Chapter 2: Numeration Systems

Chapter 2: Numeration Systems 8. In base ten, 215.687 is exactly ones, exactly tens, exactly hundreds, and exactly thousands; also, 3421 is exactly tenths and exactly hundredths. In base ten, 215.687 is exactly 215.687 ones, exactly 21.5687 tens, exactly 2.15687 hundreds, and exactly 0.215687 thousands; also, 3421 is exactly 34210 tenths and exactly 342100 hundredths. 28. 524 eight = ten 340 ten 29. 287 ten = four 10133 four 40. 200.3 five = ten 50.6 ten, or ten (If you used item 39, you might ask how the two answers are related. 55. Sketch the base blocks that show 1203 seven, and give the English words for the base ten value of each different sized piece. With large dot representing a small cube as the unit 76. 3 five 2 five = five 11 five 83. 241 six + 135 six 420 six 84. 127 nine 58 nine 58 nine Page 9

88. Use drawings of multibase blocks to illustrate 231 ten + 87 ten. Answer using a small square/block as the unit:........ Place together and then trade ten longs for a flat:........ 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. Page 10

Chapter 3: Understanding Whole Number Operations 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 six more than the Reese s Cups. There were four fewer Reese s Cups than Hershey Bars. There were 12 Milky Ways and 28 Hershey Bars. How many Tootsie Rolls did Marge buy? List five quantities involved in this problem. Sketch a diagram to show the relevant sums and differences in this situation. Solve the problem. 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). 6. Here are two word problems. How do they differ conceptually? Silvia had 14 books and then received four more books. How many books does she have now? Silvia has 14 books on one shelf and four books on another. How many books are on the two shelves? In the first, there is an action implied. In the second there is not. Because it is harder to act out the second problem, it may be more difficult for some young children. 10. 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 2) missing addend 3) 250 175 = 75 (75 circled) B) 1) subtraction 2) comparison 3) 6 1 5 9 = 4 (4 circled) C) 1) addition 2) join 3) 4 + 3 = 7 (7 circled) Page 24

26. 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 37. Make up a story problem about a bake sale, so that the problem could be solved A) by (Notice the order.) B) by. 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? 49. Consider this problem situation, which would involve dividing by 3: You are putting reading books on three empty shelves in your classroom. So the books look neat, you put the same number on each shelf. How many reading 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. The reading books are pretty big, so your assistants can carry only three at a time from the storage closet. How many trips to the storage closet will your assistants need? Page 25

46. Under a repeated-subtraction interpretation, means. The quotient is. Verify and explain your answer with a sketch. how many 1 s are in, or make,? The answer is. The sketch should show the answer, of one 1, is in. 50. 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. (Repeated subtraction, or measurement) The big bag has 48 plastic cars, to be put into bags holding six 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 six youngsters. How many cars will each youngster get? 60. Consider the following work of a student: 84 x 45 20 400 160 320 900 A) There is an error with the 20. B) There is an error with the 400. C) There is an error with the 160. D) There is an error with the 320. E) There is no error with this student s work. D 61. Use a nonstandard algorithm to calculate 128 67. Various methods, giving 8576 as the product. We usually get the long version (six partial products). Page 26

Chapter 4: Some Conventional Ways of Computing 1. Show 3335 23 with a scaffolding algorithm and then by the standard algorithm. Show how each number in the standard algorithm is associated with number in the scaffolding algorithm. 145 23 23 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 23 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. 5. Use a nonstandard algorithm to find 240 five + 314 five, but showing all partial sums. This work is all done in base five 240 + 314 4 100 1000 1104 Page 43

16. Consider this arithmetic problem: A) Write a story problem where the answer would be 6. B) Write a story problem where the answer would be 7. C) Write a story problem where the answer would be 1. D) Write a story problem where the answer would be 6 A) 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) 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) 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) Possible: Carolyn had 25 yards of fabric to make four identical costumes for a play. How much fabric did she allocate for each costume? Page 44

Chapter 5: Using Numbers in Sensible Ways 6. Show how you would mentally compute: A) 0.75 36 B) 34 12 + 34 8 C) 3458 1734 400 + 1734 A) 27, from 3/4 of 36 B) 680, from 34 (12 + 8) C) 3058, from 3458 400 (the other terms give 0) 12. Tell how one might mentally compute the following: A) 25 104 B) 25% of 104 C) 200% of 104 Each can be done in a variety of ways. Here are some possibilities: A) 25 100 + 25 x 4 is 2500 + 100 is 2600 or x 104 = 100 x 26 = 2600 B) of 104 is of 100 + of 4 is 25 + 1 is 26. C) Twice 104 is 208. NOTE: Only a few ways are shown to estimate a calculation in the problems that follow. Other ways may also be correct. Calculators should not be allowed for items involving either estimation or mental computation. 16. For each of the following, mentally obtain an estimate of the answer and write it in the blank. Use number sense. Then write enough to make clear how you thought. A) 34% discount on an $89 suitcase. B) 0.26 43,135 C) 61 334 D) 74.35% 1195 0.9837 E) (1201.794 0.25) + 0.0423 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) About 4800. Thinking: 1200 1/4. Definitely take off points for answers such as 4800.0423.) 17. 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. Page 48

19. 0.7614987 159.23842 is about A) 1.2. B) 12. C) 120. D) 1200. E) None of A D C (from ¾ of 160) 36. Write in scientific notation. A) 38,000,000,000 B) 382.45 C) 0.000 000 000 456 A) 3.8 10 10 B) 3.8245 10 2 C) 4.56 10-10 37. Write the following values in scientific notation. A) The earth is 150,000,000,000 meters from the sun. 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 10 11 B) 3 10 8 C) 7.53 10-10 38. How many seconds does it take light to reach the earth from the sun? Express your answer in scientific notation. (See item 37 for relevant data.) (1.5 10 11 ) (3 10 8 ) = (15 10 10 ) (3 10 8 ) = 5 10 2 or 500 seconds; a little over 8 minutes. Page 49

Chapter 6: Meanings for Fractions 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 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 of a discrete whole. B) Make a drawing of of a continuous whole. 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. 16. Use drawings with rectangles to show that A) B) C) A) A drawing of a rectangle divided into 16 equal parts: 10 parts will be shaded for, but only 7 parts for B) This will require a rectangle divided into 12 equal parts. will require 9 of the parts, whereas will require only 8 of the parts C) Two rectangles each divided into 4 equal parts: will require 7 of those parts, whereas will require only 6 of the parts. 18. Name three ways of thinking about the symbol. 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 is treated as a part-whole fraction, with discrete quantities. E.g., Jorge had five candy bars. Three of them were Snickers. What fraction of his candy bars were Snickers? Page 56

32. What is the exact decimal equivalent of A) B) A) 0.4 B) 37. A) Write as a fraction. B) Write 7.453 as a fraction. A) B) 43. Put these in increasing order without any hand calculation: 0.0239 0.0239 48. A) What are the rational numbers? B) How are they different from the irrational numbers? A) Rational numbers are numbers that can be expressed in the form (if only non-negative rational numbers have come up, accept ). B) Irrational numbers are numbers that cannot be expressed in that form (sufficient for now). Page 57

Chapter 7: Computing with Fractions 5. Using a rectangular region as the unit, illustrate each of the following: A) + B) + C) A) Expected way: 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 that are fifteenths), turning the 2/3 into 10/15. An additional 4/5 would entail four slivers or 12 small boxes. But there are only five small boxes left, so seven small boxes, or 7/15, 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 crossed out from the 10/8. 14. Use a drawing to help explain why is equal to. Be explicit. In the finished drawing, the unit has been cut into 4 7 equal pieces, and the answer part (the x s) is 3 5 of them. 21. A) Show 1 with a rectangular region as the whole. Write your answer as a mixed number. B) Make clear how the fraction part of your answer relates to your drawing. 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 part B. Page 70

20. A) Shade in of of this rectangle, as though you were acting it out: B) Show exactly where the is. It is of what? C) Show exactly where the is. It is of what? A) Most common is likely to be that first 2/3 is shaded and then, with cutting marks perpendicular to the first marks, 3/4 of the 2/3 is 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. Other ways are reasonable. B) 2/3 of the whole rectangle. C) 3/4 of the 2/3 part of the whole rectangle. 25. Draw a picture that represents: A. B. Point out, if needed, that although the answers are the same ( ), the meanings of the two products are different. 36. Tell what 8 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 B) How much is in each share if 8 are shared equally among three Page 71

27. 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. A) B) 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 three full 1/4s, and half of another 1/4, in 7/8. The answer is 3 1/2. B) The question is How many 3/8s are in ¼? 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. The answer is 2/3. (Note to instructor: Try to detect students who do the calculation but cannot show the meaning in the drawings.) 39. What calculation would solve this story problem? George had of a pie. He ate of what he had. What part of a whole pie did he eat? A) + B) C) D) E) None of A D C 53. Under a repeated-subtraction interpretation, means. The quotient is. Verify and explain your answer with a sketch. how many 1 s are in, or make,? The answer is. The sketch should show the answer, of one 1, is in. Page 72

Chapter 10: Expanding Our Number System 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 = I) + J) + ( ) K) 2.6 4.5 L) 3.17 2.4 A) 15 B) 2 C) 2 D) 23 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) J) 0 K) 2.6 + 4.5 = 7.1 L) 3.17 + 2.4 = 5.57 2. Reorder these numbers from smallest to largest: 7.4% 0.2 1.5 2.8 0.2 1 1 1.5 0.2 7.4% 0.2 2.8 7. 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. 8. 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 = I) J) ( ) K) 8.6 4.3 L) 1.2 2.4 A) 56 B) 15 C) 32 D) 42 E) 6 F) G) 2 H) I) 2 J) 1 K) 2 L) Page 101

11. Match the operations and the names of the 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 C) 4 + ( 3 + 1) + 2 = (4 + 3) +( 1 + 2) D) 5 + ( 8 + 0) = 5 + 8 E) -4 1 = -4 3. Multiplicative inverse property 4. Additive inverse property 5. Commutative property of addition F) 6 + (4 + 4) = 6 + 0 6. Associative property of addition G) 7. Distributive property of over + H) 3 (5 0) = (5 0) 3 8. Multiplicative identity property I) 3 + (2 + 5) = (3 + 2) + 5 9. Commutative property of multiplication A) 5 B) 7 C) 6 D) 2 E) 8 F) 4 G) 3 H) 9 I) 6 Page 102

12. 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 0 1 1 2 2 3 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? If so, what is it? G) Does 2 have a multiplicative inverse? If so, what is it? H) Do you think addition is commutative? If so, provide an example. I) Do you think multiplication is commutative? If so, provide an example. J) Do you think addition is associative? If so, provide an example. 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. Page 103

Chapter 11: Number Theory 6. Circle T if the statement is true, F if it is false. T F Every whole number is a multiple of itself. T F It is possible for an even number to have an odd factor. T F Zero is a multiple of every whole number. T F 2 50 is a factor of 100 30. T (m = 1 m) T (e.g., 12) T (0 = 0 m) T ( ) 12. Is 245 a prime number? Explain. No, 5 is a third factor. 16. Give the prime factorization of n, where n = 4 7 20 5000. If it is not possible, explain why not. 37. Circle the numbers that are prime. If a number is not prime, list at least three factors below the number. 39 2 5231211 61. 73 121 43 Only 43 is a prime. 121 = 11 11 and the sum of the digits of 5231211 add up to a multiple of 3 39. 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) and 8 is a factor of 1000 (1000 = 8 125), so whether 8 is a factor of will depend on whether 8 is a factor of efg. 45. 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 rightmost 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. Page 105

47. Write the prime factorization of each of the following. (Show your work.) A) 1485: 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) B) C) 3,, 11, 33, 99 D) 11 48. 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) B) and are two possible answers. 51. Write these numbers in simplest form: A) B) C) A) B) C) 52. A) Use the prime factorizations of 345, 264, and 495 to find the least common multiple of the three numbers. B) Compute the following:. (Leave the answer in factored form.) A) LCM is B) Page 106