Primary 3 Mathematics becomes easier to reason about when invisible relationships are made visible. A word problem may contain a whole, several parts, a comparison, repeated equal groups, a fractional relationship or a sequence of changing states. Students often become stuck not because the arithmetic is beyond them, but because too much of the structure is being held only in working memory.
This is Guide 7 in the Primary 3 Mathematics Learning Hub. It develops bar models, diagrams, tables and number sentences as tools for mathematical representation. The aim is not to make every problem look the same. The aim is to choose a representation that exposes the relationship that matters.
A representation is useful when it reduces ambiguity and makes the next mathematical move easier to see.
Why Representation Matters in Primary 3
Primary 3 introduces more multi-step problems and more situations in which the operation is not announced. A child may understand addition, subtraction, multiplication and division separately but still fail to choose or sequence them. A representation places the important quantities outside the head where they can be inspected.
- A bar model can reveal a missing part or comparison.
- An equal-group diagram can reveal multiplication or division.
- A fraction strip can reveal equivalence.
- A timeline can reveal start, finish and duration.
- A table can organise repeated data or unit relationships.
- A number sentence can preserve a relationship compactly.
Representation Is Not Decoration
A model that is copied mechanically can become another source of confusion. Students should always be able to answer: What does each part of the representation mean? Which quantity is known? Which is unknown? What relationship does the model show?
If the learner cannot explain the model, drawing more accurately will not repair the mathematical weakness.
Meaning first. Drawing second.
The Representation Toolbox
| Representation | Best used for |
|---|---|
| Part–whole bar model | Known whole, known parts and missing parts |
| Comparison bar model | Larger quantity, smaller quantity and difference |
| Equal-group model | Multiplication, sharing division and grouping division |
| Fraction strip/bar | Equal parts, equivalence, comparison, addition and subtraction |
| Timeline | Starting time, finishing time and duration |
| Table | Organising repeated values, categories, units or data |
| Number sentence | Compact symbolic statement of a relationship |
| Labelled sketch | Measurement, geometry and spatial relationships |
Part–Whole Models
A part–whole model represents a total divided into components. It is useful when the problem involves combining parts or finding a missing part.
Example: There are 425 books in a library display. 168 are fiction books. The rest are non-fiction. How many are non-fiction?
The whole is 425. One part is 168. The other part is unknown. A bar showing 425 split into 168 and an unknown segment makes the subtraction relationship visible.
425 − 168 = 257 books.
When the Whole Is Unknown
Example: A class collected 186 cans on Monday and 249 cans on Tuesday. How many cans altogether?
The two parts are known and the whole is unknown. The bar can show two adjacent sections, 186 and 249, inside one larger whole.
186 + 249 = 435 cans.
The same model family supports both addition and subtraction. The location of the unknown determines the direction.
Comparison Bar Models
Comparison models show two quantities aligned from the same starting point. The extra length of the larger bar represents the difference.
Example: Ryan has 312 cards. He has 84 more cards than Zoe. How many cards does Zoe have?
Draw Ryan’s bar as the larger quantity. Zoe’s bar is shorter. The extra section is 84. Since the larger and the difference are known, the smaller amount is 312 − 84 = 228.
Comparison Models Prevent Keyword Errors
The word “more” can tempt students to add automatically. A comparison bar makes the roles visible. If the larger quantity is already known, subtract the difference to find the smaller. If the smaller quantity is known, add the difference to find the larger.
The model does not choose the operation for you. It makes the roles clear enough that you can choose safely.
Equal-Group Models
Equal-group models represent repeated quantities of the same size. They can show multiplication and both forms of division.
Multiplication example: 7 boxes contain 8 pencils each. The model has 7 equal sections, each labelled 8. Total = 7 × 8 = 56.
Sharing division example: 56 pencils are shared equally among 7 students. The whole is 56, divided into 7 equal sections. Each section = 56 ÷ 7 = 8.
Grouping division example: 56 pencils are packed 8 per box. The whole is 56, each section has size 8, and the number of sections is unknown. 56 ÷ 8 = 7 boxes.
Three Quantities in Every Equal-Group Relationship
| Quantity | Question |
|---|---|
| Number of groups | How many equal groups? |
| Group size | How many in each group? |
| Total | How many altogether? |
Students who can identify these three roles can move between multiplication and division much more reliably.
Fraction Bars
Fraction bars represent one whole split into equal sections. They are especially useful for equivalence and comparison.
If one whole bar is split into 2 equal parts and another equal-length whole bar is split into 4 equal parts, students can see that 1/2 covers the same length as 2/4.
The model helps establish the idea before the symbolic rule is memorised.
Fractions Must Use the Same Whole
A fraction bar only supports valid comparison when the wholes are equal. A half of a short bar and a half of a long bar represent the same fraction of their own wholes but not the same absolute length. Students should learn to protect the whole before comparing fractional parts.
Using Bars to Add Related Fractions
Example: 1/2 + 1/4.
A whole divided into four equal parts shows that 1/2 is the same as 2/4. Then 2/4 + 1/4 = 3/4. The visual representation explains why the parts must first be expressed in the same-sized units.
Timelines for Time Problems
Time is naturally represented on a line because events occur in sequence. A timeline can mark the starting time, friendly landmarks such as the next hour, the finishing time and the intervals between them.
Example: A lesson starts at 9:45 a.m. and finishes at 11:10 a.m.
- 9:45 → 10:00 = 15 min
- 10:00 → 11:00 = 1 h
- 11:00 → 11:10 = 10 min
Total duration = 1 h 25 min.
Labelled Sketches for Measurement and Geometry
A labelled sketch is useful when the physical arrangement matters. In perimeter questions, label known side lengths and infer missing horizontal or vertical lengths only from valid relationships. In geometry, mark right angles, parallel lines or perpendicular lines only when stated or derived.
The sketch should make the conditions visible, not invent properties that merely look true.
Tables Organise Repeated Information
Tables are useful when several related values must be compared or generated. They reduce the chance that one value is attached to the wrong category.
| Boxes | Pencils per box | Total pencils |
|---|---|---|
| 1 | 8 | 8 |
| 2 | 8 | 16 |
| 3 | 8 | 24 |
| 7 | 8 | 56 |
This can make repeated multiplication visible as a growing pattern.
Number Sentences Are Compact Models
A number sentence is itself a representation. It compresses the relationship into symbols.
- 146 + 238 = □
- 384 − 146 = □
- 7 × 8 = □
- 56 ÷ 7 = □
- □ + 68 = 213
Students should learn to move from words to a diagram or bar model, and then from that representation to a number sentence. Over time, some simple relationships can move directly from words to symbols without a drawn model.
When to Skip the Bar Model
Not every problem needs a bar model. If the relationship is already clear and a number sentence can be written safely, drawing a full model may add time without adding understanding.
A strong learner asks: Will this representation help me see something that is currently unclear? If yes, use it. If no, choose a simpler representation.
Multi-Step Models | Show Dependencies, Not Every Sentence
A multi-step model should reveal which quantity must be found first. It does not need to reproduce the entire paragraph as a picture.
Example: A shop has 6 cartons with 35 bottles each. It sells 48 bottles. How many remain?
Representation 1: show 6 equal groups of 35 to find the starting total. Representation 2: show the starting total split into 48 sold and an unknown remainder.
6 × 35 = 210. Then 210 − 48 = 162 bottles.
The representation sequence mirrors the problem dependency: total first, remainder second.
Update the Model After Each Step
Students sometimes draw one model and then ignore it after Step 1. A better habit is to treat the intermediate answer as new known information. Update the representation or label the new state before continuing.
Solve → label the new value → update the state → continue.
Reverse Problems and Models
Models are especially useful when the story moves one direction but the question asks for an earlier state.
Example: After giving away 76 stickers, Jamal has 184 left. How many did he have at first?
A part–whole bar shows the original whole divided into 76 given away and 184 remaining. The whole is unknown, so add: 76 + 184 = 260 stickers.
The word “away” no longer tricks the student into subtracting because the model shows which role is missing.
Representation Can Expose Irrelevant Information
If a number cannot be placed meaningfully into the relationship model, ask whether it is relevant. This is a useful way to resist the belief that every number in a word problem must be used.
Bar Graphs Are Representations Too
A bar graph represents data using bar length or height. Students should not confuse a problem-solving bar model with a statistical bar graph. A bar model represents relationships among quantities in a problem. A bar graph represents data categories and values using axes and scales.
| Bar model | Bar graph |
|---|---|
| Shows part-whole, comparison or equal-group relationships | Shows data values by category |
| Usually not drawn to a numerical scale | Depends on an explicit axis scale |
| Helps choose or sequence operations | Helps read, compare and calculate from data |
Common Representation Misconceptions
- “Every problem needs a bar model.” Use the simplest representation that clarifies the relationship.
- “Longer drawn bar means exact numerical scale.” Problem-solving bar models are often relational rather than scale drawings.
- “If the model is neat, it must be correct.” Meaning matters more than appearance.
- “A comparison bar always means subtraction.” The unknown may require addition.
- “Equal groups always mean multiplication.” Division is needed when total and one equal-group quantity are known.
- “A fraction picture can use unequal parts.” Fraction models require equal partitions.
- “The diagram proves a geometric property.” Only stated or derived conditions establish the property.
Diagnostic Questions
- Draw a part–whole model for 425 = 168 + □.
- Draw a comparison model where 312 is 84 more than an unknown quantity.
- Represent 56 pencils shared among 7 students.
- Represent 56 pencils packed 8 per box. Explain how the unknown differs.
- Use a fraction bar to show why 1/2 = 2/4.
- Use a timeline to find the duration from 14:35 to 16:05.
- Explain when a number sentence is enough and a bar model is unnecessary.
- Explain the difference between a problem-solving bar model and a bar graph.
How to Practise Representation
Separate representation practice from calculation. Give a problem and ask the student to draw or choose a representation without solving. Then compare two different representations and discuss which makes the relationship clearest.
Another useful exercise is to give a finished bar model and ask the learner to create a matching word problem. This reverses the process and shows whether the relationship has been understood.
Change the Surface, Keep the Structure
A strong representation should survive a change of story. A part–whole relationship can describe books, money, distance or stickers. A comparison model can describe scores, masses or quantities. An equal-group model can describe packets, rows, boxes or teams.
This transfer is important because it moves the child away from memorising worksheet appearances.
A Representation Decision Routine
| Question | Possible representation |
|---|---|
| Is there a whole split into parts? | Part–whole bar |
| Are two quantities being compared? | Comparison bars |
| Are there repeated equal quantities? | Equal-group diagram or bar |
| Are fractional parts involved? | Fraction strip or bar |
| Does time move from start to finish? | Timeline |
| Are several values organised by category? | Table or bar graph |
| Is the relationship already clear? | Number sentence may be enough |
Exam Craft | Draw Small, Label Clearly
In an assessment, the model should serve the mathematics, not consume the page. Draw it large enough to read but small enough to remain efficient. Label known quantities, mark the unknown clearly and avoid decorative details.
When the model has done its job, move to the number sentence and calculation. Do not keep redrawing the same relationship.
See the relationship → represent it clearly → solve it efficiently.
Checkpoint | Can the Student Represent Before Solving?
- Can the learner choose a representation based on structure?
- Can the student build part–whole and comparison bars?
- Can the learner represent multiplication and both division meanings?
- Can the student use fraction bars for equivalence?
- Can the learner use a timeline for time intervals?
- Can the student label a geometry or measurement sketch accurately?
- Can the learner move from a model to a number sentence?
- Can the student decide when a model is unnecessary?
- Can the learner update a representation after an intermediate step?
- Can the student explain every part of the model in words?
How This Connects to the Primary 3 Mathematics System
Representation connects directly to Guide 4: Word Problems, Models and Bar Graphs and deepens the mathematical-language work in Guide 6: Mathematical Language, Comparison, Relationships and Inverse Thinking. It also supports fraction equivalence from Guide 2 and measurement/time reasoning from Guide 3.
Final Thought
A good representation does something important: it turns a hidden relationship into an object the learner can inspect. That reduces guessing, protects working memory and makes operation choice more deliberate.
Do not draw because the worksheet says “model”. Draw because the mathematics becomes clearer when you can see it.
Return to the Primary 3 Mathematics Learning Hub.
