Primary 3 Methods and Model Drawing
Keywords
- Comparison model
- Aligned bars that show how quantities compare.
- Equal parts
- Bar sections that represent the same amount.
- Difference
- The amount by which one quantity is greater than another.
- Total
- The sum of all the quantities.
- Systematic listing
- An ordered list that covers every possible case once.
- One unit
- The value represented by one equal section of a bar.
- Guess and check
- Test a possible answer against the given conditions, then adjust the next guess.
- Condition
- A requirement that a correct answer must satisfy.
Comparison model: three quantities
Draw one bar for each quantity. Align the equal parts, mark the difference, and use a bracket to show the total. Subtract the known quantity and the extra part before dividing the remaining amount into equal parts.
Ali has 24 stickers. Ben has 12 more stickers than Chen. The three children have 96 stickers altogether. How many stickers does Ben have?
Read the question. We know Ali’s amount, the difference between Ben and Chen, and the total.
See worked example
Ali has 24 stickers. Ben has 12 more stickers than Chen. The three children have 96 stickers altogether. How many stickers does Ben have?
- Draw Ali’s bar and label it 24 stickers. Draw equal base parts for Ben and Chen, with an extra 12 stickers on Ben’s bar.
- Bracket all three bars to show the total of 96 stickers.
- Ben and Chen have 96 − 24 = 72 stickers altogether.
- Subtract Ben’s extra 12 stickers: 72 − 12 = 60 stickers in two equal parts.
- Chen has 60 ÷ 2 = 30 stickers.
- Ben has 30 + 12 = 42 stickers. Check: 24 + 30 + 42 = 96.
Answer: Ben has 42 stickers.
Try together
Ava has 18 stickers. Bo has 10 more stickers than Cy. They have 82 stickers altogether. How many stickers does Bo have?
Reveal the answer
82 − 18 = 64; 64 − 10 = 54; 54 ÷ 2 = 27; 27 + 10 = 37. Bo has 37 stickers. Check: 18 + 27 + 37 = 82.
Systematic listing
Use systematic listing when a problem asks for every possible combination or result. Organise the cases in a fixed order to check that none are missed or repeated.
A bookmark costs $3 and a keychain costs $5. Maya buys exactly 3 items, choosing only these two kinds. What are all the possible amounts she could pay?
| Bookmarks | Keychains | Working | Total cost |
|---|---|---|---|
| 3 | 0 | 3 × $3 = $90 × $5 = $0$9 + $0 = $9 | $9 |
| 2 | 1 | 2 × $3 = $61 × $5 = $5$6 + $5 = $11 | $11 |
| 1 | 2 | 1 × $3 = $32 × $5 = $10$3 + $10 = $13 | $13 |
| 0 | 3 | 0 × $3 = $03 × $5 = $15$0 + $15 = $15 | $15 |
All possible amounts: $9, $11, $13 and $15.
The number of bookmarks plus the number of keychains must equal 3.
See worked example
At a school fair, a bookmark costs $3 and a keychain costs $5. Maya buys exactly 3 items, choosing only bookmarks and keychains. What are all the possible amounts she could pay?
- The number of bookmarks plus the number of keychains is 3.
- Start with 3 bookmarks and 0 keychains. Bookmarks: 3 × $3 = $9. Keychains: 0 × $5 = $0. Total: $9 + $0 = $9.
- Then list 2 bookmarks and 1 keychain. Bookmarks: 2 × $3 = $6. Keychain: 1 × $5 = $5. Total: $6 + $5 = $11.
- Next list 1 bookmark and 2 keychains. Bookmark: 1 × $3 = $3. Keychains: 2 × $5 = $10. Total: $3 + $10 = $13.
- Finish with 0 bookmarks and 3 keychains. Bookmarks: 0 × $3 = $0. Keychains: 3 × $5 = $15. Total: $0 + $15 = $15.
- The bookmark counts 3, 2, 1, 0 cover every possibility. The possible amounts are $9, $11, $13 and $15.
Answer: $9, $11, $13 and $15.
Try together
At a bakery, a bun costs $2 and a muffin costs $5. Noah buys exactly 4 items, choosing only buns and muffins. What are all the possible amounts he could pay?
Reveal the answer
List the bun/muffin counts: 4/0, 3/1, 2/2, 1/3, 0/4. The possible amounts are $8, $11, $14, $17 and $20.
Units and parts
Use equal units to represent quantities related by a multiple. Count all the units to find the value of one unit, then find the quantity asked for.
Ahmad and Bala have 600 marbles altogether. Ahmad has 3 times as many marbles as Bala. How many marbles does Ahmad have?
altogether
Ahmad = 3 units; Bala = 1 unit.
3 units + 1 unit = 4 units = 600 marbles.
1 unit = 600 ÷ 4 = 150 marbles.
Ahmad = 3 × 150 = 450 marbles. Check: 450 + 150 = 600.
See worked example
Ahmad and Bala have 600 marbles altogether. Ahmad has 3 times as many marbles as Bala. How many marbles does Ahmad have?
- Represent Ahmad with 3 equal units and Bala with 1 equal unit.
- The total is 3 + 1 = 4 units, so 4 units = 600 marbles.
- 1 unit = 600 ÷ 4 = 150 marbles.
- Ahmad has 3 × 150 = 450 marbles. Check: Bala has 150, and 450 + 150 = 600.
Answer: Ahmad has 450 marbles.
Try together
Ali and Ben have 240 stickers altogether. Ali has twice as many stickers as Ben. How many stickers does Ali have?
Reveal the answer
Ali has 2 units and Ben has 1 unit. 3 units = 240, so 1 unit = 80. Ali has 2 × 80 = 160 stickers.
Guess and check
Use guess and check when two conditions must both be satisfied. Make a possible guess that meets the first condition, calculate its result, and compare it with the second condition. Use whether the result is too high or too low to improve the next guess.
There are 12 chickens and cows altogether. They have 34 legs altogether. A chicken has 2 legs; a cow has 4 legs. How many chickens are there?
Condition 1: Chickens + cows = 12 animals.
Condition 2: Chicken legs + cow legs = 34 legs.
| Number of chickens | Number of cows | Chicken legs | Cow legs | Total legs | Check |
|---|---|---|---|---|---|
| 6 | 6 | 6 × 2 = 12 | 6 × 4 = 24 | 12 + 24 = 36 | ✗ 36 > 34 Too many legs |
| 8 | 4 | 8 × 2 = 16 | 4 × 4 = 16 | 16 + 16 = 32 | ✗ 32 < 34 Too few legs |
| 7 | 5 | 7 × 2 = 14 | 5 × 4 = 20 | 14 + 20 = 34 | ✓ 34 = 34 Both conditions met |
Answer: There are 7 chickens. Check: 7 + 5 = 12 animals and 14 + 20 = 34 legs.
Find the two conditions that every correct guess must satisfy.
See worked example
There are 12 chickens and cows on a farm. Altogether, they have 34 legs. How many chickens are there?
- Condition 1: The number of chickens plus the number of cows is 12. Condition 2: Their total number of legs is 34.
- Half of 12 is 6. First guess 6 chickens and 6 cows. Chicken legs: 6 × 2 = 12. Cow legs: 6 × 4 = 24. Total: 12 + 24 = 36 legs. This is 2 too many.
- Try more chickens and fewer cows: 8 chickens and 4 cows. Chicken legs: 8 × 2 = 16. Cow legs: 4 × 4 = 16. Total: 16 + 16 = 32 legs. This is 2 too few.
- The answer is between the two guesses. Try 7 chickens and 5 cows. Chicken legs: 7 × 2 = 14. Cow legs: 5 × 4 = 20. Total: 14 + 20 = 34 legs.
- Check both conditions: 7 + 5 = 12 animals and 14 + 20 = 34 legs.
Answer: There are 7 chickens.
Try together
A farm has 13 chickens and cows altogether. They have 36 legs. How many chickens are there?
Reveal the answer
There are 8 chickens and 5 cows. Check: 8 + 5 = 13 animals; 8 × 2 = 16 chicken legs and 5 × 4 = 20 cow legs; 16 + 20 = 36 legs.
Working backwards
Use working backwards when the final amount is known but the amount at first is unknown. Record the changes in the order they happened, then start with the final amount and apply the inverse operations in reverse order.
Zara had some stickers. She gave 8 stickers to Mei, received 6 stickers from her uncle, then gave half of her stickers to Ana. She had 11 stickers left. How many stickers did Zara have at first?
| Event | Stickers and working |
|---|---|
| At first | Z = ? |
| Gave 8 stickers to Mei | −8 +8 |
| After giving 8 | ? |
| Received 6 stickers | +6 −6 |
| After receiving 6 | ? |
| Gave half to Ana | ÷2 ×2 |
| At the end | 11 |
Before Zara gave half: 11 × 2 = 22.
Before Zara received 6: 22 − 6 = 16.
Before Zara gave 8: 16 + 8 = 24. So Z = 24.
Check forwards: 24 − 8 = 16; 16 + 6 = 22; 22 ÷ 2 = 11.
Answer: Zara had 24 stickers at first.
Follow the solid arrows down from Z to 11. Then start at 11 and follow the dotted arrows up.
See worked example
Zara had some stickers. She gave 8 stickers to Mei, received 6 stickers from her uncle, then gave half of her stickers to Ana. She had 11 stickers left. How many stickers did Zara have at first?
- Write the events in order: Z stickers at first → give 8 → receive 6 → give half → 11 stickers left.
- Before Zara gave half, she had 11 × 2 = 22 stickers.
- Before Zara received 6, she had 22 − 6 = 16 stickers.
- Before Zara gave 8, she had 16 + 8 = 24 stickers. So Z = 24.
- Check forwards: 24 − 8 = 16; 16 + 6 = 22; half of 22 is 11.
Answer: Zara had 24 stickers at first.
Try together
Omar had some cards. He gave 5 cards away, received 9 cards, then gave half of his cards away. He had 10 cards left. How many cards did he have at first?
Reveal the answer
Work backwards: 10 × 2 = 20; 20 − 9 = 11; 11 + 5 = 16. Omar had 16 cards at first. Check: 16 − 5 = 11, 11 + 9 = 20, and half of 20 is 10.
Before and after method
Draw separate models for the amounts before and after a change. Use the equal starting amounts and the difference in what was given away to find one equal part in the after model.
Amy and Ben each had the same number of stickers at first. Amy gave away 14 stickers and Ben gave away 22 stickers. After that, Amy had twice as many stickers as Ben. How many stickers did each have at first?
Before: equal amounts
Both complete bars represent the same number at first.
After: Amy has twice Ben’s amount
Extra after unit = 22 − 14 = 8 stickers.
The amounts at first were equal. Draw two equal bars.
Mark 14 given away by Amy and 22 given away by Ben on the before bars.
After the change, Amy has 2 equal units and Ben has 1 equal unit.
Ben gave away 22 − 14 = 8 more stickers. Their starting amounts were equal, so Amy has 8 more after. The extra unit is 8.
1 unit = 8 stickers. Amy has 2 × 8 = 16 stickers after; Ben has 8 stickers after.
Answer: Amy and Ben each had 30 stickers at first.
See worked example
Amy and Ben each had the same number of stickers at first. Amy gave away 14 stickers and Ben gave away 22 stickers. After that, Amy had twice as many stickers as Ben. How many stickers did each have at first?
- Draw two equal bars for Amy and Ben at first. Each bar represents an unknown but equal starting number.
- Show the changes on the before model: Amy gave away 14 stickers and Ben gave away 22 stickers.
- Draw a separate after model. Amy's amount is 2 equal units and Ben's amount is 1 equal unit.
- Ben gave away 22 − 14 = 8 more stickers. Because their starting amounts were equal, Amy has 8 more stickers after the change. This is 1 equal unit.
- 1 unit = 8 stickers. Amy had 2 × 8 = 16 stickers after; Ben had 8 stickers after.
- At first, Amy had 16 + 14 = 30 stickers and Ben had 8 + 22 = 30 stickers. Check: 30 − 14 = 16 and 30 − 22 = 8; 16 = 2 × 8.
Answer: Amy and Ben each had 30 stickers at first.
Try together
Nina and Omar each had the same number of tokens at first. Nina used 16 tokens and Omar used 26 tokens. Nina then had twice as many tokens as Omar. How many tokens did each have at first?
Reveal the answer
The difference in tokens used is 26 − 16 = 10, so one after unit is 10. Nina had 20 and Omar had 10 tokens after. At first, Nina had 20 + 16 = 36 and Omar had 10 + 26 = 36.
Group concept
Use the group concept when a fixed number of items is sold together for one price. Find how many whole groups the money can buy, then use any money left for single items.
Buns are sold at 3 for $5 or at $2 each. Mei buys some buns and spends exactly $12. What is the greatest number of buns Mei can buy?
Answer: Two groups give 6 buns for $10. The remaining $2 buys 1 more bun. Mei can buy at most 7 buns.
See worked example
Buns are sold at 3 for $5 or at $2 each. Mei buys some buns and spends exactly $12. What is the greatest number of buns Mei can buy?
- One group has 3 buns and costs $5. Three single buns cost $6, so the group costs less. Three groups cost $15, more than $12; Mei can buy at most two groups.
- Buy the first group: $12 − $5 = $7 left, with 3 buns bought.
- Buy another group: $7 − $5 = $2 left, with 6 buns bought.
- Buy one single bun: $2 − $2 = $0 left. The total number is 2 × 3 + 1 = 7 buns.
- Two groups give 6 buns for $10. The remaining $2 buys one more bun, so the greatest number is 7 buns.
Answer: Mei can buy at most 7 buns.
Practice together
Buns are sold at 3 for $5 or at $2 each. Ravi buys some buns and spends exactly $14. What is the greatest number of buns Ravi can buy?
Reveal the answer
Three groups cost $15, more than $14. Buy two groups: $14 − $5 = $9, then $9 − $5 = $4. Two single buns use the final $4, giving 2 × 3 + 2 = 8 buns.