Mathematics • Year 7 • Unit 2 • Lesson 3
Substitution in the Real World
Use substitution to read off real values from algebraic formulas — taxi fares, video game points, phone plans, simple geometry. Show that algebra is just a recipe waiting for ingredients.
1. Word problems
Each problem gives you an expression or formula and asks you to substitute a real value. Show your steps: write the formula, substitute with brackets, then evaluate using BODMAS.
1.1 — Taxi fare. A taxi charges $4 flag-fall plus $2 per kilometre travelled. The formula for the cost is C = 4 + 2k, where k is the number of kilometres.
(a) How much will a 7 km trip cost?
(b) How much will a 12 km trip cost? 2 marks
1.2 — Video game score. In a game, your final score is given by S = 10L + 50C, where L is the number of levels completed and C is the number of coins collected.
(a) What is the score for finishing 8 levels with 12 coins?
(b) What is the score for finishing 5 levels with 0 coins? 2 marks
1.3 — Phone plan. Your monthly phone bill in dollars is given by B = 30 + 0.25t, where t is the number of text messages sent over the included limit.
(a) How much do you pay if you stay within the limit (t = 0)?
(b) How much do you pay if you send 40 extra texts? 2 marks
1.4 — Rectangle perimeter. The perimeter of a rectangle is P = 2L + 2W, where L is the length and W is the width.
(a) Find the perimeter of a rectangle with L = 8 cm and W = 3 cm.
(b) Find the perimeter of a square (where L = W = 5 cm). 2 marks
1.5 — Temperature change. A weather formula gives the temperature change from morning to night as T = E − M, where E is the evening temperature and M is the morning temperature (both in °C).
(a) Find T when M = 18 °C and E = 24 °C.
(b) Find T when M = 12 °C and E = 5 °C. What does the negative answer tell you?
(c) Find T when M = −2 °C and E = 4 °C. (Watch the double sign!) 3 marks
2. Explain your thinking
This question is about why substitution rules matter. Use full sentences. 4 marks
2.1 Lee is asked to evaluate 2x² when x = −3 and writes: "2x² = 2(−3)² = (2 × −3)² = (−6)² = 36". The final number 36 is wrong. (i) Identify where Lee's working goes off the rails, (ii) state the correct order of operations, (iii) work out the correct value, and (iv) explain in one sentence how to remember the rule.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Taxi fare
(a) C = 4 + 2(7) = 4 + 14 = $18.
(b) C = 4 + 2(12) = 4 + 24 = $28.
1.2 — Video game score
(a) S = 10(8) + 50(12) = 80 + 600 = 680 points.
(b) S = 10(5) + 50(0) = 50 + 0 = 50 points.
1.3 — Phone plan
(a) B = 30 + 0.25(0) = 30 + 0 = $30.
(b) B = 30 + 0.25(40) = 30 + 10 = $40.
1.4 — Rectangle perimeter
(a) P = 2(8) + 2(3) = 16 + 6 = 22 cm.
(b) P = 2(5) + 2(5) = 10 + 10 = 20 cm.
1.5 — Temperature change
(a) T = 24 − 18 = 6 °C (it warmed up by 6 degrees).
(b) T = 5 − 12 = −7 °C. The negative tells us it COOLED — the evening was 7 degrees lower than the morning.
(c) T = 4 − (−2) = 4 + 2 = 6 °C. (Subtracting a negative becomes adding — it warmed up by 6 degrees, going from −2 °C up to 4 °C.)
2.1 — Lee's error (sample response)
(i) Lee's working goes wrong in step 2, where they write "2(−3)² = (2 × −3)²". This applies the multiplication BEFORE the square, which breaks the order of operations.
(ii) Correct order (BODMAS): Brackets first, then Orders (powers), then Multiplication. So in 2x², you must square x FIRST, then multiply by 2.
(iii) Correct working: 2(−3)² = 2 × 9 = 18.
(iv) Remember rule: "the square only sticks to what's right next to it." In 2x², the ² is attached to the x, not to the 2x. If you wanted both squared you would have to write (2x)².
Marking: 1 for spotting where the order was broken; 1 for correct BODMAS statement; 1 for the correct value 18; 1 for a clear memory aid.