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Primary Mathematics: Visualising, Representing and Seeing the Same Mathematics | Worked Learning Guide

A representation is useful when it makes the relationship easier to see without changing the mathematics. A good learner can move from words to diagram, diagram to number, number to table and back again while keeping the same quantity and structure intact.

Many Primary Mathematics difficulties are not failures of arithmetic. The learner knows the operation but cannot see what the quantities represent. A fraction is mistaken for two whole numbers separated by a bar. A ratio is treated as two unrelated counts. A rectangle is rotated and suddenly its height appears to have “changed”. A graph is read by visual steepness without checking the scale. A bar model is copied from a familiar worksheet even though its bars no longer match the stated relationship.

Visualisation and representation are the bridge between a mathematical relationship and the surface on which it appears. The bridge works in both directions. The learner must be able to create a model from a problem and recover the relationship from a model created by someone else.

This guide develops that transfer across number, fractions, multiplication, ratio, measurement, geometry, data and word problems. It does not prescribe one representation for every learner. A representation earns its place when it makes a relevant relationship visible and can be checked against the original quantities.

Same quantity, different form · Choose a representation · Translate between forms · Visualise mentally · Resist misleading pictures · 24 questions · Worked solutions

1. The representation changes; the quantity must not

Consider the number 0.75. It can be represented as:

• seventy-five hundredths,
• 75/100,
• 3/4,
• 75%,
• a point three-quarters of the way from 0 to 1 on a number line,
• 75 shaded cells in a 100-cell square.

These forms look different, but they represent the same quantity. Translation is successful only when value is preserved. If a learner changes 0.75 to 3/5 because both have a 3 and a 5 somewhere in the notation, the appearance has driven the reasoning instead of the quantity.

Worked example: rename without changing

Represent 3/4 as a decimal and percentage.

3/4 = 75/100 = 0.75 = 75%.

Check with a bar divided into four equal parts. Shade three. Now divide the same-length bar into one hundred equal conceptual parts: seventy-five hundredths cover exactly the same fraction of the whole.

The whole must stay visible

Three quarters of a small chocolate bar and three quarters of a large chocolate bar are the same fraction but different amounts. A fraction diagram is meaningful only when the represented whole is defined. A ratio diagram is meaningful only when the equal unit size is common across the compared quantities.

2. Arrays reveal multiplication structure

18 × 5 can be pictured as eighteen rows of five or five rows of eighteen. Rotate the array and the total remains ninety. The two orientations make the commutative relationship visible: 18 × 5 = 5 × 18.

Now split eighteen into twenty minus two. A 20 × 5 rectangle contains 100 units. Remove a 2 × 5 strip containing 10. The remaining area is 90. This area picture explains the mental calculation 18 × 5 = 20 × 5 − 2 × 5.

Worked example: 23 × 14

Split 23 into 20 + 3 and 14 into 10 + 4. The area model creates four parts:

• 20 × 10 = 200
• 20 × 4 = 80
• 3 × 10 = 30
• 3 × 4 = 12

Total = 322.

The model shows why every part of one factor must multiply every part of the other. It is not a decorative box around a memorised algorithm.

3. Number lines reveal order, distance and change

A number line is especially useful when the important relationship is position or distance.

Worked example: compare decimals

Place 0.45 and 0.7 between 0 and 1. Rename 0.7 as 0.70. Since 0.45 lies left of 0.70, 0.45 < 0.7.

This representation resists the digit-count trap. The number 45 is larger than 7 as a whole number, but 0.45 is not “45” on the decimal number line.

Worked example: subtraction as distance

Find 5.2 − 1.86.

On a number line, count from 1.86 to 2.00 (+0.14), then to 5.00 (+3.00), then to 5.20 (+0.20). Total distance = 3.34.

This complements the written subtraction algorithm and provides an independent check.

4. Bar models reveal part-whole and comparison relationships

A bar model is useful when lengths can represent quantities proportionally. It should encode the relationship stated by the problem, not merely fill space under the question.

Worked example: additive comparison

A has 18 more marbles than B. Together they have 74.

Draw B as one unknown bar. Draw A as an equal unknown bar plus an extra segment of 18. Two equal unknown parts plus 18 total 74. Remove 18 to get 56, then split equally: B = 28 and A = 46.

The model makes the fixed difference visible.

Worked example: ratio units

A:B = 3:5 and the total is 64.

Draw three equal units for A and five equal units for B. Eight units represent 64, so one unit is 8. A = 24 and B = 40.

The bar model is useful because the ratio parts are equal units. If the drawn units are visibly unequal, the picture contradicts the ratio it is supposed to represent.

5. Tables reveal repeated relationships

A table is often better than a long paragraph when one variable changes repeatedly.

Number of notebooks	Total cost at $3 each
1	$3
2	$6
3	$9
4	$12

The table exposes a constant rate: total cost = 3 × number of notebooks. A learner can move from repeated addition to multiplication and then to a general rule.

Worked example: speed table

At a constant speed of 60 km/h, distance after 1, 2, 3 and 4 hours is 60, 120, 180 and 240 km. The table makes proportional growth visible. If a fixed 20 km starting distance were added, the relationship would no longer be direct proportion even though distance would still grow linearly.

6. Graphs add shape to a table

A graph can make trend and rate visible, but only when axes and scales are read carefully.

Suppose two graphs both rise five vertical grid squares. In Graph A, each vertical grid square represents 2 units. In Graph B, each represents 20 units. The pictures show equal visual height but numerical changes of 10 and 100.

The representation helps only if its encoding rules are read.

Worked example: final value versus change

Student X’s pulse rises from 70 to 120. Student Y’s rises from 90 to 130. Y has the higher final pulse, but X has the larger increase: 50 compared with 40.

A graph can show both quantities at once, yet the question determines which comparison matters.

7. Diagrams simplify geometry without changing properties

A rectangle rotated forty-five degrees is still a rectangle. Its opposite sides remain parallel and equal; its four angles remain right angles. Page orientation is not a geometric property.

Worked example: perpendicular height

A triangle’s area depends on a chosen base and the perpendicular height to that base. A sloping side drawn upward on the page is not automatically “the height”. Mark the right angle or construct the perpendicular distance.

Representations should reveal mathematical properties, not encourage visual shortcuts that disappear when a diagram is rotated.

8. Coordinates separate position from shape

Plot a rectangle with vertices (1,1), (5,1), (5,4), (1,4). Translate every point three units right. The new vertices are (4,1), (8,1), (8,4), (4,4).

The position changes, but side lengths, angles, area and perimeter remain the same. A coordinate representation makes the transformation measurable.

Worked example: reflect across a vertical line

Reflect point (2,3) across x = 5. It lies three units left of the mirror line, so its image lies three units right: (8,3). The reflection rule is about equal perpendicular distance from the mirror, not about changing a sign automatically.

9. Unit cubes reveal hidden volume

A 3 × 4 × 5 cuboid contains 60 unit cubes. A drawing may show only the exposed outer cubes, but volume counts every cube filling the space, including hidden interior cubes.

For a 3 × 3 × 3 cube, 27 unit cubes exist. Only some are visible in a perspective drawing. The representation is partial; the mathematical model supplies the hidden structure.

Worked example: remove one corner cube

A 3 × 3 × 3 solid contains 27 cubes. Remove one corner cube. The new volume is 26 unit cubes. The surface-area change is different and requires separate face accounting. Do not use the volume picture as though it directly answers surface area.

10. Choose a representation for the relationship

Relationship	Useful representation
Place value	place-value chart, bundles, expanded form
Additive comparison	bar model, number line
Equal groups	array, grouping diagram
Fraction of a whole	fraction bar, area model, number line
Ratio	equal-unit bars, table
Repeated rate	table, graph
Geometry	labelled diagram, coordinates
Exhaustive counting	organised table, labelled cases

These are suggestions, not fixed rules. A number line can also show fractions. An area model can also show multiplication. A table can also organise geometry counts. The question is whether the representation makes the needed relationship easier to see and verify.

11. Translate in both directions

Strong representation skill is reversible.

Words to model

“A is three times B and together they total 56.” Draw one equal unit for B and three equal units for A. Four units equal 56, so one unit is 14. B = 14 and A = 42.

Model to words

A diagram shows one bar of four equal units and another of seven equal units. The two bars differ by 18. Translate: three equal units correspond to 18, so one unit is 6. The quantities are 24 and 42.

The arithmetic becomes meaningful only after the units in the diagram are interpreted.

12. Translate one idea through several forms

Take 25% of 80.

• Fraction: 25% = 1/4, so 80 ÷ 4 = 20.
• Decimal: 0.25 × 80 = 20.
• Bar model: split 80 into four equal parts; one part is 20.
• Percentage grid: 25 out of 100 equal percentage units correspond to one quarter of the whole.

A learner who can move between these forms is less dependent on one memorised procedure.

13. Visualisation continues after the diagram is removed

Visualisation means mentally representing a mathematical object or transformation. It can be trained through brief “look, hide, reconstruct” tasks.

Worked example: rotate a shape mentally

Imagine an L-shape made from three unit squares: two vertical with one attached to the right of the lower square. Rotate it 90 degrees clockwise. Which square becomes the new leftmost square?

A learner can physically rotate the drawing first, then later perform the same transformation mentally. The goal is not artistic skill; it is preserving adjacency and orientation through change.

Worked example: unfold a cube net

When predicting opposite cube faces from a net, identify which faces share edges before imagining the fold. Visualisation should be anchored by explicit adjacency information, not guessed from the flat picture alone.

14. Externalise working memory

A diagram can reduce mental load by placing relationships on the page. This is especially useful in multi-step problems.

Suppose a journey has three stages: 120 km at 60 km/h, a 30-minute stop, then 90 km at 45 km/h. A table with columns for distance, speed and time keeps the stages separate. Stage 1 time = 2 h. Stop = 0.5 h. Stage 2 time = 2 h. Total elapsed time = 4.5 h.

Trying to hold all stages mentally increases the chance of dropping the stop or averaging the two speeds incorrectly.

15. A diagram can mislead

Mathematical diagrams are often not drawn to scale. A triangle that looks isosceles is not necessarily isosceles unless equality is stated or marked. A line that looks perpendicular is not necessarily 90 degrees without a right-angle mark or deducible condition.

Likewise, a bar model is only accurate if its equal units really represent equal quantities. A beautiful but proportionally incorrect bar can create false reasoning.

Worked example: graph scale trap

One graph starts its vertical axis at 95 instead of zero. A change from 96 to 100 may look enormous because it fills most of the plot height. Read the scale before judging magnitude.

Worked example: perspective trap

In a 3D drawing, parallel edges may appear to converge. Use the stated cuboid or prism structure, not the perspective sketch, to infer equal lengths and parallel faces.

16. Representation is not proof

A picture can suggest a relationship strongly. It can even make an argument convincing. But a single picture usually represents one case.

Drawing two odd numbers as pairs plus one dot each shows that their sum can be rearranged into complete pairs. Because the same construction works for any pair of odd numbers, the representation can support a general proof. The generality comes from the structure, not from one numerical drawing.

17. Common representation errors

• Choosing a familiar diagram before identifying the relationship.
• Changing the value while converting between fraction, decimal and percentage.
• Drawing unequal ratio units.
• Treating a sketch as drawn to scale.
• Using page orientation as a geometric property.
• Reading a graph without axes and units.
• Counting only visible cubes in a solid.
• Copying a bar model whose parts do not match the problem.
• Using a representation so complicated that it hides the relationship it was meant to reveal.

18. Practice: 24 original questions

Questions 1–8: Translate quantities

1. Represent 3/5 as a decimal and percentage.
2. Place 0.45 and 0.7 on the same conceptual number line and state which is larger.
3. Show 18 × 5 as 20 × 5 minus a rectangular strip. Find the product.
4. Use an area decomposition to calculate 24 × 13.
5. A:B = 2:5 and total is 63. Draw equal-unit reasoning and find A and B.
6. A is 14 more than B and total is 68. Use a bar relationship to find both quantities.
7. Write 35% as a fraction in simplest form and as a decimal.
8. Represent 2.4 L in litres-and-millilitres form.

Questions 9–16: Read and choose representations

9. A table shows 1,2,3,4 books costing $4,$8,$12,$16. State the relationship and predict the cost of 9 books.
10. A graph rises four grid squares. Each grid square represents 25 units. What numerical increase is shown?
11. Student X rises from 72 to 120 beats per minute. Student Y rises from 90 to 130. Who has the larger final value and who has the larger increase?
12. A 4×6 rectangle is rotated 90 degrees. What happens to its area and perimeter?
13. Translate point (2,3) three units right. Give the new coordinate.
14. A 3×4×5 cuboid is built from unit cubes. How many cubes are present, including hidden ones?
15. Choose a useful representation for “three-fifths of 45” and solve.
16. Choose a useful representation for an exhaustive count of all rectangles in a small grid and explain why.

Questions 17–24: Transfer and critique

17. A learner draws a ratio 3:4 using bars of visibly different unit widths. What is wrong with the representation?
18. A triangle looks isosceles but has no equal-side marks and no stated equality. What may you conclude?
19. A graph begins at 98 on the vertical axis and rises to 100. Why might the visual change appear misleadingly large?
20. Explain why a rectangle rotated on the page remains a rectangle.
21. A bar model shows A as three equal units, B as one equal unit, and their difference as 26. Find A and B.
22. Represent 25% of 80 in two different ways and solve.
23. A three-stage journey contains 120 km at 60 km/h, a 30 min stop, then 90 km at 45 km/h. Design a table and find total elapsed time.
24. Give one example where a picture suggests a general rule but additional explanation is required before the rule is proved.

19. Worked solutions

Solutions 1–8

1. 3/5 = 0.6 = 60%. Each representation names the same part of the same whole.

2. 0.7 is larger. Rename it as 0.70; 0.45 lies to its left on the number line.

3. A 20×5 rectangle has area 100. Remove a 2×5 strip of 10. Product = 90.

4. Split 24 into 20+4 and 13 into 10+3: 200+60+40+12=312.

5. Seven equal units represent 63, so one unit is 9. A=18, B=45.

6. Two equal B-sized parts plus 14 total 68. Remove 14: 54. B=27 and A=41.

7. 35%=35/100=7/20=0.35.

8. 2.4 L = 2 L 400 ml.

Solutions 9–16

9. Cost = $4 per book. Nine books cost $36.

10. 4×25=100 units. Grid-square count without scale would be incomplete.

11. Y has larger final value, 130 versus 120. X has larger increase, 48 versus 40.

12. Nothing changes in area or perimeter: area remains 24 square units and perimeter 20 units. Orientation changes only.

13. (5,3). Only the x-coordinate increases by three.

14. 3×4×5=60 unit cubes.

15. Example: divide 45 into five equal groups and take three. One group is 9; three groups are 27.

16. Label horizontal and vertical boundary lines or use a height-width table. The representation organises all possible cases without relying on visual memory.

Solutions 17–24

17. Ratio units must represent equal-sized units before their counts 3 and 4 can encode the ratio. Unequal unit widths change the represented quantities.

18. Nothing about equal side lengths from appearance alone. Use only marked or stated properties and valid deductions.

19. A vertical range of only two units fills the plotting space, magnifying the visual height of a small numerical change. Read the axis values.

20. Rotation preserves side lengths, parallelism and right angles. Those properties define the rectangle, not its page orientation.

21. Difference between three units and one unit is two units = 26, so one unit=13. A=39 and B=13.

22. As a fraction, 25%=1/4 and 80÷4=20. As a decimal, 0.25×80=20.

23. Stage times: 2 h, 0.5 h stop, 2 h. Total elapsed time=4.5 h.

24. Example: several drawings show odd+odd=even. The pictures suggest the rule; a general explanation is needed that every odd number is pairs plus one, so the two leftover ones form another pair.

20. Representation laboratory: solve one problem three ways

Problem: A tank is 3/5 full. After 24 L is added, it is 4/5 full. Find its capacity.

Fraction-unit representation: the increase from 3/5 to 4/5 is 1/5 of the tank. Therefore 1/5=24 L and the whole is 120 L.

Bar representation: draw five equal units for the tank. Shade three initially and four finally. The one newly shaded unit represents 24 L, so five units represent 120 L.

Equation representation: let capacity be C. Then 4C/5−3C/5=24, so C/5=24 and C=120.

The three methods are not three different answers. They are three external forms of the same relationship. The equation is compact, the bar is visual, and the fraction difference highlights the unit change.

21. Parent and tutor guide

When a learner cannot start, ask “Can you show me the quantities another way?” before revealing an operation. Offer one representation only if needed: a number line for order, an array for equal groups, a bar for comparison, a table for repeated change, or a labelled diagram for geometry.

Then require translation back into words. A child who draws a correct bar but cannot say what one unit represents may be copying the appearance rather than using the model. A child who explains the relationship correctly but draws a rough sketch may understand more than the artwork suggests.

Use rotation, relabelling and altered scale to test independence from surface features. If a rectangle stops being recognised when tilted, or a ratio is lost when the bars change colour, the representation has become the concept rather than a carrier of the concept.

The first weak link

Classify failures separately: choosing the wrong representation, drawing it inaccurately, reading someone else’s model incorrectly, translating it into the wrong operation, or failing to check it against the original statement. Each needs a different repair.

For exploration before model choice, return to Exploring, Noticing and Finding Structure. For general statements suggested by representations, continue to Conjecturing and Generalising.

22. Delayed return

Three days later, give four short prompts with no suggested method: a fraction comparison, a ratio problem, a geometry relationship and a data trend. Ask the learner to choose a representation, solve, and then translate the answer into a different representation.

The goal is not to use the same model every time. It is to preserve the mathematics while the form changes.

23. Mastery receipt

• I preserve quantity when changing representations.
• I choose models because of relationships, not familiarity.
• I use arrays and area models for multiplication structure.
• I use number lines for position, order and distance.
• I use bar models for part-whole, comparison and ratio.
• I read table and graph scales before interpreting patterns.
• I do not trust geometry sketches merely because of appearance.
• I can translate a problem into a model and translate the model back into words.

Sources and scope

University of Cambridge NRICH — Thinking Mathematically, Primary Students identifies visualising and representing as a core strand of mathematical thinking alongside exploring, conjecturing and proving. NRICH — Picture Your Method is a related example of connecting calculation strategies to visual representations.

For official Singapore subject documents, use the MOE Primary curriculum and syllabus page and the learner’s current school instructions. This guide extends the BTT worked-learning collection and is not an official assessment.

Continue the Working Mathematically collection

• Exploring, Noticing and Finding Structure
• Conjecturing, Generalising and Testing a Rule
• Explaining, Convincing and Proving
• BTT Primary Mathematics Learning Hub

The Quiet Return

A representation has done its job when the learner can put it down and still carry the relationship forward. See it, draw it, name it, transform it—and make sure the mathematics survives every change of form.