8. part i and ii test bank

8/17/2019 8. PART I and II Test Bank

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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 fortests 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?

Reasoning About Numbers and Quantities Test-Bank Items with Answers page 1


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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.

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 aStarbucks 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, minutesTime 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

…



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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

roommate Cassie had finished her morning classes and was headed home about the sametime. 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.



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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 oftuna sandwiches and number of ham sandwiches…

b. There are other possible praiseworthy drawings possible, but the following suggests

the solution (for the total number of sandwiches) pretty easily.

48

8 4

#TS #HS

#VP#PP

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 unitof 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:

A butcher had two pieces of bologna, A and B, with A weighing 3 and times as much as

B. After the butcher cut 1.8 pounds off A, A was still 21

3 times as heavy as B. How many

pounds does piece B weigh?



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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?

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: 1.8 pounds After cut:

Piece A Piece B 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.



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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 only1/40 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 already25 of the way home when I started.

If I walk at my usual speed, can I catch my sister before she gets home?

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

403224168

302418126Me

Mysister

HomeSchool

(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.



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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

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



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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

5. What metric prefix means one-hundredth? ____________ centi

6. If a Pascal is some unit of measure, use your knowledge of metric prefixes to complete:

4 kiloPascals = ____Pascals. 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 i!e " #ampers will fit her$ (%lternati&ely, use the '$' pounds for " kilogram to change the kg* kg range to +$+ pounds"$' pounds$)

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.



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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 armsand four without, what is the total price they pay?

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 + 4x($45 – $8.50) + 4x ($45 – $8.50) = $382

c. Joe 8 miles

Jim



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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 7 st 29 students in all

4th Grade

Grade 3 has 18 students, Grade 4 has 11 students.



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Chapter 2 Numeration Systems

2.1 Ways of Expressing Values of Quantities, and

2.2 Place Value


1. Is the old Greek numeration system (α = 1, β = 2, γ = 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.)



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11. 34.597 has 345 whole tenths in it. True False True

12. 56 has 560 tenths in it. True False True

13. 23 has 230 hundredths in it. True False

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 B

19. How many $100 bills would make $45 billion? 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 __



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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

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


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



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6. 524 eight = __________ ten 340 ten

7. 287ten = __________ four 10133 four

8. 1012five

= ______________ in base ten. 132 ten

9. 32ten = _______________ in base four. 200four

10. 2.31four = ________________ as a mixed number in base ten. 21316 ten

11. 6 23 in base ten = __________ in base three. 202

10 three, or 20.2three

12. 1ten = ____twelve 1 twelve

13. 214.3five = ___ in base ten 59.6 ten or 5935 ten

14. 29ten = ___ in base three 1002three

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 5035 ten

19. Write 49ten in base seven. 100 seven

20. Do the "translations" in parts A-D. Show your work.

A. 3102five = _____________ten 402

B. 310.2five = _____________ten 80.4 or 8025

C. 203.6ten = ____________five 1303.3

D. (base six pieces with small

block as the unit) = ______________ten 336



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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,

1234.20 = (1× φιϖε 3) + ( 2 × φιϖε 2 ) + ( 3 × φιϖε ) + σοµ ετηινγ λεσστηαν φιϖε

ορ = ((1 × φιϖε 2 ) × φιϖε )+((2× φιϖε ) × φιϖε ) + ( 3 × φιϖε ) + (σµαλλ )

= ((1× φιϖε 2 ) + ( 2 × φιϖε ) + 3 ) × φιϖε + (σµαλλ )

=123φιϖε × φιϖε + (σµαλλ )

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.



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29. Consider x = 0.3147eight and y = 0.3147nine. Which of x and y is greater? Explain. (Be

careful.)

x is larger. Consider only x = 0.3eight and y = 0.3nine. 0.3eight =38

in base 10 and

0.3nine =3

9

in base 10. 3

8

is larger than 3

9

. Extending this reasoning, x is larger.

30. Write an algebraic expression for 204b. Answer: 2b2 + 4 (ορ, 2β 2 + 0β + 4)

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, or1

64 in base ten

32. Base eight pieces, with the small cube (a dot here) is unitas 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.

(small cube = 1; no longs)

33. large cube = 73 = 343 , or three

hundred forty-threeflat = 72 = 49 , or forty-nine

small “cube” = 1, or one

34. Write the base b numeral for 2b4 + b

2 + 3b + 1. 20131b

35. Write out 32004m in the algebraic form of the last item.

Answer: 3m4 + 2 µ 3 + 4, ορ3 µ 4 + 2 µ 3 + 0 µ 2 + 0 µ + 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.



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True False

False. Best would be the penny, the dime, and the silver dollar

*38.If 10000ten + 10b = 10023ten, what is base b?

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

41. 53six names the same number as which of these base ten numerals?

A. 186 B. 183 C. 12 D. 85 E. 33 E

42. In base ten, 111five would be written...

A. 421 B. 155 C. 31 D. 21 E. None of A-D C

43. The base b numeral 321b means...

A. 3.b2 + 2.b1 + 1 B. 3.b3 + 2.b2 + 1.b1 C. 6b A

D. 3.b + 2.b + 1 E. None of A-D

44. In base five, 32ten would be written... B

A. 152five B. 112five C. 62five D. 17five E. None of A-D

45. The base two numeral 100two equals the base ten numeral... D

A. 1100100 B. 1011100 C. 8 D. 4 E. None of A-D



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46. In base ten, 32four would be written...

D

A. 400 B. 200 C. 122 D. 14 E. 8

47. The base four numeral 11.1four could be written in base ten as... D

A. 3314 B. 33

110 C. 11

14 D. 5

14 E. None of A-D

48. The base ten decimal 18.5 could be written in base six as ... C

A. 10.5six B. 20.3six C. 30.3six D. 128.5six E. None of A-D

49. The base ten fraction 14 equals which base eight numeral? A

A. 0.2eight B. 0.14eight C. 0.02eight D. 1.4eight E. None of A-D

*50. If 31b = 28ten, then b = ... D

A. 4 B. 5 C. 7 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. 19

16B. 1

3

4C. 1

21

100D. 1

3

5E. 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 ? 11123 five or 1

613 ten



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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.)

B. How would you illustrate this with the base five blocks using drawings and showing

the intermediate steps?

A. 112five-$ .ith the small cue 0 ", first drawing shows ' longs, small cues and

longs, small cues$ 1ext, fi&e of the se&en small cues are traded for a long,gi&ing six longs and ' remaining small cues$ 1ext, fi&e longs are traded for a flat,

gi&ing " flat, " remaining long, and the ' remaining small cues, or ""' fi&e$

6. A. 0.5ten = ________ eight. B. 312.2four + 22.3four= ________ four

C. 84 ten = ________ three D. 2five x 43five = ________ five

E. 33.3six = ________ ten

%$ 2$eight -$ "22"$"four 3$ "22"2three 4$ ""fi&e E$ '"$5 ten

*7. Determine the possible value(s) for base b: 321b

– 234 b

43 b

b = six (11 b – 4 b = 3 b, or 3 b + 4 b = 11 b )

8. To the right is a partially completed addition, written in connection

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?(Drawings or word descriptions are okay.)

Finish the numerical calculation. (You do not have to draw

the wooden pieces for the rest of the work.)

1

2 1 4five

+ 3 3five 2

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



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+ 135six 420six

10. 127nine 58nine

− 58nine

11. 4.4five 13.2 five

+ 3.3five

12. 0.24seven 0.06 seven

– 0.15seven

13. 21six 100.2 six + 35.2six

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

. . . . . . .

. . .



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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: 13 +13=26

A. Base six B. Base seven C. Base eight D. Base ten A

E. It works in all of these bases.



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16. What step is wrong in the following (base eight):

13Eight x 12Eight 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.



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Chapter 3 Understanding Whole Number Operations

3.1 Additive Combinations and Comparisons

"$ wo asketall coaches, % and -, 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 theyhave unnecessarily (probably) ignored their drawing in arriving at their answer.

?8"

4"

6"

B2TA2TATBT

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.



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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)

12

(18)

46

(24) 28#HB#RC#MW

#TR

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 ofcheese 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.


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 suggeststhe solution (for the total number of pizza servings and sandwiches) pretty easily.

c. The number of sandwiches is (60 + 6) – 8 = 58.



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6

8

#RB

#VP

#PB

#CP

60

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.


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)

6

2

B

D

A

C46

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?



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Silvia has 14 books on one shelf and 4 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, 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

$ "', ecause there are the two separate amounts6 this situation in&ol&es an additi&ecomparison$

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?



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a$ ") sutraction ') missing addend ) '52 7 "85 0 85 (85 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)

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?



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(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...

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…7

15

+3 10

+5( add 3 to get to 10, then 5 more to get to15) So, 15 – 7 = 8

________________________

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.



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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 lemonadewas left after the party?

c. 312 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 baby-

sitting?

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?

4 0 2 5 0 6

– 3 9 – 1 4 9 … 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.



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2. Perform the following using the "equal additions" method, as used by one student in the

samples in your textbook.

4 3 2 7 ' + 8

4 13 12

7 ' 9 + 81 4 5

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 samemethod as the student, and (ii) comment on the student's method in terms of the "number

sense" exhibited.

a. 7 3 2 (i) 8 3 4 1 (ii)

-2 4 5 - 4 5 6 7

5 1 3

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) Thisstudent 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.



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(given) (child does)

675 677

– 198 – 200

477

second calculation (or explanation if not ok)

453

– 295

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

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:

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.


413

–248

–5

-30

200

165

9456

–3789

–3

-30

-300

6000

5667


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7. A second-grade boy is asked to subtract 64 – 55, written vertically. The child thinksabout 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

have to do tens. 20 plus 80 equals 100 and places that under the 110. She then adds 8 and6 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

3012 40 17 – 9 – 1 0 82 0 3 2 0 9

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 tothe ones place.

10. Find 21 + 49 using an empty number line. Answer below for one way.

Reasoning About Numbers and Quantities Test-Bank Items with Answers page 3221 61 65 70

+ 4

+ 40

+ 5


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11. Find 509 – 239 using an empty number line. Answer below for one way.

12. A student wrote the following answer to her problem 95 – 34:

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 12 × 6 and 6 × 12

a. The 3 × 6 drawing should show 3 groups of 6 things, such as 3 six-packs of a soft-drink, or 3 sets of 6 objects of some kind. The 6 × 3 sketch should clearly show 6 sets of3 things, or a 3 × 6 array that turned on its side is 6 × 3.

b. The 12 × 6 should clearly show 6 objects, with12 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


270 300 500 505 509

– 30

– 5 – 4

– 200


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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, 1pair 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

a. by 34 ×12. (Notice the order.)

b. by 12 × 34 .

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, ormilk 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 × 12 . (Choose your own context.)

Be attentive to the order of the factors.



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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.

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.



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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 thatshe 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 by16 ×

18 .

c. Can be solved by 34 × 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,34

÷1 12 means

_______________________________________________

The quotient is ____________. Verify and explain your answer with a sketch.

…how many 1 12 s are in, or make,34 ? The answer is

12 . The sketch should show the

answer, 12 of one 112 , is in

34 .

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.



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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.

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 312 acre orchard of orange trees....



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a. 23 × 312

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.

(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?



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__ 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?

__ 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, shownto the right], why do I have to put in the 0 [points to the zero in

980]?”

49

× 23

147

980



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What would you say to Antonio? 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

42040

364

a) Is Hiro's work correct or incorrect? Correct __ Incorrect __

b) If the work is incorrect, please explain how.

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.

6. Consider the following work of a student: D

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

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.



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a$ "++ 5"8 or "+* 5"

$ *"' 7 '9 or *" 7

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 b. 280 ÷ 70 = 40

225

980 c. 480 ÷ 0.4 = 120

375

1895

a. The ones' column adds to 10, not something ending in 5.

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.



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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 showhow 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

115 069 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.



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Similar to 1. Student’s work should show an awareness of the scaffolding algorithm

format to an abbreviated form to the usual US form.

4. Do 32 × 467 using the method of writing all partial products. What does this algorithmhave 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

4100

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 student-

generated, they can make sense to them; encourages a “make sense” view of

mathematics; can be more efficient in selected calculations. Negatives—Time away fromconventional 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).



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Initially second

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-leftalgorithm, 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?



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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 aboutdivision from the way of thinking in your story problem in part b.

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. 226 + 49

B. 226 + 49

C. 226 – 124

D. 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



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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 multiplicationof whole numbers.)

3 1 3 x

4

2

12 2

2 660

0

00

14

4 61

1

3

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 of 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 tensfrom 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



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Possible: Carolyn had 25 yards of fabric to make 4 identical costumes for a play. How

much fabric did she allocate for each costume?



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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 = ________ Thinking:

b. (25 × 29) + (25 × 11) = ________ 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. 1025

× 25 will give the same number, and 15 of 25 is 5.

5. Show how you would mentally compute the exact results:

A. 3000 -

8. part i and ii test bank

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