Ratio and Proportion With Bar Models

A bar model makes ratio and proportion visible by giving each equal ratio part one equal box. If the ratio is 2:3, draw two equal boxes for the first quantity and three for the second. The boxes show the multiplicative relationship before any arithmetic begins: there are five equal parts altogether, the second quantity has one-and-a-half times as many parts as the first, and every box must have the same value within that problem.

That simple picture helps a child decide what to divide and what to multiply. It is especially useful when a question gives a total, one quantity or a difference. England’s Year 6 mathematics programme includes problems about the relative sizes of two quantities, using integer multiplication and division facts to find missing values. Its non-statutory guidance also refers to recognising proportionality in contexts such as recipes and recording ratios with notation such as a:b.[2]

Contents

What a ratio bar model shows

Build the model before calculating

Worked example: a total is known

Worked example: one quantity is known

Worked example: the difference is known

Worked example: unequal sharing

How to check proportional reasoning

Missteps the model can expose

A practical parent routine

Prompts that reveal understanding

Next step

What a ratio bar model shows

Take the ratio red counters : blue counters = 2:3. A compact model is:

Red: [ ][ ] Blue: [ ][ ][ ]

Each box represents the same number of counters. The model does not say there are only two red and three blue counters. It says that for every group of two equal parts of red, there are three equal parts of blue. Possible matching amounts include 2 and 3, 4 and 6, 10 and 15, or 18 and 27.

This is the central multiplicative idea. Moving from 2:3 to 10:15 multiplies both quantities by 5. Adding the same number to both sides would not preserve the ratio: 2:3 is not equivalent to 7:8.

A ratio compares quantities. The colon keeps their order clear. If a question says cats:dogs = 3:5, the three-part bar belongs to cats and the five-part bar belongs to dogs. Reversing the labels changes the ratio.

The Education Endowment Foundation describes manipulatives and representations as tools whose use matters. Its guidance says a representation should have a clear mathematical purpose and should reveal structure while supporting pupils towards independent mathematics.[3] Here, the boxes have one job: to expose equal multiplicative units. They are not decoration, and they can be faded once the child can reason from the structure without drawing every box.

Build the model before calculating

Use the same sequence for most ratio bar-model questions:

Read the order. Write the two quantity labels in the order given.

Draw equal parts. A ratio of 3:4 needs three equal boxes on one line and four on the other.

Place the known information. A total belongs across all boxes; one known quantity belongs across its own bar; a difference belongs across the unmatched boxes.

Find one part. Divide the known amount by the number of equal boxes it covers.

Scale to the required quantity. Multiply the one-part value by the relevant number of boxes.

Check the original relationship. Reconstruct the total, known quantity or difference and simplify the final ratio.

The phrase “one part” matters. It is a unit of the ratio model, not necessarily one physical object or one pound. In one problem a box might be worth 5 counters; in another it might be worth £12 or 70 grams.

Worked example: a total is known

There are red and blue counters in the ratio 2:3. There are 25 counters altogether. How many are blue?

Red: [ ][ ] Blue: [ ][ ][ ] Total: 25 across 5 equal parts

There are 2 + 3 = 5 parts altogether.

One part = 25 ÷ 5 = 5 counters

Blue occupies three parts:

Blue = 3 × 5 = 15 counters

Red occupies two parts:

Red = 2 × 5 = 10 counters

Check both pieces of information:

10 + 15 = 25, so the total is correct.

10:15 simplifies to 2:3 after dividing both terms by 5.

A frequent wrong route is 25 ÷ 3, because the question asks for blue and blue has three parts. The total of 25, however, spans both bars, so it must be divided by all five parts first.

Worked example: one quantity is known

A recipe uses flour and sugar in the ratio 4:3. It uses 280 g of flour. How much sugar is needed?

Flour: [ ][ ][ ][ ] = 280 g Sugar: [ ][ ][ ]

The known 280 g covers the four flour parts, not all seven parts.

One part = 280 ÷ 4 = 70 g

Sugar occupies three parts:

Sugar = 3 × 70 = 210 g

Check by recovering the ratio:

280:210 = 4:3, because both terms divide by 70.

You can also check multiplicatively. Sugar is three quarters of the flour amount, and 3/4 of 280 = 210. The bar model and the calculation describe the same relationship.

If the recipe is scaled again, both quantities must change by the same factor. Doubling 280 g to 560 g means doubling 210 g to 420 g. Changing only one amount would alter the recipe’s ratio.

Worked example: the difference is known

Cats and dogs at a rescue centre are in the ratio 3:5. There are 12 more dogs than cats. How many cats and dogs are there?

Cats: [ ][ ][ ] Dogs: [ ][ ][ ][ ][ ] Difference: the 2 unmatched dog parts = 12

The longer dog bar has 5 − 3 = 2 extra parts. Those two parts represent the known difference of 12.

One part = 12 ÷ 2 = 6 animals

Now scale each bar:

Cats = 3 × 6 = 18 Dogs = 5 × 6 = 30

Check:

30 − 18 = 12, so the difference is correct.

18:30 simplifies to 3:5, after dividing both terms by 6.

The known number is not always a total. Asking “Which boxes does 12 describe?” prevents the common mistake of dividing 12 by all eight parts.

Worked example: unequal sharing

Two children share £84 in the ratio 3:4. How much does each receive?

First share: [ ][ ][ ] Second share: [ ][ ][ ][ ] Total: £84 across 7 equal parts

There are 3 + 4 = 7 parts.

One part = £84 ÷ 7 = £12

So:

First share = 3 × £12 = £36 Second share = 4 × £12 = £48

Check:

£36 + £48 = £84.

36:48 simplifies to 3:4, after dividing both terms by 12.

This is unequal sharing, but it is still fair according to the stated ratio: each ratio part is worth the same £12. “Equal parts” does not mean the two people receive equal amounts; it means every box in the model has equal value.

As another quick example, a group has adults and children in the ratio 5:7, with 72 people altogether. The 12 ratio parts are worth 72 ÷ 12 = 6 people each, giving 30 adults and 42 children. The check is 30 + 42 = 72, and 30:42 = 5:7 after dividing by 6.

How to check proportional reasoning

A reliable answer should pass more than one check.

Check the known amount. If the total was given, add the two quantities. If one bar was given, rebuild that bar. If the difference was given, subtract the smaller quantity from the larger.

Check the ratio. Simplify the calculated pair. For a 2:3 problem, 14:21 is valid because both terms divide by 7; 14:20 is not.

Check the direction. In a 3:5 ratio, the quantity represented by five parts must be larger. If the answer makes it smaller, the labels or multiplication have probably been reversed.

Check the scale factor. Equivalent ratios multiply or divide both quantities by the same non-zero factor. From 4:3 to 20:15, both terms are multiplied by 5.

Check divisibility in whole-object contexts. If 25 counters are split in the ratio 2:3, each of five parts is 5 counters. If the total were 26 indivisible counters, an exact whole-counter split in that ratio would not be possible. That is useful information, not a reason to force a rounded answer.

Missteps the model can expose

Treating ratio as a difference

A child may think 2:3 means “one more”, then preserve the gap by writing 4:5. But 2:3 and 4:5 are not equivalent. The matching ratio after doubling is 4:6. Ask the child to point to the equal scale factor on both bars.

Dividing by the wrong number of parts

Decide what the known value spans. In the 25-counter example, 25 covers five parts. In the recipe example, 280 covers four parts. In the rescue-centre example, 12 covers the two extra parts.

Drawing unequal boxes

Different-looking widths suggest different values. Draw the units as evenly as practical and state that every box has equal value. The model needs to preserve structure, not physical scale to the nearest millimetre.

Counting boxes without labels

The diagram only works when the bars remain attached to their quantities. Write the labels first, then draw. Finally, read the answer with its unit: counters, grams, animals, people or pounds.

A practical parent routine

Keep the first few sessions short and use counters before expecting a polished sketch.

Build a base ratio. Put down two counters of one colour and three of another. Say “two to three”, then write 2:3.

Scale both groups. Make 4:6 and 6:9. Ask what multiplier changed both quantities.

Replace counters with boxes. Draw two and three equal boxes, leaving them unnumbered at first.

Attach one known value. Try a total such as 20. Ask which boxes the 20 spans before calculating 20 ÷ 5 = 4 per part.

Change the information position. Give one bar’s value, then a difference, so the child must interpret the model rather than repeat “add the ratio numbers” automatically.

Fade the support. Move from counters to a quick bar sketch, then to a child explaining the equal-part structure without drawing when the numbers are manageable.

Pause if the child cannot explain what one box represents. Return to a small ratio with physical counters. The goal is not a beautifully ruled diagram; it is a correct multiplicative relationship.

This routine applies the sources’ curriculum content and representation principles to home practice. The EEF report is school-facing guidance for Key Stages 2 and 3; it does not test this exact routine or establish a NeurofiED outcome.[3]

Prompts that reveal understanding

Instead of asking only “What is the answer?”, try:

What do the boxes represent here?

Which amount is spread across these boxes?

How many equal parts does the known number cover?

Why are you dividing by 5 rather than by 3?

What is one part worth, including its unit?

How will you rebuild the given total or difference?

Does your pair simplify to the original ratio?

If both quantities doubled, what would stay the same?

Could the answer be exact if the objects cannot be split?

These questions keep the focus on ratio and proportion. They do not turn the activity into a general reading strategy for every word problem, and they do not depend on teaching fraction equivalence as a separate topic.

Next step

Choose whether the known number is a total, one quantity or a difference. Ask your child to draw the labelled equal parts, mark exactly where the known value belongs, find one part and verify the finished ratio.

Try a Maths lesson

Sources

[2] https://www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study

[3] https://educationendowmentfoundation.org.uk/education-evidence/guidance-reports/maths-ks-2-3