Mathematics • Year 8 • Unit 2 • Lesson 6

Linear in the Real World

Decide if everyday situations — taxi fares, phone plans, falling objects, savings accounts — are linear. Use the first-differences test where you have data, and the "is the rate constant?" test where you don't.

Apply · Real-World Maths

1. Word problems

Each scenario describes a relationship between two quantities. Decide whether it is linear and back up your answer with either a constant-rate argument or first differences. Show your working — final-answer-only earns half marks.

1.1 — Phone plan. A SIM-only plan costs $30 per month with no extra fees. Let C be the total cost after n months.

(a) Build a table for n = 1, 2, 3, 4, 5.
(b) Calculate the first differences in C.
(c) Is the relationship linear? State the gradient. 3 marks

Stuck? "Per month" is a strong hint for linear — every extra month adds the same $30.

1.2 — Taxi fare. A taxi charges $3 flag fall plus $2 per kilometre. Let C be the cost for d kilometres.

(a) Write the equation linking C and d.
(b) Build a table for d = 0, 1, 2, 3, 4.
(c) Is the relationship linear? What is the gradient and what does it represent? 3 marks

Stuck? The gradient is the cost added by ONE more km — that is the "per km" rate.

1.3 — Square garden. A square garden bed has side length s metres and area A square metres.

(a) Write the rule connecting A and s.
(b) Build a table for s = 1, 2, 3, 4, 5.
(c) Find the first differences in A. Is the relationship linear? Justify. 3 marks

Stuck? A = s² — does each extra metre of side add the same amount of area, or a bigger amount each time?

1.4 — Savings account. Amir starts with $20 saved and adds $15 to his account every week. Let S be his savings after w weeks.

(a) Write the equation for S in terms of w.
(b) Calculate S after 1, 2, 3 and 4 weeks and list the first differences.
(c) Is the relationship linear? Identify both the gradient and the starting value (y-intercept). 3 marks

Stuck? The fixed $20 starting amount is the y-intercept; the $15/week is the gradient.

1.5 — Falling apple. A physics class measured how far an apple had fallen each second after being dropped. The table read d (m): 0, 5, 20, 45, 80 at t (s): 0, 1, 2, 3, 4.

(a) Calculate the first differences in d.
(b) Is the relationship linear? Justify using your differences.
(c) In one sentence, explain why this is what you'd expect physically. 3 marks

Stuck? A falling object speeds up — so the distance added each second grows. That's a changing rate of change.

2. Explain your thinking

This question is about communication, not just answers. Use full sentences. 4 marks

2.1 A classmate says: "Any time both x and y are increasing, the relationship must be linear." In your own words, explain (i) why this is wrong, (ii) give one concrete counterexample from real life or from a table, and (iii) state the actual test you would use to decide if a relationship is linear. Use the phrase "constant rate of change" somewhere in your answer.

Stuck? Revisit lesson § "Common Pitfalls" — both quantities can rise without the rate being constant (e.g. area of a growing square).

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

1.1 — Phone plan

(a) C = 30, 60, 90, 120, 150 for n = 1..5.
(b) First differences: 30, 30, 30, 30.
(c) Linear; gradient m = 30 (dollars per month).

1.2 — Taxi fare

(a) C = 2d + 3.
(b) C = 3, 5, 7, 9, 11 for d = 0..4.
(c) Linear; gradient m = 2 ($ per km). This represents the cost of each extra km travelled.

1.3 — Square garden

(a) A = s².
(b) A = 1, 4, 9, 16, 25 for s = 1..5.
(c) First differences: 3, 5, 7, 9 — not all equal, so not linear. Each extra metre of side adds MORE area than the last (the differences grow).

1.4 — Savings account

(a) S = 15w + 20.
(b) S = 35, 50, 65, 80 for w = 1..4. First differences: 15, 15, 15.
(c) Linear. Gradient m = 15 ($/week); starting value (y-intercept) = $20.

1.5 — Falling apple

(a) First differences: 5 − 0 = 5, 20 − 5 = 15, 45 − 20 = 25, 80 − 45 = 35.
(b) Not linear. Differences are 5, 15, 25, 35 — they grow by 10 each time, so the rate of change isn't constant.
(c) The apple speeds up as it falls (gravity), so it covers more distance each second.

2.1 — Explain your thinking (sample response)

The classmate has confused "increasing" with "increasing at a constant rate". A relationship is linear only when y has a constant rate of change as x increases — both quantities can rise together without the rate being constant. A good counterexample is the area of a square as its side grows: side 1, 2, 3, 4, 5 gives area 1, 4, 9, 16, 25, both rising, but the first differences (3, 5, 7, 9) are NOT equal, so the relationship is non-linear. The correct test is to check the first differences in a table (all equal = linear) or to graph the points and confirm they fall on a perfectly straight line.

Marking: 1 mark for explaining the flaw; 1 mark for a valid counterexample with numbers; 1 mark for naming the first-differences / straight-line test; 1 mark for using "constant rate of change" in a sentence that makes sense.