019i - Problem Solving with the Four Operations

Lesson 19 — Problem Solving: Multi-Step Problems with the Four Operations

Strand: Number | Descriptor: AC9M8N04 | Duration: 45 minutes

Learning Intentions

• To solve multi-step applied problems involving integers and rational numbers, choosing efficient strategies.
• To translate a worded, real-world situation into a single correctly ordered mathematical expression.

Success Criteria

I can:

1. Identify the sequence of operations described in a worded problem and represent it as one expression.
2. Choose efficient mental, written or digital strategies appropriate to the numbers involved.
3. Solve a multi-step problem and justify each step of my reasoning.
4. Check a solution against the context to see whether it is reasonable.

Warmup

(6 minutes — “Always, sometimes, never”, pairs)

Decide whether each statement is always, sometimes, or never true. Give a supporting example or counterexample.

1. In a worded problem, the order the numbers are mentioned matches the order they should be calculated.
2. A negative answer to a real-world problem means a mistake has been made.
3. Estimating before solving a multi-step problem helps catch errors.
4. A multi-step problem can be written as a single expression using brackets.

Answers: 1. Sometimes — often true, but not when a later action (e.g. “then double the total”) must be grouped with brackets around earlier steps. 2. Sometimes — a negative answer can be entirely correct in context (e.g. a depth below sea level, a financial loss), but can also flag an error; context decides. 3. Always — a sensible habit for catching arithmetic and order-of-operations slips. 4. Always — this is the goal of today’s lesson.

Activities

Activity 1 — Guided Practice: Translating Words into Expressions (12 min)

Explicit reminder, then practice.

Remind students of the routine from Lesson 17: identify each action in order, decide what must be grouped in brackets, then evaluate.

I do: “A account balance starts at 45$60$ is added, then the balance is halved due to a shared account split.”

Final balance: 7.50$ per person.

You do: Write a single expression for each, then evaluate:

1. “Start at . Subtract , then multiply the result by .”
2. “A temperature of rises by every hour for hours, then drops suddenly by .”
3. “A number is increased by of itself, then the result is divided by .” (Use to test.)

Activity 2 — Applied Problem-solving (19 min)

Pairs. Write the full expression before evaluating. Every answer must include a sentence checking reasonableness.

Problem 1. A hiker starts at an elevation of m. She descends into a valley, losing m, then climbs back up times that loss. What is her final elevation?

Problem 2. A company’s profit changes by 1,250-$8603$ times the total change of Jan + Feb). What is the total profit change over the three months?

Problem 3. A water tank starts full. Each day it loses of a full tank to evaporation, and every third day it is topped up with of a full tank. After days (including one top-up), what fraction of the tank is full?

Problem 4. A submarine at depth m rises at m per minute. After minutes it must dive again, descending at twice the rate for minutes. Find its final depth, and determine whether it is now above or below its starting depth.

Socratic scaffolding for Problem 4:

Prompt	Purpose
Understand: what is being asked?	The final depth, and a comparison to the starting depth of m — two things to report.
Devise a plan: list the events in order	Start at ; rise for 8 min at 6 m/min; then descend for 5 min at double the rate.
What is the rate for the second phase?	Twice m/min m/min.
Write a single expression	.
Carry out the plan	.
Looking back — is the final depth above or below the start?	m is lower (deeper) than m, since — despite rising first, the second, faster descent outweighed the rise.
Looking back — does the size of the answer make sense?	Net change: m overall, consistent with ending 12 m deeper than the start. ✓

Answers: 1 — m; reasonable, since climbing three times the descent should end well above the start. 2 — Jan+Feb ; March ; total , so overall 8,440\frac34\frac{1}{12}\frac34-\frac{1}{12}=\frac{9}{12}-\frac{1}{12}=\frac{8}{12}=\frac23\frac1{12}\frac23-\frac1{12}=\frac{8}{12}-\frac1{12}=\frac{7}{12}\frac1{12}\frac16=\frac{2}{12}\frac{7}{12}-\frac1{12}+\frac2{12}=\frac{8}{12}=\frac23-132-120$ m.

Checks for Understanding

(8 minutes — exit ticket, collected)

1. Write a single expression for: “Start at . Add , then multiply the result by .” Evaluate it.
2. A lift starts at level . It descends levels, then descends the same amount again. Write an expression and find the final level.
3. A recipe needs cup of sugar per batch. A baker makes fewer batches than the planned , then doubles the reduced amount. How many cups of sugar are needed? Show your expression.
4. Reasoning. A student’s answer to a “change in temperature” problem is for a single day’s temperature change. Explain why this answer, although arithmetically possible, should prompt the student to re-check their working.

Answers: 1. ; 2. , level ; 3. cups; 4. A temperature change of in a single day is physically implausible in almost any real context, so while the arithmetic may be valid, the size of the result should trigger a check of the original numbers or expression for an error.

Common Misconceptions

Misconception	How to pre-empt it
Calculating each step separately without combining into one checked expression.	Require the full expression to be written before any evaluation begins, as modelled in Activity 1.
Missing brackets where a worded phrase like “the result” or “the total so far” signals a grouped earlier step.	Underline or highlight such phrases in the problem text before writing the expression.
Assuming a negative final answer must be wrong.	Discuss context explicitly: depths, temperatures, and financial losses are naturally negative.
Not checking an answer’s reasonableness against the real-world context.	Make a one-sentence “does this make sense?” check compulsory, as required in every applied answer this lesson.
Losing track of units (m, °C, $, cups) partway through a multi-step solution.	Model carrying the unit through every line of working, not just the final answer.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). A lift is at level . It rises levels every time a bell rings and descends level between rings. After rings (with a descent after each of the first 4), what level is it at?

Answer

Final level: 3.

E2 (Kangaroo style). A number is tripled, then is subtracted, then the result is halved, giving . What was the original number?

Answer

Working backwards: ; ; . The original number was 1.

E3 (Challenge). A tank starts full ( whole tank). Each hour it loses of what remains (not of the original full amount). After hours, what fraction of the tank is full?

Answer

Losing of what remains each hour means remains after each hour: .

E4 (Investigation). A submarine’s depth changes by the same pattern every day: rises m, then dives m. Starting at m, after how many full days does it first reach a depth of m or deeper?

Answer

Net change per day: m. Depth after days: . Solve : , so . It first reaches m or deeper after 10 full days.

Homework

1. Write a single expression and evaluate: “Start at . Subtract , then multiply by .”
2. A hot air balloon at m descends m, then rises times that descent. Write an expression for its final height and evaluate it.
3. A budget starts at 1203$45$20$ is taken twice. Write a single expression for the final balance and evaluate it.
4. A tank is full. It loses of a full tank over a week, then gains back . What fraction is left? Show your working.
5. Reasoning. Explain why “write the full expression first” is a more reliable strategy than calculating step by step from the words, especially for problems involving “the result is then doubled” or similar phrases.
6. Challenge. A number is increased by of itself, then decreased by of the new total. If the final result is , find the original number. (Hint: work backwards using fractions of “1 whole” at each stage.)

Answers: Q1 — . Q2 — m. Q3 — , so 25\frac56-\frac14+\frac1{12} = \frac{10}{12}-\frac{3}{12}+\frac{1}{12}=\frac{8}{12}=\frac23\frac14\frac54n\frac13\frac23 \times \frac54n = \frac{5}{6}n\frac56 n = 20n = 20 \div \frac56 = 20 \times \frac65 = 2424$.