Primary 6 Mathematics Learning Guide | Stack Model, Split Model and Two-Dimensional Visual Reasoning

Wait, What? Some Bar Models Need More Than One Row

Standard bar models are powerful because they turn words into visible lengths. But some Primary 6 questions contain several linked comparisons, repeated groups or quantities that need to be aligned in two directions at once. A single horizontal bar may become crowded or hide the relationship.

This guide develops stack models and split models as extensions of the familiar Singapore model method. The goal is not decorative drawing. The goal is to organise several relationships so that equal units, totals, gaps and transfers remain visible at the same time.

A good model is a visual equation: every segment must represent a quantity, and every alignment must represent a relationship.

Quick Answer

A reliable visual-modelling routine is:

NAME THE QUANTITIES → CHOOSE A COMMON ALIGNMENT → STACK RELATED BARS OR SPLIT A WHOLE INTO CONTROLLED PARTS → MARK EQUAL UNITS → LABEL KNOWN GAPS OR TOTALS → SOLVE THE VISUAL RELATIONSHIP → TRANSLATE THE MODEL INTO ARITHMETIC OR ALGEBRA → CHECK THAT EVERY BAR SEGMENT HAS MEANING.

1. What Is a Stack Model?

A stack model places several bars in parallel rows so their lengths can be compared vertically. The left edges usually share one reference point, while differences in right endpoints show gaps between quantities.

Stacking is useful when three or more quantities must be compared without losing a common baseline.

2. What Is a Split Model?

A split model begins with one whole bar and partitions it into meaningful sections. One section may be a fixed amount, another repeated unit, and another remainder. The split should follow the mathematical structure, not merely the order of sentences.

Split models are especially useful in remainder, percentage, ratio and multi-stage total problems.

3. Worked Example: Three Stacked Quantities

A has 20 more than B. C has 15 less than B. Altogether they have 215. Find each amount.

Use B as the reference bar. Stack A above B with an extra 20 segment. Stack C below B with a missing 15 segment.

1. Let B=x.
2. A=x+20.
3. C=x−15.
4. Total: 3x+5=215.
5. 3x=210, so x=70.
6. A=90, B=70, C=55.

The visual stack makes the net fixed difference +20−15=+5 immediately visible.

4. Why a Shared Baseline Matters

If bars representing A, B and C are not aligned to a common reference, visual comparison becomes ambiguous. A stack model should make “same starting point” or another chosen invariant explicit.

Alignment is the visual equivalent of using the same zero on a number line.

5. Worked Example: Split a Whole by Ratio and Fixed Amount

A total of 180 is split between A and B. A has $30 more than B.

Draw one whole bar of 180. Split off the extra 30 belonging to A. The remaining 150 represents two equal base parts, one for A and one for B.

1. Remove difference: 180−30=150.
2. Split equally: 150÷2=75.
3. B=75.
4. A=75+30=105.

The split model turns “total plus difference” into a visible balancing operation.

6. Stack Models and Ratio Units

Three quantities with ratio 2:3:5 can be drawn as stacked rows of two, three and five equal units. If a difference or total is known, the visual unit structure becomes immediately available.

Stacking is especially useful when one ratio quantity must also be compared with a fixed amount or another external condition.

7. Split Models and Remainders

If 1/4 of a quantity is removed and then 2/5 of the remainder is used, the original bar can first split into four equal sections, then the three-section remainder can be redrawn or repartitioned into five equal parts.

The visual model must allow the unit system to change when the active whole changes.

8. Two-Dimensional Reasoning Means Managing Two Relationships at Once

In some problems, horizontal length shows quantity while vertical stacking shows category or state. This creates a simple two-dimensional information map.

For example, rows may represent before and after states while segments within each row represent ratio units. The learner can compare across both direction and state.

9. Worked Example: Before-and-After Stack

A:B = 3:5. After A receives 20, A:B = 5:5. B is unchanged.

Draw two stacked states. In the before row, A=3 units and B=5. In the after row, B remains the same 5-unit length, while A becomes 5 units. The added 20 corresponds visually to the two-unit extension.

Thus 2 units=20, one unit=10, B=50.

10. Models Should Reduce, Not Increase, Complexity

A model with too many decorative boxes can become harder than the original question. Every segment should answer one of three jobs: represent a quantity, show equality, or show difference/change.

If a drawing does not help one of those jobs, simplify it.

11. Stack Models and Constant Difference

If A and B both increase by the same amount, stack their before and after bars so the equal extensions line up. The original gap then remains visually unchanged.

This is useful in age problems and same-change comparisons.

12. Stack Models and Constant Total

For internal transfer, draw the same total-length bar in both states, but move the internal dividing boundary. The fixed outer length shows constant total; the moving partition shows redistribution.

This is one of the clearest visual representations of transfer.

13. Split Models and Percentage

A 100% bar can be split into known and unknown percentage segments. If 35% is used, show 35% and 65% as complementary parts. If a later action applies only to the 65%, redraw that remainder as its own whole rather than forcing the original 100% scale through every stage.

14. Stack Models and Repeated Identity

Chained ratios A:B and B:C can be represented on stacked rows by aligning the repeated B bar to the same length. Scaling the rows until B matches is the visual equivalent of finding a common multiple.

15. Worked Example: Shared Middle Bar

A:B=2:3 and B:C=4:5. Draw A:B first. Draw B:C below it. Scale both drawings so the B bars have equal physical length. The result visually produces A:B:C=8:12:15.

The drawing is not merely illustrative; it performs the ratio alignment.

16. Translate the Model Into Arithmetic

After the relationship becomes visible, write the corresponding arithmetic. The model should not remain an isolated picture. If three equal base bars plus a net extra 5 total 215, write 3 units +5 =215.

This translation builds the bridge from model method to algebra.

17. Model Integrity Checks

• Equal units must have equal drawn length.
• Different unit systems should be clearly separated.
• Transfers should not change total bar length when total is constant.
• Remainders should be relabelled when they become new wholes.
• Known differences should be shown only once.
• All labelled segments should correspond to the wording.

18. Common Error Families

Error	What it looks like	Repair
Unequal equal-units	Draws supposed equal units at different lengths	Use consistent unit scale within a state
State collapse	Mixes before and after on one bar without labels	Use separate stacked rows
Overdrawing	Model contains decorative segments with no role	Keep only relational structure
Wrong whole	Uses original bar after a remainder becomes new whole	Redraw the active whole
Hidden double counting	Shows a fixed difference in two places	Map every known amount once
No arithmetic bridge	Can draw but cannot solve	Translate model into equations or operations

19. A First-Weak-Link Diagnostic

1. Quantity mapping: Can each bar be assigned a clear meaning?
2. Alignment: Can a useful common baseline be chosen?
3. Unit fidelity: Are equal units represented consistently?
4. State separation: Can before and after remain distinct?
5. Split logic: Can totals and remainders be partitioned correctly?
6. Translation: Can the picture become arithmetic?
7. Simplification: Can unnecessary visual detail be removed?

20. Examination Control

• Use a stack only when several quantities share a comparison baseline.
• Use a split when one whole contains several meaningful components.
• Separate different states vertically.
• Label every fixed amount and every equal unit.
• Do not force one unit scale across changed wholes.
• Translate the final model into arithmetic before calculating.

21. What Parents Can Ask

• “What does each row represent?”
• “Which bars are supposed to have equal units?”
• “What is the common baseline?”
• “Has the whole changed between these stages?”
• “Can you explain every segment in the drawing?”
• “What arithmetic sentence does this model show?”

22. What Tutors Should Protect

• Visual fidelity. Drawing must encode real relationships.
• Economy. Models should reduce cognitive load.
• State control. Use separate rows for changed states.
• Unit discipline. Equal visual units need equal mathematical meaning.
• Algebra bridge. Translate pictures into symbolic relationships.
• Prompt reduction. Let learners choose stack versus split representation.

23. Official Process Connection

The Singapore Primary Mathematics framework emphasises representation, connections, reasoning and problem solving. Stack and split models extend the model method by making multi-quantity and multi-stage relationships visually explicit.

Official reference: MOE Mathematics Syllabus — Primary One to Six.

Continue the Primary 6 Mathematics Series

• Primary 6 Mathematics Learning Hub
• Repeated Identity, Chained Ratios and Shared Middle Quantities
• Equal Stage Problems and End-State Reconstruction
• Three-Quantity Comparison Models and Multi-Bar Alignment

The Quiet Return

Stack and split models are useful when one bar no longer carries enough structure. Their purpose is to make complexity visible without making the drawing complicated.

The mature Primary 6 habit is to ask: what alignment makes these relationships easiest to see?