Mathematics Standard • Year 11 • Module 1 • Lesson 8

Formula and Equation Synthesis

Build fluency choosing the right first move — substitute, solve, rearrange or build — for mixed practical algebra questions.

Build · Skill Drill

1. Quick recall — name the strategy

For each prompt, write S (substitute), E (solve equation), R (rearrange) or B (build). 1 mark each

Q1.1 "Use C = 12 + 4r to find C when r = 7." Strategy: ____________

Q1.2 "12 + 4r = 40. Find r." Strategy: ____________

Q1.3 "d = st. Find s when d = 135 and t = 3." Strategy: ____________

Q1.4 "A table shows outputs going up by 5 each step. Write a formula." Strategy: ____________

Stuck? Revisit lesson § Choose the First Move — match the question type to the best first move.

2. Worked example — choose then solve

Follow each line of working. Note how the first decision (which strategy?) drives every step that follows.

Problem. A printer charges $25 setup plus $2 per page. A job costs $81. How many pages were printed?

Step 1 — Choose strategy.

Total known, pages unknown ⇒ Solve an equation.

Reason: we have the output ($81) and want the input (number of pages).

Step 2 — Define the variable.

Let p = number of pages.

Step 3 — Write and solve the equation.

25 + 2p = 81 ⇒ 2p = 56 ⇒ p = 28

Reason: subtract 25 from both sides, then divide both sides by 2.

Step 4 — Reasonableness check.

Check: 25 + 2(28) = 25 + 56 = 81 ✓

Conclusion. 28 pages were printed.

3. Faded example — rearrange first

Use d = st to find the average speed when d = 135 km and t = 3 h. Fill in each blank. 4 marks

Step 1 — Choose strategy:

Speed is not the subject of d = st, so we ____________ before substituting.

Step 2 — Rearrange:

d = st ⇒ s = ____ ÷ ____

Step 3 — Substitute:

s = ____ ÷ ____ = ____________

Conclusion sentence. The average speed is ____________ km/h.

Stuck? Revisit lesson § Worked Example 2 — Rearrange before substituting. To make s the subject, divide both sides by t.

4. Graduated practice — pick the strategy, then solve

For every question, name the strategy first (S / E / R / B), then show the working.

Foundation — clean numbers (4 questions)

Q	Problem	Strategy & answer
4.1 1	Use C = 12 + 4r to find C when r = 7.
4.2 1	Solve 3x + 5 = 23 for x.
4.3 1	Use A = lw. Find l when A = 60 and w = 5.
4.4 1	A table: input 0,1,2,3 → output 4,9,14,19. Write a formula.

Standard — typical HSC difficulty (6 questions)

Show choice of strategy, working, and a units sentence in the conclusion.

4.5 A gym charges $20 plus $15 per class. Find the number of classes if the total is $110. 2 marks

4.6 Use A = s² to find A when s = 9.5 m. 2 marks

4.7 Rearrange A = bh to find h. Then find h when A = 72 cm² and b = 9 cm. 2 marks

4.8 Write a formula for outputs 8, 13, 18, 23 for inputs 0, 1, 2, 3. Test it using input 3. 3 marks

4.9 A hire company charges $35 + $18 per hour. The total cost is $143. Find the hire time. 2 marks

4.10 Use d = st to find time when d = 210 km and s = 70 km/h. Rearrange first. 2 marks

Extension — strategy + reasonableness (2 questions)

4.11 A stopping-distance model is D = 0.01v² + 0.3v. A student claims that at 60 km/h the stopping distance is 540 m. Calculate D at v = 60 and explain what is wrong with the student's answer. 3 marks

4.12 A table shows distance vs time: at t = 0 h, d = 12 km; at t = 1 h, d = 92 km; at t = 2 h, d = 172 km. Write a formula linking d and t, then use it to predict d at t = 3.5 h. 4 marks

Stuck on 4.11? Substitute v = 60 carefully and compare. The student likely multiplied 0.01 by 60 then added v² in the wrong order.

5. Self-check the easy 3

Tick the first three once you've checked your method works.

For 4.1 I substituted (S) to get C = 12 + 4(7) = 40.

For 4.2 I solved (E): 3x = 18, x = 6.

For 4.3 I rearranged (R) A = lw → l = A/w = 60/5 = 12.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

Q1.1–Q1.4 — Strategy names

Q1.1: S (Substitute — output unknown). Q1.2: E (Solve equation — input unknown). Q1.3: R (Rearrange — wrong subject). Q1.4: B (Build a formula from the table pattern).

Q3 — Faded speed example

Step 1: We rearrange before substituting.
Step 2: s = d ÷ t.
Step 3: s = 135 ÷ 3 = 45.
Conclusion: The average speed is 45 km/h.

Q4.1 — C = 12 + 4r at r = 7 (S)

C = 12 + 4(7) = 12 + 28 = 40.

Q4.2 — Solve 3x + 5 = 23 (E)

3x = 18, x = 6.

Q4.3 — A = lw, find l (R)

l = A/w = 60/5 = 12.

Q4.4 — Build formula (B)

Start = 4, repeated increase = +5 per input step. y = 4 + 5x. Check x = 3: 4 + 5(3) = 19 ✓.

Q4.5 — Gym (E)

Let c = classes. 20 + 15c = 110 → 15c = 90 → c = 6 classes.

Q4.6 — A = s² at s = 9.5 (S)

A = (9.5)² = 90.25 m².

Q4.7 — A = bh, find h (R)

h = A/b = 72/9 = 8 cm.

Q4.8 — Build formula (B)

Start = 8 (at input 0), repeated increase = +5. y = 8 + 5x. Test x = 3: 8 + 5(3) = 23 ✓.

Q4.9 — Hire company (E)

Let h = hours. 35 + 18h = 143 → 18h = 108 → h = 6 hours.

Q4.10 — d = st, find t (R)

t = d/s = 210/70 = 3 h.

Q4.11 — Reasonableness check on D = 0.01v² + 0.3v at v = 60

D = 0.01(60)² + 0.3(60) = 0.01(3600) + 18 = 36 + 18 = 54 m. The student's 540 m is ten times too large — likely 0.1 used instead of 0.01, or a misplaced decimal point.

Q4.12 — Build and use a formula (B + S)

Start (t = 0): d = 12. Repeated change: +80 km per hour. Formula: d = 12 + 80t. Predict t = 3.5: d = 12 + 80(3.5) = 12 + 280 = 292 km.