Chapter 8: Ways of Thinking about Multiplication
Gia Elise Barboza
Gia Barboza
MATH 201
Department of Mathematics, Michigan State University
April 12, 2005
1
Ways of Thinking About Multiplication
How would you solve the following problems?:
1. One kind of cheese costs $2.19 a pound. How much will a package weighing 3 pounds cost?
2. One kind of cheese costs $2.19 a pound. How much will a package weighing .73 pounds cost?
1. Most individuals do not realize they need to multiply in the second situation. They either subtract or
divide. (There is about 35% - 45% less success on the second problem.
• Why? Is people’s thinking totally misguided when they think of problem #2 in terms of subtraction
or division? What about the part-of-an-amount interpretation?
2. Multiplication is often introduced in terms of addition, and the repeated addition interpretation is the
only interpretation that students retain.
3. A “times as large” interpretation is often more powerful and more general, i.e., it includes the idea of
repeated addition as a special case.
4. To demonstrate the greater generality, interpret each of the above problems in terms of a piece of cheese
that is x times as large as the one-pound piece.
• The new package is 3 times as large as the one-pound package.
• The new package is .73 times as large as the one-pound package
5. Overemphasis on repeated addition leads to the overgeneralization that multiplication always makes bigger.
• What other ways have we seen that may help clarify the confusion?
1/ Multiplicative comparisons
2/ “Copies-of” interpretation
→ Note: Both of these ideas include repeated addition as a special case
→ Example: 4 × 12 can mean either 4 groups of 12 (repeated addition), 4 times as large as a
quantity with value 12 (multiplicative comparison) or 4 copies of 12 (“copies-of”).
6. Whenever the first factor is a whole number, both the repeated addition and times-as-large interpretation
makes sense, but when the first factor is not a whole number, the times-as-large interpretation as well as
the “copies-of” makes sense while the repeated addition interpretation does not.
7. We know that it does not make bigger when the first factor is a number less than one.
1
1.1
Exercises for Section 8: 1, 2, 5c,e,6
1.1
2
SECTION 8.1 NOTES FOR TEACHING
Exercises for Section 8: 1, 2, 5c,e,6
• Will multiplying by a whole number (> 1) by any fraction always result in a product smaller than the
whole number we started with? When does multiplication make bigger.
→ No. When the fraction is greater than one multiplication does make bigger.
• Here are the ingredients in a recipe that serves 6:
a. You want enough for 8 people. What amounts should you use?
→ Since 8 is one-third more than 6, multiply each quantity by 1 13 or
4
3
to get the right amounts.
b. You want enough for 4 people (and no left-overs). What amounts should you use?
→ Since 4 is two-thirds less than 6, multiply each quantity by
2
2
3
to get the right amounts.
Section 8.1 Notes for Teaching
1. Extended exposure to repeated addition as the only way to think of multiplication has lead to what
researchers say is the reason for the incorrect idea that multiplication always makes bigger.
2. Some suggest that a “copies of” interpretation has appeal. This accommodates both the repeated addition
and part-of view. For example, 5×17.3 can be interpreted as 5 copies of 17.3 and 65 ×28 can be interpreted
as 56 copies of 28.
3. This can be applied across all types of situations (i.e. the “copies-of” interpretation unifies the types of
multiplication:
Y = tX
Y = X + X + ... ⇒ Y = tX
Y = tth X ⇒ Y = tX
(1)
(2)
(3)
(Eqn 1) Equation 1 means that Y is t times as large as X. What view of multiplication does this idea support?
Answer. Copies-of or multiplicative comparisons.
(Eqn 2) Equation 2 means that X is added t times. What view of multiplication does this idea support?
Answer. Repeated Addition.
(Eqn 3) Equation 3 means that Y is a tth part of X. What view of multiplication does this idea support?
Answer. Part-of view of multiplication.
⇔ The point: Each of these can be interpreted as follows: Y is t copies of X.
2.1
Exercises for Section 8.1: 1, 2, 4,5,10,12
1 The star player scored four times as many points during her senior year as during her freshman year.
→ The number of points scored by the star player during her senior year was 4 copies of the number of
points she scored during her freshman year.
2 The apartment rents for 3 times as much as it did 15 years ago.
→ The apartment rents for 3 copies of what it did 15 years ago.
4 The shortest player was 90% as tall as the middle sized player.
→ The shortest player is .90 copies of the middle sized player.
2
2
SECTION 8.1 NOTES FOR TEACHING
2.2
Section 8.2: Division in Multiplicative Settings
5 After extensive conditioning, the student could jog 4 21 times as far as she could before the conditioning.
→ After conditioning, the student could jog 4.5 copies as long as she could before the conditioning.
10 The class paid for 32 tickets to the play.
→ The whole class paid 32 copies of the price of one ticket.
12 In making the new budget, the old budget had to be cut by 12%.
→ The old budget was .12 copies of the new budget.
2.2
Section 8.2: Division in Multiplicative Settings
Would you act these two problems out in the same way:
1. You have some cupcakes and plan to put them into bags of 7 cupcakes each. How many bags will you
need?
2. You decide instead to take the same cupcakes and put them into 7 bags (with the same number in each
bag). How many cupcakes will there be in each bag?
1. Note that we have seen this problem before, it is review.
2. The first question connotes a repeated subtraction view of division. Here, we visualize a situation where
we keep removing groups of 7 cupcakes until the supply of cupcakes is exhausted. 42 ÷ 7 ⇒ 42 = n7
3. In the second question, we can conceptualize separating the cupcakes so that they are in seven groups of
equal size. This view is the partitive or sharing interpretation of division. 42 ÷ 7 ⇒ 42 = 7n
4. We can think of division, then, as the operation called for when solving a multiplication equation with a
missing factor.
5. In both cases, since 42 ÷ 7 = n then 42 = 7n. Instead of thinking in terms of division then, we can think
of these problems in terms of multiplication.
Would you draw the same picture for these two problems?:
1. The highway crew has several miles of highway to repair. Past experience suggests that they can repair
2.5 miles in a day. How many days will it take them to repair the stretch of highway?
2. The highway crew has the same stretch of highway to repair. But they must do it in 2.5 days. How
many miles will they have to repair each day to finish the job on time?
1. In the first question, we need an unknown number of copies of quantities of 2.5 miles.
• Example: Suppose we had 10 miles of highway to repair. We can repair it in 2.5 days. The unknown
number of copies of quantities of 2.5 miles in this situation is 4, since 4 × 2.5 = 10. We need 4 copies
of 2.5. We ask ourselves, how many copies of 2.5 will give us 10?
2. In the second question, we need 2.5 copies of an unknown quantity of miles.
• Example: Suppose we had 10 miles of highway to repair. We must do it in 2.5 days. Here, we need
2.5 copies of 4 miles in order to do the job in 10 days. 2.5 copies of X = 10 or 2.5X=10. Here, 2.5
× 4 = 10.
3
2.3
Exercises for Section 8.2: 1, 6, 7,12bc
3
CHAPTER 9: MULTIPLICATIVE COMPARISONS
Are these problems the same?:
1. You have a large piece of cheese, and you want to cut off chunks each weighing about
many pieces will you get?
3
4
pounds. How
2. You have a large piece of cheese. It is 34 of the whole cheese from which it was cut. How many pounds
were in the whole cheese before it was cut?
1. In number one, we need an unknown number of copies of quantities of
3
4
pounds.
• Let’s say that the large piece of cheese weighs 3 pounds. Then an unknown number of copies of 34 =
3. Denote the unknown quantity by X, then we have X × 43 = 3 so that X = 4. Here, 34 + 34 + 43 ...
how many times?
2. In number two, we need
and X = 4.
3
4
copies of an unknown quantity. Call the unknown quantity X. Then
3
4
×X = 3
We can visualize this as follows:
|1llb1llb1llb
{z } 1llb or, ...
3
4
2.3
Exercises for Section 8.2: 1, 6, 7,12bc
1. Write down how you would go about answering this question: How many times as large as the population
of Des Moines is the population of NYC together with that of Houston?
→ (NYC + HOUS) = X×DESMOINES. Assume the population of NYC = 7,500,000, Houston =
1,700,000 and DesMoines = 200,000. Then we have 7500000 + 1700000/200000 = X.
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3.1
Chapter 9: Multiplicative Comparisons
Section 9.1: Compound Units and Technical Definitions
Read this very short section (1 paragraph) on your own. The ideas are represented in section 9.2, so to avoid
redundancy we will cover these ideas in the next section.
3.2
Section 9.2: The Fundamental Counting Principle
Example:
You hear that Ed, Fred, Guy, Ham, Ira, and Jose ran a race. You know there were no ties, but you do not
know who was first, second, and third. In how many ways could the first three places in the race have turned
out?
How many ways are there for first place? Second? Third?
Note, in the past, we used a tree diagram to represent the outcomes. The problem with a tree diagram is it
quickly gets messy.
Example:
You have 3 blouses and 2 pairs of pants, how many different blouse-pants outfits do you have?
This is much easier to represent with a tree diagram. However, we can also solve this by recognizing that
choosing a blouse can be done in 3 ways and then choosing a pair of pants can be done in 2 ways. Therefore
3 × 2 = the number of possible outfits.
4
3
CHAPTER 9: MULTIPLICATIVE COMPARISONS
3.3
Exercises for Section 9.2: 1ab, 2, 3, 5, 6, 7, 9, 10,11
Situations which can be thought of as a sequence of acts, with a number of ways for each act to occur, can be
counted efficiently by the Fundamental Principle of Counting (aka the Multiplication Rule).
The Fundamental Principle of Counting: If an act can be performed in n ways and another act in m
ways then both acts can be performed in m × n ways.
3.3
1.
Exercises for Section 9.2: 1ab, 2, 3, 5, 6, 7, 9, 10,11
a. How much does it cost to light a 100-watt bulb for a year, if electricity costs $0.099 per kilo-watt
hour?
→ First, we must convert to kilo-watt hours
100/1000 = .1
.1 × 0.099 = .0099per/hour
.0099 × 24 × 365 = $86.72per/year
(4)
(5)
(6)
b. Suppose that the 100 watt bulb in part a. is replaced by a 40-watt bulb after 8 months and the 40
watt bulb is lit the rest of the year. How much money is saved that year?
→ For 1/3 of the year the 100 watt bulb costs
$86.72/3 = $28.91
.04 × .099 = 0.00396per/hour
0.00396 × 24 × 365 = $34.69per/year
$34.69/3 = $11.56
$28.91 − $11.56 = $17.35
(7)
(8)
(9)
(10)
(11)
2. Do on your own.
3. A 3-person crew is assigned to do a job that once took 5 people 10 hours each to do. How long will it take
the smaller crew to do the job? Answer. It took 5 people 10 hours to do the job, which is 50 man-hours.
How many hours will it take 3 men to do the job? 50/3 = 16 32 hours.
5. Answer the following questions.
a. You flip a spinner that has four differently-colored regions (red, white, blue and green), and toss one
die and count the dots on top (1 through 6 possible). How many color-dots outcomes are possible?
Answer. 4(6) = 24.
b. A couple is thinking of a name for their expected baby girl. They have thought of 3 acceptable first
names and 4 acceptable middle names (all different). How many baby girls could they have without
repeating the whole name!? Answer. 3(4)=12.
c. In a sixth-grade election, Raoul, Silvia, Tien, Vena and Wally are running for president; Angela and
Ben are running for vice president. How may ways can the election come out? Answer. 5(2) = 10.
d. In a game you toss a red die and a green die, and count the number of dots on top of each one – e.g.,
R2, G3. (The numbers of dots are not added in this game.) In how many different ways can a toss
of the dice turn out? Answer. 6(6) = 36.
6. A hamburger chain once advertised that they made 256 different kinds of hamburgers. Explain how this
claim is possible. (Hint: One can choose 1 patty or 2, mustard or not, catsup or not, cheese or not, ...).
Answer. 2n = 256 ⇒ n = 8. In general, do you know how to solve for n?
7. An ice-cream store has 31 different kinds of ice cream, and 2 kinds of cone.
a. How many different kinds of single-scoop ice-cream cones can be ordered at the store? Answer. 31(2)
= 62.
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4
PRACTICE PROBLEMS FOR FINAL EXAM
b. How many different kinds of double-scoop ice-cream cones are there? (Decision: Is vanilla on top of
chocolate the same as chocolate on top of vanilla?) Assume that vanilla on top of choc. is different
than choc. on top of vanilla. Answer. 31(31)(2) = 1922. If they are the same the answer is 31(30)(2)
= 1860.
9. Ronnie R. Thomas can write the letter “R” in 3 ways and the letter “T” in 5 ways.
a. In how many ways can Ronnie write the initials “RT”? Answer. 3(5)=15.
b. In how many ways can Ronnie write the initials “RRT” without repeating the style or “R” used?
Answer. 3(2)(5) = 30.
10. You have become a car dealer! One kind of car you will sell comes in 3 body styles, 4 colors, and 3
interior-accessory “packages,” and costs you, on average, $12,486. If you wanted to keep on your lot an
example of each type (style with color with package) that a person could buy, how much money would
this part of your inventory represent? Answer. (3)(4)(3)(12486) = $449,496.
11. License-plate numberes come in a variety of styles.
a. Why might a 2-letter followed by 4-digits style be better than a 6-digits style? Answer. There
are more possibilities with a 2-letter followed by a 4-digit style. In that case, there are (262 )(104 )
possibilities vs. only 106 possibilities with a 6-digits style.
b. How many times as large is the number of license plates possible with a 6-letters style as the number
possible with a 6-digits style? Answer. With a 6-letters style we have 266 possible combinations.
With a 6-digit style we have 106 possible combinations. To figure out how many times larger one is
over the other, we simply divide:
266
=
106
(10 × 2.6)6
=
106
(10)6 × (2.6)6
=
106
6
2.6 ≈ 308.91
4
(12)
(13)
(14)
(15)
Practice Problems for Final Exam
1. Suppose a stadium has 9 gates. Gates A, B, C and D are on the north side while gates E, F, G, H, and
I are on the south side. In how many ways can you enter the stadium through a north gate and leave
through a south gate? Demonstrate your understanding by illustrating the solution as a rectangular array
in which each entry is an ordered pair.
Exit Gate (South)
E F G H I
Entry Gate (North)
A
B
C
D
2. Prof. Klein is giving her history class a quiz with 5 questions. Since Angie has not done her homework,
she has to guess. The quiz has two multiple-choice questions with choices A, B, C, and D, and three
true-false questions.
a. How many possible ways are there for Angie to answer all five questions?
b. What is the probability that Angie will get all the questions correct?
6
4
PRACTICE PROBLEMS FOR FINAL EXAM
3. Prof. Ramirez has written a chapter test. It has three multiple-choice questions, each with m possible
answers, two multiple-choice questions each with n possible answers, and 5 true-false questions. How
many ways are there to answer the questions?
4. State the Fundamental Principle of Counting.
5. Each of 20 questions on a quiz can be answered true or false. Suppose a student guesses randomly.
a. How many ways are there of answering the test?
b. What are the chances of getting all of the answers correct?
6. Suppose Ms. Manetti gives a quiz that has two questions with x choices and three true-false questions.
Give the number of ways to answer the items on the test with an expression.
7. If a test has 5 multiple choice questions each with q possible answers and 5 true-false questions, how many
different ways would there be to answer the test?
8. Radio station call letters, such as WNEW, must start with W or K.
a. How many choices are there for the first letter?
b. How many choices are there for the second letter?
c. How many different 4-letter station names are possible?
9. Telephone area codes consist of 3 digits. Prior to 1989, they fit the following rule: The first digit must
be chosen from 2-9, the second digit must be 0 or 1, and the third digit cannot be 0. In 1989, this policy
changed so that the first and third digits could be any digit 0-9. (The second digit must still be 0 or 1).
a. How many area codes were possible before 1989?
b. How many area codes were possible after 1989?
c. How many area code possibilities were added by the policy change?
10. The Cayuga Indians played a game called Dish using a wooden bowl and six pits from peaches. The
pits were blackened on one side and uncolored on the other. A player scored 5 points if six black or six
uncolored sides landed up.
a. How many possible ways could the pits land?
b. What is the probability that a player would score 5 points with one toss?
11. Suppose a password consists of 4 characters, the first 2 being letters in the English alphabet and the last
2 being digits. Find the number, n, of:
a. passwords;
b. passwords beginning with a vowel.
12. Suppose a code consists of 2 letters followed by 3 digits. Find the number of:
a. codes;
b. codes with distinct letters; and
c. codes with the same letters.
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