Grade 6 Workbook — Chapter 6: Problem Solving with Fractions

SOL 6.CE.1 d–e · Companion to Textbook Chapter 6

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 6 · Problem Solving with Fractions

Standard 6.CE.1 d–e

In this chapter you will:

Estimate answers to fraction problems using benchmarks
Solve addition and subtraction problems in context, with and without regrouping
Solve multiplication and division problems in context
Solve multistep problems and justify your answers

Words to know: estimate · benchmark fraction · compatible numbers · least common denominator · regroup · simplest form · reciprocal · justify

Every answer on these pages goes in simplest form. All denominators are 12 or less.

PAGE 2 — Benchmarks

6.1 Estimating with Fractions

FIGURE: fig1-benchmark-fractions.png (full width) [Number line 0 to 1 labeled at 0, 1/2, and 1, with 1/12, 5/12, 7/12, and 11/12 plotted and arrows pointing each to its nearest benchmark.]

Compare the numerator to HALF the denominator.

Round each fraction to $0$ , $\tfrac{1}{2}$ , or $1$ .

Fraction	$\tfrac{2}{9}$	$\tfrac{5}{9}$	$\tfrac{8}{9}$	$\tfrac{1}{12}$	$\tfrac{7}{12}$	$\tfrac{11}{12}$
Benchmark

Round each mixed number to the nearest whole number.

Mixed number	$5\tfrac{2}{9}$	$7\tfrac{3}{4}$	$10\tfrac{1}{2}$	$3\tfrac{1}{12}$
Rounds to

Two rules to remember.

Multiplying by a number less than 1 makes the result ____________.

Dividing by a number less than 1 makes the result ____________.

PAGE 3 — Estimation practice

Estimate Before You Solve

Estimate each. You do not need the exact answer.

Problem	Estimate
$\tfrac{3}{4} + \tfrac{7}{8}$
$\tfrac{11}{12} - \tfrac{1}{8}$
$6\tfrac{1}{8} + 2\tfrac{11}{12}$
$9\tfrac{7}{8} - 4\tfrac{1}{6}$
$\tfrac{1}{3} \times 14\tfrac{3}{4}$
$20\tfrac{1}{4} \div \tfrac{3}{4}$

Reasonable or not? Circle one. If not reasonable, write the correct answer.

Statement	Reasonable?	Correct answer
$\tfrac{5}{6} + \tfrac{3}{4} = \tfrac{8}{10}$	yes / no
$\tfrac{3}{4} \times 12 = 9$	yes / no
$8 \div \tfrac{1}{2} = 4$	yes / no

Apply it. Nia needs $2\tfrac{3}{4}$ cups of flour and $1\tfrac{7}{8}$ cups of flour. She has a 5-cup bag.

Estimate: ______ Enough? ______ Exact total: ______ Left over: ______

PAGE 4 — Exit ticket 6.1

Exit Ticket · Lesson 6.1

Name: ________________________ Date: ____________

Round $\tfrac{3}{7}$ to $0$ , $\tfrac{1}{2}$ , or $1$ . ______
Estimate $6\tfrac{7}{8} + 3\tfrac{1}{12}$ . ______
Estimate $\tfrac{1}{3} \times 26$ using a compatible number. ______
How does an estimate show that $\tfrac{1}{2} + \tfrac{1}{3} = \tfrac{2}{5}$ must be wrong?

PAGE 5 — Adding and subtracting fractions

6.2 Addition and Subtraction in Context

FIGURE: fig2-fraction-bars-addition.png (full width) [Three bars of equal length: 2/3 shaded, 1/4 shaded, and 11/12 shaded, with vertical guide lines at the twelfths showing 2/3 = 8/12 and 1/4 = 3/12.]

Like denominators: add or subtract the numerators. Unlike denominators: rewrite with the least common denominator first.

Compute. Write every answer in simplest form.

a) $\tfrac{3}{10} + \tfrac{4}{10}$	b) $\tfrac{5}{8} - \tfrac{3}{8}$	c) $\tfrac{1}{2} + \tfrac{1}{3}$
d) $\tfrac{3}{4} - \tfrac{1}{6}$	e) $\tfrac{2}{5} + \tfrac{1}{2}$	f) $\tfrac{7}{8} - \tfrac{1}{2}$

Word problems.

A tank held $\tfrac{7}{10}$ gallon, then $\tfrac{1}{5}$ gallon was added. How much is in the tank? ______
A border uses pieces of $2\tfrac{1}{3}$ ft, $1\tfrac{1}{2}$ ft, and $\tfrac{3}{4}$ ft. Total length: ______

PAGE 6 — Regrouping

Regrouping Mixed Numbers

FIGURE: fig3-regrouping-mixed-number.png (right half of page) [Circle models showing 8 and 2/8 rewritten as 7 and 10/8 by trading one whole circle for eight eighths.]

$8\tfrac{1}{4} = 8\tfrac{2}{8} = 7\tfrac{10}{8}$

Compute. Circle the problems where you had to regroup.

a) $4\tfrac{3}{8} + 2\tfrac{1}{8}$	b) $5\tfrac{2}{3} - 1\tfrac{1}{3}$	c) $3\tfrac{5}{6} + 1\tfrac{1}{2}$
d) $7\tfrac{1}{5} - 2\tfrac{3}{5}$	e) $8 - 3\tfrac{5}{12}$	f) $6\tfrac{1}{4} - 2\tfrac{5}{6}$

Apply it. A water jug held $5\tfrac{1}{8}$ gallons. A team drank $1\tfrac{3}{4}$ gallons, then a coach added $2\tfrac{1}{2}$ gallons.

Step 1 (after drinking): ______ Step 2 (after adding): ______

Explain. How do you know before you start whether a subtraction needs regrouping?

PAGE 7 — Exit ticket 6.2

Exit Ticket · Lesson 6.2

Name: ________________________ Date: ____________

$\tfrac{5}{12} + \tfrac{1}{4} =$ ______
$4\tfrac{1}{6} - 1\tfrac{5}{6} =$ ______
Lin has $3\tfrac{1}{2}$ yards of ribbon and uses $1\tfrac{7}{8}$ yards. Left: ______
How do you decide whether a mixed-number subtraction needs regrouping?

PAGE 8 — Choosing the operation

6.3 Multiplication and Division in Context

FIGURE: fig4-area-model-multiplication.png (right half of page) [A square split into 4 columns and 3 rows; 3 columns shaded one way for 3/4, 2 rows shaded another way for 2/3, and the 6 overlapping parts shaded solid to show 2/3 x 3/4 = 1/2.]

Match each situation to its operation. Write M for multiply or D for divide.

Situation	M or D
$\tfrac{2}{3}$ of a $\tfrac{3}{4}$ -cup measure
How many $\tfrac{3}{4}$ -cup servings are in 6 cups
$4\tfrac{1}{2}$ pounds shared equally among 6 bags
$3\tfrac{1}{2}$ batches at $2\tfrac{2}{3}$ cups per batch

Compute. Simplest form.

a) $\tfrac{1}{3} \times \tfrac{3}{5}$	b) $\tfrac{5}{6} \times \tfrac{2}{5}$	c) $\tfrac{2}{3} \div \tfrac{1}{6}$
d) $\tfrac{3}{4} \div 2$	e) $1\tfrac{1}{2} \times 2\tfrac{2}{3}$	f) $4\tfrac{1}{2} \div 1\tfrac{1}{2}$

PAGE 9 — Division in two flavors

How Many Groups? How Big Is Each Group?

FIGURE: fig5-measurement-division.png (full width) [Six bars, each one cup divided into quarters, with 8 brackets each covering three quarter-segments, showing 6 divided by 3/4 equals 8.]

Word problems. Show the division you used.

How many $\tfrac{1}{4}$ -cup scoops are in $3\tfrac{1}{2}$ cups? ______
A rehearsal is 45 minutes. Warm-ups take $\tfrac{2}{5}$ of it. Warm-up time: ______
$6\tfrac{3}{4}$ pounds of apples go into 9 equal boxes. Each box: ______
A rope $7\tfrac{1}{2}$ ft long is cut into $1\tfrac{1}{4}$ -ft pieces. Number of pieces: ______
A garden bed is $2\tfrac{1}{2}$ yd by $1\tfrac{3}{5}$ yd. Area: ______

Explain. Why is $12 \div \tfrac{2}{3}$ greater than 12?

PAGE 10 — Exit ticket 6.3

Exit Ticket · Lesson 6.3

Name: ________________________ Date: ____________

$\tfrac{3}{5} \times \tfrac{2}{9} =$ ______
$9 \div \tfrac{3}{4} =$ ______
A cook has $2\tfrac{1}{2}$ lb of cheese and uses $\tfrac{5}{8}$ lb per pizza. Pizzas: ______
Why is $\tfrac{3}{4}$ of 8 less than 8?

PAGE 11 — The four-step routine

6.4 Multistep Problems

FIGURE: fig6-multistep-bar-model.png (full width) [A bar labeled 5 1/4 cups of flour split into segments of 1 2/3, 3/4, and an unknown leftover, with a brace under the first two labeled "amount used."]

1. Understand · 2. Estimate · 3. Solve one step at a time · 4. Justify

Work the example with the routine.

Jamal starts with $5\tfrac{1}{4}$ cups of flour, uses $1\tfrac{2}{3}$ cups for bread and $\tfrac{3}{4}$ cup for muffins.

Understand — I need to find: _______________________________________________

Estimate: ______

Step 1 (total used): ______

Step 2 (amount left): ______

Justify: _______________________________________________

Your turn.

A pitcher has $2\tfrac{1}{2}$ L. Someone pours out $\tfrac{3}{4}$ L twice. Left: ______
Sam reads $1\tfrac{1}{3}$ hr Monday, $\tfrac{3}{4}$ hr Tuesday, $1\tfrac{1}{2}$ hr Wednesday. Total: ______
Find $\tfrac{3}{4}$ of $5\tfrac{1}{3}$ , then subtract $1\tfrac{1}{4}$ . Result: ______

PAGE 12 — Error hunt

Find and Fix the Errors

Three of these are FALSE. Mark each and rewrite the false ones correctly.

Statement	True or false?	Correction
$\tfrac{1}{2} + \tfrac{1}{4} = \tfrac{2}{6}$
$3\tfrac{1}{2} - 1\tfrac{3}{4} = 2\tfrac{1}{4}$
$\tfrac{2}{3} \times \tfrac{3}{4} = \tfrac{1}{2}$
$5 \div \tfrac{1}{2} = 2\tfrac{1}{2}$
$2\tfrac{1}{4} + 1\tfrac{1}{2} = 3\tfrac{3}{4}$

Apply it. Kai has $4\tfrac{1}{2}$ hours. He spends $1\tfrac{1}{4}$ hr on homework and $\tfrac{2}{3}$ hr at practice, then splits the rest evenly between 2 chores.

Time used: ______ Time left: ______ Each chore: ______

Explain. Rosa says $\tfrac{3}{4} \times 8\tfrac{2}{3} = 26$ . Why is that unreasonable, and what is the correct product?

PAGE 13 — Exit ticket 6.4

Exit Ticket · Lesson 6.4

Name: ________________________ Date: ____________

$6\tfrac{1}{2} - 2\tfrac{1}{3} - 1\tfrac{1}{2} =$ ______
A $4\tfrac{1}{2}$ -lb melon is cut into 6 equal pieces. Two pieces weigh: ______
Estimate $5\tfrac{7}{8} + 2\tfrac{1}{12}$ : ______ Exact sum: ______
What does it mean to justify an answer?

PAGE 14 — Chapter 6 review, part 1

Chapter 6 Review

Part A · Estimating and reasonableness

Round to $0$ , $\tfrac{1}{2}$ , or $1$ : a) $\tfrac{1}{10}$ ______ b) $\tfrac{7}{12}$ ______ c) $\tfrac{9}{10}$ ______
Estimate $8\tfrac{1}{8} - 2\tfrac{7}{8}$ . ______
Estimate $\tfrac{1}{4} \times 19\tfrac{3}{4}$ . ______
Is $\tfrac{3}{5} + \tfrac{5}{6} = \tfrac{8}{11}$ reasonable? ______ Correct sum: ______

Part B · Addition and subtraction in context

$\tfrac{5}{8} + \tfrac{1}{6} =$ ______
$7\tfrac{1}{4} - 3\tfrac{2}{3} =$ ______
Dogs weigh $12\tfrac{3}{4}$ lb and $9\tfrac{1}{2}$ lb. Combined: ______ Difference: ______
Books are $2\tfrac{5}{6}$ in. and $1\tfrac{3}{4}$ in. wide. Shelf width used: ______

PAGE 15 — Chapter 6 review, part 2

Chapter 6 Review (continued)

Part C · Multiplication and division in context

$\tfrac{4}{9} \times \tfrac{3}{8} =$ ______
$5\tfrac{1}{4} \div \tfrac{3}{4} =$ ______
A $2\tfrac{1}{2}$ -hour rehearsal spends $\tfrac{3}{5}$ of the time on one song. Time on that song: ______
Number of $\tfrac{3}{4}$ -ft bows from $10\tfrac{1}{2}$ ft of ribbon: ______

Part D · Multistep problems and justification

$9\tfrac{1}{2} - \left(1\tfrac{1}{3} + 2\tfrac{5}{6}\right) =$ ______
$\tfrac{2}{3}$ of $6\tfrac{3}{4}$ , then subtract $1\tfrac{1}{2}$ : ______
Which is greater, $\tfrac{3}{4}$ of 8 or $\tfrac{2}{3}$ of 9? ______

Justify: _______________________________________________
A student writes $5\tfrac{1}{4} - 2\tfrac{3}{4} = 3\tfrac{1}{2}$ . Estimate: ______ Correct difference: ______

Why the estimate shows the error: _______________________________________________

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
Fraction rendering: set mixed numbers with a full-height numeral and a stacked fraction; do not use slashed inline fractions in headings
Figure widths: fig1-benchmark-fractions.png and fig5-measurement-division.png full width; fig3-regrouping-mixed-number.png and fig4-area-model-multiplication.png half width
Answer blanks: keep every blank on the same line as its prompt so the Canva text box does not reflow
Work space: pages 6, 9, and 11 need a 1.5 in blank band under each problem set for shown work