Addition and Subtraction

Addition and subtraction are two of the four fundamental operations in mathematics. In Key Stage 1, children move from using physical objects to working with mental strategies and, by Year 2, beginning written methods.

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What This Topic Covers

• Year 1: Number bonds within 20; adding and subtracting to 20; one-step problems
• Year 2: Recall facts to 20 and derive facts to 100; add and subtract two-digit numbers; commutativity; inverse relationship

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Year 1: Addition and Subtraction to 20

The + - = Symbols

Children learn to read and write mathematical statements using these three symbols:

• + means add / plus / and
• - means subtract / take away / minus
• = means equals / is the same as / makes

Examples:

• 5 + 3 = 8
• 12 - 4 = 8
• 7 = 16 - 9 (the equals sign does not have to come at the end!)

Number Bonds

Number bonds are pairs of numbers that add together to make a given total. Knowing them off by heart is fundamental.

Number bonds to 10:

0 + 10	1 + 9	2 + 8	3 + 7	4 + 6
5 + 5	6 + 4	7 + 3	8 + 2	9 + 1

Number bonds to 20:

0 + 20	1 + 19	2 + 18	3 + 17
4 + 16	5 + 15	6 + 14	7 + 13
8 + 12	9 + 11	10 + 10

Related Subtraction Facts

Every addition fact generates two subtraction facts:

Addition	Subtraction 1	Subtraction 2
9 + 7 = 16	16 - 7 = 9	16 - 9 = 7
6 + 8 = 14	14 - 8 = 6	14 - 6 = 8

Counting On with a Number Line (Subtraction)

Instead of "taking away", subtraction can mean finding the difference. Count forwards from the smaller number to the larger.

Example: 13 − 8 (count from 8 to 13)

8 ---+1---> 9 ---+1---> 10 ---+1---> 11 ---+1---> 12 ---+1---> 13

We counted 5 jumps, so 13 − 8 = **5**

This "counting on" method is very useful when the two numbers are close together.

Missing Number Problems

Children encounter problems where the unknown is in different positions:

• 5 + ? = 12, so ? = 7
• ? - 4 = 8, so ? = 12
• 7 = ? - 9, so ? = 16

A part-whole model (bar model) helps children see missing number problems clearly:

Example: 5 + ? = 12

+-------------------+
|        12         |    ← the WHOLE
+--------+----------+
|   5    |    ?     |    ← the PARTS
+--------+----------+

The whole is 12. One part is 5. The missing part is 12 − 5 = 7.

Example: ? − 4 = 8

+-------------------+
|        ?          |    ← the WHOLE (unknown)
+--------+----------+
|   8    |    4     |    ← the PARTS
+--------+----------+

Both parts are known (8 and 4). The whole is 8 + 4 = 12.

These build algebraic thinking at a very early stage.

flowchart TD
  Whole[Whole: 12]
  Whole --> Part1[Part: 8]
  Whole --> Part2[Part: 4]
  Part1 -.->|8 + 4 = 12| Whole
  Part2 -.->|12 - 4 = 8| Part1
  Part1 -.->|12 - 8 = 4| Part2

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Year 2: Extending to Two-Digit Numbers

Mental Strategies

Strategy	Example
Add 1s	35 + 4 = 39
Add 10s	35 + 40 = 75
Add two 2-digit numbers	35 + 24 = 59
Add three 1-digit numbers	4 + 7 + 6 = 17

Adding two 2-digit numbers using partitioning:

Split each number into tens and ones, add the tens, add the ones, then combine.

Example: 36 + 47

• Step 1: Split into tens and ones: 36 = 30 + 6, and 47 = 40 + 7
• Step 2: Add the tens: 30 + 40 = 70
• Step 3: Add the ones: 6 + 7 = 13
• Step 4: Combine: 70 + 13 = 83

Tip for adding three numbers: Look for pairs that make 10 first.

• 4 + 7 + 6: spot 4 + 6 = 10, then 10 + 7 = 17

Deriving Facts to 100

Children use known facts to derive new ones:

Known fact	Derived fact
3 + 7 = 10	30 + 70 = 100
10 - 7 = 3	100 - 70 = 30
8 + 5 = 13	80 + 50 = 130

Commutativity of Addition

Addition is commutative — you can add numbers in any order and get the same result.

• 5 + 3 = 3 + 5 = 8
• 14 + 23 = 23 + 14 = 37

Subtraction is NOT commutative:

• 8 - 3 = 5, but 3 - 8 does not equal 5

The Inverse Relationship

Addition and subtraction are inverse (opposite) operations. Children use this to check their work:

• If 34 + 27 = 61, then 61 - 27 = 34
• If 56 - 18 = 38, then 38 + 18 = 56

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Problem-Solving Language

Children encounter problems phrased using many different words. They need to recognise these as addition or subtraction:

Addition words	Subtraction words
add, plus	subtract, take away
altogether, total	difference between
more than	less than, fewer than
put together	distance between
sum, increase	decrease, reduce

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Key Vocabulary

Term	Meaning
add / addition	combine two or more numbers
subtract / subtraction	find the difference or take away
number bond	a pair that adds to a given total
sum	the result of addition
difference	the result of subtraction
commutative	order does not change the result
inverse	the reverse/opposite operation
equation	a mathematical statement with =

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Common Misconceptions

• "Subtraction means take away" — Subtraction also means finding the difference between two numbers. 12 - 9 can be solved by counting up from 9 to 12.
• "The bigger number must come first" — With subtraction this is true (when working with whole numbers), but children may overgeneralise and think it applies to addition too.
• "Equals means write the answer" — The = sign means the same as. 7 = 16 - 9 is equally valid.

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Worked Example for Parents and Teachers

Addition and subtraction in KS1 should be taught using multiple representations: physical objects, ten-frames, number lines, part-whole models and bar models. Each representation reveals a different aspect of the operation. Imagine teaching a Year 2 lesson on adding two-digit numbers crossing a tens boundary — for example, 36 + 47.

Step 1 — Concrete with base-10 (Dienes). Set out two piles. The first pile contains 3 ten-rods and 6 unit cubes (= 36). The second pile contains 4 ten-rods and 7 unit cubes (= 47). Push the ones together. Children count: 6 + 7 = 13 ones. Hold up the 13 ones — "Can we make a ten?" Exchange 10 ones for 1 ten-rod. We now have 7 ten-rods plus 3 ones, plus the original 3 ten-rods. Push the tens together: 8 ten-rods plus 3 ones = 83.

The exchange is the heart of place value. Let children physically swap 10 ones for 1 ten — not skip the step. This is where understanding of carrying is built.

Step 2 — Pictorial with a part-whole model. Draw two part-whole models on the board:

36 = 30 + 6
47 = 40 + 7

Add the parts: 30 + 40 = 70, and 6 + 7 = 13. Combine: 70 + 13 = 83. This partitioning method is mental rather than written, and exactly what the curriculum expects in Year 2.

Step 3 — Abstract with a number line. Draw an empty number line. Start at 36. Jump +40 to land on 76. Jump +4 to reach 80 (a bridge through 80). Jump +3 to reach 83. This shows the same calculation in a different way — children see that bridging through a multiple of 10 is efficient.