Connections & financial literacy
Link mathematical ideas and connect to real / other-subject contexts; money → interest, discounts, budgeting.
Mathematics is not a stack of separate topics — it is a web, and the same idea reappears in many disguises. This page is about making those links, and about the place where number sense meets everyday money: interest, discounts and budgeting.
What it means
To connect mathematical ideas is to notice that a method you learned in one place solves a problem that looks different. A fraction, a decimal and a percentage are three costumes for one number: ½ = 0.5 = 50%. Once you see that, “half price”, “0.5 of the total” and “a 50% discount” are the same calculation.
Percent means “per hundred”, so x% of an amount is x/100 multiplied by it. That single tool drives most everyday money maths.
- A discount subtracts a percentage: pay
(100 − x)%of the price. - Interest adds a percentage over time. Simple interest adds the same amount each period. Compound interest adds a percentage of the new total each period, so growth speeds up because you earn interest on past interest.
- A budget is arithmetic with a constraint: money in must cover money out. It is addition, subtraction and percentages used to plan.
The connecting habit is the point. Multiplication, ratio and repeated growth are one family of ideas wearing different clothes in the shop, the bank and the household.
Worked examples
Discount. A £40 coat is 25% off. The discount is 25/100 × 40 = £10, so you pay £40 − £10 = £30. Faster: pay 75%, and 0.75 × 40 = 30.
Simple vs compound interest on £100 at 10% per year for 3 years:
| Year | Simple (+£10 each) | Compound (×1.10 each) |
|---|---|---|
| 1 | £110 | £110 |
| 2 | £120 | £121 |
| 3 | £130 | £133.10 |
Compound pulls ahead because year 2 earns interest on £110, not £100.
Budget. Pocket money is £20. Spend 40% on a gift (0.40 × 20 = £8) and save the rest: £20 − £8 = £12 saved.
The generative-art connection
Compound growth is a loop that multiplies then redraws — exactly how generative art is built. Start with a value, multiply by a fixed factor, draw the result, repeat. Feed × 1.1 into the size of a shape each step and you get a spiral fanning outward; the visible growth is compound interest. The internal hue-pulse tool runs this multiply-and-redraw engine on a pulsing dot, so a child can watch the same repeated-factor rule that makes savings grow. Percentages, too, are pictures: shade 75% of a grid and the discounted price becomes an area you can see.
Common misconceptions
- “Percent off then percent on cancels out.” 20% off then 20% on does not return to the start:
£100 → £80 → £96, because each percentage acts on a different amount. - “Interest is just adding a fixed number.” That is only simple interest; compound interest multiplies, so it grows faster and faster.
- “A percentage is a fixed quantity.” 50% is a fraction of something — 50% of £10 and 50% of £100 are very different amounts.
What a learner should be able to solve
The benchmark for mastering this concept — the problems a child at this stage is expected to be able to work through.
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Show you can connect the three costumes of one number. A shop sign says "`1/4` off". Rewrite `1/4` as a decimal and as a percentage, then explain in one sentence why "quarter off", "`0.25` of the price taken away" and "`25%` discount" are the same instruction.
Answer
`1/4 = 0.25 = 25%`. All three say: multiply the price by `1/4` to find the amount removed (or pay `3/4 = 0.75 = 75%` of it). Demonstrable behaviour: fluently moves between fraction, decimal and percentage for the same value and links them to one real action.
Art hook Colour-wheel connector: draw a ring split into 4 equal arcs; clicking an arc shades it and simultaneously prints the same slice as `1/4`, `0.25` and `25%` in three labels, so the child sees one quantity in three costumes.
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Find a percentage of an amount in a real context. A game costs `£60`. Work out `15%` of `£60`, then say what the game costs after a `15%` discount.
Answer
`15% of 60 = 0.15 × 60 = £9`. Discounted price `= 60 − 9 = £51` (or directly `0.85 × 60 = £51`). Success: computes the percentage and applies it to solve the money problem.
Art hook Bar shrinker: draw a horizontal bar 60 units long; a slider removes 15% of its length, and the shaded-away piece is labelled £9 while the remaining bar reads £51.
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Compare simple and compound interest and say which is bigger and why. You save `£200` at `5%` per year for `2` years. Find the total under simple interest and under compound interest, then explain the difference.
Answer
Simple: `+£10` each year, so `200 → 210 → 220`. Compound: `×1.05` each year, so `200 → 210 → 220.50`. Compound gives `£220.50`, which is `£0.50` more, because year 2's interest is `5%` of `£210`, not `£200` — you earn interest on last year's interest. Behaviour: distinguishes the add-a-fixed-amount rule from the multiply-the-new-total rule.
Art hook Two-line grower: plot two dots stepping rightwards year by year; the simple dot rises by equal jumps, the compound dot by ×1.05 jumps, so the child watches the compound curve peel upward away from the straight line.
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Plan a small budget with a constraint. You get `£25` a week. You want to spend `30%` on treats, save `40%`, and keep the rest for bus fares. How much goes to each, and does it add up to `£25`?
Answer
Treats `0.30 × 25 = £7.50`; savings `0.40 × 25 = £10`; the rest is `100% − 70% = 30%`, so bus fares `0.30 × 25 = £7.50`. Check: `7.50 + 10 + 7.50 = £25`. Behaviour: uses percentages under a total-must-balance constraint and verifies it closes.
Art hook Budget pie: split a circle into arcs sized 30% / 40% / 30% (108°, 144°, 108°) in three colours, each labelled with its pound value, so the whole £25 is one full turn.
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Spot the trap in a two-step percentage claim. A shop says: "`20%` off, then we add `20%` at the till — so you pay the original price!" Is this true for a `£50` item? Show what actually happens.
Answer
Not true. `£50 → 20% off → £40 → add 20% of £40 → £40 + £8 = £48`. You pay `£48`, not `£50`, because each `20%` acts on a different amount (£50 then £40). Behaviour: reasons that percentages are relative and critiques a plausible-sounding claim.
Art hook Down-then-up animator: a bar drops 20% (to 40 units) then grows 20% of its new length (+8), landing visibly short of the start line, which stays marked at 50.
Training exercises
Practice problems, easier first, that build toward the bar above. Each doubles as a seed for an interactive artwork.
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Match the costumes. Connect each fraction to its decimal and its percentage: `1/2`, `1/4`, `3/4`, `1/10`. The choices are `0.5, 0.25, 0.75, 0.1` and `50%, 25%, 75%, 10%`.
Answer
`1/2 = 0.5 = 50%`; `1/4 = 0.25 = 25%`; `3/4 = 0.75 = 75%`; `1/10 = 0.1 = 10%`.
Art hook Three-column matcher: fraction dots on the left, decimals in the middle, percents on the right; a correct drag lights all three the same colour.
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Percent means "per hundred". A `10 × 10` grid has `100` squares. If you shade `30` of them, what percentage is shaded? What fraction and decimal is that?
Answer
`30/100 = 30% = 0.3`. 30 shaded out of 100 squares.
Art hook Hundred-grid painter: a 10×10 grid where each tapped cell increments a live counter showing the count, the fraction /100, the decimal and the percent together.
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Find a friendly percentage of an amount. What is `50%` of `£18`? What is `10%` of `£18`? What is `25%` of `£18`?
Answer
`50% of 18 = £9` (half). `10% of 18 = £1.80` (÷10). `25% of 18 = £4.50` (quarter).
Art hook Number-line splitter: an £18 line where buttons for 50%, 10%, 25% drop a marker at the matching point and label the amount.
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Apply a discount. A `£30` pair of shoes has `20%` off. How much is taken off, and what is the new price?
Answer
`20% of 30 = £6` off, so new price `= 30 − 6 = £24` (or `0.80 × 30 = £24`).
Art hook Price-tag redraw: the old price £30 shown struck through as a shaded bar loses a 20% slice, the remaining bar reading £24.
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Link percentages to measuring. A drink is `250 ml` and `40%` of it is juice. How many millilitres of juice is that?
Answer
`40% of 250 = 0.40 × 250 = 100 ml` of juice.
Art hook Liquid fill: a beaker outline 250 units tall fills to the 100-unit mark in juice colour, labelled 40%.
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Reason about which is more. Would you rather have `25%` of `£40` or `50%` of `£30`? Work out both and say which is bigger.
Answer
`25% of 40 = £10`; `50% of 30 = £15`. `£15` is bigger, so 50% of £30 is the better deal, even though `50 > 25` and `40 > 30` — the amount matters, not just the percentage.
Art hook Two-bar duel: two shaded bars side by side, one showing £10 of £40, the other £15 of £30, so the taller shaded piece wins visibly.
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Build simple interest step by step. You put `£100` in a jar and add `£5` (that's `5%`) every year. Fill in the totals after year 1, year 2 and year 3.
Answer
Year 1: `£105`. Year 2: `£110`. Year 3: `£115`. Equal `+£5` jumps each year.
Art hook Stacking coins: a column that adds one £5 coin per year-tick, the running total shown beside a straight staircase line.
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Now compound it. You put `£100` in a bank that multiplies the total by `1.05` (that's `5%`) every year. Find the total after year 1, year 2 and year 3, rounding to the nearest penny.
Answer
Year 1: `100 × 1.05 = £105`. Year 2: `105 × 1.05 = £110.25`. Year 3: `110.25 × 1.05 = £115.76` (to the nearest penny).
Art hook Multiply-and-redraw dot: a dot whose size is multiplied by the same 1.05 factor each tick, tracing a widening spiral (the same repeated-factor engine as hue-pulse), with the running total printed each step so the growth on screen IS the compound total.
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Spot the mistake. Sam says: "`30%` of `£50` is `£30`, because you just drop the percent sign." Explain what Sam did wrong and give the correct answer.
Answer
Wrong: Sam read the number as pounds. `30%` means `30/100 = 0.30`, so `30% of £50 = 0.30 × 50 = £15`. A percentage is a fraction of an amount, not a fixed number of pounds.
Art hook Grid proof: shade 30 of a 100-square grid representing £50, so each square is £0.50 and the shaded part reads £15, not £30.
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Two-step discount reasoning. A `£80` jacket is `10%` off, and members get another `10%` off the reduced price. What does a member pay? Is it the same as `20%` off `£80`?
Answer
`80 → 10% off → £72 → 10% off → £72 − £7.20 = £64.80`. Straight 20% off would be `0.80 × 80 = £64`. The two-step way costs `£0.80` more, because the second 10% acts on £72, not £80. So they are not the same.
Art hook Chained bars: bar A drops 10% then 10% again landing at £64.80; bar B drops 20% once landing at £64, side by side so the small gap is visible.
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Open budget challenge. You have `£40` for a party. Sandwiches are `£15`, and you want to spend the rest as `50%` on drinks and `50%` on decorations. How much is the rest, and how much goes to each? Show it balances to `£40`.
Answer
Rest `= 40 − 15 = £25`. Drinks `50% of 25 = £12.50`; decorations `£12.50`. Check: `15 + 12.50 + 12.50 = £40`. Balances.
Art hook Balancing budget pie: a £40 circle with one fixed sandwich slice (£15) and the remaining arc split evenly into two, each labelled £12.50, filling exactly one turn.
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Reason about reversing a discount (challenge). A toy is `£30` after a `25%` discount. Was the original price `£40`? Check by taking 25% off `£40`, and explain your answer.
Answer
Yes. `25% of 40 = £10`, and `40 − 10 = £30`, which matches. So the original price was `£40`. (The £30 is 75% of the original, and `30 ÷ 0.75 = 40`.)
Art hook Rewind bar: a bar shown at £30 = 75%, with a ghost extension growing it back to the full 100% length labelled £40.