Decimal & fraction arithmetic
Decimals to thousandths; +, −, ×, ÷ decimals; add/subtract fractions with unlike denominators; multiply a fraction/whole by a fraction; multiplication as scaling; terminating vs repeating decimals.
Once you can name fractions and decimals, the next step is to compute with them — add, subtract, multiply and divide — and to see why a fraction sometimes turns into a decimal that stops and sometimes into one that repeats forever.
What it means
A decimal extends place value past the ones. Each step to the right divides the value by ten: tenths (0.1), hundredths (0.01), thousandths (0.001). So 0.375 means 3 tenths + 7 hundredths + 5 thousandths.
Adding and subtracting decimals works because you can only combine matching place values — tenths with tenths, hundredths with hundredths. Line the numbers up by the decimal point (equivalently, line up the columns) and add as with whole numbers. 0.3 + 0.375 = 0.675.
Multiplying decimals: multiply as if there were no points, then count the total number of digits after the point in both factors and put that many after the point in the answer. For 0.4 × 0.375, compute 4 × 375 = 1500; there are 1 + 3 = 4 decimal places, so the answer is 0.1500 = 0.15. Dividing decimals: scale both numbers by the same power of ten until the divisor is a whole number, then divide normally. For 4.5 ÷ 0.5, multiply both by 10 to get 45 ÷ 5 = 9; the answer is unchanged because you scaled both sides equally.
Adding fractions with unlike denominators. You can only add parts of the same size, so first rewrite both fractions over a common denominator — a number both denominators divide into. 1/2 + 1/3: rewrite over 6 as 3/6 + 2/6 = 5/6.
Multiplying fractions is simpler than adding: multiply the numerators, multiply the denominators. 2/3 × 4/5 = 8/15. A whole number is just a fraction over 1: 6 × 2/3 = 6/1 × 2/3 = 12/3 = 4.
Multiplication as scaling. Multiplying by a number greater than 1 stretches a quantity; multiplying by a number between 0 and 1 shrinks it. × 3/4 makes something three-quarters as big — smaller, without any subtracting.
Terminating vs repeating decimals. Every fraction equals a decimal (divide top by bottom). Put the fraction in lowest terms first; then if the denominator’s only prime factors are 2 and 5, the decimal terminates: 3/8 = 0.375. Otherwise it repeats forever in a fixed cycle: 1/3 = 0.333…, 1/7 = 0.142857142857…. (Lowest terms matters: 3/6 looks like it has a 3 in the denominator, but it reduces to 1/2 = 0.5, which stops.)
Worked examples
1 — Subtract decimals. 1.2 − 0.45. Line up by the point: 1.20 − 0.45 = 0.75.
2 — Add unlike fractions.
3/4 + 1/6
common denominator 12: 9/12 + 2/12
= 11/123 — Multiply, as scaling. 2/3 × 9 = 18/3 = 6. Two-thirds of 9 is 6 — smaller than 9, because the operator is less than 1.
4 — Terminating or repeating?
3/8 = 3 ÷ 8 = 0.375 (8 = 2×2×2) terminates
5/6 = 5 ÷ 6 = 0.8333… (6 = 2×3) repeats (the 3)The generative-art connection
Multiplication-as-scaling is the beating heart of generative art. Give a shape one rule — “each copy is 3/4 the size of the last, rotated a little” — and repeat it: the copies spiral inward and shrink toward a point, drawing a fractal. Because 3/4 is less than 1 the picture visibly contracts; swap in 5/4 and it explodes outward. The child reads the size of the fraction straight off the drawing.
Repeating decimals become visible too. 1/7 = 0.142857… cycles through six digits forever; map each digit to an angle or a colour and you get a closed, endlessly looping pattern, while a terminating decimal like 3/8 simply stops. The dot-multiplier tool turns “one dot × N” into a symmetric starburst — an array you can literally count — making a whole-number or fraction product something you build rather than memorise.
Common misconceptions
- “To add fractions, add the tops and add the bottoms.”
1/2 + 1/3is not2/5. You must make the pieces the same size first (a common denominator), then add only the numerators. - “Multiplying always makes things bigger.” Only when you multiply by more than 1.
× 3/4or× 0.5makes the result smaller. - “Line up the last digits when adding decimals.” Align by the decimal point, not the right-hand edge:
1.2 + 0.45means1.20 + 0.45 = 1.65, not1.2 + 0.45shoved right.
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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Work these two decimal calculations, lining the numbers up by the decimal point: `2.6 + 1.85` and `3.4 − 1.27`.
Answer
`2.60 + 1.85 = 4.45` and `3.40 − 1.27 = 2.13`. Adding a zero to fill the empty place keeps the columns matching: tenths with tenths, hundredths with hundredths.
Art hook A decimal number line from 0 to 5 on Canvas. The learner types each addend; draw a coloured arrow of that length starting where the last ended, so `2.6` then `+1.85` chains two arrows and lands the dot at `4.45`. Tick marks every `0.1`.
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Multiply and divide: `0.6 × 0.05`, then `1.8 ÷ 0.4`.
Answer
`0.6 × 0.05`: `6 × 5 = 30`, and `1 + 2 = 3` decimal places give `0.030 = 0.03`. `1.8 ÷ 0.4`: scale both by 10 → `18 ÷ 4 = 4.5`.
Art hook A 10×10 grid = one whole. Shade a `0.6`-wide by `0.05`-tall rectangle; the overlapped little cells are the product `0.03`. Sliders set each factor and the shaded overlap redraws live.
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Add and subtract these fractions, choosing a common denominator first: `5/6 + 3/8` and `7/10 − 2/15`.
Answer
`5/6 + 3/8`: common denominator 24 → `20/24 + 9/24 = 29/24 = 1 5/24`. `7/10 − 2/15`: common denominator 30 → `21/30 − 4/30 = 17/30`.
Art hook Two circular pie dials sharing a common slice-count (24 or 30). Each fraction fills its arc; press + to sweep the arcs together into one dial, showing the sum's arc — and when it passes a full turn, a second ring starts (the whole number 1).
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Multiply, and say whether the answer is bigger or smaller than the starting number and why: `3/5 × 20` and `2/3 × 3/4`.
Answer
`3/5 × 20 = 60/5 = 12` — smaller than 20 because `3/5` is less than 1, so it scales 20 down. `2/3 × 3/4 = 6/12 = 1/2` — smaller than `3/4` for the same reason.
Art hook A square shrinking under repeated scaling: draw a shape, then each frame multiply its side by `2/3`. Nested squares contract toward the centre. A toggle to `5/4` makes them grow outward instead, so scaling < 1 vs > 1 is visible at a glance.
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Without dividing all the way, decide which of these become a decimal that stops and which repeat forever, then check one: `7/20`, `4/15`, `9/40`.
Answer
`7/20`: `20 = 2×2×5`, only 2s and 5s → terminates (`= 0.35`). `4/15`: `15 = 3×5`, has a 3 → repeats (`0.2666…`). `9/40`: `40 = 2×2×2×5` → terminates (`= 0.225`). Denominators with only prime factors 2 and 5 terminate.
Art hook For each fraction, run long division and map each successive digit to a dot placed around a circle by angle. A terminating decimal draws a short finite trail that stops; a repeater loops back and retraces the same closed loop forever — the eye sees 'stops' vs 'cycles'.
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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Copy the sum lining the numbers up by the decimal point, filling the gap with a zero: `0.7 + 0.25`. What is the total?
Answer
`0.70 + 0.25 = 0.95`. The zero keeps the tenths under tenths and hundredths under hundredths.
Art hook Base-ten style: a strip split into 100 squares. Light `0.7` of it one colour and `0.25` another; the lit fraction reads `0.95`.
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Fill in the missing place value: `0.408` means `4` tenths, `___` hundredths, and `8` thousandths.
Answer
`0` hundredths. Reading left to right past the point: tenths, hundredths, thousandths, so `0.408 = 4` tenths + `0` hundredths + `8` thousandths.
Art hook Three stacked bars labelled tenths/hundredths/thousandths. Type a decimal and each bar fills to that digit's height — a tiny place-value equaliser.
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Subtract, lining up by the point: `5.6 − 2.35`.
Answer
`5.60 − 2.35 = 3.25`.
Art hook A number line with a dot at `5.6`; a leftward arrow of length `2.35` slides the dot to `3.25`.
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Multiply as if the points weren't there, then place the point: `0.3 × 0.7`.
Answer
`3 × 7 = 21`; `1 + 1 = 2` decimal places → `0.21`.
Art hook A 10×10 grid; shade a `0.3` by `0.7` rectangle and count the `21` overlapped small cells = `0.21`.
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Divide by scaling both numbers so the divisor is a whole number: `2.4 ÷ 0.6`.
Answer
Multiply both by 10: `24 ÷ 6 = 4`. Scaling both sides equally leaves the answer unchanged.
Art hook Show `2.4` as 24 dots; group them into equal piles of `0.6` (6 dots each). The number of piles, 4, is the quotient.
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Rewrite both over a common denominator and add: `1/4 + 2/5`.
Answer
Common denominator 20: `5/20 + 8/20 = 13/20`.
Art hook A strip of 20 cells (a 1×20 bar, or a 4×5 grid read in one running order). Fill the first `5` cells blue for `1/4 = 5/20`, then the NEXT `8` cells green for `2/5 = 8/20`; because the colours never overlap, the filled run is exactly `13` cells = `13/20`.
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Subtract these unlike fractions: `5/6 − 1/4`.
Answer
Common denominator 12: `10/12 − 3/12 = 7/12`.
Art hook Two clock-like dials divided into 12; sweep `5/6` (10 slices) then erase `1/4` (3 slices), leaving a `7/12` arc.
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Multiply the numerators and the denominators: `3/4 × 5/6`. Simplify if you can.
Answer
`(3×5)/(4×6) = 15/24 = 5/8` (divide top and bottom by 3).
Art hook Fold a rectangle into 4 columns, shade 3; then into 6 rows, shade 5. The doubly-shaded cells (`15` of `24`) are the product `15/24 = 5/8`.
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A photo `12 cm` wide is scaled by `2/3`. Is the new width more or less than `12 cm`? Find it.
Answer
Less, because `2/3` is under 1. `2/3 × 12 = 24/3 = 8 cm`.
Art hook A rectangle that repeatedly scales by `2/3` each click, nesting inward — the shrink is visible; a slider swaps the factor above 1 to watch it grow.
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Spot the mistake: a pupil writes `1/3 + 1/4 = 2/7`. What went wrong, and what is the correct answer?
Answer
They added tops and bottoms; you can only add same-size pieces. Common denominator 12: `4/12 + 3/12 = 7/12`.
Art hook Side-by-side dials: the wrong `2/7` arc next to the true `7/12` arc, so the mismatch is obvious to the eye.
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Predict then check: does `3/8` give a decimal that stops or repeats? Does `2/9`? Use the denominator's prime factors.
Answer
`3/8`: `8 = 2×2×2` → stops (`0.375`). `2/9`: `9 = 3×3` → repeats (`0.222…`). Only prime factors 2 and 5 terminate.
Art hook Long-division digits plotted as dots around a circle: `3/8` traces a short trail that halts; `2/9` loops the same dot forever.
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Order these three from smallest to largest by turning each into a decimal: `2/5`, `0.36`, `3/8`.
Answer
`2/5 = 0.4`, `3/8 = 0.375`, `0.36`. Order: `0.36 < 3/8 (0.375) < 2/5 (0.4)`.
Art hook Drop each value as a dot onto a number line from 0 to 1; the sort is done visually by where the dots land.