Grade 7 Workbook — Chapter 4: Operations with Rational Numbers in Context

SOL 7.CE.1 · Companion to Textbook Chapter 4

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/. Every numbered item is the same problem as in the textbook, so one answer key serves both.

PAGE 1 — Chapter opener

Chapter 4 · Operations with Rational Numbers in Context

Standard 7.CE.1

In this chapter you will:

Add and subtract positive and negative fractions, mixed numbers, and decimals in real situations
Multiply and divide those same numbers to solve rate, scaling, and sharing problems
Estimate before you compute, and use the estimate to judge whether an answer is reasonable
Solve multistep problems that need more than one operation
Justify a solution by explaining your estimate, your operations, and what the answer means

Words to know: rational number · proper fraction · improper fraction · mixed number · thousandths place · common denominator · reciprocal · estimate · bracket · compatible numbers · multistep problem · justify

Calculator note: a calculator is allowed in this chapter. It will faithfully compute a mistyped number, so every page asks you to estimate first.

PAGE 2 — Signed changes on a number line

4.1 Adding and Subtracting Rational Numbers

FIGURE: fig1-temperature-drop.png (full width)

Fill in the blanks.

Adding a positive number moves you to the ________ on a number line.

Adding a negative number moves you to the ________.

To subtract, add the ______________: $a - b = a + ($ ______ $)$ .

Decimals in this chapter go no further than the ______________ place.

Write each situation as a signed number.

Situation	Signed number
A deposit of $45.50
A temperature fall of $12.25$ degrees
Descending $6.25$ ft underwater
A withdrawal of $18.60
Rising $9.75$ ft toward the surface

PAGE 3 — Fractions need common denominators

Common Denominators in Context

FIGURE: fig5-fraction-difference.png (full width)

$3\tfrac{1}{4} = \frac{\ \ \ \ }{8} \qquad 1\tfrac{5}{8} = \frac{\ \ \ \ }{8} \qquad \text{difference} = \frac{\ \ \ \ }{8} = \underline{\qquad}$

Guided practice

Find $-7.4 + 12.9$ . Sign of the answer: ________ Size: ________ Answer: ________
Find $2\tfrac{1}{4} + \left(-5\tfrac{3}{4}\right)$ . ________
Rewrite $6.2 - (-3.85)$ as addition: _______________ Answer: ________
At 5 p.m. the temperature is $-2.5$ °C. It drops $4.75$ °C overnight. Morning temperature: ________
Devon's balance is $34.20. He writes a check for $52.75. New balance: ________

What does the sign of your answer mean?

PAGE 4 — Independent practice 4.1

Practice · Adding and Subtracting

FIGURE: fig2-elevation-dive.png (right third of page)

Evaluate.

a) $-9.6 + 4.25$	b) $-3\tfrac{2}{3} + 1\tfrac{1}{4}$
c) $7.5 - 11.375$	d) $-\tfrac{5}{6} - \left(-\tfrac{1}{3}\right)$

A hiker starts at $1{,}240.5$ ft, descends $386.75$ ft, then climbs $152.25$ ft. Final elevation: ________
An account is at $-\$42.15$ . A $120.00 deposit arrives; charges of $18.60 and $35.75 post. Final balance: ________
A jug holds $7\tfrac{1}{2}$ qt of cider. Two pitchers take $2\tfrac{3}{4}$ qt and $1\tfrac{5}{8}$ qt. Remaining: ________
Daily price changes: $-1.25$ , $+0.875$ , $-2.5$ , $+3.125$ . Total change: ________ Higher or lower? ________
Application. At 6 a.m. it is $-6.5$ °F. It rises $13.25$ degrees by noon, then falls $8.75$ degrees by 9 p.m.

Noon: ________ 9 p.m.: ________
Reasoning. Jordan writes $-4.6 - 2.8 = -1.8$ . Correct answer: ________

What did Jordan misread?

PAGE 5 — Exit ticket 4.1

Exit Ticket · Lesson 4.1

Name: ________________________ Date: ____________

Find $-8.25 + 3.5$ . ________
Find $1\tfrac{5}{6} - 4\tfrac{1}{3}$ . ________
A balance of $-\$16.40$ receives a $25.00 deposit. New balance: ________
Using a situation of your choice, explain why subtracting a negative number makes a result larger.

PAGE 6 — Sign rules

4.2 Multiplying and Dividing Rational Numbers

Complete the sign table.

Factors	Sign of product or quotient
positive $\times$ positive
positive $\times$ negative
negative $\times$ positive
negative $\times$ negative

Continue the pattern to see why two negatives give a positive.

$(-4.5)(2) = \underline{\qquad} \quad (-4.5)(1) = \underline{\qquad} \quad (-4.5)(0) = \underline{\qquad} \quad (-4.5)(-1) = \underline{\qquad}$

Each time the second factor drops by 1, the product goes ________ by $4.5$ .

Two kinds of division. Label each problem sharing or measuring.

Problem	Sharing or measuring?
A $73.50 charge split among 5 people
A $12\tfrac{1}{2}$ -ft ribbon cut into $\tfrac{5}{8}$ -ft pieces
$6\tfrac{3}{8}$ cups poured into $\tfrac{3}{4}$ -cup bowls
$146.20 divided among 4 friends

PAGE 7 — Guided practice 4.2

Guided Practice · Multiplying and Dividing

To divide by a fraction, multiply by its reciprocal.

$\frac{51}{8} \div \frac{3}{4} = \frac{51}{8} \times \frac{\ \ \ \ }{\ \ \ \ } = \frac{\ \ \ \ }{\ \ \ \ } = \underline{\qquad}$

$(-7)(2.5)$ Sign: ________ Answer: ________
$\left(-\tfrac{3}{5}\right)\left(-\tfrac{10}{9}\right)$ in simplest form: ________
$-20.4 \div 6 =$ ________
$5\tfrac{1}{4} \div \left(-1\tfrac{1}{2}\right)$

As improper fractions: ______ $\div$ ______ Answer: ________
A well drops $0.375$ ft each day. Change after 8 days: ________

PAGE 8 — Independent practice 4.2

Practice · Multiplying and Dividing

Evaluate.

a) $(-1.2)(-4.5)$	b) $\left(-\tfrac{7}{8}\right)\left(\tfrac{2}{3}\right)$
c) $-15.75 \div (-2.5)$	d) $\left(-2\tfrac{2}{3}\right) \div \tfrac{4}{5}$

Four friends split a $146.20 bill equally. Each pays: ________
A recipe uses $2\tfrac{1}{3}$ cups of flour per batch. Flour for $2\tfrac{1}{4}$ batches: ________
A scuba tank loses $8.75$ psi each minute. Change after $7.2$ minutes: ________
A submarine descends $12.5$ ft per minute for $6.4$ minutes. Change in elevation: ________
Application. $10\tfrac{1}{2}$ cups of chili served in $1\tfrac{1}{2}$ -cup portions. Portions: ________

How do you know none is left over?
Reasoning. Why is the quotient of two negatives positive? Use a real situation, not just the rule.

PAGE 9 — Exit ticket 4.2

Exit Ticket · Lesson 4.2

Name: ________________________ Date: ____________

$(-6)(-3.5) =$ ________
$\left(-\tfrac{3}{4}\right)\left(\tfrac{8}{9}\right) =$ ________
$-22.5 \div 4.5 =$ ________
A quotient contains exactly one negative number. What is its sign, and how do you know?

PAGE 10 — Three ways to estimate

4.3 Estimating to Judge Reasonableness

FIGURE: fig3-estimation-bracket.png (full width)

Complete the three strategies for $6.85 \times 2.40$ .

Strategy	Work	Estimate
Rounding	$7 \times 2.40$
Low estimate (round both down)	$6.5 \times 2$
High estimate (round both up)	$7 \times 2.5$

Exact product: ________ Does it land inside the bracket? ________

Two errors a calculator cannot catch:

a misplaced ______________ ______________
a dropped ______________ ______________

Warning. Dividing by a number less than 1 makes the result ______________.

PAGE 11 — Guided practice 4.3

Guided Practice · Estimating

$8.9 \times 3.1$ Estimate: ________ Exact: ________
$-19.6 \div 4.9$ Compatible numbers: ______ $\div$ ______ Estimate: ______ Exact: ______
$5\tfrac{1}{8} + 2\tfrac{7}{8}$ Estimate: ________ Exact: ________
A student reports $6.4 \times 0.52 = 33.28$ .

Estimate: ________ Reasonable? ________ Correct answer: ________
$3.7 \times 4.6$ Low estimate: ______ High estimate: ______ Exact: ______

Inside the bracket? ________

PAGE 12 — Independent practice 4.3

Practice · Estimating and Judging

Estimate, then find the exact value.

	Estimate	Exact
a) $12.3 \times 4.8$
b) $-44.8 \div 5.6$
c) $9\tfrac{3}{4} + 4\tfrac{1}{8}$
d) $7\tfrac{4}{5} \times 2$

$8.3 \times 6.7$ Low: ______ High: ______ Exact: ______
Reasonable or not? Give a one-line estimate and correct any that are wrong.

Reported answer	Estimate	Reasonable?	Correction
a) $4.9 \times 5.1 = 24.99$
b) $63.9 \div 7.1 = 90$
c) $-3.2 \times 6.5 = 20.8$
d) $19.6 + (-4.85) = 14.75$

Application. Gas costs $3.199 per gallon. Estimate the cost of $11.4$ gallons: ________

Exact, rounded to the nearest cent: ________
Application. Five friends split $97.35. Estimate: ________ Exact: ________
Application. Four tickets at $13.75 plus $8.50 parking. Is $60 enough? ________ Exact total: ________
Reasoning. Why does rounding both factors up always overshoot a product of two positives, and why is a low-and-high pair more useful than one estimate?

PAGE 13 — Exit ticket 4.3

Exit Ticket · Lesson 4.3

Name: ________________________ Date: ____________

$7.8 \times 4.2$ Estimate: ________ Exact: ________
Is $-56.4 \div 7.05 = 8$ reasonable? ________ Correct answer: ________
$6.4 \times 3.7$ Low: ______ High: ______ Exact: ______
How can an estimate catch a misplaced decimal point that a calculator will not?

PAGE 14 — The four-move plan

4.4 Multistep Problems in Context

FIGURE: fig4-bar-model-split.png (full width)

The plan.

Understand — what is asked, and in what ______________?
Estimate — a rough answer with friendly numbers.
Solve — steps in order, each labeled.
Check — compare to the estimate and work ______________.

Order matters. For a $78.60 bill with a $12.60 coupon split 3 ways:

Subtract first: $78.60 - 12.60 =$ ________ then divide: $\div\ 3 =$ ________

Why can't you divide first?

Rate over time.

$\text{final value} = \text{starting value} + (\text{rate} \times \text{time})$

$4.5 + (-1.75)(6) =$ ________

PAGE 15 — Guided practice 4.4

Guided Practice · Multistep Problems

A $44.80 bill less a $4.80 discount, split 4 ways.

After discount: ________ Each pays: ________
Temperature $-3.5$ °C, rising $2.25$ °C per hour for 4 hours.

Total change: ________ Final temperature: ________
Cheese at $6.40 per pound. Cost of $2\tfrac{3}{4}$ pounds: ________
A 20-gallon tank losing $1.375$ gal per hour for 6 hours.

Water lost: ________ Water remaining: ________
A $12\tfrac{1}{2}$ -ft ribbon cut into $\tfrac{5}{8}$ -ft pieces. Number of pieces: ________

PAGE 16 — Independent practice 4.4

Practice · Multistep Problems

$24.99 per month plus $0.045 per extra minute, with 60 extra minutes.

Extra-minute charge: ________ Total bill: ________
A hiker at $2{,}150.5$ ft descends $128.25$ ft per hour for 6 hours.

Total descent: ________ Final elevation: ________
Three rooms at $1\tfrac{3}{4}$ gal each; you have $4\tfrac{1}{2}$ gal.

Paint needed: ________ More paint required: ________
Balance $-\$28.35$ ; a $150.00 deposit; four withdrawals of $22.50.

Withdrawals total: ________ Final balance: ________
Twenty shares bought at $18.75; the price falls $0.625 per day for 6 days.

Total price drop: ________ New price per share: ________ Total loss on 20 shares: ________
Application. 27 students; $8.75 admission each; $148.50 bus shared equally.

Admission total: ________ Grand total: ________ Each student pays: ________
Reasoning. For item 49, a student divided by 4 first and then subtracted $4.80 from each share.

That student's answer: ________ Why is the order wrong?

PAGE 17 — Exit ticket 4.4

Exit Ticket · Lesson 4.4

Name: ________________________ Date: ____________

A $56.40 bill less a $6.40 gift card, split 5 ways. Each pays: ________
Temperature $2.5$ °F falling $1.25$ °F per hour for 6 hours. Final: ________
$8\tfrac{2}{3}$ cups of soup in $\tfrac{2}{3}$ -cup bowls. Bowls filled: ________
Describe the check you would use to confirm item 62 is reasonable.

PAGE 18 — The three parts of a justification

4.5 Justifying a Solution

A complete justification has three parts.

An ______________ showing roughly where the answer had to land, including its ______________.
The ______________ you chose, and why each matches the situation.
An answer ______________ with units, plus a ______________ when one is available.

Model it. Three friends split a $47.25 bill after a $5.25 coupon.

Part	Your writing
Estimate
Operations and why
Computation
Answer sentence
Check (work backward)

Error analysis is justification in reverse. A student evaluates $-12.6 + 4.2 \times 3$ as $-25.2$ .

What did the student do first? _______________________

Correct value: ________

PAGE 19 — Guided practice 4.5

Guided Practice · Justifying

Write a one-sentence estimate justifying $9.8 \times 5.2 = 50.96$ .
A $6\tfrac{1}{4}$ -lb bag of rice repackaged into $\tfrac{5}{8}$ -lb bags.

Operation: ______________ Why: _______________________

Number of bags: ________
A student writes $-5.6 - (-2.4) = -8.0$ . Error: _______________________

Correct answer: ________
Temperature $1.5$ °F falling $2.5$ °F per hour for 3 hours.

Why must the answer be negative? _______________________

Answer: ________
Five friends split $81.25 evenly.

Estimate: ________ Operation and reason: _______________________

Answer sentence: _______________________

PAGE 20 — Independent practice 4.5

Practice · Justifying and Error Analysis

A student writes $\tfrac{2}{3} \times \left(-\tfrac{3}{4}\right) = -\tfrac{1}{12}$ .

Error: _______________________ Correct product: ________
A student reports $18.6 \div 0.3 = 0.62$ .

Estimate showing it is wrong: ________ Correct quotient: ________
Seven friends split a $103.60 bill after a $5.25 coupon. Write a complete justification.

Part	Your writing
Estimate
Operations and why
Computation
Answer sentence
Check

Application. A pool loses $0.375$ in of water per day. Change after 12 days: ________

Justify the sign and the size: _______________________
Application. $1\tfrac{1}{3}$ cups of sugar for 8 servings. Sugar for 20 servings: ________

Scaling factor and why: _______________________
Reasoning. Two students answer item 72 with $14.05 and $14.50. Which is right, and how does estimation tell you without redoing the division?
Reasoning. Why is "I checked by working backward" stronger than "I used a calculator"?

PAGE 21 — Exit ticket 4.5

Exit Ticket · Lesson 4.5

Name: ________________________ Date: ____________

Write an estimate justifying $-6.1 \times 3.9 = -23.79$ , covering both size and sign.

A student writes $3\tfrac{1}{2} \div \tfrac{1}{2} = 1\tfrac{3}{4}$ . Error: _______________ Correct: ________
Justify why four people splitting $63.80 each pay $15.95. Include a check.

Name the three parts of a complete justification.

PAGE 22 — Chapter 4 review, part 1

Chapter 4 Review

Part A · Adding and subtracting

Evaluate.

a) $-14.75 + 6.5$	b) $-2\tfrac{3}{8} - 1\tfrac{5}{6}$
c) $9.6 - (-3.45)$	d) $5\tfrac{1}{4} + \left(-7\tfrac{1}{2}\right)$

Balance $-\$52.80$ ; $140.00 deposit; $36.45 withdrawal. Final balance: ________
Temperature $1.5$ °F falls $9.25$ degrees, then rises $3.75$ degrees. Final: ________

Part B · Multiplying and dividing

Evaluate.

a) $(-8)(2.75)$	b) $\left(-\tfrac{3}{5}\right)\left(-\tfrac{5}{6}\right)$
c) $-33.6 \div (-4.2)$	d) $6\tfrac{2}{3} \div \left(-1\tfrac{1}{3}\right)$

A $9\tfrac{3}{4}$ -ft rope cut into $\tfrac{3}{4}$ -ft pieces. Pieces: ________
A tank drains $2.125$ gal per minute for 8 minutes. Change in volume: ________

PAGE 23 — Chapter 4 review, part 2

Chapter 4 Review (continued)

Part C · Estimating

Estimate, then find the exact value.

	Estimate	Exact
a) $11.8 \times 5.1$
b) $-71.4 \div 8.4$
c) $6\tfrac{7}{8} + 3\tfrac{1}{8}$

$5.6 \times 7.3$ Low: ______ High: ______ Exact: ______ Inside? ______
Reasonable or not? Correct any that are wrong.

Reported answer	Estimate	Reasonable?	Correction
a) $4.05 \times 19.8 = 80.19$
b) $12.6 \div 0.3 = 4.2$
c) $-7.5 + 2.25 = 9.75$

Part D · Multistep problems

$96.40 catering less a $14.40 coupon, split 4 ways. Each pays: ________
24 students; $6.25 admission; $114.00 bus shared. Each pays: ________
$3\tfrac{1}{2}$ cups of stock for 6 servings. Stock for 15 servings: ________
A drone at $96.5$ ft descends $14.25$ ft per second for 6 seconds, then rises $8.5$ ft. Final height: ________

PAGE 24 — Chapter 4 review, part 3

Chapter 4 Review (continued)

Part E · Reasoning and justification

A student evaluates $-18.4 + 6.2 \times 2$ as $-24.4$ .

Error: _______________________ Correct value: ________
Six friends split a $77.70 bill evenly. Write a complete justification.

Part	Your writing
Estimate
Operation and why
Computation
Answer sentence
Check

Why is estimating first worth the time even when a calculator is allowed? Name one specific error an estimate catches and a calculator does not.

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
Figure widths: fig1-temperature-drop.png, fig3-estimation-bracket.png, fig4-bar-model-split.png, and fig5-fraction-difference.png are landscape and run full width; fig2-elevation-dive.png is portrait and sits in the right third of page 4
Money: every dollar amount is typeset with an escaped dollar sign so it renders as currency, not math
Multistep pages: pages 15, 16, 20, and 24 give one blank per intermediate step on purpose — students should show the running total, not only the final number
Answer blanks: keep every blank on the same line as its prompt so the Canva text box does not reflow
Justification tables: the five-row tables on pages 18, 20, and 24 need at least 0.6 in of row height for handwriting