Test bank for
and
Below we have listed assessment items for most sections of Part I: Reasoning About Numbers
and Quantities and Part II: Reasoning About Algebra and Change. The items were selected from
those that instructors used while using the materials over several years at San Diego State
University. For some sections, there are very few items. The number of items is related both to
the number of times that the particular section of the module was piloted and to the emphasis
given to the material. Space here, of course, is reduced from that provided on actual tests or
quizzes.
The test-bank is a Word document rather than a PDF document so that you can select items for
tests without having to re-enter them.
Some test items are similar to previous test items. They provide the opportunity to use slightly
different items on different versions of tests. Also, some test items are more difficult than others,
and they are marked with an asterisk. However, you may not agree, and thus it is important that
you check each item to be sure that it is of the level of difficulty that you wish to have in your
examination.
Request: The items are different for Parts I and II. Part I has text items for sections whereas Part
II items are listed for each chapter. Also, Part I answers are embedded in the set of test items,
whereas answers for Part II are at the end of the set of test items. We would appreciate
information from users concerning which format is more useful.
Please note: We often use the following directions for true/false items on exams, and it should be
included in the directions of any exam given that contains such items–
For each of the statements below indicate whether the statement is True or False by CIRCLING the
proper word. IF THE STATEMENT IS FALSE, THEN BRIEFLY EXPLAIN WHY IT IS FALSE
OR RESTATE IT SO THAT IT IS TRUE.
1.1 What Is a Quantity? and
1.2 Quantitative Analysis
1. What is a quantity? Give an example. What is a possible value for your example?
E.g., The length of this room is a quantity. A possible value is 20 feet.
2. Name 5 quantities that you have dealt with so far today.
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E.g., distance from home to class, time spent traveling from home to class, amount of
gasoline purchased, amount of milk drunk at breakfast, amount of money spent on a
Starbucks coffee, etc.
3. Name 3 quantities that relate to you, and tell how they are measured.
E.g., weight (in pounds or kilograms), height (in inches), arm span (inches), shoe size
(standard sizes for shoes), waist size (in inches), etc.
4.
Would student motivation be difficult or easy to quantify? Explain.
Tell how you might go about quantifying student motivation in this class.
Probably difficult. Factors influencing motivation might include need for a passing grade,
desire to understand content, parental pressure, peer pressure, etc. A scale (such as 1 low
to 10 high) could be designed to measure these factors.
5.
List at least five relevant quantities that are involved with this problem situation. For
each quantity, if the value is given write it next to the quantity. If the value is not given,
write the unit you would use to measure it.
Pat and Li left the starting line at the same time running in opposite directions on a 400
meter oval-shaped race track. Pat was running at a constant rate of 175 meters per
minute. They met each other for the first time after they had been running for 1.5 minutes.
How far had Pat run when Li completely finished one lap?
Sample answers (quantity, value or unit if value unknown; other units possible—e.g.,
Length of track, 400 meters
Pat’s speed, 175 meters per minute
Time until they meet for first time, 1.5 minutes
Distance Pat has traveled when they meet for first time, meters
Distance Li has traveled when they meet for the first time, meters
Li’s speed, meters per minute
Time for Li to run one lap, minutes
Time for Pat to run one lap, minutes
Distance Pat has run when Li finished one lap, meters
(The above are relevant to one solution, but the following are quantities in
the situation as well.)
Difference in times for one lap for Pat and Li, seconds or minutes
Difference in speeds, Pat and Li, meters per minute
…
6.
Carry out a quantitative analysis of the following problem situation by answering each of
the questions that follow.
Jennie got on the freeway at 2:00 PM, using the entrance closest to her home, and
traveled at 55 mph to the College Avenue exit, where she turned off at 2:12 PM. Her
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time. Cassie entered the freeway from the College Avenue entrance at 2:08 PM, and
traveled to the home exit at 60 mph. What time did Cassie arrive at the exit ramp to go
home?
a. What quantities here are critical?
b. What quantities here are related?
c. What quantities do I know the value of?
d. What quantities do I need to know the value of?
a. Jennie’s starting time, Jennie’s exit time, time Jennie traveled, speed Jennie traveled,
distance Jennie traveled, Cassie’s starting time, distance Cassie traveled, speed Cassie
traveled, time Cassie traveled.
b. All in part a are related, but in different ways.
c. Jennie’s starting time, Jennie’s exit time, speed Jennie traveled, Cassie’s starting time,
speed Cassie traveled
d. Time Jennie traveled, distance Jennie traveled (= distance Cassie traveled), time Cassie
traveled, to get Cassie’s exit ramp time.
7.
Consider this problem situation:
The school cafeteria is ready to serve two kinds of sandwiches, tuna and ham, and two
kinds of pizza, pepperoni and vegetarian. There are 48 servings of pizza prepared.
There are 8 more tuna sandwiches prepared than there are servings of pepperoni pizza.
There are 4 fewer ham sandwiches prepared than there are servings of vegetarian pizza.
Altogether, how many sandwiches are prepared?
a. List 8 quantities involved in this problem.
b. Sketch a diagram to show the relevant sums and differences in this situation.
c. Solve the problem.
It will be difficult for your students to avoid algebra or trial-and-error on this problem;
decide whether you wish to prohibit the use of algebra. You might also consider omitting
part c.
a. E.g., number of kinds of sandwiches, number of kinds of pizza, number of servings of
pizza prepared, difference in number of tuna sandwiches prepared vs number of
servings of pepperoni pizza, difference in number of ham sandwiches prepared vs
number of servings of vegetarian pizza, total number of sandwiches prepared, number
of tuna sandwiches prepared, number of ham sandwiches prepared, number of servings
of pepperoni pizza, number of servings of vegetarian pizza, difference in number of
tuna sandwiches and number of ham sandwiches…
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b. There are other possible praiseworthy drawings possible, but the following suggests the
solution (for the total number of sandwiches) pretty easily.
48
#VP
#PP
4
8
#HS
#TS
c. There are 52 sandwiches prepared in all. (8 + 48 – 4)
8.
Consider the following problem situation:
Two trains leave from different stations and travel toward each other on parallel tracks.
They leave at the same time. The stations are 217 miles apart. One train travels at 65
mph and the other travels at 72 mph. How long after they leave their stations do they
meet each other?
List six quantities in the problem (note that you are not asked to solve this problem). If a
value is given, write it next to the quantity. If no value is given, write an appropriate unit
of measure.
Samples (quantity, value or unit if value unknown)…
Distance between stations, 217 miles
Speed of one train, 65 miles per hour
Speed of other train, 72 miles per hour
Total speed of the two trains, miles per hour
Time until trains meet, hours (or minutes)
Distance first train has traveled when they meet, miles
Distance second train has traveled when they meet, miles
9.
Carry out a quantitative analysis of the following problem situation by answering each of
the questions that follow, and then solve the problem:
1
A butcher had two pieces of bologna, A and B, with A weighing 3 and 3 times as much as
B. After the butcher cut 1.8 pounds off A, A was still 2 13 times as heavy as B. How many
pounds does piece B weigh?
a. What quantities here are critical?
b. What quantities here are related?
c. What quantities do I know the value of?
d. What quantities do I need to know the value of?
e. What is the weight of B, in pounds?
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a. Weights of pieces A and B before cut; comparison (ratio) of pieces before cut,
comparison (ratio) of pieces after cut, weight of piece cut from A.
b. Same as a, along with weight of B.
c. Weight of A, weight cut from A, ratio of A to B before cut, ratio of A to B after cut.
d. Weight of B.
e.
Before cut:
Piece A
1.8 pounds
Piece B
After cut:
Piece A
Piece B
Piece B must weigh 1.8 pounds.
10. Consider the following problem situation:
Two boats simultaneously left a pier and traveled in opposite directions. One traveled at
a speed of 18 nautical miles per hour and the other at 22 nautical miles per hour. How
far apart were they after 2.5 hours?
List five relevant quantities that are involved in this problem. For each quantity, if a value
is given, write it next to the quantity. If the value is not given, write the unit you would
use to measure it, and its value if possible.
Speed of first boat; 18 nautical miles per hour
Speed of second boat, 22 nautical miles per hour
Distance traveled by first boat in 2.5 hours, nautical miles; 2.5  18 = 45 n.m.
Distance traveled by second boat in 2.5 hours, nautical miles 2.5  22 = 55 n. m.
Total distance between boats at 2.5 hours nautical miles: 45 n.m. + 55 n.m. = 100 n. m.
After 2.5 hours they are 100 nautical miles apart.
11. My brother and I go to the same school. My brother takes 50 minutes to walk to school,
and I take 40 minutes. If he gets a 49-minute head start one day, can I catch him before
he gets to school? Explain, without referring to any short-cut in your explanation. (Hint:
Do not do a lot of calculation.)
No. Brother needs only 1 more minute to get to school, and in 1 minute, I can travel only
1/40 of the distance to school.
12. My sister can walk from school to home in 40 minutes.
I can walk from school to home in 30 minutes. But today I stayed for some extra help,
and my sister was already 25 of the way home when I started.
If I walk at my usual speed, can I catch my sister before she gets home?
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If "Yes," exactly what fraction of the trip have they covered when I catch her?
If "No," exactly what fraction of the trip have I covered when my sister gets home?
In either case, write enough (words, numbers, drawings) to make your thinking clear.
_____ (yes/no) Explanation, including fraction of the trip:
No, I cannot catch up with my sister. My sister has 3/5 of the way to go, which should
take 3/5 of 40 minutes, or 24 minutes before arriving at home. But in 24 minutes, I can
cover only 24/30 or 5/6 of the way to home.
On the diagram, my sister is at the second colored dot (16 minutes) when I begin, and has
24 minutes of walking before arrival. In those 24 minutes, I can walk only to X.
Location after traveling so many minutes
My
sister
8
16
24
32
40
Home
6
12
18
24
30
School
Me
(Comment: The Brother and I exercise in Section 1.2 is usually much more difficult than
either #11 or #12, so either of these is reasonable for a timed test.)
13. The big dog weighs 5 times as much as the little dog. The little dog weighs 2/3 as much
as the medium sized dog. The medium sized dog weighs 9 pounds more than the little
dog. How much does the big dog weigh?
a. List 3 quantities associated with this problem. If possible, give the associated value.
b. Draw a diagram to represent the quantities in this problem.
c. This diagram was provided by a 5th grader. Tell why it is not helpful.
Large dog
Small dog
Medium dog
d. Solve the problem and explain your solution process.
a. E.g., weight of large dog, weight of medium dog, weight of small dog, large dog's
weight in terms of small dog's weight; small dog's weight in terms of medium dog's
weight.
b.
Large dog: 5S
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Small dog: 1S (which is 2/3 M)
Medium dog: (M is 3/2 of S )
c. The diagram does not tell anything about their sizes other than whichwas larger and
smaller than the medium dog.
d. If the medium dog is 9 pounds more than the small dog, then the medium dog
weighs 27 pounds, and the small dog weights 18 pounds. The large dog weighs
five times as much as the small dog, so is 5 x 18 = 90 pounds.
14. Give two quantities that one could have in mind when he/she says, "That's a big athlete!"
Height, weight, popularity, …
15. Give two quantities that one could have in mind when he/she says, “This has been a good
day.”
Outside temperature, amount of work accomplished, amount of time spent playing ball
and/or picnicking, .....
1.3 Values of Quantities
ON TRUE/FALSE ITEMS, ASK FOR AN EXPLANATION IF FALSE.
1.
The label on a can of chicken broth claims that its weight is 1.4 kg. Use your metric
knowledge to tell how many milligrams this would be.
1,400,000 mg
2.
The larger the unit of measure used to express the value of a quantity, the larger its
numeric value will be.
True
False
False. The larger the unit, the smaller the numerical value will be, for describing the
same measurement.
3.
Using benchmarks, find an estimate of the following and explain how you did it.
The length of this line in metric units:
(NOTE to instructor: measure this line as it is printed out before using it in a test.)
4.
Complete the following:
a) 2.3 km = ___ m
2300
b) 2 cm = ___ km
0.00002
c) 2.14 g = ___ kg
0.00214
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5.
What metric prefix means one-hundredth? ____________
6.
If a Pascal is some unit of measure, use your knowledge of metric prefixes to complete:
4 kiloPascals = ____Pascals.
centi
4000
7.
Name a metric unit that is analogous to a quart. Which is larger? A liter is slightly larger.
8.
Name a metric unit that is analogous to a yard. Which is larger? A meter is slightly
larger.
9.
Size 1 Pampers fit babies who weigh 4 to 6 kg. Maggie weighs 11 pounds. Will Size 1
Maggie weighs 5 kilograms and Size 1 Pampers will fit her. (Alternatively, use the 2.2
pounds for 1 kilogram to change the 4 kg-6 kg range to 8.8 pounds-13.2 pounds.)
10. What are some advantages to using the metric system of measurement?
The metric system allows easy conversion of units because units differ by powers of ten.
It is used in science for this reason, in all countries. Most countries use it for all measures.
1.4 Issues for Learning
1.
Some children, when asked to solve a story problem, try different operations on the
numbers, and then decide which one seems to give the best answer. What is the danger of
solving problems in this way?
Students will not know what the answer means. They do not understand the problem, thus
they try to find an acceptable answer for the teacher, even though they cannot explain it.
2.
Many teachers teach “key words” for solving word problems. What are the limitations of
this strategy?
Same answer as above. Also, key words can be misleading—they only work part of the
time.
3. Use diagrams to solve the following problems. (Hint: Use strip diagrams such as in the
exercises for 1.4.)
a. Jesse collects stamps. He now has 444 stamps. He has three times a many stamps from
European countries as he does from Asian countries. How many of his stamps are from
European countries?
b. Silvia and Juan are buying a new table and new chairs for their dining area. Chairs with
arm rests are \$45; those with no arms are \$8.50 cheaper The table is 4 times as much
as a chair without arm rests. If they buy a table and six chairs, two with arms and four
without, what is the total price they pay?
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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
a.
European
444 in all, so 111 Asian and 333 European
Asian
\$45
b. Chairs with arm rests
\$8.50
Chairs without arm rests
Table
2 x \$45 + 4 x (\$45 – \$8.50) + 4 x (\$45 – \$8.50) = \$382
c. Joe
8 miles
Jim
2 miles
Jim drives 10 miles one way, so 20 miles both ways, and Joe drives 16
miles both ways, so together they drive 36 miles per day.
4th
7 st
29 students in all
page 9
Chapter 2 Numeration Systems
2.1 Ways of Expressing Values of Quantities, and
2.2 Place Value
ON TRUE/FALSE ITEMS, ASK FOR AN EXPLANATION IF FALSE.
1. Is the old Greek numeration system ( = 1,  = 2,  = 3, etc.) a place-value system?
Explain. No. A place-value system requires that the placement of the symbol have
meaning, which is not true of the Greek system.
2. Most present-day societies use the Hindu-Arabic numeration system. True False
True
3. How many tens are in 7654? How many whole tens are in 7654?
765.4, 765
4. How many hundreds are in 23? How many whole hundreds?
0.23,
0
5. How many tenths are in 1.03? How many whole tenths?
10.3,
10
6.
How many ones are in 4352.678? How many whole ones?
4352.678, 4352
7. In base ten, 3421 is exactly __________ ones, is exactly __________ tens, is exactly
___________ hundreds, is exactly ___________ thousands; also, 3421 is exactly
___________ tenths, is exactly ___________hundredths.
In base ten, 3421 is exactly 3421 ones, is exactly 342.1 tens, is exactly 34.21
hundreds, is exactly 3.421 thousands; also, 3421 is exactly 34210 tenths, is exactly
342100 hundredths.
8.
In base ten, 215.687 is exactly __________ ones, is exactly ____________ tens, is
exactly ____________ hundreds, is exactly _____________ thousands; also, 3421 is
exactly ___________ tenths, is exactly ___________hundredths.
In base ten, 215.687 is exactly 215.687 ones, is exactly 21.5687 tens, is exactly
2.15687 hundreds, is exactly 0 .215687 thousands; also, 215.687 is exactly 2156.87
tenths, is exactly 21568.7 hundredths.
9. (Roman numerals) IX = ____________ten and XI = ____________ten 9, 11
10. 34,597 has 345 whole thousands in it.
True
False
False. 34,597 has 34 thousands in it. (Or, if you have emphasized describing the exact
number, 34,597 has 34.597 thousands in it.)
11. 34.597 has 345 whole tenths in it.
True
False
True
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12. 56 has 560 tenths in it.
True
False
13. 23 has 230 hundredths in it.
True
False
True
False. It has 2300 hundredths in it.
14. 45 has 4500 hundredths in it.
True
False
True
15. 632.1 has 632.1 ones in it.
True
False
True
16. A soap factory packs 100 bars of soap in each box for shipment. If the factory makes
15,287 bars of soap, how many full boxes will they have for shipment? Explain.
152, because there are 152 hundreds in 15,287.
17. How many \$10 bills could one get for \$10 million?
A. 1,000,000
B. 100,000
C. 10,000
D. 1000
E. None of A-D
A
18. How many \$100 bills could one get for a billion dollars?
A. 100,000,000
B. 10,000,000
C. 1,000,000
D. 100,000
E. None of A-D
19. How many \$100 bills would make \$45 billion?
B
450,000,000
20. Judy says, "Well, hundredths are smaller than tenths. So 0.36 is smaller than 0.4."
Comment on Judy’s reasoning.
Although Judy does choose the smaller number correctly, her reasoning is risky. If the
numbers were 0.56 and 0.4, using just her reasoning would give an incorrect choice for
the smaller number.
21. Grady thinks that 0.36 is bigger than 0.4 because 36 is bigger than 4. Comment on
Grady 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
is correct or incorrect. If it is incorrect, please explain.
a) Is Dale's answer correct or incorrect? Correct __
Incorrect __
Pat’s answer: 423.1 could be thought about as 400 ones and 23.1 tenths
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a) Is Pat's answer correct or incorrect? Correct __
Incorrect __
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 __
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 __
Pat’s answer: a) Incorrect b) 400 ones and 231 tenths, or…
Jan’s answer: a) Incorrect b) 420 ones and 31 tenths, or…
2.3 Bases Other Than Ten
ON TRUE/FALSE ITEMS, ASK FOR AN EXPLANATION IF FALSE.
1. For whole numbers, any two-digit numeral in base five represents a smaller number than
the same two-digit numeral in base twenty. True
False
True
2. In base b there are b – 1 different digits. True
False
False. There are b digits: 0, 1, 2, 3, ... b–1.
3. These are the digits that are needed for a base seven place-value system: 0, 1, 2, 3, 4, 5, 6,
7.
True
False
False; 7 is not a digit used in a base seven place-value system.
4. In base b, 3 + 2b3 + b would be written ________________.
2013b
5. A place-value, base-twenty system would require _____ digits.
20
6. 524 eight = __________ ten
340 ten
7. 287ten = __________ four
10133 four
8. 1012five = ______________ in base ten.
132 ten
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9. 32ten = _______________
in base four.
200four
10. 2.31four = ________________ as a mixed number in base ten.
11. 6 23 in base ten = __________ in base three.
13
2 16
ten
20 102 three , or 20.2 three
12. 1ten = ____twelve
1 twelve
13. 214.3five = ___ in base ten
59.6 ten or 59 53 ten
1002three
14. 29ten = ___ in base three
15. 7 ten = ___ in base nine
7 nine
16. 203.6 ten = ____________ five
1303.3five
17. 2003five = _____________ten
253 ten
18. 200.3five = _____________ten
50.6 ten, or 50 53 ten
19. Write 49ten in base seven.
100 seven
20. Do the "translations" in parts A-D. Show your work.
A. 3102five = _____________ten
B. 310.2five = _____________ten
C. 203.6ten = ____________five
D. (base six pieces with small
block as the unit)
402
80.4 or 80 25
1303.3
= ______________ten
336
21. You are living and working on a planet that uses only base five. How many five-dollar
bills can you get for \$1234.20five? Write your answer in base five since you are living on
the planet. Write enough (numbers, words,...) to make your thinking clear.
123. As in base ten, 1230five = 123x10five = 123 fives. More symbolically,
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1234.20  (1  five3 )  (2  five2 )  (3  five)  something less than five
 ((1  five2 )  five)  ((2  five)  five)  (3  five)  (small)
or
= ((1  five2 )  (2  five)  3)  five  (small)
=123 five  five  (small)
So: in base 5, \$1234.20five is: 123 fives, + something small)
22. In base five, the two whole numbers immediately before 2001five are _________five and
____________ five.
1444 and 2000 (either order)
23. If you are counting in base five, what would be the next six numerals after 2314five?
2320, 2321, 2322, 2323, 2324, and 2330
24. If you have been counting in base five, what would the five numerals before 2314five
have been?
2304, 2310, 2311, 2312, 2313
25. Write how many fingers you have, in base five. In base two. In base ten. In base…
20five. 1010two. 10ten. …
26 . Which is larger? 21four or 21 five? Explain.
When a numeral has more than one digit, it will vary in value if written in different bases
because the place values will differ. 21four = 9 ten and 21 five = 11 ten
27. Consider: x = 81765fifteen and y = 81765thirteen. Which of x and y is greater? Explain.
x because each digit other than the one’s place represents more in base fifteen than in
base thirteen.
28. Consider: x = 74213sixteen and y = 74213fourteen. Which is greater, x or y (or are they
equal)? Explain. x because each digit other than the one’s place represents more.
29. Consider x = 0.3147eight and y = 0.3147nine. Which of x and y is greater? Explain. (Be
careful.)
x is larger.
0.3nine =
3
9
Consider only x = 0.3eight and y = 0.3nine. 0.3eight =
in base 10.
3
8
is larger than
3
9.
3
8
in base 10 and
Extending this reasoning, x is larger.
page 14
Answer: 2b2  4 (or, 2b2  0b  4)
30. Write an algebraic expression for 204b.
31. If a base-eight flat = 1, the numeral ______________ would give the numerical value of
the small cube. (You may give your answer either in base eight or in base ten--just make
clear which.)
0.01eight, or
1
64
in base ten
32. Base eight pieces, with the small cube (a dot here) is
asunit
the unit.
= ______________ten
1220ten
33. Sketch the wooden pieces that show 1203seven, and give the English words for the base
ten value of each different sized piece of wood.
Answer: with large dot representing a small cube, as the unit.
large cube = 7 3  343 , or three
hundred forty-three
flat = 72  49 , or forty-nine
small “cube” = 1, or one
33.
(small cube = 1; no longs)
34. Write the base b numeral for 2b4 + b2 + 3b + 1.
20131b
35. Write out 32004m in the algebraic form of the last item.
Answer: 3m4  2m3  4, or 3m4  2m3  0m2  0m  4
36. The best coins to use in thinking about the first three whole-number place values in base
five would be the penny, the nickel, and the quarter.
True
False
True
37. The best coins to use in thinking about the first three whole-number place values in base
ten would be the penny, the dime, and the half-dollar.
True
False
False. Best would be the penny, the dime, and the silver dollar
*38. If 10000ten + 10b = 10023ten, what is base b?
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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.
large cube = seven hundred
twenty-nine;
flat = eighty-one
small cube = one
61.
41. 53six names the same number as which of these base ten numerals?
A. 186
B. 183
C. 12
D. 85
E. 33
E
E. None of A-D
C
42. In base ten, 111five would be written...
A. 421
B. 155
C. 31
D. 21
43. The base b numeral 321b means...
A. 3.b2 + 2.b1 + 1
B. 3.b3 + 2.b2 + 1.b1
D. 3.b + 2.b + 1
E. None of A-D
C. 6b
A
44. In base five, 32ten would be written...
A. 152five
B. 112five
B
C. 62five
D. 17five
E. None of A-D
45. The base two numeral 100two equals the base ten numeral...
A. 1100100
B. 1011100
C. 8
D. 4
D
E. None of A-D
46. In base ten, 32four would be written...
A. 400
B. 200
C. 122
D
D. 14
E. 8
47. The base four numeral 11.1four could be written in base ten as...
A. 33 14
B. 33 101
C. 11 14
D. 5 14
D
E. None of A-D
page 16
48. The base ten decimal 18.5 could be written in base six as ...
A. 10.5six
B. 20.3six
49. The base ten fraction
A. 0.2eight
1
4
C. 30.3six
D. 128.5six
C
E. None of A-D
equals which base eight numeral?
B. 0.14eight
C. 0.02eight
A
D. 1.4eight E. None of A-D
*50. If 31b = 28ten, then b = ...
A. 4
B. 5
C. 7
D
D. 9
E. This is impossible for any whole number b.
51. What base does the following counting work in:
1, 2, 3, 4, 10, 11, 12, 13, 14, 20, 21, …
Base two
Base four
Base five
Base six
Base five
52. Which of the following is the base ten fraction representation for 1.21 four?
A. 1
9
16
B. 1
3
4
C. 1
21
100
D. 1
3
5
E. None of these.
A
2.4 Operations in Different Bases
1. Write an addition equation for (# fingers) + (# toes) = (answer) in some base other than
base ten.
(Samples) Base five: 20 + 20 = 40
Base three: 101 + 101 = 202
Base eight: 12 + 12 = 24
2. 3five  2five = _______five
11 five
3. What is 34 five ÷ 23 five ?
6
1 11
23 five or 1 13 ten
4. Show 3four x 21 four using drawings of base four materials (cubes, flats, longs, singles).
Show all the steps involved, including the intermediate steps. Make clear what your
choice of one is.
Using the small cube as the unit: First show three groups, each with 2 longs and 1 small
cube. Combine the three small cubes, and then the six longs, finally trading four longs
for 1 flat, leaving 2 longs and the 3 small cubes. 123four
5. A. Add 24five + 33five in base five. (The numbers are already written in base five, so
there should be no conversions done.)
page 17
B. How would you illustrate this with the base five blocks using drawings and showing
the intermediate steps?
A. 112five
B. With the small cube = 1, first drawing shows 2 longs, 4 small cubes and 3
longs, 3 small cubes. Next, five of the seven small cubes are traded for a long,
giving six longs and 2 remaining small cubes. Next, five longs are traded for a flat,
giving 1 flat, 1 remaining long, and the 2 remaining small cubes, or 112 five.
6. A. 0.5ten = ________ eight.
C. 84 ten = ________ three
B. 312.2four + 22.3four= ________ four
D. 2five x 43five = ________ five
E. 33.3six = ________ ten
A. 0.4eight
B. 1001.1four
C. 10010three
*7. Determine the possible value(s) for base b:
D. 141five
E. 21.5 ten
321b
– 234 b
43 b
b = six (11 b – 4 b = 3 b, or 3 b + 4 b = 11 b )
1
8. To the right is a partially completed addition, written in connection
2
1 4five
with wooden pieces. At the time of the work to the right, what
pieces of wood would be displayed, if the small block is the unit?
+ 3 3five
(Drawings or word descriptions are okay.)
2
Finish the numerical calculation. (You do not have to draw
the wooden pieces for the rest of the work.)
Two flats, five longs, and 2 small cubes, at the time of the work. (The trade of the five
longs for a flat is not reflected in the work yet.) Final sum: 302
9.
241six
+ 135six
420six
10. 127nine
 58nine
58nine
11. 4.4five
+ 3.3five
13.2 five
12. 0.24seven
0.06 seven
page 18
– 0.15seven
13. 21six
+ 35.2six
100.2 six
13. Use drawings of multibase blocks to illustrate 231ten + 87ten
Answer using a small square/block as the unit:
.
.......
Place together then trade ten longs for a flat:
........
Answer: Using small squares (dots here) as the unit: put the ones together to form 8 ones;
put the tens (longs) together to form 11 tens; trade 10 tens for a 100 (flat). One would
now have three hundreds (flats), one ten (long), and eight ones. The answer is 318.
14. Use drawings of multibase blocks to illustrate 32five + 23five
Answer: the small square is being used as the unit
..
...
.....
The five small squares can be traded for a five (long) leaving 0 ones. There are now six
longs. Five would be traded for a flat of twenty-five, leaving one five (long). The answer
is therefore 110 five
15. What base does the following addition NOT work in:
A. Base six
B. Base seven
C. Base eight
13 +13=26
D. Base ten
A
E. It works in all of these bases.
page 19
16. What step is wrong in the following (base eight):
13Eight
x12Eight
6
20
30
100
156
A. 6
B. 20
C. 30
D. 100
E. None
E
17. A. Subtract the following in base five. Show all your work:
2 2 1 five
–
4 2 five
11
2 2 11
–
4 2
1 2 4
B. Use your work in part A to explain how the way we regroup in base five subtraction is
similar to the way that we regroup in base ten subtraction.
We first consider the ones. Regrouping may be necessary to subtract, as in A, where we
regrouped to make six ones, and again when we regrouped to make seven fives.
18. Use drawings of base ten blocks to show that 3 x 15 = 45
Using the long as the unit (although another unit could be selected here)
Group ten longs to get a fourth hundred, and an answer of 45.
page 20
Chapter 3 Understanding Whole Number Operations
1. Two basketball coaches, A and B, are talking.
A says to B: "Your tallest player is 6 inches taller than my tallest player!"
B says to A: "Yes, but your second-tallest player is 8 inches taller than my second-tallest
player."
A says to B: "Hmm. My second-tallest player is 4 inches shorter than my tallest player."
Make a drawing, and tell the difference in heights of Coach B's two tallest players.
Drawing:
Diff. in heights, Coach B's _____
There are several possible arrangements. Below is one that helps to see that the
difference asked for (BT vs B2T) is 18 inches. Students may assign an arbitrary number
to the height of B’s tallest player rather than rely on their drawing. Point out that they
have unnecessarily (probably) ignored their drawing in arriving at their answer.
6"
4"
?
8"
BT
AT
A2T
B2T
2.
To determine how much older your father is than you, you need to make an additive
comparison of his and your ages.
True
False
True
3.
Marge bought several types of candy for Halloween: Milky Ways, Tootsie Rolls, Reese's
Cups, and Hershey Bars. Milky Ways and Tootsie Rolls together were 6 more than the
Reese's Cups. There were 4 fewer Reese's Cups than Hershey Bars. There were 12
Milky Ways and 28 Hershey Bars. How many Tootsie Rolls did Marge buy?
List 5 quantities involved in this problem.
Sketch a diagram to show the relevant sums and differences in this situation.
Solve the problem.
page 21
The five quantities are usually easy: E.g., the number for each type of candy, and some of
the explicit comparisons mentioned. Here is a diagram, with the deduced numbers of bars
in parentheses, giving 18 TRs (start with the HBs, then determine the RCs, then the
TR+MW total and finally the TRs).
(30)
6
4
#TR
(18)
#MW
12
#RC
(24)
#HB
28
4. The school cafeteria is ready to serve two kinds of sandwiches, roast beef and peanut
butter, and two kinds of pizza, cheese and vegetarian. There are 60 servings of pizza
prepared. There are 8 fewer roast beef sandwiches prepared than there are servings of
cheese pizza. There are 6 more peanut butter sandwiches prepared than there are servings
of vegetarian pizza. All together, how many servings of sandwiches are prepared?
a. List 8 quantities involved in this problem.
b. Sketch a diagram to show the relevant sums and differences in this situation.
c. Solve the problem.
Again, depending on whether you have used the earlier, similar problem, many of your
students will use algebra or trial-and-error on this problem; we suggest, for now,
prohibiting algebra. You might also consider omitting part c. But the problem can be
solved with the use of a drawing, as seen below.
a. Number of kinds of sandwiches, number of kinds of pizza, number of servings of pizza
prepared, difference in number of roast beef sandwiches prepared vs number of servings
of cheese pizza, difference in number of peanut butter sandwiches prepared vs number of
servings of vegetarian pizza, total number of sandwiches prepared, number of roast beef
sandwiches prepared, number of peanut butter sandwiches prepared, number of servings
of cheese pizza, number of servings of vegetarian pizza, difference in number of roast
beef sandwiches and number of peanut butter sandwiches, total number of servings of
pizza and sandwiches,…
b. There are other possible praiseworthy drawings possible, but the following suggests the
solution (for the total number of pizza servings and sandwiches) pretty easily.
c. The number of sandwiches is (60 + 6) – 8 = 58.
page 22
60
#CP
#VP
8
6
#RB
#PB
5. A local community college has two sections of Math 210 (Sections A and B), and two
sections of Math 211 (Sections C and D). Together, Sections C and D have 46 students.
Section A has 6 more students than Section D. Section B has 2 fewer students than
Section C. How many students are there in Section A and Section B all together?
a. For each given value write the quantity next to it.
b. Sketch a diagram to show the relevant sums and differences in this situation.
c. Solve the problem. Show all your work here.
a. 46 students, total number of students in C and D
6 students, difference in numbers of students in A and D
2 students, difference in numbers of students in B and C
b. (sample drawing)
46
C
D
2
6
B
A
c. (46 – 2) + 6 = 50 students for Sections A and B together
1.
Here are two word problems. How do they differ conceptually?
Silvia had 14 books, and then received 4 more books. How many books does she have
now?
Silvia has 14 books on one shelf and 4 books on another. How many books are on the two
shelves?
page 23
In the first, there is an action implied. In the second there is not. Because it is harder to
“act out” the second problem, in may be more difficult for some young children.
2. A first grade teacher always reads subtraction statements such as “7 – 5 = 2” to his class
as “seven take away five is two.” That is, he always reads the minus sign as “take away.”
Comment on why this might not be a good practice.
Reading “–“ only as “take away” ignores the fact that other situations—comparison and
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:
a. How many more candy bars did Jack have than Bill?
b. How many toothpicks would you need to act the problem out? Explain your answer.
What type of subtraction is this?
a. 4
b. 12, because there are the two separate amounts; this situation involves an additive
comparison.
5. For a, b, and c below, state:
1) the operation you would use to answer the question,
2) the situation in which the problem fits, and
a. Susan has \$175. She wants to go on a skiing trip that costs \$250. How much more
money does she need?
b. John is 6 ft 1 in. tall and Steve is 5 ft 9 in. tall. How much taller than Steve is John?
c. Karen has four fish in her aquarium. She puts three more in. How many fish are in
the aquarium now?
a. 1) subtraction
b. 1) subtraction
2) comparison
2) join
3) 250 – 175 = 75 (75 circled)
3) 6’1” – 5’9” = 4” (4” circled)
3) 4 + 3 = 7 (7 circled)
page 24
6. Rita is given this problem: Zetta has \$39, but she needs \$78 to buy a jacket she wants.
How much more does she need?
Rita's reply: "79 minus 40 is 39, so she needs \$39."
Explain Rita's reasoning. What is your reaction to this method of doing the problem?
Rita has increased the minuend and subtrahend by the same amount, so the difference
stays the same. (Think of a comparison subtraction drawing, even though this is a
7. Filene is asked “What is 79 minus 32?" She responded: “8 to 40, 30 to 70, and 9 more, so
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
Josh received a nickel (to 40 cents), a dime (to 50 cents) then probably two quarters to
make \$1, then a ten dollar bill, which would add up to \$11.65. This was not correct; he
should have received \$12.65. After the coins, Josh should have been given another dollar.
“40, 50, 88, 89, 90, and 100” is probably what the cashier said, distinguishing coins from
bills as she handed them out.
9. a. Make drawings of circular "pizzas" to illustrate 6 – 2, take-away view.
The drawing should show 6 circles, with 2 being removed by arrows or otherwise marked
out in some way.
b. A child is shown 9 apples and 6 oranges, and asked “How many more apples than
oranges?" She says that apples and oranges are different things, and so she doesn’t
understand the question. What might you do to help her?
(One possible way....) Line up the apples then the oranges below, and ask how many
apples don’t have an orange partner, then ask whether there are more apples than oranges,
and how many more.
10. Finish the story so that your question could be answered by the given calculation, and so
that your story involves the view given.
a. 6 – 2.5, missing addend. "The two joggers decided to run at the beach...
page 25
b. 6 – 2.5, comparison. "The two joggers decided to run at the beach...
a. They usually run 6 miles. How much farther do they have to run, if they have already
run 2.5 miles?
b. One runs 6 miles and the other runs 2.5 miles. How much farther does the first jogger
run than the second jogger does?
11. In each of the following, which way of thinking about subtraction is involved?
a. This story problem: "Basketball score: Aztecs 82, Opponents 69. By how many
points did the Aztecs win?"
b. The following thinking/drawing strategy (sometimes used with children having trouble
with their basic subtraction facts):
For 15 – 7, think
of going "up the
hill," going to 10
along the way…
+5
+3
15
( add 3 to get to 10, then 5 more to get to
15) So, 15 – 7 = 8
10
________________________
7
a. comparison
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 .
b. For the same problem, finish the problem to so that it involves a take-away subtraction.
c. . For the same problem, finish the problem to so that it involves a missing-addend
subtraction.
Samples a. .....and 3 12 gallons of Kool-Aid. How much more lemonade than Kool-Aid
did you make?
b. At the party, the people drank 3 12 gallons of the lemonade. How much of the lemonade
was left after the party?
page 26
c. 3 12 gallons were made from frozen lemonade and the rest from fresh lemons. How
14. a. Finish this story problem so that it involves a comparison subtraction that could be
solved by 26–12.
Laresa had \$26 when she went into the store....
b. For the same problem, finish the problem to so that it involves a take-away subtraction.
c. For the same problem, finish the problem to so that it involves a missing-addend
subtraction.
Samples
a. .....and her friend Tisha had \$12. How much more did Laresa have than Tisha?
b. She bought a wallet for \$12. How much did she have left.
c. She had \$12 and then received cash for baby-sitting. How much did she earn babysitting?
15. Give the rest of the "family of facts" for k – 3 = p.
Any order:
3+p=k
p+3=k
k–p=3
3.3 Children’s Ways of Adding and Subtracting
1. Following is only the start of a child's work (in base ten). What seems lacking in this
child's understanding?
402
– 39
506
–149
…7
…3
The child seems to be unaware of what subtraction means, and is just working with the
subtraction of the digits in the column without regard to the order. She does not have
place-value understanding.
2. Perform the following using the "equal additions" method, as used by one student in the
432
–287
4 1 3 12
–3 2 98 7
1 4 5
page 27
OR: Adding a ten to the 2 to make 12, and a ten to the 80 to make 90, then ten tens to the
30 to make 13 tens, and 1 hundred to the 200 to make 300, allows all the subtractions (12
– 7, 13 – 9, 4 – 3), giving 145.
3. The work of two students is shown below. Each student "invented" the method used, that
is, it was not taught to the student. For each student figure out what the student was
thinking while doing the problem. Then (i) work the second problem using the same
method as the student, and (ii) comment on the student's method in terms of the "number
sense" exhibited.
a.
732
-2 4 5
513
(i) 8 3 4 1
-4567
(ii)
b. 19 x 35. Well, 20 x 35 is like 10 x 35 two times, so that's 350 two times, which
is 700. But that's 20 35s and I only want 19 of them. So 700 minus 30 is 670
minus 5 is 665.
(i) 21 x 43
(ii)
a.
(i) 4226
(ii) This student is calculating larger – smaller in each place value,
ignoring what is being subtracted from what. The student is showing no number
sense, or awareness of what subtraction means.
b.
(i) 903: 20  43 = 860: 860 + 43. 860 + 40 = 900. 900 + 3 = 903. (ii) This
student is showing excellent number sense (and operation sense, in that he/she
knows that 21 forty-threes can be obtained by adding 20 forty-threes and another
43), in working with place values independently, and in a fashion that shows
awareness of “easy numbers.”
4 Is this child's thinking all right? If it is, complete the second calculation using the child's
method. If the thinking is not all right, explain why not.
(given)
675
– 198
(child does)
second calculation (or explanation if not ok)
677
453
– 200
– 295
477
Okay. Second calculation (in columns): 458 – 300 = 158
5.
A visitor to a first-grade classroom saw a teacher ask a child to solve this problem: Jaime
gets \$5 a week for keeping the yard in good shape. He is saving his money for the country
fair. After 4 weeks, how much has he saved?
She thinks to herself: This is a multiplication problem, and first-graders have not yet been
taught multiplication, so they can’t answer this problem..” But after a few minutes Li-Li
say that the answer is 20. She explains how she did this problem and she did not do any
page 28
formal multiplication, much to the visitor’s surprise. What did she most likely do to find
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
She probably subtracted 2 from 8 four times, until she reached 0, then counted the
number of times she subtracted 2–– 4 times.
6.
Felisha was asked to find 413 – 248. Here is how she did this problem:
413
–248
–5
-30
200
165
Is her answer correct? Explain what she was doing. Find 9456 -3789 using this method.
Yes, her answer is correct. She was finding partial addends, using negative numbers, then
9456
–3789
–3
-30
-300
6000
5667
7.
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
page 29
have to do tens. 20 plus 80 equals 100 and places that under the 110. She then adds 8 and
6 and writes 14 and writes that below the 100. Adding, she says it is 224.
Where does Zenaida go wrong? Discuss her place value understanding.
Zenaida is on the right track and appears to have some knowledge of place value, but it is
not strong enough to carry her through this problem. She adds from left to right,
indicating that either she does know the standard algorithm, or just prefers this method.
Her major error was to write ten hundred as 110, probably thinking ten and a hundred is
10 hundred.
9.
Here is Ben's work:
2
3
3/ 0 2
– 9
203
4/ 0 17
–108
209
1
Is Ben’s method correct? Make up another subtraction problem that would lead Ben to
apply his same method. Then finish the calculation as Ben would. Show the work Ben
would do.
Various. The story problem should involve a 0 in the middle of what will be the
minuend. Ben will incorrectly rename or “borrow” from the hundreds place directly to
the ones place.
10. Find 21 + 49 using an empty number line.
+ 40
+4 +5
21
61 65
11. Find 509 – 239 using an empty number line.
70
– 200
– 30
–5
270
300
–4
500 505 509
12. A student wrote the following answer to her problem 95 – 34:
page 30
95 – 34 =
95 – 35 = 60
60 – 1 = 59
Analyze this student’s thinking.
The student noticed that 35 is compatible with 95, but after subtracting 35 the answer
would have to be adjusted. Unfortunately, she does not realize that in subtracting 35, she
has already subtracted 1 too much, and that 1 should be added to 60, not subtracted from
it.
3.4 Ways of Thinking About Multiplication
1. a. Make sketches for 3  6 and 6  3 and contrast them.
b. Make sketches of
1
2
 6 and 6 
1
2
a. The 3  6 drawing should show 3 groups of 6 things, such as 3 six-packs of a softdrink, or 3 sets of 6 objects of some kind. The 6  3 sketch should clearly show 6 sets of
3 things, or a 3  6 array that turned on its side is 6  3.
b. The 12  6 should clearly show 6 objects, with 12 of them designated either as three
objects or as half of each of the six objects, such as 6 circles of which 3 are shaded. The 6
 12 should show 6 objects that are halved, such as 6 semicircles.
2. A designer of women’s “mix and match” clothing designs 3 styles of skirts, 2 pairs of
pants, 3 types of tops, and 4 styles of jackets. How many different outfits could be
purchased, if each outfit has a skirt OR pants, a top, and a jacket? (Assume that a woman
will not wear a skirt and a pair of pants at the same time.)
(3+2)  3  4 = 60, or possibly (3  3  4) + (2  3  4) = 36 + 24 = 60.
3. A clothes designer designs women's "mix and match" wardrobe with 2 styles of skirts, 1
pair of pants, 3 types of tops, and 2 styles of jackets. How many different outfits could be
purchased, if each outfit has a skirt or pants, a top, and a jacket?
(2+1)  3  2 = 18
4. Mitchell decides to get his car painted and to buy new hubcaps. He selects 5 colors he
likes and 3 styles of hubcaps. Then he decides to paint the roof a different color than the
body. He decides to let his wife make the final decision. How many choices does she
(5  4)  3 = 60, assuming color compatibility.
5. Make up a story problem about a bake sale, so that the problem could be solved
page 31
a. by
3
4
 12. (Notice the order.)
b. by 12  43 .
a. Various possibilities. Each should involve 3/4 of some quantity with 12 as its
numerical value. Example: There were 12 chocolate cakes, and 3/4 of them were sold
by 10:00. How many chocolate cakes were sold by 10:00?
b. Various possibilities, but each should involve 12 amounts, each with numerical value
3/4. Example: They had 12 cakes, and by 10:00 they had sold 3/4 of each cake. How
much cake had they sold by 10:00?
6. Make drawings of circular "pizzas" to illustrate each of the following.
a. 3  4, array
b. 13  6, fractional part of an amount
a. Three rows, or sets, each with 4 pizzas. (NOT 4 sets of 3 each)
b. Six pizzas, with sets of two delineated and one of those sets indicated, OR with 1/3 of
each pizza indicated. (Contrast continuous pizzas with discrete children, say.)
7. Give our label (e.g., take-away, etc.) for the situation, and write the equation for solving
this problem.
"A coffee shop has 4 kinds of pastries that you like. You always drink coffee, tea, or milk
with your pastry. In how many ways could you place a pastry-plus-drink order?"
Fundamental counting principle. 4  3 = n
8. Make up a story problem that could be solved by 16 
Be attentive to the order of the factors.
1
2
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.
page 32
True
False
False. If m is a (positive) fraction less than 1 (and n is a positive number—we don’t
usually take off if this is omitted because at this stage only non-negative numbers have
been the focus), the product mn will be less than n. 1/2 x 6 is 3 and 3 < 6.
3.5 Ways of Thinking About Division
1.
a. This is a typical problem from an elementary textbook:
Jasmine works in a book store. Today three boxes of Harry Potter books arrived. There
are 144 books in each box. Jasmine is told to stack the books in piles in an area of the
book store. She is told to put the books into 16 piles. How many piles can she make?
What interpretation of division is represented in this problem?
b. What if the question changes to She is told to put 27 books in each pile? What
interpretation of division is now represented?
a. There are 432 books. She could do this problem by putting one book down 16 times,
then a second book on top 16 times, etc. This is the partitive or equal sharing
interpretation of division.
b. This time Jasmine would put 27 books in a pile, then 27 in another pile, etc. She is
“taking-away” 27 books at a time, and she can do this 16 times. This is the quotitive or
measurement or repeated subtraction interpretation of division.
(Admittedly, if Jasmine is working in a bookstore, she probably knows enough to simply
divided 432 by 27 or by 16. But to do that, she must have some ideas about division that
she learned in school, doing problems of both types.)
2.
Make up a story problem involving quantities of ice cream in an ice-cream store, so that
the problem could be solved by the calculations given:
a. Can be solved by 2  18 .
b. Can be solved by 16  18 .
c. Can be solved by 43  24 .
Samples: a. How many 1/8 quart servings can you get from 2 quarts?
b. The store puts 1/8 quart on each cone. How many pints would they use for 16 cones?
c. The store stocks 24 different kinds of ice cream. Three-fourths of them are changed
every month. How many kinds are changed every month?
3.
If a is any number other than 0, then 1 ÷ a is less than 1.
True
False
False. If a is a (positive) fraction less than 1, then 1 ÷ a is greater than 1
4. Under a repeated-subtraction interpretation, 43  1 12 means
_______________________________________________
page 33
The quotient is ____________. Verify and explain your answer with a sketch.
…how many 1 12 s are in, or make,
answer, 12 of one 1 12 , is in 43 .
3
4
? The answer is 12 . The sketch should show the
5. a. Decide which type of division the following word problem is depicting and explain
Mr. Burke's class of 24 fourth graders is doing a project on keeping the environment
clean. There are 6 different topics the students need to explore, and Mr. Burke wants the
same number of students to explore each topic. How many students will be in each
group where each group explores a different topic?
b. Write another word problem that illustrates the other type of division using the same
context as the problem above.
a. Sharing equally, or partitive, division. The 24 students are to be put into 6 equal sized
groups.
b. (Repeated subtraction, or measurement, division) The 24 students are to be put into
teams of 6 to work on projects about keeping the environment clean. How many
projects will Mr. Burke have to grade?
6.
Write a word problem for 37 ÷ 5 for which the answer would be 2.
Sample: Thirty-seven children want to play a game that involves teams of 5 players.
How many children won’t be on a team (but may get to be substitutes)?
7.
Consider this problem situation, which would involve dividing by 3:
"You are putting reading books on 3 shelves in your classroom. So the books look neat,
you put the same number on each shelf. How many books will be on each shelf?"
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.
page 34
Samples: (Repeated subtraction, or measurement) The big bag has 48 plastic cars, to be
put into bags holding 6 cars each. How many bags of cars will there be? (Partitive, or
sharing) The big bag has 48 plastic cars, to be split fairly among 6 youngsters. How
many cars will each youngster get?
9. Write a word problem for 37 ÷ 5 for which the answer would be 7.
Sample: Thirty-seven children want to play a game that involves teams of 5 players.
How many teams can be formed?
10. Circle each which is undefined:
0 ÷ 6,
6 ÷ 0,
0 ÷ 0, and explain why any
undefined one(s) is undefined. If the symbol is defined, tell what it equals.
Undefined: 6÷0 and 0÷0. Explanations should reflect your emphasis in class, most likely
through examination of a related multiplication “check.” 0÷6 = 0.
11. Finish the following story to make story problems that could be solved by the indicated
calculation.
"The farmer has a 3 12 acre orchard of orange trees....
a. 23  3 12
b. 3 12  5
c. 3 12  0.8
Samples: a. …She and her workers have harvested 2/3 of the orchard. How many acres
have they harvested?
b. …If she wants to replace all the trees over a 5-year period, how many acres should she
plan to replace each year?
c. …One sprayerful of fertilizer can cover 0.8 acre. How many sprayerfuls will she need
to cover the whole orchard?
12. Make drawings of circular "pizzas" to illustrate 4 12 ÷ 3, sharing equally.
There should be 4 1/2 pizzas shown, with marks to show how each of three equal shares
“gets” 1 1/2 pizza. Just showing 4 1/2 pizzas and then 1 1/2 pizza is not a good answer.
3.6 Children Find Products and Quotients
1. Is this child's thinking all right? If it is, complete the second calculation using the child's
method. If the thinking is not all right, explain why not.
page 35
(child's work)
124
 15
1000
200
40
500
100
20
1860
second calculation (or explanation if not ok)
132
 14
a. Okay. Second calculation (in columns): 1000 + 300 + 20 + 400 + 120 + 8,
sum = 1848.
2. Fiesha finds 32  54 as follow:
54
 32
1500
120
100
_ 8
1728
a) Which is true of Fiesha’s mathematical steps?
__ Fiesha’s steps are
mathematically correct.
__ Fiesha’s steps are
mathematically
flawed.
__ I cannot tell if Fiesha’s steps are
mathematically correct or
flawed.
b) Understanding of multiplication
__ Fiesha doesn't appear to
understand multiplication.
__ Fiesha may or may not
understand
multiplication.
__ Fiesha shows good understanding
of multiplication.
c) If Fiesha’s steps are mathematically correct, use her way of thinking to solve 24  53.
If they are not, explain how Fiesha’s reasoning is flawed.
Fiesha: a) Reasoning okay
b) Shows good understanding of multiplication.
c) Probably (in columns) 1000 + 60 + 200 + 12 in some order; sum = 1272
3. Amy finds 32  54 as follows:
54 is 4 more than 50, so find 32  50 and add 4 back to get 1728.
a) Which is true of Amy’s mathematical steps?
page 36
__ Amy’s steps are
mathematically correct.
__ Amy’s steps are
mathematically
flawed.
__ I cannot tell if Amy’s steps are
mathematically correct or
flawed.
b) Does Amy show understanding of multiplication?
__ Amy doesn't appear to
understand multiplication.
__ Amy may or may not
understand
multiplication.
__ Amy shows good understanding
of multiplication.
c) If Amy’s steps are mathematically correct, use her way of thinking to solve 24  53. If
they are not, explain how Amy’s reasoning is flawed.
Amy: a) Reasoning probably okay, although the phrasing is not perfect (should be “add
four 32s back”).
b) Shows good understanding of multiplication.
c) Probably 20  53, plus 4  53.
4. Antonio asks, “When I multiply [for example, 49  23, shown
to the right], why do I have to put in the 0 [points to the zero in
980]?”
What would you say to Antonio?
49
23
147
980
1127

The 980 comes from 20  49, so it is a number of tens.
5.
Following is an example of a child's work. You are to study the work and then to judge
the student’s understanding.
Hiro was asked to divide 4240 by 6. His work is shown below.
Hiro's work:
7 6 R4
6 4240
42
040
36
4
a) Is Hiro's work correct or incorrect? Correct __
b) If the work is incorrect, please explain how.
Incorrect __
a) Hiro’s work is incorrect. b) In considering the 04 (the number of tens left), Hiro
forgot to note in the quotient space that there are 0 tens for 40 ÷ 6.
page 37
6.
Consider the following work of a student:
84
A. There is an error with the 20
x 45
B. There is an error with the 400
20
C. There is an error with the 160
400
D. There is an error with the 320
160
E. There is no error with this student’s work
D
320
900
7.
Use a nonstandard algorithm to calculate 128 x 67.
Various methods, giving 8576 as the product. We usually get the long version (six partial
products).
3.7 Issues for Learning: Developing Number Sense
1.
In each pair, choose the larger. Explain your reasoning. Your justification should appeal
to number and operation sense, not to computation.
a. 1838 + 517
or
1836 + 514
b. 612 – 29
or
613 – 34
c. 0.578
or
0.002 + 0.0328
a. 1838 + 517 because each addend is larger than the corresponding one in 1836 + 514
b. 613 – 34 because 612 – 29 is the same as 613 – 30, OR 613 is only 1 more than 612,
but subtracting 34 rather than 29 more than overcomes that.
c. 0.578 because the sum of the addends in the second sum will not reach 0.5.
2. I am a number with 21 tens, 14 ones, and 11 tenths. What number am I?
225.1
3. Tell why the following are incorrect:
a.
310
225
980
375
1895
b. 280 ÷ 70 = 40
c. 480 ÷ 0.4 = 120
a. The ones' column adds to 10, not something ending in 5.
page 38
b. There are only 4 seventies in 280.
c. There are 480 ones in 480, so there will be far more 0.4s than that in 480.
page 39
Chapter 4 Some Conventional Ways of Computing
4.1 Operating on Whole Numbers and Decimal Numbers
1.
Show 3335 ÷ 23 with a scaffolding algorithm, then by the standard algorithm and show
how each number in the standard algorithm is associated with number in the scaffolding
algorithm.
145
23 3335
23 3335
2300
100
23
1035
103
460
20
92
575
115
460
20
115
115
0
69
3
46
2
46
___
0
145
In the second algorithm, the 23 actually is 2300, yielding 100 in the quotient. The 103
is actually 1030, from which 920 (that is 23x 40, which 460 twice, making the first
division easier) is subtracted, leaving 115 in both algorithms. In the first algorithm, 115÷
23 is done in two steps, and in one step in the second algorithm, both times yielding 5. The
first scaffolding algorithm could be done in multiple ways yielding the same result.
2. Use the scaffolding method to compute 5883 ÷ 17.
Something along the lines of the following, which unnecessarily gives the best guesses
for each place value (one of the talking points for the scaffolding algorithm):
17 5883 |
5100 | 300
783 |
680 | 40
103 |
102 | 6
1 346
3. Show, using 324 ÷ 28, how to work from the scaffolding algorithm to the standard
algorithm.
Similar to 1. Student’s work should show an awareness of the scaffolding algorithm
format to an abbreviated form to the usual US form.
page 40
4. Do 32  467 using the method of writing all partial products. What does this algorithm
have to offer that the standard algorithm does not?
In some order, 12000 + 1800 + 210 + 800 + 120 + 14 (= 14944). This algorithm should
“make sense” since it takes into account the place value of each digit.
5. Use a nonstandard algorithm to find 240five + 314 five , but showing all partial sums.
This work is all done in base five
240
+ 314
4
100
1000
1104
6.
Name two positive and two negative aspects of learning nonstandard algorithms.
Samples: Positives—Practice reasoning about the operations and place values; if studentgenerated, they can make sense to them; encourages a “make sense” view of
mathematics; can be more efficient in selected calculations. Negatives—Time away from
conventional algorithms, which always work; students who attempt to just memorize the
techniques without understanding them will likely garble them.
7.
Make a drawing of base ten materials that shows the initial set-up for 3  130.2. Make
clear what = 1. Do not take time to draw all the later steps of the calculation with the
base ten materials.
With the long = 1, the drawing should show first three groups (rows are nice), with 1
large cube, 3 flats, and 2 small cubes in each group.
8.
Draw how one would act out 200 – 62 (take-away view) to support the usual right-left
algorithm, with base ten materials. Make a separate drawing for each step (add steps if
you need them).
Initially
second
page 41
third
fourth
(etc. as needed)
Initially, 2 flats. Second, trade a flat for 10 longs. Third, trade a long for 10 small cubes,
giving 1 flat, 9 longs, and 10 small cubes. Fourth, take away (x-out, say) 2 small cubes
and then 6 longs, leaving 1 flat, 3 longs, and 8 small cubes.
9.
Draw how one would act out 200 – 62 (comparison view) to support the usual right-left
algorithm, with base ten materials. Make a separate drawing for each step (add steps if
you need them).
Initially, 2 flats, and in a row below 6 longs and 2 small cubes. In the top row trade 1 flat
for 10 longs, and then one of the longs for 10 small cubes, giving 1 flat, 9 longs, and 10
small cubes above the 6 longs and 2 small cubes. Comparing the two rows, starting with
the small cubes, shows that (for the conventional right-left algorithm) the top row has 8
more small cubes, 3 more longs, and 1 more flat.
10. You decide to introduce your fourth-graders to the long-division algorithm with one-digit
divisors, using base ten materials and 96 ÷ 3. You also want to use a story problem that
they would find interesting as the basis for their work.
a. From the practical standpoint of acting out the calculation, which way of thinking
problem for 96 ÷ 3?
b. Write such a story problem (involving 96 ÷ 3).
c. Show how you would act out your story problem, with drawings of the base ten
materials.
d. Write a second story problem for 96 ÷ 3, involving a different way of thinking about
division from the way of thinking in your story problem in part b.
page 42
e. Answer a-d if you were to use 960 ÷ 320 instead of 96 ÷ 3.
a. Sharing equally is more practical for 96 ÷ 3, since acting out that calculation with
repeated subtraction of 3s would be unwieldy.
b. Various. The 96 should be put into 3 (equal-sized) amounts or groups.
c. With 9 longs and 6 small cubes, “deal” them out to three locations equally. You
d. Various. This time the situation should call for how many groups of size 3 are in, or
make, a group of 96. Notice that the units for the 3 and the 96 should be the same.
e. For 960 ÷ 320, repeated subtraction is much more practical, etc.
11. A student places multibase blocks on the table as follows:
••••••
then
••••••
Write which calculation that the student might be doing, with an explanation:
A.
B.
C.
D.
226 + 49
226 + 49
226 – 124
226 – 118
We first see 226. To add 49, the 12 ones would first be placed together and replaced by
one long and 2 ones. That is not done here. To subtract 49, the first step would be to break
a long into 10 ones, but that is not done here. To subtract 124, I can remove 4 ones, but I
need to change one flat to 10 longs before I can subtract tens. This is done here. To
subtract 118, the first step would be to change one long to 10 ones. That is not done here.
Thus, C is correct.
12. What would the next line be, in a Russian peasant calculation of 23  624?
You do not have to do the complete algorithm.
23  624
11  1248
13. The Russian peasant method for multiplying uses two basic processes:
doubling and _______________________.
…halving.
14. Below is a worked-out calculation of 313  42, using the lattice method for
multiplication. Explain why the method does give the correct number in the tens place
(the circled 4). (Note: Some current textbooks use this algorithm to teach multiplication
of whole numbers.)
page 43
3 1 3 x
1 12 04 1 2 4
3 06 0 2 0 6 2
1 4 6
The circle 4 comes from the diagonal 2+0+2. The top 2 comes from 3  4(0) = 12(0), so
that 2 is describing a number of4 tens in the product. The 0 comes from 3  2 = 06, and
shows that that partial product does not contribute a whole number of tens to the product.
The bottom 2 comes from 1(0)  2 = 2(0), showing that it is counting the number of tens
from that partial product.
15. Write a word problem that would require solving 540 ÷ 4.
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
b. Write a story problem where the answer would be 7.
Possible: Twenty-five children where going on a field trip. Parents escorting the
children allowed no more than four children in each car. How many cars were needed?
c. Write a story problem where the answer would be 1.
Possible: Jake was buying school supplies for his four children. He bought a pack of
25 pens. After dividing them evenly among his children, with each child getting the
maximum amount possible, how many pens did he have left for himself?
d. Write a story problem where the answer would be 6 14
Possible: Carolyn had 25 yards of fabric to make 4 identical costumes for a play. How
much fabric did she allocate for each costume?
page 44
Chapter 5 Using Numbers in Sensible Ways
5.1 Mental Computation
1. Describe 3 different ways that you could MENTALLY calculate 16  25.
Some possible ways: 4  4  5  5 = 4  5  4  5 = 20  20 = 400
(10  25) + (6  25) = 250 + 150 = 400
(16  20) + (16  5) = 320 + 80 = 400
16  100
4 = 4  100 (after dividing 16 by 4) = 400
2. For each of the following, MENTALLY calculate the EXACT ANSWER and write it in
the blank. Use EXCELLENT NUMBER SENSE. Then write enough to make clear
how you thought.
a.
3618 + 2472 – 2618 – 472 = ________
b.
(25  29) + (25  11) = ________ Thinking:
Thinking:
a. 3000 Thinking: 3618 – 2618 = 1000. 1000 + 2472 = 3472. 3472 – 472 = 3000
b. 1000 Thinking: Given = 25  (29 + 11) = 25  40 = 25  4  10 = 100  10.
3. Give the exact answer mentally: 73.8 + 511.37 + 24 – 73.8. Write how you thought.
535.37, taking advantage of the subtraction of 73.8 and the addend 73.8, then working
with the 11 + 24.
4. Describe how you would MENTALLY compute the EXACT result in each of the
following without using the standard algorithm: YOUR DESCRIPTION SHOULD BE
CONCISE AND INCLUDE THE EXACT RESULT.
A.
234 – 119
Description:
B.
12% of 150
Description:
C.
25 
2
5
Description:
Samples:
A. 115 Change to 235 – 120, then work left-to-right.
B. 18 2% of 150 is 3, and 12% of 150 is 6 times as much as that.
C. 10 25  25 will give the same number, and 15 of 25 is 5.
5. Show how you would mentally compute the exact results:
A.
3000 - 2575
B.
0.75 x 24
page 45
C.
24  13 + 24  7
Samples: A. 425
B. 18
C. 480
One way: 2575 + 25 = 2600, plus 400 to get to 3000.
Second way: Add 25 to each to change to 3025 – 2600.
3/4 of 24. 1/4 of 24 is 6. 3  6 = 18
Given = 24  (13 + 7) = 24  20.
IF YOUR STUDENTS DO NOT KNOW ORDER OF OPERATIONS, REFER THEM
TO THE APPENDIX WITH A REVIEW OF PROCEDURES.
6.
Show how you would mentally compute:
A. 0.75  24
B. 34  12 + 34  8
C. 3458 – 1734 – 400 + 1734
A. 18, from 3/4 of 24
B. 680, from 34  (12 + 8)
C. 3058, from 3458 – 400 (the other terms give 0)
7.
Give the exact answer by mental calculation. Then write down your thinking. You are
asked for a second way of thinking in parts B and C.
a. 479.38 + 18.9 + 2.4 – 479.38 = ____________ Thinking:
b. 12  125 = ________
One way of thinking:
c. 1714 – 897 = ________
A second way of thinking:
One way of thinking: A second way of thinking:
d. 20% of 45 = ____________ Thinking:
e. 24  750 = ___________
Thinking:
a. 21.3 Thinking: The first and last give 0, so it is only a matter of finding 18.9 +
2.4, which = 20 + 13 tenths, or 20 plus 1 and 3 tenths.
b. 1500 One way: 3  4  125 = 3  (4  125). A second way: 3  4  25  5 =
3  (4  25)  5. Students often offer 10  125 = 1250 plus 2  125 = 250, so
that’s why we ask for two ways here.
c. 817 One way: Change to 1717 – 900 by adding 3 to each. Second way
(missing addend): 897 + 3 = 900. Plus 814 more (perhaps in steps). 817.
d. 9
Thinking: 1/5 of 45.
page 46
e. 18 000 Thinking: 24  750 = 24  75  10. 24  75 = 6  4  75 = 6  300 =
1800.
1800  10 = 18,000. Of course, there are many other ways, such as
24(1000 – 250) = 24,000 – 24 x 250 but 24 x 250 = 6 x 4 x 250 = 6 x 1000 =
6000. Finally, 24,000 – 6000 is 18,000.
8.
A person can reasonably calculate the exact answer to 1563 – 198 mentally by...
A. counting on his/her fingers
B. calculating 1565 – 200
C. calculating 1565 – 200 – 2
D. calculating 1600 – 200
E. This calculation is impossible to do mentally.
B
9. A person who is calculating the exact answer to 18  15 mentally starts by calculating 2
 15. The person would finish the mental calculation by calculating...
A. 9  30
D. 60  4.5
B. 27  10
C. 36  7.5
E. None of A-D works.
A
10. For each, describe two different strategies for performing the following computation
mentally.
a.
29 + 58
c. 8  15
b. 74 – 28
a. 30 + 57 = 87; 20 + 50 is 70, plus 9 is 79, plus 1 is 80, plus 7 is 87
b. 76 – 30 = 46; 74 – 24 is 50, – 4 more is 46
c. 4  30 = 120; 8  10 + 8  5 is 80 + 40 is 120; 8  5 = 40, 40  3 = 120
11. Determine the following mentally, writing enough to make your mental work clear.
a. 40% of 80
b. 15% of 300
c. 20% of 14
d. 100% of 71
e. 5% of 60
f. 120% of 20
g. 15 is 25% of ?
h. 14 is 50% of ?
a. 32
b. 45
c. 2.8
d. 71
e. 3
f. 24
g. 60
h. 28
12. Tell how one might mentally compute the following:
a. 25 x 104
b. 25% of 104
c. 200% of 104
d. 200% larger than 12
e. 200% as large as 12
Each can be done in a variety of ways. Here are some possiblilities:
a. 25 x 100 + 25 x 4 is 2500 + 100 is 2600 or
100
4
x 104 = 100 x 26 = 2600
page 47
b.
1
4
of 104 is
1
4
of 100 +
1
4
of 4 is 25 + 1 is 26.
c. Twice 104 is 208.
d. 100% larger than 12 is 24, so 200% larger than 12 would be 36.
e. Twice as large as 12, so 24.
5.2 Computational Estimation
NOTE: Only a few ways are shown to estimate a calculation. Other ways may also
be correct. Calculators should not be allowed for items involving estimation or
mental computation.
1.
Show how you would estimate:
a.
b.
c.
d.
391  612
0.74  798
32% of 19
196% of 25
a. Round 391 to 400 and 612 to 600; 400  600 = 240,000
b. Around 3/4 of 800, which is 600
c. 6 is a good estimate because a third of18 is 6.
d. Just under 50, because 2  25. (Also, since it is 4% of 25, or 1, less than that, the exact
2
Using benchmarks, find an estimate of each of the following and explain your reasoning.
a. 7.5 % of \$594.
b. 0.32147  (67.557% of 89.4853)
c. 2 89  ( 13
24  0.99184)
d. 5.8 ÷ 12
a. 45. 7.5% is 3/4 of 10%; 10% of 600 is 60, and 3/4 of \$60 is about \$45.
b. 20. The estimate should be about 1/3 of 2/3 of 90, or about 1/3 of 60.
d. About 1/2. 6 ÷ 12 is 1/2.
3. Joe Blue was estimating 92 x 31. He said that rounding 31 to 30 then taking 92 x 30
(which is 2760) is a better estimate than rounding 92 to 90 and then taking 90 x 31 (which
is 2790), because in the first case you lost only 1 by rounding 31 to 30, but in the second
case you lost 2 by rounding 92 to 90. Explain how Joe's reasoning is incorrect. Your
Estimating 92  31 via 92  30 will be “off” by 92. Estimating 92  31 via 90  31 will
be “off” by only 2  31, or 62.
page 48
4. For each of the following, MENTALLY obtain an ESTIMATE of the answer and write it
in the blank. Use EXCELLENT NUMBER SENSE.
Then write enough to make
clear how you thought.
a. 34% discount on an \$89 suitcase. _________ Thinking:
b. 0.26  43,135 ≈ ___________ Thinking:
c. 61  334 ≈ ______________ Thinking:
d. 74.35%  1195  0.9837 ≈ ______________ Thinking:
e. (1201.794 ÷ 24.3%) + 0.0423 ≈ ______________ Thinking:
a. About \$30. Thinking: 34% is about 1/3, and \$89 is about \$90. 1/3 of \$90 is \$30.
b. About 11,000. Thinking: 0.26 is about 1/4; round 43,135 to 44,000.
c. About 20 000. Thinking: 60  1/3 of 1000 = 20 000.
Thinking: 3/4  1200  1
e. 4800 Thinking: 1200 ÷ 1/4. Definitely take off points for answers such as
4800.0423.)
5. Is 40 ÷ 1.99 less than, equal to, or greater than 20? Explain, showing your understanding
of a meaning of division.
. Greater than 20, since there will be more 1.99s in 40 than there are 2s.
6. Why is this student NOT showing good number sense, in estimating
249.738 + 48.246?
"Well, 249 is about 250, and 48 is about 50. 250 + 50 is 300. Then .738 + .246 is about
1. So, my estimate is 301."
In using 250 and 50 for the whole number parts, the student is already making an overestimate. Working with the decimals would only make the estimate farther off.
7.
A. 1.2
B. 12
C. 120
D. 1200 E. None of A-D
C (from 3/4 of 160)
8.
1.334496 x 301.66 is closest to...
A. 400
9.
B. 391
C. 390
D. 40
E. 39
A
A. 7
B. 70
C. 700
D. 7000
E. None of A-D
10. Of the estimates listed, which is best for 0.3347876 x 629.847291143?
page 49
B
A. 1800
B. 180
C. 200
D. 210
E. 240
D
11. Use number sense in locating the decimal point in the answer. Explain your thinking
briefly.
a. 77.5  2.84 = 2 2 0 7 Explanation:
b. 1002.6 ÷ 3.6 = 2 7 8 5 Explanation:
a. 220.7 Explanation: The product is about 80  3 = 240.
b. 278.5 Explanation: The quotient is about 1000 ÷ 4 = 250.
12. You are using a calculator that can show only 8 digits of an answer. When you finish,
you realize that you forgot to press the decimal points in the original numbers. Show
where the decimal points go, and explain how you know, using number sense (no paperpencil calculation).
a. 9.87429  637.21945 = 6 2 9 2 0 8 9 6 Thinking:
b. 413.69824 ÷ 24.92617 = 1 6 5 9 6 9 4 4 Thinking:
a. 6292.0896 Thinking: The product should be around 10  630 = 6300.
b. 16.596944 Thinking: The quotient should be around 400 ÷ 25, or 16 (how many
25s make 400).
13. Show excellent number sense in estimating the following. Write enough to make your
thinking clear.
a. 75.48%  883.375  567
566
Is your estimate (less than, equal to, greater than) the exact answer? (Don't figure the
b. 32% of \$595.45
c. 60% of \$271
d. 87  52
a. About 660: 3/4  880  1. Since each of the new factors is less than the original, 660
will be less than the actual answer (Aside--from a calculator: The exact answer is a
bit more than 667.949).
b. About 1/3 of \$600 is \$200
c. \$150 : 3/5 of \$250, or \$180 from 2/3 of 270.
d. 4500 from 90  50
14. Company X bought 17,569 truckloads worth \$24,598 each. Company Y bought 24,598
truckloads worth \$17,569 each. Which company paid more, and how do you know?
The companies paid the same. 17569  24598 = 24598  17569.
15. Betty was asked to give her best estimate for 25% of 7991.8.
She estimates by taking 14 of 8000, which is 2000, and taking
Therefore, her estimate is 2000.2
1
4
of 0.8, which is 0.2.
page 50
Comment on Betty’s reasoning.
Betty does not seem to realize that in using 8000 for 7991.8, the 0.8 is already involved.
(She likely used the 8000 for just the 7991.) Using 8000 introduces an error that would
only be exaggerated by adding a number-senseless 1/4 of 0.8. Another point that may not
have come up in class is that reporting an estimate with tenths, like 2000.2, implies that
the estimate is quite close, down to the nearest tenth. That clearly is not the case here.
16. 42,189 ÷ 511,264 is about how many percent?
A. 8% B. 12.5%
C. 80%
D. 125%
E. None of A-D
A from 40 ÷ 500 = (40 ÷ 5) ÷ 100 = 8 ÷ 100 = 8%
17. Estimate the following with a brief explanation:
a. 65% of 37
b. 140% of 52
c. 18% of 971
d. 43 % of 120 = ___ (decimal)
a. 24 from 2/3 of 36
b. 70 from 7/5 of 50
c. 195 from 1/5 of 975 (200 probably okay, from 1/5 of 1000)
d. 1.2 from 0.01 from 3/4 being close to 1, a number that is easy to calculate with.
18. Use benchmarks to estimate the following. Explain how you estimated.
a. 60% of \$271
b. 87  52
c. 32% of \$595.45
a. \$150 3/5 of \$250, or \$180 from 2/3 of 270.
b. 4500 from 90  50
c. About \$200. 1/3 of \$600
19. ESTIMATE the indicated quantity and tell how you did it.
a. 15% of \$51.07
b. 25% of 1998
c. 125% of 47
d. 48% of 212
e. 4% of 201
a. 10% of 50 is 5, and 5% of 50 is 2.5, so the answer is about 7.5.
b. 14 of 2000 is 500.
c. 100% of 47 is 47; 14 of 48 is 12, so the answer is about 59.
d. Half of 212 is 106 (or half of 200 is 100)
e. 4 x 200 is 800, so 8; or 1% of 200 is 2 so 4% is 4  2 is 8.
page 51
5.3 Estimating Values of Quantities
1. Approximately how long is this line segment, in inches or in centimeters? Be sure to name
the measuring unit you use.
_________________________________________________________
What benchmark did you use?
Measure this line segment before you give it to students.
2. Name an item that costs approximately \$500,000.
A. A car
C.
B. a jet plane
C. a nice home
D. A stereo
3. Name two types of estimates you use in your daily life.
Could be length of time to drive to campus, amount of money to fill tank with gas,
amount of money needed at cash register, cost of meals for a week, etc.
4.
Using benchmarks, find an estimate of the time it would take, averaging 50 mph for 8
hours a day, to drive across the United States.
Estimating that the distance across the US is about 3000 miles, the trip would take 3000 ÷
50 = 60 hours, or 60 ÷ 8 = 7 1/2 driving days of 8 hours each.
5.4 Using Scientific Notation for Estimating Values of Very Large and Very
Small Quantities
1. Write in scientific notation.
2.
a. 38,000,000,000
b. 382.45
c. 0.000000000456
a. 3.8  1010
b. 3.8245  102
c 4.56  10-10
a. The earth is 150,000,000,000 meters from the sun. Write this in scientific notation.
b. The speed of light is 300,000,000 meters per second.
c. A dust particle is 0.000 000 000 753 kg.
a. 1.5  1011
b. 3  108
c. 7.53  10-10
page 52
3. How many seconds does it take light to reach the earth from the sun? Express your answer
in scientific notation. (See item 2 for relevant data.)
1.5  1011 ÷ 3  108 = 15  1010 ÷ 3  108 = 5  102 or 500 seconds; a little over 8
minutes.
page 53
Chapter 6 Meanings for Fractions
6.1 Understanding the Meanings of
1.
a
b
Use diagrams to show that 43 is the same as 3 ÷ 4. LABEL THE RELEVANT PARTS OF
YOUR DIAGRAM AND EXPLAIN THE EQUIVALENCE.
3
4
could be drawn as one whole broken into 4 equal size pieces, where three of the
fourths would be indicated in some fashion. The sketch for 3 ÷ 4 should show 3 wholes
with, for sharing division, each marked into fourths. Each share would get 1 fourth from
each, or a total of 3 fourths of a whole.
2.
Circle the letter of any of the following regions that is cut to show fourths. If a choice
does NOT show fourths, explain why not.
A.
B.
C.
D.
A, B, and D do not show fourths. In each of these cases, the four pieces do not all have
the same area.
3. Shown to the right is 1 43 yards of carpet. Sketch (fairly
accurately) 1 yard of carpet and 3 13 yards of carpet. If the
piece of carpet shown sells for \$28, how much should 9
yards cost?
1 yard would be 4/7 of the rectangular region, then 3 1/3 would be 3 1/3 of those.
9 yards should cost \$144 (if 7/4 yards cost \$28, each 1/4 yard costs \$4, so 1 yard cost
\$16).
4.
T F Fractions are always less than 1. (Explain if F.)
False, although the part-whole interpretation might lead one to think so. A fraction such
as 54 is more than one.
5. To change 3 14 to a fraction, a common rule is to calculate 3  4, add 1, and write that
(3 4)1
answer over 4: 3 14 =
= 13
4
4 . Use a number line (and words as you need them) to
explain why that rule makes sense.
page 54
0
1
2
3
Cut each unit into four equal pieces. Then the 3 =
6.
(3 4)
4
4
, plus the 14 , give 3 14 =
(3 4)1
4
.
Make a drawing that shows 2 45 , if the shape to the
right is 1.
Just 2 regions like this, along with 4/5 of another




7. If the hearts shown represent only 25 of the chocolate
hearts you gave away on Valentine's Day, how many
chocolate hearts did you give away in all?
A. 8
8.
B. 28
D. 50
E. None of A-D
If the apples shown are 43 of the apples used for pies,
how many apples were used for pies in all?
A. 3
9.
C. 30
B. 9
C. 16
D. 24
D
¾
¾ ¾¾
¾
¾ ¾
¾
¾
¾
¾
¾
E. None of A-D
C
If 43 of the apples shown in #8 above were red apples, how many of the apples were red?
A. 3
B. 9 C. 16
D. 24
E. None of A-D
B
10. Mark
3
5
on this number line as A. Mark
0
1
2
5
3
on the number line as B.
3
4
Look for gross inaccuracies. If the number line shows 5, the item is "trickier."
11. Is Ardis’s reasoning correct? Explain.
Shannon: "I still have half my spelling words to learn and 43 of my vocabulary words to
learn."
Ardis: “Well, 43 is more than a half because 12  24 . So, you have more vocabulary words
than spelling words still to learn."
page 55
No, unless the number of spelling words was the same as the number of vocabulary
words, giving units of the same size.
12. Bill and Tom have pieces of land that are the same size. Each plants flowers on part of
his land, as shown in the sketches.
flowers.
Tom's flowers.
flowers
flowers
Who has more land in flowers?
A. Bill and Tom have the same amount of land in flowers.
B. Bill has more land in flowers.
C. Tom has more land in flowers.
D. One cannot determine who has more land in flowers.
C. Tom has more land in flowers than Bill, because he has 3/7 of his land and Bill has
only 3/8 of his (equal-sized piece of) land.
13. Pat and Dana like to argue with each other about mathematics problems. They discuss
the figure below:
Pat: The shaded region is one-and-a-half times as much as the unshaded region.
Dana: Wait! I think that the unshaded region is 23 of the shaded region.
Who is correct? Why?
Both are correct. They are using different units—Pat’s is the unshaded part, and Dana’s
14. a. Make a drawing the shows
b. Make a drawing of
2
5
2
5
of a discrete whole.
of a continuous whole.
page 56
a. The whole must consist of 5 (or 10, or…) objects, with 2 (or 4 in two pairs, or…)
designated in some way.
b. The whole must be continuous, such as a line or rectangle, divided into 5 equal parts,
with 2 designated in some way.
50
53
50
15. Given 10
and 13
, why are we able to “cancel”the zeros in 10
, but we are not able to
53
“cancel” the 3s in 13
?
50
The 0s represent factors of 10, 10
 510
110 , and a common factor can be ignored. But the 3s
53
in 13
do not represent common factors.
16. Use drawings with rectangles to show that
7
a. 58  16
b. 43  23
c. 1 43  64
a. A drawing of a rectangle divided into 16 equal parts: 10 parts will be shaded for
only 7 parts for
2
3
, but
7
16
b. This will require a rectangle divided into 12 equal parts.
whereas
5
8
3
4
will require 9 of the parts,
will require only 8 of the parts
c. Two rectangles each divided into 4 equal parts: 1 43 will require 7 of those parts,
whereas
6
4
will require only 6 of the parts.
17 Use the part-whole notion of fractions to explain why
The first fraction has smaller pieces than the second.
7
13
 127
18. Name three ways of thinking about the symbol “ 87 ”.
Some examples: as part of a whole, 7 of 8 equal parts of a circular region; as 7 ÷ 8; as a
ratio; as a probability; 7 of 8 discrete objects; ...
19. Write a story problem in which 53 is treated as a part-whole fraction, with discrete
quantities.
e.g., Jorge had 5 candy bars. 3 of them were Snickers. What fraction of his candy bars
were Snickers?
6.2 Equivalent (Equal) Fractions.
1.
 58 = which one or ones of the following?
400
10
50
20
 500
i. 500
ii. 12
iii. 64
iv. 20
Explain:
800
15  16
80  80
25  32
All are equal, since the fractions in them are equal to those in the original.
2.
How would you convince a child with visual evidence that
4
5
2
3
 46 ?
page 57
We usually expect a drawing of 2/3 of a region, and then added marks to that or to a copy
cutting each third into two equal pieces, giving 4/6 for the same amount as the 2/3.
3. A child says, "My teacher says to put = between 43 and 68 , but I think 68 is bigger."
How might the child be thinking? What would you do next?
The child is probably focusing on the larger numerator and larger denominator in 6/8.
One could make a drawing to show that the two fractions are equal.
4. How would you convince a child with visual evidence that 12  126 ?
We usually expect a drawing of 1/2 of a region, and then added marks to that or to a copy
cutting each half into six equal pieces, giving 6/12 for the same amount as the 1/2.
5. Use sketches to show that:
20
A. 15
 1 13
B. 73  219
A. Show 1 1/3, mark the thirds to make 4/3 visible, then cut each third into 5 equal
pieces, giving fifteenths.
B. Show 3/7 and then cut each seventh into 3 equal pieces, giving twenty-firsts.
6. Name three common denominators for the fractions
7
5
7
9
7
a. 83 and 12
.
b. 72
and 144
c. 40
and 60
a 24, 48, 72...
b. 144, 288, 432, ....
c. 120, 240, 360, ...
7. Write the simplest fraction form for each:
zy3 x 2
a.
91618
12815
b. y4 z5
a.
9
5
2
b. x 4
yz
8. Put these in order, smallest to largest, using the symbols <, >, and =
15
19
3
8
300
400
<
11
16
11
300
16 < 401
300
401
<
300
400
3
8
6
8
=
6
8
< 15
19
200
303
9. Put these in order, smallest to largest:
200
303
6
9
203
300
10. a. Give a number between
b. Give a number between
c. Give a number between
6
9
9
12
203
300
9
12
3
4
7 and 7 .
8
8
111 and 113 .
7
8
114 and 113 .
page 58
d. Which number is closer to 1:
a.
b.
c.
d.
9
10
and
10
9
?
3
4
1
3
6
7 is less than half and 7 is greater than half, so 2 is between; OR 7 = 14
4
8
7
3
4
7 = 14 so 14 is between 7 and 7 .
8
112
7
8
7
8
8
114 < 114 . 114ths are smaller than 113ths, so 114 < 114 < 113 .
9
1
10
1
1
1
9
10 is 10 away from 1, and 9 is 9 away from 1. 10 < 9 so 10 is closer to
11. Which of these is a rationale for comparing
3
7
and
7
11 ?
and
one.
(You can choose more than
a. They are the same because 7 – 3 = 4 and 11 – 7 = 4.
7
7
b. 11
is more than 12 and 73 is less than 12 so 73 < 11
c.
3
7
=
33
77
and
7
11
=
49
77
so
3
7
is smaller.
d. Sevenths are bigger than elevenths so 73 is larger.
e. Equal because each fraction is four parts away from one.
b. and c.
6.3 Relating Fractions, Decimals, and Percents
1. Fill in the blank cells in the table below so that the numbers in each row are equivalent to
the given one:
Fraction
a.
Decimal
Percent
13
20
b.
0.00089
c.
d.
43
4
a.
0.65
b. 89/100,000
c. 1225/10000 (might be simplified to 49/400) 0.1225
d.
10.75
2. What is the exact decimal equivalent of
A. 0.4
12.25%
A.
2
5
?___
65%
0.089%
1075%
B.
2
7
?___
B. 0.285714
page 59
3. What is the fraction equivalent of
A. 651/1000
A. 0.651? ___ B. 0.44444 ...? ___
B. 4/9
4. What is the percent equivalent of
A. 240%
A. 12
? ___
5
B. 0.00035? ___
B. 0.035%
5. Circle each fraction that has a terminating decimal, without calculating the decimal.
A.
749327
2x106
B.
3000005
6x108
C.
21000042000014
350
A and C have terminating decimals (7 is a factor of numerator and denominator in C and
so that fraction can be simplified.) C may be more difficult than should be on an
examination unless similar ones have been used in class.
6. T F
F
7.
3
18
can be written as a terminating decimal. (Explain if F.)
The fraction = 1/6, which has factor of 3 in only the denominator.
a. Write 56.38 as a fraction.
a. 5582
99
8.
b.
b. Write 7.453 as a fraction.
7453
1000
T F If the following is false, correct it so that it is true in some non-trivial way:
If D = 0.742 , then 100D = 74.200.
F …then 100D = 74.242 .
9.
As a fraction, 1.4 =
A. 14
B. 57
C.
14
100
D.
41
10
E. None of A-D
B
10. NUMBER SENSE ITEM--no calculator or hand calculation and no decimals. Put these
121
in order. Explain your decision only for 101
300 and 360 .
141
280
smallest
1.3%
1000
3000
_____
121
360
101
300
_____
101
300
1.3%
_____
121
360
_____
1000
3000
_____ largest
141
280
page 60
121
Thinking for 101
300 and 360 :
121
360
and
101
300
are both greater than 1/3, by 1/360 and 1/300,
respectively. Since 1/300 is greater than 1/360,
101
300
is greater than
11. Put these in order, smallest to largest: 0.03 0.0295
0.0295
0.03
0.1
12. Put these in order, smallest to largest:
8.5% 110%
15
11
1.7
0.1
121
360
.
0.13999
0.13999
8.5% 1.7
15
11
14
8
110%
14
8
13. Put these in increasing order without any hand calculation:
11
211
200
81.2%
0.0239
40
392
31
0.0239
11
40
211
392
200
31
81.2%
14. Give three decimals between 2.3456 and 2.3457. If it is not possible, explain why not.
Samples: 2.34561, 2.34562, 2.345601 (infinitely many possibilities)
#
15. Write each of the following as a fraction in the whole
whole # form, if it is possible. If it is not
possible, explain why not.
__
A. 9.6534 = _________
B. 96.6534 = _________
A.
96534
10000
(may be simplified to 48267
5000 )
B.
956869
9900
16. Give three decimals between 0.301 and 0.302. If it is not possible, explain why not.
Samples: 0.3011, 0.3012, 0.30189 (many other possibilities)
17. For each comparison, use number sense to order these fractions and decimals. Use only
the symbols <, = and >.
0.1
17
35
0.1
1
8
1
8
1
5
2
3
17
35
1
5
59%
0.85
2
3
9
10
0.85
59%
9
10
18. a. What are the “rational” numbers?
page 61
b. How are they different from the “irrational” numbers?
a. Rational numbers are numbers that can be expressed in the form
integer
non-zero integer
(if only
whole number
non-negative rational numbers have come up, accept non-zero
).
whole number
b. Irrational numbers are numbers that cannot be expressed in that form. (Sufficient for
now.)
19. As a decimal, 0.525% is...
A. 525.
B. 52.5
C. 5.25
D. 0.525
E. 0.0525
F. None of A-E
F
20. As a percent, 5.35 is...
A. 0.535% B. 5.35% C. 53.5% D. 535% E. None of A-D
D
21. As a decimal, 1.775% is...
A. 0.01775 B. 0.1775
22. 1.5% = ...
A. 150
C. 1.775 D. 17.75 E. None of A-D
B. 15
C. 1.5
D. 0.15
A
E. 0.015
E
E. None of A. – D.
D
23. 15% = ...
A. 150
B. 15
C. 1.5
D. 0.15
6.4 Estimating Fractional Values
1. Using benchmarks, find an estimate of each of the following and explain how you did it.
a. The sum of
b.
7
15
c.
7
8
1
7
23 , 15
,
5
9
, and
13
12 .
My thinking:
11
 12
My thinking:
 109
My thinking:
d. 7 67  14
My thinking:
a. About 2. The first fraction is negligible compared to the others, the second two
fractions are about 1/2 each, and the last fraction is about 1.
b. A little under 1 1/2. My thinking: 7/15 is a bit smaller than 1/2, and 11/12 is a bit
smaller than 1
page 62
c. A little less than 2. Both fractions are slightly less than 1.
d. 32 8  4
2. A fraction is close to
True
1
2
when the numerator is close to two times the denominator.
False
False….when the denominator is close to two times the numerator, OR when the
numerator is about half the denominator.
*3. A fraction ba is close to
denominator.
True
3
4
when three times the numerator is close to four times the
False
False. …when four times the numerator is close to three times the denominator.
Or,…when the numerator is about 3 times one-fourth of the denominator.
*4. How can you tell when a fraction is close to 14 ? Do not refer to decimal numbers.
When the numerator is about 1/4 of the denominator, OR when the denominator is close
to 4 times the numerator.
5. Show how you would mentally compute the exact results:
a.
b.
c.
3
7
4
1
7874
1
3
4
7
7  10  7  10
18
80
21
24  2  42  100%
My thinking:
My thinking:
= ___
d. 114 – 24 = ___
3
8
My thinking:
My thinking:
a. 2 1/8 First + third = 1; other two = 1 1/8.
b. 1 57 The second and fourth fractions give 1.
c. Exactly 15. The first two fractions give a product of 30 (3/4 of 40). 30 x 1/2 = 15.
d, Exactly 89 5/8. 114 – 24 is 90; 90 – 3/8 =89 5/8.
6. For each of the following, obtain an estimate of the answer and write it in the blank. Use
excellent number sense. Write enough to make clear how you thought about the
problem.
a. 68 43 ÷ 2 15
16 = ___
My thinking:
7
11
b. 4 16
+ 5 12
– 2 18 = ___
My thinking:
c. 301 ÷
7
16
= ___
My thinking:
page 63
d. The sum of
11
3
23 , 27
, and
5
16
My thinking:
a. About 23, from 69 ÷ 3
b. About 8 1/4, from under 4 1/2 plus under 6, less 2 1/8. 10 1/2 – 2 = 8 1/2 might be
close enough.
c. About 600, from how many 1/2s make 300.
d. About 3/4 from 1/2 + negligible + 1/4
7
7. 112
 2 115  3 125  137 is about...
A. 8 12
B. 8
C. 7 12
24
D. 6 24
48
E. None of A-D
B (the first and third fractions give another 1 exactly)
8. 20% of 150 is 13 of ...
A. 10 B. 90 C. 120
D. 250E. 900
B. 20% of 150 is 30, and 30 is 1/3 of 90
9
9. Which is larger, 97 or 11
? Provide a justification for your choice that does not refer to
decimals.
7/9 is 2/9 less than 1, and 9/11 is 2/11 less than 1. Since ninths are larger than elevenths,
2/9 is larger than 2/11. So 7/9 is farther “under” 1 than 9/11 is, and 9/11 is larger.
8
10. Which is closer to 12 : 15
or 73 ? Explain your reasoning.
8/15 is half a fifteenth more than 1/2, and 3/7 is half a seventh less than 1/2. Since
sevenths are larger than fifteenths, 3/7 will be farther away from 1/2 than 8/15 is, that is,
8/15 will be closer to 1/2.
11. Circle each that is close to, or equal to, 13 . (No explanation is required.)
3
40
102
175
297
0.2six
5
119
301
600
106
0.2six is 13 in base 10.
*12. Which fraction is larger,
denominators.
38
48
or
47
60
? Explain without using decimals or common
The first fraction is 2/48, or 1/24, more than 3/4, and the second is 2/60, or 1/30, more
than 3/4. Since 1/24 > 1/30, the first fraction, 38/48, is larger.
13.Which fraction is closer to 0: 18 or 19 ? Explain (without referring to decimals).
1/9, because cutting a whole into 9 equal parts will give smaller pieces than cutting the
whole into 8 equal parts will give.
14. Which fraction is closer to 1:
1
8
or
1
9
? Explain (without referring to decimals).
page 64
1/8, from the reasoning in #13, 1/8 will be closer to 1 because 1/9 is closer to 0.
15. Which fraction is closer to one-half: 18 or 19 ? Explain (without referring to decimals).
1/8, from the reasoning above, and the fact that both are less than 1/2.
16. Which fraction is closer to 1: 87 and 89 .
Focusing on the “missing” part shows that 7/8 is farther below 1 than 8/9 is, so 7/8 is
closer to 1.
29
*17. Which value is larger: 17
40 pound, or 64 pound? Explain your thinking, showing your
understanding of fractions and of benchmarks (not decimals or common denominators).
Using a benchmark of 1/2 pound, 17/40 is 3/40 under 1/2 and 29/64 is 3/64 under 1/2.
Since fortieths are larger than sixty-fourths, 3/40 > 3/64, so 17/40 will be more under 1/2
than 29/64 will be. 29/64 pounds is larger
18. Which is the greater value: 169 of a cup, or 15
28 of a cup? Explain, showing your grasp of
fractions and benchmarks and without using decimals or complicated common
denominators.
Using a benchmark of 1/2 cup, 9/16 is 1/16 more than 1/2, and 15/28 is 1/28 more than
1/2. Since sixteenths are larger than twenty-eighths, 9/16 is larger than 15/28.
*19. Why is
2
3
a better benchmark than 1 is, for answering this question: "Which is larger,
or 17
(You do not have to answer the question in quotation marks.)
21 ?"
With a benchmark of 1, one has to compare 7/30 and 4/21—different sizes of basic
“pieces” as well as different numbers of them (the 7 smaller pieces might or might not be
larger than the 4 larger pieces). So 1 is not a good benchmark. But with 2/3 as a
benchmark, it is a matter of comparing 3/20 and 3/14, for which only the size of
twentieths vs fourteenths need be involved.
23
30
20. Name good fraction benchmarks for the following percents:
a. 48%
b. 165%
c. 9%
a. 1/2
b. 1 2/3 or 5/3
c. 1/10 (possibly 1/11 with really good number sense)
11
13
21. Which is larger, 13
or 15
finding common denominators.
13/15 is larger. 11/13 is 2/13 less than 1, and 13/15 is 2/15 less than 1. Since fifteenths
are smaller than thirteenths, 2/15 is smaller than 2/13, so 13/15 is less far under 1 than
11/13 is.
page 65
22. For each of the following, tell which is larger by referring to benchmarks. For each
problem, show the benchmark used.
a. 43 or
4
9
b.
10
41
or
3
11
a. 3/4, benchmark 1/2
b. 3/11, benchmark 1/4 (3/11 > 3/12 = 1/4, 10/41 < 1/4)
23. For each of the following, find a fraction between the two given fractions and show how
you found it. Do not use decimal numbers or common denominators.
a.
5
21
and
11
40
b.
7
13
and
8
11
a. 1/4 5/21 < 5/20 = 1/4, and 11/40 > 10/40 = 1/4
b. 2/3. 7/13 is barely over 1/2, and 8/11 > 8/12 = 2/3.
24. In each case tell which is larger by referring to benchmarks. Explain what you did
making clear what your benchmark is. (Do not use common denominators.)
a.
7
15
or
4
7
of the distance?
b.
8
27
or
7
18
of those voting?
a. 4/7 of the distance, since 4/6 > 1/2 but 7/15 < 1/2.
b. 7/18 of those voting, since 8/27 < 9/27 = 1/3, but 7/18 > 6/18 = 1/3.
page 66
Chapter 7 Computing with Fractions
1.
Make up a story problem that could be solved by 3 12  2 13 and illustrates the
view of subtraction.
comparison
Various possibilities, but in each the two quantities should be distinct, as in, “One recipe
calls for 3 12 cups of sugar, and another calls for 2 13 cups of sugar. How much more sugar
does the first recipe call for than the second?”
2.
Make up a story problem that could be solved by calculating 123 – 48 13 .
Various possibilities, e.g., Velma had 123 feet of ribbon and used 48 13 for package bows.
How much ribbon does she have left?
3.
Following is an example of a child's work. You are to study the work and then to judge the
student’s understanding.
Rona was asked to subtract 2 58 from 4 18 . Her work is shown below.
Rona's work:
4 18  3 11
8
2 58  2 58
1 68
a) Is Rona's work correct or incorrect? Correct __
b) If the work is incorrect, please explain how.
Incorrect __
a) Incorrect b) In changing a 1 from the 4 into eighths, Rona did the usual base ten place
value “borrowing” rather than the correct 8/8 + 1/8.
4. Four identical pizzas are shared among 3 people. Show 2 ways that the pizzas could be cut.
i. Cut each into three parts; each person gets 1/3 of each pizza: 4/3 or 1 1/3 pizzas
ii. Cut one into three parts. Each person get a whole one (3/3) plus another third; 1 1/3
pizzas. (There are many more ways, but perhaps complex.)
5. Using a rectangular region as the unit, illustrate each of the following:
2
4
a. 3 + 5
7
3
b. 5 + 2
10
1
c. 8 – 2
perpendicular to the marks for thirds to give slivers that are fifths (and small boxes
page 67
fifteenths). 4/5 would entail 4 slivers or 12 small boxes. But there are only 5 small boxes
left, so 3 small boxes, or 3/15, of another whole is needed.
b. Similar to part A, except that each addend is more than one whole. 2 9/10 when all the
work is done.
c. Easiest is to recognize that 10/8 = 5/4 = 1 1/4, but that may not be regarded as fair.
Show 10/8 with parallel cutting marks and shading on the two rectangular regions. That
should make recognizable that 1/2 is 4/8, which can then be x-ed out from the 10/8.
6. Explain a strategy you could use to mentally compute the exact answer:
a. 2 83 + 5 14 + 3 87 + 2
b. 2 45 +3 16 + 5 23 – 1.8
c. 3 12 – 2 83
a. The whole numbers give 12, and 3/8 + 7/8 give 1 1/4 which, with the other 1/4 gives
1/2. 13 1/2.
1
b. 2 4/5 – 1.8 is 2.8 – 1.8 is 1. 1/6 + 2/3 is 1/6 + 4/6 is 5/6. Now: 1 + 3 + 5 + 5/6 is 9 5/6.
c. 1/2 is 4/8. So the difference is 1 1/8.
7. Why is a common denominator needed to add and subtract fractions?
Because finding equivalent fractions that allow a common denominator d allow us to add
and subtract d1 units; that is, we have the same unit to use for each fraction.
8.
Three children took a hike, carrying one heavy backpack. One child carried the backpack
for 83 of the hike, and a second carried it for 16 of the hike. For what part of the hike did
the third child carry the backpack? Which child carried the backpack for the greatest part
of the hike?
3
8
9.
 16  1324 , so the third child carried it
11
24
of the hike, the greatest part.
Jody mows 18 of the yard and then takes a break. After the break, Jody mows another 13 of
the yard and stops for lunch. After lunch, Jody mows another 16 of the yard and then goes
swimming. What part of the yard does Jody still have to mow?
1
8
 13  16 
3
24
 248  244 
15
24

5
8
mowed, so there is still
3
8
of the yard to mow.
10. The truck farmer has planted 13 of a field in cherry tomatoes, 15 of the field in large
tomatoes, and 27 of the field in popcorn. The farmer plans to plant sweet corn in the
remaining part of the field. About what part of the field will be in sweet corn?
page 68
35
21
30
86
Planted already is 13  15  27  105
of the field, leaving
 105
 105
 105
sweet corn, about a fifth of the field.
19
105
of the field for
11. Emmy wants to read an assigned book in four days. She reads 23 of the book the first day
but only 151 of the book the second day and another 151 of the book on the third day.
a. What part of the book must she read the fourth day?
b. On which day did Emmy read the most?
2
3
2
 151  151  10
15  3 , so she must read
1
3
of the book the fourth day.
b. She read the greatest part of the book on the first day.
12. Judy has a 10-page paper to write. The first day she writes 1 12 pages, takes a break, and
then writes 3 43 pages more. The next day she writes another 2 13 pages. How many pages
does she still have to write?
She has written 1 12  3 43  2 13  7 127 pages, so she still has 10  7 127  2 125 pages to write.
13. a. On the number line below, show where
b. On the same number line, show where
4
7+
4
7 –
1
2
1
2
would be. Label it A.
would be. Label it B.
0
1
2
0B
1A
2
7.2 Multiplying by a Fraction, and
7.3 Dividing by a Fraction
1.
Use a drawing to help explain why
3
4
In the finished drawing, the unit has
been cut into 47 equal pieces, and
the answer part is 35 of them.
2.
35
. Be explicit.
 57 is equal to 47
x x x x x
x x x x x
x x x x x
If two numbers are multiplied, their product is greater than (or equal to) each number.
(Explain if F.)
True
False
page 69
F If (the factors are positive and) one of the factors is a number less than 1, then the
product will be less than the other number.
3.
Donny is asked to solve the following problem: If cheese is \$1.89 per pound, how much
does 0.67 pound cost?
Circle the expression that correctly represents the problem.
A. 1.89 + 0.67
B. 1.89 – 0.67
C. 0.67 – 1.89
D. 0.67  1.89
E. 1.89 ÷ 0.67
F. 0.67 ÷ 1.89
G. None of A–F
D
4.
Which of A-E would best locate the point for 0.89  n on the number line (n positive)?
A
•
5.
C D
•
n
B
•
65%
of n
E
•
C
Which of A-E would best locate the point for n ÷ 0.89 on the number line (n positive)?
A
B
E
C D
•
•
•
•
65%
n
of n
E
A
B C D
E
F
G
6.
0
1
Which letter is on the most likely place for
a. 12  2?
b. 58 of 12 ?
c. 43  23
d. 1 ÷
a. F
b. B
3
4
c. D
of
2
3 of
d. G
2
3
e.
1
4
 13
e. E
this rectangle, as though you were acting it out:
b. Show exactly where the
2
3
is. It is
c. Show exactly where the 43 is. It is
2
3 of
3
4
what? __________
of what? __________
page 70
a. Most common should be, first 2/3 is shaded, then with cutting marks perpendicular to
the first marks, 3/4 of the 2/3 double-shaded. Continuing the marks for the 3/4 of the
2/3 cuts the whole rectangle into twelfths, and the double-shaded part is clearly 6/12.
b. 2/3 of the whole rectangle.
c. 3/4 of the 2/3 part.
8.
a. Show 1 ÷
number.
3
8
with a rectangular region as the whole. Write your answer as a mixed
a.-b. Using a rectangular region cut into eighths, mark 3/8, then another 3/8. The
remaining part is not enough for another 3/8, but it is 2/3 of another 3/8. So 1 ÷ 3/8 is
2 and 2/3. Be sure to look for evidence for b.
9.
The product of a positive number n by another positive number m is always greater than n.
True
False
False If m is less than 1, then the product is less than n.
10. Three-fourths of seven ones is the same amount as seven three-fourths of one.
True
False
11. Draw a picture that represents A.
True
 25
3
4
B.
2
5

3
4
10. A.
B.
x x
x x
x x
12. Draw a picture that represents:
A.
4
3
 25
x
x
x
x
B.
x
x
2
5

4
3
Similar to #11. In each of #11 or #12, point out, if needed, that although the answers are
the same, the meanings of the two products are different. You may wish to review
“commutative property of multiplication” language.
page 71
13. Working on his summer tan, Zonker leaves 45 of his skin exposed. He has only enough
lotion to cover half of his exposed skin. What part of his body will be unprotected?
Draw a neatly labeled diagram displaying your solution.
It should be clear from the drawing that 4/5 was shown first, then 1/2 of that, then added
segments to cut the whole region into (10) equal pieces. 4/10, or 2/5, of his body will not
be protected.
14. Use these circles to show each division. For each part tell what question is being asked.
Tell what your answer is in each case, and show how you obtained it.
7
1
8 ÷ 4
a.
1
b. 4
3
÷ 8
a. The question is, how many 1/4s are in 7/8. So 7/8 of the circular region should be
shown, with 1/4s then being marked off. There will be 3 full 1/4s, and half of another
1/4, in 7/8.
b. The question is, how many 3/8s are in 1/4. So 1/4 of the circular region should be
shown. There is not a whole 3/8 in that 1/4, only a part of a 3/8. “Ghosting” in the rest
of a 3/8 and the 1/8 markings should show that there is 2/3 of a 3/8 in 1/4.
Try to detect students who do the calculation but cannot show the meaning in the
drawings.
15. Why is division of fractions confusing to people? Give two reasons.
Division of fractions does not always “make smaller.” Why a division problem becomes
a multiplication is sometimes a mystery, particularly for those who think multiplication
“makes bigger.” Which fraction to invert is also confusing, perhaps at a later time.
16. Is one meaning of
1
4
 13 the following: How many ones is 14 of
1
3
of 1?
If so, how much of a one or how many ones?
Yes:
1
12
of a one.
17. Is one meaning of
1
4
 13 the following: How many 13 s of 1 are in 14 of 1?
If so, how many _____ thirds or how much of a third of 1 are/is in one-fourth of 1?
page 72
Yes, there is
3
4
of one-third of 1 in
18. One meaning of
45
59
45
59
16
19
45
59
of
35
17
of 1?
False
True
35
is: How many
 17
True
20. One meaning of
of 1.
35
is: How many ones is
 17
True
19. One meaning of
1
4
35
17
of 1 are in
45
59
of 1?
False
True
25
is: How many
 17
False. It should read, How many
25
17
25
17
of 1 are in
of 1 are in
16
19
16
19
of
25
17
?
of 1?
21. Use excellent number sense in estimating (1201.794 ÷ 24.3%) + 0.0423 gives ____.
4800 (NOT 4800.0423)
22. The stage-coach robbers have a 12-mile head start on the sheriff's posse. But the sheriff's
posse has faster horses, so the posse catches up by 43 mile every hour. How many hours
will it take the posse to catch the robbers? Show your work.
The question becomes, how many 3/4 miles are there in 12 miles? (Each one of those will
indicate an hour of catch-up time.) So, 12 ÷ 43 = 16. It will take16 hours to catch up.
23. Tell what 8 87 ÷ 3 means...
a. with the repeated subtraction or measurement view:
b. with the partitive or equal sharing view:
a. how many 3s are in, or make, 8 87 .
b. how much is in each share if 8 87 are shared equally among 3.
24. True or False: Division (involving only positive numbers) always gives a smaller answer
than the number being divided. (If False, explain.)
False. Division by a number less than 1 gives a quotient larger than the dividend.
25. What calculation would solve this story problem?
"Cheese was \$2.55 a pound. A woman bought a 0.85 pound package of the cheese.
How much did she pay?"
A. 2.55 + 0.85
B. 2.55 – 0.85 C. 0.85 x 2.55 D. 2.55 ÷ 0.85
C
26. What calculation would solve this story problem?
page 73
A.
4
5
+
4
5
of a pie. He ate
2
3
B. 45 –
2
3 of
2
3
what he had. What part of a whole pie did he eat?"
C.
2
3

4
5
D.
4
5
÷ 23
E. None of A-D C
27. What calculation would solve this story problem?
"She paid \$4.80 for 43 pound of candy. How much does the candy cost, per pound?"
3
4
A. (4.80 ÷ 4) x 4
B. 4.80 x
D. 4.80 + 0.75
E. None of A-D
C. 4.80 ÷
3
4
C
28. Tuna costs 70¢ a can at Store X and 80¢ a can at Store Y. How much will 8 cans of tuna
cost, if 14 of the cans are bought at Store X and the rest at Store Y?
A. \$1.40 B. \$4.80 C. \$5.60
D. \$6.20 E. None of A-D
D
29. One day Joe's old car used 58 quarts of oil. The next day it used only 23 as much oil. How
many quarts of oil did the car use on the second day?
A. 58  23
B. 23  58
C. 58  23
D. 23  58
E. None of A-D
B
30. On day 3 the car used 1 14 quarts of oil! But on day 4 it used only 60% as much as it did
on day 3. How many quarts of oil did the car use on day 4?
60
A. 1 14  0.60 B. 1 14  0.60 C. 53  1 14 D. 1 14  100
E. None of A-D
C
31. How many 23 pound portions can be obtained from 30 pounds?
A. 20
B. 45
C. 60
D. 90
E. None of A-D
B
32. Following are three students’ solutions to 169 of 48.
a. For each student, first evaluate the student’s mathematical reasoning. Check
whether the steps are mathematically correct or flawed or indicate that you cannot tell.
b. Next mark each student's work with "doesn't appear to understand," "may or may
not understand," or "shows good understanding of" multiplication of fractions.
9
1
1
1
Jessica: 16 is 16 more than 2 , and half of 48 is 24 and 16 of 48 is 3, so 27.
a. Mathematical steps: Choose one.
__ Jessica’s steps are
mathematically correct.
__ Jessica’s steps are
mathematically
flawed.
__ I cannot tell if Jessica’s steps
are mathematically correct or
flawed.
b. Understanding of multiplication of fractions: Choose one.
__ Jessica doesn't appear to
understand multiplication
of fractions.
__ Jessica may or may not
understand
multiplication of
fractions.
__ Jessica shows good
understanding of
multiplication of fractions.
page 74
c. If Jessica’s steps are mathematically correct, use her way of thinking to solve
If they are not, explain how Jessica’s reasoning is flawed.
3
8
 32 .
3
Justin:
9
16
 48 
9
16

48
1

27
1
 27
a. Mathematical steps: Choose one.
__ Justin’s steps are
mathematically correct.
__ Justin’s steps are
mathematically flawed.
__ I cannot tell if Justin’s steps
are mathematically correct
or flawed.
b. Understanding of multiplication of fractions: Choose one.
__ Justin doesn't appear to
understand multiplication
of fractions.
__ Justin may or may not
understand
multiplication of
fractions.
9
__ Justin shows good
understanding of
multiplication of fractions.
9
Stacy: 16 of 48? 9  16 = 144; 144 ÷ 48 = 3; 9  3 = 27. So 16 of 48 is 27.
a. Mathematical steps: Choose one.
__ Stacy’s steps are
mathematically correct.
__ Stacy’s steps are
mathematically flawed.
__ I cannot tell if Stacy’s steps
are mathematically correct
or flawed.
b. Understanding of multiplication of fractions: Choose one.
__ Stacy doesn't appear to
understand multiplication
of fractions.
__ Stacy may or may not
understand
multiplication of
fractions.
__ Stacy shows good
understanding of
multiplication of
fractions.
Jessica: a) mathematically correct
b) Jessica may or may not understand multiplication of fractions, but she does show
number sense.
c) Possibly, 1/4 of 32 is 8, and half of that is 4, so 12.
Justin: a) mathematically correct
b) Justin may or may not understand multiplication of fractions.
Stacy: a) mathematically flawed
b) doesn’t appear to understand multiplication of fractions.
33. Finish the following story problem so that your question could be answered by the
calculation 18 ÷ 12 : "You buy 18 muffins for an after-school faculty meeting . .
Sample: …They are large, so you cut them into 1/2-muffin servings. How many servings
will you get?
C
A
B
D
E
34. For this number line
q
0 p
2
1
page 75
i) the point for p + q would be closest to A? B? C? D?
ii) the point for p  q would be closest to A? B? C? D?
(i) C
E? (circle)
E? (circle)
(ii) A (since both p and q are less than 1)
35. Karen and Sue go bike riding. Karen biked for 3 13 hours, which was 23 as many hours as
Sara biked. How many hours did Sara bike?
A. 2 29 B. 2 23
C. 4
D. 5
E. None of A-D
D
36. After the family had gone 140 miles, the children asked, "Are we getting close?" The
driver said, "We've gone only 27 of the way." How far did the family still have to go?
A. 40 miles B. 80 miles C. 350 miles D. 490 miles E. None of A-D
C
37. A farmer keeps 40% of his land uncultivated. Two–thirds of the cultivated portion is
planted with corn. What fraction of the entire land is planted with corn? MAKE A
40% uncultivated gives 60% that is cultivated, and 60% = 3/5 makes an easy drawing.
2/3 of the 3 cultivated pieces will give the part of the field planted in corn: 2/5.
38. Alex ate 12 of a pizza. Tandy ate 13 of what was left. Finally, Tabby ate
left. How much of the whole pizza did Tabby eat?
1
2
of what was
Alex leaves 1/2 of the pizza. Tandy ate 1/3 of that, which is 1/6 of the pizza, leaving 2/6
of the pizza. Tabby ate 1/2 of the 2/6 of pizza, so Tabby ate 1/6 of the pizza.
39. Consider this number line:
F
H
0
J
A
K L
B1
Points A and B are between 0 and 1 as indicated. Which of the remaining points
best represent
a. A x B
b. A + B
c. A – B
d. B – A
a. H
b. K
c. F
d. J
40. Under a repeated-subtraction interpretation, 43  1 14 means
_______________________________________________
page 76
The quotient is ____________. Verify and explain your answer with a sketch.
…how many 1 14 s are in, or make,
answer, 53 of one 1 14 , is in 43 .
3
4
3
5
. The sketch should show the
page 77
Chapter 8 Multiplicative Comparisons and Multiplicative Reasoning
8.1 Quantitative Analysis of Multiplicative Situations, and
8.2 Fractions in Multiplicative Comparisons
1. A ratio is the result of comparing two quantities to determine how much larger one is than
the other.
True
False
False …to determine how many times as large one is, compared to the other.
Note for 2-5. Refer to the situation described and fill in the blanks with the appropriate
numbers. If not enough information is provided to determine the number that should go in the
blank, write “impossible to determine.”
2.
At Riverdale Middle School, 18 of the students are in the band. Two out of every three
students in the band are girls.
a. The number of boys in the band is __________ times the number of girls in the
band.
b. What fraction of the students who play in the band are boys?
c. What fraction of the students at Riverdale are boys who play in the band?
d. The number of girls in the band is ________ times the number of students in the
school.
e. What is the ratio of girls who do not play in the band to the boys who do not play in
the band?
(A diagram may be helpful for parts C, D, and E.)
a. 1/2 b. 1/3
c. 1/24
d. 1/12 (or 2/24)
3.
e. impossible to determine
Five of every 6 students interviewed favored a change in library hours.
a. Among those interviewed, what is the ratio of those who favor a change to those who
do not favor a change? _______
b. Among those interviewed, there are ____ times as many students who favor a change
as there are students who do not favor a change.
c. Among those interviewed, there are ____ times as many students who do not favor a
change as there are students who do favor a change.
d. What fraction of the students interviewed favor a change in the library hours? _____
page 78
a. 5:1 b. 5
4.
c. 1/5 d. 5/6
According to a U.S. News/CNN poll, three out of 10 people went away on vacation in
August.
a. What is the ratio of those who went away on vacation in August to those who didn't?
b. What percent of people did not go away on vacation in August?
c. The number of people who did not go on vacation is ___ times the number who did.
a. 3:7
b. 70%
c. 2 1/3
5.
Maxine polled her entire class concerning the date for the next test. Twelve people
preferred that the test be given on Wednesday. Twenty-eight voted to have the test on
Friday.
a. What fraction of the class wanted the test on Wednesday?
b. What is the ratio of those who prefer Wednesday to Friday?
c. Those who want the test on Friday are how many times as much as those who prefer
the test on Wednesday?
a. 12/40 (or 3/10)
b. 12:28 (someone may give 3:7)
c. 2 1/3
6.
For the situation below, answer the questions and fill in the blanks. If you cannot answer
a question, explain why. If the question asks for an explanation, give an explanation. If
the question instructs you to draw a relevant diagram, be sure to label it.
Ms. Collin's class has 15 girls and 18 boys. 25 of the children have pets. 20 of the
children have dogs.
a. The ratio of boys to girls is: ___ .
b. The girls make up what fractional part of the class? ___
c. The number of boys is ___ times the number of girls.
d. The number of girls is ___ times the number of boys.
e. The ratio of the number of children who have pets to the number of children who do
not have pets is ___
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f. The number of children who have dogs is what fractional part of the number of
children who have pets? ___
g. Of the children who have pets, the number of children who do not have dogs is what
fractional part of the number of children who have dogs? ___ Draw a relevant
diagram and label it.
h. The number of girls who have dogs is ___?
i
If I said 8 of the children have cats, is this inconsistent with the information that I
a. 18:15 (possibly 6:5)
b. 15/33, or 5/11
c. 1 1/5
d. 5/6
e. 25:8
f. 20/25, or 4/5
g. 1/4 (look for diagram)
h. impossible to
determine
i. No, some children may have both a dog and a cat as pets.
7. Two landscapers mowed the lawn of a wealthy family. When they finished, landscaper A
had mowed only 73 as much as the more experienced landscaper B mowed.
A
B
a. Mark on the drawing of the lawn to show how the mowing might have been done.
b. A's part is _____ times as much as B's part.
c. A's part is what part of the lawn?
d. What is the ratio of A's part to B's part?
e. If they are paid \$150 for mowing the lawn, what would be a fair split of the \$150?
a. The region should be marked into 10 equal pieces, with 3 labeled for A and the others
for B.
b. 3/7
c. 3/10
d. 3:7
e. \$45 for A; \$105 for B (\$150 for the 10 pieces is a rate of \$15 per piece)
8. If town A had 12,000 more people than it does, its population would be 1 23 times as big as
town B is now. Town B has 45,000 people currently. What is the current population of
town A?
63 000, The larger A will have 1 23  45000  75000 people, so the current A has 75000 –
12000 = 63000 people. Alternatively, with the additional 12 000, A:B = 5:3, so for every
3 people B would have, A would have 5. Since B has 45 000 = 315000 people, the
larger A would have 515000 or 75 000 people. The current A has 75 000 – 12 000 = 63
000 people.
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9.
Alan and Bob started mowing a rectangular lawn. After mowing 43 of the whole lawn they
got tired and stopped. When they stopped Alan had mowed 27 as much as Bob had
mowed.
a. Mark the diagram of the lawn below to show how much of the lawn each boy mowed
and label the parts with each boy’s initial to indicate clearly the parts mowed by each.
b. The area of lawn that Bob mowed is ______ times as large as the area that Alan
mowed.
c. Alan mowed ________ of the total lawn area.
d. The ratio of the area of lawn Bob mowed to the area Alan mowed is ________.
e. Together they were paid \$12 for the work they did. How much money did each boy
get if they were paid proportionally to the amount they worked? Alan ___ Bob___
a. 3/4 of the lawn should be shaded. Then just that portion should be cut into 9 equal
parts, since the A:B ratio 2:7 says 2 parts for Alan for every 7 parts for Bob. So 2 of
those parts should be labeled A and 7 B.
b. 3 1/2 (either from the drawing, or from B:A = 7:2)
c. 1/6 (more marks show 6/36)
d. 7:2
e. Alan: \$2.67, Bob: \$9.33 (from \$12 for 9 pieces is a rate of \$1 1/3 per piece. \$2 2/3
and \$9 1/3 would be exact but do not fit money)
10. Tim worked 30 hours last week, which was
a. How many hours did Robert work?
5
3
times as many hours as Robert worked.
b. Who worked more hours?
c. Which type of comparison does the question in part (b) address, additive or
multiplicative?
a. 18 (a drawing should make clear the 5:3 comparison; Tim’s 30 hours means each of
his 5 pieces could be thought of as 6 hours, so Robert’s time is 18 hours.)
b. Tim (without part A, just from the given sentence)
c. Either. If a follow-up question had been “How many more?” that would have been an
page 81
11. Below is a diagram of a candy bar that is being shared between two people:
a. Cut the bar into two parts so that part A is 43 of part B.
b. Part B is
times as large as part A.
c. Part B is how much of the bar?
d. What is the ratio of part B to part A?
a. Cutting the bar to reflect the 3:4 ratio gives 7 equal pieces (A 3, B 4)
b. 1 1/3 (either from the drawing or from the B:A = 4:3 relationship)
d. 4/7
c. 4:3
12.
a. Cut the rectangle into two regions, A and B, so that A is
b. What fraction of the whole rectangle is A?
3
2
as large as B.
a. Cut the rectangle into 5 equal pieces: A is three of the pieces, B is 2 of the pieces.
b. A is 3/5 of the entire rectangle.
13.
4
5
of an amount is 40.
a. How many parts is the whole split into?
b. How many parts is the 40 split into?
c. How much is in each part?
d. What is the full amount?
a. 50
b. 40
c. 1/50
d. 50/50 or 1.
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Pat and Ron split a cake. Pat's share is 2/3
as large as Ron's share.
12.
a. Sketch fairly accurately on the "cake" to show
Pat's and Ron's shares. Label them P and R.
b. Ron's share is ____ times as large as Pat's share.
c. Pat's share is what fractional part of the whole cake? _____
d. What is the ratio of Ron's share to Pat's share? ______
a. 5 pieces, with 2 for P and 3 for R.
b. 1 1/2
c. 2/5
d. 3:2
13. John baked a cake for his son’s birthday party. At the party,
was eaten. (A diagram may be helpful.)
2
7
as much was left over as
a. The remaining portion is what fractional part of the whole cake?
b. What is the ratio of the amount of cake eaten to the whole cake?
The diagram should have 9 equal pieces, to reflect the L:E = 2:7 ratio.
a. 2/9
b. 7:9
14. Two painters paint a wall from opposite ends. When they finish, painter P has painted
only 23 as much as the more experienced painter Q painted.
P
Q
a. Mark on the drawing of the wall to show how the painting might have been done.
b. Q's part is ___ times as much as P's part.
c. Q's part is what part of the wall? ___
d. What is the ratio of P's part to Q's part? ___
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e. If they are paid \$100 for painting the wall, what would be a fair split of the \$100?
a. 5 equal pieces, 2 for P and 3 for Q
b. 1 1/2
c. 3/5
d. 2:3
e. P \$40; Q \$60 (each of the 5 pieces would be worth \$20)
15. Consider the region below as a pizza that is being shared by two people.
a. Divide the pizza into two parts so that part A is 53 of part B.
b. Part A is how much of the pizza?
c. What is the ratio of part B to part A?
a. 8 equal pieces, 3 for A and 5 for B
b. 3/8
c. 5:3
page 84
Chapter 9 Ratios, Rates, Proportions, and Percents
9. 1 Ratio as a Measure, and
9.2 Using Proportions to Compare Ratios and Solve for Missing Values
1.
Any driveway ramp that is 3.205 ft high is less steep than any other ramp that is 3.98 ft
high.
True
False
False The steepness also depends on another dimension, usually the horizontal length of
the driveway.
2.
One recipe for 8 people requires 5 cups of flour. Al expects only 6 people and so wants
to cut the recipe so that it will serve 6.
b. If Al is thinking multiplicatively, how much flour will he use? Explain your answer.
c. Which way is better, Al's or your part B method? Explain.
a. If Al is using additive thinking: 8 – 6 = 2, so he should use 5 – 2 = 3 cups of flour.
b. The recipe calls for 5/8 cup of flour per person, so for 6 people Al should use 6(5/8),
or 3 3/4, cups of flour.
c. The part B method is better, because to taste the same, recipes should use the same
proportions of ingredients. Rates allow that.
3.
Ruben and Ofilia are painting the walls of a large lecture hall. They mixed 2 gallons of
blue paint with 5 gallon of white paint for a total of 7 gallons of paint. They ran out of
paint. They estimated that they needed one half gallon to finish. Find the portion of the
room which has been painted.
a. Find the portion of the room which has been painted. Draw a neatly labeled diagram
b. If they mix 1/4 gallon of blue paint with 1 gallon of white paint, will the paint colors
c. Suggest what amounts of blue paint and white paint would finish the room and match
the color, perhaps with a little left over for touch-ups.
a. The diagram should show the walls (probably one rectangle), marked into 15 equal
pieces, since the 7 gallons is 14 half-gallons. They have painted 14/15 of the room
with the 7 gallons.
b. No. The original rate was 2:5 which is the same as 2/5:1. But if 1 gallon of white paint
is used with ¼ gallon of blue paint, the ratio is 1/4:1 or 1:4 which is not equivalent to
2/5:1.
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c. We need a ratio a:b where a:b is equivalent to 2:5 and where a+b = 1/2. ¼ gallon blue
paint and 5/8 gallon of white paint would provide the correct ratio and the sum is
slightly more than ½, but close enough for this situation.
4.
María was asked to draw an isosceles triangle (a triangle with two equal sides) as part of a
homework assignment. She had a straight edge to draw straight lines, but didn't have a
ruler to measure the length of her sides. Thus, all she could do was eyeball the correct
lengths. She decided to draw more than one triangle so that she could go to school a little
early, get a ruler, measure all of her triangles, and select the "best" isosceles triangle to
turn in to the teacher.
Suppose María drew the following triangles:
1.1 in
1.2 in
1.6 in
1.7 in
2.0 in
2.2 in
a. Which triangle should she select to turn in to the teacher? Why?
b. Suppose María drew 20 triangles like the ones shown above, but of various sizes.
Once she measures the lengths of the sides, how should she go about selecting which
triangle is the "best" isosceles triangle?
a. Actually none of the triangles is isosceles. The best-looking one will have its ratio of
designated sides closest to 1. So it is a matter of deciding which of 1.1:1.2, 1.6:1.7,
and 2.0:2.2 is closest to one. Using fraction versions makes the comparison fairly
easy: 11/12, 16/17, 20/22 = 10/11. Fraction considerations show that 16/17 is closest
to 1, so the triangle with sides 1.6 in. and 1.7 in. will be the best-looking one.
(Remark: As with the squareness measure, students may insist on focusing on the
additive comparison. An example of a triangle with sides 0.1 in. and 0.2 in. may
convince them that an additive comparison is not the best way to tell.)
b. She should find the one with the ratio of the supposedly equal-length sides closest to
1.
5.
There are four water slides at the Six Flags Atlantis water park. The following are the
measurements of the four water slides. Circle the letter of the steepest slide.
A. Length: 100 ft.; height: 81 ft.
B. Length: 60 ft.; height: 45 ft.
C. Length: 80 ft.; height: 70 ft.
D. Length: 10 ft.; height: 7 ft.
page 86
C. Steepness can be measured by the height:length ratio. Using the fraction forms and
number sense, 70:80 is the largest of the four ratios.
6.
a. 3 bags of tea are used with 4 cups of water in one teapot, and 10 bags of tea with 12
cups of water in another teapot. Which pot will have the stronger tea, or will the two
b. 5 scoops of coffee are used with 8 cups of water in one coffee pot, while 7 scoops of
coffee are used with 10 cups of water in another coffee pot. Which pot will have the
c. The two problems above were given to a sixth grade class. Which one do you think
was more difficult, and why?
a. The 10 bags in 12 cups will be stronger, since the first mixture has 3/4 bag per cup of
water and the second 10/12, or 5/6, bag per cup of water, and 5/6 > 3/4.
b. The 7 scoops, 10 cups of water coffee will be stronger, because 7/10 scoop per cup
of water gives more coffee taste than 5/8 scoop per cup of water will.
c. Part B almost certainly would be more difficult for sixth graders, because the
additive comparisons for the two recipes are the same, a difference of 3. And, there
is not an easy relationship between the quantities in part B, as there is in part A: 12
cups is 3 times as much water as 4 cups, so repeating the first recipe 3 times would
mean 9 bags for 12 cups, a weaker tea than with the 10 bags for 12 cups.
7.
Two painters on a large project want to paint different areas the same color. Painter A
mixes 3 quarts of red paint with 2 gallons of white paint, and Painter B mixes 5 quarts of
the same kind of red paint with 4 gallons of white paint. Painter A says the two mixtures
will be the same color and Painter B says his mixture will be redder than Painter A’s.
Explain the thinking of each one. Which one, if either, is correct? Explain your decision.
Painter A is using an additive comparison; painter B is looking at the larger amount of red
paint in his mixture. Neither reasoning is correct, since it is the ratio of red to white that
is important. For A, R:W = 3:2 = (1 1/2):1, for B, R:W = 5:4 = (1 1/4):1, so A’s mixture
will be redder.
8.
A friend needs a serious operation and is considering two hospitals that have the
following records for the surgery:
Successes
Failures
Hospital A
20
8
Hospital B
50
18
The friend sees that B has more successes, but A has fewer failures, and asks you for help
in deciding. What will you say?
page 87
(Assume that in either case the number of operations attempted is enough to give the staff
sufficient experience.) There are different acceptable approaches, but each depends on
the multiplicative comparison in some form. One way is to check the rate of # of failures
per success—Hospital A, F:S = 8:20 = 0.4:1; Hospital B, F:S = 18:50 = 0.36:1, so B
looks slightly better, having a lower failure to success rate. Or one could compare the #
successes: # failures rates. Or one could calculate the percents of successes in all the
cases (A: 20/28  71%; B: 50/68  74%).
9.
Terry uses 1 cup of Mr. Spiffy in 1 12 gallons of water to clean the kitchen floor. What
percent of the cleaning solution is Mr. Spiffy? (Note: 16 cups = 1 gallon) SHOW
The 1 1/2 gallons, or 24 cups, of water, plus the cup of Mr. Spiffy gives 25 cups of
cleaning solution. 1:25 or 1/25 = 4%
10. Make a sketch, and give your explanation, to illustrate this situation:
After a bake sale, there were 3 identical white cakes left. You and your friend split them,
but you took only 14 as much as your friend took. Label your part M (for “me”) and your
friend’s part F.
How much of a cake (or how many cakes) did your friend take? (Give the numerical
Rectangular cakes may be easier to deal with. In any case, the multiplicative comparison
of the two shares is 1:4, so the cakes should be cut into 5 equal amounts, with 1 part for
M and 4 parts for F. This cutting can be done in two ways. The first is easier—cut each
cake into 5 equal pieces. The second is slightly harder to sketch and then to name the
fractions—cut the total of the 3 cakes into 5 equal pieces. In either case, F gets 12/5, or 2
2/5, cakes (12/5 makes immediate sense with the first method; 2 2/5 may be more natural
for the second method).
11. Make a drawing and give an explanation to illustrate and answer this situation:
Karen and Sara both jog. Sara jogged 2 12 miles, which is
many miles did Karen jog?
2
3
as far as Karen jogged. How
Each 1/3 of Karen’s distance is 1 1/4 miles, so Karen distance was 3 3/4 miles. Here is
one sketch (others are possible, of course):
2
1
2
mi.
S's jog
K's jog
page 88
12. If the Browns were to save an additional \$14,000, they would have 1 25 times as much
money as the amount the Jones have in their savings account. The Jones have \$65,000.
How much do the Browns currently have in their savings account? SHOW YOUR
1 2/5 times as much as \$65,000 = \$91,000. So the Browns must have \$91,000 – \$14,000
= \$77,000 in their savings account
13. A group of 37 persons goes to a holiday camp for 35 days. They need to buy enough
sugar for the trip. They read that the average consumption of sugar is 2.2 kg per week for
10 persons. How much sugar do they need? SHOW YOUR WORK CLEARLY AND
One way: The 35 days is 5 weeks, so they will need 5  2.2 = 11 kg for every 10 persons
for the whole period. Since there are 37 persons, they will need 3.7  11 = 40.7 kg of
sugar for the camp.
14. The pollster noticed that for every 40 men who were in favor of X, there were 28 women
who were in favor of X. According to these figures, if 280 men were in favor of X, how
many women were in favor of X?
A. 400
B. 196
C. 168
D. 40
E. None of A-D
B
15. A machine can make 700 bolts in 40 minutes. At that rate, how many bolts can the
machine make in one hour?
A. 900
B. 1050
C. 2800
D. 28,000
E. None of A-D
B
16. Miguel runs 200 meters in 40 seconds; Paul runs 150 meters in 12 minute. Who runs
faster?
A. Miguel
B. Paul C. They run at the same speed. D. More information is needed.
C (Each runs 300 m/min., or 5 m/s.)
17. John drives 7 103 miles to campus each day, while Vaneta drives only 4 miles to campus.
John drives how many times as far as Vaneta?
(7 3/10):4 translates into 1 33/40 times as far.
Box 1 BBW
drawing a B ball from Box 1 (to the right) is the same
Box 2 BBBBBBWWW
as for Box 2.
A child says, "They say 2 to 1 gives the same probability as 6 to 3 does, but I don't see it.
I think Box 2 has a lot more chances of giving a B."
page 89
How would you try to convince the child that 2:1 is indeed equal to 6:3, using the balls in
the boxes (not just symbolically with numbers)?
In Box 2, draw rings to show three BBW groups. (Re-drawing so that the Ws are under
the Bs makes that easier.)
19. A child says that the two situations below would give the same "chocolatey-ness," since
"Each way has one more spoonful of chocolate sprinkles."
Situation 1 3 spoonfuls of chocolate sprinkles on 2 scoops of vanilla ice cream.
Situation 2 4 spoonfuls of chocolate sprinkles on 3 scoops of vanilla ice cream.
Give a drawing (not just calculations) that should help the child see that the "chocolateyness" in the two situations would be different. (Use words as needed.)
Draw to show that in Situation 1, each scoop gets 1 1/2 spoonfuls, and in Situation 2,
each scoop gets 1 1/3 spoonfuls.
20. A store sells 48 pecan pies for every 15 banana pies it sells. At this rate, if the store sells
60 banana pies, how many pecan pies will it sell?
A. 12
B. 93
C. 120
D. 192
E. None of A-D
D
21. The pollster noticed that for every 8 men who were in favor of a ballot measure, there
were 5 women who were in favor of the measure. According to these figures, if 1200
women were in favor of the measure, how many men were in favor of the measure?
A. 750
B. 1197
C. 1920
D. 2100
E. None of A-D
22. Clarissa runs 240 meters in 45 seconds; Doña runs 160 meters in
faster?
A. Clarissa
B. Doña
1
2
C
minute. Who runs
C. They run at the same speed.
A. Clarissa: She runs 360 meters in one minute; Dona runs 320 meters in one minute.
23. Use a three-column table to solve this problem. Label the columns Picture Frames, Cost,
and Notes.
If a box of 36 picture frames (all the same kind) cost \$86.40, how much would 5 frames
cost? (Assume no sale price, discount for volume, sales tax, etc.)
Picture Frames
Cost
Notes
page 90
36
\$86.40
given
18
\$43.20
halved each
9
\$21.60
halved each
3
\$ 7.20
found a third of each
1
\$ 2.40
found a third of each
5
\$12.00
5 (2 + 0.40) = \$10 + \$2 =\$12.00
25. Felicia can run a mile in 4.7 minutes. How long would it take her to run 8 miles?
This is not a proportion problem because she cannot conceivably continue running at such
a speed for 8 miles.
9.3 Percents in Comparisons and Changes
1.
15% of a given amount is the same as 10% of the amount plus 10% of twice the amount.
True
False
False ….the same as 10% of the amount plus 10% of half the amount.
2.
Terry uses 1 cups of Mr. Spiffy in 1 12 gallons of water to clean the kitchen floor. What
percent of the cleaning solution is Mr. Spiffy? (Note: 16 cups = 1 gallon)
The 1 1/2 gallons, or 24 cups, of water, plus the cup of Mr. Spiffy gives 25 cups of
cleaning solution. 1:25 or 1/25 = 4% (This item is repeated from earlier.)
3.
What does "the sales tax rate is 7 43 %" mean?
The tax will be \$7.75 for every \$100 of goods (or 7 3/4 ¢ for every 1 dollar of goods).
4.
You receive 12 points out of 20 on the first quiz, 15 points out of 20 on the second quiz
and 60 points out of 75 on the first midterm.
a. On which quiz or test did you perform best? Show your work.
b. How would you find your average score?
c. Estimate what percent that average represents.
a. 12/20 = 60%, 15/20 = 75%, 60/75 = 80%, so you did best on the midterm.
page 91
b-c. Assuming that all points are weighted equally, add the total number of points
received, and compare that to the total number of points possible:
(12+15+60)/(20+20+75) = 87/115  75.6% (via estimation: 87/115  87/111 =
(9x87)/999  783/1000 = 78.3%). In contrast, assuming that a quiz has equal weight
as a midterm (unlikely), (60%+75%+80%)/3  71.7% (via estimation: 215%÷3 
71%). You might wish to ask your students why that second average is lower
(equating the weights makes the 60% quiz performance more influential than if the
average were based on equal values for each point).
5.
Kathee owed her dad \$80 and then paid off \$24 from her tip earnings.
a. What percent of the original debt had she paid?
b. What percent was still owed?
c. What was her new debt after the payment?
The following evening she paid him an additional \$20 from her tip earnings.
d. What percent of her new debt was paid off?
e. What was her debt after the last payment?
f. What percent of her original debt is now paid off?
a. 30%
b. 70%
c. \$56
d. 35.7%
e. \$36
f. 55%
6. If this box represents 75% of something, modify the box so that it represents 125% of the
same thing.
Divide the box into 3 equal parts, then add an additional two parts of the same size.
7.
The median cost of housing rose 25% in one city in a particular year. The median price
was then \$360,000. What was the median cost of housing at the end of the previous year?
8.
Gala apples are on sale today, at \$1.30 per pound. This is a discount of 30%. What was
the cost before the sale?
9.
The Nasdaq closed at 1,590 today, off by 0.09%. What the Nasdaq at yesterday’s closing?
About 1591.4 (the main idea is to catch incorrect thinking or poor number sense)
page 92
10. Estimate each of the following and explain how you arrived at your estimate:
a. 79% of \$119
b. 0.5% of 89 kilograms
c.
31% of 21,343 voters
d. 121% of \$29
Here are some possible ways to estimate:
a. About 80 % of \$120: 10% is \$12; 8 x \$12 is \$96
b. 1% is 0.9, half of that is 0.45 kilograms
c. About a third of 21,000, so 7000 voters
page 93
Chapter 10 What’s to the Left of 0?
10.1 Adding and Subtracting Signed Numbers
1.
Complete the following. For each subtraction problem, first rewrite it as an addition
problem.
a. – 7 +– 8 =
b. 3 + –5 =
c 16 + –14 =
d. –21 + –2 =
e. 17 – –3 =
f. –2 – –5 =
g. –13 – 9 =
h. 14 – –16 =
k. –2.6 – 4.5
l. –3.17 – 2.4
c. 2
d. –23
i.
5
4
+ – 12
j.
a. –15
b. –2
e. 17 – –3 = 17 + 3 = 20
f. –2 – –5 = –2 + 5 = 3
g. –13 – 9 = –13+ –9 = – 22
h. 14 – –16 = 14 + 16 = 30
i.
2.
– 11
11
17 + ( 17 )
3
4
k. –2.6 + –4.5 = –7.1
j. 0
l. –3.17 + –2.4 = –5.57
Reorder these numbers from smallest to largest:
7.4%
21
7
–1 5
–
4
11
6
5
4
–
11
6
–1.5
0.2
–0.2
–1.5
2.8
7.4%
0.2
–0.2
5
4
–1 5
4
2.8
21
7
3. Complete this “fact family” table:
3 + –5 = – 2
4.
3 + –5 = – 2
–2 – –5 = 3
–5 + 3 = –2
–2 – 3 = –5
Is the statement a + b =/a/ – /b/ always true, sometimes true, or never true? Explain your
Sometimes true. If a is positive and b is negative and |a| > |b| then a + b =/a/ – /b/ but if a
is positive and b is negative a + b = – ( /b/ – /a/).
page 94
5.
Explain how one could use white (positive) and dark (negative) chips to model the
following
a. 4 + (–6)
b. 5 – (–2)
a.
b.
Adding two white and two dark does not change the value in the first box. Then remove
two darks to represent subtracting –2. This leaves seven whites, or +7.
4.
Which properties does each of the following involve?
a. (-2 + 3) + -5 = (3 + -2) + -5
b. (-2 + 3) + -5 = -5 + (3 + -2)
c. (-2 + 3) + -5 = -2 +( 3 + -5)
a and b. Commutativity of addition
10.3 Multiplying and Dividing Rational Numbers
1. Does 0 have a multiplicative inverse? Explain.
No, 0 does not have a multiplicative inverse. There would have to be a number n such that
0  n = 1, but 0  n always = 0. So no number would work as the multiplicative inverse of
0.
2.
Complete the following:
a. – 7  – 8 =
b. 3  –5 =
c 16  –2 =
d. –21  –2 =
e. 18 ÷ –3 =
f. –2 ÷ –5 =
g. –18 ÷ 9 =
h. 4 ÷ –16 =
k. –8.6 ÷ 4.3
l. –1.2 ÷ –2.4
i.
5
4
÷ – 12
j.
11
17
 (– 17
11 )
page 95
2.
a. 56
b. –15
c. –32
d. 42
e. – 6
f.
2
5
g. –2
h. – 14
i. –2 12
j. –1
k. –2
l.
1
2
Continue the following pattern of multiplication of integers with six more products in the
pattern. The multipliers decrease by _____ each time. The multiplicand is always -4.
The product ____creases by _______ each time. For this pattern to continue working,
the product of two negative integers must be _____________________________
3  –4 = –12
2  –4 = –8. . . . . . . .
1  –4 = –4
0  –4 = 0
–1  –4 = 4
–2  –4 = 8
–3  –4 = 12 –4  –4 = 16.
The multipliers decrease by __1___ each time. The multiplicand is always –4. The
product __in__ creases by __4_____ each time. For this pattern to continue working, the
product of two negative integers must be ___positive_________________
3.
Give the exact answer to each of the following, taking advantage of properties. Then tell
which property or properties enabled you to answer them so easily.
a. If 213 five  142 five = 41401 five, then 142 five  213 five = ______five.
Property(ies):
b.
179
196

0
343
Property(ies):
c. 84.96  100% = _________
Property(ies):
d. (548 + –967) + –548 = _______
Property(ies):
a. 41401five
b. 179/196
c. 84.96
d. –967
3.
 ________
Commutative property of multiplication
Identity property of multiplication
Commutative property of addition, associative property of
Match the operations and the names of properties by placing the correct number to the left
of the letters A-E. (Not all properties on the right will necessarily be used; some may be
used more than once.)
__a. –3  (2 + 5) = –3  (5 + 2)
1. associative property of multiplication
__b. 3  (2 + 5) = (3  2) + (3  5)
page 96
__c. 4 + (–3 + –1) + 2 = (4 + –3) +( –1 + 2) 3. multiplicative inverse property
__d. 5 + ( 8 + 0) = 5 + 8
__e -4  1 = -4
__f. 6 + (4 + –4) = 6 + 0
__g. 23  23  1
7. distributive property of  over +
__h. 3

(5
 0) = (5 
0)

3
8. multiplicative identity property
__i. 3 + (2 + –5) = (3 + 2) + –5
a. 5
b. 7
c. 6
d. 2
e. 8
9. commutative property of multiplication
f. 4
g. 3
h. 9
i. 6
10.4 Other Number Systems
1.
Consider clock arithmetic using a clock with four numbers: 0, 1, 2, and 3.
a. Complete these tables:
+
0
1
2
3
0
1
2
3

0
1
2
3
0
1
2
3
b. Do you think the set of numbers, 0, 1, 2, 3, is closed under addition? If not, provide an
example that shows it is not.
c. Do you think the set of numbers, 0, 1, 2, 3, is closed under multiplication? If not,
provide an example that shows it is not.
d. Is there an additive identity? If so, what is it?
e. Is there a multiplicative identity? If so, what is it?
f. Does 3 have an additive inverse? Is so, what is it?
g. Does 2 have a multiplicative inverse? If so, what is it?
h. Do you think addition commutative? Is so, provide an example.
i. Do you think multiplication is commutative? If so, provide an example.
j. Do you think addition associative? Is so, provide an example.
page 97
k. Do you think multiplication is associative? If so, provide an example.
l. Do you think multiplication is distributive over addition? If so, provide an example.
a.
+
0
1
2
3
0
1
2
3

0
0
1
2
3
0
0
0
0
0
1
1
2
3
0
1
0
1
2
3
2
2
3
0
1
2
0
2
0
2
3
3
0
1
2
3
0
3
2
1
b. The set is closed under addition. All sums are 0, 1, 2, or 3.
c. The set is closed under multiplication. All products are 0, 1, 2, or 3.
d. 0 is the additive identity: any number plus 0 is that number.
e. 1 is the multiplicative identity: any number times 1 is that number.
f. Yes. The additive inverse of 3 is 1: 3 + 1 = 0, which is the additive identity.
g. No. There is no number which, if multiplied by 2, is 0. (All products with 2 as a factor
are either 0 or 2.)
h. Yes, addition is commutative. Example: 1 + 3 = 0 and 3 + 1 = 0
i. Yes, multiplication is commutative. Example: 2  3 = 2 and 3  2 = 2
j. Yes, addition is associative. Example: 2 + (3 + 1) = (2 + 3) + 1 because
the left side is 2 + 0 = 2 and the right side is 1 + 1 = 2.
k. Yes, multiplication is associative. Example: 2  (3  1) = (2  3)  1 because
the left side is 2  3 = 2 and the right side is 2  1 = 2.
l. Yes, multiplication is distributive over addition. Example: 3  (2 + 3) =
(3  2) + (3  3) because 3  1 = 3 and 2 + 1 = 3.
page 98
Chapter 11 Number Theory
11.1 Factors and Multiples, Primes and Composites, and
11.2 Prime Factorization
1.
The following sounds all right, but it is not always true. Give a counterexample.
"Suppose that k is not a factor of m and k is not a factor of n. Then k is not a factor of
m+n."
Counterexample:
Many possibilities. Sample: 3 is not a factor 4 and 3 is not a factor of 2. But 3 is a factor
of 4+2 = 6.
2.
What, if anything, can you say about the oddness or evenness of m…
a. when 5063338  m is an even number
b. when 5063338 + m is an even number
a. Nothing can be said about m (the product will be even no matter whether m is even or
odd).
b. m must be even.
3.
T F 5 is a multiple of 0. (Explain if F.)
F All multiples of 0 equal 0, since m0 = 0.
4.
Say the same thing as the following sentence, but use the word "multiple."
360 is a factor of N.
N is a multiple of 360.
5.
Rephrasing:
If n = 43759462138999999249 + 764321572, then is n an even number, or is n an odd
n is even. Since the square of an odd number is odd, the sum of the odd and odd will be
even
6.
7.
Circle T if the statement is true, F otherwise.
T F
Every whole number is a multiple of itself.
T (m = 1m)
T F
It is possible for an even number to have an odd factor.
T (e.g., 12)
T F
Zero is a multiple of every whole number.
T (0 = 0m)
T F
250 is a factor of 10030.
T ( 100  22  52 , so 10030  260  560 )
_________ because
page 99
Yes, every number m is a factor of 0, since 0 = 0m.
8.
9.
Circle T if the statement is true, F otherwise.
T F
Every whole number is a factor of itself.
T
T F
It is possible for an odd number to have an even factor.
F
T F
Zero is a factor of every whole number.
T F
520 is a factor of 5012.
F (0 is a factor only of 0)
T ( 50  2  52 , so 5012  212  524 )
Of what numbers, if any, is 0 a multiple? Explain your answer.
0 is a multiple of every number m, since 0m = 0
10. Suppose 7 is not a factor of n. Can 21 be a factor of n? If 21 can be a factor of n, give an
example for n. If 21 cannot be a factor of n, give an explanation from basic principles.
No. If 21 were a factor of x, x = (some #)21. But 21 = 37, so then
x would = (some #)37 and 7 would have to be a factor of x.
11. Determine whether m and n are primes. Write only enough to make your decisions clear.
a. m = 23  29 (= 667) ______________ because
b. n = 133 ____________ because
a. Not a prime (23 or 29 is a third factor)
b. Prime, because none of 2, 3, 5, 7, 11, 13, etc., is a factor (Instructor: If you have
introduced the n bound for testing of primes, your student should stop with 11.)
12. Is 245 a prime number? Explain.
______ because
No, 5 is a third factor.
13. T F There are no values of b and c for which 27b = 9c. (Explain, if F.)
F 27b  (33 )b  33b and 9 c  (32 )c  32c and all that is required is that 3b = 2c, for which
there are infinitely many solutions—e.g., b=2, c=3; b=4, c=6; b=6, c=9;…
14. T F There are no values of r and s for which 11r = 9s. (Explain, if F.)
T (unique factorization into primes)
15. Is there a whole number M which would make this true? If so, tell what M is. If not, tell
why not.
page 100
a. 35.52.173 = 34.174.M
b. 24.72.118.22 = 25.7.116.M
It may be a good idea to cite the relevance of the unique factorization into primes result.
a. Not possible because there are already too many 17s on the right-hand side.
b. 7 113 (Notice that the left-hand side is not in its prime factorization form.)
16. Give the prime factorization of n, where n = 4 x 720 x 5000. If it is not possible, explain
why not.
25  5 4  720
17. Is it possible to find a non-zero whole number m so that 14m = 260  759 ? ____
Explain.
No. Since 14 = 27, 14 to any power will have the same number of 2s as 7s.
18. KNOWLEDGE OF NUMBER THEORY QUESTION; NO CALCULATORS. Without
calculation, explain why Romeo and Juliet can or cannot both be correct, when they are
talking about the same large number:
Romeo: "The number is 7  11  172  37  67  97."
Juliet: "The number is 3  11  212  37  67  89."
____________ because
No, Juliet’s number has 3 as a factor, but Romeo’s does not. It may be a good idea to cite
the relevance of the unique factorization into primes result.
19. It is correct that 3721164 = 12  172  29  37.
Give the prime factorization of 372116400 (notice the extra two zeros).
Hint: Do not work too hard.
2 4  3 52 172  29  37 (Students may overlook the 12 in the original factorization; despite
the hint, some may not recognize that the target number is just 100 times the original
one.)
20. Is this all right? If it is, explain why. If it is not, give a counterexample.
"If a number has n factors (n > 1), then the square of the number has 2n factors."
No. Students should have no trouble finding a counterexample (the item is just testing
ability to read an if-then and know what a counterexample is).
page 101
21. When you were a spy, two of your paid informants gave you the following information
Informant 1: "The code-number is 33  70  some odd number."
Informant 2: "The code-number is 35  66  some even number."
What can you tell from your informants' information?
Their information is inconsistent. Although the known numbers do give the same prime
factorization, Informant 2’s “even number” would involve another factor of 2 that
Informant 1’s odd number could not.
22. Fill in the blanks to make a true sentence. If no number or algebraic expression will
make the sentence true, say so.
a. An example of a number which has an odd number of factors is _______.
b. If n = 138.1710, then the prime factorization of 26.n is _____________.
c. 3 will be a factor of 140000000?000000014 if the missing digit, ?, is __ or __ or __.
a. Sample: 4
b. 2 139 1710
c. 2 or 5 or 8
23. THEORY QUESTION; NO CALCULATORS. Is it possible, for some choice of positive
whole numbers m and n, that 45m = 15n? Justify your answer.
No. Comparing 45m  32m  5m , and 15n  3n  5n , shows that m would have to equal n
because of the 5s and then the 2m ≠ n (m and n are to be positive).
24. Every two different prime numbers are relatively prime.
True
False
True
25. T F If we write the first 10,000 numbers in 6 columns, as started below, then 9999 would
be in the 5th column. Write enough (numbers, words) to make your thinking clear.)
1
2
3
4
5
6
7
8
9 10 11 12
etc.
False 10000 ÷ 6 = 1666 R 4, so 10,000 numbers would occupy 1666 full rows and only
4 into the next row. 10,000 would be in the 4th column so 9999 would be in the third
have the form 6n +1, the second column 6n +2, the third column 6n + 3, the fourth
page 102
column 6n + 4, the fifth column 6n + 5, and the sixth column 6n.
so 9999 would be in the third column.
9999 = 6 x 1666 + 3
26. Tell the difference between (a) "give a prime factor of 350" vs "give a prime factorization
of 350," and the difference between (b) "give a number that has an odd factor" vs "give a
number that has an odd number of factors."
a. “Give a prime factor of 350” means to identify only one of the prime factors of 350,
whereas “give a prime factorization of 350” means to give a product of primes equal
to 350.
b. “Give a number that has an odd factor” means to find a number that has a factor that
is 3, 5, 7, 9, etc., but “give a number that have an odd number of factors” means to
find a number such that when you find all of its factors, there are an odd number of
them.
27. Say the same thing as the following sentence, but use the word "factor."
M is a multiple of 240.
Rephrasing:
240 is a factor of M.
28. Put 0 and 2 (one of each) into the blanks to make a true statement, and explain. If it is not
possible, explain why.
_____ is a multiple of _____ because _________________________
0, 2, because 0 = 0  2
29 Circle T if the statement is always true, F if it is always false, and D if it depends on the
value of a variable. The variable n represents a positive whole number.
T F D
If 27 is a factor of n, then n is a multiple of 27.
T
30. Give the prime factorization of n, where n = 137 x 3000.
2 3  3 5 3 137
31. In each part, find a whole number for m to make the equality true. If it is not possible,
explain why. For credit, your work should show an understanding of number theory.
a. 52  103  176 = 23  176  m
b. 52  76  114 = 5  356  114  m
a. m = 5 5
b. Not possible, since there are already more 5s on the right-hand side than appear on
the left-hand side.
page 103
32. Explain, without extensive calculation, why the following equation can or cannot be
correct:
172  192  375 = 184  414
It is _______________________ because (In your explanation the grader will look for a
clear reference to a major theoretical result.)
It cannot be correct, since 17 appears as a factor on the left-hand side, but not on the
right-hand side. Unique factorization into primes says this is impossible.
33. (THEORY QUESTION; NO CALCULATORS) Is it possible, for some choice of
positive whole numbers m and n, that 35m = 25n? Justify your decision.
No, since 35 m involves factors of 7, which cannot appear in the factorization of 25 n .
34. Name the number of factors of each of these numbers and list them, in factored form.
a. 52  173
b. 35
a. 12: 50  170
50  171
50  172
50  173
51  170
51  171
51  172
51  173
52  170
52  171
30
32
b. 6.
31
52  172
33
34
35
52  173
(1, 3, 9, 27, 81, 243)
35. What is the largest prime number that you need to text to check for divisibility of
a. 173
b. 982
a. 13 because 13  13 = 169 which is the largest square smaller than 173.
31 because 31  31 = 961 is the largest square less than 982.
b.
36. When the number 540 is written as a product of its prime factors in the form a 2b 3c , what
is the numerical value of a + b + c ? Choose one of the following:
A. 30
B. 5
C. 6
D. 7
C.
11.3 Divisibility Tests to Determine Whether a Number Is Prime
1.
Circle the numbers that are prime. If a number is not prime, list at least three factors
below the number.
392
5231211
61. 73
121
43
page 104
Only 43 is a prime. 121 = 11x11 and the sum of the digits of 5231211 add up to a
multiple of 3
2.
Circle the numbers below that divide 11220.
2,
3,
4,
5,
6,
8,
9,
10,
12,
15,
18,
20
All except 8, 9, and 18 divide 11220.
3.
a. State a divisibility test for 8.
b. Explain why your test in part a will definitely work, using the general 7-"digit"
a. 8 is a factor of n if and only if 8 is a factor of the number named by the rightmost
three digits.
b. abcdefg  (abcd) 1000  efg, and 8 is a factor of 1000 (1000 = 8125), so whether 8
is a factor of abcdefg will depend on whether 8 is a factor of efg.
4.
KNOWLEDGE OF NUMBER THEORY QUESTION; NO CALCULATORS
a. Circle any which is a factor of 62296715880, which is equal to
23  32  5  7  472  31  192.
15
16
21
75
94
n=19312
217
b. Explain how you know that your answers in part A are correct, even without
calculation.
a. 15, 21, 94 (=247), and 217 (=731) are the only ones.
b. By unique factorization into primes, the given factorization gives those prime factors,
and only those prime factors, that are possible for the given number, as well as the
number of appearances of each prime factor. Hence, the prime factors of any
composite factor of the given number can involve only the primes given, and no more
than the number available.
5.
KNOWLEDGE OF NUMBER THEORY QUESTION; NO CALCULATORS
Circle any which is a factor of 80000000005332:
3
4
5
6
8
9
12
15
3, 4, 6, 12. Not 5, 8, 9, and 15
6.
For each part, give an example, if one exists. If there is no example, explain why not.
page 105
a. A whole number which has 15, 21, and 1000 as factors, but does not have 9 as a
factor.
b. A prime number that has 7 and 19 as factors, and is not a perfect square.
a. The number must have 23, 3, 53, and 7 as factors, so 23 x 3 x 53 x 7 would have 15,
21, and 1000 as factors but 9 would not be a factor. (On tests, usually without
calculators permitted, we allow an answer to be left in factored form.)
b. Not possible, if 7 and 19 are factors, there will be more than two factors since 1 and
the number itself are already two factors.
7.
KNOWLEDGE OF NUMBER THEORY QUESTION; NO CALCULATORS.
Circle each of the given choices that is a factor of the given number n.
n = 22.103.711.135
Choices:
8
14
21
28
35
Only 21 is not a factor. (Students may overlook that n is not quite in prime-factored
form.)
8.
If it is possible, give a whole number that is relatively prime to 24. If it is not possible,
explain why.
There are many possibilities, so long as the number avoids prime factors that appear in
the prime factorization of 24: 2 and 3. Examples: 5, 11, 13, 35, 49, 53,…
9.
State a divisibility test for 4, and explain why it works.
4 is a factor of n if and only if 4 is a factor of the number named by the right-most two
digits. The test works because a number can be expressed as a certain number of 100s,
plus whatever is named by the right-most two digits. Since 4 is a factor of 100,
divisibility of the whole number will depend exclusively on whether 4 is a factor of the
number named by the right-most two digits.
10. KNOWLEDGE OF THEORY QUESTION; NO CALCULATORS)
Is it possible, for some choice of positive whole numbers m and n, that 75m = 25n?
No, since 75 m involves factors of 3, which cannot appear in the factorization of 25 n .
11.4 Greatest Common Factor, Least Common Multiple
1. Write the prime factorization of each of the following. (Show your work.)
a. 1485:
page 106
b. 792:
c. Name all common factors of 1485 and 792 (They can be in factored form).
d. What is the greatest common prime factor of 1485 and 792?_________
a. 33  5 11
2.
b. 2 3  32 11
c. 3, 32 , 11, 33, 99
d. 11
a. What is the least common multiple of 1485 and 792 (in factored form)?
b. Write two other common multiple of 1485 and 792.
a. 2 3  33  5 11
3.
b. 2 4  33  5 11 and 2 5  33  5 11 are two possible answers.
If it is possible, give a whole number that is relatively prime to 24. If it is not possible,
explain why.
There are many possibilities, so long as the number avoids prime factors that appear in
the prime factorization of 24: 2 and 3. Examples: 5, 7, 19, 35, 49, 53,… (This item is
repeated from an earlier section.)
4.
Suppose K = 25  7 11 , L = 2 3  7 1113
M = 2  29 2
and N = 4 11132  29
Name the least common multiple of each of the following (in factored form).
a. K and L
b. M and N
c. K and M
d. K, L and N
a. 2 5  7 1113
5.
b. 22 11132  292
c. 2 5  7 11 29 2
d. 2 5  7 11132  29
Using K, L, M, and N as defined above, name the greatest common factor of each of the
following (in factored form).
a. K and N
b. K and L
c. M and N
d. K, L, and M
a. 2 2 11
b. 2 3  7 11
c. 2  29
d. 2
6. Write these numbers in simplest form:
page 107
a. 26
b. 616
c. 129
a. 2
b. 7
c. 3
792
65
9
5
7.
215
5
a. Use the prime factorizations of 345, 264, and 495 to find the least common multiple
of the three numbers.
b. Compute the following:
345
495
250 (Leave the answer in factored form.)
 264
a. 345  3 5  23
264  2 3  32  5 11 23
264  2 3  32  5 11 23
b.
8.
345
495
250
 264

3523
32 511
 2325

311
3
23
311
 22 5311 
3
495  32  5 11 LCM is
22 23
22 311
 22 5311 
3
22 235 3
22 311
Two neighboring satellites send out signals at regular intervals. One sends a signal every
180 seconds, and the other sends a signal every 280 seconds. If both satellites send out a
signal at 12:00 midnight on January 1, when will be the next time that they both send out
a signal at the same time?
12:42 a.m. The next simultaneous occurrence will happen when a multiple of 180
seconds next coincides with a multiple of 280 seconds, that is, at the least common
multiple of 180 and 280. This will first happen 2520 seconds, or 42 minutes, later.
9. Hamburger patties come in packages of 16, and hamburger buns come in bags of 12. How
many of each do you need to buy so that you have the same number of buns as you do of
hamburgers?
The LCM of 16 and 12 is 48, so buy 3 packages of patties and 4 dozen buns.
10. As a charitable service, your class undertakes a project where they fill backpacks with
school supplies for recent immigrants. The donations include 135 notebooks, 216 pencils,
and 81 pens. You want to use all the donations and include the same number of each item
in each backpack. What is the largest number of backpacks you can fill and how many
items will be in each backpack?
5 notebooks, 8 pencils, 3 pens in 27 backpacks: GCF is the number of backpacks.
11. Two football players are working out by running around a track. The first can run the
track in 3 minutes, and the second one can run the track in 4 minutes. If they begin at the
starting point at the same time and run in the same direction at the same rates, when will
the both be at the starting point again?
12 minutes
page 108
12. The band has been invited to march at the Rose Parade and need to make money to cover
the expenses. The divide up into three teams and shovel snow from driveways for four
days before Christmas. The first team made \$315, the second \$240, and the third \$210. If
they charged a flat rate for each driveway, what was that rate?
GCF (315,240,210) = 15, so \$15
Chapter 12
1. Completion.
A. As the demand for housing increases, the price of housing ________________.
B. Say that C = 100 – 20t describes the Celsius temperature C vs time t in hours, as a liquid
cools down. The –20 tells you this about the situation: ___________________
2. Complete each sentence with "increases," decreases," "doesn't change," or "can't say anything
definite," as appropriate.
As the semester goes on, the number of days until final exams _____________.
As a person's peanut-butter consumption increases, her miles traveled to
work_______________.
As the speed of a car increases, the stopping distance of the car ____________.
As the number of calculations increases, the probability of an error ____________.
3. What coordinate graphing conventions are being violated by the following graph of data
showing candy-bar sales for three children who sold for different amounts of time?
page 109
No. of
candy bars
sold
14
13
12
11
10
3
6
5
No. of hours
spent
4. Below is a graph for Candle 1.
Candle 2 (graph not given) burns according to h = – 2t + 25, where h is the height in cm t
minutes after being lit.
Height (cm)
30
27
24
21
18
15
12
Candle 1
9
6
3
0
3
6
9
12 15 18 21 24 27 30 33 36 39 42 45 48 51
Time (min.)
A. Which candle, 1 or 2, burns at a faster rate? Explain.
Candle ___ because
B. Write an equation that describes Candle 1's height h vs time burning t.
page 110
h = _________________
C. On the above coordinate system, draw the graph for the burning of Candle 2.
( h = – 2t + 25 ) What information does the 25 give?
D. Exactly how many minutes will Candle 2 last, after it is lit?
_________ minutes
E. If the two candles, 1 and 2, are lit at the same time, will they ever be the same height
(before they both burn out)? If "Yes," tell when (approx.). If "No," tell why not.
5. The bowling place is under new ownership! The new owners plan to charge for shoes-ball
rental and of course for each game, so that their dollar income, I, from a bowler who rents
shoes-ball and who bowls n games is
I = 2n + 2.5.
A. How much are they planning to charge for shoes-ball rental? ______
That plan did not work, so they
want to try a new plan (graph to
the right).
Cost (\$)
B. How much will they charge per
game, under the new plan?
8
______
6
C. Will they make more money
under the new plan? Explain.
4
2
1
6. A bakery keeps records and makes graphs
of its cookie production. To the right are
partial graphs for two days.
2
3
4
5
6 # of
games
Number of
Tuesday
10 000
A. Write a story that might tell what
happened on Tuesday. Do not
introduce any numbers besides
those indicated for Tuesday.
Friday
0
0
8 Number of
hours
page 111
B. What is the slope for Friday's graph, and what does that slope mean about the cookie
Slope: ________
Meaning:
7. A new candle is 12 inches long and burns 3 inches every 20 minutes. In a graph showing the
height of the candle as it burns minute by minute, what is the slope? What does that slope
mean in this situation?
Chapter 13
1. Completion.
A. At a speed of 80 m/min, Dude can go 170 meters in ___minutes _____ sec (exactly).
B. For a turtle trip of n feet in 10 seconds, turtle's speed is _____________.
C. A trip of m feet at a speed of 25 feet per second takes _________ seconds.
2. T F This is an acceptable story for the
Distance
travelled
graph to the right: "I rode my bike up
a hill and then down the other side."
time
3.
Story: Wiley Coyote, Jr., left school at a slow but steady pace, heading for his cave.
When he was one-third of the way to the cave, he realized that he had forgotten his math
book, so he ran back to school to get it. While he was at school he played for a while
with some other coyotes. He then realized that he would be late for supper, so he started
jogging at a steady rate toward the cave. About half-way to the cave, he thought that he
was still going to be late, so he ran faster and got to the cave in time for supper.
Draw two qualitative graphs for the story, one showing Wiley's distance from the cave vs.
time since he left school. The second graph should show Wiley's total distance traveled vs.
time since he left school. The two graphs should be coordinated so that corresponding times
line up.
page 112
Dis tance from
cave
0
0
Total distance
traveled
0
0
Time since
left s chool
Time since
left s chool
4. (not the same as #3--involves speed)
Story: Wiley Coyote left school at a slow but steady pace, heading for his cave. When he
was one-third of the way to the cave, he realized that he had forgotten his math book, so he
ran back to school to get it. While he was at school he played checkers for a while with
some other coyotes. He then realized that he would be late for supper, so he started jogging
at a steady rate toward the cave. About half-way to the cave, he thought that he was still
going to be late, so he ran faster and got to the cave in time for supper.
Draw two qualitative graphs for the story, one showing Wiley's distance from the cave vs.
time since he left school. The second graph should show Wiley's speed vs. time since he left
school. The two graphs should be coordinated so that corresponding times line up. Do not
page 113
Distance from
cave
0
Time since
left school
0
Speed
Time since
left school
0
0
5. When is the speed greater--at A or at B?
distance
travelled
B
Greater at ____ because
A
time
6. Complete a qualitative distance-time graph for the following story. Make clear what your
distance refers to.
"Wiley is in his cave and then walks slowly toward a canyon, planning to make a
trap for Roadrunner. Halfway there, he stops for a short rest. Then he jogs on to
make up for lost time. When he gets to the canyon, he realizes that it is almost time
for Animal Planet on TV, so he runs as fast as he can back to the cave."
7. "Wiley walked out of the cave toward Roadrunner's usual resting spot, and after a while
started slowly crawling to sneak up on Roadrunner. Roadrunner saw Wiley, however, and
with Wiley close behind ran very fast straight to a secret hiding place right next to Wiley's
cave."
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Make qualitative graphs for the story on the following three coordinate systems. The
systems are aligned so that your graph can show events at the same time for the different
quantities.
Wiley's distance
from cave
time
Wiley's total
distance
time
from the cave
time
8. Give the indicated companion graph, in each case.
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A. distance
B.
speed
time
time
speed
distance
time
time
9. A. Write a short story that would yield this graph.
distance
time
B. Write a short story that would yield the graph in part A, but with speed on the vertical
10. Below is a flask for water: Sketch a graph to show the relationship of the volume of the
water in the flask and the height of the water as water is poured into the empty flask in a
height
volume
page 116
Chapter 14
1. Put the letter of the graph from the right
that would be most likely to belong to
the given equation.
___
___
___
___
A
y
B
D
C
y = 2x + 7
y = – 2x + 7
y = 22
x
33
41
y = 50
x
E
x
2. Write an equation for each of these, and a story for the graph in part B.
A. Our dog eats 12 pound of dog food every day. We buy a bag that weighs 24 pounds.
What amount, A, remains after d days?
Equation: A =
y
B.
25
Equation: y =
20
10
1
2
3
x
Story for the graph:
3. Ordinarily it takes Brother 20 minutes to go from home to school, and it takes Sister 16
minutes to go from home to school. Each child walks at a steady rate. Today Brother left
home at 8:00 a.m. and so got a 3-minute head start on Sister, who left at 8:03.
page 117
A. Draw a graph to show the percent of the distance to school that Sister has covered vs.
the time. Label the graph S.
Percent of dis tance
to s chool (p)
100
50
10
8:00
8:05
8:10
8:20
8:15
B. On the coordinate system above, draw a second graph to show the percent of the
distance to school that Brother has covered. Label the graph B.
C. According to your graphs, will Sister catch up with Brother? If so, how do you know,
and when does she catch up? If not, how do you know?
D. What is the slope of Brother's graph, and what does that slope mean, in this situation?
Slope:
Means:
4. Ordinarily it takes Brother 20 minutes to go from home to school, and it takes Sister 16
minutes to go from home to school. Today Brother got a 2-minute head start on Sister.
home
school
page 118
Time
A. Draw a graph to show the percent of the distance to school that Sister has covered vs.
the time that Sister travels. Label the graph S.
Percent of dis tance
to school (p)
100
50
10
0
5
10
Time Sis ter
travels (t)
15
B. On the coordinate system above, draw a second graph B to show the percent of the
distance to school that Brother has covered, using Sister's travel-time scale as the
clock. The first point, taking into account Brother's head start, is shown. (Be sure to
notice that the horizontal axis scale is for Sister's travel time.)
C. According to your graphs, will Sister catch up with Brother? If so, how do you know,
and when does she catch up? If not, how do you know?
5.
Mr. Cool joins Rabbit and Turtle in an Over-and-Back race, 200 meters each way.
Rabbit: Speed over--50 m/s; time back--10 seconds
Mr. Cool: Time over--8 seconds; speed back--40 m/s
Turtle: The same speed both ways, but he rested for 5 seconds after the
first 200 meters.
A. Who of Rabbit and Mr. Cool finished first? ______________
Work:
B. What was Rabbit's average speed for the race? _____________
Work:
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C. What was Turtle's speed when he was moving, if Turtle tied Rabbit (remember that
Turtle rested for 5 sec)?
Work:
6. Chicken joins Turtle and Rabbit in Over and Back. They run an Over-and-Back race, with
known data as in the drawing below.
Turtle
60 feet per second over -->
<--7.5 sec. to go back
300 ft.
one way
Rabbit
300 ft.
one way
25 ft./sec. for the first 100 ft. over, 6 sec
for the next 200 feet over -->
Chicken
<-- 75 feet per second back
300 ft.
one way
A. Going the same speed over and back, Rabbit just barely won the race. What was
B. What was Chicken's average speed for the whole over-and-back trip? Show your
work.
7. Wiley joined Rabbit and Turtle in an Over-and-Back trip, 200 m each way. Show your work.
Rabbit: Over--20 m/s; back--4 s
Wiley: Over--8 s; back--40 m/s
Turtle: The same speed both ways.
A. Who of Rabbit and Wiley finished first?
B. What was Turtle 's speed, if he tied Rabbit?
C. What was Rabbit 's average speed? ______
D. Use one coordinate system to show qualitative speed vs time graphs for Rabbit 's
(mark with ____ ), Wiley's (mark with ......), and Turtle 's (mark with x x x x ) speeds
over and back.
page 120
speed (m/s)
time (s)
8. A chicken and a squirrel join the turtle and the rabbit in Over and Back. They run an Overand-Back race, with known data as in the drawing below.
4 sec. for the first 100 ft. over, 6 sec. for the next 200 ft over
Chicken
300 ft.
<--75 feet per second back
Squirrel
40 feet per second over -->
<--5 seconds to go back
300 ft.
Turtle
60 feet per second over-->
<--7 1/2 seconds to go back
300 ft.
Rabbit
300 ft.
A. Going the same speed over and back, the rabbit just barely won the race. What was
the rabbit's speed? Explain your thinking.
B. Who came in second, third, and fourth? (Indicate ties, if any. No explanation is
required.)
Second____________
Third___________ Fourth___________
C. At the half-way mark of the race (300 feet), what was the order of the animals?
(Indicate ties, if any. No explanation is required.)
Second__________ Third________
D. What was the chicken's average speed over?
Fourth________
________
page 121
9. A new "Over" distance is set at 100 feet. The Rabbit takes 5 12 seconds to go over, and comes
back at 40 feet per second. What should the Turtle's speed be, so that the animals tie? Write
enough so that your thinking is clear.
Rabbit's speed _______
10.
A. Someone buys a few candy bars at 55¢ each and several others on sale at 35¢ each. Is
the average price for the candy bars 45¢? Explain briefly.
______________ because
B. A store sells two sizes of soft drinks--a large for \$1.10, and a small for \$0.90. Friday
the store sold 236 soft drinks. The manager reasons, "The \$1.10 and \$0.90 give an
average of \$1. So we must have taken in \$236 from soft drinks on Friday." Is the
manager correct/incorrect? Explain in some detail.
______________ because
11.
Tell whether each is correct, and explain why or why not.
A. Fund-raiser: "We go two donations of \$5000 and two of \$1000 each! That's an
average of \$3000 per donation!"
______________ because
B. Quality-control trainee: "One shipment had 1% of the items defective, and another
shipment had 5% defective. That's an average of 3% defective."
______________ because
12. There is a (common) error, or some confusion, in each of the following situations. Find the
error/confusion, and explain why some thinking was "off," even though the person thought the
thinking was all right.
A. Before summer school a student had completed 96 units, with a gpa of 2.9. The
student takes two 3-unit courses in summer school and gets an A and a B, a 3.5 gpa.
2.9 + 3.5
The student is pleased on calculating
and finding the new gpa to be 3.2.
2
B. A teen-ager says she never has anything nice to wear. Her parents tell her she can buy
4 new blouses for her birthday, but to keep the average price at \$25 (or less). The girl
page 122
finds some \$40 blouses she really likes and calculates from
40 + x
= 25 that x = 10.
2
So a \$40 blouse and a \$10 blouse would give an average of \$25. She is happy to find
an acceptable \$10 blouse on a sale table. She buys it and three of the \$40 blouses in
different colors.
C. A person bought a used car and wants to check its gas mileage. For the first few fillups, she covered 823 miles and got 24.4 miles per gallon. The next fill-up, after 240
24.4+40
more miles, took 6 gallons, so she got 40 miles per gallon. She calculates
=
2
64.4
2 = 32.2 mpg and feels much better.
D. Abe got 94 out of 100 on one test, but only 26 out of 50 on a second. He figures he
has 94 + 26, or 120, out of 150, and finds that to be 80%. Hence, he cannot
understand why his teacher's grade book shows a different result: 94%, 52%, which
94%+52%
146%
give an average of
=
= 73%.
2
2
(Choose carefully, based on your class work and/or discussions.)
13. Express the general property in algebra that each illustrates, using variables.
A. (400 + 25) x 3 = (400 x 3) + (25 x 3)
B.
C.
(57 x 16) + (43 x 16) = (57 + 43) x 16
(600 + 32) ÷ 4 = (600 ÷ 4) + (32 ÷ 4)
75  18 75 18


D.
3
3
3
14. Name the property or properties that justify each of the following.
A. (700 + 60 + 3) + (200 + 30 + 5) = (3 + 5) + (60 + 30) + (700 + 200)
B. 5 x (17 + 3) = (5 x 17) + (5 x 3)
C. 984 + (717 + 563) = 984 + (563 + 717)
D. (56 x 89) x
113
113
= 56 x 89
E.  75  (  23  75)   23 (Hint: More than one property!)
F.
1
3
 95  13  4  13  (95  4)
G.  26 
52
52
  26
H. 5 x (17 x 3) = 5 x (3 x 17)
page 123
15. Test each of the following algebra statements to see whether they appear to be true in general.
If a statement appears always to be true, draw a diagram to justify the statement. Add an
explanation if the diagram is not self-explanatory. If a statement is not true in general, give a
counterexample.
A.
a
b c
 ba 
a
c
B. (a  b)  (c  d)  (a  c)  (b  d)
C. (a  b)2  a 2  b 2
D. (a  b)3  a 3  b 3
E. (a  b)  c  a  (b  c)
16. In each part use the given, correct algebraic statements to answer the calculation.
A. (x  4)(2x  1)  2x 2  9x  4 implies that 14seventeen x 21seventeen = _________ seventeen.
B. (2x  3)(2x  1)  4x 2  8x  3 implies that 23eleven x 21eleven = ________eleven.
C. (Bonus) What other bases could be used in parts A and B?
17.
A. It is correct that 12nine x 32nine = 384nine. How might that inform (x+2)(3x+2)?
B. Explain the "might" in part A by considering 4nine x 13nine = 53nine. Give another
calculation that would mis-inform an algebraic expression. Bonus: Why does 4ninex13nine
= 53nine, or your calculation, give an incorrect idea for algebra?
18. Calculate the sum and product of the pairs of polynomials.
A. 4x 2   7x  3 and 3x  5
B.
5x   6 and  3x  7
19. Make a drawing to justify a(b  c  d)  ab  ac  ad .
20. T F The "balance" diagram to the right shows
x + 2 = x3. (If it is not true, give a correct equation.
21. Make a drawing of a balance for the following equation. Solve the equation by showing
actions with the balance, one step at a time.
4x + 5 = 5x + 3
1
22. Use specific values for m and n in (a m )n  a mn to give a basis for justifying that a 2  a .
1
22'. Use a m  a n  a m n to justify defining a 2  a .
23. Use
1
am
 a m n as the basis for defining a 0  1 and a  n  n for non-zero values for a.
n
a
a
page 124
24. Show your mastery of the conventional order of operations by evaluating each.
A. 6 + 3 x 7 – (2 + –1)5
B. 10 – 4 ÷ 3 x 2 + 1
C. 10 – 4 ÷ 3 x (2 + 1)
D. f(3), for f(x) = 4x2 – 7x + –2
E. g(–2), for g(x) = 9 – 3x – 5x3
25. Finish each story problem so that it can be described by the given equation.
A. 50 – n = 16
"Jamal had 50 pieces of paper…."
B. 3n + 16.99 = 37 "Jose went shopping and bought a CD for \$16.99…"
Chapter 15
1. Give the 100th and the nth entries for these lists, assuming the patterns continue.
A. 12, 22, 32, 42, 52, …
100th ________
nth _________
B. 3, 5, 7, 9, 11, …
100th________
nth _________
C. 2 12 , 4, 5 12 , 7, 8 12 , 10,… 100th ________
nth _________
2. Does each of the following give a function? Explain your decisions.
A. Associate with each whole number n its third power n3.
B. Assign to each person in the town his/her current last name.
C. Assign to the last names of people in town, the first names.
3. Find a likely function rule for each of the following. Show your work.
A.
B.
x
2
3
4
5
…
f(x)
15
19
23
27
…
f(x) =
x
1
2
3
4
…
y
14
11
8
5
…
y=
page 125
C.
D.
x
4
3
1
2
…
g(x)
29
21
5
13
…
x
0
1
2
3
4
…
g(x) =
y
5
6
9
14
21
…
y=
5. Two students have been looking for a function rule for the data to the right below.
n | f(n) .
1 3
2 5
3 8
4 12
Akeena: "I got f(n) = 12 n(n+1) + 2."
Bea: "Yes, but my mom worked a long time on it and said
f(n) = 12 n(n+1) + 2 + (n–1)(n–2)(n–3)(n–4). Let's ask the
teacher."
You (the teacher):
6. A child is making “space modules with antennas” from toothpicks:
etc.
1-room
module
(6 t-picks)
2-room
module
(11 t-p)
3-room
module
(16 t-p)
The child wonders, “How many toothpicks would it take to make a 100-room module
with antennas?!”
page 126
7. (take-home. Instructor: This is exercise 25 in 15.1.) Figure out a short-cut for squaring a
number ending in 5. Here are some free data:
452 = 2025
352 = 1225
752 = 5625
152 = 225
8. (take-home) Here are examples of a short-cut for mentally squaring a number:
2
Example 1. 76 . Go to the closest multiple of 10--here, for 76, plus 4 to 80. Then go the
opposite way from 76 but the same amount, minus 4, to 72. Multiply 80 and 72 mentally:
2
5760. Add the square of the up-down number 4. 5760 + 16 = 5776. 76 = 5776.
2
Example 2. 62 . Go to 60 (down 2). Then go up from 62 by 2: 64. 60 x 64 = 3840.
2
Add the square of 2. 62 = 3844.
2
2
2
Example 3. 57 . Go to 60, then 54. 60 x 54 = 3240. Add 3 . 57 = 3249.
2
2
2
Example 4. 198 . 200 x 196 = 39200. Add 2 . 39204 (= 198 ).
2
A. Use the shortcut to calculate 37 mentally, and then write the mental steps you did.
B. Give a justification that the method works for squaring any n. Label the up-down
number x. (Instructor: Caution--solution requires a little algebra.)
9.
The Hikers' Supply Company's
Snack Mix Recipe
Ingredients: 3 cups of nuts
1 cup of raisins
2 cups of M&Ms
Combine and mix well. Yield: 9 packages of snack mix.
A. How many cups of ingredients are in 1 package of snack mix? ______ Explain.
B. What fractional part of each package of snack mix is M&Ms? _____ (No explanation
required.)
C. What fractional part of 2 packages of snack mix is M&Ms? _____ (No explanation
required.)
D. How many cups of M&Ms are in 2 packages of trail mix? _____ Explain.
E. If you need 24 packages of snack mix, what quantities of nuts, raisins, and M&Ms do
you need? (No explanation required, but show your work.)
page 127
________ of nuts; ________ of raisins; ________ of M&Ms
10.
A new machine gives 5 choco-nut
hearts for every 4 chocolate bars and 12 cup
choc
bars
nuts
of nuts put into the machine (nothing is lost
in the machine).
Solve by reasoning. Your reasoning may
involve calculations, but not by just
calculating with a proportion.
choco-nut
hearts
 

A. How many hearts would the machine give for 18 bars and the right amount of nuts?
_____ Reasoning:
B. How many cups of nuts would be the right amount, for 18 bars?
_____ Reasoning:
C. An order for 48 hearts comes in. How many bars will that order take?
_____ Reasoning:
11. What is the difference between an arithmetic sequence and a geometric sequence?
12. Suppose g(x) = 3x – 2, and h(x) is defined by Machine X to
the right. Give the output if 10 is the input to the
combination…
A. first h(x), then g(x).
input
and then double
that sum.
B. first g(x), then h(x).
output
13. What does it mean to say, "'Combination' of functions is not commutative"?
14. Illustrate with drawings of "machines" that "combination" of functions is associative.
Chapter 12
1. A. increases.
B. the temperature goes down 20 Celsius degrees each hour.
2. decreases; doesn't change; increases; increases
page 128
3. Mainly, the scale on the hours-spent axis is not in increasing order. That the scale on the
vertical axis does not start at 0 is sometimes done, so that is less serious.
4.
A. Candle 2 because it burns at a rate of 2 cm per minute, but Candle 1 burns at a rate of
only 13 cm per minute.
B. h = – 13 t + 15
C. (Graph should start at (0, 25) and have slope –2. It goes through (12.5, 0).)
D. 12.5
E. Yes, they will be the same height (13 cm, not required) after 6 minutes (from either
the graph or algebraic work). A solution from the graph may be off a bit.
5.
A. \$2.50
6.
7.
B. \$3.50 (using the clearest graph point \$14 for 4 games)
C. Yes, if the bowler bowls more than 1 game. But if the bowler bowls just 1 game, the
first plan makes more money. (The graphs cross between 1 and 2 games.)
A. (sample) On Tuesday, the bakery had 10 000 cookies at the start and baked more
B. –1250 This slope means that the bakery's cookie inventory went down by 1250 cookies
every hour, on Friday.
slope: – 203 That slope means that the candle burns by 203 inch every minute.
Chapter 13
1.
A. 2 minutes 7.5 seconds
B. 10n feet per second (look for unit)
C. 25m or m ÷ 25
2. F (Distance traveled would not decrease; the story-writer may have the graph-as-picture
misconception.)
3. Graphs something like the following. Judge relative distances according to the standards
page 129
Distance from
cave
C
B
A
D
E
0
0
Total distance
traveled
Time since
left school
Pos sible fine points :
B s hould involve s ame
dis tance as A.
C flat; any duration
D dis tance slightly less
than A,B distance.
E s ame dis tance as D.
Overall: non-decreas ing
E
D
C
B
A
0
0
Pos sible fine points :
A should be 1/3 of the way
down to 0.
B s hould s how faster speed
than A.
C flat; any duration
D down to about 1/2 of the
way to 0.
E faster s peed than D.
Time since
left school
4. Graphs something like the following; negative slopes not called for.
Distance from
cave
A
B
C
D
E
0
0
Time since
left school
Speed
E
D
B
A
0
C
0
Time since
left school
Fine points (s peed):
Recall: Negative slopes
have emphasized.
B height should be greater
than height for A.
D vs B height ?, but
E height s hould be
greater than height for D.
5. B because for a given interval of time, more distance is covered.
page 130
6. Here is a graph for distance-to-cave vs time. A graph for distance-from-cave vs time is also
allowed, but in either case the student label the vertical axis.
Dis tance to
cave
Time
7.
Wiley's distance
from cave
Wiley's total
dis tance
R-runner's distance
from cave
page 131
8.
9.
A. Speed-time graph should have two horizontal segments, with the first one having
greater height. (Adjust, if you have emphasized negative speeds.)
B. Distance-time graph should have a line segment for first part (starting at origin), with
second part made up of a curve upward.
A. (sample) Wiley was out by his look-out point when he thought he saw Roadrunner,
so Wiley ran steadily farther away from the cave until he was sure. It was Roadrunner!
Then Wiley ran steadily back to the cave to get a net, and then ran even faster back to his
lookout point. Even though he looked and looked, he did not see Roadrunner.
B. (sample) Wiley ran out of his cave at a steadily increasing speed, and then slowed
down at a constant rate, looking for Roadrunner. He saw Roadrunner and stopped, but
immediately ran at an faster and faster speed until he reached his starting speed, and then
kept running at that speed.
10. The graph should be curved upward, with tangents having increasing slopes as the volume
increases. ( If your students should expect a post-filling continuation, then the height stays
constant after the flask is filled.)
Chapter 14
1. In order: A, B, E, D
2.
3.
4.
5.
A. A = 24 – 12 d, or algebraic equivalent
B. y = 5x + 15 Sample story: Joe has \$15 and can clear \$5 an hour painting an elderly
neighbor's fence. How much will Joe have after x hours?
A. Sister's linear graph S should start at (8:03, 0) and go to (8:19, 100)
B. Brother's linear graph B should start at (8:00, 0) and go to (8:20, 100).
C. The graphs should intersect around (8:15, 75), showing that Sister catches up at 8:15
(12 minutes after starting)
D. The slope for B is 5, meaning that Brother covers 5% of the distance to school every
minute.
A. Graph S should be straight, joining (0, 0) and (16, 100).
B. Graph B should be straight, joining (0, 10) and (18, 100).
C. The graphs should intersect around (8, 50), showing that Sister catches up 8 minutes
after she started (at the half-way point).
A. Mr. Cool (13 seconds vs. 14 seconds for Rabbit)
4
B. 400
14  28 7 m/s (Look for unit.)
4
C. 400
9  44 9 m/s (Look for unit.)
page 132
6.
7.
A. Turtle's time was 12.5 seconds (and Chicken's 14 seconds), giving Turtle an average
speed of 48 ft/s. So Rabbit must have been going slightly faster than 48 ft/s.
6
B. 600
14  42 7 ft/s
A. Wiley (13 seconds vs 14 seconds for Rabbit)
4
B. 400
14  28 7 m/s (Look for unit.)
C. 28 47 m/s
D. s peed (m/s )
R
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
W
T
R: 10 s + 4 s
W: 8 s + 5 s
T: 7 s + 7 s
W
R
time (s)
8.
A. Chicken takes 14 seconds; Squirrel, 12.5 seconds; and Turtle, 12.5 seconds. So
600
 48 ft/s.
Rabbit was going slightly faster than 12.5
B. Squirrel and Turtle tie for second, and Chicken is fourth.
C. After 300 feet, T was first (5 s); R second (6.25 s); S third (7.5 s); and C fourth (10 s).
D. 30 ft/s
9. 25 ft/s (5 12 s + 2 12 s = 8 s for the 200 ft)
10.
11.
12.
A. No, because the numbers for the kinds of candy bars bought are different.
B. Incorrect, because the numbers sold of the sizes were likely different.
A. Correct, because there is the same number of each size of donation.
B. Incorrect probably, because the shipments might have involved different numbers of
items.
A. The error lies in not recognizing that the individual gpas are based on different
numbers of units. The new gpa is only slightly greater than the 2.9 (about 2.93).
B. Again, the error lies in not taking into account the different numbers of types of
 32.50 dollars per blouse.
blouses bought. The average is actually 10120
4
C. The 40 mpg is based on only 240 miles, whereas the 24.4 mpg is based on 823 miles,
so the average she calculated does not recognize that the mpg based on the greater
distance will have a greater influence on the overall average than the mgp based on the
1063
240 miles. (The overall average is 823240
33.76  39.7  26.8 mpg.)
page 133
D. The teacher chose to weight the two exams the same and so used the percents. Abe's
method would have the 50-point exam contribute only 50 of 150 total points, rather than
100% of a possible 200%.
13. Choices of letters may vary from those here, of course, and quantifiers may be included,
A.
B.
C.
D.
14.
15.
(a + b) x c = (a x c) + (b x c), or (a + b)c = (ac) + (bc) or just ac + bc
(a x b) + (c x b) = (a + c) x b
(a + b) ÷ c = (a ÷ c) + (b ÷ c)
ab
a
b
c  c  c (compare parts C and D)
A. Commutativity and associativity of addition
B. Distributivity (of multiplication over addition)
C. Commutativity of addition (not associativity)
D. 1 is the multiplicative identity
E. Commutativity and associativity of addition; additive inverse property; 0 is the
F. Distributivity (of multiplication over addition)
G. 1 is the multiplicative identity
H. Commutativity of multiplication (not associativity)
A. Not true in general; student should have shown a counterexample
B.
True. Sample diagram:
a+b
a
a
b
b
then
c
c+d
C.
D.
E.
16.
17.
c
d
(a + b) – (c + d)
a–c
d
b–d
Not true in general; student should have shown a counterexample.
Not true in general; student should have shown a counterexample.
Not true in general; student should have shown a counterexample.
A. 294seventeen (Let x = 17 in the algebraic equation.)
B. 483eleven
C. Any bases in which 9 (part A) or 8 (part B) are legitimate digits.
A. (x+2)(3x+2) might equal 3x2 + 8x + 4.
B. The given base nine expression would imply (x+2)(3x+2) = 5x + 3. A second
example should suggest an incorrect algebraic result also. The problem is that 4nine x 3nine
= 13 nine, and the 1 influences the next place value. Notice that in part A, there is no such
page 134
influence from any of the base nine multiplications 2 x 2 = 4, 2 x 3 = 6, 1 x 3 = 3, and 1 x
2 = 2, with 6+2 = 8 still involving only one place value.
A. Sum 4x 2   4x  8 ; product 12x 3  x 2   26x  15 , or algebraic equivalents.
B. Sum 2x + 1; product –15x2 + 53x + –42, or algebraic equivalents
18.
19. One way: Rectangular region with width a and length b + c + d. The ab, ac, and ad
subregions can be marked.
20.
F
x + 2 = 3x
21. One pan should have 4 boxes, preferably with each labeled "x", and 5 singles. The other pan
should have 5 boxes and 3 singles. The balance work should show the removal of 4x and 3
singles from each pan (either order), with x = 2 the solution.
Let m 
22.
1
a 2 must be
1
1
1
22'. a 2 a 2  a 2
 21
1
2
25.
A.
B.
1
2
1
a.
1
 a1  a , so a 2 must equal a .
23. Letting m = n, 1 
24.
1
and n  2. Then (a 2 )2  a 2  a1  a. So a 2 , when squared, gives a. That is,
am
a0
1
m m
0
n
0 n
.
And
so
with
m
=
0,

a

a
a

a

 n
m
n
a
a
a
26
8 13
C. 6
D. 13
E. 55
Samples: A. "…After a week of school, he had 16 pieces left. How many pieces did he
use?" Or (comparison subtraction), "…Jamal had 16 pieces more than Jerry. How many
pieces did Jerry have?"
B. "…and 3 pairs of socks. He spent exactly \$37 on the items. How much was each pair
of socks?"
Chapter 15
1.
2.
A. 1002; 10n + 2
B. 201; 2n + 1
C. 151; 1 12 n + 1
A. Yes; each whole number is assigned to exactly one perfect cube.
B. Yes; each person is assigned exactly one last name at a time.
C. No; a given last name might be assigned to more than one first name. For example,
Smith might be assigned to Joseph and also Kendra.
page 135
3.
A. f(x) = 4x + 7
B. y = –3x + 17
C. g(x) = 8x – 3
D. y = x2 + 5
4. Some other rule might also describe these data.
5. "Both function rules are correct for the data given. We need more data to see which one, or
perhaps even some other one, is better."
6. 501 (the n-room module would take 5n + 1 toothpicks). One justification: For the 100-room
module, the antennas take 2x100 toothpicks, the tops and bottoms of the rooms take 2 x 100,
and the room dividers take 100 + 1. So the total is 2 x 100 + 2 x 100 + (100 + 1), or 501. A
similar argument gives 2n + 2n + (n + 1), or 5n + 1, for the n-room module.
7. Reports will vary, and some will likely use an example. Here is one version: Take the
number of tens and multiply it by the next larger number. Write that down, and write 25 after
that. In another version, one squares the number of tens, adds the number of tens to that,
writes the sum down, and writes 25 after that. It is to be hoped that students ask for a
justification. A justification can be based on (10n + 5)2 = 100n(n+1) + 25, or 100(n2 + n) + 25.
40 x 34 = 1360, 32 = 9, 1360 + 9 = 1369
(n + x)(n – x) + x2 = n2 – x2 + x2 = n2
8.
A.
B.
9.
A. 6 cups of ingredients make 9 packages, so 1 package has 69 , or 23 , cup of ingredients.
B. Because M&Ms make up 26  13 of the ingredients, 13 of each package will be M&Ms.
C. 13
D. 29 of 2 cups = 49 cup of M&Ms .
E. ( 249 = 2 23 recipes needed) 8 cups of nuts, 2 23 cups of raisins, 5 13 cups of M&Ms
4 12  5  22 12 hearts (18 bars would make 4 12 recipes)
A.
4 12  12  2 14 cups
B.
C.
48 hearts would be 48 ÷ 5 = 9 53 recipes, so 9 53  4  38 25
chocolate bars will be needed.
10.
11. In an arithmetic sequence, each entry after the first is obtained by adding the same number to
the previous entry. In a geometric sequence, each entry after the first is obtained by
multiplying the previous entry by the same number.
12.
A. 82
B. 64
13. The order in which a 'combination' of functions is carried out can give different results.
(E.g., see #12.)
page 136
14. Look for three "machines" combined differently. (Machine 1 after Machine 2) after Machine
3 versus Machine 1 after (Machine 2 after Machine 3).
page 137