Topics · Primary Maths · Word Problems
Word problems and the bar model
The hard part of a word problem is never the arithmetic. It is working out which sum to do — and a bar model is how you find out, by turning each sentence into a length you can see.
- 569 exam papers analysed
- appears in 97% of P2 Maths papers
- ~4 min read
01 Draw first, calculate last
A word problem read straight through is a wall of words. Drawn, it becomes a picture with one piece missing, and the missing piece usually tells you the operation without your having to decide.
The order that works is always the same: read one sentence, draw one thing. Do not read to the end and then try to hold it all in your head.
> Ali has 3 units of stickers. Bala has the same, and 48 more. Together they have 240.
Two bars, drawn the same width where the sentence says they are the same, and one extra piece for the 48.
02 The four shapes a problem comes in
Almost every primary word problem is one of these four, and knowing which one you are looking at is most of the work.
- Part and whole. Two or more parts making a total. Draw them end to end in one bar. Missing total → add. Missing part → subtract.
- Comparison. Two quantities, one bigger. Draw two bars aligned on the left; the extra sticking out is the difference.
- Equal groups. The same amount several times. Draw equal units. This is multiplication one way and division the other.
- Before and after. Something changes. Draw two models, one for each moment, and line them up so what stayed the same is obvious.
03 Working with units
When a problem gives no numbers for the parts — only how they compare — draw units. Each unit is the same unknown amount, and the whole method rests on drawing them the same size.
Once the picture is drawn, find what one unit is worth. Everything else follows from it by multiplying.
04 Checking the answer is possible
The last mark is lost more often than the first. Two checks take a few seconds each and catch most of it.
- Put the answer back into the picture. If Ali's bar is now 96 and Bala's is 144, does 96 + 144 give the 240 the question stated? If not, the model or the arithmetic is wrong.
- Ask whether the size is sensible. An answer bigger than the total, or a negative number of children, means something went in the wrong direction.
- Answer the question that was asked. Finding one unit is usually not the answer — the question wanted Bala's stickers, or the difference, or how many are left. Read the last sentence again before writing.
- Write the unit. Dollars, kilograms, children. A bare number is not an answer to a question about money.
05 How to say them
These are words most children have never said out loud, and a word you cannot say is a word you avoid using. Capitals mark the syllable you lean on. Tap the speaker to hear it.
- bar modelmore than onebar models
A drawing that shows quantities as lengths, so a comparison you have to read becomes one you can see. The standard Singapore method for word problems.
- unitmore than oneunits
One equal share in a bar model, standing for an amount you do not know yet. Find what one unit is worth and everything else follows.
- part-wholemore than onepart-whole models
A model where two or more parts sit end to end to make a total.
- differencesayDIF-uh-runssmore than onedifferences
How much bigger one quantity is than another — the piece sticking out when two bars are lined up on the left. Found by subtracting.
- remaindersayri-MAYN-duhmore than oneremainders
What is left over after sharing equally. In a word problem it often has to be rounded up rather than down — five children still need a sixth box.
06 What each level is tested on
The same topic, asked differently as it goes up. The percentage is the share of papers at that level in which it appears.
- P2
One-step problems in words: adding and taking away, with a simple part-whole picture.
Practise P2 Word Problems — 10 questions →97% of P2 papers
- P3
Two steps, and the first comparison models with a difference.
Practise P3 Word Problems — 10 questions →89% of P3 papers
- P4
Multi-step problems mixing the four operations, and equal-group models.
Practise P4 Word Problems — 10 questions →92% of P4 papers
- P5
Units where the parts are given only as a comparison, and fraction or ratio quantities inside the story.
Practise P5 Word Problems — 10 questions →86% of P5 papers
- P6
Before-and-after problems where a quantity changes, and models where the total stays fixed while the parts move.
Practise P6 Word Problems — 10 questions →93% of P6 papers
07 Mistakes worth knowing about
Each of these is wrong. Decide why before you open it.
“I read the whole question, then started drawing.”
Draw as you read, one sentence at a time. Holding four facts in your head and then drawing is where facts get dropped.
“The question said ‘more’, so I added.”
Keyword-spotting fails constantly. “Ali has 48 more than Bala. Ali has 200. How many has Bala?” says “more” and needs a subtraction. The picture tells you; the keyword does not.
“My units are different sizes but it is only a sketch.”
Then it is not a bar model. Equal units drawn unequal is exactly the information the drawing exists to carry, and once it is lost the picture cannot help you.
“I found one unit is 40, so the answer is 40.”
Almost never. One unit is the step, not the destination. Go back and read what the question actually asked for.
“240 ÷ 7 = 34 remainder 2, so 34 boxes.”
The two left over still need a box, so it is 35. Whether a remainder rounds up or is discarded depends on the story, not on the arithmetic.
Now try it
50 questions across 5 levels, marked as you go with the working shown.