Primary 4 Method and Model Drawing
Stacking model
Use a stacking model when a question gives the total for several items and tells you that one type costs more or less than another. The model helps you compare the items and find an unknown amount.
Maya spent $3.50 on 3 pencils and 1 eraser. Each pencil cost $0.50 more than the eraser. How much did the eraser cost?
Compare one pencil and one eraser
The green parts match. One pencil has an extra $0.50.
Stack all four items
Four green parts match. Each pencil has one orange $0.50 extra part.
The eraser cost $0.50.
Try together
Kai spent $4.50 on 3 pens and 1 eraser. Each pen cost $0.50 more than the eraser. How much did the eraser cost?
Reveal the answer
Three extra parts cost 3 × $0.50 = $1.50. Then $4.50 − $1.50 = $3.00 for four equal parts. $3.00 ÷ 4 = $0.75, so the eraser cost $0.75.
Systematic listing
Use an ordered list when you need to test possible numbers against two conditions. Start with the smallest possible number of boxes, add one box at a time, and stop at the first total that fits both conditions.
At a bakery, Nora packs cookies into boxes. If she puts 4 cookies in each box, 5 cookies are left. If she puts 6 cookies in each box, she needs 3 more cookies. (a) What is the smallest number of cookies Nora has? (b) How many boxes does she use?
| Number of boxes | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| 4 cookies in each box | 4 | 8 | 12 | 16 |
| 5 cookies left add 5 | 9 | 13 | 17 | 21 |
| 6 cookies in each box | 6 | 12 | 18 | 24 |
| 3 cookies short subtract 3 | 3 | 9 | 15 | 21 |
Check: 4 × 4 + 5 = 21. Also, 6 × 4 − 3 = 21.
Guess and check
Use guess and check when two totals must both be true. Start with a sensible guess, calculate the second total, then use “too many” or “too few” to choose a better guess. Keep each trial in the table.
A toy shop has 34 toy motorcycles and toy cars altogether. The toys have 106 wheels altogether. Each toy motorcycle has 2 wheels and each toy car has 4 wheels. How many of each toy are there?
Condition 1: Toy motorcycles + toy cars = 34 toys.
? + ? = 34 toys
Condition 2: Motorcycle wheels + car wheels = 106 wheels.
| Toy motorcycles | Toy cars | Motorcycle wheels | Car wheels | Total wheels | Check |
|---|---|---|---|---|---|
| 17 | 17 | 17 × 2 = 34 | 17 × 4 = 68 | 34 + 68 = 102 | ✗ Too few wheels |
| 16 | 18 | 16 × 2 = 32 | 18 × 4 = 72 | 32 + 72 = 104 | ✗ Still too few |
| 15 | 19 | 15 × 2 = 30 | 19 × 4 = 76 | 30 + 76 = 106 | ✓ Both conditions met |
Answer: 15 toy motorcycles and 19 toy cars. Check: 15 + 19 = 34 toys, and 15 × 2 + 19 × 4 = 106 wheels.
Alternative method: Assumption method
Use the same 34 toy motorcycles and toy cars with 106 wheels. Start by pretending that every toy is a motorcycle, then correct the missing wheels.
- 1
Assume all 34 toys are motorcycles.
34 × 2 = 68 wheels - 2
Compare with the real total. We are short of wheels.
106 − 68 = 38 wheels short - 3
Replace one motorcycle with one car. Find the wheel difference for one swap.
4 − 2 = 2 more wheels - 4
Find how many motorcycles must be replaced by cars.
38 ÷ 2 = 19 toy cars - 5
Find the motorcycles left from the 34 toys.
34 − 19 = 15 toy motorcycles - 6
Check both conditions.
15 + 19 = 34 toys; 15 × 2 + 19 × 4 = 106 wheels
Assume all 34 toys are motorcycles. Show the first step when you are ready.
Answer: 19 toy cars and 15 toy motorcycles.
Try a simpler question with pictures
A shelf has 8 toy motorcycles and toy cars with 22 wheels altogether. A motorcycle has 2 wheels and a car has 4 wheels. How many of each toy are there?
- 1
Assume all 8 toys are motorcycles.
8 × 2 = 16 wheels - 2
Compare with the 22 wheels in the question.
22 − 16 = 6 wheels short - 3
A car has 2 more wheels than a motorcycle.
4 − 2 = 2; 6 ÷ 2 = 3 swaps - 4
Change one motorcycle to a car.
18 wheels; 4 still short - 5
Change a second motorcycle to a car.
20 wheels; 2 still short - 6
Change a third motorcycle to a car.
22 wheels; no shortage - 7
Count and check both kinds of toy.
5 motorcycles + 3 cars = 8 toys; 5 × 2 + 3 × 4 = 22 wheels
Look at the 8 toy spaces. First, pretend every toy is a motorcycle.
Answer: 5 toy motorcycles and 3 toy cars.
Try together
Another box has 28 toy motorcycles and toy cars with 88 wheels altogether. How many of each toy are there?
Reveal the answer
Start with 14 of each: 84 wheels, which is 4 too few. Swap two motorcycles for two cars. There are 12 motorcycles and 16 cars; 12 × 2 + 16 × 4 = 88 wheels.
Before and after method
Draw two models when an amount changes. Label one “before” and one “after”. A transfer changes both people’s bars but keeps their combined total the same.
At first, Maya had 18 more stickers than Ben. She then gave 15 stickers to Ben. After that, Ben had twice as many stickers as Maya. How many stickers did each of them have at first?
Before
After
- 1
Draw the before model. Maya has 18 more stickers than Ben.
Before: Maya − Ben = 18 - 2
Split Maya’s extra 18 into the 15 she gives Ben and the part she keeps.
18 − 15 = 3 stickers remain - 3
Draw the after model using the same starting bar: Maya has 3 extra and Ben has the 15 she gave him.
After: Maya = common part + 3; Ben = common part + 15 - 4
Compare the new extra parts. Ben has 12 more after the transfer.
15 − 3 = 12 stickers - 5
Ben has twice as many as Maya, so their after difference is one Maya amount.
Maya: 12; Ben: 2 × 12 = 24 - 6
Reverse the transfer to find how many each had before.
Maya: 12 + 15 = 27; Ben: 24 − 15 = 9 - 7
Check both clues and the unchanged total.
27 − 9 = 18; 27 − 15 = 12; 9 + 15 = 24; 24 = 2 × 12
Read the before, transfer and after clues. Show the first step to draw the before model.
Answer: Before, Maya had 27 stickers and Ben had 9 stickers.
Working backwards
Working backwards helps us find what two people had at first when we know what they had at the end. Read the story in order and show each change in the table. Then follow the arrows back up, using the opposite operation for each change. When one person gives something to the other, both amounts change.
At first, Aisha and Ryan each had some trading cards. Aisha gave 6 cards to Ryan. Ryan then gave half of his cards to Aisha. Finally, Aisha gave 4 cards to her sister. In the end, Aisha had 22 cards and Ryan had 12 cards. How many cards did each of them have at first?
| Event | Aisha | Ryan |
|---|---|---|
| At first | ? | ? |
| Aisha gave 6 to Ryan | −6+6 | +6−6 |
| After giving 6 | ? | ? |
| Ryan gave half to Aisha | +half−12 | ÷2×2 |
| After giving half | ? | ? |
| Aisha gave 4 to her sister | −4+4 | No change |
| At the end | 22 | 12 |
Before Aisha gave 4 away: 22 + 4 = 26 cards. Ryan still had 12.
Before Ryan gave half: 12 × 2 = 24 cards for Ryan. He gave 12 to Aisha, so she had 26 − 12 = 14.
Before Aisha gave 6: 14 + 6 = 20 for Aisha; 24 − 6 = 18 for Ryan.
Check forwards: 20, 18 → 14, 24 → 26, 12 → 22, 12.
Answer: At first, Aisha had 20 cards and Ryan had 18 cards.
Read the question one clue at a time, then follow the solid arrows down the table.
Sets concept
A set is one complete group that repeats in the same way. Find what one set contains and costs. Then use the total cost to find the number of sets and the number of items asked for.
Sharon bought some donuts and cupcakes. For every 3 donuts, she bought 2 cupcakes. Each donut cost $2 and each cupcake cost $3. She spent $48 altogether. How many cupcakes did she buy?
One set
$48 ÷ $12 = 4 sets
4 sets × 2 cupcakes = 8 cupcakes
- 1
Put 3 donuts and 2 cupcakes in one set.
One set = 3 donuts + 2 cupcakes - 2
Find the cost of each kind of item in the set.
3 × $2 = $6; 2 × $3 = $6 - 3
Add both costs to find the price of one set.
$6 + $6 = $12 per set - 4
Use the total spent to find how many complete sets Sharon bought.
$12 × 4 = $48, so $48 ÷ $12 = 4 sets - 5
Each set has 2 cupcakes. Count the cupcakes in 4 sets.
4 × 2 = 8 cupcakes - 6
Check both the item counts and the total cost.
12 donuts × $2 = $24; 8 cupcakes × $3 = $24; $24 + $24 = $48
Read the question. First, group the donuts and cupcakes that are bought together.
Answer: Sharon bought 8 cupcakes.
Constant Difference Concept
When two ages increase by the same number of years, their difference remains constant. Represent the later ages with equal units and match the extra units to the known age difference.
In how many years will Father be 3 times as old as Nora?
Nora is 8 years old and her father is 36 years old. In the future, Father will be 3 times as old as Nora.
Read the ages and the future relationship before drawing the bars.
See worked example
Nora is 8 years old and her father is 36 years old. In how many years will her father be 3 times as old as Nora?
- Draw the current ages: Father 36 years and Nora 8 years.
- Find their age difference: 36 − 8 = 28 years. This difference does not change.
- For the future ages, draw 3 equal units for Father and 1 equal unit for Nora. The difference is 2 units.
- 2 units = 28 years, so 1 unit = 28 ÷ 2 = 14 years. Nora will be 14 years old.
- Find the time passed: 14 − 8 = 6 years. Check: Father will be 36 + 6 = 42 years old, and 42 = 3 × 14.
Answer: In 6 years.
Try together
Ben is 10 years old and his aunt is 34 years old. In how many years will his aunt be twice as old as Ben?
Reveal the answer
The difference is 34 − 10 = 24 years. In the future, the difference is 1 unit, so Ben will be 24 years old. It will take 24 − 10 = 14 years. Check: 34 + 14 = 48 and 48 = 2 × 24.