Model Stacking

When three people are linked by different clues, drawing three unrelated bars makes the question feel larger than it is. Put the bars on one baseline. A shared base quantity becomes repeated equal units, while a fixed “more” or “less” piece stays visibly separate.

Parents: ask your child to choose whose amount should be the base and predict how many equal units the total will contain before revealing the stack.

By the end you will be able to choose a base quantity, stack three related bars, keep fixed pieces separate, and solve the common unit from a total.

Choose a base quantity

Model stacking fits when several quantities can be described using one amount as a base. If one person has “12 more” than another, draw the shared base and a separate 12 piece. If a third person has “twice as many”, copy the base bar twice.

It does not fit when the relationships use unrelated wholes or when no common base can be identified. A comparison model can be enough for two quantities; stacking is useful when a third linked quantity joins the picture.

Stack the related bars

Here is the full question:

Amir, Bella and Chen have 100 stamps altogether. Bella has 12 more stamps than Amir. Chen has twice as many stamps as Amir. How many stamps does each child have?

Let Amir's collection be 1 unit. Bella has the same 1 unit plus a fixed 12-stamp piece. Chen has two copies of Amir's unit. Align all three left edges so every 1u block has the same width.

Translate the stack

$$\text{Amir} = 1u$$

$$\text{Bella} = 1u + 12\text{ stamps}$$

$$\text{Chen} = 2u$$

Remove the fixed piece before dividing

The three amounts total 4u plus the fixed 12 stamps. The total is 100, so first remove the fixed piece. Only the remaining 88 stamps are carried by four equal units.

Find one unit

$$4u + 12 = 100$$

$$4u = 100 - 12 = 88$$

$$1u = 88 \div 4 = 22\text{ stamps}$$

$$\text{Amir} = 1 \times 22 = 22\text{ stamps}$$

$$\text{Bella} = 22 + 12 = 34\text{ stamps}$$

$$\text{Chen} = 2 \times 22 = 44\text{ stamps}$$

🔬

Model stack: find the shared unit

🤔 Predict first: What should be removed before dividing the total?

The fixed 12 stampsOne unitChen's whole bar

The answers are Amir 22, Bella 34 and Chen 44. The model gives one base value, then each bar is read from its own relationship.

Check every clue

$$22 + 34 + 44 = 100\text{ stamps}$$

$$34 - 22 = 12\text{ stamps}$$

$$44 = 2 \times 22\text{ stamps}$$

Fixed pieces are not units

A rectangle labelled 12 stamps has a known value. It is not automatically the same length as 1u. If you counted it as another unit, you would write 5u = 100 and get the wrong answer. The drawing must show the difference between a repeated base piece and a fixed add-on.

Watch out — easily mixed up

Quick recap

Graduated practice

1. Three card collections

Ana, Bo and Cara have 87 cards altogether. Bo has 7 more cards than Ana. Cara has three times as many cards as Ana. Find each amount.

Show solution

Use Ana as 1u. Bo is 1u plus 7, and Cara is 3u. There are five units and a fixed 7 cards.

Practice 1

$$5u + 7 = 87$$

$$5u = 80$$

$$1u = 80 \div 5 = 16\text{ cards}$$

$$\text{Ana} = 16$$

$$\text{Bo} = 16 + 7 = 23$$

$$\text{Cara} = 3 \times 16 = 48$$

Check: 16 + 23 + 48 = 87, Bo is 7 more and Cara is triple Ana.

2. A shorter first bar

Dev has 5 fewer stickers than Eva. Farah has twice as many stickers as Eva. Together they have 135 stickers. Find each amount.

Show solution

Let Eva be 1u. Dev is 1u minus 5, and Farah is 2u. The total has four units minus 5.

Practice 2

$$4u - 5 = 135$$

$$4u = 140$$

$$1u = 140 \div 4 = 35\text{ stickers}$$

$$\text{Dev} = 35 - 5 = 30$$

$$\text{Eva} = 35$$

$$\text{Farah} = 2 \times 35 = 70$$

Check: 30 + 35 + 70 = 135 and Dev is 5 fewer than Eva.

3. A comparison built on a second bar

Hari has some beads. Gita has twice as many as Hari. Iman has 9 more beads than Gita. Together they have 129 beads. Find each amount.

Show solution

Use Hari as 1u. Gita is 2u and Iman is 2u plus 9, giving five units and a fixed 9 beads.

Practice 3

$$5u + 9 = 129$$

$$5u = 120$$

$$1u = 120 \div 5 = 24\text{ beads}$$

$$\text{Hari} = 24$$

$$\text{Gita} = 2 \times 24 = 48$$

$$\text{Iman} = 48 + 9 = 57$$

Check: 24 + 48 + 57 = 129 and Iman is 9 more than Gita.

🎯 Mastery check

Answer all 7 — your progress is saved on this device.

1. Which amount is the useful base in the stamp problem?

   Amir's amountBella's fixed 12The final total

2. Amir is 1u, Bella is 1u + 12 and Chen is 2u. How many true units are in the stack?

   3u4u5u

3. Why is Bella's 12-stamp piece kept separate?

   It is a known fixed amountIt is another copy of AmirIt must be doubled for Chen

4. From 4u + 12 = 100, what equation finds the four unit blocks?

   4u = 100 - 124u = 100 + 125u = 100

5. In the stamp problem, what is one unit?

   12 stamps22 stamps44 stamps

6. If Ana is 1u, Bo is 1u + 7 and Cara is 3u, how many units are in their total?

   4u5u6u

7. Which check tests Chen's comparison in the main example?

   44 = 2 × 2222 + 34 + 44 = 10034 − 22 = 12

See my score