The unitary method, made visual with PhET Unit Rates

The unitary method is one of the quietly important techniques in Australian secondary maths. If three kilos of apples cost $9, how much does one kilo cost? Three dollars. Five kilos? Fifteen.

It sounds trivial written like that. In a classroom, it's the strategy that lets students solve percentage problems, rate problems, currency conversions, and recipe scaling without memorising six different formulas. Find the "per one" amount, then multiply.

PhET's Unit Rates simulation is purpose-built for this. Worth knowing about, especially if your child is at the stage where word problems start to land harder than calculations.

What the simulation does

Unit Rates is a free, browser-based PhET sim. It opens to a market scene where you can buy fruit, candy, or paint at a given price per unit. You can:

• Adjust the price and quantity and see the cost update
• Use a double number line to scale quantities up or down
• Switch between three scenarios (shopping, racing, baking)
• Try challenge problems with hidden unit rates

No login, no download. Works on a phone but the dragging is much better on a tablet or laptop.

Three things it teaches well

1. The "per one" mental anchor

The whole sim is built around this. Every quantity has a corresponding price, and the price-per-unit is shown as a separate number that doesn't change as you scale.

For students who hit a word problem and freeze ("but how do I know what to do?"), the unitary method gives them a reliable first move: find what one of them costs (or weighs, or takes). Unit Rates turns this into muscle memory in twenty minutes.

2. Scaling as multiplication

Buy 2 oranges at 80c each: $1.60. Buy 6 oranges: $4.80. Buy 0.5 oranges (for some reason): 40c.

The double number line down the bottom of the sim shows two parallel scales — quantity on one, price on the other — moving in lockstep. By the time a student has worked through a few scenarios, "scale by multiplying" stops being a rule and becomes the obvious move.

3. Working backwards is the same operation in reverse

The challenge problems include questions like "if 3 oranges cost $2.40, how much does 1 orange cost?" The student divides to find the unit rate, then multiplies to find the answer they actually want.

This forward-backwards symmetry is the heart of the unitary method. Once a student trusts it in the sim, they trust it in word problems.

Where it falls short

A few honest limitations:

• Always money or distance. All three scenarios in the sim are framed around price-per-unit or speed. Other real rate contexts (population density, concentration, frequency, flow rate) don't get any time. You'll need a different resource for those.
• No algebra. By Year 9, the unitary method is supposed to feed into setting up proportional equations (3/x = 9/15, solve for x). The sim doesn't go there. Mathspace handles the algebraic move properly.
• Limited challenge ceiling. The sim is excellent for Year 7-8 and good revision for Year 9. By Year 10 it's mostly background skill.

For a complete unitary method unit you'd combine this sim with practice across varied contexts (Khan Academy has decent word problems), the algebraic version of the strategy (Mathspace works for this), and a teacher reviewing where the student is actually getting stuck.

How we use it in the weekly review

In the No Drama Koala program, Unit Rates tends to show up in two specific moments.

The first is for students who hit word problems and freeze. When Mathspace data shows them getting calculations right but stumbling whenever the question wraps the maths in a real situation, the unitary method is usually the missing strategy. I'll record a short Loom video using the sim to walk through the "find one first" move, then send a couple of follow-up practice questions.

The second is the "anything else" section of the weekly review email. Something like: "open this sim, set up a scenario, and write down three different ways the unit rate would help you. Tell me what you find." Open-ended enough to keep them thinking, specific enough that they don't drift.

Both work because the sim is free, no login, and renders cleanly on any device.

Try this with your child

If your child is in Year 7 to 9 and you want a quick test of whether the unitary method is in their toolkit, try this:

1. Open Unit Rates on their device.
2. Set the apple price to 50c each.
3. Ask: how much would 7 apples cost?

If they answer $3.50 and you ask how they worked it out, listen for "I worked out one apple, then times seven." That's the strategy.

If they answer with a guess, or freeze on the question, or reach for a calculator without showing any working — that's the gap. Not a crisis, but worth fixing. The unitary method shows up in percentage problems, rate problems, recipe scaling, currency conversion, and dozens of senior maths topics. A student who never installs it ends up memorising a new formula for each of those, which is exhausting and brittle.

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Want a qualified Australian teacher reviewing your child's actual rate and proportion work each week and setting next week's tasks calibrated to their specific gaps? That's exactly what the weekly review program does. Start with a free discovery check-in, I'll send back a personalised video diagnostic of where they're at.

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Related reading: PhET ratios and rates: building rate fluency · Teaching percentages with PhET · PhET decimals and the number line