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.