Addition and Subtraction
KS1MA-Y2-D002
Pupils recall and use addition and subtraction facts to 20 fluently, derive related facts up to 100, add and subtract two-digit numbers using multiple methods, and understand commutativity and the inverse relationship.
National Curriculum context
In Year 2, addition and subtraction develop substantially in scope and sophistication from Year 1. Pupils extend their understanding of the language of addition and subtraction to include sum and difference, and practise addition and subtraction to 20 to become increasingly fluent in deriving related facts — for example, using 3 + 7 = 10; 10 – 7 = 3 to calculate 30 + 70 = 100; 100 – 70 = 30. They check their calculations by adding to check subtraction and adding numbers in a different order to check addition (for example, 5 + 2 + 1 = 1 + 5 + 2 = 1 + 2 + 5), which establishes commutativity and associativity of addition. Recording addition and subtraction in columns supports place value and prepares pupils for formal written methods with larger numbers in Key Stage 2. The statutory requirements now demand that pupils work with a two-digit number and ones, a two-digit number and tens, two two-digit numbers, and adding three one-digit numbers — a substantial progression from the within-20 focus of Year 1.
4
Concepts
2
Clusters
3
Prerequisites
4
With difficulty levels
Lesson Clusters
Recall and use addition and subtraction facts to 20 and 100
introduction CuratedFluent recall of number facts and the procedural skill of adding/subtracting two-digit numbers are tightly linked (C004 and C005 mutually co-teach). This pairing drives fluency across the rest of the domain.
Understand properties and relationships of addition and subtraction
practice CuratedCommutativity, associativity and the inverse relationship are the conceptual scaffolding that underlies mental calculation strategies. C006 co-teaches with C007.
Teaching Suggestions (1)
Study units and activities that deliver concepts in this domain.
Addition and Subtraction Within 100
Mathematics Worked Example SetPedagogical rationale
Y2 extends addition and subtraction to two-digit numbers, which introduces the critical concept of exchanging (regrouping). Pupils must understand that 10 ones can be exchanged for 1 ten and vice versa. This is best taught concretely with Dienes blocks: physically swapping 10 unit cubes for a ten-stick makes the exchange visible and tangible. The inverse relationship between addition and subtraction is a key reasoning concept: if 34 + 27 = 61, then 61 - 27 = 34. This is taught through the part-whole model.
Access and Inclusion
2 of 4 concepts have identified access barriers.
Barrier types in this domain
Recommended support strategies
Prerequisites
Concepts from other domains that pupils should know before this domain.
Concepts (4)
Recall of addition and subtraction facts to 20 and derived facts to 100
knowledge AI FacilitatedMA-Y2-C004
By Year 2, pupils should know number bonds to 20 and be able to use them to derive related facts to 100 — for example, knowing 3 + 7 = 10 leads directly to 30 + 70 = 100; 300 + 700 = 1000. This bridging from known facts to derived facts is a critical mental arithmetic skill. Mastery means pupils recall facts to 20 instantly without counting, and fluently apply place value reasoning to extend them to larger numbers.
Teaching guidance
Ensure bonds to 10 and 20 are fully automatic through regular rapid-fire oral and written practice before working on derived facts. When pupils know 6 + 4 = 10, show them explicitly how this gives 60 + 40 = 100 — use base-ten materials to make the structural connection visible. The curriculum specifies checking by adding numbers in different orders (5 + 2 + 1 = 1 + 5 + 2 = 1 + 2 + 5), establishing commutativity and associativity. Practise all three forms of each bond (a + b = c, c – b = a, b = c – a) so that derived subtraction facts are as automatic as addition facts.
Common misconceptions
Many pupils know addition bonds but fail to derive the related subtraction facts automatically. They may know 6 + 14 = 20 but not immediately know 20 – 14 = 6. The extension from bonds to 10/20 to derived facts to 100 requires explicit teaching; pupils often do not make this connection spontaneously. When checking calculations by reordering numbers, pupils sometimes change the calculation itself rather than just verifying.
Difficulty levels
Recalling addition bonds to 10 using a ten frame and counters.
Example task
Put 6 counters on the ten frame. How many empty spaces? What bond to 10 does this show?
Model response: 4 empty spaces. 6 + 4 = 10.
Instant recall of all bonds to 20 without concrete support, including related subtraction facts.
Example task
What is 8 + 7? What is 15 – 8?
Model response: 8 + 7 = 15. 15 – 8 = 7.
Deriving facts to 100 from known bonds to 10 and 20 using place value reasoning.
Example task
You know 6 + 4 = 10. What is 60 + 40? What is 56 + 4?
Model response: 60 + 40 = 100. 56 + 4 = 60.
Using known facts to check addition and subtraction by adding numbers in a different order.
Example task
Is this correct: 37 + 25 = 62? Check by adding in a different order.
Model response: Check: 25 + 37 = 25 + 30 + 7 = 55 + 7 = 62. Yes, it is correct.
CPA Stages
concrete
Children use ten frames with counters and Numicon plates to practise bonds to 10 and 20 until recall is automatic. Double ten frames make bonds to 20 visible. Cubes are grouped in tens to bridge from bonds to 10/20 to derived facts to 100.
Transition: Child recalls all bonds to 10 instantly without the ten frame and begins to derive bonds to 100 using Dienes blocks, stating the connection: 'If 7 + 3 = 10, then 70 + 30 = 100.'
pictorial
Children use part-whole models to record fact families within 20 and derived fact families to 100. They draw ten frame diagrams from memory and use number lines to show how known facts extend to larger numbers.
Transition: Child writes complete fact families for any bond within 20 and draws diagrams showing the derived facts to 100 or beyond, explaining the place value connection.
abstract
Children recall all addition and subtraction facts to 20 instantly and derive related facts to 100 fluently using place value reasoning. They check calculations by adding in a different order (commutativity) without any visual support.
Transition: Child recalls any fact within 20 in under 3 seconds and derives related facts to 100 instantly, explaining the place value reasoning: 'I know 6 + 4 = 10, so 60 + 40 = 100.'
Delivery rationale
Primary maths (Y2) with concrete stage requiring physical manipulatives (Ten frames, Double ten frames). AI delivers instruction; facilitator sets up materials.
Access barriers (2)
Recall of addition and subtraction facts to 20 is a fluency target — the curriculum expects 'instant recall'. Timed testing of these facts creates significant anxiety for children with processing speed difficulties or maths anxiety.
Deriving facts to 100 from known facts to 20 (e.g. if 3+7=10 then 30+70=100) requires holding the known fact, applying the place value transformation, and producing the derived fact. This is a three-step mental process.
Adding and subtracting two-digit numbers
Keystone skill AI FacilitatedMA-Y2-C005
Pupils in Year 2 add and subtract with two-digit numbers using concrete objects, pictorial representations and mental methods — working with a two-digit number and ones, a two-digit number and tens, and two two-digit numbers. They also add three one-digit numbers. Mastery means pupils can perform these calculations accurately and efficiently using appropriate strategies, selecting between mental methods (using known facts and partitioning) and recording in columns as preparation for formal written methods.
Teaching guidance
Use base-ten materials (Dienes) to model calculations concretely before using pictorial representations (drawn tens and ones, number lines, bar models). Teach multiple strategies: counting on from the larger number for small additions, partitioning for larger additions (43 + 25 = 40 + 20 + 3 + 5 = 65), and using number bonds. The curriculum specifies recording in columns to support place value and prepare for formal methods — introduce column recording as a pictorial/abstract stage that mirrors the concrete Dienes work. Adding three one-digit numbers is best approached by first finding a pair that makes 10 (e.g. for 4 + 7 + 6, notice 4 + 6 = 10, so the answer is 10 + 7 = 17).
Common misconceptions
When adding two two-digit numbers, pupils often add all four digits without regard to place value (e.g. 34 + 25 = 50 + 9 instead of 50 + 9 correctly — but sometimes written as 50 + 59 by combining wrongly). When crossing ten boundaries (e.g. 37 + 6 = 43), pupils often give 33 or 43 depending on whether they count on correctly. Subtracting a two-digit number from a two-digit number causes confusion when the ones digit of the first number is smaller than the ones digit being subtracted (e.g. 32 – 7); pupils may subtract in the wrong direction (7 – 2 = 5) giving 35 instead of 25.
Difficulty levels
Adding a one-digit number to a two-digit number without crossing 10, using Dienes blocks.
Example task
Use Dienes blocks to work out 34 + 5.
Model response: 34 + 5 = 39. I kept the 3 tens and added 5 ones to the 4 ones to get 9 ones.
Adding and subtracting two-digit numbers and ones, tens, or two-digit numbers using a number line, including crossing tens boundaries.
Example task
Work out 47 + 26 using a number line.
Model response: Start at 47. Jump 20 to get 67. Jump 6 to get 73. 47 + 26 = 73.
Adding and subtracting combinations of two-digit numbers using efficient mental or recorded methods, and solving word problems.
Example task
There are 38 children in one class and 27 in another. How many altogether? Show your method.
Model response: 38 + 27 = 65. Method: 38 + 20 = 58, then 58 + 7 = 65.
CPA Stages
concrete
Children use Dienes blocks on a place value mat to add and subtract two-digit numbers physically. For addition, they combine tens and ones, exchanging 10 ones for 1 ten when the ones total exceeds 9. For subtraction, they exchange 1 ten for 10 ones when they cannot subtract the ones directly.
Transition: Child adds and subtracts two-digit numbers using Dienes blocks, correctly exchanging 10 ones for 1 ten (or vice versa) when needed, and states the result with place value explanation.
pictorial
Children use number lines with drawn jumps to add (jump forwards) and subtract (jump backwards), partitioning the second number into tens and ones for efficient jumping. They begin to record in columns, mirroring the Dienes work with drawn tens and ones.
Transition: Child uses a number line or column recording to add and subtract two-digit numbers, correctly partitioning and carrying/exchanging, arriving at the right answer consistently.
abstract
Children add and subtract two-digit numbers using efficient mental methods (partitioning, bridging through 10, using known facts) or recorded column methods. They select the best strategy for each calculation and solve word problems independently.
Transition: Child adds and subtracts two-digit numbers using mental or written methods without concrete support, choosing an efficient strategy and getting the correct answer including when crossing tens boundaries.
Delivery rationale
Primary maths (Y2) with concrete stage requiring physical manipulatives (Dienes blocks (ten-sticks and unit cubes), Place value mats). AI delivers instruction; facilitator sets up materials.
Access barriers (2)
Adding and subtracting two-digit numbers involves multi-step procedures: partition into tens and ones, add/subtract each part, recombine. Column methods add further steps: align digits, work right to left, carry/exchange. Each step must be completed correctly for the whole procedure to work.
Two-digit addition/subtraction requires holding partial results while computing the next step. When carrying or exchanging, the child must remember the carried digit while processing the next column.
Commutativity and associativity of addition; non-commutativity of subtraction
knowledge AI FacilitatedMA-Y2-C006
Commutativity means that the order of addends does not affect the sum (3 + 7 = 7 + 3). Associativity means that when adding three or more numbers, the grouping does not affect the result ((2 + 3) + 4 = 2 + (3 + 4)). Subtraction is not commutative (7 – 3 is not equal to 3 – 7). These properties are stated explicitly in the Year 2 curriculum and are essential for developing efficient calculation strategies. Mastery means pupils can apply these properties fluently to choose efficient calculation orders and justify their choices.
Teaching guidance
Demonstrate commutativity concretely by showing that two groups of objects combined give the same total regardless of order. Use arrays: a 3 × 4 array can be read as 3 rows of 4 or 4 rows of 3 — both give 12. For addition checking, require pupils to add numbers in a different order to verify (curriculum example: 5 + 2 + 1 = 1 + 5 + 2). Show that this works for addition but not subtraction: 7 – 3 = 4 but 3 – 7 cannot be 4. Connect to the number line: counting on 7 then 3 lands at the same place as counting on 3 then 7. Establish these as mathematical rules, not just tricks.
Common misconceptions
Pupils frequently try to apply commutativity to subtraction, writing 3 – 7 = 7 – 3 = 4, or stating that subtraction 'doesn't matter which way round'. They may apply the principle correctly in addition but have no understanding of why it works. Some pupils understand that 3 + 7 = 7 + 3 but do not use this knowledge to choose the more efficient calculation order.
Difficulty levels
Demonstrating commutativity with concrete objects: showing that 3 + 5 gives the same total as 5 + 3.
Example task
Put 3 red cubes and 5 blue cubes together. Count the total. Now put 5 blue cubes and 3 red cubes together. Is the total the same?
Model response: 3 + 5 = 8. 5 + 3 = 8. Yes, the total is the same both times.
Using commutativity to check addition calculations and recognising that subtraction is not commutative.
Example task
You worked out 14 + 8 = 22. How can you check this using commutativity? Does the same trick work for 14 – 8?
Model response: Check: 8 + 14 = 22. Yes, it matches. No, 14 – 8 = 6 but 8 – 14 is not 6, so subtraction is not commutative.
Using commutativity and associativity to add three numbers efficiently by choosing the easiest pair first.
Example task
Work out 7 + 8 + 3. Which two numbers would you add first to make it easier? Why?
Model response: Add 7 + 3 = 10 first, then 10 + 8 = 18. This is easier because 7 + 3 makes a number bond to 10.
CPA Stages
concrete
Children demonstrate commutativity by combining two groups of objects in both orders and observing the total is the same. They push 3 red cubes and 5 blue cubes together, count the total, then push 5 blue cubes and 3 red cubes together and count again. For subtraction, they show that reversing the order gives a different result.
Transition: Child demonstrates with objects that swapping the order of addition gives the same total, and explains why subtraction does not work the same way: 'You can't take 7 from 4 because there aren't enough.'
pictorial
Children use bar models and number lines to show that addition in both orders arrives at the same total. They draw fact families and use the commutative property to check addition by reordering. They add three numbers and show that rearranging gives the same sum.
Transition: Child draws bar models or number line diagrams to verify commutativity, and reorders three addends to find number bonds to 10 before adding the third number.
abstract
Children apply commutativity and associativity strategically: they choose the most efficient addition order (putting the largest number first, pairing numbers that make 10) and check calculations by adding in a different order. They state the rule as a mathematical principle.
Transition: Child spontaneously reorders addends to find efficient pairs (bonds to 10) and checks calculations by adding in a different order, explaining: 'Addition can be done in any order but subtraction cannot.'
Delivery rationale
Primary maths (Y2) with concrete stage requiring physical manipulatives (Interlocking cubes in two colours, Counters). AI delivers instruction; facilitator sets up materials.
Inverse relationship between addition and subtraction
knowledge AI FacilitatedMA-Y2-C007
Addition and subtraction are inverse operations — each undoes the other. If 7 + 5 = 12, then 12 – 5 = 7 and 12 – 7 = 5. This relationship allows pupils to check calculations (add to check subtraction, subtract to check addition) and to solve missing number problems by working backwards. Mastery means pupils fluently move between addition and subtraction facts, use the inverse relationship to check their work, and understand why these operations are linked.
Teaching guidance
Use the bar model (part-whole model) as the primary representation — it makes the part-part-whole structure visible and shows how the same three numbers give both an addition and two subtractions. The curriculum specifically states pupils should use the inverse to check calculations and solve missing number problems: if ? – 9 = 7, pupils add 7 + 9 = 16 to find the missing number. Practise presenting fact families: 7 + 5 = 12, 5 + 7 = 12, 12 – 5 = 7, 12 – 7 = 5. Require pupils to write the full fact family when given one fact.
Common misconceptions
Pupils often know the inverse relationship in principle but fail to apply it spontaneously to check calculations. They may write 12 – 5 = 7 and then check it by repeating 12 – 5 rather than using 7 + 5. Missing number problems that require using the inverse — such as ? – 5 = 7 — are consistently harder than problems where the unknown is at the end of the equation.
Difficulty levels
Using a bar model to see that the same three numbers make one addition and two subtractions.
Example task
The bar model shows: whole = 12, parts = 5 and 7. Write an addition and a subtraction using these numbers.
Model response: 5 + 7 = 12. 12 – 7 = 5.
Using the inverse relationship to check a calculation: adding to check subtraction and subtracting to check addition.
Example task
You think 15 – 9 = 6. Check it using addition.
Model response: Check: 6 + 9 = 15. Yes, it is correct.
Using the inverse to solve missing number problems and explain why the strategy works.
Example task
Find the missing number: ? – 8 = 13. Explain how you found it.
Model response: The missing number is 21. I used the inverse: 13 + 8 = 21. This works because if I take 8 away from the mystery number and get 13, then putting the 8 back must give me the mystery number.
CPA Stages
concrete
Children use bar models (part-whole models) with physical cubes to see that the same three numbers form both an addition and two subtractions. They place cubes in the 'whole' and 'parts' sections and physically move them to generate all four related facts.
Transition: Child generates all four facts in a fact family from a bar model with cubes and explains: 'If I know the addition, I can work out the subtraction because they are inverse operations.'
pictorial
Children use drawn bar models to check calculations by using the inverse: checking subtraction by adding, and checking addition by subtracting. They solve missing number problems by drawing bar models and reasoning backwards.
Transition: Child draws bar models to check calculations using the inverse operation and solves missing number problems by identifying the unknown part or whole, explaining their reasoning.
abstract
Children use the inverse relationship fluently without drawing: checking subtraction by adding, adding to check subtraction, and solving missing number problems by working backwards. They explain why the inverse method works as a mathematical principle.
Transition: Child uses the inverse to check and solve calculations mentally, explaining: 'If I subtract and get the answer, I can add the answer back to check. Addition and subtraction are inverse operations.'
Delivery rationale
Primary maths (Y2) with concrete stage requiring physical manipulatives (Part-whole model trays, Interlocking cubes). AI delivers instruction; facilitator sets up materials.