Deck 7: Computing With Fractions

Question 1

Make up a story problem that could be solved by
and illustrates the comparison view of subtraction.

Answer 1

There are various possibilities, but in each the two quantities should be distinct. For example, one recipe calls for
cups of sugar, and another calls for
cups of sugar. How much more sugar does the first recipe call for than the second?

Question 2

Make up a story problem that could be solved by calculating 123 - $48 \frac { 1 } { 3 }$

Answer 2

There are various possibilities. For example, Velma had 123 feet of ribbon and used 48 $\frac { 1 } { 3 }$
for package bows. How much ribbon does she have left?

Question 3

Following is an example of a child's work. You are to study the work and then judge the student's understanding.
?
Rona was asked to subtract $2 \frac { 5 } { 8 }$
from $4 \frac { 1 } { 8 }$
. Her work is shown below.
$\begin{array} { r } \text { Rona's work:}&4 \frac { 1 } { 8 } = 3 \frac { 11 } { 8 } \\\\&\underline{- 2 \frac { 5 } { 8 } = - 2 \frac { 5 } { 8 } }\\&\quad\quad\quad\quad 1 \frac { 6 } { 8 }\end{array}$ ?

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

Answer 3

A) Incorrect
B) In changing a 1 from the 4 into eighths, Rona did the usual base-ten place value, borrowing 10 rather than the correct 8/8 + 1/8.

Question 4

Four identical pizzas are shared among three people. Show two ways that the pizzas could be cut.

Question 5

Using a rectangular region as the unit, illustrate each of the following.

A) $2 / 3 + 4 / 5$
B) $7 / 5 + 3 / 2$
C) $10 / 8 - 1 / 2$

Question 6

Explain a strategy you could use to mentally compute the exact answer to the following.

A) $2 \frac { 3 } { 8 } + 5 \frac { 1 } { 4 } + 3 \frac { 7 } { 8 } + 2$

B) $2 \frac { 4 } { 5 } + 3 \frac { 1 } { 6 } + 5 \frac { 2 } { 3 } - 1.8$

C) $3 \frac { 1 } { 2 } - 2 \frac { 3 } { 8 }$

Question 7

Why is a common denominator needed to add and subtract fractions?

Question 8

Three children took a hike, carrying one heavy backpack. One child carried the backpack for 3/8 of the hike, and a second carried it for 1/6 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?

Question 9

Jody mows 1/8 of the yard then takes a break. After the break, Jody mows another 1/3 of the yard then stops for lunch. After lunch, Jody mows another 1/6 of the yard and then goes swimming. What part of the yard does Jody still have to mow?

Question 10

The vegetable farmer has planted 1/3 of a field in cherry tomatoes, 1/5 of the field in large tomatoes, and 2/7 of the field in popcorn. The farmer plans to plant sweet corn in the remaining part of the field. What part of the field will be in sweet corn? Use number sense to describe this part with a more familiar fraction.

Question 11

Emmy wants to read an assigned book in four days. She reads 2/3 of the book the first day, only 1/15 of the book the second day, and another 1/15 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?

Question 12

Judy has a 10-page paper to write. The first day she writes $1 \frac { 1 } { 2 }$
pages, takes a break, and then writes $3 \frac { 3 } { 4 }$
pages more. The next day, she writes another $2 \frac { 1 } { 3 }$
pages. How many pages does she still have to write?

Question 13

A) On the number line below, show approximately where 4/7 + 1/2 would be using number sense. Label it
B) On the same number line, show approximately where 4/7 - 1/2 would be, using number sense. Label it b.

Question 14

Use a drawing to help explain why
is equal to
. Be explicit.

Question 15

If two numbers are multiplied, their product is greater than (or equal to) each number.

Question 16

Donny is asked to solve the following problem: If cheese is $1.89 per pound, how much does 0.67 pounds cost? Which computation will answer the question?

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

Question 17

Which point of A-E would BEST locate the point for 0.89 × n on the number line (n is positive)?

Question 18

Which point of A-E would BEST locate the point for n ÷ 0.89 on the number line (n is positive)?

Question 19

Which location on the number line indicated by the letter P, Q, R, S T, U, or W represents each of the following?

A) 1/2 × 2
B) 5/8 of 1/2
C) 3/4 of 2/3
D) 1 ÷ 2/3
E) 1/4 ÷ 1/3

Question 20

A) Imagine you were to shade in 3/4 of 2/3 of this rectangle. What would it look like?
B) Show exactly where 2/3 is. It is 2/3 of what?
C) Show exactly where 3/4 is. It is 3/4 of what?

Question 21

A) Show 1 ÷ 3/8 with a rectangular region as the whole. Write your answer as a mixed number.
B) Clearly explain how the fraction part of your answer relates to your drawing.

Question 22

The product of a positive number n and another positive number m is always greater than n.

Question 23

Three-fourths of seven ones is the same amount as seven 3/4 of one.

Question 24

Draw a picture that represents the following:

A) 3/4 × 2/5
B) 2/5 × 3/4

Question 25

Draw a picture that represents the following:

A) 4/3 × 2/5
B) 2/5 × 4/3

Question 26

Working on his summer tan, Zonker leaves 4/5 of his skin exposed. He has only enough lotion to cover 1/3 of his exposed skin. What part of his body will be unprotected? Draw a neatly labeled diagram displaying your solution.

Question 27

Use these circles to show each division. For each part tell what question is being asked. Tell what your answer is in each case, and show how you obtained it.

A) 7/8 ÷ 1/4
B) 1/4 ÷ 3/8

Question 28

Why is the division of fractions confusing to people? Give two reasons.

Question 29

Is one meaning of 1/4 × 1/3 the following: How many 1s is 1/4 of 1/3 of 1? If it is one meaning, how much of a 1 or how many 1s?

Question 30

Is one meaning of 1/4 ÷ 1/3 the following: how many 1/3s of 1 are in 1/4 of 1?
If it is one meaning, how many 1/3s or how much of 1/3 of 1 are/is in 1/4 of 1?

Question 31

One meaning of 45/59 × 35/17 is, "How many ones is 45/59 of 35/17 of 1?"

Question 32

One meaning of 45/59 ÷ 35/17 is, "How many 35/17of 1 are in 45/59 of 1?"

Question 33

One meaning of 16/19 ÷ 25/17 is, "How many 25/17 of 1 are in 16/19 of 25/17?"

Question 34

Use number sense in estimating that (1201.794 ÷ 24.3%) + 0.0423 is approximately _____.

Question 35

The stagecoach 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 3/4 of a mile every hour. How many hours will it take the posse to catch the robbers? Show your work.

Question 36

Tell what 8
$\frac { 7 } { 8 }$
÷ 3 means

A) with the repeated subtraction or measurement view.
B) with the partitive or equal-sharing view.

Question 37

Division involving only positive numbers always gives a smaller answer than the number being divided.

Question 38

What calculation would solve this story problem?
Cheese was $2.55 a pound. A woman bought 0.85 pounds of the cheese. How much did she pay?

A) 2.55 + 0.85
B) 2.55 - 0.85
C) 0.85 × 2.55
D) 2.55 ÷ 0.85

Question 39

What calculation would solve this story problem?
George had 4/5 of a pie. He ate 2/3 of what he had. What part of a whole pie did he eat?

A) 4/5 + 2/3
B) 4/5 - 2/3
C) 2/3 × 4/5
D) 4/5 ÷ 2/3

Question 40

What calculation would solve this story problem?
A woman paid $4.80 for 3/4 of a pound of candy. How much does the candy cost, per pound?

A) (4.80 ÷ 4) × 4
B) 4.80 × 3/4
C) 4.80 ÷ 3/4
D) 4.80 + 0.75

Question 41

Tuna costs 70¢ a can at Store X and 80¢ a can at Store Y. How much will eight cans of tuna cost if one-quarter 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

Question 42

One day, Joe's old car used 5/8 of a quart of oil. The next day, it used only 2/3 as much oil. How many quarts of oil did the car use on the second day? Which computation will answer the question?

A) 5/8 + 2/3
B) 2/3 × 5/8
C) 5/8 ÷ 2/3
D) 2/3 ÷ 5/8

Question 43

One day, Joe's old car used 5/8 of a quart of oil. The next day, it used only 2/3 as much oil. On day three, the old car used 1 $\frac { 1 } { 4 }$
Quarts of oil! But on day four, it used only 60% as much as it did on day three. How many quarts of oil did the car use on day four? Which computation will answer the question?

A) $1 \frac { 1 } { 4 } + 0.60$

B) $1 \frac { 1 } { 4 } - 0.60$

C) $\frac { 3 } { 5 } \times 1 \frac { 1 } { 4 }$ $1 \frac { 1 } { 4 } \times 0.60$

D) $1 \frac { 1 } { 4 } \div \frac { 60 } { 100 }$

Question 44

How many portions of 2/3 a pound can be obtained from 30 pounds?

A) 20
B) 45
C) 60
D) 90

Question 45

Following are three students' solutions to 9/16 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.
Jessica: 9/16 is 1/16 more than1/2. Half of 48 is 24, and 1/16 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 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.
C) If Jessica's steps are mathematically correct, use her way of thinking to solve 3/8 × 32. If they are not, explain how Jessica's reasoning is flawed.
Justin:

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 multiplication of fractions: choose one.
- Justin doesn't appear to understand multiplication of fractions.
- Justin may or may not understand multiplication of fractions.
- Justin shows good understanding of multiplication of fractions.

Stacy: 9/16 of 48? 9 × 16 = 144; 144 ÷ 48 = 3; and 9 × 3 = 27. So 9/16 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 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.

Question 46

Finish the following story problem so that your question could be answered by the calculation 18 ÷1/2.
You buy 18 muffins for an after-school faculty meeting …

Question 47

For this number line,

A) the point for p + q would be closest to which letter?
B) the point for p × q would be closest to which letter?

Question 48

Karen and Sara go for a bike ride. Karen biked for $3 \frac { 1 } { 3 }$ Hours, which was 2/3 as many hours as Sara biked. How many hours did Sara bike?

A) 2 $\frac { 2 } { 9 }$

B) 2 $\frac { 2 } { 3 }$

C) 4

D) 5

Question 49

After the family had gone 140 miles, the children asked, "Are we getting close?" The driver said, "We've gone only 2/7 of the way." How far did the family still have to go?

A) 40 miles
B) 80 miles
C) 350 miles
D) 490 miles

Question 50

A farmer keeps 40% of his land UN-cultivated. Two-thirds of the cultivated portion is planted with corn. What fraction of the entire land is planted with corn? Make a drawing and show your work clearly.

Question 51

Alex ate 1/2 of a pizza. Tandy ate 1/3 of what was left. Finally, Tabby ate 1/2 of what was left. How much of the whole pizza did Tabby eat?

Question 52

Consider this number line:
Points R and S represent numbers between 0 and 1 as indicated. Which of the remaining points best represent the following?

A) $R \times S$
B) $R + S$
C) $R - S$
D) $1 \div R$

Question 53

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

Question 54

A recipe calls for a 1/2 cup of sugar. You have 2/3 of a cup of sugar. If you use all of your sugar, how many recipes can you make? Solve by means of a diagram.
Referent unit for the 1/2: _____
Referent unit for the 2/3: _____
Referent unit for the answer: ــــــــــــــــــــــــــــــــــ