Mathematics • Year 8 • Unit 2 • Lesson 4

Tables in the Real World

Use tables of values to model real situations: phone plans, plant growth, taxi metres, swimming-pool filling, and currency conversion. Apply the Linear Test to spot which relationships will graph as a straight line.

Apply · Real-World Maths

1. Word problems

For each scenario, build a small table, apply the Linear Test, and answer the question. Show your working.

1.1 — Phone plan. A phone plan costs $20 per month plus $0.10 per minute of calls.

(a) Build a table for total monthly cost C against minutes used m, using m = 0, 50, 100, 150, 200.
(b) Apply the Linear Test on the C-values for these equally-spaced m-values.
(c) Write a rule connecting C to m. 3 marks

Stuck? For each extra 50 minutes the cost goes up by $0.10 × 50 = $5. That's a constant first difference → linear.

1.2 — Bean-plant growth. A student measures her bean plant each day. Day 0: 2 cm, Day 1: 5 cm, Day 2: 8 cm, Day 3: 11 cm, Day 4: 14 cm.

(a) Show first differences and state whether the growth is linear.
(b) Predict the plant height on Day 10 using a rule of the form height = an + b.
(c) Write one sentence about why real plant growth probably wouldn't actually follow this rule forever. 3 marks

Stuck? Constant +3 per day → linear, h = 3n + 2.

1.3 — Taxi metre. A taxi metre shows: 1 km → $6, 2 km → $9, 3 km → $12, 4 km → $15.

(a) Apply the Linear Test to confirm it's linear.
(b) Write the rule for cost C as a function of distance k.
(c) Predict the cost for a 12 km trip. 3 marks

Stuck? Differences are all +3 → linear, C = 3k + 3 (a flag-fall of $3 plus $3 per km).

1.4 — Swimming-pool filling. A pool is being filled from a hose. The table below records the depth in cm after t minutes:

t (min): 0 2 4 6 8

d (cm): 10 14 18 22 26

(a) Apply the Linear Test (note: t jumps by 2 each time, so compare Δd over Δt).
(b) Write the rule for depth d in terms of time t.
(c) When will the depth reach 80 cm? 3 marks

Stuck? Each 2-minute step adds 4 cm → 2 cm/min, with initial depth 10. Rule: d = 2t + 10.

1.5 — Currency conversion. The table converts Australian dollars (A) into US dollars (U) at a fixed rate:

A: 10 20 30 40 50

U: 6.5 13 19.5 26 32.5

(a) Show that U = 0.65A (use first differences over Δ A = 10 to argue linearity, then verify).
(b) Use the rule to convert A$200 to USD. 3 marks

Stuck? Each +A$10 adds +US$6.5 — that's a constant rate of 0.65 USD per AUD, with no fixed fee, so the rule passes through (0, 0).

2. Explain your thinking

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

2.1 A classmate looks at a table and says: "All the y-values are increasing, so the relationship must be linear." In your own words, explain (i) why this reasoning is wrong, (ii) what the actual test for linearity is, and (iii) give one example of a non-linear table where every y-value still increases. Use the phrase "first differences must be equal" somewhere in your answer.

Stuck? Revisit lesson § "The Linear Test". y can keep going up but at a faster and faster (or slower and slower) rate.

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) m = 0, 50, 100, 150, 200 → C = 20, 25, 30, 35, 40.
(b) First differences: 5, 5, 5, 5 → all equal → linear (constant rate $5 per 50 minutes, i.e. $0.10/min).
(c) Rule: C = 0.10m + 20.

1.2 — Bean plant

(a) Differences: 3, 3, 3, 3 → linear, rate +3 cm per day.
(b) h = 3n + 2. Day 10: 3(10) + 2 = 32 cm.
(c) Real plants slow down and eventually stop growing (and limited light/water/space), so a constant-rate rule eventually breaks down.

1.3 — Taxi metre

(a) Differences: 3, 3, 3 → all equal → linear.
(b) Rate 3 → C = 3k + c. At k = 1, C = 6 → 3 + c = 6 → c = 3. Rule: C = 3k + 3.
(c) k = 12: C = 3(12) + 3 = $39.

1.4 — Swimming-pool filling

(a) Each 2-min step adds 4 cm → rate of 2 cm/min, constant. So linear.
(b) At t = 0, d = 10 → d = 2t + 10.
(c) 2t + 10 = 80 → 2t = 70 → t = 35 minutes.

1.5 — Currency conversion

(a) Each +A$10 adds +US$6.5, a constant rate of 0.65 USD per AUD. Rule passes through (0, 0) (no AUD ↔ no USD). So U = 0.65A. Check A = 30: 0.65 × 30 = 19.5 ✓.
(b) A = 200: U = 0.65 × 200 = US$130.

2.1 — Explain your thinking (sample response)

"All y-values increasing" only shows the relationship is going UP — it says nothing about HOW FAST it's going up. The actual test for linearity is that the first differences must be equal (assuming equally spaced x-values). For example, the squares y = 1, 4, 9, 16, 25 are all increasing, but the first differences are 3, 5, 7, 9 — which are growing themselves, so the relationship is NOT linear (it's a curve). A table where y keeps going up but the gap between consecutive values changes is the classic non-linear case.

Marking: 1 mark for spotting that "increasing" doesn't imply linear; 1 mark for stating "first differences must be equal"; 1 mark for a valid non-linear example whose y-values still increase; 1 mark for clear communication.