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PSLE Mathematics · Primary 5 and 6

Model Drawing for Word Problems

Model drawing uses bars of equal units to show the quantities in a word problem, so you can see the relationships before you calculate. It is the main way PSLE word problems on whole numbers, fractions and ratio are solved at primary level, and it prepares you for algebra, where a letter plays the role of one unit. This guide covers the four models you will use most: part-whole, comparison, before-and-after, and equal fractions. It ends with a worked example and three practice problems.

What the syllabus expects

  • Draw part-whole models to show a total made up of several parts
  • Draw comparison models to show 'more than', 'less than' and 'times as many'
  • Draw before-and-after models when quantities are added, removed or transferred
  • Use equal units to solve problems where a fraction of one quantity equals a fraction of another
  • Find the value of one unit and use it to find the quantities asked for
  • Check answers by substituting them back into the conditions of the problem

Units and part-whole models

A model is a set of bars drawn to show the quantities in a problem. The idea that makes models work is the unit: every unit in your drawing must stand for the same amount, so draw units the same length. In a part-whole model, one long bar is the total and it is split into parts. If 3/5 of a class are girls and there are 40 pupils, draw a bar of 5 units, shade 3 units for the girls, and label the whole bar 40. Then 1 unit = 40 ÷ 5 = 8 and the girls are 3 × 8 = 24. Label every bar with a name, and write the given numbers in the right places, including question marks for what you need to find.

Comparison models

A comparison model draws one bar for each person, lined up at the left so you can see the difference. 'Sam has 4 times as many cards as Tim' means Tim's bar is 1 unit and Sam's bar is 4 units. If they have 135 cards altogether, 5 units = 135, so 1 unit = 27 and Sam has 108. 'Ann has 18 more beads than Ben' means Ann's bar is the same as Ben's plus an extra piece labelled 18. If they have 100 beads altogether, take away the extra 18 to leave 82, which is 2 equal bars, so Ben has 41 and Ann has 59. Note that '4 times as many' is not the same as '4 more'.

Before-and-after and transfer models

When quantities change, draw the bars before the change, then draw them again after it. Before you draw, decide what stays the same. If only one person spends money, the other person's amount does not change. If one person gives some to another, the total does not change. If both receive the same amount, the difference does not change. For a transfer, remember that giving away an amount changes the difference by twice that amount. If Ali gives Ben 15 stickers and they then have the same number, Ali must have had 2 × 15 = 30 more than Ben at the start, because Ali loses 15 and Ben gains 15.

Equal fractions of two quantities

Some problems say that a fraction of one quantity equals a fraction of another, for example '2/5 of the boys is equal to 1/2 of the girls'. Make the numerators the same, because equal numerators mean equal pieces. 1/2 = 2/4, so 2/5 of the boys equals 2/4 of the girls. Each of those 2 pieces is the same size, so the boys are 5 units and the girls are 4 units. If there are 360 pupils, 9 units = 360, 1 unit = 40, so there are 200 boys and 160 girls. Check: 2/5 of 200 = 80 and 1/2 of 160 = 80. This idea links model drawing with ratio, because the boys and girls are in the ratio 5 : 4.

Worked example

Ahmad and Ben had $400 altogether. After Ahmad spent 1/3 of his money and Ben spent $50, they had the same amount of money left. How much money did Ahmad have at first?

  1. Draw Ahmad's bar as 3 units, because he spent 1/3 of it. He spent 1 unit and has 2 units left.
  2. Ben has the same amount left as Ahmad, so Ben's bar is 2 units plus the $50 he spent.
  3. Altogether at first: 3 units + 2 units + $50 = 5 units + $50 = $400.
  4. 5 units = $400 − $50 = $350.
  5. 1 unit = $350 ÷ 5 = $70.
  6. Ahmad at first = 3 units = 3 × $70 = $210.
  7. Check: Ahmad had $210 and spent $70, leaving $140. Ben had $400 − $210 = $190 and spent $50, leaving $140. They have the same amount.

Answer: Ahmad had $210 at first.

Common mistakes

  • Drawing units of different lengths, so that bars look equal when they are not. Keep every unit the same size, or the model will mislead you.
  • Using units from two different wholes as if they were the same size, such as 1/3 of Ahmad's money and 1/3 of the total. Decide what each unit stands for before you draw.
  • Reading '4 times as many' as '4 more'. 4 times as many means 4 units against 1 unit, while 4 more means an extra piece labelled 4.
  • Forgetting that a transfer changes the difference by twice the amount given. If one person gives 15 to another and they become equal, the difference at first was 30.
  • Answering with the value of 1 unit instead of the quantity asked for. Look back at the question mark on your model before writing the answer.
  • Not labelling the bars. Unlabelled bars make it easy to mix up who is who in before-and-after problems.

Exam tips

  • Read the whole question once, then draw the bars sentence by sentence, labelling each one with a name and the given numbers.
  • In before-and-after problems, decide what stays the same (one quantity, the total or the difference) before you draw anything.
  • Finish by checking your answer against every sentence in the question, not just the last one.

Practice questions

1. Easy: Sam has 4 times as many cards as Tim. They have 135 cards altogether. How many cards does Sam have?

Show answer

108 cards. Working: Tim has 1 unit and Sam has 4 units, so 5 units = 135 and 1 unit = 27. Sam has 4 × 27 = 108.

2. Medium: There are 84 pens and pencils in a box. 2/3 of the pens is equal to 1/2 of the pencils. How many pencils are there?

Show answer

48 pencils. Working: 1/2 = 2/4, so 2/3 of the pens equals 2/4 of the pencils. Pens are 3 units and pencils are 4 units, so 7 units = 84 and 1 unit = 12. Pencils = 4 × 12 = 48. Check: 2/3 of 36 = 24 and 1/2 of 48 = 24.

3. Harder: Lily had 3 times as much money as Mona. After Lily spent $45 and Mona received $15, they had the same amount of money. How much money did Lily have at first?

Show answer

$90. Working: Mona is 1 unit and Lily is 3 units. After the changes, 3 units − $45 = 1 unit + $15, so 2 units = $45 + $15 = $60 and 1 unit = $30. Lily had 3 × $30 = $90. Check: $90 − $45 = $45 and $30 + $15 = $45.

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