Turning Words into Sums

The idea in one sentence

Most word problems aren’t hard because of the arithmetic — they’re hard because it isn’t obvious what to calculate, and drawing the situation is what makes it obvious.

Draw the bar

A bar model turns a sentence into a picture. One bar stands for the whole, split into parts.

Sam has 45 stickers. Ali has 17 fewer. How many altogether?

Draw Sam’s bar. Draw Ali’s shorter bar beside it, with the 17 gap marked. Now you can see that Ali has \(45 - 17 = 28\), and the question wants both bars: \(45 + 28 = 73\).

The picture does two things at once. It shows which operation is needed, and it shows that there are two steps — the commonest reason an answer is wrong is stopping after the first one.

Keywords help, until they don’t

“Altogether” often means add. “Left” often means subtract. “Each” often means multiply or divide.

But these are hints, not rules, and problems are frequently written to defeat them. “Sam has 12 more than Ali, who has 30” uses “more” for a sum, while “how many more does Sam have?” uses the same word for a difference. Keyword-hunting without reading is how a learner ends up adding two numbers that should have been compared.

The reliable method is to ask what is actually happening, then check whether the answer should be bigger or smaller than what you started with.

A method that works

1. Read it twice. The first read is for the story, the second for the numbers.
2. Say what you’re looking for, in words. “How many Ali has.”
3. Draw it — a bar, a number line, a quick sketch.
4. Estimate before calculating. Roughly 45 and roughly 30 is about 75.
5. Calculate.
6. Answer the question that was asked, with units, and check it against the estimate.

Step 6 catches more errors than any other. Plenty of correct arithmetic gets marked wrong because it answered a different question than the one at the bottom of the problem.

Multi-step problems

Most real problems need more than one operation, and the trick is to name the missing middle quantity.

A box of 6 pens costs £4.50. How much for 15 pens?

You can’t get there in one move. The middle quantity is the price of one pen: \(4.50 \div 6 = 0.75\). Then \(15 \times 0.75 = £11.25\).

Writing down “price of one pen” before calculating turns an impossible-looking problem into two easy ones.

Common mix-ups

• Grabbing the numbers and guessing an operation. Read first.
• Trusting keywords over the actual situation.
• Stopping halfway on a two-step problem.
• Leaving off the units, or answering a different question.

Check yourself

A shop sells notebooks at £3 each. Priya buys 4 and pays with a £20 note. What’s the change — and what is the middle quantity?

The middle quantity is the total cost: \(4 \times 3 = £12\). Then the change is \(20 - 12 = £8\).