Grade 6 Workbook — Chapter 5: Multiplying and Dividing Fractions

SOL 6.CE.1 a–c · Companion to Textbook Chapter 5

Each page below is one Canva page. Headings are sized for direct paste: page title as H1, section labels as H2. Figures referenced by filename live in ../figures/.

PAGE 1 — Chapter opener

Chapter 5 · Multiplying and Dividing Fractions

Standard 6.CE.1 a, b, c

In this chapter you will:

• Model fraction multiplication and division with area models, set models, and strips
• Multiply fractions and mixed numbers, in simplest form
• Divide fractions and mixed numbers, in simplest form
• Explain what happens when you multiply or divide by a number between 0 and 1

Words to know: factor · product · area model · set model · fraction strip · simplest form · dividend · divisor · quotient · improper fraction · reciprocal

Calculator note: parts a, b, and c of 6.CE.1 are assessed without a calculator. Do all of this chapter by hand.

Number limits for this chapter: denominators of 12 or less · proper fractions, improper fractions, and mixed numbers · every answer in simplest form

PAGE 2 — The area model

5.1 Modeling Fraction Multiplication

FIGURE: fig1-area-model-half-third.png (left half) [square in six cells, one column and one row hatched, overlap cell solid, showing 1/2 x 1/3 = 1/6]

"Of" means multiply. The double-shaded part of the square is the product.

Fill in the blanks.

In an area model, the number of equal cells equals the ____________ times the ____________.

The number of double-shaded cells equals the ____________ times the ____________.

Shade and solve. For each, write how many cells the square has, how many are double-shaded, and the product in simplest form.

PAGE 3 — Set models and strips

Groups and Strips

FIGURE: fig3-set-model-three-fourths-of-eight.png (full width) [eight circles in four groups of two, six filled, showing 3/4 of 8 = 6]

Fraction of a set.

$\tfrac{3}{4}$ of $8$: each group has ______, take ______ groups, answer ______

$\tfrac{1}{3}$ of $9$: each group has ______, take ______ groups, answer ______

$\tfrac{2}{3}$ of $12$: each group has ______, take ______ groups, answer ______

$\tfrac{3}{4}$ of $16$: each group has ______, take ______ groups, answer ______

FIGURE: fig4-strip-repeated-addition.png (full width) [three segments of 2/5 laid end to end on a number line from 0 to 2, total 6/5]

Repeated addition. Write each as a sum, then as a mixed number in simplest form.

$3 \times \tfrac{2}{5} =$ ______________________ $=$ ______

$2 \times \tfrac{3}{8} =$ ______________________ $=$ ______

$5 \times \tfrac{2}{3} =$ ______________________ $=$ ______

$4 \times \tfrac{3}{10} =$ ______________________ $=$ ______

PAGE 4 — Exit ticket 5.1

Exit Ticket · Lesson 5.1

Name: ________________________ Date: ____________

1. $\tfrac{1}{2} \times \tfrac{2}{5} =$ ______ (simplest form)

2. $\tfrac{3}{4}$ of $16 =$ ______

3. $4 \times \tfrac{3}{10} =$ ______ (mixed number, simplest form)

4. Explain how an area model shows that $\tfrac{1}{3} \times \tfrac{1}{4} = \tfrac{1}{12}$.

PAGE 5 — The multiplication rule

5.2 Multiplying Fractions and Mixed Numbers

Multiply the numerators. Multiply the denominators. Simplify.

$$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$

Multiply. Show the unsimplified product, then the simplest form.

Convert to improper fractions.

$$1\tfrac{1}{2} = \frac{\phantom{0}}{2} \qquad 2\tfrac{1}{4} = \frac{\phantom{0}}{4} \qquad 2\tfrac{5}{8} = \frac{\phantom{00}}{8} \qquad 3\tfrac{1}{2} = \frac{\phantom{0}}{2}$$

PAGE 6 — Mixed numbers and estimation

Multiplying Mixed Numbers

Estimate first, then compute. Write answers in simplest form.

Apply it. A trail is $2\tfrac{2}{3}$ miles long. Jonah walks $\tfrac{3}{4}$ of it.

Estimate: ______ Exact distance: ______ miles

Predict. Will $\tfrac{7}{8} \times \tfrac{4}{5}$ be greater or less than $\tfrac{4}{5}$? ______ Compute: ______

PAGE 7 — Exit ticket 5.2

Exit Ticket · Lesson 5.2

Name: ________________________ Date: ____________

Simplest form for every answer.

1. $\tfrac{4}{9} \times \tfrac{3}{8} =$ ______

2. $1\tfrac{1}{2} \times 2\tfrac{2}{3} =$ ______

3. $6 \times \tfrac{5}{12} =$ ______

4. Why do you convert a mixed number to an improper fraction before multiplying?

PAGE 8 — How many fit?

5.3 Modeling Fraction Division

FIGURE: fig5-strip-division-three-by-fourth.png (full width) [three whole strips each cut into fourths, twelve parts numbered, showing 3 divided by 1/4 = 12]

Read division as "how many of the divisor fit into the dividend?"

Count the pieces.

How many $\tfrac{1}{4}$ fit into 3? ______ so $3 \div \tfrac{1}{4} =$ ______

How many $\tfrac{1}{3}$ fit into 2? ______ so $2 \div \tfrac{1}{3} =$ ______

How many $\tfrac{1}{8}$ fit into 1? ______ so $1 \div \tfrac{1}{8} =$ ______

How many $\tfrac{1}{2}$ fit into 4? ______ so $4 \div \tfrac{1}{2} =$ ______

How many $\tfrac{1}{3}$ fit into 5? ______ so $5 \div \tfrac{1}{3} =$ ______

Common denominator strategy. Rename, then divide the numerators.

$$\tfrac{1}{2} \div \tfrac{1}{8} = \tfrac{\phantom{0}}{8} \div \tfrac{1}{8} = \underline{\phantom{000}}$$

$$\tfrac{2}{3} \div \tfrac{1}{6} = \tfrac{\phantom{0}}{6} \div \tfrac{1}{6} = \underline{\phantom{000}}$$

$$\tfrac{3}{4} \div \tfrac{1}{8} = \tfrac{\phantom{0}}{8} \div \tfrac{1}{8} = \underline{\phantom{000}}$$

PAGE 9 — When part of one more fits

Quotients That Are Not Whole Numbers

FIGURE: fig6-strip-five-sixths-by-third.png (full width) [strip showing 5/6 shaded with thirds marked beneath, two whole thirds plus half of a third]

Complete.

In $\tfrac{5}{6} \div \tfrac{1}{3}$, ______ whole thirds fit, using up $\tfrac{\phantom{0}}{6}$.

The leftover is $\tfrac{\phantom{0}}{6}$, which is ______ of a third.

So $\tfrac{5}{6} \div \tfrac{1}{3} =$ ______

Divide.

Apply it. Nadia has $4\tfrac{1}{2}$ feet of ribbon and cuts $\tfrac{3}{4}$-foot pieces. Number of pieces: ______

Explain. Why is $3 \div \tfrac{1}{4}$ larger than 3?

PAGE 10 — Exit ticket 5.3

Exit Ticket · Lesson 5.3

Name: ________________________ Date: ____________

1. $2 \div \tfrac{1}{5} =$ ______

2. $\tfrac{3}{4} \div \tfrac{1}{8} =$ ______

3. $\tfrac{5}{6} \div \tfrac{1}{3} =$ ______

4. How many $\tfrac{1}{3}$-cup scoops fill 2 cups? ______ What does that have to do with division?

PAGE 11 — Reciprocals

5.4 Dividing Fractions and Mixed Numbers

Two numbers are reciprocals when their product is 1. To divide by a fraction, multiply by its reciprocal.

Complete the reciprocal table.

Check one. $\tfrac{3}{4} \times \tfrac{4}{3} = \tfrac{\phantom{00}}{12} =$ ______

Which number has no reciprocal? ______ Why?

Divide. Rewrite as multiplication first.

$$\tfrac{3}{4} \div \tfrac{2}{5} = \tfrac{3}{4} \times \frac{\phantom{0}}{\phantom{0}} = \underline{\phantom{0000}}$$

$$\tfrac{5}{8} \div \tfrac{5}{6} = \tfrac{5}{8} \times \frac{\phantom{0}}{\phantom{0}} = \underline{\phantom{0000}}$$

$$\tfrac{1}{2} \div \tfrac{3}{4} = \tfrac{1}{2} \times \frac{\phantom{0}}{\phantom{0}} = \underline{\phantom{0000}}$$

$$\tfrac{2}{3} \div 4 = \tfrac{2}{3} \times \frac{\phantom{0}}{\phantom{0}} = \underline{\phantom{0000}}$$

PAGE 12 — Division practice

Divide and Check

Divide. Simplest form.

Check by multiplying. Multiply your answer to (a) by $\tfrac{2}{5}$. Do you get $\tfrac{3}{5}$ back? ______

Show the work: _______________________________________________

Apply it. A recipe uses $1\tfrac{1}{4}$ cups of flour per batch. Priya has $7\tfrac{1}{2}$ cups.

Full batches: ______

PAGE 13 — Exit ticket 5.4

Exit Ticket · Lesson 5.4

Name: ________________________ Date: ____________

Simplest form for every answer.

1. $\tfrac{4}{5} \div \tfrac{2}{3} =$ ______

2. $3\tfrac{1}{2} \div \tfrac{7}{8} =$ ______

3. $\tfrac{5}{8} \div 5 =$ ______

4. Reciprocal of $\tfrac{3}{7}$: ______ of $5$: ______ of $1\tfrac{1}{2}$: ______

PAGE 14 — Bigger or smaller?

5.5 Multiplying and Dividing by a Number Between 0 and 1

FIGURE: fig7-number-line-scaling.png (full width) [number line 0 to 20 with arrows from 12 to 9 for times 3/4, from 12 to 16 for divided by 3/4, and a loop at 12 for times 1]

Complete the rule table.

Predict, then compute.

PAGE 15 — Sorting and disproving

Sort and Explain

Sort each expression by its result compared to 9.

$$9 \times \tfrac{1}{3} \qquad 9 \times \tfrac{3}{2} \qquad 9 \times \tfrac{5}{6} \qquad 9 \times 1 \qquad 9 \div \tfrac{3}{4} \qquad 9 \div 3$$

Compute and compare.

$2\tfrac{1}{2} \times \tfrac{4}{5} =$ ______ which is ____________ than $2\tfrac{1}{2}$

$2\tfrac{1}{2} \div \tfrac{4}{5} =$ ______ which is ____________ than $2\tfrac{1}{2}$

Apply it. A photo is 8 inches wide.

Scaled by $\tfrac{3}{4}$: ______ inches. Scaled by $\tfrac{5}{4}$: ______ inches.

Which factor enlarges the photo? ______ Why?

Disprove it. Jaden says multiplying always makes a number bigger and dividing always makes it smaller. Give one counterexample of each kind.

Multiplication: _______________ Division: _______________

What condition did he leave out? _______________________________________________

PAGE 16 — Exit ticket 5.5

Exit Ticket · Lesson 5.5

Name: ________________________ Date: ____________

1. Is $16 \times \tfrac{5}{8}$ more or less than 16? ______ Answer: ______

2. Is $16 \div \tfrac{5}{8}$ more or less than 16? ______ Answer: ______

3. $\tfrac{3}{4} \times \tfrac{2}{3} =$ ______ which is ____________ than $\tfrac{3}{4}$

4. Why does multiplying by $\tfrac{5}{8}$ shrink a number while multiplying by $\tfrac{8}{5}$ grows it?

PAGE 17 — Chapter 5 review, part 1

Chapter 5 Review

Part A · Modeling

1. Area model for $\tfrac{2}{3} \times \tfrac{1}{2}$. Product: ______

2. Strip model for $\tfrac{3}{4} \div \tfrac{1}{8}$. Quotient: ______

3. Repeated addition: $4 \times \tfrac{2}{3} =$ ______ (mixed number)

4. Set model: $\tfrac{3}{5}$ of $10 =$ ______

Part B · Multiplying and dividing

Simplest form for every answer.

5. a) $\tfrac{3}{8} \times \tfrac{2}{3}$ ______ b) $\tfrac{5}{9} \times \tfrac{3}{10}$ ______ c) $\tfrac{7}{12} \times \tfrac{6}{7}$ ______

6. a) $1\tfrac{1}{2} \times 2\tfrac{2}{5}$ ______ b) $2\tfrac{1}{3} \times 1\tfrac{1}{2}$ ______

7. a) $\tfrac{3}{4} \div \tfrac{3}{8}$ ______ b) $\tfrac{5}{6} \div \tfrac{5}{12}$ ______ c) $\tfrac{2}{5} \div 4$ ______

8. a) $4\tfrac{1}{2} \div \tfrac{3}{4}$ ______ b) $3\tfrac{1}{3} \div 1\tfrac{2}{3}$ ______

9. $8 \div \tfrac{2}{3} =$ ______

PAGE 18 — Chapter 5 review, part 2

Chapter 5 Review (continued)

Part C · Multiplying and dividing by a number between 0 and 1

10. More or less than 24, then compute: a) $24 \times \tfrac{5}{6}$ ______ , ______ b) $24 \div \tfrac{5}{6}$ ______ , ______

11. Why is $\tfrac{1}{2} \times \tfrac{1}{2}$ less than $\tfrac{1}{2}$?

    ---

12. Which results are greater than $\tfrac{7}{8}$? Compute each.

    $\tfrac{7}{8} \times \tfrac{4}{3} =$ ______ $\tfrac{7}{8} \div \tfrac{4}{3} =$ ______ $\tfrac{7}{8} \times \tfrac{2}{3} =$ ______

    Greater than $\tfrac{7}{8}$: _______________

Part D · Application and reasoning

13. Each bag holds $\tfrac{3}{4}$ pound. $2\tfrac{1}{2}$ bags hold ______ pounds.

14. A board $5\tfrac{1}{4}$ ft long is cut into $\tfrac{3}{4}$-ft pieces. Number of pieces: ______

15. $\tfrac{5}{6} \div \tfrac{1}{6} = 5$ but $\tfrac{5}{6} \times \tfrac{1}{6} = \tfrac{5}{36}$. Why so different?

    ---

16. Check $\tfrac{3}{4} \div \tfrac{2}{5}$ using multiplication. Show the check and explain why it works.

    ---

Canva production notes

• Page size: 8.5 × 11 in, 0.75 in margins
• Type: headings 24–28 pt, body 12–14 pt, answer blanks 14 pt with 1.5 line spacing
• Area models: print each square at least 1.75 in on a side so a 3 by 4 grid of cells stays countable; use two distinct hatch directions rather than two colors so the pages photocopy cleanly
• Blank area-model grids: page 2 needs five unshaded squares in the margin, one per table row, for students to shade
• Strip diagrams: keep all strips on a page the same length so equal wholes stay visually equal; this is the whole point of the model
• Figure widths: strip and number-line figures full width; single-square area models half width
• Answer blanks: keep every blank on the same line as its prompt so the Canva text box does not reflow