Cardinal Math

Grades 3–5 · 3.NF.3, 4.NF.1, 4.NF.2, 4.NF.3, 5.NF.1 · the long version

How do I help my child with fractions?

Fractions: equal parts, equivalence, adding

Your child could shade three of four boxes and call it three quarters. Then the homework asked for 1/4 + 2/4 and got 3/8, or which is bigger, 1/3 or 1/8, and got 1/8. That's where fractions stop being pictures and start needing an idea.

The idea: a fraction counts pieces of a particular size. The bottom number says what size; the top counts them. 1/4 + 2/4 is three pieces, all still fourths. Stuck children aren't stuck on the pictures; the two numbers look like ordinary numbers, and ordinary numbers get added, and bigger means more.

What it looks like with a paper strip

Fold a paper strip in half, then in half again: four equal pieces. Shade one, then two more. Three pieces, each a fourth: 3/4. Fold a second strip once more, into eighths, and hold them side by side: two eighths cover one fourth. Everything in this guide is that strip, folded.

The five ways it goes wrong

These mistakes aren't random. Nearly every wrong answer comes from one of five rules that almost work.

  1. Adds the tops and adds the bottoms

    Adds the numerators and adds the denominators: 1/4 + 2/4 is 3/8.

    1/4 + 2/4 they write 3/8 it is 3/4

    Why children reach for it. It's what the eye does with two numbers over two numbers, and it even looks tidy. The bottom number isn't yet the name of the piece to them. The most famous fraction mistake there is.

    What replaces it. Fold a strip into fourths. Shade one, then two more, and ask how many pieces and what size. Nothing became an eighth. Later try a half and a fourth: the half becomes two fourths first.

  2. Adds the same number to top and bottom

    Makes an equivalent fraction by adding to both parts: 2/3 becomes 3/4.

    Equal to 2/3 they write 3/4 it is 4/6

    Why children reach for it. "Do the same to both" kept things balanced before, so the child does it here. But a fraction survives multiplying both parts, not adding to both. 3/4 is more than 2/3.

    What replaces it. Fold thirds, then fold the same strip into sixths. Each third is now two sixths, 2/3 is 4/6, and nothing was added: every piece got cut. Ask "what did we do to every piece?"

  3. Bigger numbers mean a bigger fraction

    Orders fractions by their numbers: 1/8 is bigger than 1/3 because 8 is bigger than 3.

    Which is bigger, 1/3 or 1/8? they write 1/8 it is 1/3

    Why children reach for it. For six years a bigger number meant more. Fractions are the first place a bigger number on the bottom means less, and that takes a while to believe.

    What replaces it. Share one chocolate bar between three people and another between eight, and ask which share they want. Then write the shares as fractions: the bottom counts the people sharing.

  4. Judges by the gap to one

    Compares by how many pieces are missing from a whole: 5/6 and 7/8 are both one short, so they must be equal.

    Which is bigger, 5/6 or 7/8? they write the same it is 7/8

    Why children reach for it. The child has noticed something real: both are one piece short. They miss that an eighth is smaller than a sixth, so 7/8 is short by less. A good mistake; they're thinking about size.

    What replaces it. Draw both on strips the same length. The missing eighth is visibly smaller than the missing sixth; ask which strip has less missing. The thinking is already right.

  5. Takes the smaller piece from the larger

    In 3 1/4 − 1 3/4, takes 1/4 from 3/4 because that's the way round that works, and gets 2 2/4.

    3 1/4 − 1 3/4 they write 2 2/4 it is 1 2/4

    Why children reach for it. Smaller-from-larger, the classic subtraction mistake, reborn with fractions. There wasn't enough on top, so the child swapped the order instead of opening a whole into fourths.

    What replaces it. Same move as regrouping with tens: 3 1/4 is two wholes and five fourths. Build it with paper fourths and take three away; the whole that had to open is right there on the table.

The five-minute check

A short run of problems on a phone or tablet, chosen as your child answers. If one of the five patterns is behind their mistakes, the check names it. Nothing to install, nothing stored.

Hand the phone over, or play it here. When it says all done, take it back and tap see what we found.

Start the five-minute check

What the school is probably doing

Area models
Circles and bars cut into equal parts and shaded. Still the best picture for equal parts.
Fraction strips
Paper strips folded into halves, fourths, eighths. Where equivalence becomes obvious instead of a rule.
Number lines
A fraction as a point between 0 and 1. Schools lean on this from grade 3 because it makes a fraction a number, not a picture of pizza.
Equivalence by multiplying
Top and bottom multiplied by the same number: folding every piece again, written down.

They're not the new way versus the right way, just one idea at different distances from the objects. Whichever way you learned, insist on knowing what size the pieces are.

Questions parents ask

What grade is this?

Equal parts and unit fractions start in grade 3 in the US (3.NF), equivalence and adding with the same denominator are grade 4 (4.NF.1, 4.NF.3), and adding with different denominators is grade 5 (5.NF.1).

Why can't you add the bottoms?

The bottom number isn't a quantity, it's the name of the piece. Two fourths and one fourth are three fourths, the way two apples and one apple are three apples.

Is 2/4 the same as 1/2?

Same amount, different name. Fold a strip in half and shade one half, then fold it again: the shaded part is now two fourths and nothing changed.

Pizza or number line?

Both, in that order. Pizza for equal parts and adding same-size pieces; the number line for seeing 3/4 as a number between 0 and 1, and for comparing.

They say 1/8 is bigger than 1/3.

The bottom number is still a quantity to them. The chocolate bar shared three ways and eight ways settles it fast; the words take longer.

How long should we practise?

Five to ten minutes most days beats an hour on Sunday. Three or four problems done slowly with a folded strip are worth more than a page.

Cardinal Math watches for all of these, every day.

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