Unit 4 Topic 1 · underpins compound interest, loans, annuities, depreciation and half the sequences work · week 1 of the internal run
The concept in one sentence: multiplying by r = 1 + i keeps the whole amount AND adds the growth, and every finance topic in the course is that single move wearing a different costume.
The thing that makes it click: the $100 story. At 4% interest you keep your $100 (that is the 1) and gain $4 (that is the 0.04), so you multiply by 1.04. Multiplying by 0.04 alone would turn $100 into $4.
1 · Prerequisite check (30 seconds)
Ask: "Write 4% as a decimal" and "If A = 200, what is 1.05 × A?"
If either wobbles: detour to the percentages penny-drop on foundations.html before anything else. Ten minutes there saves the whole week. Do not teach recurrence on top of shaky percentage-to-decimal conversion.
2 · The teaching path (concrete → picture → notation)
Concrete: the $100 story out loud, with real numbers. Then $100 at 4% for TWO years: 104, then 108.16. Let him notice the second year earned more ("interest earning interest" is compound interest defined, without the definition).
Picture: a hop-table on paper. Balance → ×1.04 → balance → ×1.04 → balance. Arrows labelled with the multiplier. This IS the recurrence before it has letters.
Notation last: only once the hops feel obvious, write the formula-book version:
Aₖ₊₁ = r · Aₖ ± d where r = 1 + i · A₀ = the start
Then the shortcut for many hops at once: A = P(1 + i)ⁿ. Sell it as "the lazy version of pressing = twenty-four times".
The one engine, four costumes line: + d is money going in (investment with deposits), − d is money coming out (loan repayments), no d is pure compound growth, and for decay (depreciation) the multiplier drops below 1: keep 80% of a 20% loser, so × (1 − i) = × 0.8. Same engine downhill.
The 1 keeps the money. The i adds the interest.
The Ans key IS the recurrence. Type the start, press =, then Ans × 1.008 − 750, and every press of = is one month passing.
"Do that to $100. What happens?" ($4.) "Where did your hundred go?" Let the absurdity do the teaching, then rebuild: keep + add
⭐ Compounds once for a multi-year question
Treats interest as a one-off event, not a repeated process
Back to the hop-table: one arrow per period, count the arrows. Then n = number of arrows
⭐ n = years when compounding is monthly
The question SAYS years, so n feels like years
"n counts compoundings, not birthdays." Ritual: circle the compounding word in every question before touching the calculator
⭐ Reverses present and future value
Multiplying feels like the default move
"Going forwards in time = multiply. Coming backwards = divide." Draw the timeline arrow both ways. And once annuities arrive, add the formula-choice rule: paying off a debt or drawing down a pot = the PV formula, saving up to a target = the FV formula. His starred error may be formula CHOICE, not just direction
Subtracts the repayment before charging interest
Paying first feels virtuous
"The bank charges you BEFORE your payment lands. Interest on, repayment off, in that order." Show both orders on Halim month 1 and compare
Rounds every line, answer drifts
Tidiness instinct
"Round last. The exam wants the cents at the END, not along the way." Keep full digits on the calculator via Ans
4 · Questioning ladder (run it in this order)
Rung
Ask
Listen for
Recall
"6% p.a. monthly. What is i? What is r?"
0.005 and 1.005, no hesitation
Do
Halim's first month by hand (50 000, i = 0.0055, repay 750)
× 1.0055 first, then − 750 → $49 525
Explain back
"Why is there a 1 in 1.0055? Teach me like I'm the student"
Any version of "the 1 keeps what you already have"
Transfer
"A ute loses 20% a year. What is the multiplier?"
× 0.8, ideally with "keep 80%" reasoning. Decay = same engine downhill
He owns the concept when he can do the transfer rung WITHOUT being told it is the same idea.
5 · How QCAA examines it
SF form: "develop a recurrence relation" then "use it to determine the value after..." (the sample's Q1 and Q2, about 7.5 of 40 marks). Marks are for i, for the recurrence, and for iterated values
CF form: compare two loans, "evaluate the reasonableness of the claim" (Pippa). The judgement SENTENCE carries a mark: claim + evidence + verdict
Formula book letters: i = rate per period, r = 1 + i (the multiplier), A₀ = start, d = payment. Note the book uses i for rate and r for multiplier, the OPPOSITE of some textbooks
Also printed in the book, the annuity pair: for loans A_PV = d[1 − (1+i)⁻ⁿ] ÷ i, for savings A_FV = d[(1+i)ⁿ − 1] ÷ i. The choice rule: paying off a debt or drawing money down = PV formula; saving up to a target = FV formula
Effective annual rate i_effective = (1 + i)ᵏ − 1 (k = compounding periods per year) is also printed, and it is THE tool for comparing two loans quoted with different compounding (Pippa-style questions)
Technology expectation: the Ans-key iteration is the intended method on a scientific calculator. No finance solver exists in General Maths
6 · Stuck scripts (instead of re-explaining louder)
What do we keep? What do we add?
Say the rate as money on $100.
Is money going IN or coming OUT each period? So is d plus or minus?
How many arrows on the hop-table? That is your n.
If two scripts in a row don't unstick him, stop the question and drop a rung on the ladder instead. Re-explaining the same way, louder and slower, teaches him that being stuck is a performance problem. It is an entry-point problem.
7 · ADHD delivery notes (Josh-specific)
One worked example + one faded example per sitting is the ceiling. Depth beats coverage
The Ans-key trick gives a fast physical win early, use it in the first ten minutes
Name his diagnostic error as "the classic trap" (it starred in the drills for everyone), never as his mistake
Between sessions the drills do the spaced retrieval for you: 3 drills a week is the whole homework ask
Cooked night fallback: just the recall rung + one Ans-key iteration. Zero is the only fail
Cooked WEEK fallback (not just cooked night): one drill, done anywhere, still counts