Math // Ratio & Proportion

Bar Models for Ratios

Represent ratio and proportion problems using bar models (rectangular strips divided into equal parts labelled with quantities) and tape diagrams (segmented strips showing part-to-part and part-to-whole relationships); use these visual models to set up and solve unequal sharing, scaling, and percentage problems — drawing the diagram first, then reading off the answer

Ages
9 to 12
Type
Representation
Builds on
0 topics
Leads to
5 topics

Quick check

If the learner needs to share 30 stickers between two friends in a 2:3 ratio, can they draw a bar model — splitting a strip into 5 equal boxes — and use it to work out each share without just guessing?

Ask this to see whether the topic is secure.

Topic // Mastery

What mastery looks like

What a learner can do once this topic is secure.
  • Draw a bar model to represent a ratio problem — e.g. sharing £20 in the ratio 3:2 by drawing 5 equal blocks
  • Use a bar model to solve a proportion problem and explain each step
  • Compare bar models with other representations (tables, double number lines) and explain when each is most useful

Topic // Map

Where this fits

What to learn first and what this topic opens up, with the reason for each link. Required links come first.
  • RequiredMust come first
  • HelpfulHelps, but not essential

Before this

0 topics

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Bar Models for Ratios

Ages 9 to 12 // Representation

Leads to

5 topics
  • Ratio ProblemsRequired

    Ratio & Proportion // Ages 10 to 11

    Solving relative-sizes problems is most reliably scaffolded by drawing a bar model to set up the relationship

  • Understanding fractionsRequired

    Ratio & Proportion // Ages 10 to 11

    Unequal sharing problems are the canonical use case for tape diagrams — the visual model makes the structure of the problem transparent

  • Dividing Quantities by RatioRequired

    Ratio & Proportion // Ages 11 to 13

    Dividing a quantity in a given ratio is the natural next step from the bar model representation

  • Calculating PercentagesHelpful

    Ratio & Proportion // Ages 10 to 11

    Percentage-of-amount problems can be set up as bar models showing 100% divided into parts

  • Scale and similar shapesHelpful

    Ratio & Proportion // Ages 10 to 11

    Scale factor problems for similar shapes can be represented as ratio bar models

Topic // Research-informed

Teaching notes

From the Marble Skill Taxonomy. These notes have not been validated in classrooms, so treat them as a starting point for your own judgement.

The approach

Teach the bar model (tape diagram) as the workhorse for ratio and proportion. A ratio like 3:2 becomes 3 equal blocks beside 2 equal blocks; once the value of ONE block is known, every share, scaling and percentage answer is read off the picture. Build three habits: (1) draw bars matching the relationship, (2) find the value of one block (known amount ÷ number of blocks it covers — e.g. total ÷ all blocks when the total is given, or one share ÷ that share's blocks when a part is given), (3) read off what is asked — with the equal-unit convention (every block the same size) made firm first. Progress from concrete cubes/counters to a drawn, labelled bar to the numeric calculation, and always draw before calculating so problem structure is visible rather than guessed from key words. Constructing a model is the core objective; reading a prepared model is a valuable scaffold toward it.

Common misconceptions

  • Draws blocks of unequal size, so the model no longer represents equal units
  • Sets up part-to-whole when the problem is part-to-part (draws a bar of 3 for a 3:2 ratio and treats the 2 as part of it), or vice versa
  • Jumps to arithmetic without finding the value of one block, so the diagram is decorative rather than a solving tool
  • Maps the wrong quantity to the wrong bar, or forgets which bar is which after drawing

Useful representations

  • Part-to-part twin bars (e.g. 3 equal blocks : 2 equal blocks) with a brace showing the total
  • Part-to-whole single bar split into the total number of parts, one section shaded
  • Cubes/counters as the concrete precursor to drawn blocks
  • Ratio table and double number line as complementary models to contrast
  • One labelled block annotated with its value (the pivot of the three-move routine)

Teaching practices

  • Establish the equal-unit convention explicitly: every block in the bars is the SAME size and stands for the same amount — the model breaks if blocks are drawn unequal.
  • Teach the three-move routine: (1) draw bars matching the ratio/parts, (2) find the value of one block (total ÷ number of blocks), (3) read off the required quantity — make step 2 the pivot.
  • Contrast part-to-part (two separate bars, e.g. 3 and 2) with part-to-whole (one bar of 5, or a whole bar split) so children pick the right layout for the question.
  • Progress concrete → pictorial → abstract: build the ratio with coloured cubes, then draw the labelled bar, then connect to the numeric calculation.
  • Label the bars fully — the total (a brace over the whole), the known value, and the '?' — so the diagram carries the problem's information.
  • Compare the bar model with double number lines and ratio tables on the same problem so children can say when each is most useful (bar for sharing a fixed total; double number line/table for scaling up a rate).

Activities to try

  1. 1

    Build the tape diagram: draw, find one block, read off

    Why it works: Each block is an equal unit standing for the same amount; dividing the whole by the number of blocks gives one block's value, and from that single fact every share and proportion can be read straight off the picture without a memorised rule.

    1. Given '£20 shared in the ratio 3:2', build two bars — 3 equal blocks beside 2 equal blocks — keeping every block the same width.
    2. Place a brace spanning all 5 blocks and label it with the total, £20.
    3. Find the value of one block by dividing the total by the number of blocks: £20 ÷ 5 = £4.
    4. Fill every block with £4, then read off the two shares: 3 × £4 = £12 and 2 × £4 = £8; check they add back to £20.
    5. Toggle the same ratio between part-to-part twin bars (3 and 2) and a part-to-whole single bar of 5, and say which layout the question needs.
  2. 2

    Concrete first: build the ratio with cubes, then draw it

    Why it works: The drawn block is a shorthand for an equal physical unit; moving from cubes to the drawn bar preserves the equal-unit meaning rather than inventing a new rule.

    1. Lay out coloured cubes or counters for a part-to-part ratio — e.g. 1 cube of concentrate for every 4 cubes of water.
    2. Count the total cubes and share a given total across them so each cube (block) gets an equal value.
    3. Redraw the same arrangement as a labelled bar diagram, matching one block to one cube.
    4. State in words what each block stands for before doing any arithmetic.
  3. 3

    Read a worked bar model (scaffold)

    Why it works: Fluent reading of a prepared diagram — knowing what every block, brace and shaded section stands for — supports constructing your own model, which this scaffold then leads into.

    1. Study a labelled '£20 shared 3:2' diagram and, part by part, state what each feature means: a single block, the 3-section, the 2-section, the brace, any shading.
    2. Reveal and confirm: one block = £20 ÷ 5 = £4; the 3-section = £12; the brace shows the whole = £20.
    3. Explain back, in your own words, how the one-block value was found and how each share follows from it.

Math // Ratio & Proportion

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