Solving Multi-Step Problems with Rational Numbers

Grade 7 | 7.EE.B.3

In this lesson:

• Plan before you compute — read the problem structure first
• Choose the right number form for cleaner arithmetic
• Use properties of operations as shortcuts

What You Will Be Able to Do

By the end of this lesson:

1. Identify operations and sequence of steps in multi-step problems
2. Select the best number form for each step
3. Convert between forms strategically to simplify computation
4. Apply properties of operations to calculate efficiently
5. Use estimation to assess whether an answer is reasonable
6. Communicate a solution strategy, explaining each step's purpose

What Steps Do We Need?

Maria earns $12.50 per hour. After 6.5 hours, she buys lunch for $8.75.

How much money does she have left?

Before you calculate anything — what do we need to find? What information do we have?

The Plan → Compute → Check Framework

Before every problem, write the plan:

• Read: What is the question? What do we know?
• Plan: List the steps — no arithmetic yet
• Compute: Execute each step, tracking signs
• Check: Does the answer make sense?

The plan is the hard part. Arithmetic is execution.

Submarine Problem: Plan Then Compute

Problem: Submarine at −340 ft rises 25 ft/min for 8 min, then descends 60 ft. Final depth?

Plan: (1) find rise: → (2) add to depth → (3) subtract descent

Compute:

Inventory Problem: Plan Then Compute

Problem: A store has 144 items. is sold; 27 new items arrive. How many remain?

Plan: (1) find items sold: → (2) subtract → (3) add new stock

Compute:

Quick Check: Write the Plan

Problem: A car travels 2 hours at 45 mph, then 1.5 hours at 60 mph.

What is the total distance traveled?

Your task: Write the plan — list the steps in words with no arithmetic.

Think about it, then advance to check your plan.

Check: Car Problem Plan and Solution

Plan: (1) → (2) → (3) add both distances

Key: distance = speed × time for each leg.

From Planning to Form Choice

You now have a plan. The next question is: which form of each number makes the arithmetic simplest?

The same value can be written as:

• A fraction:
• A decimal:
• A mixed number:

Choosing the right form isn't luck — it's strategy.

Reference: Choosing the Right Number Form

Minimize conversions — choose the form the problem already uses

Worked Example: Towel Bar Problem

Problem: A 9¾-inch bar is centered on a 27½-inch door. Distance from each edge?

Fraction path:

Decimal check:

Hiker Problem: Converting to Match Dominant Form

Problem: Hiker walks 2.4 km, km, then 1.8 km. Total?

Two numbers are decimals → convert to 0.75.

Your Turn: Simplify This Mixed Expression

Simplify exactly:

Step 1: Which form should you use? (Hint: 1.5 = )

Step 2: Convert all to that form and simplify.

Why would converting to a decimal cause a problem?

Quick Check: Decimals and Precision

Why should we keep as a fraction rather than converting to a decimal?

Consider: what is ?

Rule: Keep repeating decimals (, , ...) as fractions when an exact answer is required.

Rearranging Computation Before You Calculate

You know how to plan and choose forms.

Now: can you rearrange to make the arithmetic easier?

• Commutative: change the order
• Associative: change the grouping
• Distributive: factor out or expand

Look for a shortcut before you compute.

Reference: Three Properties as Computation Tools

• Commutative: — reorder to find friendly pairs
• Associative: — regroup for easier arithmetic
• Distributive: — factor out shared multipliers

Worked Example: Field Trip Cost

Problem: 28 students each owe $3.25 for the trip plus $0.75 for supplies. Total?

Without properties:

With distributive property:

Same answer — but one path is dramatically faster.

Applying Two Different Properties as Shortcuts

Distributive:

(Rewrite 4.8 as to create a friendly factor)

Commutative + Associative:

(Regroup the two sixths together — they cancel cleanly)

Your Turn: Apply a Property

Simplify:

Step 1: What is inside the parentheses?

Step 2: Multiply.

Which property makes this easy? Name it.

Quick Check: Name the Property

Which property is used here, and why?

Why rearrange the terms before computing?

Estimation as a Required Reasonableness Check

You've planned, chosen forms, and applied properties.

Final step: Is your answer reasonable?

• Round to the nearest whole number
• Use benchmark fractions: , ,
• Use compatible numbers that divide evenly

A good estimate catches major errors — not optional.

Three Estimation Strategies with Examples

Maria Problem: Estimate Then Verify

Problem: $12.50/hr × 6.5 hours, minus $8.75 lunch. Money remaining?

Estimate: $13 × 6 = $78; minus $9 ≈ $69

Exact:

$72.50 is close to $69 — ✓ reasonable

Car Speed: Estimation Catches a Factor-of-10 Error

Problem: Car travels 215 miles in 3.5 hours. Average speed?

Estimate: mph (compatible numbers)

Student A: 61.4 mph. Student B: 6.14 mph.

Student B is wrong — 6.14 is off by a factor of 10 from our estimate.

Jacket Discount: Use Estimation to Check Answers

Problem: $180 jacket discounted 35%. Sale price?

Estimate: 35% of $200 = $70 → sale price ≈ $130

Student A says $117. Student B says $63.

Check: $117 is near $130 ✓. $63 implies ~65% off — wrong.

Your Turn: Estimate Before Computing

Problem: A hiker carries a 4.8 kg pack for 6.5 hours. Estimate the "effort score" = weight × hours.

Step 1: Estimate using round numbers.

Step 2: Compute exactly.

Step 3: Compare — is the exact answer in the expected range?

Write your estimate before computing.

Quick Check: What Is Good Estimation?

Complete the sentence:

"Estimation is _____ (disciplined approximation / random guessing)."

Name one strategy and describe when you'd use it.

Four Habits of Multi-Step Problem Solvers

✓ Plan first — list steps before computing

✓ Choose forms — minimize conversions

✓ Apply properties — look for a shortcut

✓ Estimate — ballpark before accepting an answer

Watch Out: Four Common Calculation Errors

"Total" ≠ always add — read the structure

Never drop a negative sign mid-calculation

Keep , as fractions for exact answers

Distribute to every term:

Preview: Writing and Solving Equations

Next lesson: 7.EE.B.4

• Planning a problem structure → writing an equation for it
• The form-choice and property skills from today apply directly
• Rational number fluency is the arithmetic engine for every equation

Multi-step thinking today is the foundation for algebra.