014i - Problem Solving with Integers

Lesson 14 — Problem Solving and Consolidation: Integers

Strand: Number | Descriptor: AC9M7N07 | Duration: 45 minutes

Learning Intentions

• To solve multi-step practical problems involving addition and subtraction of integers.
• To interpret integer answers correctly in context.

Success Criteria

I can:

1. Translate a worded context into an integer expression.
2. Evaluate expressions with several integer additions and subtractions.
3. Interpret a negative answer sensibly in its context (debt, depth, temperature drop).
4. Check that an answer is reasonable against the situation described.

Warmup

(6 minutes — “Which is the odd one out?”, pairs)

Each set contains three expressions with the same value and one that differs. Find the odd one out.

Set A: ; ; ;

Set B: ; ; ;

Answers: A — the first three all equal ; is the odd one out. B — the first three all equal ; is the odd one out.

Activities

Activity 1 — Multi-step Expressions (10 min)

Explicit reminder of the order of operations, then practice.

Additions and subtractions are evaluated left to right. Brackets first.

I do:

You do:

(Answers: , , , , , .)

Activity 2 — Contextual Problems (14 min)

Pairs. Every solution must show the integer expression, the evaluation, and a sentence answering in context.

Problem 1 — Finance. Jack’s account has 120$85$60$40$. What is his balance? What does the answer mean for Jack?

Problem 2 — Altitude. A drone takes off from a clifftop m above sea level. It climbs m, descends m, then climbs m. What is its final height relative to sea level? Is this physically possible? (Discuss: it would be below sea level, so the drone must be over the water beyond the cliff, or the scenario is unrealistic.)

Problem 3 — Temperature. At am the temperature was C. It rose C by am, a further C by pm, then fell C by midnight. Find the temperature at each time listed.

Problem 4 — Golf. In a four-round tournament, a golfer scores , , and relative to par. What is her total? A second golfer scores , , and . Who wins, and by how much? (In golf, a lower score wins.)

Socratic scaffolding for Problem 4:

Prompt	Purpose
Understand: what do the signs mean here?	Negative is below par, which in golf is good.
What is the unknown?	Two totals, then the comparison.
Devise a plan	Add each golfer’s four scores, then compare.
Carry it out (golfer 1)	.
Carry it out (golfer 2)	.
Which is the winner?	Careful — the smaller total wins. , so golfer 1 wins.
By how much?	The distance between them: strokes.
Looking back	Does the answer make sense? Golfer 1 was under par more often. ✓

Answers: 1. 15$2595 + 40 - 180 + 25 = -2020-8-47-9^\circ-7-43$ strokes.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. Evaluate: (a) (b) (c) .
2. A submarine at m rises m, then dives m. Write a single expression and find the final depth.
3. Write an integer expression for: “The temperature was C, dropped by C, then rose by C.” Evaluate it.
4. Reasoning. A bank balance of 50-50^\circ$C are both “negative fifty”. Explain what each means in its own context, and why one is far more serious than the other.
5. Two divers are at m and m. How far apart are they vertically?

Answers: 1. , , ; 2. m; 3. C; 4. 50$50-50^\circ$50-50^\circ27$ m.

Common Misconceptions

Misconception	How to pre-empt it
Evaluating left to right incorrectly, e.g. read as .	Model the left-to-right convention explicitly and require one line per step.
Interpreting a negative answer as an error.	Discuss what negative means in each context before starting. A negative depth, balance or score is meaningful.
Assuming “biggest” always means most positive.	The golf problem forces the distinction. Ask “does bigger mean better here?” for each context.
Losing track in multi-step problems.	Require a running-total table for any problem with more than three steps.
Answering with a bare number and no context.	Insist every contextual answer ends in a sentence with units.
Confusing a change with a position, e.g. “fell ” treated as “C”.	Distinguish carefully: is the change; the new temperature is the old plus the change.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). In a quiz, correct answers score , wrong answers score , and blanks score . Yuki answers questions correctly, gets wrong, and leaves blank. What is her score?

Answer

E2 (Kangaroo style). What is the value of ?

Answer

Group in pairs: . There are pairs, each equal to , giving .

E3 (Challenge). The numbers are placed one per box so that the three numbers along each straight line of a plus-shape (three across, three down, sharing the centre) sum to . What number must be in the centre?

Answer

The seven numbers total . The two lines together use all seven numbers but count the centre twice, so , giving . The centre must be .

E4 (Challenge). A lift starts at level . It makes six moves: . What is its final level, and what is the highest level it reaches during the journey?

Answer

Running totals: . Final level ; highest level reached .

Homework

1. Evaluate: (a) (b) (c) (d) .
2. A company records monthly profits (positive) and losses (negative): 4000-$1500-$2800$3200$. What is the total for the four months?
3. At the start of a card game each player has points. Over five rounds Elif scores . What is her final score? What was her lowest score at any point?
4. A hiker starts at m above sea level, descends m into a valley, then climbs m. Find her final elevation.
5. The temperature in a freezer is C. A fridge is at C. How much warmer is the fridge?
6. Two cities record minimum temperatures of C and C. Find the difference. Which is colder?
7. Reasoning. Explain why the expression and the expression always give answers that are opposites. Illustrate with , .
8. Challenge. Over days a share price changed by 3+$7-$12+$2x-$4x$.

Answers: Q1 — , , , . Q2 — 2900-5-6-4, 3, -6, -4, -510523^\circ16^\circ-23^\circa=-3, b=8a-b = -11b-a = 11-6x = +$2$.