5.2 Fractions as Quotients and Fraction Multiplication

Unit Goals

  • Students develop an understanding of fractions as the division of the numerator by the denominator, that is $a \div b = \frac{a}{b}$, and solve problems that involve the multiplication of a whole number and a fraction, including fractions greater than 1.

Section A Goals

  • Represent and explain the relationship between division and fractions.
  • Solve problems involving division of whole numbers leading to answers that are fractions.

Section B Goals

  • Connect division to multiplication of a whole number by a non-unit fraction.
  • Connect division to multiplication of a whole number by a unit fraction.
  • Explore the relationship between multiplication and division.

Section C Goals

  • Find the area of a rectangle when one side length is a whole number and the other side length is a fraction or mixed number.
  • Represent and solve problems involving the multiplication of a whole number by a fraction or mixed number.
  • Write, interpret and evaluate numerical expressions that represent multiplication of a whole number by a fraction or mixed number.
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Section A: Fractions as Quotients

Problem 1

Pre-unit

Practicing Standards:  3.NF.A.2.b

  1. Locate \(\frac{6}{4}\) on the number line.
    Number line from 0 to 2. 3 evenly spaced tick marks labeled 0, 1, 2.
  2. Explain or show why your point represents \(\frac{6}{4}\).

Solution

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Problem 2

Pre-unit

Practicing Standards:  3.NF.A.1

Shade \(\frac{3}{4}\) of the rectangle. Explain or show your reasoning.

Rectangle 

Solution

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Problem 3

Pre-unit

Practicing Standards:  4.NF.B.4.b

Explain or show why \(\frac{4}{3} = 4 \times \frac{1}{3}\).

Solution

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Problem 4

Pre-unit

Practicing Standards:  4.NF.B.4.c

Each workbook is \(\frac{3}{8}\) inch thick. How many inches thick is a stack of 5 workbooks? Explain or show your reasoning.

Solution

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Problem 5

Pre-unit

Practicing Standards:  3.OA.A.2, 4.OA.A.2

  1. There are 36 fish in 4 aquariums. There are the same number of fish in each aquarium. How many fish are in each aquarium? Show or explain your reasoning.
  2. There are 24 dogs at a shelter. There are 4 times as many dogs as cats at the shelter. How many cats are there at the shelter? Show or explain your reasoning.

Solution

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Problem 6

Pre-unit

Practicing Standards:  4.NF.B.4.c

A bottle holds \(\frac{7}{10}\) liter of water. How much water do 6 bottles hold? Explain or show your reasoning.

Solution

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Problem 7

Pre-unit

Practicing Standards:  3.MD.C.7.b

Rectangle. Horizontal side: 12 centimeters. Vertical side: 8 centimeters. 

What is the area of the rectangle? Explain or show your reasoning.

Solution

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Problem 8

  1. 3 students equally share 18 sheets of construction paper for an art project. How many sheets of paper does each student get? Explain or show your reasoning.
  2. 3 students equally share 1 tube of glue for an art project. How much glue does each student get? Explain or show your reasoning.

Solution

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Problem 9

  1. 4 hikers equally share 3 liters of water. How many liters of water does each hiker drink? Explain or show your reasoning.
  2. 4 hikers equally share 5 liters of water. How many liters of water does each hiker drink? Explain or show your reasoning.

Solution

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Problem 10

  1. Jada cuts an 11 inch strip of paper into 5 equal parts. How many inches long is each part?
  2. Jada cuts a strip of paper into 5 equal parts. Each part is \(\frac{7}{5}\) inches long. How long was the strip of paper?

Solution

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Problem 11

3 diagrams of equal lengths. 5 equal parts. 1 part shaded. Total length, 1
  1. Describe a situation that the diagram could represent.
  2. Write an equation that represents the diagram and the situation.

Solution

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Problem 12

Decide whether each equation is true or false. Explain or show your reasoning.
  1. \(3 \div 7 = \frac{3}{7}\).
  2. \(18 \div 5 = \frac{5}{18}\).
  3. \(15 \div 6 = 2 \frac{1}{2}\).

Solution

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Problem 13

Exploration

  1. Describe a situation in the classroom or at home where you share something equally with your classmates or family that results in fractional size parts.
  2. Draw a picture to represent the situation.
  3. Write a division equation to represent the situation.

Solution

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Problem 14

Exploration

Elena is traveling to visit her grandparents who live 125 miles away.

  1. Elena stops for lunch \(\frac{2}{3}\) of the way. How far has Elena traveled? Explain or show your reasoning.
  2. Elena enters the city where her grandmother lives after 110 miles. Is she more or less than \(\frac{9}{10}\) of the way there? Explain or show your reasoning.

Solution

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Problem 15

Exploration

  1. Describe a situation that represents the equation  \(4 \div 6 = \frac{4}{6}\).
  2. Draw a diagram to represent the situation.

Solution

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Section B: Fractions of Whole Numbers

Problem 1

Han cuts a 15-foot piece of rope into 4 equal parts. Decide whether each expression represents the length of each part of the rope in feet. Explain or show your reasoning.

  1. \(15 \div 4\)
  2. \(4 \times 15\)
  3. \(3 \frac{3}{4}\)

Solution

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Problem 2

Find the value of each expression.

  1. \(\frac{1}{2} \times 6\)
  2. \(\frac{1}{7} \times 6\)
  3. \(\frac{1}{8} \times 11\)
  4. \(\frac{1}{3} \times 34\)

Solution

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Problem 3

  1. Kiran ran \(\frac{1}{5}\) the length of his road, which is 9 miles long. How far did Kiran run? Show or explain your thinking.

Solution

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Problem 4

Exploration

Map of Puyallap Indian Reservation 
  1. Each square on the map represents 2,178 square feet. Make an estimate for the number of square feet shown on the map. Explain or show your reasoning.
  2. Each square represents \(\frac{1}{20}\) acre of actual land. How many square feet are in an acre? Explain or show your reasoning.

Solution

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Problem 5

Exploration

A standard rectangular sheet of paper measures \(8 \frac{1}{2}\) inches in width and 11 inches in length. How many square inches are there in a sheet of paper?

If you get stuck, consider using the grid.

Grid. 12 rows of 12 of the same size squares. 

Solution

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Section C: Area and Fractional Side Lengths

Problem 1

Area diagram. Length, 4. Width, 3. 
Area diagram. Length, 4. Width, 1 third. 
  1. How are the diagrams the same? How are they different?
  2. How is finding the area of the shaded region the same? How is it different?

Solution

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Problem 2

  1. What is the area of this rectangle? Explain or show your reasoning.

    Rectangle. Horizontal side, 12 units. Vertical side, 10 units. 
  2. What is the area of the shaded region? Explain or show your reasoning.

    Area diagram. Length, 4. Width, 3 fifths. 
  3. How are these two area calculations the same? How are they different?

Solution

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Problem 3

The shaded part of this diagram shows the top of a stove. What is the area of the stove top? Explain or show your reasoning.

Area diagram. Length, 3 and 1 half feet. Width, 2 feet. 

Solution

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Problem 4

Find the area of the shaded region. Explain or show your reasoning.

Area diagram. Length, 3 feet. Width, 4 and 1 fourth feet. 

Solution

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Problem 5

Select all of the expressions that represent the shaded area in square feet.

  1. \(3 + 5 \frac{3}{4}\)
  2. \(3 \times 5 \frac{3}{4}\)
  3. \(3 \times \left(5 + \frac{3}{4}\right)\)
  4. \((3 \times 5) + \frac{3}{4}\)
  5. \(3 \times 6 - \left(3 \times \frac{1}{4}\right)\)
Area diagram. Length, 5 and 3 fourths feet. Width, 3 feet. 

Write one more expression that represents the shaded area.

Solution

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Problem 6

Tyler says that \(9 \frac{11}{12} \times 5\) is a little less than 50. 
  1. Do you agree with Tyler? Explain or show your reasoning.
  2. What is the value of \(9 \frac{11}{12} \times 5\)?

Solution

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Problem 7

A banner at a sporting event is 8 feet long and \(2 \frac{1}{3}\) feet wide.

  1. Sketch and label a diagram of the banner.
  2. Find the area of the banner.

Solution

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Problem 8

Evaluate each expression. Explain or show your reasoning.

  1. \(3\frac{2}{5} \times 10\)
  2. \(8 \times \frac{14}{3}\)
  3. \(3 \frac{41}{100} \times 5\)

Solution

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Problem 9

Exploration

  1. A regular sheet of paper is \(8\frac{1}{2}\) inches wide and 11 inches long. How many times would you need to fold the sheet of paper in half before the area is less than 1 square inch? Explain or show your reasoning.

  2. A piece of chart paper is 23 inches wide by 33 inches long. How many times would you need to fold it in half before its area is less than 1 square inch?

Solution

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Problem 10

Exploration

Part of the rectangle is shaded.

Area diagram. Length, 6. Width, 3 and 2 fifths. 
  1. Write a multiplication expression that represents the shaded area.
  2. Write a division expression that represents the shaded area.
  3. Write any other expressions that represent the shaded area.

Solution

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Problem 11

Exploration

This is a picture of the Empire State Building:
Photograph, Empire State Building.
The base of the Empire State Building is a rectangle. What do you think the area of the rectangle is in square meters?
  1. Make an estimate that is too small.
  2. Make an estimate that is too large.
  3. The length of the rectangle is 129\(\frac{1}{5}\) meters. The width is 57 meters. What is the area of the base of the Empire State Building?

Solution

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