Multiplication and Division

Multiplication and division build on children's understanding of addition and subtraction. In Key Stage 1, children move from concrete sharing and grouping activities to learning the 2, 5 and 10 times tables by the end of Year 2.

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What This Topic Covers

• Year 1: Solve one-step problems using concrete objects, pictorial representations and arrays
• Year 2: Recall 2, 5 and 10 multiplication tables; use x, div and = symbols; understand commutativity; solve problems in context

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Year 1: Concrete Foundations

Grouping and Sharing

Children begin by physically grouping and sharing objects — the conceptual basis of multiplication and division.

Grouping (for multiplication):

"Put 12 counters into groups of 3. How many groups do you have?" 12 counters: 4 groups of 3, so 4 x 3 = 12

Sharing (for division):

"Share 12 apples equally between 4 children. How many does each child get?" 12 / 4 = 3 apples each

Arrays

An array is an arrangement of objects in rows and columns. Arrays make multiplication visual and concrete.

* * * *
* * * *
* * * *

This array shows 3 rows of 4 = 3 x 4 = 12

Arrays also demonstrate commutativity — the same array can be read as 3 x 4 or 4 x 3:

* * *         * * * *
* * *         * * * *
* * *         * * * *
* * *
4 x 3 = 12    3 x 4 = 12

Doubling and Halving

Doubling is the same as multiplying by 2. Halving is the same as dividing by 2.

Number	Double	Half
2	4	1
6	12	3
10	20	5
20	40	10

flowchart LR
  A[3 groups of 4]
  A --> B[4 + 4 + 4]
  B --> C[12]
  A -.same as.-> D[3 x 4]
  D --> C

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Year 2: Times Tables and Formal Notation

The x, div and = Signs

Year 2 children learn to write and interpret statements using formal symbols:

• x means multiply / times / groups of / lots of
• div means divide / share equally / group into
• = means equals / is the same as

Examples:

• 3 x 5 = 15
• 20 / 4 = 5
• 6 x 2 = 12

The 2 Times Table

Fact	Answer
1 x 2	2
2 x 2	4
3 x 2	6
4 x 2	8
5 x 2	10
6 x 2	12
7 x 2	14
8 x 2	16
9 x 2	18
10 x 2	20

The 2 times table is the same as counting in 2s and relates to even numbers. An even number can be split into two equal groups with nothing left over.

The 5 Times Table

Fact	Answer
1 x 5	5
2 x 5	10
3 x 5	15
4 x 5	20
5 x 5	25
6 x 5	30
7 x 5	35
8 x 5	40
9 x 5	45
10 x 5	50

The 5 times table connects to telling the time — the minute hand counts in 5s around the clock.

The 10 Times Table

Fact	Answer
1 x 10	10
2 x 10	20
3 x 10	30
4 x 10	40
5 x 10	50
6 x 10	60
7 x 10	70
8 x 10	80
9 x 10	90
10 x 10	100

The 10 times table connects directly to place value — multiplying by 10 moves every digit one place to the left.

Odd and Even Numbers

Even (divisible by 2)	Odd (not divisible by 2)
2, 4, 6, 8, 10	1, 3, 5, 7, 9
12, 14, 16, 18, 20	11, 13, 15, 17, 19
Always end in 0, 2, 4, 6 or 8	Always end in 1, 3, 5, 7 or 9

Commutativity of Multiplication

Multiplication is commutative — the order of the numbers does not matter:

• 4 x 5 = 5 x 4 = 20

Division is NOT commutative:

• 20 / 5 = 4, but 5 / 20 does not equal 4

The Inverse Relationship

Multiplication and division are inverse operations:

Multiplication	Related division 1	Related division 2
4 x 5 = 20	20 / 5 = 4	20 / 4 = 5
3 x 10 = 30	30 / 10 = 3	30 / 3 = 10

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Key Vocabulary

Term	Meaning
multiply / multiplication	repeated addition / groups of
divide / division	share equally or group into
product	the result of multiplication
quotient	the result of division
array	objects arranged in rows and columns
factor	a number that multiplies to give a product
double	multiply by 2
halve	divide by 2
odd	cannot be divided equally by 2
even	can be divided equally by 2
times table	a list of multiplication facts for a number

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Common Misconceptions

• "Division always means sharing" — Division is also grouping. 12 / 3 can mean "how many groups of 3 in 12?" (= 4 groups), not just "share 12 into 3 groups" (= 4 each).
• "Multiplication always makes things bigger" — True for whole numbers greater than 1, but this assumption causes problems later.
• "The times tables are just for memorising" — Understanding why 3 x 5 = 5 x 3 (commutativity) and how arrays show this is as important as recall.

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Worked Example for Parents and Teachers

The most powerful tool for teaching multiplication and division in KS1 is the array — a rectangular arrangement of objects. Arrays connect grouping, sharing, multiplication, division and commutativity in a single visual structure.

Imagine a Year 2 lesson with the objective: "Children will use arrays to derive multiplication and related division facts."

Step 1 — Build an array (concrete). Give each pair 20 counters or unifix cubes. Say, "We are going to build an array of 3 rows of 4." Model the language as you build: "Row 1, 2, 3. Each row has 4. So I have 3 lots of 4." Children copy with their own counters. Ask, "How many altogether? How did you count?" Some may count in 1s (12), others in 4s (4, 8, 12), others may already know 3 × 4 = 12.

Step 2 — Read the array two ways. Tilt the array so children look at it with the rows running horizontally. Then turn the page 90° so the rows become columns. Ask, "What can we see now?" They should notice it now looks like 4 rows of 3. Say it together: "3 lots of 4 is 12. 4 lots of 3 is also 12. The order does not matter — that is called commutativity."

Step 3 — Generate the four facts. From a single 3 × 4 array, write all four related facts on the board:

3 × 4 = 12
4 × 3 = 12
12 ÷ 3 = 4
12 ÷ 4 = 3

This is the fact family — sometimes called the triangle of facts. Children draw an array, write all four facts, and explain how the array shows each one. "The 12 ÷ 3 = 4 means: I have 12 counters in 3 equal rows; each row has 4."

Step 4 — Apply to known tables. Build a 5 × 2 array. Children should already be counting in 5s and 2s. Use the array to demonstrate that 5 × 2 = 2 × 5 = 10, and 10 ÷ 5 = 2, 10 ÷ 2 = 5. Repeat with the 10 times table.

Verbal prompts to use:

• "How many rows? How many in each row?"
• "What is the multiplication fact?"
• "What division fact does this array show?"
• "How is this array the same as that one? How is it different?"