Deck 3: Understanding Whole Number Operations

Question 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.

Answer 1

There are several possible arrangements. One of these arrangements is illustrated by the figure below, which helps show that the difference asked for (BT vs. B2T) is 18 inches. (BT is Coach B's tallest player and B2T is Coach B's second-tallest player.) 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.

Question 2

To determine how much older than you your father is, you need to make an additive comparison of his and your ages.

Answer 2

True

Question 3

Marge bought several types of candy for Halloween: Milky Ways, Tootsie Rolls, Reese's Cups, and Hershey Bars. The Milky Ways and Tootsie Rolls together were six more than the Reese's Cups. There were four fewer Reese's Cups than Hershey Bars. There were 12 Milky Ways and 28 Hershey Bars. How many Tootsie Rolls did Marge buy?

A) List five quantities involved in this problem.
B) Sketch a diagram to show the relevant sums and differences in this situation.
C) Solve the problem.

Answer 3

A) The five quantities are usually easy: for example, the number for each type of candy, and some of the explicit comparisons mentioned.
B) The diagram below shows the deduced numbers of bars in parentheses. (HB stands for Hershey Bars and MW for Milky Ways, two given quantities, whereas the number of Tootsie Rolls and Reese's Cups is deduced.)
​

Question 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 eight fewer roast beef sandwiches prepared than there are servings of cheese pizza. There are six more peanut butter sandwiches prepared than there are servings of vegetarian pizza. Altogether, how many servings of sandwiches are prepared?

A) List eight quantities involved in this problem.
B) Sketch a diagram to show the relevant sums and differences in this situation.
C) Solve the problem.

Question 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 six more students than Section D. Section B has two fewer students than Section C. How many students are there in Section A and Section B altogether?

Question 6

In which word problem (A or B) is there an action involved that may make it easier for young children to solve?

A) Silvia had 14 books and then received 4 more books. How many books does she have now?
B) Silv ia has 14 books on one shelf and 4 books on another. How many books are on the two shelves?

Question 7

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.

Question 8

Write a missing-addend word problem using $35.95 and $19.50.

Question 9

Suppose you are using toothpicks to act out the following story problem:
Jack had eight candy bars. Bill had four.

A) How many more candy bars did Jack have than Bill?
B) How many toothpicks would you need to act the problem out?
C) What type of subtraction is this?

Question 10

For A, B, and C below, state:
1) the operation you would use to answer the question,
2) the situation in which the problem fits, and
3) an expression that yields the answer.

A) Susan has $175. She wants to go on a ski 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?

Question 11

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 replies, "79 minus 40 is 39, so she needs $39."
Explain Rita's reasoning. What is your reaction to this method of doing the problem?

Question 12

Filene is asked, "What is 79 minus 32?" She responds, "8 to 40, 30 to 70, and 9 more, so the answer is 47." Explain Filene's reasoning. What is your reaction to this method of doing the problem?

Question 13

A bill for school supplies was $87.35. Josh paid with two $50 bills. Rikki, at the cash register (one that did not tell the change to be given to the buyer), counted Josh's change: "40, 50, $1, and $10 makes $100." How much change did Josh receive? In what currency? Is that what he should have received?

Question 14

A) Make drawings of circular "pizzas" to illustrate 6 - 2, take-away view.
B) A child is shown nine apples and six 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?

Question 15

In each story problem, the calculation 6 - 2.5 can be used to solve the problem. Which story problem involves the "missing-addend" situation and which involves "comparison"?

A) Two joggers decided to run at the beach. They usually run for 6 miles. How much farther do they have to run if they have already run 2.5 miles?
B) Two joggers decided to run at the beach. One runs 6 miles and the other runs 2.5 miles. How much farther than the second jogger does the first jogger run?

Question 16

In each scenario, which way of thinking about subtraction is involved?

A) Story problem: Basketball score: Aztecs 82, Opponents 69. By how many points did the Aztecs win?
B) 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…

Question 17

Which way of thinking about subtraction is involved in each story problem? 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?

Question 18

For each story problem, 5 - 3

can be used to solve the problem. Label each of the story problems as comparison subtraction, take-away subtraction, or missing-addend subtraction.

A) You made 5 gallons of lemonade for a school party and 3 gallons of Kool-Aid. How much more lemonade than Kool-Aid did you make?
B) You made 5 gallons of lemonade for a school party. At the party, the guests drank 3
gallons of the lemonade. How much of the lemonade was left after the party?
C) You made 5 gallons of lemonade for a school party. 3

gallons were made from frozen lemonade and the rest from fresh lemons. How much of the lemonade was made from fresh lemons?

Question 19

For each story problem, 26 - 12 can be used to solve the problem. Label each of the story problems as comparison subtraction, take-away subtraction, or missing-addend subtraction.

A) Laresa had $26 when she went into the store, and her friend Tisha had $12. How much more did Laresa have than Tisha?
B) Laresa had $26 when she went into the store. She bought a wallet for $12. How much did she have left?
C) Laresa had $26 when she went into the store because she had $12 already and then received cash for babysitting. How much did she earn babysitting?

Question 20

Give the rest of the "family of facts" for k - 3 = p.

Question 21

Following is only the start of a child's work (in base ten). What seems lacking in this child's understanding? (Assume, perhaps unfairly, that the child is not thinking of negative numbers.)

Question 22

Perform the following subtraction using the "equal additions" method.

Question 23

The work of two students is shown below. Each student "invented" the method used; that is, it was not taught to the student. Figure out what each student was thinking while doing the problem. Then work the second problem (i) using the same method as the student, and (ii) comment on the student's method in terms of the "number sense" exhibited.
A)
$\begin{array} { l } 732\quad\quad \text { (i) } 8341 \\\underline{ - 245 } \quad \quad \underline{- 4567 }\\513\end{array}$
B) 19 × 35. Well, 20 × 35 is like 10 × 35 two times, so that's 350 two times, which is 700. But that's twenty 35s and I only want nineteen 35s. So 700 minus 30 is 670 minus 5 is 665.
(i) 21 × 43

Question 24

Is this child's thinking correct? If it is, complete the second calculation using the child's method. If the thinking is not correct, explain why not. $\begin{array} { r r } \text { (given) } & \text { (child does) } \\675 & 677 \\- 198 & - 200 \\& 477\end{array}$
Second calculation (or explanation if not correct):
$\begin{array} { r } 453 \\- 295 \\\hline\end{array}$

Question 25

A visitor to a first-grade classroom sees a teacher ask a student to solve this problem:
Jaime gets $5 a week for keeping the yard in good shape. He is saving his money for the county fair. After four weeks, how much has he saved?

A) The visitor thinks to himself, "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, the first-grader says that the answer is 20. The student explains how she did this problem, and, much to the visitor's surprise, she did not do any formal multiplication. What did she most likely do to find the answer?
B) This visitor also sees the students work another problem: 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? The visitor thinks, "This is a division problem and first-graders have not yet learned to divide." But then one student says that it would take four days. How do you suppose she explained this answer without using division?

Question 26

Felisha was asked to find 413 - 248. Here is how she did this problem: $\begin{array} { r } 413 \\\underline{- 248 }\\ - 5 \\- 30 \\\underline{ 200 }\\165 \end{array}$
Is her answer correct? Explain what she was doing. Find 9456 - 3789 using this method.

Question 27

A second-grade boy is asked to subtract 64 - 55, written vertically. The child thinks about the problem and then writes 9. He explains his thinking by saying, "6 take away 5 is 1. I mean, 60 take away 50 is 10. 5 take away 4 is 1, and 10 take away 1 is 9." Is he correct? Use his thinking to find 243 - 124.

Question 28

Zenaida is asked to add 428 and 686 in vertical form. She begins by saying, "Six thousand plus four thousand..." The interviewer then asks her what column the six and four are in, and she identifies it as the hundreds column. She begins again by saying, "Four hundred plus six hundred is ten hundred," and writes 110 below the line. She then says we have to do tens. She does 20 plus 80 equals 100 and places that under the 110. She then adds 8 and 6 and writes 14 and then writes that below the 100. Adding, she says it is 224. Where does Zenaida go wrong? Discuss her place value understanding.

Question 29

Here is Ben's work:
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.

Question 30

Find 21 + 49 using an empty number line.

Question 31

Find 509 - 239 using an empty number line.

Question 32

A student wrote the following answer to the problem 95 - 34:
95 - 34 =
95 - 35 = 60
60 - 1 = 59
So the answer is 59
Analyze this student's thinking.

Question 33

A) Make sketches for 3 × 6 and 6 × 3 and contrast them.
B) Make sketches of 1/2 × 6 and 6 × 1/2.

Question 34

A designer of women's "mix and match" clothing designs three styles of skirts, two pairs of pants, three types of tops, and four 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.)

Question 35

A clothes designer comes up with women's "mix and match" items with two styles of skirts, one pair of pants, three types of tops, and two styles of jackets. How many different outfits could be purchased if each outfit has a skirt or pants, a top, and a jacket?

Question 36

Mitchell decides to get his car painted and to buy new hubcaps. He selects five colors he likes and three styles of hubcaps. Then he decides to paint the roof a different color than the body. He decides to let his wife make the final decision. How many choices does she have?

Question 37

Make up a story problem about a bake sale so that the problem could be solved by:

A) 3/4 × 12. (Notice the order.)
B) 12 × 3/4 .

Question 38

Using a circle to represent 1, make drawings to illustrate each of the following.

A) 3 × 4, array
B) 1/3 × 6, fractional part of an amount

Question 39

Give the label (e.g., take-away, etc.) for the situation, and write the equation for solving this problem:
A coffee shop has four 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?

Question 40

A) Can the following word problem be solved by 16 × 1/2?
B) Does the word problem illustrate a situation involving 16 halves or half of 16?
Word problem: Don has sixteen one-half dollars. How much money does Don have?

Question 41

Finish each story so that your question can 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 one free topping of nuts, sprinkles, or chocolate sauce (you don't have to take one, of course) with each cup of ice cream...
C) 6 × 1/2, repeated addition. You work in a candy shop...

Question 42

The product of a number n by any other number m different from 0 is always greater than n.

Question 43

A) This is a typical problem from an elementary textbook:
Jasmine works in a bookstore. 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 store. She is told to put the books into 16 piles. How many piles can she make?
What interpretations of multiplication and division are represented in this problem? Explain.
B) What if the question changes to "She is told to put 27 books in each pile"? What interpretation of division is now represented? Explain.

Question 44

Make up a story problem involving quantities of ice cream in an ice cream store so that the problem can be solved by the calculations given:

A) 2 ÷ 1/8
B) 16 × 1/8
C) 3/4 × 24

Question 45

If a is any number other than 0, then 1 ÷ a is less than 1.

Question 46

Under a repeated-subtraction interpretation, $\frac { 3 } { 4 } \div 1 \frac { 1 } { 2 }$
means _____. The quotient is _____. Verify and explain your answer with a sketch.

Question 47

Which type of division does each word problem depict?

A) Mr. Burke's class of 24 fourth-graders is doing a project on keeping the environment clean. There are six 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 if each group explores a different topic?
B) Mr. Burke's class of 24 fourth-graders are to be put into teams of six to work on projects about keeping the environment clean. How many projects will Mr. Burke have to grade?

Question 48

Write a word problem for 37 ÷ 5 for which the answer would be 2.

Question 49

Consider this problem situation, which would involve dividing by 3:
You are putting books on three empty shelves in your classroom. To make them look neat, you put the same number of books on each shelf. How many books will be on each shelf?
Write another problem situation about the books so that your problem involves another way of thinking about division by 3.

Question 50

Label each word problem with the type of division being depicted.

A) The big bag has 48 plastic cars, to be put into bags holding six cars each. How many bags of cars will there be?
B) The big bag has 48 plastic cars, to be split fairly among six youngsters. How many cars will each youngster get?

Question 51

Write a word problem for 37 ÷ 5 for which the answer would be 7.

Question 52

Of the expressions 0 ÷ 6, 6 ÷ 0, and 0 ÷ 0, which is/are undefined? Explain why any undefined choice(s) is/are undefined. If the expression is defined, tell what it equals.

Question 53

Provide a calculation that would be used to solve each story problem.

A) A farmer has a $3 \frac { 1 } { 2 }$ -acre orchard of orange trees. She and her workers have ha of the orchard. How many acres have they harvested?
B) A farmer has a $3 \frac { 1 } { 2 }$ -acre orchard of orange trees. If she wants to replace all the 5 -year period, how many acres should she plan to replace each year?
C) A farmer has a $3 \frac { 1 } { 2 }$ -acre orchard of orange trees. One sprayerful of fertilizer $\mathrm { c }$ acre. How many sprayerfuls will she need to cover the whole orchard?

Question 54

Which calculation and type of division does this drawing illustrate?

A) 4 $\frac { 1 } { 2 }$ ÷ 3, sharing equally

B) 4 $\frac { 1 } { 2 }$ ÷ 3, repeated subtraction

C) 4 $\frac { 1 } { 2 }$ • 3, repeated addition

Question 55

Is this child's thinking correct? If it is, complete the second calculation using the child's method. If the thinking is not correct, explain why not.

Question 56

Fiesha finds 32 × 54 as follows:
$\begin{array} { r } 54 \\\underline{\times \quad 32} \\ 1500 \\120 \\100 \\\underline{8}\\1728\end{array}$

A) Which is TRUE of Fiesha's mathematical steps?
i. Fiesha's steps are mathematically correct.
ii. Fiesha's steps are mathematically flawed.
iii. It cannot be determined whether Fiesha's steps are mathematically correct or flawed.
B) Which is TRUE of Feisha's understanding of multiplication?
i. Fiesha doesn't appear to understand multiplication.
ii. Fiesha may or may not understand multiplication.
iii. Fiesha shows good understanding of multiplication.

Question 57

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?
i. Amy's steps are probably mathematically correct, although the phrasing is not perfect (should be "add four 32s back").
ii. Amy's steps are mathematically flawed.
iii. It cannot be determined whether Amy's steps are mathematically correct or flawed.
B) Which is TRUE of Amy's understanding of multiplication?
i. Amy doesn't appear to understand multiplication.
ii. Amy may or may not understand multiplication.
iii. Amy shows good understanding of multiplication.

Question 58

Antonio asks, "When I multiply [e.g., 49 × 23, shown below], why do I have to put in the 0 [points to the zero in 980]?" What would you say to Antonio?

Question 59

Following is an example of a child's work. Study the work and then judge the student's understanding.
Hiro was asked to divide 4240 by 6. His work is shown below.
Hiro's work:

A) Is Hiro's work correct or incorrect?
B) If the work is incorrect, please explain how.

Question 60

Consider the following work of a student:
$\begin{array} { r } 84 \\\underline{\times \quad 45} \\ 20 \\400 \\160 \\\underline{320 }\\ 900\end{array}$

A) There is an error with the 20.
B) There is an error with the 400.
C) There is an error with the 160.
D) There is an error with the 320.
E) There is no error with this student's work.

Question 61

Use a nonstandard algorithm to calculate 128 × 67.

Question 62

In each pair, choose which is GREATER.

A) 1838 + 517 or 1836 + 514
B) 612 - 29 or 613 - 34
C) 0.578 or 0.002 + 0.0328

Question 63

I am a number with 21 tens, 14 ones, and 11 tenths. What number am I?

Question 64

Tell why the following calculations are INCORRECT.

A)
$\begin{array}{llcc} \text { Sum: } & 310 \\ \text { } &225\\ \text { } &980\\ \text { } &\underline{375}\\ \text { } &1895\\\end{array}$

B) 280 ÷ 70 = 40
C) 480 ÷ 0.4 = 120

Question 65

The large rectangle is divided into two smaller rectangles. Write a number sentence involving multiplication that shows that the area of the large rectangle is equal to the sum of the areas of the two smaller rectangles.

Question 66

In a board game, a murder occurs. There are six possible suspects (Plum, White, Green, Mustard, Peacock, Scarlet), six possible weapons (candlestick, knife, lead pipe, revolver, rope, wrench), and nine possible rooms where the murder can take place. At the beginning of the game, three cards are chosen at random (one suspect, one weapon, one room).

A) How many different ways can these three cards be chosen?
B) If the game maker decided to add another possible suspect, Orange, to cover all the colors in the rainbow, how many more possibilities are there?