Mathematics • Year 8 • Unit 2 • Lesson 3

Patterns in the Real World

Use number patterns where they actually live: stacking chairs at a school assembly, paying for taxi rides, growing matchstick shapes, training for a fun run, and saving for a phone. Then explain how a position-to-term rule beats listing every value.

Apply · Real-World Maths

1. Word problems

For each scenario find the rule, then use it to predict. Show your working — a final answer without working only earns half marks.

1.1 — Stacking chairs. A single chair is 90 cm tall. Each EXTRA chair stacked on top adds 12 cm to the total height.

(a) Write a rule for the height H (in cm) of a stack of n chairs.
(b) Use the rule to find the height of a stack of 8 chairs.
(c) The classroom ceiling is 240 cm. What is the tallest stack you could make without hitting it? 3 marks

Stuck? n = 1 gives just one chair (90 cm). Adding (n − 1) extra chairs adds 12 cm each — so H = 90 + 12(n − 1), which also equals 12n + 78.

1.2 — Taxi fare. A taxi charges a $4 "flag fall" (fixed fee) plus $2 per kilometre travelled.

(a) Write a rule for the cost C (in dollars) of a trip of k kilometres.
(b) Find the cost of a 7 km trip.
(c) A passenger paid $24. How many kilometres did they travel? 3 marks

Stuck? The "starting value" is the flag fall ($4). The "per step" amount is the per-km charge.

1.3 — Matchstick triangles. One triangle uses 3 matchsticks. Adding each next triangle (attached side-by-side) uses 2 more sticks.

(a) Complete the table: n = 1 → 3, n = 2 → 5, n = 3 → ___, n = 4 → ___.
(b) Write a position-to-term rule for the number of matchsticks M when there are n triangles.
(c) How many matchsticks are needed for 25 triangles? 3 marks

Stuck? Differences are +2 each time → try 2n. n = 1 gives 2, but the actual value is 3 → add 1. So M = 2n + 1.

1.4 — Training plan. A runner runs 2 km in week 1 and adds 0.5 km each week.

(a) Write a rule for the distance D (in km) run in week n.
(b) How far does she run in week 10?
(c) In which week does she first reach a 7 km run? 3 marks

Stuck? Common difference is 0.5, so try D = 0.5n + (something). For week 1 the rule must give 2, so the constant is 1.5.

1.5 — Saving for a phone. A student starts with $30 and saves $15 each week.

(a) Write a rule for the total savings S after w weeks.
(b) How much does the student have after 6 weeks?
(c) The phone costs $300. In which week will the student first have enough to buy it? 3 marks

Stuck? "Starts with $30" = the constant. "$15 each week" = the coefficient of w. So S = 15w + 30.

2. Explain your thinking

This question is about communication. Use full sentences. 4 marks

2.1 A classmate says: "To find the 100th term of a sequence, you just have to keep adding the rule 100 times — there's no quicker way." In your own words, explain (i) why this method works but is slow, (ii) what a position-to-term rule is and why it's faster, and (iii) demonstrate the difference using the sequence 5, 8, 11, 14, 17, … to find the 100th term. Use the phrase "position-to-term rule" somewhere in your answer.

Stuck? Revisit lesson § Key Terms — "position-to-term rule". The whole point is to jump to any term in ONE calculation.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

1.1 — Stacking chairs

(a) Rule: H = 12n + 78 (equivalent to H = 90 + 12(n − 1)). Check n = 1: 12(1) + 78 = 90 ✓.
(b) n = 8: H = 12(8) + 78 = 96 + 78 = 174 cm.
(c) 12n + 78 ≤ 240 → 12n ≤ 162 → n ≤ 13.5. Largest whole n is n = 13 chairs (height = 12(13) + 78 = 234 cm).

1.2 — Taxi fare

(a) C = 2k + 4.
(b) k = 7: C = 2(7) + 4 = $18.
(c) 2k + 4 = 24 → 2k = 20 → k = 10 km.

1.3 — Matchstick triangles

(a) n = 3 → 7; n = 4 → 9.
(b) Common difference 2 → 2n. For n = 1, 2n = 2 but value is 3 → add 1. Rule: M = 2n + 1.
(c) n = 25: M = 2(25) + 1 = 51 matchsticks.

1.4 — Training plan

(a) Common difference 0.5; week 1 = 2 → constant 1.5. Rule: D = 0.5n + 1.5.
(b) Week 10: D = 0.5(10) + 1.5 = 5 + 1.5 = 6.5 km.
(c) 0.5n + 1.5 = 7 → 0.5n = 5.5 → n = 11. She first reaches 7 km in week 11.

1.5 — Saving for a phone

(a) S = 15w + 30.
(b) w = 6: S = 15(6) + 30 = 90 + 30 = $120.
(c) 15w + 30 ≥ 300 → 15w ≥ 270 → w ≥ 18. So week 18 is the first week she has at least $300 (S = $300 exactly).

2.1 — Explain your thinking (sample response)

Adding the rule 100 times does work — that's the term-to-term method — but it takes 99 separate additions and you'd very likely make a small arithmetic slip along the way. A position-to-term rule is a formula that jumps straight from a position number n to the value of that term in a single calculation, no listing required. For the sequence 5, 8, 11, 14, 17, … the common difference is 3 and the first term is 5, giving the rule term = 3n + 2. To find the 100th term you just substitute n = 100: term = 3(100) + 2 = 302. That's one multiplication and one addition instead of 99 separate steps — much faster, and far less prone to error.

Marking: 1 mark for acknowledging the slow method works but is error-prone; 1 mark for correctly defining the position-to-term rule; 1 mark for deriving term = 3n + 2; 1 mark for showing the 100th term = 302 with the rule.