Problem-solving method

The equal stage method, explained for parents

For problem sums that end with "and then they had the same". That sentence is the most useful one in the question, and most children read straight past it. Here is what it buys you, and where the marks get lost.

Used from
P5
Taught at
P5–P6

What this method is

A way of deciding where to start drawing. These questions hand you a total, change each share differently, then say the two ended up level. A child who starts at "at first" is drawing the part with all the unknowns in it, and stalls there.

Equal stage starts at the other end. The moment they are equal is the one place you already know both bars match, so draw that first, call each 1 unit, and let the changes tell you what the start must have been.

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.

  • “they had equal amounts”

    The one length in the question you are given for free. Draw it before anything else and make each person 1 unit.

  • “they each had the same number”

    Same signal wearing different clothes. It does not matter what the quantity is, only that the two are level at one named moment.

  • “spent half of his money”

    A fraction of that person's OWN amount, so the bar left at the end is that fraction of the start. Halved means their starting bar is 2 units, not 1.

  • “had $210 altogether at first”

    The total belongs to the START. You can only use it once both starting amounts are written in units, which is the step children skip.

Watch it work

Aisyah and Wei Ming had $210 altogether. Aisyah was then given $30, and Wei Ming spent half of his money. In the end they each had the same amount. How much money did Wei Ming have at first?

Both moments, drawn on one scale. Wei Ming is two equal boxes at first and one at the end, because spending half is what halves the bar. Aisyah's bar only reaches the same length once the money she was given is added on. The two end bars matching is the fact the question hands you for free, and it is where the drawing starts.

  1. Draw the end, not the beginning

    at the end: Aisyah = 1 unit, Wei Ming = 1 unit

    The question says they finished level, so both bars are the same length. This is the only length you are handed, and it costs nothing to write it down first.

  2. Undo the halving

    Wei Ming: 1 unit at the end came from 2 units at first

    He spent half, so what is LEFT is the unit. Working forwards tempts a child to make him 1 unit at the start. Working back from the end makes it 2, which is the whole trick.

  3. Undo the gift

    Aisyah: 1 unit at the end, so 1 unit less $30 at first

    She was given money, so she had less before it arrived. Subtract it to get back to her starting bar. Adding it here is the second most common slip.

  4. Now the total is usable

    2 units + 1 unit − $30 = $210, so 3 units = $240 and 1 unit = $80

    The $210 was always a starting total, which is why it could not be used until both starting amounts were in units. One equation, and the unit falls out.

  5. Answer the person who was asked about

    Wei Ming at first = 2 × $80 = $160

    Check both ways. Aisyah started with $50, and $50 plus $30 is $80. Wei Ming started with $160, and half of that is $80. Level, as promised.

The one it gets confused with

Three methods sit close to this one, and the question that separates them is always about what the question actually told you.

  • Before and after

    Does the question name a moment where the two are LEVEL? If it only says something changed, you still need two pictures, but there is no equal length to anchor them to.

  • Constant total, difference and part

    Is something unchanged the whole way through? Those methods anchor on the thing that never moved. This one anchors on a moment the two MATCH, which is not the same idea.

  • Units and parts

    Units and parts is how you turn the equal moment into numbers once you have it. Equal stage is the earlier decision about where the 1 unit belongs in the first place.

How can you tell it has clicked?

Give them a question ending in "and then they had the same" and watch where the pencil lands. A child who draws the END has the method. One who writes "at first" and stops is trying to work forwards through the unknowns.

The sharper test: ask which person is 2 units. With a reason, "the one who spent half, because the half left is the unit", they have it. With a shrug, they copied the shape from a similar question.

One thing to try tonight

One question, before any drawing: which sentence tells you the two are the same? Have them underline it. In most of these questions it is the second to last line, and it is the line that gets skimmed.

Then one more: so how many units is each of them at the start? That is where the method actually lives. Everything after it is arithmetic they can already do.

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.

Try it free