Section A Practice Problems

Problem 1

What fraction of the rectangle is shaded? Explain how you know.

Diagram. 6 equal parts, 4 parts shaded.

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Solution
46\frac{4}{6}, because there are 6 equal pieces in the rectangle and 4 of them are shaded.

Problem 2

  1. Locate and label 34\frac{3}{4} and 64\frac{6}{4} on the number line.

    Number line. Scale 0 to 2, by 1's.

  2. Explain why your points represent 34\frac{3}{4} and 64\frac{6}{4}.

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Solution

  1. Number line

  2. Sample response: I divided each whole into 4 equal parts so each part is 14\frac{1}{4}. There are 3 of those parts to get to 34\frac{3}{4} and 6 of them to get to 64\frac{6}{4}.

Problem 3

Write a multiplication expression for each image. Explain your reasoning.

  1.  

    Array. 4 rows of 5 dots.
    ​​​​​​

  2.  

    Diagram. Rectangle partitioned into 3 rows of 7 of the same size squares.

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Solution
  1. 4×54 \times 5 (or 5×45 \times 4), because there are 4 rows of 5 (or 5 columns of 4).
  2. 3×73 \times 7 (or 7×37 \times 3), because there are 3 rows of 7 (or 7 columns of 3).

Problem 4

Here are the lengths of some lizards in inches. Use the lengths to complete the line plot.

  • 2142\frac{1}{4}
  • 1121\frac{1}{2}
  • 2242\frac{2}{4}
  • 3
  • 3243\frac{2}{4}
  • 2
  • 2142\frac{1}{4}
  • 2142 \frac{1}{4}
  • 2342\frac{3}{4}
  • 2
  • 2142\frac{1}{4}
  • 3

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Solution
Sample response:

dot plot
Dot plot titled Length of Lizards from 0 to 4 by 1’s. Horizontal axis lizard length, in inches. Hash marks at fourths. Beginning at 1 and two fourths, the number of X’s above each one fourth increment is 1, 0, 2, 4, 1, 1, 2, 0, 1.

Problem 5

Write an expression that matches each diagram. Then find the value of each expression.

a.
Diagram. 5 equal parts each labeled 1 half.

b.
4 diagrams of equal length. 3 equal parts. 1 part shaded. Total length, 1.

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Solution
  1. 5×125 \times \frac{1}{2}. The value is 52\frac{5}{2}.
  2. 4×134 \times \frac{1}{3}. The value is 43\frac{4}{3}.

Problem 6

There are 5 friends that go on a hike. They each bring 14\frac{1}{4} cup of nuts.

  1. If the shaded parts represent the amount of nuts the friends bring on their hike, which diagram matches the story? Explain your reasoning.

    A
    4 diagrams of equal length. 5 equal parts. 1 part shaded. Total length, 1 cup.

    B
    5 diagrams of equal length. 4 equal parts. 1 part shaded. Total length, 1 cup.

  2. How many cups of nuts do the friends bring on the hike?
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Solution
  1. Diagram B matches the story. There are 5 portions of nuts, one for each friend, and each portion is 14\frac{1}{4} cup.
  2. The friends bring 54\frac{5}{4} cups of nuts on the hike. 

Problem 7

Kiran’s cat eats 12\frac{1}{2} cup of food each day.

  1. How much food does Kiran’s cat eat in a week?
  2. Draw a diagram to represent the situation.

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Solution
  1. Kiran’s cat eats 72\frac{7}{2} cups of food in a week.
  2.  

    7 diagrams of equal length. 2 equal parts. 1 part shaded. Total length, 1 cup.

Problem 8

  1. Draw a diagram to show 3×783 \times \frac{7}{8}.
  2. How does the diagram help you find the value of the expression 3×783 \times \frac{7}{8}?
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Solution

  1. Sample response:

    tape diagram solution

  2. Sample response: I can see that there are 3 groups of seven 18\frac{1}{8}s, so that is 218\frac{21}{8} altogether.

Problem 9

Find the number that makes each equation true. Draw a diagram if it is helpful.

  1. 103=‾ ×13\frac{10}{3} = \underline{\hspace{0.7cm}} \times \frac{1}{3}

  2. 103=‾ ×23\frac{10}{3} = \underline{\hspace{0.7cm}} \times \frac{2}{3}

  3. 103=‾ ×53\frac{10}{3} = \underline{\hspace{0.7cm}} \times \frac{5}{3}

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Solution
  1. 103=10×13\frac{10}{3} = 10 \times \frac{1}{3}
  2. 103=5×23\frac{10}{3} = 5 \times \frac{2}{3}
  3. 103=2×53\frac{10}{3} = 2 \times \frac{5}{3}

Problem 10

Each bead weighs 58\frac{5}{8} gram. How much do 7 beads weigh? Explain or show your reasoning.

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Solution
358\frac{35}{8} grams. They weigh 7×587 \times \frac{5}{8} grams, which is 358\frac{35}{8}, because 7×5=357 \times 5 = 35.

Problem 11

  1. Measure how thick your book is to the nearest 18\frac{1}{8} inch.
  2. If your classmates stacked their books together, how tall would the stack be? Explain or show your reasoning.
  3. Check your answer by measuring the stack, if possible.
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Solution

Sample responses:

  1. 38\frac{3}{8} inch
  2. There are 25 students in my class so that would be 25×3825 \times \frac{3}{8} or 758\frac{75}{8} inches. 
  3. It was a little less than 9 inches or 728\frac{72}{8} inches, so my answer was close.

Problem 12

Diego walks the same number of miles to school each day. He says he walks 485\frac{48}{5} miles in total, but he does not say how many days that distance includes.

What are some possible numbers of days Diego walks to school? What is the distance he walks each of those days?

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Solution

Sample responses:

number of days miles each day
1 485\frac{48}{5}
2 245\frac{24}{5}
3 165\frac{16}{5}
number of days miles each day
4 125\frac{12}{5}
6 85\frac{8}{5}
8 65\frac{6}{5}