Multiplication becomes much more flexible when students see that a difficult product can be broken into easier products without changing the answer. This is the reasoning behind arrays, partial products, decomposition and the distributive structure that later supports written algorithms, area models and mental calculation.
This is Guide 66 in the Primary 3 Mathematics Learning Hub. It is an enrichment and transfer guide that develops multiplication properties through age-appropriate language: turn factors around, split a factor, regroup equal groups, use zero and one sensibly, connect arrays to partial products, and choose efficient mental strategies.
Start Here | Four Structural Moves
- Turn it: 7 × 8 = 8 × 7.
- Split it: 7 × 24 = 7 × 20 + 7 × 4.
- Group it: 2 × (3 × 5) can be seen as 6 groups of 5.
- Anchor it: multiplying by 1 preserves a quantity; multiplying by 0 gives zero.
The goal is not to memorise property names. The goal is to see which changes preserve the product.
Turn the Factors Around
Multiplication can be turned around without changing the product. A 4-by-7 array and a 7-by-4 array contain the same 28 objects.
- 4 × 7 = 28
- 7 × 4 = 28
This is useful when one orientation connects more easily to a known fact.
Split a Factor to Use Easier Facts
Suppose a learner wants to find 6 × 27. Instead of treating 27 as one block, split it by place value:
- 27 = 20 + 7.
- 6 × 20 = 120.
- 6 × 7 = 42.
- 120 + 42 = 162.
This is the same structure later compressed inside the written multiplication algorithm.
Arrays Make Splitting Visible
A 6-by-27 rectangle can be cut into a 6-by-20 rectangle and a 6-by-7 rectangle. The total area or number of objects remains unchanged because the two smaller parts exactly reconstruct the original array.
This connects directly to Guide 60: Unit Squares, Arrays & Area Models.
Worked Example 1 | 8 × 34
- 34 = 30 + 4.
- 8 × 30 = 240.
- 8 × 4 = 32.
- 240 + 32 = 272.
Worked Example 2 | Split Around a Friendly Number
Find 7 × 19.
- 19 = 20 − 1.
- 7 × 20 = 140.
- Subtract 7 × 1 = 7.
- 140 − 7 = 133.
Splitting does not have to be only by tens and ones. A nearby friendly number can also create an efficient route.
Zero Groups
0 × 8 means zero groups of 8. There are no objects, so the product is 0. The same is true for 8 × 0.
This should be understood from the meaning of multiplication, not memorised as an isolated exception.
One Group
1 × 8 means one group of 8, so the total remains 8. Multiplying by 1 preserves the quantity.
Grouping Groups
Consider 2 groups, each containing 3 groups of 5 counters. There are 6 groups of 5 altogether, giving 30 counters.
- 2 × (3 × 5) = 2 × 15 = 30.
- (2 × 3) × 5 = 6 × 5 = 30.
At Primary 3 level, the important idea is that regrouping equal groups can preserve the total. Formal property terminology is optional enrichment; the structure matters more than the label.
Worked Example 3 | Double One Factor, Halve the Other
For some products, a compensation strategy can make the calculation easier.
- 4 × 18 = 72.
- Double 4 to 8 and halve 18 to 9.
- 8 × 9 = 72.
This is enrichment reasoning, not a required Primary 3 method. It is useful because it demonstrates that some transformations preserve a product.
Why Compensation Works Here
If one factor doubles while the other halves exactly, the total number of objects can remain unchanged. Four groups of 18 can be reorganised into eight groups of 9.
Do Not Apply Compensation Blindly
Changing only one factor changes the product. 4 × 18 is not equal to 8 × 18. A compensation move works only when the corresponding change preserves the total structure.
Mental Multiplication With Tens
Place-value decomposition makes multiplication by tens manageable:
- 6 × 30 = 6 × 3 tens = 18 tens = 180.
- 8 × 40 = 8 × 4 tens = 32 tens = 320.
The zero is not simply “added at the end”; the multiplication is acting on tens as units.
Worked Example 4 | 6 × 43
- 43 = 40 + 3.
- 6 × 40 = 240.
- 6 × 3 = 18.
- 240 + 18 = 258.
Worked Example 5 | 9 × 26 Through Ten Groups
- 10 × 26 = 260.
- Subtract one group of 26.
- 260 − 26 = 234.
This combines the nine-times-table structure from Guide 65 with multi-digit decomposition.
Worked Example 6 | Compare Two Routes
Find 8 × 19.
- Route A: 8 × 10 + 8 × 9 = 80 + 72 = 152.
- Route B: 8 × 20 − 8 = 160 − 8 = 152.
Both are correct. Route B is shorter for many learners because 20 is a friendly benchmark.
Properties Support Error Checking
If 7 × 24 is calculated as 148, split it independently: 7 × 20 = 140 and 7 × 4 = 28, giving 168. A second representation exposes the error.
Properties Support Written Algorithms
The standard written method for 7 × 24 compresses the same partial products. When students understand the split 20 + 4, regrouping inside the algorithm has meaning rather than appearing as mysterious carried digits.
Connect this to Guide 58: Multiplication & Division Algorithms.
Properties Support Area Reasoning
A 7-by-24 rectangle can be split into 7-by-20 and 7-by-4 pieces. The areas 140 and 28 recombine to 168. Multiplication structure and geometry therefore reinforce each other.
Properties Support Mental Calculation
Flexible mental multiplication means choosing a decomposition that fits the numbers. 6 × 49 may be easier as 6 × 50 − 6 than as 6 × 40 + 6 × 9.
There is no prize for using the same decomposition every time. Strategy quality depends on accuracy, clarity and efficiency.
Common Property Misconceptions
- Turning factors changes the answer. Arrays show why it does not.
- Splitting one factor means splitting the answer arbitrarily. Each part must still be multiplied by the other factor.
- 7 × (20 + 4) = 7 × 20 + 4. The 4 must also be multiplied by 7.
- Multiplying by zero leaves the number unchanged. Zero groups produce zero.
- Multiplying by one doubles the number. One group preserves it.
- Any change to factors can be compensated somehow. Only structure-preserving changes work.
Worked Error Analysis | 6 × 24 = 124
A student writes 6 × 20 = 120 and then adds 4 instead of 6 × 4. The decomposition is correct but the distributive step is incomplete. The repair is to show the original 6-by-24 array split into two rectangles: both pieces still have height 6.
Worked Error Analysis | 0 × 9 = 9
The student may be confusing “multiply by zero” with “add zero”. Return to equal groups: zero groups of nine objects contain no objects.
Diagnostic Set
- Show why 6 × 8 = 8 × 6 with an array.
- Split 7 × 32 into two easier products.
- Find 9 × 27 using ten groups minus one.
- Explain why 1 × 46 = 46.
- Explain why 0 × 46 = 0.
- Compare two mental routes for 8 × 19.
- Use an area model to explain 5 × 23.
Student Route | Change the Form, Preserve the Product
When a multiplication problem feels difficult, ask: Can I turn it? Split it? Use ten and adjust? Double a known fact? The transformation is useful only if the total product remains unchanged.
Parent Route | Ask “Why Is It Still the Same?”
If a child says 8 × 19 = 8 × 20 − 8, ask why. The explanation should connect 19 groups to 20 groups with one group removed. This checks whether the strategy is understood rather than copied.
Teacher Route | Compare Structures, Not Just Answers
Present 7 × 24 as an array, expanded multiplication and written algorithm. Ask students to identify what each representation preserves. Then present two decompositions and discuss which is more efficient.
Diagnostic Map
| Observed behaviour | Likely weak link | Repair |
|---|---|---|
| does not turn factors | commutative connection | rotate arrays |
| splits but multiplies only one part | distribution across decomposition | use two-part area model |
| cannot use 20 or 50 as anchors | benchmark multiplication | practise tens-based products |
| zero/one facts confused | equal-groups meaning | return to concrete groups |
| strategy correct but inefficient | method selection | compare routes for same product |
Practice Progression
- turn factors around;
- split two-digit factors by place value;
- use friendly-number compensation;
- connect arrays to partial products;
- use zero and one meaningfully;
- compare two valid routes;
- apply structure to area models;
- connect mental methods to written algorithms.
Exam Craft | Choose a Decomposition That Reduces Risk
For a mental multiplication question, choose a split that produces facts you know well. A shorter strategy is valuable only if it stays reliable. Write one clear intermediate line when needed so that the decomposition remains visible.
Next Route
Continue with Guide 65: Derived Multiplication Facts, Guide 41: Mental Computation, Guide 58: Algorithms, and Guide 60: Area Models.
Return to the Primary 3 Mathematics Learning Hub.
