4.3 Extending Operations to Fractions

Unit Goals

  • Students learn that a fraction $\frac{a}{b}$ is a product of a whole number $a$ and a unit fraction $\frac{1}{b}$, or $\frac{a}{b} = a \times \frac{1}{b}$, and that $n \times \frac{a}{b} = \frac{(n \space \times \space a)}{b}$. Students learn to add and subtract fractions with like denominators, and to add and subtract tenths and hundredths.

Section A Goals

  • Recognize that $n \times \frac{a}{b} = \frac{(n \space \times \space a)}{b}$.
  • Represent and explain that a fraction $\frac{a}{b}$ is a multiple of $\frac{1}{b}$, namely $a \times \frac{1}{b}$.
  • Represent and solve problems involving multiplication of a fraction by a whole number.

Section B Goals

  • Create and analyze line plots that display measurement data in fractions of a unit ($\frac18, \frac14, \frac12$).
  • Represent and solve problems that involve the addition and subtraction of fractions and mixed numbers, including measurements presented in line plots.
  • Use various strategies to add and subtract fractions and mixed numbers with like denominators.

Section C Goals

  • Reason about equivalence to add tenths and hundredths.
  • Reason about equivalence to solve problems involving addition and subtraction of fractions and mixed numbers.
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Section A: Equal Groups of Fractions

Problem 1

Pre-unit

Practicing Standards:  3.NF.A.1

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

Diagram. 6 equal parts, 4 parts shaded.

Solution

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Problem 2

Pre-unit

Practicing Standards:  3.NF.A.2

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

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

  2. Explain why your points represent \(\frac{3}{4}\) and \(\frac{6}{4}\).

Solution

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Problem 3

Pre-unit

Practicing Standards:  3.OA.A.1

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.

Solution

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Problem 4

Pre-unit

Practicing Standards:  3.MD.B.4

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

  • \(2\frac{1}{4}\)
  • \(1\frac{1}{2}\)
  • \(2\frac{2}{4}\)
  • 3
  • \(3\frac{2}{4}\)
  • 2
  • \(2\frac{1}{4}\)
  • \(2 \frac{1}{4}\)
  • \(2\frac{3}{4}\)
  • 2
  • \(2\frac{1}{4}\)
  • 3

Solution

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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.

Solution

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Problem 6

Five friends go on a hike. They each bring \(\frac{1}{4}\) 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.

    A4 diagrams of equal length. 5 equal parts. 1 part shaded. Total length, 1 cup.
    B5 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?

Solution

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Problem 7

Kiran’s cat eats \(\frac{1}{2}\) 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.

Solution

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Problem 8

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

Solution

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Problem 9

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

  1. \(\frac{10}{3} = \underline{\hspace{0.7cm}} \times \frac{1}{3}\)

  2. \(\frac{10}{3} = \underline{\hspace{0.7cm}} \times \frac{2}{3}\)

  3. \(\frac{10}{3} = \underline{\hspace{0.7cm}} \times \frac{5}{3}\)

Solution

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Problem 10

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

Solution

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Problem 11

Exploration

  1. Measure how thick your workbook is to the nearest \(\frac{1}{8}\) inch.
  2. If all of your classmates stacked their workbooks together, how tall would the stack be? Explain or show your reasoning.
  3. Check your answer by measuring, if possible.

Solution

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Problem 12

Exploration

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

What are some possible number of days Diego counted and the distance he walked each of those days?

Solution

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Section B: Addition and Subtraction of Fractions

Problem 1

  1. Write \(\frac{4}{3}\) in as many ways as you can as a sum of fractions.
  2. Write \(\frac{9}{8}\) in at least 3 different ways as a sum of fractions.

Solution

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Problem 2

  1. Draw “jumps” on the number lines to show two ways to use fourths to make a sum of \(\frac{7}{4}\).

    Number Line. Scale 0 to 2, by 1’s. Evenly spaced tick marks. First tick mark, 0. Fifth tick mark, 1. Last tick mark, 2.   

    Number Line. Scale 0 to 2, by 1’s. Evenly spaced tick marks. First tick mark, 0. Fifth tick mark, 1. Last tick mark, 2.   
  2. Represent each combination of jumps as an equation.

Solution

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Problem 3

  1. Number line. 

    Explain how the diagram represents \(\frac{13}{5} - \frac{4}{5}\).

    Use the diagram to find the value of \(\frac{13}{5}- \frac{4}{5}\).

  2. Use a number line to represent and find the difference \(\frac{9}{4} - \frac{3}{4}\).

    Number line. Scale from 0 to unlabeled. 13 evenly spaced tick marks. First tick mark, 0. Fifth tick mark, 1.

Solution

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Problem 4

Show two different ways to find the difference: \(2 - \frac{3}{4}\)

Solution

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Problem 5

Elena is making friendship necklaces and wants the chain and clasp to be a total of \(18\frac{1}{4}\) inches long. She is going to use a clasp that is \(2\frac{3}{4}\) inches long. How long does her chain need to be? Explain or show your reasoning.

Solution

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Problem 6

For each of the expressions, explain whether you think it would be helpful to decompose one or more numbers to find the value of the expression.

  1. \(\frac{4}{3} + \frac{5}{3}\)
  2. \(5\frac{1}{5} - 2\frac{2}{5}\)
  3. \(9\frac{5}{6} - 6\frac{1}{6}\)

Solution

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Problem 7

The lengths of the shoes of a dad and his two daughters are shown.

Image of 3 pairs of shoes and their lengths. Pink shoes, 8 and 5 eighths inches. Cat shoes, 3 and 6 eighths inches. Dad's shoes, 12 and 1 eighth inches.

For each question, show your reasoning.

  1. How much longer is the older daughter’s shoes than her sister’s?
  2. Which is longer, the dad’s shoes or the combined lengths of his daughters’ shoes?

Solution

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Problem 8

Exploration

A chocolate chip cookie recipe calls for \(2\frac{3}{4}\) cups of flour. You only have a \(\frac{1}{4}\)-cup measuring cup and a \(\frac{3}{4}\)-cup measuring cup that you can use. 

  1. What are different combinations of the measuring cups that you can use to get a total of \(2\frac{3}{4}\) cups of flour?
  2. Write each of the combinations as an addition equation.

Solution

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Problem 9

Exploration

The table shows some lengths of different shoe sizes in inches.

U.S. shoe size insole length
1 \(7\frac{6}{8}\)
1.5 8
2 \(8\frac{1}{8}\)
2.5 \(8\frac{2}{8}\)
3 \(8\frac{4}{8}\)
3.5 \(8\frac{5}{8}\)
4 \(8\frac{6}{8}\)
4.5 9
5 \(9\frac{1}{8}\)
5.5 \(9\frac{2}{8}\)
6 \(9\frac{4}{8}\)
6.5 \(9\frac{5}{8}\)
7 \(9\frac{6}{8}\)
  1. What do you notice about the insole lengths as the size increases?
  2. What will the insole length increase be from size 7 to 7.5? What is the insole length of a size 7.5 shoe?
  3. Predict the insole length for sizes 9, 10, and 12. Explain your prediction. Then solve to find out if your prediction is true.

Solution

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Section C: Addition of Tenths and Hundredths

Problem 1

Andre is building a tower out of different foam blocks. These blocks come in three different thicknesses: \(\frac{1}{2}\)-foot, \(\frac{1}{4}\)-foot, and \(\frac{1}{8}\)-foot. 

Andre stacks two \(\frac{1}{2}\)-foot blocks, two \(\frac{1}{4}\)-foot blocks, and two \(\frac{1}{8}\)-foot blocks to create a tower. What will the height of the tower be in feet? Explain or show how you know.

Solution

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Problem 2

Find the value of each of the following sums. Show your reasoning. Use number lines if you find them helpful.

  1. \(\frac{1}{10} + \frac{3}{100}\)
    Number line. Scale 0 to 1. Evenly spaced by tenths.
  2. \(\frac{24}{100} + \frac{4}{10}\)
    Number line. Scale 0 to 1. Evenly spaced by tenths.
  3. \(\frac{7}{10} + \frac{13}{100}\)
    Number line. Scale 0 to 1. Evenly spaced by tenths.

Solution

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Problem 3

Is the value of each expression greater than, less than or equal to 1? Explain how you know.

  1. \(\frac{3}{10} + \frac{7}{100}\)
  2. \(\frac{13}{10} + \frac{7}{100}\)
  3. \(\frac{30}{100} + \frac{7}{10}\)

Solution

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Problem 4

Diego and Lin continued to play with their coins.

Diego said that he has exactly 3 coins whose thickness adds up to \(\frac{50}{100}\) cm. What coins does Diego have? Explain or show your reasoning.

coin thickness in cm
1 centavo \(\frac{12}{100}\)
10 centavos \(\frac{22}{100}\)
1 peso \(\frac{16}{100}\)
2 pesos \(\frac{14}{100}\)
5 pesos \(\frac{2}{10}\)
20 pesos \(\frac{25}{100}\)

Solution

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Problem 5

Exploration

A chocolate cake recipe calls for 2 cups of flour. You gather your measuring cups and notice you have these sizes: \(\frac{1}{2}\) cup, \(\frac{1}{3}\) cup, \(\frac{1}{4}\) cup, and \(\frac{1}{6}\) cup.

  1. What are the different ways you could use all 4 measuring cups to measure 2 cups of flour?
  2. What are other ways you could use just some of the 4 measuring cups to measure exactly 2 cups of flour?

Solution

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Problem 6

Exploration

A dime is worth \(\frac{1}{10}\) of a dollar and a penny is worth \(\frac{1}{100}\) of a dollar. 

  1. If I have \(\frac{89}{100}\) of a dollar, how many different combinations of dimes and pennies could I have? Use equations to show your reasoning.
  2. A nickel is worth \(\frac{5}{100}\) of a dollar. How many different combinations of dimes, nickels and pennies could I have if I still have \(\frac{89}{100}\) of a dollar? Use equations to show your reasoning.

Solution

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