Solving Real-World Problems: All Four Operations

In this lesson:

• Select and justify the right operation from context
• Apply all four operations to integers, fractions, and decimals
• Solve multistep problems and validate answers

Learning Objectives for This Lesson

By the end of this lesson, you should be able to:

1. Select the right operation for real-world problems
2. Apply all four operations, including with negative values
3. Solve multistep problems by chaining operations
4. Interpret and validate answers in context
5. Convert between fractions, decimals, and mixed numbers

Same Numbers, Four Different Situations

Context Clues Identify the Operation

Keywords signal which operation to use:

• Addition: combining, total, net gain, rises
• Subtraction: difference, change, how far apart
• Multiplication: rate × time, per unit for a count
• Division: shared equally, how many groups

Identify the clue and name the operation before computing

Using a Four-Step Problem-Solving Structure

For every real-world problem, follow these steps:

1. Identify — name the operation and justify it in words
2. Write — write the number sentence (with signs)
3. Compute — solve with correct sign rules
4. Interpret — state what the answer means in context

Worked Example: Scuba Diver's New Depth

• Identify: She dives additional depth → add the two depths (both negative)
• Write:
• Compute:
• Interpret: She is 25.5 meters below the surface ✓

Temperature Change Always Uses Subtraction

Problem: Temperature changes from −3.5°F to 11°F. What is the change?

Identify: "Change" = new − old → subtraction

Interpret: A 14.5°F increase ✓

Reasonableness: crossed zero, so the change exceeds 11° — checks out.

Elevator Descent: Rate Times Time

Problem: An elevator descends floors/min. How far in min?

Identify: rate × time → multiplication

Interpret: Descended 3 floors below starting position ✓

Convert first:

Dividing a Loss Among Investors

Problem: A loss of 3/4 dollar is shared equally among 3 investors.

Identify: shared equally → division

Interpret: Each investor loses 1/4 dollar ✓

Sharing a loss gives each person a negative share

Your Turn: Two Practice Problems

Identify the operation before computing:

1. A hiker's elevation changes from 1,250 m to 840 m. What is the change?

2. Temperature drops 2.5°C per hour for 3 hours. Total change?

For each: identify → write → compute → interpret

Answers: Elevation Change and Temperature Drop

Problem 1 — change = new − old:

Descended 410 m ✓

Problem 2 — rate × time:

Temperature fell 7.5°C total ✓

Choosing Fractions or Decimals Strategically

Choose the form that makes computation cleaner:

• Fractions — better when cancellation is possible
• Decimals — better for addition/subtraction with messy denominators
• Always state your form choice explicitly

Both 1.875 and are correct — match the context

Chaining Steps in Multistep Problems

The answer to one step feeds the next:

• Label each step (Step 1, Step 2, ...)
• Write intermediate results with signs before continuing
• Check reasonableness at each step, not just the end

A sign error in Step 1 cascades through every step after

Credit Card Balance: Plan the Steps First

Balance: −245. Payment: 120. Purchases: 67.50. Final balance?

Identify before computing:

• Step 1: payment reduces debt → add positive
• Step 2: purchases increase debt → add negative

Both steps use addition — signs differ

Credit Card Balance: Two-Step Solution

Step 1 — After the 120-dollar payment:

Reduced the debt ✓

Step 2 — After 67.50 in purchases:

More negative than after payment, less than the start ✓

Final: Balance is −192.50 (overdrawn by 192.50)

Boat Position: Plan Before Computing

Boat: −8 km (south). North 3/2 km/h × 6 h, then south 2 km/h × 3/4 h.

Steps:

• Step 1: distance north → add to position
• Step 2: distance south → subtract from position

North = positive; south = negative

Boat Position: Number Line Journey

All three arrows converge on the final position: −0.5 km (south of port)

Calculating the Boat's Final Position

Step 1 — North leg:

Step 2 — South leg:

Final: 0.5 km south of port ✓

Dropping Signs Causes Multistep Errors

Fix: Box the sign; copy result with sign to start Step 2.

Your Turn: Temperature Rise Then Fall

Problem: Store temperature: −4°C. Rises 3/2°C/h for 6 h. Then falls 1.5°C/h for 4 h. Final temperature?

Step 1 (given):

Your turn — Step 2:

Temperature Practice: Answer and Reasoning

Step 2:

Final: −1°C ✓

Reasonableness: Started at 5°C, cooled 6° → below zero ✓

Double-check: net change from −4°C is +3°, so −4 + 3 = −1 ✓

Validate Every Answer with Four Questions

Before writing a final answer, check:

1. Estimate — is the size roughly expected?
2. Sign — does positive/negative make sense in context?
3. Units — are the units correct?
4. Scale — is the value plausible? (−500°F is absurd)

Validation is part of the solution, not optional polish

A Computation Is Right, Interpretation Wrong

Problem: Submarine goes from −150 m to −85 m. How far did it ascend?

✓ Correct: Ascended 65 m (−85 is less negative than −150 → rose toward surface)

Wrong: "Descended 65 m" — positive change means upward, not downward

Recipe Scaling with Mixed Number Forms

Problem: Recipe uses cups vinegar/batch. For 2/3 of a batch?

Identify: fraction of a batch → multiplication

Convert mixed number:

Interpret: 5/6 cup removed ✓

Compound Finance Problem with Two Steps

Problem: Account starts at −180 dollars. Three 45-dollar deposits, then split 4 ways.

Step 1:

Step 2:

Interpret: Each person's share is −11.25 (a loss) ✓

Finding the Average Temperature over Five Days

• Sum:
• Average: ✓

Strategy Checklist for Complex Problems

Plan before computing:

1. What operations do I need?
2. In what order?
3. Fraction or decimal form?
4. Does the answer make sense?

Your Turn: Submarine Depth Problem

Submarine at −90 m. Rises 3/4 m/min for 20 min, then descends 1.2 m/min for 15 min.

Plan first:

• Step 1: distance up (rate × time) → add
• Step 2: distance down → subtract

Solve, then advance for the answer

Complete Solution to Submarine Depth Problem

Step 1 — Up:

Step 2 — Down:

Final depth: −93 m (net −3 m from start) ✓

Key Takeaways and Misconception Warnings

✓ Read context first — context decides the operation
✓ Four steps: identify → write → compute → interpret
✓ Box intermediate signs; check reasonableness at each step

Read before computing — don't jump to arithmetic
Positive change in depth means ascending, not descending

Preview of What Comes Next

In 7.EE.B.3, you'll solve the same real-world problem types with an algebraic approach:

• Same four-step structure and sign rules
• New tool: variables and equations to model the situation
• Emphasis on estimation and reasonableness by substitution

Fluency from 7.NS.A.3 is the foundation for all of 7.EE