5.6 More Decimal and Fraction Operations

Unit Goals

  • Students solve multi-step problems involving measurement conversions, line plots, and fraction operations, including addition and subtraction of fractions with unlike denominators. They also explain patterns when multiplying and dividing by powers of 10 and interpret multiplication as scaling by comparing products with factors.

Section A Goals

  • Explain patterns when multiplying and dividing by powers of 10.
  • Solve multi-step problems involving measurement conversions.

Section B Goals

  • Add and subtract fractions with unlike denominators.
  • Create line plots to display fractional measurement data, and use the information to solve problems.
  • Solve problems involving addition and subtraction of fractions.

Section C Goals

  • Interpret multiplication as scaling (resizing).
  • Make generalizations about multiplying a whole number by a fraction greater than, less than and equal to 1.
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Section A: Measurement Conversions and Powers of 10

Problem 1

Pre-unit

Practicing Standards:  4.NF.A.1

Find the number that makes each equation true. Explain or show your reasoning.

  1. \(\frac{\boxed{\phantom{\frac{0}{000}}}}{12} = \frac{2}{3}\)
  2. \(\frac{5}{6} = \frac{\boxed{\phantom{\frac{0}{000}}}}{12}\)

Solution

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Problem 2

Pre-unit

Practicing Standards:  4.MD.A.1

  1. The road around a lake is 15 kilometers long. How many meters is that?
  2. The length of an alligator is 4 meters. How many centimeters is that?

Solution

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Problem 3

Pre-unit

Practicing Standards:  4.NBT.A.1

The value of the 6 in 618,204 has how many times the value as the 6 in 563? Explain or show your reasoning.

Solution

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Problem 4

Pre-unit

Practicing Standards:  4.NF.B.3.c

Find the value of each sum.

  1. \(\frac{3}{8} + \frac{9}{8}\)
  2. \(3\frac{1}{5} + \frac{3}{5}\)
  3. \(2\frac{4}{10} + 1\frac{7}{10}\)

Solution

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Problem 5

Pre-unit

Practicing Standards:  4.OA.A.1

Lin spent 5 minutes reading the story. Noah spent 3 times as long as Lin. How long did Noah spend reading the story? Explain or show your reasoning.

Solution

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Problem 6

Pre-unit

Practicing Standards:  5.NF.B.4

  1. Write a multiplication expression for the shaded area and find the value of the expression. Explain or show your reasoning.

    Diagram, square. Length and width, 1. Partitioned into 5 rows of 6 of the same size squares. In the top left corner, 3 rows of 5 squares shaded. 
  2. Find the value of \(\frac{5}{7} \times \frac{10}{3}\).

Solution

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Problem 7

Pre-unit

Practicing Standards:  4.MD.B.4

The line plot shows the lengths of some toothpicks in inches.

Dot plot titled Toothpick Measurements from 0 to 3 by 1’s.
  1. How many measurements are there?
  2. What is the difference between the longest and shortest toothpicks?

Solution

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Problem 8

  1. Write a multiplication equation relating the values of 0.5 and 0.05.
  2. Write a division equation relating the values of 0.5 and 0.05.

Solution

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Problem 9

Write each number using exponential notation.

  1. \(10 \times 10 \times 10\)
  2. \(100 \times 100\)
  3. 100,000
  4. 1,000,000,000

Solution

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Problem 10

  1. How many centimeters are in each measurement?

    0.12 m

    3.5 m

    19 m

  2. How many millimeters are in each measurement?

    3 m

    37 m

    1,915 m

  3. How does a whole number of meters change when it is converted to millimeters?

Solution

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Problem 11

  1. How many meters are in each measurement?

    16 millimeters

    1,375 millimeters

    57 millimeters

  2. How does a whole number of millimeters change when you express the measurement in meters?

Solution

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Problem 12

A track is 366 meters around. An athlete runs 15 laps. How many kilometers is that? Explain or show your reasoning.

Solution

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Problem 13

Clare drinks 8 glasses of water each day. There are 235 milliliters in each glass. How many liters of water does Clare drink each day? Explain or show your reasoning.

Solution

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Problem 14

A track is 400 yards around. How many full laps does Tyler need to run if he wants to run at least 2 miles?

Solution

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Problem 15

Exploration

  1. Write each of these numbers using exponential notation.
    1. 1,000,000,000 (the approximate population of Africa in 2009)
    2. 100,000,000,000 (estimated number of stars in Milky Way)
    3. 1,000,000,000,000 (amount of dollars added to United States debt each year recently)
    4. 100,000,000,000,000 (denomination of a bill in Zimbabwe)

      Photograph of 1 hundred trillion dollars bill from Zimbabwe.
    5. 1,000,000,000,000,000,000,000 (estimated number of stars in universe)
    6. 10,000,000,000,000,000,000 (estimated number of grains of sand on the earth)
    7. 100,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
      000,000,000,000,000,000,000,000,000,000 (estimated number of atoms in the universe!)
  2. How is exponential notation helpful for writing these numbers?

Solution

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Problem 16

Exploration

You have a piggy bank with 1 kg of coins inside. Which coins would you like to be in the piggy bank? Explain or show your reasoning.

coin approximate weight (grams)
penny 2.5
nickel 5
dime 2.3
quarter 5.7

Solution

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Section B: Add and Subtract Fractions with Unlike Denominators

Problem 1

  1. Find the value of each sum. Explain or show your reasoning.

    1. \(\frac{5}{6} +\frac{2}{6}\)
    2. \(\frac{5}{6} + \frac{2}{3}\)
  2. How were the calculations the same? How were they different?

Solution

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Problem 2

  1. Explain why the expressions \(\frac{2}{3} - \frac{7}{12}\) and \(\frac{8}{12} - \frac{7}{12}\) are equivalent.
  2. How is the expression \(\frac{8}{12} - \frac{7}{12}\) helpful to find the value of \(\frac{2}{3} - \frac{7}{12}\)?

Solution

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Problem 3

Find the value of each expression. Explain or show your reasoning.

  1. \(\frac{1}{4} + \frac{1}{5}\)
  2. \(\frac{10}{9} - \frac{3}{4}\)

Solution

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Problem 4

  1. Find the value of \(2\frac{3}{4} - \frac{1}{3}\). Explain or show your reasoning.
  2. Find the value of \(3\frac{2}{7} - \frac{4}{5}\). Explain or show your reasoning.

Solution

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Problem 5

Jada picked \(4 \frac{2}{3}\) cups of blackberries. Andre picked \(3\frac{5}{8}\) cups of blackberries. 

  1. How many cups of blackberries did Jada and Andre pick together? Explain or show your reasoning.
  2. How many more cups of blackberries did Jada pick than Andre? Explain or show your reasoning.

Solution

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Problem 6

Find the value of each expression. Explain or show your reasoning.

  1. \(\frac{7}{8} + \frac{4}{13}\)
  2. \(\frac{7}{8} - \frac{3}{20}\)

Solution

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Problem 7

Here are the lengths of some pieces of ribbon measured in inches:

  • \(3\frac{1}{4}\)
  • \(4\frac{1}{8}\)
  • \(3\frac{6}{8}\)
  • \(3\frac{1}{8}\)
  • \(2\frac{5}{8}\)
  • \(3\frac{2}{4}\)
  • \(3\frac{1}{4}\)
  • \(3\frac{7}{8}\)
  • \(4\frac{1}{8}\)
  • \(3\frac{1}{2}\)
  • \(2\frac{7}{8}\)
  • \(4\frac{1}{8}\)
  • \(3\frac{3}{4}\)
  • \(3\frac{2}{8}\)

  1. Complete the line plot with the ribbon lengths.

    Horizontal line. Not labeled.

  2. What is the sum of the lengths of the ribbons that measure more than 4 inches? Explain or show your reasoning.

Solution

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Problem 8

Han is making a line plot of the seedlings his class grew. This is what he has done so far.

Dot plot titled Growing Plants from 0 to 3 by 1’s.

Use this information to complete the line plot. Explain or show your reasoning.

  • There are 15 seedlings altogether.
  • The tallest seedling is \(2\frac{1}{8}\) taller than the shortest seedling.
  • There are 3 seedlings of the shortest height.

Solution

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Problem 9

Exploration

  1. Put the numbers 2, 3, 4, and 5 in the four boxes so that the expression is as close to 1 as possible. \(\frac{\boxed{\phantom{\frac{0}{000}}}}{\boxed{\phantom{\frac{0}{000}}}} + \frac{\boxed{\phantom{\frac{0}{000}}}}{\boxed{\phantom{\frac{0}{000}}}}\)
  2. Put the numbers 2, 3, 4, and 5 in the four boxes so that the expression is as close to 1 as possible. \(\frac{\boxed{\phantom{\frac{0}{000}}}}{\boxed{\phantom{\frac{0}{000}}}} - \frac{\boxed{\phantom{\frac{0}{000}}}}{\boxed{\phantom{\frac{0}{000}}}}\)

Solution

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Problem 10

Exploration

Make a line plot of seedling heights so that each of these statements is true.

  • There are 12 measurements.
  • The largest measurement is \(2 \frac{3}{8}\) inches more than the smallest measurement.
  • The sum of the measurements is \(18\frac{3}{8}\) inches.
Horizontal line, not labeled.

Explain how you made the line plot.

Solution

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Section C: The Size of Products

Problem 1

  1. Andre ran \(\frac{4}{5}\) of a 7 mile trail. Did Andre run more or less than 7 miles? Explain or show your reasoning.
  2. Clare ran \(\frac{\boxed{\phantom{\frac{0}{000}}}}{\Large{10}}\) of a 7 mile trail. She ran more than 7 miles. Choose a number that could go in the box. Explain or show your reasoning.

Solution

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Problem 2

The point J on the number line shows how many miles Jada ran. Label the points on the number line to show how far each of these students ran.

  1. Clare ran \(\frac{8}{5}\) as far as Jada.
  2. Tyler ran \(\frac{4}{3}\) as far as Jada.
  3. Lin ran \(\frac{1}{2}\) as far as Jada.
Number line. Tick mark, labeled 0. 4 dots plotted. From left to right, blank line, J, blank line, blank line. 

Solution

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Problem 3

The point A is labeled on the number line.

Number line. Tick mark, labeled 0. Point, labeled A to the right. 

Label each of these points on the number line.

  • \(\frac{2}{5} \times \text{A}\)
  • \(\frac{13}{10} \times \text{A}\)
  • \(\frac{7}{4} \times \text{A}\)

Solution

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Problem 4

Use the equation \(\frac{5}{7} = \left(1 - \frac{2}{7}\right)\) to explain why \(\frac{5}{7} \times \frac{11}{3} < \frac{11}{3}\).

Solution

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Problem 5

Explain why multiplying a fraction by a number less than 1 makes the fraction smaller.

Solution

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Problem 6

Exploration

A point P is labeled on the number line.

Number line. Tick mark, labeled 0. Point to the right of 0. Labeled P.

  1. P is \(\frac{3}{4}\) of a number A. Plot A on the number line. Explain or show your reasoning.
  2. P is \(\frac{5}{9}\) of a number B. Plot B on the number line. Explain or show your reasoning.

Solution

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Problem 7

Exploration

  1. About \(10^6\) people live in Michigan. About \(10^4\) of the people in Michigan live in Flint.

    1. How many times as many people live in Michigan as in Flint?
    2. How many times as many people live in Flint as in Michigan?

  2. There are about \(10^{11}\) stars in the Milky Way. There are about \(10^{21}\) stars in the universe.

    1. How many times as many stars are there in the universe than in the Milky Way?
    2. How many times as many stars are there in the Milky Way than in the universe?

Solution

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