Lesson 16 | Fraction Operations | 4th Grade Mathematics

Objective

Multiply a whole number by a fraction.

Common Core Standards

Core Standards

The core standards covered in this lesson

4.NF.B.4.B — Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)

Number and Operations—Fractions

4.NF.B.4.B — Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Multiply a whole number by a fraction using a visual model.
Generate a general method for multiplying a whole number by a unit fraction, i.e., $$n \times \frac{1}{b}=\frac{n}{b}$$ (MP.8).
Generate a general method for multiplying a whole number by a non-unit fraction, i.e., $${n\times{{a\over b}}={{n\times a}\over b}}$$ (MP.8).
Understand a multiple of $${a\over b}$$ as a multiple of $${1\over b}$$.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

You might decide to stretch this lesson over two days. One way you could do this would be to have Anchor Tasks 1 and 2 on the first day and Anchor Task 3 on the second day. Divvy up Problem Set and Homework problems accordingly.

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Anchor Tasks

Tasks designed to teach criteria for success of the lesson, and guidance to help draw out student understanding

Problem 1

a. Here are two tables with expressions. Find the value of each expression. Use a model if it's helpful. Leave the last two rows of each table blank for now.

Set A

Set B

expression	value
$$1\times\frac{1}{8}$$
$$2\times\frac{1}{8}$$
$$3\times\frac{1}{8}$$
$$4\times\frac{1}{8}$$
$$5\times\frac{1}{8}$$
$$6\times\frac{1}{8}$$

expression	value
$$2\times \frac{1}{3}$$
$$2\times \frac{1}{4}$$
$$2\times \frac{1}{5}$$
$$2\times \frac{1}{6}$$
$$2\times \frac{1}{7}$$
$$2\times \frac{1}{8}$$

b. Study your completed tables. What patterns do you see in how the expressions and values are related?

c. In the last two rows of the table of Set A, write $$\frac{11}{8}$$ and $$\frac{13}{8}$$ in the "value" column. Write the expressions with that value.

d. In the last two rows of the table of Set B, write $$\frac{2}{12}$$ and $$\frac{2}{15}$$ in the "value" column. Write the expressions with that value.

Guiding Questions

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References

Illustrative Mathematics Grade 4 Unit 3 Lesson 3 Activity 1

Grade 4 Unit 3 Lesson 3 Activity 1, accessed on Nov. 11, 2022, 4:07 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Problem 2

Draw a model to represent $$\frac{2}{3}$$.
Show how you would use or adjust the model above to represent $$4\times \frac{2}{3}$$.
Based on your model in Part (ii), what is the value of $$4\times \frac{2}{3}$$?

b. Find the value of each expression below. Draw a model if it would be helpful.

$$2\times \frac{1}{5}$$
$$2\times \frac{3}{5}$$
$$2\times \frac{4}{5}$$
$$6\times \frac{4}{5}$$

c. Willa said that to multiply a whole number by a fraction, she’d multiply the whole number and the numerator and use the same denominator. Do you agree with Willa? Explain your reasoning.

Guiding Questions

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References

Illustrative Mathematics Grade 4 Unit 3 Lesson 4 Activity 2

Grade 4 Unit 3 Lesson 4 Activity 2, accessed on Nov. 11, 2022, 4:14 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Modified by Fishtank Learning, Inc.

Problem 3

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

$$\frac{12}{5} = 12\space \times $$ _____	$$\frac{12}{5} = 3\space \times $$ _____
$$\frac{12}{5} = 6\space \times $$ _____	$$\frac{12}{5} = 2\space \times $$ _____
$$\frac{12}{5} = 4\space \times $$ _____	$$\frac{12}{5} = 1\space \times $$ _____

b. Here are two sets of numbers:

Set A	Set B
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	$$\frac17, \frac27,\frac37,\frac47,\frac57,\frac67,\frac77$$

Choose a number from Set A and a number from Set B to complete this equation and make it true:

$$\frac67 =$$ _____ $$\times$$ _____

Choose a different number from Set A and a number from Set B to complete the equation to make it true:

$$\frac67 =$$ _____ $$\times$$ _____

c. Explain or show how you know that the two equations you wrote are both true.

Guiding Questions

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References

Illustrative Mathematics Grade 4 Unit 3 Lesson 5 Activity 1

Grade 4 Unit 3 Lesson 5 Activity 1, accessed on March 25, 2022, 9:32 a.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Problem Set

Problem Set

Answer Keys

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Discussion of Problem Set

What did Angie do to solve? Will her strategy work all the time?
How did you record your solutions to #6?
For which problems in #6 would repeated addition be a particularly inefficient strategy? Why?
Look at your answers for #6(c) and #6(d). Convert each answer to a mixed number. What do you notice? How are the expressions in #6(c) and #6(d) similar?
How did Goldie solve in #7? How is this similar to showing $${{3\over4}+{3\over4}+{3\over4}}$$?

Target Task

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Problem 1

Which of the following is equivalent to $${7\times {3\over8}}$$? There are three correct answers.

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Problem 2

Explain how to find $${3 \times {4\over5}}$$ using a number line. Find the product.

Student Response

An example response to the Target Task at the level of detail expected of the students.

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Additional Practice

The Extra Practice Problems can be used as additional practice for homework, during an intervention block, etc. Daily Word Problems and Fluency Activities are aligned to the content of the unit but not necessarily to the lesson objective, therefore feel free to use them anytime during your school day.

Extra Practice Problems

Extra Practice Problems

Answer Keys

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Word Problems and Fluency Activities

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Lesson 15

Lesson 17

Lesson Map

Topic A: Building, Adding, and Subtracting Fractions Less Than or Equal to 1

Decompose fractions as a sum of unit fractions and as a multiple of a unit fraction.

4.NF.B.3.B 4.NF.B.4.A

Decompose fractions as a sum of fractions with the same denominator in more than one way.

4.NF.B.3.B 4.NF.B.4.A

Add and subtract fractions within 1 with the same units.

4.NF.B.3.A 4.NF.B.3.C

Solve word problems that involve the addition and subtraction of fractions where the total is less than or equal to one.

4.NF.B.3.D

Topic B: Building, Adding, and Subtracting Fractions Less Than 2

Decompose non-unit fractions less than 2 as a sum of unit fractions, as a sum of non-unit fractions, and as a whole number times a unit fraction.

4.NF.B.3.B 4.NF.B.4.A

Add and subtract fractions that require regrouping where the total is less than 2.

4.NF.B.3.A 4.NF.B.3.C

Add two fractions where one denominator is a multiple of the other using the denominators 2, 3, 4, 5, 6, 8, 10, and 12.

4.NF.B.3.A 4.NF.B.3.C

Topic C: Building, Adding, and Subtracting Fractions Greater Than or Equal to 2

Decompose and compose non-unit fractions greater than two as a sum of unit fractions, as a sum of non-unit fractions, and as a whole number times a unit fraction.

4.NF.B.3.B

Convert fractions greater than 1 to mixed numbers.

4.NF.B.3.B 4.NF.B.3.C

Convert mixed numbers to fractions greater than 1.

4.NF.B.3.B 4.NF.B.3.C

Compare and order fractions greater than 1 using various methods.

4.NF.B.3.C

Add fractions and mixed numbers where the total is greater than or equal to 2.

4.NF.B.3.C

Subtract fractions and mixed numbers where the total is greater than or equal to 2.

4.NF.B.3.C

Add and subtract mixed numbers using a variety of mental strategies.

4.NF.B.3.C

Solve word problems involving addition and subtraction of fractions.

4.NF.B.3.D

Topic D: Multiplication of Fractions

Multiply a whole number by a fraction.

4.NF.B.4.B

Multiply a whole number by a mixed number.

4.NF.B.4.B

Solve word problems involving multiplication of fractions.

4.NF.B.4.C

Solve word problems involving addition, subtraction, and multiplication of fractions.

4.NF.B.3.D 4.NF.B.4.C

Topic E: Line Plots

Make a line plot (dot plot) representation to display a data set of measurements in fractions of a unit.

4.MD.B.4

Solve problems using information presented in line plots.

4.MD.B.4

Fraction Operations

Objective

Common Core Standards

Core Standards

Number and Operations—Fractions

Criteria for Success

Tips for Teachers

Anchor Tasks

Problem 1

Guiding Questions

References

Problem 2

Guiding Questions

References

Problem 3

Guiding Questions

References

Problem Set

Discussion of Problem Set

Target Task

Problem 1

Problem 2

Student Response

Additional Practice

Extra Practice Problems

Word Problems and Fluency Activities

Lesson Map

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