem, and first-graders have not yet been taught multiplication, so they can’t answer this problem..” But after a few minutes Li-Li say that the answer is 20. She explains how she did this problem and she did not do any Reasoning About Numbers and Quantities Test-Bank Items with Answers page 28 formal multiplication, much to the visitor’s surprise. What did she most likely do to find the answer? She probably used repeated addition: 5 and 5 is 10, and 5 more is 15, and 5 more is 20. This visitor also saw another problem the children worked: “8 miles of highway are being paved. If the workers pave 2 miles a day, how long will it take them to pave all 8 miles?" She thought: “This is a division problem and first graders have not yet learned to divide.” But then Belinda said that it would take 4 days. How do you suppose she explained this answer, without using division? She probably subtracted 2 from 8 four times, until she reached 0, then counted the number of times she subtracted 2–– 4 times. 6. Felisha was asked to find 413 – 248. Here is how she did this problem: 413 –248 –5 -30 200 165 Is her answer correct? Explain what she was doing. Find 9456 -3789 using this method. Yes, her answer is correct. She was finding partial addends, using negative numbers, then adding the partial addends. 9456 –3789 –3 -30 -300 6000 5667 7. A second-grade boy is asked to subtract 64 – 55, written vertically. The child thinks about the problem and then writes 9. He explains his thinking by saying, "6 take-away 5 is 1, I mean, 60 take-away 50 is 10. 5 take-away 4 is 1 and 10 take-away 1 is 9.” Is he correct? Use his thinking to find 243 – 124. He is correct. 200 – 100 is 100. 40 – 20 is 20. 4 – 3 is 1. 120 – 1 is 119. 8. Zenaida is asked to add 428 and 686, in vertical form. She begins by saying “Six thousand plus four thousand...The interviewers then asked her what column the six and four are in, and she identifies it as the hundreds column. She begins again by saying 4 hundred plus 6 hundred is ten hundred and writes below the line: 110. She then says we Reasoning About Numbers and Quantities Test-Bank Items with Answers page 29 have to do tens. 20 plus 80 equals 100 and places that under the 110. She then adds 8 and 6 and writes 14 and writes that below the 100. Adding, she says it is 224. Where does Zenaida go wrong? Discuss her place value understanding. Zenaida is on the right track and appears to have some knowledge of place value, but it is not strong enough to carry her through this problem. She adds from left to right, indicating that either she does know the standard algorithm, or just prefers this method. Her major error was to write ten hundred as 110, probably thinking ten and a hundred is 10 hundred. 9. Here is Ben's work: 2 3 3/ 0 2 – 9 203 4/ 0 17 –108 209 1 Is Ben’s method correct? Make up another subtraction problem that would lead Ben to apply his same method. Then finish the calculation as Ben would. Show the work Ben would do. Various. The story problem should involve a 0 in the middle of what will be the minuend. Ben will incorrectly rename or “borrow” from the hundreds place directly to the ones place. 10. Find 21 + 49 using an empty number line. Answer below for one way. + 40 +4 +5 21 61 65 11. Find 509 – 239 using an empty number line. 70 Answer below for one way. – 200 – 30 –5 270 300 –4 500 505 509 12. A student wrote the following answer to her problem 95 – 34: Reasoning About Numbers and Quantities Test-Bank Items with Answers page 30 95 – 34 = 95 – 35 = 60 60 – 1 = 59 So the answer is 59 Analyze this student’s thinking. The student noticed that 35 is compatible with 95, but after subtracting 35 the answer would have to be adjusted. Unfortunately, she does not realize that in subtracting 35, she has already subtracted 1 too much, and that 1 should be added to 60, not subtracted from it. 3.4 Ways of Thinking About Multiplication 1. a. Make sketches for 3 6 and 6 3 and contrast them. b. Make sketches of 1 2 6 and 6 1 2 a. The 3 6 drawing should show 3 groups of 6 things, such as 3 six-packs of a softdrink, or 3 sets of 6 objects of some kind. The 6 3 sketch should clearly show 6 sets of 3 things, or a 3 6 array that turned on its side is 6 3. b. The 12 6 should clearly show 6 objects, with 12 of them designated either as three objects or as half of each of the six objects, such as 6 circles of which 3 are shaded. The 6 12 should show 6 objects that are halved, such as 6 semicircles. 2. A designer of women’s “mix and match” clothing designs 3 styles of skirts, 2 pairs of pants, 3 types of tops, and 4 styles of jackets. How many different outfits could be purchased, if each outfit has a skirt OR pants, a top, and a jacket? (Assume that a woman will not wear a skirt and a pair of pants at the same time.) (3+2) 3 4 = 60, or possibly (3 3 4) + (2 3 4) = 36 + 24 = 60. 3. A clothes designer designs women's "mix and match" wardrobe with 2 styles of skirts, 1 pair of pants, 3 types of tops, and 2 styles of jackets. How many different outfits could be purchased, if each outfit has a skirt or pants, a top, and a jacket? (2+1) 3 2 = 18 4. Mitchell decides to get his car painted and to buy new hubcaps. He selects 5 colors he likes and 3 styles of hubcaps. Then he decides to paint the roof a different color than the body. He decides to let his wife make the final decision. How many choices does she have? Explain your answer. (5 4) 3 = 60, assuming color compatibility. 5. Make up a story problem about a bake sale, so that the problem could be solved Reasoning About Numbers and Quantities Test-Bank Items with Answers page 31 a. by 3 4 12. (Notice the order.) b. by 12 43 . a. Various possibilities. Each should involve 3/4 of some quantity with 12 as its numerical value. Example: There were 12 chocolate cakes, and 3/4 of them were sold by 10:00. How many chocolate cakes were sold by 10:00? b. Various possibilities, but each should involve 12 amounts, each with numerical value 3/4. Example: They had 12 cakes, and by 10:00 they had sold 3/4 of each cake. How much cake had they sold by 10:00? 6. Make drawings of circular "pizzas" to illustrate each of the following. a. 3 4, array b. 13 6, fractional part of an amount a. Three rows, or sets, each with 4 pizzas. (NOT 4 sets of 3 each) b. Six pizzas, with sets of two delineated and one of those sets indicated, OR with 1/3 of each pizza indicated. (Contrast continuous pizzas with discrete children, say.) 7. Give our label (e.g., take-away, etc.) for the situation, and write the equation for solving this problem. "A coffee shop has 4 kinds of pastries that you like. You always drink coffee, tea, or milk with your pastry. In how many ways could you place a pastry-plus-drink order?" Fundamental counting principle. 4 3 = n 8. Make up a story problem that could be solved by 16 Be attentive to the order of the factors. 1 2 . (Choose your own context.) Various. Look for situations involving 16 halves, not half of 16. 9. Finish each story so that your question could be answered by the given calculation, and so that your story involves the view given. Be alert to the order of the factors. a. 6 4, repeated addition. "You are looking in a photo album... b. 4 8, fundamental counting principle. The ice-cream shop offers for free one of nuts, sprinkles, or chocolate sauce (you don't have to take one, of course) with each cup of ice cream... c. 6 12 , repeated addition. "You work in a candy shop... Samples: a. …and notice that 4 pictures fit on a page. How many pictures would be on 6 pages? b. …You have 8 favorite kinds of ice cream. In how many ways could you order a cup of ice cream? c. One customer bought half-pound boxes for gifts for 6 colleagues. How many pounds did the customer get? 10. The product of a number n by any other number m different from 0 is always greater than n. Reasoning About Numbers and Quantities Test-Bank Items with Answers page 32 True False False. If m is a (positive) fraction less than 1 (and n is a positive number—we don’t usually take off if this is omitted because at this stage only non-negative numbers have been the focus), the product mn will be less than n. 1/2 x 6 is 3 and 3 < 6. 3.5 Ways of Thinking About Division 1. a. This is a typical problem from an elementary textbook: Jasmine works in a book store. Today three boxes of Harry Potter books arrived. There are 144 books in each box. Jasmine is told to stack the books in piles in an area of the book store. She is told to put the books into 16 piles. How many piles can she make? What interpretation of division is represented in this problem? b. What if the question changes to She is told to put 27 books in each pile? What interpretation of division is now represented? a. There are 432 books. She could do this problem by putting one book down 16 times, then a second book on top 16 times, etc. This is the partitive or equal sharing interpretation of division. b. This time Jasmine would put 27 books in a pile, then 27 in another pile, etc. She is “taking-away” 27 books at a time, and she can do this 16 times. This is the quotitive or measurement or repeated subtraction interpretation of division. (Admittedly, if Jasmine is working in a bookstore, she probably knows enough to simply divided 432 by 27 or by 16. But to do that, she must have some ideas about division that she learned in school, doing problems of both types.) 2. Make up a story problem involving quantities of ice cream in an ice-cream store, so that the problem could be solved by the calculations given: a. Can be solved by 2 18 . b. Can be solved by 16 18 . c. Can be solved by 43 24 . Samples: a. How many 1/8 quart servings can you get from 2 quarts? b. The store puts 1/8 quart on each cone. How many pints would they use for 16 cones? c. The store stocks 24 different kinds of ice cream. Three-fourths of them are changed every month. How many kinds are changed every month? 3. If a is any number other than 0, then 1 ÷ a is less than 1. True False False. If a is a (positive) fraction less than 1, then 1 ÷ a is greater than 1 4. Under a repeated-subtraction interpretation, 43 1 12 means _______________________________________________ Reasoning About Numbers and Quantities Test-Bank Items with Answers page 33 The quotient is ____________. Verify and explain your answer with a sketch. …how many 1 12 s are in, or make, answer, 12 of one 1 12 , is in 43 . 3 4 ? The answer is 12 . The sketch should show the 5. a. Decide which type of division the following word problem is depicting and explain your reasoning. Mr. Burke's class of 24 fourth graders is doing a project on keeping the environment clean. There are 6 different topics the students need to explore, and Mr. Burke wants the same number of students to explore each topic. How many students will be in each group where each group explores a different topic? b. Write another word problem that illustrates the other type of division using the same context as the problem above. a. Sharing equally, or partitive, division. The 24 students are to be put into 6 equal sized groups. b. (Repeated subtraction, or measurement, division) The 24 students are to be put into teams of 6 to work on projects about keeping the environment clean. How many projects will Mr. Burke have to grade? 6. Write a word problem for 37 ÷ 5 for which the answer would be 2. Sample: Thirty-seven children want to play a game that involves teams of 5 players. How many children won’t be on a team (but may get to be substitutes)? 7. Consider this problem situation, which would involve dividing by 3: "You are putting reading books on 3 shelves in your classroom. So the books look neat, you put the same number on each shelf. How many books will be on each shelf?" Write another problem situation about the reading books, so that your problem involves another way of thinking about division by 3. Sample: The reading books are pretty big, so your assistants can carry only 3 at a time from the storage closet. How many trips to the storage closet will your assistants need? 8. Write two word problems about cars, so that the first problem shows the repeated subtraction meaning of division, while the second problem shows the partitive or sharing meaning of division. Reasoning About Numbers and Quantities Test-Bank Items with Answers page 34 Samples: (Repeated subtraction, or measurement) The big bag has 48 plastic cars, to be put into bags holding 6 cars each. How many bags of cars will there be? (Partitive, or sharing) The big bag has 48 plastic cars, to be split fairly among 6 youngsters. How many cars will each youngster get? 9. Write a word problem for 37 ÷ 5 for which the answer would be 7. Sample: Thirty-seven children want to play a game that involves teams of 5 players. How many teams can be formed? 10. Circle each which is undefined: 0 ÷ 6, 6 ÷ 0, 0 ÷ 0, and explain why any undefined one(s) is undefined. If the symbol is defined, tell what it equals. Undefined: 6÷0 and 0÷0. Explanations should reflect your emphasis in class, most likely through examination of a related multiplication “check.” 0÷6 = 0. 11. Finish the following story to make story problems that could be solved by the indicated calculation. "The farmer has a 3 12 acre orchard of orange trees.... a. 23 3 12 b. 3 12 5 c. 3 12 0.8 Samples: a. …She and her workers have harvested 2/3 of the orchard. How many acres have they harvested? b. …If she wants to replace all the trees over a 5-year period, how many acres should she plan to replace each year? c. …One sprayerful of fertilizer can cover 0.8 acre. How many sprayerfuls will she need to cover the whole orchard? 12. Make drawings of circular "pizzas" to illustrate 4 12 ÷ 3, sharing equally. There should be 4 1/2 pizzas shown, with marks to show how each of three equal shares “gets” 1 1/2 pizza. Just showing 4 1/2 pizzas and then 1 1/2 pizza is not a good answer. 3.6 Children Find Products and Quotients 1. Is this child's thinking all right? If it is, complete the second calculation using the child's method. If the thinking is not all right, explain why not. Reasoning About Numbers and Quantities Test-Bank Items with Answers page 35 (child's work) 124 15 1000 200 40 500 100 20 1860 second calculation (or explanation if not ok) 132 14 a. Okay. Second calculation (in columns): 1000 + 300 + 20 + 400 + 120 + 8, sum = 1848. 2. Fiesha finds 32 54 as follow: 54 32 1500 120 100 _ 8 1728 a) Which is true of Fiesha’s mathematical steps? __ Fiesha’s steps are mathematically correct. __ Fiesha’s steps are mathematically flawed. __ I cannot tell if Fiesha’s steps are mathematically correct or flawed. b) Understanding of multiplication __ Fiesha doesn't appear to understand multiplication. __ Fiesha may or may not understand multiplication. __ Fiesha shows good understanding of multiplication. c) If Fiesha’s steps are mathematically correct, use her way of thinking to solve 24 53. If they are not, explain how Fiesha’s reasoning is flawed. Fiesha: a) Reasoning okay b) Shows good understanding of multiplication. c) Probably (in columns) 1000 + 60 + 200 + 12 in some order; sum = 1272 3. Amy finds 32 54 as follows: 54 is 4 more than 50, so find 32 50 and add 4 back to get 1728. a) Which is true of Amy’s mathematical steps? Reasoning About Numbers and Quantities Test-Bank Items with Answers page 36 __ Amy’s steps are mathematically correct. __ Amy’s steps are mathematically flawed. __ I cannot tell if Amy’s steps are mathematically correct or flawed. b) Does Amy show understanding of multiplication? __ Amy doesn't appear to understand multiplication. __ Amy may or may not understand multiplication. __ Amy shows good understanding of multiplication. c) If Amy’s steps are mathematically correct, use her way of thinking to solve 24 53. If they are not, explain how Amy’s reasoning is flawed. Amy: a) Reasoning probably okay, although the phrasing is not perfect (should be “add four 32s back”). b) Shows good understanding of multiplication. c) Probably 20 53, plus 4 53. 4. Antonio asks, “When I multiply [for example, 49 23, shown to the right], why do I have to put in the 0 [points to the zero in 980]?” What would you say to Antonio? 49 23 147 980 1127 The 980 comes from 20 49, so it is a number of tens. 5. Following is an example of a child's work. You are to study the work and then to judge the student’s understanding. Hiro was asked to divide 4240 by 6. His work is shown below. Hiro's work: 7 6 R4 6 4240 42 040 36 4 a) Is Hiro's work correct or incorrect? Correct __ b) If the work is incorrect, please explain how. Incorrect __ a) Hiro’s work is incorrect. b) In considering the 04 (the number of tens left), Hiro forgot to note in the quotient space that there are 0 tens for 40 ÷ 6. Reasoning About Numbers and Quantities Test-Bank Items with Answers page 37 6. Consider the following work of a student: 84 A. There is an error with the 20 x 45 B. There is an error with the 400 20 C. There is an error with the 160 400 D. There is an error with the 320 160 E. There is no error with this student’s work D 320 900 7. Use a nonstandard algorithm to calculate 128 x 67. Various methods, giving 8576 as the product. We usually get the long version (six partial products). 3.7 Issues for Learning: Developing Number Sense 1. In each pair, choose the larger. Explain your reasoning. Your justification should appeal to number and operation sense, not to computation. a. 1838 + 517 or 1836 + 514 b. 612 – 29 or 613 – 34 c. 0.578 or 0.002 + 0.0328 a. 1838 + 517 because each addend is larger than the corresponding one in 1836 + 514 b. 613 – 34 because 612 – 29 is the same as 613 – 30, OR 613 is only 1 more than 612, but subtracting 34 rather than 29 more than overcomes that. c. 0.578 because the sum of the addends in the second sum will not reach 0.5. 2. I am a number with 21 tens, 14 ones, and 11 tenths. What number am I? 225.1 3. Tell why the following are incorrect: a. 310 225 980 375 1895 b. 280 ÷ 70 = 40 c. 480 ÷ 0.4 = 120 a. The ones' column adds to 10, not something ending in 5. Reasoning About Numbers and Quantities Test-Bank Items with Answers page 38 b. There are only 4 seventies in 280. c. There are 480 ones in 480, so there will be far more 0.4s than that in 480. Reasoning About Numbers and Quantities Test-Bank Items with Answers page 39 Chapter 4 Some Conventional Ways of Computing 4.1 Operating on Whole Numbers and Decimal Numbers 1. Show 3335 ÷ 23 with a scaffolding algorithm, then by the standard algorithm and show how each number in the standard algorithm is associated with number in the scaffolding algorithm. 145 23 3335 23 3335 2300 100 23 1035 103 460 20 92 575 115 460 20 115 115 0 69 3 46 2 46 ___ 0 145 In the second algorithm, the 23 actually is 2300, yielding 100 in the quotient. The 103 is actually 1030, from which 920 (that is 23x 40, which 460 twice, making the first division easier) is subtracted, leaving 115 in both algorithms. In the first algorithm, 115÷ 23 is done in two steps, and in one step in the second algorithm, both times yielding 5. The first scaffolding algorithm could be done in multiple ways yielding the same result. 2. Use the scaffolding method to compute 5883 ÷ 17. Something along the lines of the following, which unnecessarily gives the best guesses for each place value (one of the talking points for the scaffolding algorithm): 17 5883 | 5100 | 300 783 | 680 | 40 103 | 102 | 6 1 346 3. Show, using 324 ÷ 28, how to work from the scaffolding algorithm to the standard algorithm. Similar to 1. Student’s work should show an awareness of the scaffolding algorithm format to an abbreviated form to the usual US form. Reasoning About Numbers and Quantities Test-Bank Items with Answers page 40 4. Do 32 467 using the method of writing all partial products. What does this algorithm have to offer that the standard algorithm does not? In some order, 12000 + 1800 + 210 + 800 + 120 + 14 (= 14944). This algorithm should “make sense” since it takes into account the place value of each digit. 5. Use a nonstandard algorithm to find 240five + 314 five , but showing all partial sums. This work is all done in base five 240 + 314 4 100 1000 1104 6. Name two positive and two negative aspects of learning nonstandard algorithms. Samples: Positives—Practice reasoning about the operations and place values; if studentgenerated, they can make sense to them; encourages a “make sense” view of mathematics; can be more efficient in selected calculations. Negatives—Time away from conventional algorithms, which always work; students who attempt to just memorize the techniques without understanding them will likely garble them. 7. Make a drawing of base ten materials that shows the initial set-up for 3 130.2. Make clear what = 1. Do not take time to draw all the later steps of the calculation with the base ten materials. With the long = 1, the drawing should show first three groups (rows are nice), with 1 large cube, 3 flats, and 2 small cubes in each group. 8. Draw how one would act out 200 – 62 (take-away view) to support the usual right-left algorithm, with base ten materials. Make a separate drawing for each step (add steps if you need them). Initially Reasoning About Numbers and Quantities second Test-Bank Items with Answers page 41 third fourth (etc. as needed) Initially, 2 flats. Second, trade a flat for 10 longs. Third, trade a long for 10 small cubes, giving 1 flat, 9 longs, and 10 small cubes. Fourth, take away (x-out, say) 2 small cubes and then 6 longs, leaving 1 flat, 3 longs, and 8 small cubes. 9. Draw how one would act out 200 – 62 (comparison view) to support the usual right-left algorithm, with base ten materials. Make a separate drawing for each step (add steps if you need them). Initially, 2 flats, and in a row below 6 longs and 2 small cubes. In the top row trade 1 flat for 10 longs, and then one of the longs for 10 small cubes, giving 1 flat, 9 longs, and 10 small cubes above the 6 longs and 2 small cubes. Comparing the two rows, starting with the small cubes, shows that (for the conventional right-left algorithm) the top row has 8 more small cubes, 3 more longs, and 1 more flat. 10. You decide to introduce your fourth-graders to the long-division algorithm with one-digit divisors, using base ten materials and 96 ÷ 3. You also want to use a story problem that they would find interesting as the basis for their work. a. From the practical standpoint of acting out the calculation, which way of thinking about division--repeated subtraction or sharing equally--should you use in your story problem for 96 ÷ 3? b. Write such a story problem (involving 96 ÷ 3). c. Show how you would act out your story problem, with drawings of the base ten materials. d. Write a second story problem for 96 ÷ 3, involving a different way of thinking about division from the way of thinking in your story problem in part b. Reasoning About Numbers and Quantities Test-Bank Items with Answers page 42 e. Answer a-d if you were to use 960 ÷ 320 instead of 96 ÷ 3. a. Sharing equally is more practical for 96 ÷ 3, since acting out that calculation with repeated subtraction of 3s would be unwieldy. b. Various. The 96 should be put into 3 (equal-sized) amounts or groups. c. With 9 longs and 6 small cubes, “deal” them out to three locations equally. You would start with the longs to illustrate the usual algorithm. d. Various. This time the situation should call for how many groups of size 3 are in, or make, a group of 96. Notice that the units for the 3 and the 96 should be the same. e. For 960 ÷ 320, repeated subtraction is much more practical, etc. 11. A student places multibase blocks on the table as follows: •••••• then •••••• Write which calculation that the student might be doing, with an explanation: A. B. C. D. 226 + 49 226 + 49 226 – 124 226 – 118 We first see 226. To add 49, the 12 ones would first be placed together and replaced by one long and 2 ones. That is not done here. To subtract 49, the first step would be to break a long into 10 ones, but that is not done here. To subtract 124, I can remove 4 ones, but I need to change one flat to 10 longs before I can subtract tens. This is done here. To subtract 118, the first step would be to change one long to 10 ones. That is not done here. Thus, C is correct. 12. What would the next line be, in a Russian peasant calculation of 23 624? You do not have to do the complete algorithm. 23 624 11 1248 13. The Russian peasant method for multiplying uses two basic processes: doubling and _______________________. …halving. 14. Below is a worked-out calculation of 313 42, using the lattice method for multiplication. Explain why the method does give the correct number in the tens place (the circled 4). (Note: Some current textbooks use this algorithm to teach multiplication of whole numbers.) Reasoning About Numbers and Quantities Test-Bank Items with Answers page 43 3 1 3 x 1 12 04 1 2 4 3 06 0 2 0 6 2 1 4 6 The circle 4 comes from the diagonal 2+0+2. The top 2 comes from 3 4(0) = 12(0), so that 2 is describing a number of4 tens in the product. The 0 comes from 3 2 = 06, and shows that that partial product does not contribute a whole number of tens to the product. The bottom 2 comes from 1(0) 2 = 2(0), showing that it is counting the number of tens from that partial product. 15. Write a word problem that would require solving 540 ÷ 4. Example: A grandmother had 4 grandchildren. She had $540 to give as Christmas gifts to the children, who all received the same amount. How much did each grandchild receive? 16. Consider this arithmetic problem: 4 25 a. Write a story problem where the answer would be 6. Possible: Jake was buying school supplies for his four children. He bought a pack of 25 pens. If each child received the same number of pens, how many could each child receive? b. Write a story problem where the answer would be 7. Possible: Twenty-five children where going on a field trip. Parents escorting the children allowed no more than four children in each car. How many cars were needed? c. Write a story problem where the answer would be 1. Possible: Jake was buying school supplies for his four children. He bought a pack of 25 pens. After dividing them evenly among his children, with each child getting the maximum amount possible, how many pens did he have left for himself? d. Write a story problem where the answer would be 6 14 Possible: Carolyn had 25 yards of fabric to make 4 identical costumes for a play. How much fabric did she allocate for each costume? Reasoning About Numbers and Quantities Test-Bank Items with Answers page 44 Chapter 5 Using Numbers in Sensible Ways 5.1 Mental Computation 1. Describe 3 different ways that you could MENTALLY calculate 16 25. Some possible ways: 4 4 5 5 = 4 5 4 5 = 20 20 = 400 (10 25) + (6 25) = 250 + 150 = 400 (16 20) + (16 5) = 320 + 80 = 400 16 100 4 = 4 100 (after dividing 16 by 4) = 400 2. For each of the following, MENTALLY calculate the EXACT ANSWER and write it in the blank. Use EXCELLENT NUMBER SENSE. Then write enough to make clear how you thought. a. 3618 + 2472 – 2618 – 472 = ________ b. (25 29) + (25 11) = ________ Thinking: Thinking: a. 3000 Thinking: 3618 – 2618 = 1000. 1000 + 2472 = 3472. 3472 – 472 = 3000 b. 1000 Thinking: Given = 25 (29 + 11) = 25 40 = 25 4 10 = 100 10. 3. Give the exact answer mentally: 73.8 + 511.37 + 24 – 73.8. Write how you thought. 535.37, taking advantage of the subtraction of 73.8 and the addend 73.8, then working with the 11 + 24. 4. Describe how you would MENTALLY compute the EXACT result in each of the following without using the standard algorithm: YOUR DESCRIPTION SHOULD BE CONCISE AND INCLUDE THE EXACT RESULT. A. 234 – 119 Description: B. 12% of 150 Description: C. 25 2 5 Description: Samples: A. 115 Change to 235 – 120, then work left-to-right. B. 18 2% of 150 is 3, and 12% of 150 is 6 times as much as that. C. 10 25 25 will give the same number, and 15 of 25 is 5. 5. Show how you would mentally compute the exact results: A. 3000 - 2575 B. 0.75 x 24 Reasoning About Numbers and Quantities Test-Bank Items with Answers page 45 C. 24 13 + 24 7 Samples: A. 425 B. 18 C. 480 One way: 2575 + 25 = 2600, plus 400 to get to 3000. Second way: Add 25 to each to change to 3025 – 2600. 3/4 of 24. 1/4 of 24 is 6. 3 6 = 18 Given = 24 (13 + 7) = 24 20. IF YOUR STUDENTS DO NOT KNOW ORDER OF OPERATIONS, REFER THEM TO THE APPENDIX WITH A REVIEW OF PROCEDURES. 6. Show how you would mentally compute: A. 0.75 24 B. 34 12 + 34 8 C. 3458 – 1734 – 400 + 1734 A. 18, from 3/4 of 24 B. 680, from 34 (12 + 8) C. 3058, from 3458 – 400 (the other terms give 0) 7. Give the exact answer by mental calculation. Then write down your thinking. You are asked for a second way of thinking in parts B and C. a. 479.38 + 18.9 + 2.4 – 479.38 = ____________ Thinking: b. 12 125 = ________ One way of thinking: c. 1714 – 897 = ________ A second way of thinking: One way of thinking: A second way of thinking: d. 20% of 45 = ____________ Thinking: e. 24 750 = ___________ Thinking: a. 21.3 Thinking: The first and last give 0, so it is only a matter of finding 18.9 + 2.4, which = 20 + 13 tenths, or 20 plus 1 and 3 tenths. b. 1500 One way: 3 4 125 = 3 (4 125). A second way: 3 4 25 5 = 3 (4 25) 5. Students often offer 10 125 = 1250 plus 2 125 = 250, so that’s why we ask for two ways here. c. 817 One way: Change to 1717 – 900 by adding 3 to each. Second way (missing addend): 897 + 3 = 900. Plus 814 more (perhaps in steps). 817. d. 9 Thinking: 1/5 of 45. Reasoning About Numbers and Quantities Test-Bank Items with Answers page 46 e. 18 000 Thinking: 24 750 = 24 75 10. 24 75 = 6 4 75 = 6 300 = 1800. 1800 10 = 18,000. Of course, there are many other ways, such as 24(1000 – 250) = 24,000 – 24 x 250 but 24 x 250 = 6 x 4 x 250 = 6 x 1000 = 6000. Finally, 24,000 – 6000 is 18,000. 8. A person can reasonably calculate the exact answer to 1563 – 198 mentally by... A. counting on his/her fingers B. calculating 1565 – 200 C. calculating 1565 – 200 – 2 D. calculating 1600 – 200 E. This calculation is impossible to do mentally. B 9. A person who is calculating the exact answer to 18 15 mentally starts by calculating 2 15. The person would finish the mental calculation by calculating... A. 9 30 D. 60 4.5 B. 27 10 C. 36 7.5 E. None of A-D works. A 10. For each, describe two different strategies for performing the following computation mentally. a. 29 + 58 c. 8 15 b. 74 – 28 Some possible answers: a. 30 + 57 = 87; 20 + 50 is 70, plus 9 is 79, plus 1 is 80, plus 7 is 87 b. 76 – 30 = 46; 74 – 24 is 50, – 4 more is 46 c. 4 30 = 120; 8 10 + 8 5 is 80 + 40 is 120; 8 5 = 40, 40 3 = 120 11. Determine the following mentally, writing enough to make your mental work clear. a. 40% of 80 b. 15% of 300 c. 20% of 14 d. 100% of 71 e. 5% of 60 f. 120% of 20 g. 15 is 25% of ? h. 14 is 50% of ? a. 32 b. 45 c. 2.8 d. 71 e. 3 f. 24 g. 60 h. 28 12. Tell how one might mentally compute the following: a. 25 x 104 b. 25% of 104 c. 200% of 104 d. 200% larger than 12 e. 200% as large as 12 Each can be done in a variety of ways. Here are some possiblilities: a. 25 x 100 + 25 x 4 is 2500 + 100 is 2600 or Reasoning About Numbers and Quantities 100 4 x 104 = 100 x 26 = 2600 Test-Bank Items with Answers page 47 b. 1 4 of 104 is 1 4 of 100 + 1 4 of 4 is 25 + 1 is 26. c. Twice 104 is 208. d. 100% larger than 12 is 24, so 200% larger than 12 would be 36. e. Twice as large as 12, so 24. 5.2 Computational Estimation NOTE: Only a few ways are shown to estimate a calculation. Other ways may also be correct. Calculators should not be allowed for items involving estimation or mental computation. 1. Show how you would estimate: a. b. c. d. 391 612 0.74 798 32% of 19 196% of 25 Possible answers: a. Round 391 to 400 and 612 to 600; 400 600 = 240,000 b. Around 3/4 of 800, which is 600 c. 6 is a good estimate because a third of18 is 6. d. Just under 50, because 2 25. (Also, since it is 4% of 25, or 1, less than that, the exact answer, 49, is fairly easy.) 2 Using benchmarks, find an estimate of each of the following and explain your reasoning. a. 7.5 % of $594. b. 0.32147 (67.557% of 89.4853) c. 2 89 ( 13 24 0.99184) d. 5.8 ÷ 12 a. 45. 7.5% is 3/4 of 10%; 10% of 600 is 60, and 3/4 of $60 is about $45. b. 20. The estimate should be about 1/3 of 2/3 of 90, or about 1/3 of 60. c. About 3 1/2. Almost 3, plus about 1/2 of 1 d. About 1/2. 6 ÷ 12 is 1/2. 3. Joe Blue was estimating 92 x 31. He said that rounding 31 to 30 then taking 92 x 30 (which is 2760) is a better estimate than rounding 92 to 90 and then taking 90 x 31 (which is 2790), because in the first case you lost only 1 by rounding 31 to 30, but in the second case you lost 2 by rounding 92 to 90. Explain how Joe's reasoning is incorrect. Your answer should show number sense. Estimating 92 31 via 92 30 will be “off” by 92. Estimating 92 31 via 90 31 will be “off” by only 2 31, or 62. Reasoning About Numbers and Quantities Test-Bank Items with Answers page 48 4. For each of the following, MENTALLY obtain an ESTIMATE of the answer and write it in the blank. Use EXCELLENT NUMBER SENSE. Then write enough to make clear how you thought. a. 34% discount on an $89 suitcase. _________ Thinking: b. 0.26 43,135 ≈ ___________ Thinking: c. 61 334 ≈ ______________ Thinking: d. 74.35% 1195 0.9837 ≈ ______________ Thinking: e. (1201.794 ÷ 24.3%) + 0.0423 ≈ ______________ Thinking: a. About $30. Thinking: 34% is about 1/3, and $89 is about $90. 1/3 of $90 is $30. b. About 11,000. Thinking: 0.26 is about 1/4; round 43,135 to 44,000. c. About 20 000. Thinking: 60 1/3 of 1000 = 20 000. d. About 900 Thinking: 3/4 1200 1 e. 4800 Thinking: 1200 ÷ 1/4. Definitely take off points for answers such as 4800.0423.) 5. Is 40 ÷ 1.99 less than, equal to, or greater than 20? Explain, showing your understanding of a meaning of division. . Greater than 20, since there will be more 1.99s in 40 than there are 2s. 6. Why is this student NOT showing good number sense, in estimating 249.738 + 48.246? "Well, 249 is about 250, and 48 is about 50. 250 + 50 is 300. Then .738 + .246 is about 1. So, my estimate is 301." In using 250 and 50 for the whole number parts, the student is already making an overestimate. Working with the decimals would only make the estimate farther off. 7. 0.7614987 x 159.23842 is about... A. 1.2 B. 12 C. 120 D. 1200 E. None of A-D C (from 3/4 of 160) 8. 1.334496 x 301.66 is closest to... A. 400 9. B. 391 C. 390 D. 40 E. 39 A 278.4132987 x 0.2617944285 is about... A. 7 B. 70 C. 700 D. 7000 E. None of A-D 10. Of the estimates listed, which is best for 0.3347876 x 629.847291143? Reasoning About Numbers and Quantities Test-Bank Items with Answers page 49 B A. 1800 B. 180 C. 200 D. 210 E. 240 D 11. Use number sense in locating the decimal point in the answer. Explain your thinking briefly. a. 77.5 2.84 = 2 2 0 7 Explanation: b. 1002.6 ÷ 3.6 = 2 7 8 5 Explanation: a. 220.7 Explanation: The product is about 80 3 = 240. b. 278.5 Explanation: The quotient is about 1000 ÷ 4 = 250. 12. You are using a calculator that can show only 8 digits of an answer. When you finish, you realize that you forgot to press the decimal points in the original numbers. Show where the decimal points go, and explain how you know, using number sense (no paperpencil calculation). a. 9.87429 637.21945 = 6 2 9 2 0 8 9 6 Thinking: b. 413.69824 ÷ 24.92617 = 1 6 5 9 6 9 4 4 Thinking: a. 6292.0896 Thinking: The product should be around 10 630 = 6300. b. 16.596944 Thinking: The quotient should be around 400 ÷ 25, or 16 (how many 25s make 400). 13. Show excellent number sense in estimating the following. Write enough to make your thinking clear. a. 75.48% 883.375 567 is about ____________. 566 Is your estimate (less than, equal to, greater than) the exact answer? (Don't figure the exact answer out!) ______________ because b. 32% of $595.45 c. 60% of $271 d. 87 52 a. About 660: 3/4 880 1. Since each of the new factors is less than the original, 660 will be less than the actual answer (Aside--from a calculator: The exact answer is a bit more than 667.949). b. About 1/3 of $600 is $200 c. $150 : 3/5 of $250, or $180 from 2/3 of 270. d. 4500 from 90 50 14. Company X bought 17,569 truckloads worth $24,598 each. Company Y bought 24,598 truckloads worth $17,569 each. Which company paid more, and how do you know? The companies paid the same. 17569 24598 = 24598 17569. 15. Betty was asked to give her best estimate for 25% of 7991.8. She estimates by taking 14 of 8000, which is 2000, and taking Therefore, her estimate is 2000.2 Reasoning About Numbers and Quantities 1 4 of 0.8, which is 0.2. Test-Bank Items with Answers page 50 Comment on Betty’s reasoning. Betty does not seem to realize that in using 8000 for 7991.8, the 0.8 is already involved. (She likely used the 8000 for just the 7991.) Using 8000 introduces an error that would only be exaggerated by adding a number-senseless 1/4 of 0.8. Another point that may not have come up in class is that reporting an estimate with tenths, like 2000.2, implies that the estimate is quite close, down to the nearest tenth. That clearly is not the case here. 16. 42,189 ÷ 511,264 is about how many percent? A. 8% B. 12.5% C. 80% D. 125% E. None of A-D A from 40 ÷ 500 = (40 ÷ 5) ÷ 100 = 8 ÷ 100 = 8% 17. Estimate the following with a brief explanation: a. 65% of 37 b. 140% of 52 c. 18% of 971 d. 43 % of 120 = ___ (decimal) a. 24 from 2/3 of 36 b. 70 from 7/5 of 50 c. 195 from 1/5 of 975 (200 probably okay, from 1/5 of 1000) d. 1.2 from 0.01 from 3/4 being close to 1, a number that is easy to calculate with. 18. Use benchmarks to estimate the following. Explain how you estimated. a. 60% of $271 b. 87 52 c. 32% of $595.45 a. $150 3/5 of $250, or $180 from 2/3 of 270. b. 4500 from 90 50 c. About $200. 1/3 of $600 19. ESTIMATE the indicated quantity and tell how you did it. a. 15% of $51.07 b. 25% of 1998 c. 125% of 47 d. 48% of 212 e. 4% of 201 a. 10% of 50 is 5, and 5% of 50 is 2.5, so the answer is about 7.5. b. 14 of 2000 is 500. c. 100% of 47 is 47; 14 of 48 is 12, so the answer is about 59. d. Half of 212 is 106 (or half of 200 is 100) e. 4 x 200 is 800, so 8; or 1% of 200 is 2 so 4% is 4 2 is 8. Reasoning About Numbers and Quantities Test-Bank Items with Answers page 51 5.3 Estimating Values of Quantities 1. Approximately how long is this line segment, in inches or in centimeters? Be sure to name the measuring unit you use. _________________________________________________________ What benchmark did you use? Measure this line segment before you give it to students. 2. Name an item that costs approximately $500,000. A. A car C. B. a jet plane C. a nice home D. A stereo 3. Name two types of estimates you use in your daily life. Could be length of time to drive to campus, amount of money to fill tank with gas, amount of money needed at cash register, cost of meals for a week, etc. 4. Using benchmarks, find an estimate of the time it would take, averaging 50 mph for 8 hours a day, to drive across the United States. Estimating that the distance across the US is about 3000 miles, the trip would take 3000 ÷ 50 = 60 hours, or 60 ÷ 8 = 7 1/2 driving days of 8 hours each. 5.4 Using Scientific Notation for Estimating Values of Very Large and Very Small Quantities 1. Write in scientific notation. 2. a. 38,000,000,000 b. 382.45 c. 0.000000000456 a. 3.8 1010 b. 3.8245 102 c 4.56 10-10 a. The earth is 150,000,000,000 meters from the sun. Write this in scientific notation. b. The speed of light is 300,000,000 meters per second. c. A dust particle is 0.000 000 000 753 kg. a. 1.5 1011 b. 3 108 c. 7.53 10-10 Reasoning About Numbers and Quantities Test-Bank Items with Answers page 52 3. How many seconds does it take light to reach the earth from the sun? Express your answer in scientific notation. (See item 2 for relevant data.) 1.5 1011 ÷ 3 108 = 15 1010 ÷ 3 108 = 5 102 or 500 seconds; a little over 8 minutes. Reasoning About Numbers and Quantities Test-Bank Items with Answers page 53 Chapter 6 Meanings for Fractions 6.1 Understanding the Meanings of 1. a b Use diagrams to show that 43 is the same as 3 ÷ 4. LABEL THE RELEVANT PARTS OF YOUR DIAGRAM AND EXPLAIN THE EQUIVALENCE. 3 4 could be drawn as one whole broken into 4 equal size pieces, where three of the fourths would be indicated in some fashion. The sketch for 3 ÷ 4 should show 3 wholes with, for sharing division, each marked into fourths. Each share would get 1 fourth from each, or a total of 3 fourths of a whole. 2. Circle the letter of any of the following regions that is cut to show fourths. If a choice does NOT show fourths, explain why not. A. B. C. D. A, B, and D do not show fourths. In each of these cases, the four pieces do not all have the same area. 3. Shown to the right is 1 43 yards of carpet. Sketch (fairly accurately) 1 yard of carpet and 3 13 yards of carpet. If the piece of carpet shown sells for $28, how much should 9 yards cost? 1 yard would be 4/7 of the rectangular region, then 3 1/3 would be 3 1/3 of those. 9 yards should cost $144 (if 7/4 yards cost $28, each 1/4 yard costs $4, so 1 yard cost $16). 4. T F Fractions are always less than 1. (Explain if F.) False, although the part-whole interpretation might lead one to think so. A fraction such as 54 is more than one. 5. To change 3 14 to a fraction, a common rule is to calculate 3 4, add 1, and write that (3 4)1 answer over 4: 3 14 = = 13 4 4 . Use a number line (and words as you need them) to explain why that rule makes sense. Reasoning About Numbers and Quantities Test-Bank Items with Answers page 54 0 1 2 3 Cut each unit into four equal pieces. Then the 3 = 6. (3 4) 4 4 , plus the 14 , give 3 14 = (3 4)1 4 . Make a drawing that shows 2 45 , if the shape to the right is 1. Just 2 regions like this, along with 4/5 of another 7. If the hearts shown represent only 25 of the chocolate hearts you gave away on Valentine's Day, how many chocolate hearts did you give away in all? A. 8 8. B. 28 D. 50 E. None of A-D If the apples shown are 43 of the apples used for pies, how many apples were used for pies in all? A. 3 9. C. 30 B. 9 C. 16 D. 24 D ¾ ¾ ¾¾ ¾ ¾ ¾ ¾ ¾ ¾ ¾ ¾ E. None of A-D C If 43 of the apples shown in #8 above were red apples, how many of the apples were red? A. 3 B. 9 C. 16 D. 24 E. None of A-D B 10. Mark 3 5 on this number line as A. Mark 0 1 2 5 3 on the number line as B. 3 4 Look for gross inaccuracies. If the number line shows 5, the item is "trickier." 11. Is Ardis’s reasoning correct? Explain. Shannon: "I still have half my spelling words to learn and 43 of my vocabulary words to learn." Ardis: “Well, 43 is more than a half because 12 24 . So, you have more vocabulary words than spelling words still to learn." Reasoning About Numbers and Quantities Test-Bank Items with Answers page 55 No, unless the number of spelling words was the same as the number of vocabulary words, giving units of the same size. 12. Bill and Tom have pieces of land that are the same size. Each plants flowers on part of his land, as shown in the sketches. The shaded part shows Bill's flowers. The shaded part shows Tom's flowers. flowers flowers Who has more land in flowers? A. Bill and Tom have the same amount of land in flowers. B. Bill has more land in flowers. C. Tom has more land in flowers. D. One cannot determine who has more land in flowers. Explain your choice. C. Tom has more land in flowers than Bill, because he has 3/7 of his land and Bill has only 3/8 of his (equal-sized piece of) land. 13. Pat and Dana like to argue with each other about mathematics problems. They discuss the figure below: Pat: The shaded region is one-and-a-half times as much as the unshaded region. Dana: Wait! I think that the unshaded region is 23 of the shaded region. Who is correct? Why? Both are correct. They are using different units—Pat’s is the unshaded part, and Dana’s is the shaded region. 14. a. Make a drawing the shows b. Make a drawing of 2 5 2 5 of a discrete whole. of a continuous whole. Reasoning About Numbers and Quantities Test-Bank Items with Answers page 56 a. The whole must consist of 5 (or 10, or…) objects, with 2 (or 4 in two pairs, or…) designated in some way. b. The whole must be continuous, such as a line or rectangle, divided into 5 equal parts, with 2 designated in some way. 50 53 50 15. Given 10 and 13 , why are we able to “cancel”the zeros in 10 , but we are not able to 53 “cancel” the 3s in 13 ? 50 The 0s represent factors of 10, 10 510 110 , and a common factor can be ignored. But the 3s 53 in 13 do not represent common factors. 16. Use drawings with rectangles to show that 7 a. 58 16 b. 43 23 c. 1 43 64 a. A drawing of a rectangle divided into 16 equal parts: 10 parts will be shaded for only 7 parts for 2 3 , but 7 16 b. This will require a rectangle divided into 12 equal parts. whereas 5 8 3 4 will require 9 of the parts, will require only 8 of the parts c. Two rectangles each divided into 4 equal parts: 1 43 will require 7 of those parts, whereas 6 4 will require only 6 of the parts. 17 Use the part-whole notion of fractions to explain why The first fraction has smaller pieces than the second. 7 13 127 18. Name three ways of thinking about the symbol “ 87 ”. Some examples: as part of a whole, 7 of 8 equal parts of a circular region; as 7 ÷ 8; as a ratio; as a probability; 7 of 8 discrete objects; ... 19. Write a story problem in which 53 is treated as a part-whole fraction, with discrete quantities. e.g., Jorge had 5 candy bars. 3 of them were Snickers. What fraction of his candy bars were Snickers? 6.2 Equivalent (Equal) Fractions. 1. 58 = which one or ones of the following? 400 10 50 20 500 i. 500 ii. 12 iii. 64 iv. 20 Explain: 800 15 16 80 80 25 32 All are equal, since the fractions in them are equal to those in the original. 2. How would you convince a child with visual evidence that 4 5 Reasoning About Numbers and Quantities 2 3 46 ? Test-Bank Items with Answers page 57 We usually expect a drawing of 2/3 of a region, and then added marks to that or to a copy cutting each third into two equal pieces, giving 4/6 for the same amount as the 2/3. 3. A child says, "My teacher says to put = between 43 and 68 , but I think 68 is bigger." How might the child be thinking? What would you do next? The child is probably focusing on the larger numerator and larger denominator in 6/8. One could make a drawing to show that the two fractions are equal. 4. How would you convince a child with visual evidence that 12 126 ? We usually expect a drawing of 1/2 of a region, and then added marks to that or to a copy cutting each half into six equal pieces, giving 6/12 for the same amount as the 1/2. 5. Use sketches to show that: 20 A. 15 1 13 B. 73 219 There are different ways; follow your expectation from class work. Samples… A. Show 1 1/3, mark the thirds to make 4/3 visible, then cut each third into 5 equal pieces, giving fifteenths. B. Show 3/7 and then cut each seventh into 3 equal pieces, giving twenty-firsts. 6. Name three common denominators for the fractions 7 5 7 9 7 a. 83 and 12 . b. 72 and 144 c. 40 and 60 a 24, 48, 72... b. 144, 288, 432, .... c. 120, 240, 360, ... 7. Write the simplest fraction form for each: zy3 x 2 a. 91618 12815 b. y4 z5 a. 9 5 2 b. x 4 yz 8. Put these in order, smallest to largest, using the symbols <, >, and = 15 19 3 8 300 400 < 11 16 11 300 16 < 401 300 401 < 300 400 3 8 6 8 = 6 8 < 15 19 200 303 9. Put these in order, smallest to largest: 200 303 6 9 203 300 10. a. Give a number between b. Give a number between c. Give a number between 6 9 9 12 203 300 9 12 3 4 7 and 7 . 8 8 111 and 113 . 7 8 114 and 113 . Reasoning About Numbers and Quantities Test-Bank Items with Answers page 58 d. Which number is closer to 1: a. b. c. d. 9 10 and 10 9 ? 3 4 1 3 6 7 is less than half and 7 is greater than half, so 2 is between; OR 7 = 14 4 8 7 3 4 7 = 14 so 14 is between 7 and 7 . 8 112 7 8 7 8 8 114 < 114 . 114ths are smaller than 113ths, so 114 < 114 < 113 . 9 1 10 1 1 1 9 10 is 10 away from 1, and 9 is 9 away from 1. 10 < 9 so 10 is closer to 11. Which of these is a rationale for comparing one answer.) 3 7 and 7 11 ? and one. (You can choose more than a. They are the same because 7 – 3 = 4 and 11 – 7 = 4. 7 7 b. 11 is more than 12 and 73 is less than 12 so 73 < 11 c. 3 7 = 33 77 and 7 11 = 49 77 so 3 7 is smaller. d. Sevenths are bigger than elevenths so 73 is larger. e. Equal because each fraction is four parts away from one. b. and c. 6.3 Relating Fractions, Decimals, and Percents 1. Fill in the blank cells in the table below so that the numbers in each row are equivalent to the given one: Fraction a. Decimal Percent 13 20 b. 0.00089 c. d. 43 4 a. 0.65 b. 89/100,000 c. 1225/10000 (might be simplified to 49/400) 0.1225 d. 10.75 2. What is the exact decimal equivalent of A. 0.4 12.25% A. 2 5 ?___ 65% 0.089% 1075% B. 2 7 ?___ B. 0.285714 Reasoning About Numbers and Quantities Test-Bank Items with Answers page 59 3. What is the fraction equivalent of A. 651/1000 A. 0.651? ___ B. 0.44444 ...? ___ B. 4/9 4. What is the percent equivalent of A. 240% A. 12 ? ___ 5 B. 0.00035? ___ B. 0.035% 5. Circle each fraction that has a terminating decimal, without calculating the decimal. A. 749327 2x106 B. 3000005 6x108 C. 21000042000014 350 A and C have terminating decimals (7 is a factor of numerator and denominator in C and so that fraction can be simplified.) C may be more difficult than should be on an examination unless similar ones have been used in class. 6. T F F 7. 3 18 can be written as a terminating decimal. (Explain if F.) The fraction = 1/6, which has factor of 3 in only the denominator. a. Write 56.38 as a fraction. a. 5582 99 8. b. b. Write 7.453 as a fraction. 7453 1000 T F If the following is false, correct it so that it is true in some non-trivial way: If D = 0.742 , then 100D = 74.200. F …then 100D = 74.242 . 9. As a fraction, 1.4 = A. 14 B. 57 C. 14 100 D. 41 10 E. None of A-D B 10. NUMBER SENSE ITEM--no calculator or hand calculation and no decimals. Put these 121 in order. Explain your decision only for 101 300 and 360 . 141 280 smallest 1.3% 1000 3000 _____ 121 360 101 300 _____ 101 300 1.3% _____ 121 360 _____ 1000 3000 _____ largest 141 280 Reasoning About Numbers and Quantities Test-Bank Items with Answers page 60 121 Thinking for 101 300 and 360 : 121 360 and 101 300 are both greater than 1/3, by 1/360 and 1/300, respectively. Since 1/300 is greater than 1/360, 101 300 is greater than 11. Put these in order, smallest to largest: 0.03 0.0295 0.0295 0.03 0.1 12. Put these in order, smallest to largest: 8.5% 110% 15 11 1.7 0.1 121 360 . 0.13999 0.13999 8.5% 1.7 15 11 14 8 110% 14 8 13. Put these in increasing order without any hand calculation: 11 211 200 81.2% 0.0239 40 392 31 0.0239 11 40 211 392 200 31 81.2% 14. Give three decimals between 2.3456 and 2.3457. If it is not possible, explain why not. Samples: 2.34561, 2.34562, 2.345601 (infinitely many possibilities) # 15. Write each of the following as a fraction in the whole whole # form, if it is possible. If it is not possible, explain why not. __ A. 9.6534 = _________ B. 96.6534 = _________ A. 96534 10000 (may be simplified to 48267 5000 ) B. 956869 9900 16. Give three decimals between 0.301 and 0.302. If it is not possible, explain why not. Samples: 0.3011, 0.3012, 0.30189 (many other possibilities) 17. For each comparison, use number sense to order these fractions and decimals. Use only the symbols <, = and >. 0.1 17 35 0.1 1 8 1 8 1 5 2 3 17 35 1 5 59% 0.85 2 3 9 10 0.85 59% 9 10 18. a. What are the “rational” numbers? Reasoning About Numbers and Quantities Test-Bank Items with Answers page 61 b. How are they different from the “irrational” numbers? a. Rational numbers are numbers that can be expressed in the form integer non-zero integer (if only whole number non-negative rational numbers have come up, accept non-zero ). whole number b. Irrational numbers are numbers that cannot be expressed in that form. (Sufficient for now.) 19. As a decimal, 0.525% is... A. 525. B. 52.5 C. 5.25 D. 0.525 E. 0.0525 F. None of A-E F 20. As a percent, 5.35 is... A. 0.535% B. 5.35% C. 53.5% D. 535% E. None of A-D D 21. As a decimal, 1.775% is... A. 0.01775 B. 0.1775 22. 1.5% = ... A. 150 C. 1.775 D. 17.75 E. None of A-D B. 15 C. 1.5 D. 0.15 A E. 0.015 E E. None of A. – D. D 23. 15% = ... A. 150 B. 15 C. 1.5 D. 0.15 6.4 Estimating Fractional Values 1. Using benchmarks, find an estimate of each of the following and explain how you did it. a. The sum of b. 7 15 c. 7 8 1 7 23 , 15 , 5 9 , and 13 12 . My thinking: 11 12 My thinking: 109 My thinking: d. 7 67 14 My thinking: a. About 2. The first fraction is negligible compared to the others, the second two fractions are about 1/2 each, and the last fraction is about 1. b. A little under 1 1/2. My thinking: 7/15 is a bit smaller than 1/2, and 11/12 is a bit smaller than 1 Reasoning About Numbers and Quantities Test-Bank Items with Answers page 62 c. A little less than 2. Both fractions are slightly less than 1. d. 32 8 4 2. A fraction is close to True 1 2 when the numerator is close to two times the denominator. False False….when the denominator is close to two times the numerator, OR when the numerator is about half the denominator. *3. A fraction ba is close to denominator. True 3 4 when three times the numerator is close to four times the False False. …when four times the numerator is close to three times the denominator. Or,…when the numerator is about 3 times one-fourth of the denominator. *4. How can you tell when a fraction is close to 14 ? Do not refer to decimal numbers. When the numerator is about 1/4 of the denominator, OR when the denominator is close to 4 times the numerator. 5. Show how you would mentally compute the exact results: a. b. c. 3 7 4 1 7874 1 3 4 7 7 10 7 10 18 80 21 24 2 42 100% My thinking: My thinking: = ___ d. 114 – 24 = ___ 3 8 My thinking: My thinking: a. 2 1/8 First + third = 1; other two = 1 1/8. b. 1 57 The second and fourth fractions give 1. c. Exactly 15. The first two fractions give a product of 30 (3/4 of 40). 30 x 1/2 = 15. d, Exactly 89 5/8. 114 – 24 is 90; 90 – 3/8 =89 5/8. 6. For each of the following, obtain an estimate of the answer and write it in the blank. Use excellent number sense. Write enough to make clear how you thought about the problem. a. 68 43 ÷ 2 15 16 = ___ My thinking: 7 11 b. 4 16 + 5 12 – 2 18 = ___ My thinking: c. 301 ÷ 7 16 = ___ Reasoning About Numbers and Quantities My thinking: Test-Bank Items with Answers page 63 d. The sum of 11 3 23 , 27 , and 5 16 My thinking: a. About 23, from 69 ÷ 3 b. About 8 1/4, from under 4 1/2 plus under 6, less 2 1/8. 10 1/2 – 2 = 8 1/2 might be close enough. c. About 600, from how many 1/2s make 300. d. About 3/4 from 1/2 + negligible + 1/4 7 7. 112 2 115 3 125 137 is about... A. 8 12 B. 8 C. 7 12 24 D. 6 24 48 E. None of A-D B (the first and third fractions give another 1 exactly) 8. 20% of 150 is 13 of ... A. 10 B. 90 C. 120 D. 250E. 900 B. 20% of 150 is 30, and 30 is 1/3 of 90 9 9. Which is larger, 97 or 11 ? Provide a justification for your choice that does not refer to decimals. 7/9 is 2/9 less than 1, and 9/11 is 2/11 less than 1. Since ninths are larger than elevenths, 2/9 is larger than 2/11. So 7/9 is farther “under” 1 than 9/11 is, and 9/11 is larger. 8 10. Which is closer to 12 : 15 or 73 ? Explain your reasoning. 8/15 is half a fifteenth more than 1/2, and 3/7 is half a seventh less than 1/2. Since sevenths are larger than fifteenths, 3/7 will be farther away from 1/2 than 8/15 is, that is, 8/15 will be closer to 1/2. 11. Circle each that is close to, or equal to, 13 . (No explanation is required.) 3 40 102 175 297 0.2six 5 119 301 600 106 0.2six is 13 in base 10. 40/119 (about 40/120) is about 13 *12. Which fraction is larger, denominators. 38 48 or 47 60 ? Explain without using decimals or common The first fraction is 2/48, or 1/24, more than 3/4, and the second is 2/60, or 1/30, more than 3/4. Since 1/24 > 1/30, the first fraction, 38/48, is larger. 13.Which fraction is closer to 0: 18 or 19 ? Explain (without referring to decimals). 1/9, because cutting a whole into 9 equal parts will give smaller pieces than cutting the whole into 8 equal parts will give. 14. Which fraction is closer to 1: 1 8 or 1 9 ? Explain (without referring to decimals). Reasoning About Numbers and Quantities Test-Bank Items with Answers page 64 1/8, from the reasoning in #13, 1/8 will be closer to 1 because 1/9 is closer to 0. 15. Which fraction is closer to one-half: 18 or 19 ? Explain (without referring to decimals). 1/8, from the reasoning above, and the fact that both are less than 1/2. 16. Which fraction is closer to 1: 87 and 89 . Focusing on the “missing” part shows that 7/8 is farther below 1 than 8/9 is, so 7/8 is closer to 1. 29 *17. Which value is larger: 17 40 pound, or 64 pound? Explain your thinking, showing your understanding of fractions and of benchmarks (not decimals or common denominators). Using a benchmark of 1/2 pound, 17/40 is 3/40 under 1/2 and 29/64 is 3/64 under 1/2. Since fortieths are larger than sixty-fourths, 3/40 > 3/64, so 17/40 will be more under 1/2 than 29/64 will be. 29/64 pounds is larger 18. Which is the greater value: 169 of a cup, or 15 28 of a cup? Explain, showing your grasp of fractions and benchmarks and without using decimals or complicated common denominators. Using a benchmark of 1/2 cup, 9/16 is 1/16 more than 1/2, and 15/28 is 1/28 more than 1/2. Since sixteenths are larger than twenty-eighths, 9/16 is larger than 15/28. *19. Why is 2 3 a better benchmark than 1 is, for answering this question: "Which is larger, or 17 (You do not have to answer the question in quotation marks.) 21 ?" With a benchmark of 1, one has to compare 7/30 and 4/21—different sizes of basic “pieces” as well as different numbers of them (the 7 smaller pieces might or might not be larger than the 4 larger pieces). So 1 is not a good benchmark. But with 2/3 as a benchmark, it is a matter of comparing 3/20 and 3/14, for which only the size of twentieths vs fourteenths need be involved. 23 30 20. Name good fraction benchmarks for the following percents: a. 48% b. 165% c. 9% a. 1/2 b. 1 2/3 or 5/3 c. 1/10 (possibly 1/11 with really good number sense) 11 13 21. Which is larger, 13 or 15 ? Justify your answer without referring to decimal numbers or finding common denominators. 13/15 is larger. 11/13 is 2/13 less than 1, and 13/15 is 2/15 less than 1. Since fifteenths are smaller than thirteenths, 2/15 is smaller than 2/13, so 13/15 is less far under 1 than 11/13 is. Reasoning About Numbers and Quantities Test-Bank Items with Answers page 65 22. For each of the following, tell which is larger by referring to benchmarks. For each problem, show the benchmark used. a. 43 or 4 9 b. 10 41 or 3 11 a. 3/4, benchmark 1/2 b. 3/11, benchmark 1/4 (3/11 > 3/12 = 1/4, 10/41 < 1/4) 23. For each of the following, find a fraction between the two given fractions and show how you found it. Do not use decimal numbers or common denominators. a. 5 21 and 11 40 b. 7 13 and 8 11 a. 1/4 5/21 < 5/20 = 1/4, and 11/40 > 10/40 = 1/4 b. 2/3. 7/13 is barely over 1/2, and 8/11 > 8/12 = 2/3. 24. In each case tell which is larger by referring to benchmarks. Explain what you did making clear what your benchmark is. (Do not use common denominators.) a. 7 15 or 4 7 of the distance? b. 8 27 or 7 18 of those voting? a. 4/7 of the distance, since 4/6 > 1/2 but 7/15 < 1/2. b. 7/18 of those voting, since 8/27 < 9/27 = 1/3, but 7/18 > 6/18 = 1/3. Reasoning About Numbers and Quantities Test-Bank Items with Answers page 66 Chapter 7 Computing with Fractions 7.1 Adding and Subtracting Fractions 1. Make up a story problem that could be solved by 3 12 2 13 and illustrates the view of subtraction. comparison Various possibilities, but in each the two quantities should be distinct, as in, “One recipe calls for 3 12 cups of sugar, and another calls for 2 13 cups of sugar. How much more sugar does the first recipe call for than the second?” 2. Make up a story problem that could be solved by calculating 123 – 48 13 . Various possibilities, e.g., Velma had 123 feet of ribbon and used 48 13 for package bows. How much ribbon does she have left? 3. Following is an example of a child's work. You are to study the work and then to judge the student’s understanding. Rona was asked to subtract 2 58 from 4 18 . Her work is shown below. Rona's work: 4 18 3 11 8 2 58 2 58 1 68 a) Is Rona's work correct or incorrect? Correct __ b) If the work is incorrect, please explain how. Incorrect __ a) Incorrect b) In changing a 1 from the 4 into eighths, Rona did the usual base ten place value “borrowing” rather than the correct 8/8 + 1/8. 4. Four identical pizzas are shared among 3 people. Show 2 ways that the pizzas could be cut. i. Cut each into three parts; each person gets 1/3 of each pizza: 4/3 or 1 1/3 pizzas ii. Cut one into three parts. Each person get a whole one (3/3) plus another third; 1 1/3 pizzas. (There are many more ways, but perhaps complex.) 5. Using a rectangular region as the unit, illustrate each of the following: 2 4 a. 3 + 5 7 3 b. 5 + 2 10 1 c. 8 – 2 Samples: a. Start with 2/3 of a rectangular region shaded, then cut with marks perpendicular to the marks for thirds to give slivers that are fifths (and small boxes Reasoning About Numbers and Quantities Test-Bank Items with Answers page 67 fifteenths). 4/5 would entail 4 slivers or 12 small boxes. But there are only 5 small boxes left, so 3 small boxes, or 3/15, of another whole is needed. b. Similar to part A, except that each addend is more than one whole. 2 9/10 when all the work is done. c. Easiest is to recognize that 10/8 = 5/4 = 1 1/4, but that may not be regarded as fair. Show 10/8 with parallel cutting marks and shading on the two rectangular regions. That should make recognizable that 1/2 is 4/8, which can then be x-ed out from the 10/8. 6. Explain a strategy you could use to mentally compute the exact answer: a. 2 83 + 5 14 + 3 87 + 2 b. 2 45 +3 16 + 5 23 – 1.8 c. 3 12 – 2 83 a. The whole numbers give 12, and 3/8 + 7/8 give 1 1/4 which, with the other 1/4 gives 1/2. 13 1/2. 1 b. 2 4/5 – 1.8 is 2.8 – 1.8 is 1. 1/6 + 2/3 is 1/6 + 4/6 is 5/6. Now: 1 + 3 + 5 + 5/6 is 9 5/6. c. 1/2 is 4/8. So the difference is 1 1/8. 7. Why is a common denominator needed to add and subtract fractions? Because finding equivalent fractions that allow a common denominator d allow us to add and subtract d1 units; that is, we have the same unit to use for each fraction. 8. Three children took a hike, carrying one heavy backpack. One child carried the backpack for 83 of the hike, and a second carried it for 16 of the hike. For what part of the hike did the third child carry the backpack? Which child carried the backpack for the greatest part of the hike? 3 8 9. 16 1324 , so the third child carried it 11 24 of the hike, the greatest part. Jody mows 18 of the yard and then takes a break. After the break, Jody mows another 13 of the yard and stops for lunch. After lunch, Jody mows another 16 of the yard and then goes swimming. What part of the yard does Jody still have to mow? Answer: 1 8 13 16 3 24 248 244 15 24 5 8 mowed, so there is still 3 8 of the yard to mow. 10. The truck farmer has planted 13 of a field in cherry tomatoes, 15 of the field in large tomatoes, and 27 of the field in popcorn. The farmer plans to plant sweet corn in the remaining part of the field. About what part of the field will be in sweet corn? Reasoning About Numbers and Quantities Test-Bank Items with Answers page 68 35 21 30 86 Planted already is 13 15 27 105 of the field, leaving 105 105 105 sweet corn, about a fifth of the field. 19 105 of the field for 11. Emmy wants to read an assigned book in four days. She reads 23 of the book the first day but only 151 of the book the second day and another 151 of the book on the third day. a. What part of the book must she read the fourth day? b. On which day did Emmy read the most? a. She has read 2 3 2 151 151 10 15 3 , so she must read 1 3 of the book the fourth day. b. She read the greatest part of the book on the first day. 12. Judy has a 10-page paper to write. The first day she writes 1 12 pages, takes a break, and then writes 3 43 pages more. The next day she writes another 2 13 pages. How many pages does she still have to write? She has written 1 12 3 43 2 13 7 127 pages, so she still has 10 7 127 2 125 pages to write. 13. a. On the number line below, show where b. On the same number line, show where 4 7+ 4 7 – 1 2 1 2 would be. Label it A. would be. Label it B. 0 1 2 0B 1A 2 7.2 Multiplying by a Fraction, and 7.3 Dividing by a Fraction 1. Use a drawing to help explain why 3 4 In the finished drawing, the unit has been cut into 47 equal pieces, and the answer part is 35 of them. 2. 35 . Be explicit. 57 is equal to 47 x x x x x x x x x x x x x x x If two numbers are multiplied, their product is greater than (or equal to) each number. (Explain if F.) True False Reasoning About Numbers and Quantities Test-Bank Items with Answers page 69 F If (the factors are positive and) one of the factors is a number less than 1, then the product will be less than the other number. 3. Donny is asked to solve the following problem: If cheese is $1.89 per pound, how much does 0.67 pound cost? Circle the expression that correctly represents the problem. A. 1.89 + 0.67 B. 1.89 – 0.67 C. 0.67 – 1.89 D. 0.67 1.89 E. 1.89 ÷ 0.67 F. 0.67 ÷ 1.89 G. None of A–F D 4. Which of A-E would best locate the point for 0.89 n on the number line (n positive)? A • 5. C D • n B • 65% of n E • C Which of A-E would best locate the point for n ÷ 0.89 on the number line (n positive)? A B E C D • • • • 65% n of n E A B C D E F G 6. 0 1 Which letter is on the most likely place for a. 12 2? b. 58 of 12 ? c. 43 23 d. 1 ÷ a. F b. B 7. a. Shade in 3 4 c. D of 2 3 of d. G 2 3 e. 1 4 13 e. E this rectangle, as though you were acting it out: b. Show exactly where the 2 3 is. It is c. Show exactly where the 43 is. It is Reasoning About Numbers and Quantities 2 3 of 3 4 what? __________ of what? __________ Test-Bank Items with Answers page 70 a. Most common should be, first 2/3 is shaded, then with cutting marks perpendicular to the first marks, 3/4 of the 2/3 double-shaded. Continuing the marks for the 3/4 of the 2/3 cuts the whole rectangle into twelfths, and the double-shaded part is clearly 6/12. b. 2/3 of the whole rectangle. c. 3/4 of the 2/3 part. 8. a. Show 1 ÷ number. 3 8 with a rectangular region as the whole. Write your answer as a mixed b. Then make clear how the fraction part of your answer relates to your drawing. a.-b. Using a rectangular region cut into eighths, mark 3/8, then another 3/8. The remaining part is not enough for another 3/8, but it is 2/3 of another 3/8. So 1 ÷ 3/8 is 2 and 2/3. Be sure to look for evidence for b. 9. The product of a positive number n by another positive number m is always greater than n. True False False If m is less than 1, then the product is less than n. 10. Three-fourths of seven ones is the same amount as seven three-fourths of one. True False 11. Draw a picture that represents A. True 25 3 4 B. 2 5 3 4 Possible answers: 10. A. B. x x x x x x 12. Draw a picture that represents: A. 4 3 25 x x x x B. x x 2 5 4 3 Similar to #11. In each of #11 or #12, point out, if needed, that although the answers are the same, the meanings of the two products are different. You may wish to review “commutative property of multiplication” language. Reasoning About Numbers and Quantities Test-Bank Items with Answers page 71 13. Working on his summer tan, Zonker leaves 45 of his skin exposed. He has only enough lotion to cover half of his exposed skin. What part of his body will be unprotected? Draw a neatly labeled diagram displaying your solution. It should be clear from the drawing that 4/5 was shown first, then 1/2 of that, then added segments to cut the whole region into (10) equal pieces. 4/10, or 2/5, of his body will not be protected. 14. Use these circles to show each division. For each part tell what question is being asked. Tell what your answer is in each case, and show how you obtained it. 7 1 8 ÷ 4 a. 1 b. 4 3 ÷ 8 a. The question is, how many 1/4s are in 7/8. So 7/8 of the circular region should be shown, with 1/4s then being marked off. There will be 3 full 1/4s, and half of another 1/4, in 7/8. b. The question is, how many 3/8s are in 1/4. So 1/4 of the circular region should be shown. There is not a whole 3/8 in that 1/4, only a part of a 3/8. “Ghosting” in the rest of a 3/8 and the 1/8 markings should show that there is 2/3 of a 3/8 in 1/4. Try to detect students who do the calculation but cannot show the meaning in the drawings. 15. Why is division of fractions confusing to people? Give two reasons. Division of fractions does not always “make smaller.” Why a division problem becomes a multiplication is sometimes a mystery, particularly for those who think multiplication “makes bigger.” Which fraction to invert is also confusing, perhaps at a later time. 16. Is one meaning of 1 4 13 the following: How many ones is 14 of 1 3 of 1? If so, how much of a one or how many ones? Yes: 1 12 of a one. 17. Is one meaning of 1 4 13 the following: How many 13 s of 1 are in 14 of 1? If so, how many _____ thirds or how much of a third of 1 are/is in one-fourth of 1? Reasoning About Numbers and Quantities Test-Bank Items with Answers page 72 Yes, there is 3 4 of one-third of 1 in 18. One meaning of 45 59 45 59 16 19 45 59 of 35 17 of 1? False True 35 is: How many 17 True 20. One meaning of of 1. 35 is: How many ones is 17 True 19. One meaning of 1 4 35 17 of 1 are in 45 59 of 1? False True 25 is: How many 17 False. It should read, How many 25 17 25 17 of 1 are in of 1 are in 16 19 16 19 of 25 17 ? of 1? 21. Use excellent number sense in estimating (1201.794 ÷ 24.3%) + 0.0423 gives ____. 4800 (NOT 4800.0423) 22. The stage-coach robbers have a 12-mile head start on the sheriff's posse. But the sheriff's posse has faster horses, so the posse catches up by 43 mile every hour. How many hours will it take the posse to catch the robbers? Show your work. The question becomes, how many 3/4 miles are there in 12 miles? (Each one of those will indicate an hour of catch-up time.) So, 12 ÷ 43 = 16. It will take16 hours to catch up. 23. Tell what 8 87 ÷ 3 means... a. with the repeated subtraction or measurement view: b. with the partitive or equal sharing view: a. how many 3s are in, or make, 8 87 . b. how much is in each share if 8 87 are shared equally among 3. 24. True or False: Division (involving only positive numbers) always gives a smaller answer than the number being divided. (If False, explain.) False. Division by a number less than 1 gives a quotient larger than the dividend. 25. What calculation would solve this story problem? "Cheese was $2.55 a pound. A woman bought a 0.85 pound package of the cheese. How much did she pay?" A. 2.55 + 0.85 B. 2.55 – 0.85 C. 0.85 x 2.55 D. 2.55 ÷ 0.85 C 26. What calculation would solve this story problem? Reasoning About Numbers and Quantities Test-Bank Items with Answers page 73 "George had A. 4 5 + 4 5 of a pie. He ate 2 3 B. 45 – 2 3 of 2 3 what he had. What part of a whole pie did he eat?" C. 2 3 4 5 D. 4 5 ÷ 23 E. None of A-D C 27. What calculation would solve this story problem? "She paid $4.80 for 43 pound of candy. How much does the candy cost, per pound?" 3 4 A. (4.80 ÷ 4) x 4 B. 4.80 x D. 4.80 + 0.75 E. None of A-D C. 4.80 ÷ 3 4 C 28. Tuna costs 70¢ a can at Store X and 80¢ a can at Store Y. How much will 8 cans of tuna cost, if 14 of the cans are bought at Store X and the rest at Store Y? A. $1.40 B. $4.80 C. $5.60 D. $6.20 E. None of A-D D 29. One day Joe's old car used 58 quarts of oil. The next day it used only 23 as much oil. How many quarts of oil did the car use on the second day? A. 58 23 B. 23 58 C. 58 23 D. 23 58 E. None of A-D B 30. On day 3 the car used 1 14 quarts of oil! But on day 4 it used only 60% as much as it did on day 3. How many quarts of oil did the car use on day 4? 60 A. 1 14 0.60 B. 1 14 0.60 C. 53 1 14 D. 1 14 100 E. None of A-D C 31. How many 23 pound portions can be obtained from 30 pounds? A. 20 B. 45 C. 60 D. 90 E. None of A-D B 32. Following are three students’ solutions to 169 of 48. a. For each student, first evaluate the student’s mathematical reasoning. Check whether the steps are mathematically correct or flawed or indicate that you cannot tell. b. Next mark each student's work with "doesn't appear to understand," "may or may not understand," or "shows good understanding of" multiplication of fractions. 9 1 1 1 Jessica: 16 is 16 more than 2 , and half of 48 is 24 and 16 of 48 is 3, so 27. a. Mathematical steps: Choose one. __ Jessica’s steps are mathematically correct. __ Jessica’s steps are mathematically flawed. __ I cannot tell if Jessica’s steps are mathematically correct or flawed. b. Understanding of multiplication of fractions: Choose one. __ Jessica doesn't appear to understand multiplication of fractions. __ Jessica may or may not understand multiplication of fractions. Reasoning About Numbers and Quantities __ Jessica shows good understanding of multiplication of fractions. Test-Bank Items with Answers page 74 c. If Jessica’s steps are mathematically correct, use her way of thinking to solve If they are not, explain how Jessica’s reasoning is flawed. 3 8 32 . 3 Justin: 9 16 48 9 16 48 1 27 1 27 a. Mathematical steps: Choose one. __ Justin’s steps are mathematically correct. __ Justin’s steps are mathematically flawed. __ I cannot tell if Justin’s steps are mathematically correct or flawed. b. Understanding of multiplication of fractions: Choose one. __ Justin doesn't appear to understand multiplication of fractions. __ Justin may or may not understand multiplication of fractions. 9 __ Justin shows good understanding of multiplication of fractions. 9 Stacy: 16 of 48? 9 16 = 144; 144 ÷ 48 = 3; 9 3 = 27. So 16 of 48 is 27. a. Mathematical steps: Choose one. __ Stacy’s steps are mathematically correct. __ Stacy’s steps are mathematically flawed. __ I cannot tell if Stacy’s steps are mathematically correct or flawed. b. Understanding of multiplication of fractions: Choose one. __ Stacy doesn't appear to understand multiplication of fractions. __ Stacy may or may not understand multiplication of fractions. __ Stacy shows good understanding of multiplication of fractions. Jessica: a) mathematically correct b) Jessica may or may not understand multiplication of fractions, but she does show number sense. c) Possibly, 1/4 of 32 is 8, and half of that is 4, so 12. Justin: a) mathematically correct b) Justin may or may not understand multiplication of fractions. Stacy: a) mathematically flawed b) doesn’t appear to understand multiplication of fractions. 33. Finish the following story problem so that your question could be answered by the calculation 18 ÷ 12 : "You buy 18 muffins for an after-school faculty meeting . . Sample: …They are large, so you cut them into 1/2-muffin servings. How many servings will you get? C A B D E 34. For this number line q 0 p 2 1 Reasoning About Numbers and Quantities Test-Bank Items with Answers page 75 i) the point for p + q would be closest to A? B? C? D? ii) the point for p q would be closest to A? B? C? D? (i) C E? (circle) E? (circle) (ii) A (since both p and q are less than 1) 35. Karen and Sue go bike riding. Karen biked for 3 13 hours, which was 23 as many hours as Sara biked. How many hours did Sara bike? A. 2 29 B. 2 23 C. 4 D. 5 E. None of A-D D 36. After the family had gone 140 miles, the children asked, "Are we getting close?" The driver said, "We've gone only 27 of the way." How far did the family still have to go? A. 40 miles B. 80 miles C. 350 miles D. 490 miles E. None of A-D C 37. A farmer keeps 40% of his land uncultivated. Two–thirds of the cultivated portion is planted with corn. What fraction of the entire land is planted with corn? MAKE A DRAWING. SHOW YOUR WORK CLEARLY AND CIRCLE YOUR ANSWER. 40% uncultivated gives 60% that is cultivated, and 60% = 3/5 makes an easy drawing. 2/3 of the 3 cultivated pieces will give the part of the field planted in corn: 2/5. 38. Alex ate 12 of a pizza. Tandy ate 13 of what was left. Finally, Tabby ate left. How much of the whole pizza did Tabby eat? 1 2 of what was Alex leaves 1/2 of the pizza. Tandy ate 1/3 of that, which is 1/6 of the pizza, leaving 2/6 of the pizza. Tabby ate 1/2 of the 2/6 of pizza, so Tabby ate 1/6 of the pizza. 39. Consider this number line: F H 0 J A K L B1 Points A and B are between 0 and 1 as indicated. Which of the remaining points best represent a. A x B b. A + B c. A – B d. B – A a. H b. K c. F d. J 40. Under a repeated-subtraction interpretation, 43 1 14 means _______________________________________________ Reasoning About Numbers and Quantities Test-Bank Items with Answers page 76 The quotient is ____________. Verify and explain your answer with a sketch. …how many 1 14 s are in, or make, answer, 53 of one 1 14 , is in 43 . 3 4 ? The answer is Reasoning About Numbers and Quantities 3 5 . The sketch should show the Test-Bank Items with Answers page 77 Chapter 8 Multiplicative Comparisons and Multiplicative Reasoning 8.1 Quantitative Analysis of Multiplicative Situations, and 8.2 Fractions in Multiplicative Comparisons 1. A ratio is the result of comparing two quantities to determine how much larger one is than the other. True False False …to determine how many times as large one is, compared to the other. Note for 2-5. Refer to the situation described and fill in the blanks with the appropriate numbers. If not enough information is provided to determine the number that should go in the blank, write “impossible to determine.” 2. At Riverdale Middle School, 18 of the students are in the band. Two out of every three students in the band are girls. a. The number of boys in the band is __________ times the number of girls in the band. b. What fraction of the students who play in the band are boys? c. What fraction of the students at Riverdale are boys who play in the band? d. The number of girls in the band is ________ times the number of students in the school. e. What is the ratio of girls who do not play in the band to the boys who do not play in the band? (A diagram may be helpful for parts C, D, and E.) a. 1/2 b. 1/3 c. 1/24 d. 1/12 (or 2/24) 3. e. impossible to determine Five of every 6 students interviewed favored a change in library hours. a. Among those interviewed, what is the ratio of those who favor a change to those who do not favor a change? _______ b. Among those interviewed, there are ____ times as many students who favor a change as there are students who do not favor a change. c. Among those interviewed, there are ____ times as many students who do not favor a change as there are students who do favor a change. d. What fraction of the students interviewed favor a change in the library hours? _____ Reasoning About Numbers and Quantities Test-Bank Items with Answers page 78 a. 5:1 b. 5 4. c. 1/5 d. 5/6 According to a U.S. News/CNN poll, three out of 10 people went away on vacation in August. a. What is the ratio of those who went away on vacation in August to those who didn't? b. What percent of people did not go away on vacation in August? c. The number of people who did not go on vacation is ___ times the number who did. a. 3:7 b. 70% c. 2 1/3 5. Maxine polled her entire class concerning the date for the next test. Twelve people preferred that the test be given on Wednesday. Twenty-eight voted to have the test on Friday. a. What fraction of the class wanted the test on Wednesday? b. What is the ratio of those who prefer Wednesday to Friday? c. Those who want the test on Friday are how many times as much as those who prefer the test on Wednesday? a. 12/40 (or 3/10) b. 12:28 (someone may give 3:7) c. 2 1/3 6. For the situation below, answer the questions and fill in the blanks. If you cannot answer a question, explain why. If the question asks for an explanation, give an explanation. If the question instructs you to draw a relevant diagram, be sure to label it. Ms. Collin's class has 15 girls and 18 boys. 25 of the children have pets. 20 of the children have dogs. a. The ratio of boys to girls is: ___ . b. The girls make up what fractional part of the class? ___ c. The number of boys is ___ times the number of girls. d. The number of girls is ___ times the number of boys. e. The ratio of the number of children who have pets to the number of children who do not have pets is ___ Reasoning About Numbers and Quantities Test-Bank Items with Answers page 79 f. The number of children who have dogs is what fractional part of the number of children who have pets? ___ g. Of the children who have pets, the number of children who do not have dogs is what fractional part of the number of children who have dogs? ___ Draw a relevant diagram and label it. h. The number of girls who have dogs is ___? i If I said 8 of the children have cats, is this inconsistent with the information that I have already given you? Explain. a. 18:15 (possibly 6:5) b. 15/33, or 5/11 c. 1 1/5 d. 5/6 e. 25:8 f. 20/25, or 4/5 g. 1/4 (look for diagram) h. impossible to determine i. No, some children may have both a dog and a cat as pets. 7. Two landscapers mowed the lawn of a wealthy family. When they finished, landscaper A had mowed only 73 as much as the more experienced landscaper B mowed. A B a. Mark on the drawing of the lawn to show how the mowing might have been done. b. A's part is _____ times as much as B's part. c. A's part is what part of the lawn? d. What is the ratio of A's part to B's part? e. If they are paid $150 for mowing the lawn, what would be a fair split of the $150? a. The region should be marked into 10 equal pieces, with 3 labeled for A and the others for B. b. 3/7 c. 3/10 d. 3:7 e. $45 for A; $105 for B ($150 for the 10 pieces is a rate of $15 per piece) 8. If town A had 12,000 more people than it does, its population would be 1 23 times as big as town B is now. Town B has 45,000 people currently. What is the current population of town A? 63 000, The larger A will have 1 23 45000 75000 people, so the current A has 75000 – 12000 = 63000 people. Alternatively, with the additional 12 000, A:B = 5:3, so for every 3 people B would have, A would have 5. Since B has 45 000 = 315000 people, the larger A would have 515000 or 75 000 people. The current A has 75 000 – 12 000 = 63 000 people. Reasoning About Numbers and Quantities Test-Bank Items with Answers page 80 9. Alan and Bob started mowing a rectangular lawn. After mowing 43 of the whole lawn they got tired and stopped. When they stopped Alan had mowed 27 as much as Bob had mowed. a. Mark the diagram of the lawn below to show how much of the lawn each boy mowed and label the parts with each boy’s initial to indicate clearly the parts mowed by each. b. The area of lawn that Bob mowed is ______ times as large as the area that Alan mowed. c. Alan mowed ________ of the total lawn area. d. The ratio of the area of lawn Bob mowed to the area Alan mowed is ________. e. Together they were paid $12 for the work they did. How much money did each boy get if they were paid proportionally to the amount they worked? Alan ___ Bob___ a. 3/4 of the lawn should be shaded. Then just that portion should be cut into 9 equal parts, since the A:B ratio 2:7 says 2 parts for Alan for every 7 parts for Bob. So 2 of those parts should be labeled A and 7 B. b. 3 1/2 (either from the drawing, or from B:A = 7:2) c. 1/6 (more marks show 6/36) d. 7:2 e. Alan: $2.67, Bob: $9.33 (from $12 for 9 pieces is a rate of $1 1/3 per piece. $2 2/3 and $9 1/3 would be exact but do not fit money) 10. Tim worked 30 hours last week, which was a. How many hours did Robert work? 5 3 times as many hours as Robert worked. b. Who worked more hours? c. Which type of comparison does the question in part (b) address, additive or multiplicative? a. 18 (a drawing should make clear the 5:3 comparison; Tim’s 30 hours means each of his 5 pieces could be thought of as 6 hours, so Robert’s time is 18 hours.) b. Tim (without part A, just from the given sentence) c. Either. If a follow-up question had been “How many more?” that would have been an additive comparison Reasoning About Numbers and Quantities Test-Bank Items with Answers page 81 11. Below is a diagram of a candy bar that is being shared between two people: a. Cut the bar into two parts so that part A is 43 of part B. b. Part B is times as large as part A. c. Part B is how much of the bar? d. What is the ratio of part B to part A? a. Cutting the bar to reflect the 3:4 ratio gives 7 equal pieces (A 3, B 4) b. 1 1/3 (either from the drawing or from the B:A = 4:3 relationship) d. 4/7 c. 4:3 12. a. Cut the rectangle into two regions, A and B, so that A is b. What fraction of the whole rectangle is A? 3 2 as large as B. a. Cut the rectangle into 5 equal pieces: A is three of the pieces, B is 2 of the pieces. b. A is 3/5 of the entire rectangle. 13. 4 5 of an amount is 40. a. How many parts is the whole split into? b. How many parts is the 40 split into? c. How much is in each part? d. What is the full amount? a. 50 b. 40 c. 1/50 d. 50/50 or 1. Reasoning About Numbers and Quantities Test-Bank Items with Answers page 82 Pat and Ron split a cake. Pat's share is 2/3 as large as Ron's share. 12. a. Sketch fairly accurately on the "cake" to show Pat's and Ron's shares. Label them P and R. b. Ron's share is ____ times as large as Pat's share. c. Pat's share is what fractional part of the whole cake? _____ d. What is the ratio of Ron's share to Pat's share? ______ a. 5 pieces, with 2 for P and 3 for R. b. 1 1/2 c. 2/5 d. 3:2 13. John baked a cake for his son’s birthday party. At the party, was eaten. (A diagram may be helpful.) 2 7 as much was left over as a. The remaining portion is what fractional part of the whole cake? b. What is the ratio of the amount of cake eaten to the whole cake? The diagram should have 9 equal pieces, to reflect the L:E = 2:7 ratio. a. 2/9 b. 7:9 14. Two painters paint a wall from opposite ends. When they finish, painter P has painted only 23 as much as the more experienced painter Q painted. P Q a. Mark on the drawing of the wall to show how the painting might have been done. b. Q's part is ___ times as much as P's part. c. Q's part is what part of the wall? ___ d. What is the ratio of P's part to Q's part? ___ Reasoning About Numbers and Quantities Test-Bank Items with Answers page 83 e. If they are paid $100 for painting the wall, what would be a fair split of the $100? a. 5 equal pieces, 2 for P and 3 for Q b. 1 1/2 c. 3/5 d. 2:3 e. P $40; Q $60 (each of the 5 pieces would be worth $20) 15. Consider the region below as a pizza that is being shared by two people. a. Divide the pizza into two parts so that part A is 53 of part B. b. Part A is how much of the pizza? c. What is the ratio of part B to part A? a. 8 equal pieces, 3 for A and 5 for B b. 3/8 c. 5:3 Reasoning About Numbers and Quantities Test-Bank Items with Answers page 84 Chapter 9 Ratios, Rates, Proportions, and Percents 9. 1 Ratio as a Measure, and 9.2 Using Proportions to Compare Ratios and Solve for Missing Values 1. Any driveway ramp that is 3.205 ft high is less steep than any other ramp that is 3.98 ft high. True False False The steepness also depends on another dimension, usually the horizontal length of the driveway. 2. One recipe for 8 people requires 5 cups of flour. Al expects only 6 people and so wants to cut the recipe so that it will serve 6. a. If Al is thinking additively, how much flour will he use? Explain your answer. b. If Al is thinking multiplicatively, how much flour will he use? Explain your answer. c. Which way is better, Al's or your part B method? Explain. a. If Al is using additive thinking: 8 – 6 = 2, so he should use 5 – 2 = 3 cups of flour. b. The recipe calls for 5/8 cup of flour per person, so for 6 people Al should use 6(5/8), or 3 3/4, cups of flour. c. The part B method is better, because to taste the same, recipes should use the same proportions of ingredients. Rates allow that. 3. Ruben and Ofilia are painting the walls of a large lecture hall. They mixed 2 gallons of blue paint with 5 gallon of white paint for a total of 7 gallons of paint. They ran out of paint. They estimated that they needed one half gallon to finish. Find the portion of the room which has been painted. a. Find the portion of the room which has been painted. Draw a neatly labeled diagram displaying your solution. b. If they mix 1/4 gallon of blue paint with 1 gallon of white paint, will the paint colors match? Explain your answer briefly. c. Suggest what amounts of blue paint and white paint would finish the room and match the color, perhaps with a little left over for touch-ups. a. The diagram should show the walls (probably one rectangle), marked into 15 equal pieces, since the 7 gallons is 14 half-gallons. They have painted 14/15 of the room with the 7 gallons. b. No. The original rate was 2:5 which is the same as 2/5:1. But if 1 gallon of white paint is used with ¼ gallon of blue paint, the ratio is 1/4:1 or 1:4 which is not equivalent to 2/5:1. Reasoning About Numbers and Quantities Test-Bank Items with Answers page 85 c. We need a ratio a:b where a:b is equivalent to 2:5 and where a+b = 1/2. ¼ gallon blue paint and 5/8 gallon of white paint would provide the correct ratio and the sum is slightly more than ½, but close enough for this situation. 4. María was asked to draw an isosceles triangle (a triangle with two equal sides) as part of a homework assignment. She had a straight edge to draw straight lines, but didn't have a ruler to measure the length of her sides. Thus, all she could do was eyeball the correct lengths. She decided to draw more than one triangle so that she could go to school a little early, get a ruler, measure all of her triangles, and select the "best" isosceles triangle to turn in to the teacher. Suppose María drew the following triangles: 1.1 in 1.2 in 1.6 in 1.7 in 2.0 in 2.2 in a. Which triangle should she select to turn in to the teacher? Why? b. Suppose María drew 20 triangles like the ones shown above, but of various sizes. Once she measures the lengths of the sides, how should she go about selecting which triangle is the "best" isosceles triangle? a. Actually none of the triangles is isosceles. The best-looking one will have its ratio of designated sides closest to 1. So it is a matter of deciding which of 1.1:1.2, 1.6:1.7, and 2.0:2.2 is closest to one. Using fraction versions makes the comparison fairly easy: 11/12, 16/17, 20/22 = 10/11. Fraction considerations show that 16/17 is closest to 1, so the triangle with sides 1.6 in. and 1.7 in. will be the best-looking one. (Remark: As with the squareness measure, students may insist on focusing on the additive comparison. An example of a triangle with sides 0.1 in. and 0.2 in. may convince them that an additive comparison is not the best way to tell.) b. She should find the one with the ratio of the supposedly equal-length sides closest to 1. 5. There are four water slides at the Six Flags Atlantis water park. The following are the measurements of the four water slides. Circle the letter of the steepest slide. A. Length: 100 ft.; height: 81 ft. B. Length: 60 ft.; height: 45 ft. C. Length: 80 ft.; height: 70 ft. D. Length: 10 ft.; height: 7 ft. Reasoning About Numbers and Quantities Test-Bank Items with Answers page 86 C. Steepness can be measured by the height:length ratio. Using the fraction forms and number sense, 70:80 is the largest of the four ratios. 6. a. 3 bags of tea are used with 4 cups of water in one teapot, and 10 bags of tea with 12 cups of water in another teapot. Which pot will have the stronger tea, or will the two have the same strength? Explain your answer. b. 5 scoops of coffee are used with 8 cups of water in one coffee pot, while 7 scoops of coffee are used with 10 cups of water in another coffee pot. Which pot will have the stronger coffee? Explain your answer. c. The two problems above were given to a sixth grade class. Which one do you think was more difficult, and why? a. The 10 bags in 12 cups will be stronger, since the first mixture has 3/4 bag per cup of water and the second 10/12, or 5/6, bag per cup of water, and 5/6 > 3/4. b. The 7 scoops, 10 cups of water coffee will be stronger, because 7/10 scoop per cup of water gives more coffee taste than 5/8 scoop per cup of water will. c. Part B almost certainly would be more difficult for sixth graders, because the additive comparisons for the two recipes are the same, a difference of 3. And, there is not an easy relationship between the quantities in part B, as there is in part A: 12 cups is 3 times as much water as 4 cups, so repeating the first recipe 3 times would mean 9 bags for 12 cups, a weaker tea than with the 10 bags for 12 cups. 7. Two painters on a large project want to paint different areas the same color. Painter A mixes 3 quarts of red paint with 2 gallons of white paint, and Painter B mixes 5 quarts of the same kind of red paint with 4 gallons of white paint. Painter A says the two mixtures will be the same color and Painter B says his mixture will be redder than Painter A’s. Explain the thinking of each one. Which one, if either, is correct? Explain your decision. Painter A is using an additive comparison; painter B is looking at the larger amount of red paint in his mixture. Neither reasoning is correct, since it is the ratio of red to white that is important. For A, R:W = 3:2 = (1 1/2):1, for B, R:W = 5:4 = (1 1/4):1, so A’s mixture will be redder. 8. A friend needs a serious operation and is considering two hospitals that have the following records for the surgery: Successes Failures Hospital A 20 8 Hospital B 50 18 The friend sees that B has more successes, but A has fewer failures, and asks you for help in deciding. What will you say? Reasoning About Numbers and Quantities Test-Bank Items with Answers page 87 (Assume that in either case the number of operations attempted is enough to give the staff sufficient experience.) There are different acceptable approaches, but each depends on the multiplicative comparison in some form. One way is to check the rate of # of failures per success—Hospital A, F:S = 8:20 = 0.4:1; Hospital B, F:S = 18:50 = 0.36:1, so B looks slightly better, having a lower failure to success rate. Or one could compare the # successes: # failures rates. Or one could calculate the percents of successes in all the cases (A: 20/28 71%; B: 50/68 74%). 9. Terry uses 1 cup of Mr. Spiffy in 1 12 gallons of water to clean the kitchen floor. What percent of the cleaning solution is Mr. Spiffy? (Note: 16 cups = 1 gallon) SHOW YOUR WORK CLEARLY AND CIRCLE YOUR ANSWER. The 1 1/2 gallons, or 24 cups, of water, plus the cup of Mr. Spiffy gives 25 cups of cleaning solution. 1:25 or 1/25 = 4% 10. Make a sketch, and give your explanation, to illustrate this situation: After a bake sale, there were 3 identical white cakes left. You and your friend split them, but you took only 14 as much as your friend took. Label your part M (for “me”) and your friend’s part F. How much of a cake (or how many cakes) did your friend take? (Give the numerical answer that makes sense here, and add an explanation if the numerical answer is not clear from your sketch.) Rectangular cakes may be easier to deal with. In any case, the multiplicative comparison of the two shares is 1:4, so the cakes should be cut into 5 equal amounts, with 1 part for M and 4 parts for F. This cutting can be done in two ways. The first is easier—cut each cake into 5 equal pieces. The second is slightly harder to sketch and then to name the fractions—cut the total of the 3 cakes into 5 equal pieces. In either case, F gets 12/5, or 2 2/5, cakes (12/5 makes immediate sense with the first method; 2 2/5 may be more natural for the second method). 11. Make a drawing and give an explanation to illustrate and answer this situation: Karen and Sara both jog. Sara jogged 2 12 miles, which is many miles did Karen jog? 2 3 as far as Karen jogged. How Each 1/3 of Karen’s distance is 1 1/4 miles, so Karen distance was 3 3/4 miles. Here is one sketch (others are possible, of course): 2 1 2 mi. S's jog K's jog Reasoning About Numbers and Quantities Test-Bank Items with Answers page 88 12. If the Browns were to save an additional $14,000, they would have 1 25 times as much money as the amount the Jones have in their savings account. The Jones have $65,000. How much do the Browns currently have in their savings account? SHOW YOUR WORK CLEARLY AND CIRCLE YOUR ANSWER. 1 2/5 times as much as $65,000 = $91,000. So the Browns must have $91,000 – $14,000 = $77,000 in their savings account 13. A group of 37 persons goes to a holiday camp for 35 days. They need to buy enough sugar for the trip. They read that the average consumption of sugar is 2.2 kg per week for 10 persons. How much sugar do they need? SHOW YOUR WORK CLEARLY AND CIRCLE YOUR ANSWER. One way: The 35 days is 5 weeks, so they will need 5 2.2 = 11 kg for every 10 persons for the whole period. Since there are 37 persons, they will need 3.7 11 = 40.7 kg of sugar for the camp. 14. The pollster noticed that for every 40 men who were in favor of X, there were 28 women who were in favor of X. According to these figures, if 280 men were in favor of X, how many women were in favor of X? A. 400 B. 196 C. 168 D. 40 E. None of A-D B 15. A machine can make 700 bolts in 40 minutes. At that rate, how many bolts can the machine make in one hour? A. 900 B. 1050 C. 2800 D. 28,000 E. None of A-D B 16. Miguel runs 200 meters in 40 seconds; Paul runs 150 meters in 12 minute. Who runs faster? A. Miguel B. Paul C. They run at the same speed. D. More information is needed. C (Each runs 300 m/min., or 5 m/s.) 17. John drives 7 103 miles to campus each day, while Vaneta drives only 4 miles to campus. John drives how many times as far as Vaneta? (7 3/10):4 translates into 1 33/40 times as far. 18. Your answer book says that the probability of Box 1 BBW drawing a B ball from Box 1 (to the right) is the same Box 2 BBBBBBWWW as for Box 2. A child says, "They say 2 to 1 gives the same probability as 6 to 3 does, but I don't see it. I think Box 2 has a lot more chances of giving a B." Reasoning About Numbers and Quantities Test-Bank Items with Answers page 89 How would you try to convince the child that 2:1 is indeed equal to 6:3, using the balls in the boxes (not just symbolically with numbers)? In Box 2, draw rings to show three BBW groups. (Re-drawing so that the Ws are under the Bs makes that easier.) 19. A child says that the two situations below would give the same "chocolatey-ness," since "Each way has one more spoonful of chocolate sprinkles." Situation 1 3 spoonfuls of chocolate sprinkles on 2 scoops of vanilla ice cream. Situation 2 4 spoonfuls of chocolate sprinkles on 3 scoops of vanilla ice cream. Give a drawing (not just calculations) that should help the child see that the "chocolateyness" in the two situations would be different. (Use words as needed.) Draw to show that in Situation 1, each scoop gets 1 1/2 spoonfuls, and in Situation 2, each scoop gets 1 1/3 spoonfuls. 20. A store sells 48 pecan pies for every 15 banana pies it sells. At this rate, if the store sells 60 banana pies, how many pecan pies will it sell? A. 12 B. 93 C. 120 D. 192 E. None of A-D D 21. The pollster noticed that for every 8 men who were in favor of a ballot measure, there were 5 women who were in favor of the measure. According to these figures, if 1200 women were in favor of the measure, how many men were in favor of the measure? A. 750 B. 1197 C. 1920 D. 2100 E. None of A-D 22. Clarissa runs 240 meters in 45 seconds; Doña runs 160 meters in faster? A. Clarissa B. Doña 1 2 C minute. Who runs C. They run at the same speed. D. More information is needed. A. Clarissa: She runs 360 meters in one minute; Dona runs 320 meters in one minute. 23. Use a three-column table to solve this problem. Label the columns Picture Frames, Cost, and Notes. If a box of 36 picture frames (all the same kind) cost $86.40, how much would 5 frames cost? (Assume no sale price, discount for volume, sales tax, etc.) One possible answer. Picture Frames Cost Reasoning About Numbers and Quantities Notes Test-Bank Items with Answers page 90 36 $86.40 given 18 $43.20 halved each 9 $21.60 halved each 3 $ 7.20 found a third of each 1 $ 2.40 found a third of each 5 $12.00 5 (2 + 0.40) = $10 + $2 =$12.00 25. Felicia can run a mile in 4.7 minutes. How long would it take her to run 8 miles? This is not a proportion problem because she cannot conceivably continue running at such a speed for 8 miles. 9.3 Percents in Comparisons and Changes 1. 15% of a given amount is the same as 10% of the amount plus 10% of twice the amount. True False False ….the same as 10% of the amount plus 10% of half the amount. 2. Terry uses 1 cups of Mr. Spiffy in 1 12 gallons of water to clean the kitchen floor. What percent of the cleaning solution is Mr. Spiffy? (Note: 16 cups = 1 gallon) The 1 1/2 gallons, or 24 cups, of water, plus the cup of Mr. Spiffy gives 25 cups of cleaning solution. 1:25 or 1/25 = 4% (This item is repeated from earlier.) 3. What does "the sales tax rate is 7 43 %" mean? The tax will be $7.75 for every $100 of goods (or 7 3/4 ¢ for every 1 dollar of goods). 4. You receive 12 points out of 20 on the first quiz, 15 points out of 20 on the second quiz and 60 points out of 75 on the first midterm. a. On which quiz or test did you perform best? Show your work. b. How would you find your average score? c. Estimate what percent that average represents. a. 12/20 = 60%, 15/20 = 75%, 60/75 = 80%, so you did best on the midterm. Reasoning About Numbers and Quantities Test-Bank Items with Answers page 91 b-c. Assuming that all points are weighted equally, add the total number of points received, and compare that to the total number of points possible: (12+15+60)/(20+20+75) = 87/115 75.6% (via estimation: 87/115 87/111 = (9x87)/999 783/1000 = 78.3%). In contrast, assuming that a quiz has equal weight as a midterm (unlikely), (60%+75%+80%)/3 71.7% (via estimation: 215%÷3 71%). You might wish to ask your students why that second average is lower (equating the weights makes the 60% quiz performance more influential than if the average were based on equal values for each point). 5. Kathee owed her dad $80 and then paid off $24 from her tip earnings. a. What percent of the original debt had she paid? b. What percent was still owed? c. What was her new debt after the payment? The following evening she paid him an additional $20 from her tip earnings. d. What percent of her new debt was paid off? e. What was her debt after the last payment? f. What percent of her original debt is now paid off? a. 30% b. 70% c. $56 d. 35.7% e. $36 f. 55% 6. If this box represents 75% of something, modify the box so that it represents 125% of the same thing. Divide the box into 3 equal parts, then add an additional two parts of the same size. 7. The median cost of housing rose 25% in one city in a particular year. The median price was then $360,000. What was the median cost of housing at the end of the previous year? About $288,000. 8. Gala apples are on sale today, at $1.30 per pound. This is a discount of 30%. What was the cost before the sale? About $1.86 9. The Nasdaq closed at 1,590 today, off by 0.09%. What the Nasdaq at yesterday’s closing? About 1591.4 (the main idea is to catch incorrect thinking or poor number sense) Reasoning About Numbers and Quantities Test-Bank Items with Answers page 92 10. Estimate each of the following and explain how you arrived at your estimate: a. 79% of $119 b. 0.5% of 89 kilograms c. 31% of 21,343 voters d. 121% of $29 Here are some possible ways to estimate: a. About 80 % of $120: 10% is $12; 8 x $12 is $96 b. 1% is 0.9, half of that is 0.45 kilograms c. About a third of 21,000, so 7000 voters d. About $30 + $3 + $3 is about $36 Reasoning About Numbers and Quantities Test-Bank Items with Answers page 93 Chapter 10 What’s to the Left of 0? 10.1 Adding and Subtracting Signed Numbers 1. Complete the following. For each subtraction problem, first rewrite it as an addition problem. a. – 7 +– 8 = b. 3 + –5 = c 16 + –14 = d. –21 + –2 = e. 17 – –3 = f. –2 – –5 = g. –13 – 9 = h. 14 – –16 = k. –2.6 – 4.5 l. –3.17 – 2.4 c. 2 d. –23 i. 5 4 + – 12 j. a. –15 b. –2 e. 17 – –3 = 17 + 3 = 20 f. –2 – –5 = –2 + 5 = 3 g. –13 – 9 = –13+ –9 = – 22 h. 14 – –16 = 14 + 16 = 30 i. 2. – 11 11 17 + ( 17 ) 3 4 k. –2.6 + –4.5 = –7.1 j. 0 l. –3.17 + –2.4 = –5.57 Reorder these numbers from smallest to largest: 7.4% 21 7 –1 5 – 4 11 6 5 4 – 11 6 –1.5 0.2 –0.2 –1.5 2.8 7.4% 0.2 –0.2 5 4 –1 5 4 2.8 21 7 3. Complete this “fact family” table: 3 + –5 = – 2 4. 3 + –5 = – 2 –2 – –5 = 3 –5 + 3 = –2 –2 – 3 = –5 Is the statement a + b =/a/ – /b/ always true, sometimes true, or never true? Explain your answer. Sometimes true. If a is positive and b is negative and |a| > |b| then a + b =/a/ – /b/ but if a is positive and b is negative a + b = – ( /b/ – /a/). Reasoning About Numbers and Quantities Test-Bank Items with Answers page 94 5. Explain how one could use white (positive) and dark (negative) chips to model the following a. 4 + (–6) b. 5 – (–2) a. b. Adding two white and two dark does not change the value in the first box. Then remove two darks to represent subtracting –2. This leaves seven whites, or +7. 4. Which properties does each of the following involve? a. (-2 + 3) + -5 = (3 + -2) + -5 b. (-2 + 3) + -5 = -5 + (3 + -2) c. (-2 + 3) + -5 = -2 +( 3 + -5) a and b. Commutativity of addition c. Associativity of addition. 10.3 Multiplying and Dividing Rational Numbers 1. Does 0 have a multiplicative inverse? Explain. No, 0 does not have a multiplicative inverse. There would have to be a number n such that 0 n = 1, but 0 n always = 0. So no number would work as the multiplicative inverse of 0. 2. Complete the following: a. – 7 – 8 = b. 3 –5 = c 16 –2 = d. –21 –2 = e. 18 ÷ –3 = f. –2 ÷ –5 = g. –18 ÷ 9 = h. 4 ÷ –16 = k. –8.6 ÷ 4.3 l. –1.2 ÷ –2.4 i. 5 4 ÷ – 12 j. 11 17 (– 17 11 ) Reasoning About Numbers and Quantities Test-Bank Items with Answers page 95 2. a. 56 b. –15 c. –32 d. 42 e. – 6 f. 2 5 g. –2 h. – 14 i. –2 12 j. –1 k. –2 l. 1 2 Continue the following pattern of multiplication of integers with six more products in the pattern. The multipliers decrease by _____ each time. The multiplicand is always -4. The product ____creases by _______ each time. For this pattern to continue working, the product of two negative integers must be _____________________________ 3 –4 = –12 2 –4 = –8. . . . . . . . 1 –4 = –4 0 –4 = 0 –1 –4 = 4 –2 –4 = 8 –3 –4 = 12 –4 –4 = 16. The multipliers decrease by __1___ each time. The multiplicand is always –4. The product __in__ creases by __4_____ each time. For this pattern to continue working, the product of two negative integers must be ___positive_________________ 3. Give the exact answer to each of the following, taking advantage of properties. Then tell which property or properties enabled you to answer them so easily. a. If 213 five 142 five = 41401 five, then 142 five 213 five = ______five. Property(ies): b. 179 196 0 343 Property(ies): c. 84.96 100% = _________ Property(ies): d. (548 + –967) + –548 = _______ Property(ies): a. 41401five b. 179/196 c. 84.96 d. –967 3. ________ Commutative property of multiplication Identity property of addition Identity property of multiplication Commutative property of addition, associative property of addition, additive inverse property, identity property of addition Match the operations and the names of properties by placing the correct number to the left of the letters A-E. (Not all properties on the right will necessarily be used; some may be used more than once.) __a. –3 (2 + 5) = –3 (5 + 2) 1. associative property of multiplication __b. 3 (2 + 5) = (3 2) + (3 5) 2. additive identity property Reasoning About Numbers and Quantities Test-Bank Items with Answers page 96 __c. 4 + (–3 + –1) + 2 = (4 + –3) +( –1 + 2) 3. multiplicative inverse property __d. 5 + ( 8 + 0) = 5 + 8 4. additive inverse property __e -4 1 = -4 5. commutative property of addition __f. 6 + (4 + –4) = 6 + 0 __g. 23 23 1 6. associative property of addition 7. distributive property of over + __h. 3 (5 0) = (5 0) 3 8. multiplicative identity property __i. 3 + (2 + –5) = (3 + 2) + –5 a. 5 b. 7 c. 6 d. 2 e. 8 9. commutative property of multiplication f. 4 g. 3 h. 9 i. 6 10.4 Other Number Systems 1. Consider clock arithmetic using a clock with four numbers: 0, 1, 2, and 3. a. Complete these tables: + 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 b. Do you think the set of numbers, 0, 1, 2, 3, is closed under addition? If not, provide an example that shows it is not. c. Do you think the set of numbers, 0, 1, 2, 3, is closed under multiplication? If not, provide an example that shows it is not. d. Is there an additive identity? If so, what is it? e. Is there a multiplicative identity? If so, what is it? f. Does 3 have an additive inverse? Is so, what is it? g. Does 2 have a multiplicative inverse? If so, what is it? h. Do you think addition commutative? Is so, provide an example. i. Do you think multiplication is commutative? If so, provide an example. j. Do you think addition associative? Is so, provide an example. Reasoning About Numbers and Quantities Test-Bank Items with Answers page 97 k. Do you think multiplication is associative? If so, provide an example. l. Do you think multiplication is distributive over addition? If so, provide an example. a. + 0 1 2 3 0 1 2 3 0 0 1 2 3 0 0 0 0 0 1 1 2 3 0 1 0 1 2 3 2 2 3 0 1 2 0 2 0 2 3 3 0 1 2 3 0 3 2 1 b. The set is closed under addition. All sums are 0, 1, 2, or 3. c. The set is closed under multiplication. All products are 0, 1, 2, or 3. d. 0 is the additive identity: any number plus 0 is that number. e. 1 is the multiplicative identity: any number times 1 is that number. f. Yes. The additive inverse of 3 is 1: 3 + 1 = 0, which is the additive identity. g. No. There is no number which, if multiplied by 2, is 0. (All products with 2 as a factor are either 0 or 2.) h. Yes, addition is commutative. Example: 1 + 3 = 0 and 3 + 1 = 0 i. Yes, multiplication is commutative. Example: 2 3 = 2 and 3 2 = 2 j. Yes, addition is associative. Example: 2 + (3 + 1) = (2 + 3) + 1 because the left side is 2 + 0 = 2 and the right side is 1 + 1 = 2. k. Yes, multiplication is associative. Example: 2 (3 1) = (2 3) 1 because the left side is 2 3 = 2 and the right side is 2 1 = 2. l. Yes, multiplication is distributive over addition. Example: 3 (2 + 3) = (3 2) + (3 3) because 3 1 = 3 and 2 + 1 = 3. Reasoning About Numbers and Quantities Test-Bank Items with Answers page 98 Chapter 11 Number Theory 11.1 Factors and Multiples, Primes and Composites, and 11.2 Prime Factorization 1. The following sounds all right, but it is not always true. Give a counterexample. "Suppose that k is not a factor of m and k is not a factor of n. Then k is not a factor of m+n." Counterexample: Many possibilities. Sample: 3 is not a factor 4 and 3 is not a factor of 2. But 3 is a factor of 4+2 = 6. 2. What, if anything, can you say about the oddness or evenness of m… a. when 5063338 m is an even number b. when 5063338 + m is an even number a. Nothing can be said about m (the product will be even no matter whether m is even or odd). b. m must be even. 3. T F 5 is a multiple of 0. (Explain if F.) F All multiples of 0 equal 0, since m0 = 0. 4. Say the same thing as the following sentence, but use the word "multiple." 360 is a factor of N. N is a multiple of 360. 5. Rephrasing: If n = 43759462138999999249 + 764321572, then is n an even number, or is n an odd number? Explain your answer. n is even. Since the square of an odd number is odd, the sum of the odd and odd will be even 6. 7. Circle T if the statement is true, F otherwise. T F Every whole number is a multiple of itself. T (m = 1m) T F It is possible for an even number to have an odd factor. T (e.g., 12) T F Zero is a multiple of every whole number. T (0 = 0m) T F 250 is a factor of 10030. T ( 100 22 52 , so 10030 260 560 ) Does 0 have any factors? Explain your answer. Reasoning About Numbers and Quantities _________ because Test-Bank Items with Answers page 99 Yes, every number m is a factor of 0, since 0 = 0m. 8. 9. Circle T if the statement is true, F otherwise. T F Every whole number is a factor of itself. T T F It is possible for an odd number to have an even factor. F T F Zero is a factor of every whole number. T F 520 is a factor of 5012. F (0 is a factor only of 0) T ( 50 2 52 , so 5012 212 524 ) Of what numbers, if any, is 0 a multiple? Explain your answer. 0 is a multiple of every number m, since 0m = 0 10. Suppose 7 is not a factor of n. Can 21 be a factor of n? If 21 can be a factor of n, give an example for n. If 21 cannot be a factor of n, give an explanation from basic principles. No. If 21 were a factor of x, x = (some #)21. But 21 = 37, so then x would = (some #)37 and 7 would have to be a factor of x. 11. Determine whether m and n are primes. Write only enough to make your decisions clear. a. m = 23 29 (= 667) ______________ because b. n = 133 ____________ because a. Not a prime (23 or 29 is a third factor) b. Prime, because none of 2, 3, 5, 7, 11, 13, etc., is a factor (Instructor: If you have introduced the n bound for testing of primes, your student should stop with 11.) 12. Is 245 a prime number? Explain. ______ because No, 5 is a third factor. 13. T F There are no values of b and c for which 27b = 9c. (Explain, if F.) F 27b (33 )b 33b and 9 c (32 )c 32c and all that is required is that 3b = 2c, for which there are infinitely many solutions—e.g., b=2, c=3; b=4, c=6; b=6, c=9;… 14. T F There are no values of r and s for which 11r = 9s. (Explain, if F.) T (unique factorization into primes) 15. Is there a whole number M which would make this true? If so, tell what M is. If not, tell why not. Reasoning About Numbers and Quantities Test-Bank Items with Answers page 100 a. 35.52.173 = 34.174.M b. 24.72.118.22 = 25.7.116.M It may be a good idea to cite the relevance of the unique factorization into primes result. a. Not possible because there are already too many 17s on the right-hand side. b. 7 113 (Notice that the left-hand side is not in its prime factorization form.) 16. Give the prime factorization of n, where n = 4 x 720 x 5000. If it is not possible, explain why not. 25 5 4 720 17. Is it possible to find a non-zero whole number m so that 14m = 260 759 ? ____ Explain. No. Since 14 = 27, 14 to any power will have the same number of 2s as 7s. 18. KNOWLEDGE OF NUMBER THEORY QUESTION; NO CALCULATORS. Without calculation, explain why Romeo and Juliet can or cannot both be correct, when they are talking about the same large number: Romeo: "The number is 7 11 172 37 67 97." Juliet: "The number is 3 11 212 37 67 89." ____________ because No, Juliet’s number has 3 as a factor, but Romeo’s does not. It may be a good idea to cite the relevance of the unique factorization into primes result. 19. It is correct that 3721164 = 12 172 29 37. Give the prime factorization of 372116400 (notice the extra two zeros). Hint: Do not work too hard. 2 4 3 52 172 29 37 (Students may overlook the 12 in the original factorization; despite the hint, some may not recognize that the target number is just 100 times the original one.) 20. Is this all right? If it is, explain why. If it is not, give a counterexample. "If a number has n factors (n > 1), then the square of the number has 2n factors." No. Students should have no trouble finding a counterexample (the item is just testing ability to read an if-then and know what a counterexample is). Reasoning About Numbers and Quantities Test-Bank Items with Answers page 101 21. When you were a spy, two of your paid informants gave you the following information about the same secret code-number: Informant 1: "The code-number is 33 70 some odd number." Informant 2: "The code-number is 35 66 some even number." What can you tell from your informants' information? Their information is inconsistent. Although the known numbers do give the same prime factorization, Informant 2’s “even number” would involve another factor of 2 that Informant 1’s odd number could not. 22. Fill in the blanks to make a true sentence. If no number or algebraic expression will make the sentence true, say so. a. An example of a number which has an odd number of factors is _______. b. If n = 138.1710, then the prime factorization of 26.n is _____________. c. 3 will be a factor of 140000000?000000014 if the missing digit, ?, is __ or __ or __. a. Sample: 4 b. 2 139 1710 c. 2 or 5 or 8 23. THEORY QUESTION; NO CALCULATORS. Is it possible, for some choice of positive whole numbers m and n, that 45m = 15n? Justify your answer. No. Comparing 45m 32m 5m , and 15n 3n 5n , shows that m would have to equal n because of the 5s and then the 2m ≠ n (m and n are to be positive). 24. Every two different prime numbers are relatively prime. True False True 25. T F If we write the first 10,000 numbers in 6 columns, as started below, then 9999 would be in the 5th column. Write enough (numbers, words) to make your thinking clear.) 1 2 3 4 5 6 7 8 9 10 11 12 etc. False 10000 ÷ 6 = 1666 R 4, so 10,000 numbers would occupy 1666 full rows and only 4 into the next row. 10,000 would be in the 4th column so 9999 would be in the third column. Another way to think about this is that the numbers in the first column always have the form 6n +1, the second column 6n +2, the third column 6n + 3, the fourth Reasoning About Numbers and Quantities Test-Bank Items with Answers page 102 column 6n + 4, the fifth column 6n + 5, and the sixth column 6n. so 9999 would be in the third column. 9999 = 6 x 1666 + 3 26. Tell the difference between (a) "give a prime factor of 350" vs "give a prime factorization of 350," and the difference between (b) "give a number that has an odd factor" vs "give a number that has an odd number of factors." a. “Give a prime factor of 350” means to identify only one of the prime factors of 350, whereas “give a prime factorization of 350” means to give a product of primes equal to 350. b. “Give a number that has an odd factor” means to find a number that has a factor that is 3, 5, 7, 9, etc., but “give a number that have an odd number of factors” means to find a number such that when you find all of its factors, there are an odd number of them. 27. Say the same thing as the following sentence, but use the word "factor." M is a multiple of 240. Rephrasing: 240 is a factor of M. 28. Put 0 and 2 (one of each) into the blanks to make a true statement, and explain. If it is not possible, explain why. _____ is a multiple of _____ because _________________________ 0, 2, because 0 = 0 2 29 Circle T if the statement is always true, F if it is always false, and D if it depends on the value of a variable. The variable n represents a positive whole number. T F D If 27 is a factor of n, then n is a multiple of 27. T 30. Give the prime factorization of n, where n = 137 x 3000. 2 3 3 5 3 137 31. In each part, find a whole number for m to make the equality true. If it is not possible, explain why. For credit, your work should show an understanding of number theory. a. 52 103 176 = 23 176 m b. 52 76 114 = 5 356 114 m a. m = 5 5 b. Not possible, since there are already more 5s on the right-hand side than appear on the left-hand side. Reasoning About Numbers and Quantities Test-Bank Items with Answers page 103 32. Explain, without extensive calculation, why the following equation can or cannot be correct: 172 192 375 = 184 414 It is _______________________ because (In your explanation the grader will look for a clear reference to a major theoretical result.) It cannot be correct, since 17 appears as a factor on the left-hand side, but not on the right-hand side. Unique factorization into primes says this is impossible. 33. (THEORY QUESTION; NO CALCULATORS) Is it possible, for some choice of positive whole numbers m and n, that 35m = 25n? Justify your decision. No, since 35 m involves factors of 7, which cannot appear in the factorization of 25 n . 34. Name the number of factors of each of these numbers and list them, in factored form. a. 52 173 b. 35 a. 12: 50 170 50 171 50 172 50 173 51 170 51 171 51 172 51 173 52 170 52 171 30 32 b. 6. 31 52 172 33 34 35 52 173 (1, 3, 9, 27, 81, 243) 35. What is the largest prime number that you need to text to check for divisibility of a. 173 b. 982 a. 13 because 13 13 = 169 which is the largest square smaller than 173. 31 because 31 31 = 961 is the largest square less than 982. b. 36. When the number 540 is written as a product of its prime factors in the form a 2b 3c , what is the numerical value of a + b + c ? Choose one of the following: A. 30 B. 5 C. 6 D. 7 C. 11.3 Divisibility Tests to Determine Whether a Number Is Prime 1. Circle the numbers that are prime. If a number is not prime, list at least three factors below the number. 392 5231211 61. 73 Reasoning About Numbers and Quantities 121 Test-Bank Items with Answers 43 page 104 Only 43 is a prime. 121 = 11x11 and the sum of the digits of 5231211 add up to a multiple of 3 2. Circle the numbers below that divide 11220. 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20 All except 8, 9, and 18 divide 11220. 3. a. State a divisibility test for 8. b. Explain why your test in part a will definitely work, using the general 7-"digit" number, abcdefg in your explanation. a. 8 is a factor of n if and only if 8 is a factor of the number named by the rightmost three digits. b. abcdefg (abcd) 1000 efg, and 8 is a factor of 1000 (1000 = 8125), so whether 8 is a factor of abcdefg will depend on whether 8 is a factor of efg. 4. KNOWLEDGE OF NUMBER THEORY QUESTION; NO CALCULATORS a. Circle any which is a factor of 62296715880, which is equal to 23 32 5 7 472 31 192. 15 16 21 75 94 n=19312 217 b. Explain how you know that your answers in part A are correct, even without calculation. a. 15, 21, 94 (=247), and 217 (=731) are the only ones. b. By unique factorization into primes, the given factorization gives those prime factors, and only those prime factors, that are possible for the given number, as well as the number of appearances of each prime factor. Hence, the prime factors of any composite factor of the given number can involve only the primes given, and no more than the number available. 5. KNOWLEDGE OF NUMBER THEORY QUESTION; NO CALCULATORS Circle any which is a factor of 80000000005332: 3 4 5 6 8 9 12 15 3, 4, 6, 12. Not 5, 8, 9, and 15 6. For each part, give an example, if one exists. If there is no example, explain why not. Reasoning About Numbers and Quantities Test-Bank Items with Answers page 105 a. A whole number which has 15, 21, and 1000 as factors, but does not have 9 as a factor. b. A prime number that has 7 and 19 as factors, and is not a perfect square. a. The number must have 23, 3, 53, and 7 as factors, so 23 x 3 x 53 x 7 would have 15, 21, and 1000 as factors but 9 would not be a factor. (On tests, usually without calculators permitted, we allow an answer to be left in factored form.) b. Not possible, if 7 and 19 are factors, there will be more than two factors since 1 and the number itself are already two factors. 7. KNOWLEDGE OF NUMBER THEORY QUESTION; NO CALCULATORS. Circle each of the given choices that is a factor of the given number n. n = 22.103.711.135 Choices: 8 14 21 28 35 Only 21 is not a factor. (Students may overlook that n is not quite in prime-factored form.) 8. If it is possible, give a whole number that is relatively prime to 24. If it is not possible, explain why. There are many possibilities, so long as the number avoids prime factors that appear in the prime factorization of 24: 2 and 3. Examples: 5, 11, 13, 35, 49, 53,… 9. State a divisibility test for 4, and explain why it works. 4 is a factor of n if and only if 4 is a factor of the number named by the right-most two digits. The test works because a number can be expressed as a certain number of 100s, plus whatever is named by the right-most two digits. Since 4 is a factor of 100, divisibility of the whole number will depend exclusively on whether 4 is a factor of the number named by the right-most two digits. 10. KNOWLEDGE OF THEORY QUESTION; NO CALCULATORS) Is it possible, for some choice of positive whole numbers m and n, that 75m = 25n? Justify your decision. No, since 75 m involves factors of 3, which cannot appear in the factorization of 25 n . 11.4 Greatest Common Factor, Least Common Multiple 1. Write the prime factorization of each of the following. (Show your work.) a. 1485: Reasoning About Numbers and Quantities Test-Bank Items with Answers page 106 b. 792: c. Name all common factors of 1485 and 792 (They can be in factored form). d. What is the greatest common prime factor of 1485 and 792?_________ a. 33 5 11 2. b. 2 3 32 11 c. 3, 32 , 11, 33, 99 d. 11 a. What is the least common multiple of 1485 and 792 (in factored form)? b. Write two other common multiple of 1485 and 792. a. 2 3 33 5 11 3. b. 2 4 33 5 11 and 2 5 33 5 11 are two possible answers. If it is possible, give a whole number that is relatively prime to 24. If it is not possible, explain why. There are many possibilities, so long as the number avoids prime factors that appear in the prime factorization of 24: 2 and 3. Examples: 5, 7, 19, 35, 49, 53,… (This item is repeated from an earlier section.) 4. Suppose K = 25 7 11 , L = 2 3 7 1113 M = 2 29 2 and N = 4 11132 29 Name the least common multiple of each of the following (in factored form). a. K and L b. M and N c. K and M d. K, L and N a. 2 5 7 1113 5. b. 22 11132 292 c. 2 5 7 11 29 2 d. 2 5 7 11132 29 Using K, L, M, and N as defined above, name the greatest common factor of each of the following (in factored form). a. K and N b. K and L c. M and N d. K, L, and M a. 2 2 11 b. 2 3 7 11 c. 2 29 d. 2 6. Write these numbers in simplest form: Reasoning About Numbers and Quantities Test-Bank Items with Answers page 107 a. 26 b. 616 c. 129 a. 2 b. 7 c. 3 792 65 9 5 7. 215 5 a. Use the prime factorizations of 345, 264, and 495 to find the least common multiple of the three numbers. b. Compute the following: 345 495 250 (Leave the answer in factored form.) 264 a. 345 3 5 23 264 2 3 32 5 11 23 264 2 3 32 5 11 23 b. 8. 345 495 250 264 3523 32 511 2325 311 3 23 311 22 5311 3 495 32 5 11 LCM is 22 23 22 311 22 5311 3 22 235 3 22 311 Two neighboring satellites send out signals at regular intervals. One sends a signal every 180 seconds, and the other sends a signal every 280 seconds. If both satellites send out a signal at 12:00 midnight on January 1, when will be the next time that they both send out a signal at the same time? 12:42 a.m. The next simultaneous occurrence will happen when a multiple of 180 seconds next coincides with a multiple of 280 seconds, that is, at the least common multiple of 180 and 280. This will first happen 2520 seconds, or 42 minutes, later. 9. Hamburger patties come in packages of 16, and hamburger buns come in bags of 12. How many of each do you need to buy so that you have the same number of buns as you do of hamburgers? The LCM of 16 and 12 is 48, so buy 3 packages of patties and 4 dozen buns. 10. As a charitable service, your class undertakes a project where they fill backpacks with school supplies for recent immigrants. The donations include 135 notebooks, 216 pencils, and 81 pens. You want to use all the donations and include the same number of each item in each backpack. What is the largest number of backpacks you can fill and how many items will be in each backpack? 5 notebooks, 8 pencils, 3 pens in 27 backpacks: GCF is the number of backpacks. 11. Two football players are working out by running around a track. The first can run the track in 3 minutes, and the second one can run the track in 4 minutes. If they begin at the starting point at the same time and run in the same direction at the same rates, when will the both be at the starting point again? 12 minutes Reasoning About Numbers and Quantities Test-Bank Items with Answers page 108 12. The band has been invited to march at the Rose Parade and need to make money to cover the expenses. The divide up into three teams and shovel snow from driveways for four days before Christmas. The first team made $315, the second $240, and the third $210. If they charged a flat rate for each driveway, what was that rate? GCF (315,240,210) = 15, so $15 Reasoning about Algebra and Change Chapter 12 1. Completion. A. As the demand for housing increases, the price of housing ________________. B. Say that C = 100 – 20t describes the Celsius temperature C vs time t in hours, as a liquid cools down. The –20 tells you this about the situation: ___________________ 2. Complete each sentence with "increases," decreases," "doesn't change," or "can't say anything definite," as appropriate. As the semester goes on, the number of days until final exams _____________. As a person's peanut-butter consumption increases, her miles traveled to work_______________. As the speed of a car increases, the stopping distance of the car ____________. As the number of calculations increases, the probability of an error ____________. 3. What coordinate graphing conventions are being violated by the following graph of data showing candy-bar sales for three children who sold for different amounts of time? Reasoning About Numbers and Quantities Test-Bank Items with Answers page 109 No. of candy bars sold 14 13 12 11 10 3 6 5 No. of hours spent 4. Below is a graph for Candle 1. Candle 2 (graph not given) burns according to h = – 2t + 25, where h is the height in cm t minutes after being lit. Height (cm) 30 27 24 21 18 15 12 Candle 1 9 6 3 0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 Time (min.) A. Which candle, 1 or 2, burns at a faster rate? Explain. Candle ___ because B. Write an equation that describes Candle 1's height h vs time burning t. Reasoning About Numbers and Quantities Test-Bank Items with Answers page 110 h = _________________ C. On the above coordinate system, draw the graph for the burning of Candle 2. ( h = – 2t + 25 ) What information does the 25 give? D. Exactly how many minutes will Candle 2 last, after it is lit? _________ minutes E. If the two candles, 1 and 2, are lit at the same time, will they ever be the same height (before they both burn out)? If "Yes," tell when (approx.). If "No," tell why not. 5. The bowling place is under new ownership! The new owners plan to charge for shoes-ball rental and of course for each game, so that their dollar income, I, from a bowler who rents shoes-ball and who bowls n games is I = 2n + 2.5. A. How much are they planning to charge for shoes-ball rental? ______ That plan did not work, so they want to try a new plan (graph to the right). Cost ($) B. How much will they charge per game, under the new plan? 8 ______ 6 C. Will they make more money under the new plan? Explain. 4 2 1 6. A bakery keeps records and makes graphs of its cookie production. To the right are partial graphs for two days. 2 3 4 5 6 # of games Number of cookies Tuesday 10 000 A. Write a story that might tell what happened on Tuesday. Do not introduce any numbers besides those indicated for Tuesday. Reasoning About Numbers and Quantities Friday 0 0 8 Number of hours Test-Bank Items with Answers page 111 B. What is the slope for Friday's graph, and what does that slope mean about the cookie situation? Show your work. Slope: ________ Meaning: 7. A new candle is 12 inches long and burns 3 inches every 20 minutes. In a graph showing the height of the candle as it burns minute by minute, what is the slope? What does that slope mean in this situation? Chapter 13 1. Completion. A. At a speed of 80 m/min, Dude can go 170 meters in ___minutes _____ sec (exactly). B. For a turtle trip of n feet in 10 seconds, turtle's speed is _____________. C. A trip of m feet at a speed of 25 feet per second takes _________ seconds. 2. T F This is an acceptable story for the Distance travelled graph to the right: "I rode my bike up a hill and then down the other side." time 3. Story: Wiley Coyote, Jr., left school at a slow but steady pace, heading for his cave. When he was one-third of the way to the cave, he realized that he had forgotten his math book, so he ran back to school to get it. While he was at school he played for a while with some other coyotes. He then realized that he would be late for supper, so he started jogging at a steady rate toward the cave. About half-way to the cave, he thought that he was still going to be late, so he ran faster and got to the cave in time for supper. Draw two qualitative graphs for the story, one showing Wiley's distance from the cave vs. time since he left school. The second graph should show Wiley's total distance traveled vs. time since he left school. The two graphs should be coordinated so that corresponding times line up. Reasoning About Numbers and Quantities Test-Bank Items with Answers page 112 Dis tance from cave 0 0 Total distance traveled 0 0 Time since left s chool Time since left s chool 4. (not the same as #3--involves speed) Story: Wiley Coyote left school at a slow but steady pace, heading for his cave. When he was one-third of the way to the cave, he realized that he had forgotten his math book, so he ran back to school to get it. While he was at school he played checkers for a while with some other coyotes. He then realized that he would be late for supper, so he started jogging at a steady rate toward the cave. About half-way to the cave, he thought that he was still going to be late, so he ran faster and got to the cave in time for supper. Draw two qualitative graphs for the story, one showing Wiley's distance from the cave vs. time since he left school. The second graph should show Wiley's speed vs. time since he left school. The two graphs should be coordinated so that corresponding times line up. Do not worry about negative speeds. Reasoning About Numbers and Quantities Test-Bank Items with Answers page 113 Distance from cave 0 Time since left school 0 Speed Time since left school 0 0 5. When is the speed greater--at A or at B? distance travelled B Greater at ____ because A time 6. Complete a qualitative distance-time graph for the following story. Make clear what your distance refers to. "Wiley is in his cave and then walks slowly toward a canyon, planning to make a trap for Roadrunner. Halfway there, he stops for a short rest. Then he jogs on to make up for lost time. When he gets to the canyon, he realizes that it is almost time for Animal Planet on TV, so he runs as fast as he can back to the cave." 7. "Wiley walked out of the cave toward Roadrunner's usual resting spot, and after a while started slowly crawling to sneak up on Roadrunner. Roadrunner saw Wiley, however, and with Wiley close behind ran very fast straight to a secret hiding place right next to Wiley's cave." Reasoning About Numbers and Quantities Test-Bank Items with Answers page 114 Make qualitative graphs for the story on the following three coordinate systems. The systems are aligned so that your graph can show events at the same time for the different quantities. Wiley's distance from cave time Wiley's total distance time Roadrunner's distance from the cave time 8. Give the indicated companion graph, in each case. Reasoning About Numbers and Quantities Test-Bank Items with Answers page 115 A. distance B. speed time time speed distance time time 9. A. Write a short story that would yield this graph. distance time B. Write a short story that would yield the graph in part A, but with speed on the vertical scale instead of distance. 10. Below is a flask for water: Sketch a graph to show the relationship of the volume of the water in the flask and the height of the water as water is poured into the empty flask in a steady stream. height volume Reasoning About Numbers and Quantities Test-Bank Items with Answers page 116 Chapter 14 1. Put the letter of the graph from the right that would be most likely to belong to the given equation. ___ ___ ___ ___ A y B D C y = 2x + 7 y = – 2x + 7 y = 22 x 33 41 y = 50 x E x 2. Write an equation for each of these, and a story for the graph in part B. A. Our dog eats 12 pound of dog food every day. We buy a bag that weighs 24 pounds. What amount, A, remains after d days? Equation: A = y B. 25 Equation: y = 20 10 1 2 3 x Story for the graph: 3. Ordinarily it takes Brother 20 minutes to go from home to school, and it takes Sister 16 minutes to go from home to school. Each child walks at a steady rate. Today Brother left home at 8:00 a.m. and so got a 3-minute head start on Sister, who left at 8:03. Reasoning About Numbers and Quantities Test-Bank Items with Answers page 117 A. Draw a graph to show the percent of the distance to school that Sister has covered vs. the time. Label the graph S. Percent of dis tance to s chool (p) 100 50 10 8:00 8:05 8:10 8:20 8:15 B. On the coordinate system above, draw a second graph to show the percent of the distance to school that Brother has covered. Label the graph B. C. According to your graphs, will Sister catch up with Brother? If so, how do you know, and when does she catch up? If not, how do you know? D. What is the slope of Brother's graph, and what does that slope mean, in this situation? Slope: Means: 4. Ordinarily it takes Brother 20 minutes to go from home to school, and it takes Sister 16 minutes to go from home to school. Today Brother got a 2-minute head start on Sister. home Reasoning About Numbers and Quantities school Test-Bank Items with Answers page 118 Time A. Draw a graph to show the percent of the distance to school that Sister has covered vs. the time that Sister travels. Label the graph S. Percent of dis tance to school (p) 100 50 10 0 5 10 Time Sis ter travels (t) 15 B. On the coordinate system above, draw a second graph B to show the percent of the distance to school that Brother has covered, using Sister's travel-time scale as the clock. The first point, taking into account Brother's head start, is shown. (Be sure to notice that the horizontal axis scale is for Sister's travel time.) C. According to your graphs, will Sister catch up with Brother? If so, how do you know, and when does she catch up? If not, how do you know? 5. Show your work for credit. Mr. Cool joins Rabbit and Turtle in an Over-and-Back race, 200 meters each way. Rabbit: Speed over--50 m/s; time back--10 seconds Mr. Cool: Time over--8 seconds; speed back--40 m/s Turtle: The same speed both ways, but he rested for 5 seconds after the first 200 meters. A. Who of Rabbit and Mr. Cool finished first? ______________ Work: B. What was Rabbit's average speed for the race? _____________ Work: Reasoning About Numbers and Quantities Test-Bank Items with Answers page 119 C. What was Turtle's speed when he was moving, if Turtle tied Rabbit (remember that Turtle rested for 5 sec)? Work: 6. Chicken joins Turtle and Rabbit in Over and Back. They run an Over-and-Back race, with known data as in the drawing below. Turtle 60 feet per second over --> <--7.5 sec. to go back 300 ft. one way Rabbit 300 ft. one way 25 ft./sec. for the first 100 ft. over, 6 sec for the next 200 feet over --> Chicken <-- 75 feet per second back 300 ft. one way A. Going the same speed over and back, Rabbit just barely won the race. What was Rabbit's speed? Explain your thinking. B. What was Chicken's average speed for the whole over-and-back trip? Show your work. 7. Wiley joined Rabbit and Turtle in an Over-and-Back trip, 200 m each way. Show your work. Rabbit: Over--20 m/s; back--4 s Wiley: Over--8 s; back--40 m/s Turtle: The same speed both ways. A. Who of Rabbit and Wiley finished first? B. What was Turtle 's speed, if he tied Rabbit? C. What was Rabbit 's average speed? ______ D. Use one coordinate system to show qualitative speed vs time graphs for Rabbit 's (mark with ____ ), Wiley's (mark with ......), and Turtle 's (mark with x x x x ) speeds over and back. Reasoning About Numbers and Quantities Test-Bank Items with Answers page 120 speed (m/s) time (s) 8. A chicken and a squirrel join the turtle and the rabbit in Over and Back. They run an Overand-Back race, with known data as in the drawing below. 4 sec. for the first 100 ft. over, 6 sec. for the next 200 ft over Chicken 300 ft. <--75 feet per second back Squirrel 40 feet per second over --> <--5 seconds to go back 300 ft. Turtle 60 feet per second over--> <--7 1/2 seconds to go back 300 ft. Rabbit 300 ft. A. Going the same speed over and back, the rabbit just barely won the race. What was the rabbit's speed? Explain your thinking. B. Who came in second, third, and fourth? (Indicate ties, if any. No explanation is required.) Second____________ Third___________ Fourth___________ C. At the half-way mark of the race (300 feet), what was the order of the animals? (Indicate ties, if any. No explanation is required.) Leader_________ Second__________ Third________ D. What was the chicken's average speed over? Reasoning About Numbers and Quantities Fourth________ ________ Test-Bank Items with Answers page 121 9. A new "Over" distance is set at 100 feet. The Rabbit takes 5 12 seconds to go over, and comes back at 40 feet per second. What should the Turtle's speed be, so that the animals tie? Write enough so that your thinking is clear. Rabbit's speed _______ 10. A. Someone buys a few candy bars at 55¢ each and several others on sale at 35¢ each. Is the average price for the candy bars 45¢? Explain briefly. ______________ because B. A store sells two sizes of soft drinks--a large for $1.10, and a small for $0.90. Friday the store sold 236 soft drinks. The manager reasons, "The $1.10 and $0.90 give an average of $1. So we must have taken in $236 from soft drinks on Friday." Is the manager correct/incorrect? Explain in some detail. ______________ because 11. Tell whether each is correct, and explain why or why not. A. Fund-raiser: "We go two donations of $5000 and two of $1000 each! That's an average of $3000 per donation!" ______________ because B. Quality-control trainee: "One shipment had 1% of the items defective, and another shipment had 5% defective. That's an average of 3% defective." ______________ because 12. There is a (common) error, or some confusion, in each of the following situations. Find the error/confusion, and explain why some thinking was "off," even though the person thought the thinking was all right. A. Before summer school a student had completed 96 units, with a gpa of 2.9. The student takes two 3-unit courses in summer school and gets an A and a B, a 3.5 gpa. 2.9 + 3.5 The student is pleased on calculating and finding the new gpa to be 3.2. 2 B. A teen-ager says she never has anything nice to wear. Her parents tell her she can buy 4 new blouses for her birthday, but to keep the average price at $25 (or less). The girl Reasoning About Numbers and Quantities Test-Bank Items with Answers page 122 finds some $40 blouses she really likes and calculates from 40 + x = 25 that x = 10. 2 So a $40 blouse and a $10 blouse would give an average of $25. She is happy to find an acceptable $10 blouse on a sale table. She buys it and three of the $40 blouses in different colors. C. A person bought a used car and wants to check its gas mileage. For the first few fillups, she covered 823 miles and got 24.4 miles per gallon. The next fill-up, after 240 24.4+40 more miles, took 6 gallons, so she got 40 miles per gallon. She calculates = 2 64.4 2 = 32.2 mpg and feels much better. D. Abe got 94 out of 100 on one test, but only 26 out of 50 on a second. He figures he has 94 + 26, or 120, out of 150, and finds that to be 80%. Hence, he cannot understand why his teacher's grade book shows a different result: 94%, 52%, which 94%+52% 146% give an average of = = 73%. 2 2 (Choose carefully, based on your class work and/or discussions.) 13. Express the general property in algebra that each illustrates, using variables. A. (400 + 25) x 3 = (400 x 3) + (25 x 3) B. C. (57 x 16) + (43 x 16) = (57 + 43) x 16 (600 + 32) ÷ 4 = (600 ÷ 4) + (32 ÷ 4) 75 18 75 18 D. 3 3 3 14. Name the property or properties that justify each of the following. A. (700 + 60 + 3) + (200 + 30 + 5) = (3 + 5) + (60 + 30) + (700 + 200) B. 5 x (17 + 3) = (5 x 17) + (5 x 3) C. 984 + (717 + 563) = 984 + (563 + 717) D. (56 x 89) x 113 113 = 56 x 89 E. 75 ( 23 75) 23 (Hint: More than one property!) F. 1 3 95 13 4 13 (95 4) G. 26 52 52 26 H. 5 x (17 x 3) = 5 x (3 x 17) Reasoning About Numbers and Quantities Test-Bank Items with Answers page 123 15. Test each of the following algebra statements to see whether they appear to be true in general. If a statement appears always to be true, draw a diagram to justify the statement. Add an explanation if the diagram is not self-explanatory. If a statement is not true in general, give a counterexample. A. a b c ba a c B. (a b) (c d) (a c) (b d) C. (a b)2 a 2 b 2 D. (a b)3 a 3 b 3 E. (a b) c a (b c) 16. In each part use the given, correct algebraic statements to answer the calculation. A. (x 4)(2x 1) 2x 2 9x 4 implies that 14seventeen x 21seventeen = _________ seventeen. B. (2x 3)(2x 1) 4x 2 8x 3 implies that 23eleven x 21eleven = ________eleven. C. (Bonus) What other bases could be used in parts A and B? 17. A. It is correct that 12nine x 32nine = 384nine. How might that inform (x+2)(3x+2)? B. Explain the "might" in part A by considering 4nine x 13nine = 53nine. Give another calculation that would mis-inform an algebraic expression. Bonus: Why does 4ninex13nine = 53nine, or your calculation, give an incorrect idea for algebra? 18. Calculate the sum and product of the pairs of polynomials. A. 4x 2 7x 3 and 3x 5 B. 5x 6 and 3x 7 19. Make a drawing to justify a(b c d) ab ac ad . 20. T F The "balance" diagram to the right shows x + 2 = x3. (If it is not true, give a correct equation. 21. Make a drawing of a balance for the following equation. Solve the equation by showing actions with the balance, one step at a time. 4x + 5 = 5x + 3 1 22. Use specific values for m and n in (a m )n a mn to give a basis for justifying that a 2 a . 1 22'. Use a m a n a m n to justify defining a 2 a . 23. Use 1 am a m n as the basis for defining a 0 1 and a n n for non-zero values for a. n a a Reasoning About Numbers and Quantities Test-Bank Items with Answers page 124 24. Show your mastery of the conventional order of operations by evaluating each. A. 6 + 3 x 7 – (2 + –1)5 B. 10 – 4 ÷ 3 x 2 + 1 C. 10 – 4 ÷ 3 x (2 + 1) D. f(3), for f(x) = 4x2 – 7x + –2 E. g(–2), for g(x) = 9 – 3x – 5x3 25. Finish each story problem so that it can be described by the given equation. A. 50 – n = 16 "Jamal had 50 pieces of paper…." B. 3n + 16.99 = 37 "Jose went shopping and bought a CD for $16.99…" Chapter 15 1. Give the 100th and the nth entries for these lists, assuming the patterns continue. A. 12, 22, 32, 42, 52, … 100th ________ nth _________ B. 3, 5, 7, 9, 11, … 100th________ nth _________ C. 2 12 , 4, 5 12 , 7, 8 12 , 10,… 100th ________ nth _________ 2. Does each of the following give a function? Explain your decisions. A. Associate with each whole number n its third power n3. B. Assign to each person in the town his/her current last name. C. Assign to the last names of people in town, the first names. 3. Find a likely function rule for each of the following. Show your work. A. B. x 2 3 4 5 … f(x) 15 19 23 27 … f(x) = Reasoning About Numbers and Quantities x 1 2 3 4 … y 14 11 8 5 … y= Test-Bank Items with Answers page 125 C. D. x 4 3 1 2 … g(x) 29 21 5 13 … x 0 1 2 3 4 … g(x) = y 5 6 9 14 21 … y= 4. Explain why your answer to 3A might not be correct. 5. Two students have been looking for a function rule for the data to the right below. n | f(n) . 1 3 2 5 3 8 4 12 Akeena: "I got f(n) = 12 n(n+1) + 2." Bea: "Yes, but my mom worked a long time on it and said f(n) = 12 n(n+1) + 2 + (n–1)(n–2)(n–3)(n–4). Let's ask the teacher." You (the teacher): 6. A child is making “space modules with antennas” from toothpicks: etc. 1-room module (6 t-picks) 2-room module (11 t-p) 3-room module (16 t-p) The child wonders, “How many toothpicks would it take to make a 100-room module with antennas?!” Your answer, and a justification for it: Reasoning About Numbers and Quantities Test-Bank Items with Answers page 126 7. (take-home. Instructor: This is exercise 25 in 15.1.) Figure out a short-cut for squaring a number ending in 5. Here are some free data: 452 = 2025 352 = 1225 752 = 5625 152 = 225 8. (take-home) Here are examples of a short-cut for mentally squaring a number: 2 Example 1. 76 . Go to the closest multiple of 10--here, for 76, plus 4 to 80. Then go the opposite way from 76 but the same amount, minus 4, to 72. Multiply 80 and 72 mentally: 2 5760. Add the square of the up-down number 4. 5760 + 16 = 5776. 76 = 5776. 2 Example 2. 62 . Go to 60 (down 2). Then go up from 62 by 2: 64. 60 x 64 = 3840. 2 Add the square of 2. 62 = 3844. 2 2 2 Example 3. 57 . Go to 60, then 54. 60 x 54 = 3240. Add 3 . 57 = 3249. 2 2 2 Example 4. 198 . 200 x 196 = 39200. Add 2 . 39204 (= 198 ). 2 A. Use the shortcut to calculate 37 mentally, and then write the mental steps you did. B. Give a justification that the method works for squaring any n. Label the up-down number x. (Instructor: Caution--solution requires a little algebra.) 9. The Hikers' Supply Company's Snack Mix Recipe Ingredients: 3 cups of nuts 1 cup of raisins 2 cups of M&Ms Combine and mix well. Yield: 9 packages of snack mix. A. How many cups of ingredients are in 1 package of snack mix? ______ Explain. B. What fractional part of each package of snack mix is M&Ms? _____ (No explanation required.) C. What fractional part of 2 packages of snack mix is M&Ms? _____ (No explanation required.) D. How many cups of M&Ms are in 2 packages of trail mix? _____ Explain. E. If you need 24 packages of snack mix, what quantities of nuts, raisins, and M&Ms do you need? (No explanation required, but show your work.) Reasoning About Numbers and Quantities Test-Bank Items with Answers page 127 ________ of nuts; ________ of raisins; ________ of M&Ms 10. A new machine gives 5 choco-nut hearts for every 4 chocolate bars and 12 cup choc bars nuts of nuts put into the machine (nothing is lost in the machine). Solve by reasoning. Your reasoning may involve calculations, but not by just calculating with a proportion. choco-nut hearts A. How many hearts would the machine give for 18 bars and the right amount of nuts? _____ Reasoning: B. How many cups of nuts would be the right amount, for 18 bars? _____ Reasoning: C. An order for 48 hearts comes in. How many bars will that order take? _____ Reasoning: 11. What is the difference between an arithmetic sequence and a geometric sequence? 12. Suppose g(x) = 3x – 2, and h(x) is defined by Machine X to the right. Give the output if 10 is the input to the combination… A. first h(x), then g(x). input Add 4 to the input, and then double that sum. B. first g(x), then h(x). output 13. What does it mean to say, "'Combination' of functions is not commutative"? 14. Illustrate with drawings of "machines" that "combination" of functions is associative. Answers for Test Bank Items, Reasoning about Algebra and Change Chapter 12 1. A. increases. B. the temperature goes down 20 Celsius degrees each hour. 2. decreases; doesn't change; increases; increases Reasoning About Numbers and Quantities Test-Bank Items with Answers page 128 3. Mainly, the scale on the hours-spent axis is not in increasing order. That the scale on the vertical axis does not start at 0 is sometimes done, so that is less serious. 4. A. Candle 2 because it burns at a rate of 2 cm per minute, but Candle 1 burns at a rate of only 13 cm per minute. B. h = – 13 t + 15 C. (Graph should start at (0, 25) and have slope –2. It goes through (12.5, 0).) D. 12.5 E. Yes, they will be the same height (13 cm, not required) after 6 minutes (from either the graph or algebraic work). A solution from the graph may be off a bit. 5. A. $2.50 6. 7. B. $3.50 (using the clearest graph point $14 for 4 games) C. Yes, if the bowler bowls more than 1 game. But if the bowler bowls just 1 game, the first plan makes more money. (The graphs cross between 1 and 2 games.) A. (sample) On Tuesday, the bakery had 10 000 cookies at the start and baked more cookies at a steady rate during the 8-hour shift. B. –1250 This slope means that the bakery's cookie inventory went down by 1250 cookies every hour, on Friday. slope: – 203 That slope means that the candle burns by 203 inch every minute. Chapter 13 1. A. 2 minutes 7.5 seconds B. 10n feet per second (look for unit) C. 25m or m ÷ 25 2. F (Distance traveled would not decrease; the story-writer may have the graph-as-picture misconception.) 3. Graphs something like the following. Judge relative distances according to the standards your class work has set. Reasoning About Numbers and Quantities Test-Bank Items with Answers page 129 Distance from cave C B A D E 0 0 Total distance traveled Time since left school Pos sible fine points : B s hould involve s ame dis tance as A. C flat; any duration D dis tance slightly less than A,B distance. E s ame dis tance as D. Overall: non-decreas ing E D C B A 0 0 Pos sible fine points : A should be 1/3 of the way down to 0. B s hould s how faster speed than A. C flat; any duration D down to about 1/2 of the way to 0. E faster s peed than D. Time since left school 4. Graphs something like the following; negative slopes not called for. Distance from cave A B C D E 0 0 Time since left school Speed E D B A 0 C 0 Time since left school Fine points (s peed): Recall: Negative slopes not required. Adjust if you have emphasized. B height should be greater than height for A. D vs B height ?, but E height s hould be greater than height for D. 5. B because for a given interval of time, more distance is covered. Reasoning About Numbers and Quantities Test-Bank Items with Answers page 130 6. Here is a graph for distance-to-cave vs time. A graph for distance-from-cave vs time is also allowed, but in either case the student label the vertical axis. Dis tance to cave Time 7. Wiley's distance from cave Wiley's total dis tance R-runner's distance from cave Reasoning About Numbers and Quantities Test-Bank Items with Answers page 131 8. 9. A. Speed-time graph should have two horizontal segments, with the first one having greater height. (Adjust, if you have emphasized negative speeds.) B. Distance-time graph should have a line segment for first part (starting at origin), with second part made up of a curve upward. A. (sample) Wiley was out by his look-out point when he thought he saw Roadrunner, so Wiley ran steadily farther away from the cave until he was sure. It was Roadrunner! Then Wiley ran steadily back to the cave to get a net, and then ran even faster back to his lookout point. Even though he looked and looked, he did not see Roadrunner. B. (sample) Wiley ran out of his cave at a steadily increasing speed, and then slowed down at a constant rate, looking for Roadrunner. He saw Roadrunner and stopped, but immediately ran at an faster and faster speed until he reached his starting speed, and then kept running at that speed. 10. The graph should be curved upward, with tangents having increasing slopes as the volume increases. ( If your students should expect a post-filling continuation, then the height stays constant after the flask is filled.) Chapter 14 1. In order: A, B, E, D 2. 3. 4. 5. A. A = 24 – 12 d, or algebraic equivalent B. y = 5x + 15 Sample story: Joe has $15 and can clear $5 an hour painting an elderly neighbor's fence. How much will Joe have after x hours? A. Sister's linear graph S should start at (8:03, 0) and go to (8:19, 100) B. Brother's linear graph B should start at (8:00, 0) and go to (8:20, 100). C. The graphs should intersect around (8:15, 75), showing that Sister catches up at 8:15 (12 minutes after starting) D. The slope for B is 5, meaning that Brother covers 5% of the distance to school every minute. A. Graph S should be straight, joining (0, 0) and (16, 100). B. Graph B should be straight, joining (0, 10) and (18, 100). C. The graphs should intersect around (8, 50), showing that Sister catches up 8 minutes after she started (at the half-way point). A. Mr. Cool (13 seconds vs. 14 seconds for Rabbit) 4 B. 400 14 28 7 m/s (Look for unit.) 4 C. 400 9 44 9 m/s (Look for unit.) Reasoning About Numbers and Quantities Test-Bank Items with Answers page 132 6. 7. A. Turtle's time was 12.5 seconds (and Chicken's 14 seconds), giving Turtle an average speed of 48 ft/s. So Rabbit must have been going slightly faster than 48 ft/s. 6 B. 600 14 42 7 ft/s A. Wiley (13 seconds vs 14 seconds for Rabbit) 4 B. 400 14 28 7 m/s (Look for unit.) C. 28 47 m/s D. s peed (m/s ) R xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx W T R: 10 s + 4 s W: 8 s + 5 s T: 7 s + 7 s W R time (s) 8. A. Chicken takes 14 seconds; Squirrel, 12.5 seconds; and Turtle, 12.5 seconds. So 600 48 ft/s. Rabbit was going slightly faster than 12.5 B. Squirrel and Turtle tie for second, and Chicken is fourth. C. After 300 feet, T was first (5 s); R second (6.25 s); S third (7.5 s); and C fourth (10 s). D. 30 ft/s 9. 25 ft/s (5 12 s + 2 12 s = 8 s for the 200 ft) 10. 11. 12. A. No, because the numbers for the kinds of candy bars bought are different. B. Incorrect, because the numbers sold of the sizes were likely different. A. Correct, because there is the same number of each size of donation. B. Incorrect probably, because the shipments might have involved different numbers of items. A. The error lies in not recognizing that the individual gpas are based on different numbers of units. The new gpa is only slightly greater than the 2.9 (about 2.93). B. Again, the error lies in not taking into account the different numbers of types of 32.50 dollars per blouse. blouses bought. The average is actually 10120 4 C. The 40 mpg is based on only 240 miles, whereas the 24.4 mpg is based on 823 miles, so the average she calculated does not recognize that the mpg based on the greater distance will have a greater influence on the overall average than the mgp based on the 1063 240 miles. (The overall average is 823240 33.76 39.7 26.8 mpg.) Reasoning About Numbers and Quantities Test-Bank Items with Answers page 133 D. The teacher chose to weight the two exams the same and so used the percents. Abe's method would have the 50-point exam contribute only 50 of 150 total points, rather than 100% of a possible 200%. 13. Choices of letters may vary from those here, of course, and quantifiers may be included, depending on your emphasis. A. B. C. D. 14. 15. (a + b) x c = (a x c) + (b x c), or (a + b)c = (ac) + (bc) or just ac + bc (a x b) + (c x b) = (a + c) x b (a + b) ÷ c = (a ÷ c) + (b ÷ c) ab a b c c c (compare parts C and D) A. Commutativity and associativity of addition B. Distributivity (of multiplication over addition) C. Commutativity of addition (not associativity) D. 1 is the multiplicative identity E. Commutativity and associativity of addition; additive inverse property; 0 is the additive identity F. Distributivity (of multiplication over addition) G. 1 is the multiplicative identity H. Commutativity of multiplication (not associativity) A. Not true in general; student should have shown a counterexample B. True. Sample diagram: a+b a a b b then c c+d C. D. E. 16. 17. c d (a + b) – (c + d) a–c d b–d Not true in general; student should have shown a counterexample. Not true in general; student should have shown a counterexample. Not true in general; student should have shown a counterexample. A. 294seventeen (Let x = 17 in the algebraic equation.) B. 483eleven C. Any bases in which 9 (part A) or 8 (part B) are legitimate digits. A. (x+2)(3x+2) might equal 3x2 + 8x + 4. B. The given base nine expression would imply (x+2)(3x+2) = 5x + 3. A second example should suggest an incorrect algebraic result also. The problem is that 4nine x 3nine = 13 nine, and the 1 influences the next place value. Notice that in part A, there is no such Reasoning About Numbers and Quantities Test-Bank Items with Answers page 134 influence from any of the base nine multiplications 2 x 2 = 4, 2 x 3 = 6, 1 x 3 = 3, and 1 x 2 = 2, with 6+2 = 8 still involving only one place value. A. Sum 4x 2 4x 8 ; product 12x 3 x 2 26x 15 , or algebraic equivalents. B. Sum 2x + 1; product –15x2 + 53x + –42, or algebraic equivalents 18. 19. One way: Rectangular region with width a and length b + c + d. The ab, ac, and ad subregions can be marked. 20. F x + 2 = 3x 21. One pan should have 4 boxes, preferably with each labeled "x", and 5 singles. The other pan should have 5 boxes and 3 singles. The balance work should show the removal of 4x and 3 singles from each pan (either order), with x = 2 the solution. Let m 22. 1 a 2 must be 1 1 1 22'. a 2 a 2 a 2 21 1 2 25. A. B. 1 2 1 a. 1 a1 a , so a 2 must equal a . 23. Letting m = n, 1 24. 1 and n 2. Then (a 2 )2 a 2 a1 a. So a 2 , when squared, gives a. That is, am a0 1 m m 0 n 0 n . And so with m = 0, a a a a n m n a a a 26 8 13 C. 6 D. 13 E. 55 Samples: A. "…After a week of school, he had 16 pieces left. How many pieces did he use?" Or (comparison subtraction), "…Jamal had 16 pieces more than Jerry. How many pieces did Jerry have?" B. "…and 3 pairs of socks. He spent exactly $37 on the items. How much was each pair of socks?" Chapter 15 1. 2. A. 1002; 10n + 2 B. 201; 2n + 1 C. 151; 1 12 n + 1 A. Yes; each whole number is assigned to exactly one perfect cube. B. Yes; each person is assigned exactly one last name at a time. C. No; a given last name might be assigned to more than one first name. For example, Smith might be assigned to Joseph and also Kendra. Reasoning About Numbers and Quantities Test-Bank Items with Answers page 135 3. A. f(x) = 4x + 7 B. y = –3x + 17 C. g(x) = 8x – 3 D. y = x2 + 5 4. Some other rule might also describe these data. 5. "Both function rules are correct for the data given. We need more data to see which one, or perhaps even some other one, is better." 6. 501 (the n-room module would take 5n + 1 toothpicks). One justification: For the 100-room module, the antennas take 2x100 toothpicks, the tops and bottoms of the rooms take 2 x 100, and the room dividers take 100 + 1. So the total is 2 x 100 + 2 x 100 + (100 + 1), or 501. A similar argument gives 2n + 2n + (n + 1), or 5n + 1, for the n-room module. 7. Reports will vary, and some will likely use an example. Here is one version: Take the number of tens and multiply it by the next larger number. Write that down, and write 25 after that. In another version, one squares the number of tens, adds the number of tens to that, writes the sum down, and writes 25 after that. It is to be hoped that students ask for a justification. A justification can be based on (10n + 5)2 = 100n(n+1) + 25, or 100(n2 + n) + 25. 40 x 34 = 1360, 32 = 9, 1360 + 9 = 1369 (n + x)(n – x) + x2 = n2 – x2 + x2 = n2 8. A. B. 9. A. 6 cups of ingredients make 9 packages, so 1 package has 69 , or 23 , cup of ingredients. B. Because M&Ms make up 26 13 of the ingredients, 13 of each package will be M&Ms. C. 13 D. 29 of 2 cups = 49 cup of M&Ms . E. ( 249 = 2 23 recipes needed) 8 cups of nuts, 2 23 cups of raisins, 5 13 cups of M&Ms 4 12 5 22 12 hearts (18 bars would make 4 12 recipes) A. 4 12 12 2 14 cups B. C. 48 hearts would be 48 ÷ 5 = 9 53 recipes, so 9 53 4 38 25 chocolate bars will be needed. 10. 11. In an arithmetic sequence, each entry after the first is obtained by adding the same number to the previous entry. In a geometric sequence, each entry after the first is obtained by multiplying the previous entry by the same number. 12. A. 82 B. 64 13. The order in which a 'combination' of functions is carried out can give different results. (E.g., see #12.) Reasoning About Numbers and Quantities Test-Bank Items with Answers page 136 14. Look for three "machines" combined differently. (Machine 1 after Machine 2) after Machine 3 versus Machine 1 after (Machine 2 after Machine 3). Reasoning About Numbers and Quantities Test-Bank Items with Answers page 137

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# MATH 210 FINAL EXAM