Fractions as numbers & equivalence
Unit & non-unit fractions with small denominators as numbers; numerator/denominator; equivalence (½ = 2/4); compare same-denominator fractions; add/subtract same-denominator.
A fraction is a number, not just a picture of a sliced-up cake. It names an exact amount that sits between the whole numbers on the number line, and different-looking fractions can name the very same amount.
What it means
Write a fraction as a/b. The bottom number, the denominator (b), says how many equal parts one whole has been split into. The top number, the numerator (a), says how many of those parts you have taken. So 3/4 means: cut a whole into 4 equal parts, then take 3 of them.
A unit fraction has 1 on top — 1/2, 1/3, 1/4 — a single equal part. A non-unit fraction takes more than one part, like 3/4 or 2/3.
The key idea at this stage is that a fraction is a number with a home on the number line. Split the gap between 0 and 1 into b equal steps; the fraction a/b is the point you reach after a steps. 1/4 is one quarter of the way from 0 to 1; 3/4 is three quarters of the way.
Because a fraction is a number, two different fractions can be equal. 1/2 and 2/4 land on exactly the same point: halving a whole, or quartering it and taking two pieces, gives the same amount. These are equivalent fractions. You get one from another by multiplying (or dividing) top and bottom by the same number: 1/2 = (1×2)/(2×2) = 2/4. This works because you are cutting every part into the same number of smaller parts — more, thinner slices, but the same total.
When two fractions share a denominator, comparing and combining them is easy. The parts are the same size, so you just count them. 3/5 is more than 2/5 because 3 parts beat 2 parts. To add or subtract, keep the denominator and work on the numerators: 2/5 + 1/5 = 3/5.
Worked examples
- Name it. A ribbon is cut into 8 equal pieces and you take 5. That is
5/8— denominator 8 (parts in the whole), numerator 5 (parts taken). - Find an equivalent. Is
1/3the same as2/6? Multiply top and bottom of1/3by 2:(1×2)/(3×2) = 2/6. Yes — same point on the line. - Compare. Which is bigger,
4/7or5/7? Same denominator, so count sevenths: 5 beats 4, so5/7is bigger. - Add and subtract.
3/8 + 4/8 = 7/8. And7/8 − 2/8 = 5/8. The denominator never changes; only the count of parts does.
The generative-art connection
Fractions are what a rotating pattern is built from. Spin a motif around a centre and stamp it at equal turns: 4 copies means each sits 1/4 of a full turn — 90° — from the next; 6 copies means 1/6 of a turn each. The fraction literally IS the angle between arms. In the dot-multiplier tool (dot-multiplier.nico.art), setting the number of arms slices the circle into that many equal fractions, and you watch equivalence appear directly: 3 arms and 6 arms both land dots at the top, because 1/3 and 2/6 mark the same positions. A colour wheel does the same for hue — stepping by 1/12 of the wheel gives twelve evenly spaced colours. Making the art is choosing a denominator and counting numerators around a circle.
Common misconceptions
- “The bigger the denominator, the bigger the fraction.” No — a bigger denominator means smaller parts.
1/8is less than1/4, because eighths are thinner slices than quarters. - “Add across, top and bottom.”
1/4 + 1/4is not2/8. The pieces are quarters; two quarters make2/4. Add the numerators, keep the denominator. - “Equivalent fractions are a different amount.”
2/4looks like more pieces than1/2, but they are the same number — the same point on the line, just described with finer slices.
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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A pizza is cut into 6 equal slices. You eat 4 of them and your friend eats 1. Write the fraction you ate and name its numerator and denominator. Which unit fraction is a single slice? Who ate more, and how do you know?
Answer
You ate `4/6`: numerator = 4 (slices taken), denominator = 6 (equal parts in the whole). One single slice is the unit fraction `1/6`. Your friend ate `1/6`. You ate more because both fractions are counted in sixths (same-size parts), so `4/6 > 1/6` — 4 parts beat 1 part.
Art hook Draw a circle split into 6 equal pie wedges; a slider fills wedges one at a time and prints the fraction `n/6` as it grows. Each wedge is `1/6` of a full turn (60 degrees).
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Find a fraction equivalent to `3/4` that has 8 as its denominator. Show how you got it.
Answer
Multiply top and bottom by 2: `(3×2)/(4×2) = 6/8`. So `3/4 = 6/8` — the same point on the line, just cut into thinner slices.
Art hook Show two bars of equal length: the top split into 4 parts with 3 shaded, the bottom split into 8 with 6 shaded. The shaded lengths line up exactly, proving `3/4 = 6/8`.
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Put these three fractions in order from smallest to largest: `5/9`, `2/9`, `8/9`. How do you know?
Answer
`2/9`, `5/9`, `8/9`. They share the denominator 9, so the parts are the same size — just count ninths: 2 < 5 < 8.
Art hook Place 3 dots on a 0-to-1 number line divided into 9 equal steps, at steps 2, 5 and 8. Animate them sliding into sorted order along the line.
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Work these out, keeping the denominator the same: `4/10 + 3/10` and `9/10 − 5/10`.
Answer
`4/10 + 3/10 = 7/10` and `9/10 − 5/10 = 4/10`. The denominator (10) stays the same; only the count of tenths changes.
Art hook A single bar of 10 equal cells: light up 4 blue then 3 green to show 7 filled, then dim 5 to show the subtraction. The colour blocks read as the numerators.
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Is `2/3` the same amount as `4/6`? Explain using equal parts, not just a picture.
Answer
Yes. Cut every third into 2 smaller pieces: `(2×2)/(3×2) = 4/6`. Twice as many slices, each half the size, so the total amount is unchanged. They are equivalent fractions — the same point on the line.
Art hook Overlay two rings on a circle: inner ring in 3 wedges (2 shaded), outer ring in 6 wedges (4 shaded). The shaded arcs sweep the exact same angle, showing `2/3 = 4/6`.
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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A chocolate bar is broken into 5 equal squares and you take 2. Write this as a fraction. What does the bottom number 5 tell you?
Answer
`2/5`. The bottom number (denominator) 5 tells you the whole was split into 5 equal parts.
Art hook A row of 5 equal squares; clicking squares shades them and updates the fraction `n/5` live above the bar.
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Which of these are unit fractions (a single equal part): `1/4`, `3/5`, `1/7`, `2/3`?
Answer
`1/4` and `1/7` are unit fractions — they have 1 on top. `3/5` and `2/3` are non-unit fractions.
Art hook Show four circles each split into parts; light up the single-wedge ones and let the child tap which are unit fractions. Correct taps glow gold.
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The gap from 0 to 1 on a number line is split into 4 equal steps. Which step lands on `3/4`? Which lands on `1/4`?
Answer
`3/4` is the 3rd step from 0; `1/4` is the 1st step. Each step is one quarter of the way from 0 to 1.
Art hook Draw a 0-to-1 line with 4 tick marks; a marker hops one tick per click and prints the fraction `k/4` it lands on.
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Which is bigger, `2/8` or `5/8`? They share a denominator, so how do you decide?
Answer
`5/8` is bigger. Same-size eighths, so just count: 5 parts beat 2 parts.
Art hook Two stacked 8-cell bars, one filled to 2 and one to 5; the longer coloured bar visibly wins the comparison.
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Fill in the missing number so the fractions are equivalent: `1/2 = ?/10`.
Answer
`5/10`. Multiply top and bottom of `1/2` by 5: `(1×5)/(2×5) = 5/10`.
Art hook A bar split in 2 with one half shaded, morphing (animated) into a bar split in 10 with 5 shaded — same shaded length throughout.
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Add these, keeping the denominator: `1/6 + 2/6 + 2/6`.
Answer
`5/6`. Add the numerators (1 + 2 + 2 = 5) and keep the denominator 6.
Art hook Fill a 6-wedge pie in three coloured bursts of 1, 2 and 2 wedges; the running total `n/6` shows the sum reaching `5/6`.
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Spot the mistake. Sam says `1/3 + 1/3 = 2/6`. What did Sam do wrong, and what is the right answer?
Answer
Sam added the denominators too. You keep the denominator: `1/3 + 1/3 = 2/3`. The pieces are thirds, so two of them make `2/3`, not `2/6`.
Art hook Show two thirds of a pie shading in; a wrong '2/6' label appears then gets crossed out and replaced by '2/3' as the same shaded arc stays put.
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Reasoning: without drawing, which is bigger, `1/5` or `1/9`? Explain why a bigger bottom number does NOT mean a bigger fraction.
Answer
`1/5` is bigger. A bigger denominator (9) means the whole is cut into more, thinner slices, so each single slice `1/9` is smaller than `1/5`.
Art hook Split one circle into 5 wedges and another into 9; shade one wedge in each. The fifth is visibly the fatter slice, showing `1/5 > 1/9`.
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Find two different fractions that are both equivalent to `1/2` (other than `2/4`).
Answer
Many work, e.g. `3/6` and `5/10` (also `4/8`, `6/12`). Each comes from multiplying top and bottom of `1/2` by the same number.
Art hook Stack several bars of equal length, each split into a different even number of parts (4, 6, 8, 10) with exactly half shaded; the shaded ends all line up, showing `2/4`, `3/6`, `4/8`, `5/10` are all the same half.
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A ribbon is `7/12` long and you cut off `3/12`. How much ribbon is left?
Answer
`4/12`. Subtract the numerators (7 − 3 = 4), keep the denominator 12.
Art hook A 12-segment ribbon shaded to 7; scissors snip 3 segments and the remaining `4/12` stays highlighted.
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On a circle with 12 evenly spaced dots (like a clock), one dot is `1/12` of the way round. Which dot is `1/4` of the way round? Which is `1/2`?
Answer
`1/4` of the way is the 3rd dot (since `1/4 = 3/12`); `1/2` of the way is the 6th dot (`1/2 = 6/12`). Equivalent fractions point to the same dot.
Art hook Twelve dots evenly on a circle; highlight the dot for a chosen unit fraction and reveal its twelfths-equivalent, e.g. `1/4` and `3/12` both light the same dot — equivalence made visible on a clock face.
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Open-ended: choose a denominator you like (say 8). Build a design by shading a fraction of a circle split into that many wedges, then shade an equivalent fraction on a second circle with a different denominator. Which pair did you make?
Answer
Any valid equivalent pair, e.g. shade `4/8` of one circle and `1/2` of another — both sweep half the circle. Success = the two shaded arcs cover the same angle.
Art hook Two side-by-side wedge circles with independent denominator sliders; when the child's two shaded arcs match angle, the circles pulse to celebrate a found equivalence.