Deck 6: Meanings for Fractions

Question 1

Use diagrams to show that
is the same as 3 ÷ 4. LABEL THE RELEVANT PARTS OF YOUR DIAGRAM AND EXPLAIN THE EQUIVALENCE.

Answer 1

Three-quarters could be drawn as one whole broken into four equal-sized pieces, where three of the fourths would be indicated in some fashion. The sketch for 3 ÷ 4 should show three wholes with, for sharing division, each marked into fourths. Each share would get one-fourth from each whole, or a total of three-fourths of a whole.

Question 2

Which diagram does NOT show fourths. Explain why not.

Answer 2

A, B, and D do not show fourths. In each of these cases, the four pieces do not all have the same area.

Question 3

Shown below is $1 \frac { 3 } { 4 }$ yards of carpet. Sketch (fairly accurately) 1 yard of carpet and $3 \frac { 1 } { 3 }$ yards of carpet. If the piece of carpet shown sells for $28, how much should 9 yards cost?

Answer 3

One yard would be 4/7 of the rectangular region, and then $3 \frac { 1 } { 3 }$ would be $3 \frac { 1 } { 3 }$ of those. Nine yards should cost $144 (if 7/4 yards cost $28, each 1/4 yard costs $4, so 1 yard costs $16).

Question 4

Fractions are always less than 1.

Question 5

To change 3 $\frac { 1 } { 4 }$ to a fraction, a common rule is to calculate 3 × 4, add 1, and write that answer over 4: 3 $\frac { 1 } { 4 }$ = $\frac { ( 3 \times 4 ) + 1 } { 4 }$ = $\frac { 13 } { 4 }$ . Use a number line (and words as you need them) to explain why that rule makes sense.

Question 6

Make a drawing that shows 2 $\frac { 4 } { 5 }$ if the shape below is 1.

Question 7

If the hearts shown represent only 2/5 of the chocolate hearts you gave away on Valentine's Day, how many chocolate hearts did you give away in all?

A) 8
B) 28
C) 30
D) 50

Question 8

If the heart-shaped cookies shown are 3/4 of the cookies in a package, how many cookies in all are there in the package?

A) 3
B) 9
C) 16
D) 24

Question 9

If 3/4 of the heart-shaped cookies shown below are red heart-shaped cookies, how many of the cookies are red?

A) 3
B) 9
C) 16
D) 24

Question 10

Mark 3/5 on this number line as A. Mark 5/3 on the number line as B.

Question 11

Given a number line and only the points for the indicated fractions, accurately locate and label the point for the number in each part. If your markings do not make your reasoning clear, add an explanation.

A) 7/8
B) 1

Question 12

Given a number line and only the points for the indicated fractions, accurately locate and label the point for the number in each part. If your markings do not make your reasoning clear, add an explanation.

A) 0
B) 2/3

Question 13

Given a number line and only the points for the indicated fractions, accurately locate and label the point for the number in each part. If your markings do not make your reasoning clear, add an explanation.

A) 2
B) $2 \frac { 1 } { 2 }$

Question 14

Given a number line and only the points for the indicated fractions, accurately locate and label the point for the number in each part. If your markings do not make your reasoning clear, add an explanation.

A) 1
B) 2

Question 15

Given a number line and only the points for the indicated fractions, accurately locate and label the point for the number in each part. If your markings do not make your reasoning clear, add an explanation.

A) 10
B) 9

Question 16

Is Ardis's reasoning correct? Explain.
Shannon: "I still have to learn half of my spelling words and 3/4 of my vocabulary words."
Ardis: "Well, 3/4 is more than a half because 1/2 = 2/4. So, you have to learn more vocabulary words than spelling words."

Question 17

Using a number line, show the two fractions $\frac { 7 } { 4 }$ and 1 $\frac { 6 } { 8 }$ are equivalent.

Question 18

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. The shaded part shows Bill's flowers:

The shaded part shows Tom's flowers:


Who has more land in flowers? Explain your choice.

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.

Question 19

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 2/3 of the shaded region."
Who is correct? Why?

Question 20

A) Make a drawing that shows 2/5 of a discrete whole.
B) Make a drawing that shows 2/5 of a continuous whole.

Question 21

Given $\frac { 50 } { 10 }$ and $\frac { 53 } { 13 }$ , why are we able to "cancel" the zeros in $\frac { 50 } { 10 }$ but not able to "cancel" the 3s in $\frac { 53 } { 13 }$
?

Question 22

Use drawings with rectangles to show that

A) $\frac { 5 } { 8 } > \frac { 7 } { 16 }$

B) $\frac { 3 } { 4 } > \frac { 2 } { 3 }$

C) $1 \frac { 3 } { 4 } > \frac { 6 } { 4 }$

Question 23

Use the part-whole notion of fractions to explain why $\frac { 7 } { 13 } < \frac { 7 } { 12 }$

Question 24

Name three ways of thinking about the symbol $\frac { 7 } { 8 }$

Question 25

Write a story problem in which 3/5 is treated as a part-whole fraction, with discrete quantities.

Question 26

$\frac { 4 } { 5 } - \frac { 5 } { 8 }$
Is NOT equivalent to:

A) $\frac { 400 } { 500 } - \frac { 500 } { 800 }$

B) $\frac { 12 } { 15 } - \frac { 10 } { 16 }$

C) $\frac { 64 } { 80 } - \frac { 50 } { 80 }$

D) $\frac { 20 } { 25 } - \frac { 20 } { 32 }$

E) $\frac { 32 } { 40 } - \frac { 20 } { 40 }$

Question 27

How would you convince a child with visual evidence that
?

Question 28

A child says, "My teacher says to put = between $\frac { 3 } { 4 }$ and $\frac { 6 } { 8 }$ , but I think $\frac { 6 } { 8 }$ is bigger."
How might the child be thinking? What would you do next?

Question 29

How would you convince a child with visual evidence that
?

Question 30

Use sketches to show that:

A) $\frac { 20 } { 15 } = 1 \frac { 1 } { 3 }$

B) $\frac { 3 } { 7 } = \frac { 9 } { 21 }$

Question 31

Name three common denominators for the following fractions:

A) $\frac { 3 } { 8 }$ and $\frac { 7 } { 12 }$

B) $\frac { 5 } { 72 }$ and $\frac { 7 } { 144 }$

C) $\frac { 9 } { 40 }$ and $\frac { 7 } { 60 }$

Question 32

Which pair of numbers is relatively prime?

A) 35, 15
B) 12, 27
C) 28, 45

Question 33

Write the simplest fraction form for each:

A) $\frac { 9 \times 16 \times 18 } { 12 \times 8 \times 15 }$

B) $\frac { z y ^ { 3 } x ^ { 2 } } { y ^ { 4 } z ^ { 5 } }$

Question 34

Put these fractions in order, from smallest to largest, using the symbols <, >, and =.

Question 35

Put these fractions in order from smallest to largest. $\frac { 200 } { 303 }$ $\frac { 6 } { 9 }$ $\frac { 9 } { 12 }$ $\frac { 203 } { 300 }$

Question 36

A) Give a number between $\frac { 3 } { 7 }$ and $\frac { 4 } { 7 }$ .

B) Give a number between $\frac { 8 } { 111 }$ and $\frac { 8 } { 113 }$

C) Give a number between $\frac { 7 } { 114 }$ and $\frac { 8 } { 113 }$


D) Which number is closer to 1: $\frac { 9 } { 10 }$ or $\frac { 10 } { 9 }$ ?

Question 37

Which of these is a rationale for comparing $\frac { 3 } { 7 }$ And $\frac { 7 } { 11 }$ ? (You can choose more than one answer.)

A) They are the same because 7 - 3 = 4 and 11 - 7 = 4.

B) The fraction $\frac { 7 } { 11 }$ Is more than $\frac { 1 } { 2 }$ , and $\frac { 3 } { 7 }$ Is less than $\frac { 1 } { 2 }$ , so $\frac { 3 } { 7 }$ < $\frac { 7 } { 11 }$

C) Since $\frac { 3 } { 7 }$ = $\frac { 33 } { 77 }$ and $\frac { 7 } { 11 }$ = $\frac { 49 } { 77 }$ , $\frac { 3 } { 7 }$ Is smaller.

D) Sevenths are bigger than elevenths, so $\frac { 3 } { 7 }$ Is larger.

E) They are equal because each fraction is four parts away from 1.

Question 38

Fill in the blank cells in the table below so that the numbers in each row are equivalent to the given number.

Question 39

What is the exact decimal equivalent of each fraction?

A) $\frac { 2 } { 5 }$

B) $\frac { 2 } { 7 }$

Question 40

What is the fraction equivalent of each decimal amount?

A) 0.651
B) 0.44444 …

Question 41

What is the percent equivalent of each amount?

A)
B) 0.00035

Question 42

Circle each fraction that has a terminating decimal, without calculating the decimal.

A) $\frac { 749327 } { 2 \times 10 ^ { 6 } }$

B) $\frac { 3000005 } { 6 \times 10 ^ { 8 } }$

C) $\frac { 21000042000014 } { 350 }$

Question 43

Three-eighteenths can be written as a terminating decimal.

Question 44

Write each decimal amount as a fraction.

A) $56.\overline{38}$
B) 7.453

Question 45

Correct the following statement so that it is true in some nontrivial way:
If D = $0.7\overline{42}$ , then 100D = 74.200.

Question 46

As a fraction, 1.4 =

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

B) $\frac { 7 } { 5 }$

C) $\frac { 14 } { 100 }$

D) $\frac { 41 } { 10 }$

Question 47

(Number sense item, which means no calculator or hand calculation and no decimals.) Use number sense to put these amounts in order, from smallest to largest. $\frac { 141 } { 280 }$ $\frac { 101 } { 300 }$ 1.3% $\frac { 121 } { 360 }$ $\frac { 1000 } { 3000 }$

Question 48

Put these amounts in order, from smallest to largest: $\begin{array}{llcc} 0.03&0.0295&0.1&0.13999 \\\end{array}$

Question 49

Put these amounts in order, from smallest to largest:
$8.5 \% \quad 1.7 \quad \frac{15}{11} \quad \frac{14}{8} \quad 110 \%$

Question 50

Put these amounts in INCREASING order without any hand calculation: $\begin{array} { l l l l l } \frac { 11 } { 40 } & 0.0239 & 8 \overline { 1.2 } \% & \frac { 211 } { 392 } & \frac { 200 } { 31 }\end{array}$

Question 51

Give three decimals between 2.3456 and 2.3457. If it is not possible, explain why not.

Question 52

Write each decimal amount in the
$\frac { \text { whole \# } } { \text { whole \# } }$
form if possible. If it is not possible, explain why not.

A) 9.6534 = _________
B) 96.65 $\overline{34}$ = ________

Question 53

Give three decimals between 0.301 and 0.302. If it is not possible, explain why not.

Question 54

Use number sense to order these fractions and decimals from smallest to largest, using the symbol <.

$0.1\quad\frac{17}{35}\quad\frac{1}{8}\quad\frac{2}{3}\quad\frac{1}{5}\quad0.85\quad\frac{9}{10}\quad59\%$

Question 55

A) What are the "rational" numbers?
B) How are they different from the "irrational" numbers?

Question 56

As a decimal, 0.525% is:

A) 525.
B) 52.5.
C) 5.25.
D) 0.525.
E) 0.00525.

Question 57

As a percent, 5.35 is:

A) 0.535%.
B) 5.35%.
C) 53.5%.
D) 535%.

Question 58

As a decimal, 1.775% is:

A) 0.01775.
B) 0.1775.
C) 1.775.
D) 17.75.

Question 59

1.5% =

A) 150.
B) 15.
C) 1.5.
D) 0.15.
E) 0.015.

Question 60

15% =

A) 150.
B) 15.
C) 1.5.
D) 0.15.

Question 61

Sketch a line segment that is 12.5% longer than 8 inches. How long is the line segment you sketched?

Question 62

Using benchmarks, find an estimate of each of the following and explain how you did it.

A) The sum of $\frac { 1 } { 23 }$ , $\frac { 7 } { 15 }$ , $\frac { 5 } { 9 }$ , and $\frac { 13 } { 12 }$

B) $\frac { 7 } { 15 } + \frac { 11 } { 12 }$

C) $\frac { 7 } { 8 } + \frac { 9 } { 10 }$

D) $7 \frac { 6 } { 7 } \div \frac { 1 } { 4 }$

Question 63

A fraction is close to 1/2 when the numerator is close to two times the denominator.

Question 64

A fraction a/b is close to 3/4 when three times the numerator is close to four times the denominator.

Question 65

How can you tell when a fraction is close to 1/4? Do not refer to decimal numbers.

Question 66

Show how you would mentally compute the exact results. Explain your thinking.

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

B) $\frac { 1 } { 7 } + \frac { 3 } { 10 } + \frac { 4 } { 7 } + \frac { 7 } { 10 }$

C) $\frac { 18 } { 24 } \times \frac { 80 } { 2 } \times \frac { 21 } { 42 } \times$ 100% = ___

D) 114 - 24 $\frac { 3 } { 8 }$ = ___

Question 67

For each of the following, obtain an estimate of the answer and write it in the blank. Use excellent number sense. Write enough to clearly explain how you thought about the problem.

A) 68 $\frac { 3 } { 4 }$ ÷ 2 $\frac { 15 } { 16 }$ =___

B) 4 $\frac { 7 } { 16 }$ + 5 $\frac { 11 } { 12 }$ - 2 $\frac { 1 } { 8 }$ = ___

C) 301 ÷ $\frac { 7 } { 16 }$ = ___

D) the sum of $\frac { 11 } { 23 }$ , $\frac { 3 } { 27 }$ , and $\frac { 5 } { 16 }$

Question 68

$1 \frac { 7 } { 12 } + 2 \frac { 5 } { 11 } + 3 \frac { 5 } { 12 } + \frac { 7 } { 13 }$
Is about:

A)8 $\frac { 1 } { 2 }$
B) 80 miles
C) $7 \frac { 12 } { 24 }$

D) $6 \frac { 24 } { 48 }$

Question 69

20% of 150 is 1/3 of:

A) 10.
B) 90.
C) 120.
D) 250.
E) 900.

Question 70

Which is larger, 7/9 or 9/11? Provide a justification for your choice, which does not refer to decimals.

Question 71

Which is closer to 1/2-8/15 or 3/7?

Question 72

Which amount is/are close, or equal, to 1/3? There may be more than one.
0.2_six, 3/5, 40/119, 10²/10⁶, 175/301, 297/600

Question 73

Which fraction is larger, 38/48 or 47/60? Explain without using decimals or common denominators.

Question 74

Which fraction is closer to 0-1/8 or 1/9? Explain without referring to decimals.

Question 75

Which fraction is closer to 1-1/8 or 1/9? Explain without referring to decimals.

Question 76

Which fraction is closer to 1/2-1/8 or 1/9?

Question 77

Which fraction is closer to 1-7/8 or 8/9?

Question 78

Which value is larger, 17/40 pound or 29/64 pound? Explain your thinking, showing your understanding of fractions and benchmarks, without using decimals or common denominators.

Question 79

Which is the greater value, 9/16 of a cup or 15/28 of a cup? Explain showing your grasp of fractions and benchmarks, without using decimals or complicated common denominators.

Question 80

Why is 2/3 a better benchmark than 1 for answering the question: which is larger-23/30 or 17/21? (You do not have to answer the second question.)