Problem-solving method

Branching, or a fraction of the remainder, explained for parents

For problem sums where the second fraction is of what is left, not of the whole. Here is the drawing that keeps the two apart, and the three words that tell you to use it.

Used from
P5
Taught at
P5–P6

What this method is

When a second fraction applies to what is left rather than to the whole, you split the bar twice. Once into spent and remainder, then the remainder into its own pieces. The second bar covers the remainder and only the remainder, which is the whole method.

The drawing is small. The idea it protects is not: a fraction is always a fraction of something, and the something changes as you go.

When to reach for it

These are the words in a question that point at this method. You do not need to be able to solve it to spot them.

  • “of the remainder”

    The signal, and the most reliable one on this page. Whatever follows is a fraction of what is left, not of the total.

  • “of the remaining buns”

    The same instruction spelled out in the question's own nouns. Branch at this word.

  • “then she gave 13\frac{1}{3} of what was left”

    What was left is now the whole. Everything measured after this point is measured against it.

  • “after spending 25\frac{2}{5}, he spent 12\frac{1}{2} of the rest”

    Two fractions in sequence, and the second is of the rest. This is the full shape in one sentence.

  • “he had 150 left”

    The final leftover, stated as a real number. It is usually what pins the whole chain to actual amounts.

Watch it work

A baker made 900 buns. She sold 40% of them and gave 59\frac{5}{9} of the remaining buns to a charity. She then ate some buns and had 150 buns left. What percentage of the buns she made did she eat?

The top bar splits the 900 into the remainder, 540, and the 360 already sold. The lower bar splits only the remainder: charity 300, eaten 90, and the 150 left over.

  1. Split the total first

    sold = 40% of 900 = 360, so the remainder = 900 − 360 = 540

    One bar, two pieces, and nothing clever yet. The 540 is the number everything after this point will be measured against.

  2. Branch the remainder, not the total

    59\frac{5}{9} of 540 = 300 to the charity

    Five ninths of the remainder is 300. Of the total it would have been 500, and that gap is where the mark is won or lost on questions like this one.

  3. See what the branch leaves

    540 − 300 = 240 buns

    This is the second whole shrinking again. Following it on the drawing is easier than following it in a sentence, which is the argument for drawing at all.

  4. Find what she ate

    240 − 150 = 90 buns

    The 150 she was left with is given, so what she ate is simply the rest of that lower bar.

  5. Answer the question that was asked

    90 out of 900 → 10%

    Back to the original 900, because the question asked for a percentage of what she made. Children who answer 37.5%, a percentage of the remainder, have done the maths and missed the sentence.

The one it gets confused with

Each of these draws bars too. The question is how many wholes are in play, and whether they are the same whole.

  • The bar model

    Do both fractions apply to the same whole? Then one split is enough and there is nothing to branch.

  • Units and parts

    Once the remainder is split you are back to counting units. Branching only decides which bar you count them on.

  • Before, change, after

    Is the change a transfer between two people rather than a sequence of removals? Then draw two states instead of one branch.

How can you tell it has clicked?

Point at the second fraction and ask what it is a fraction of. A child with the method answers with the remainder, not the total, and can show you which part of the drawing they mean.

On paper, check how long the second bar is. If it runs the full length of the first one, the fraction has been taken of the total, and the misunderstanding is sitting there in the drawing before it ever reaches the arithmetic.

One thing to try tonight

Read the question aloud together and stop at every fraction to ask one thing: of what?

That is the whole intervention. It takes a minute, it targets exactly the step that fails, and it works even if you have not touched fractions in twenty years.

Where your child will meet it

Want the bigger picture on how these methods fit together? Maths heuristics, explained for parents

See which methods your child reaches for

StudyLah names the method in every worked solution, so you can see whether the problem was the method or the arithmetic.

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