Addition and Subtraction
KS1MA-Y1-D002
Pupils learn to read, write and interpret addition and subtraction statements, use number bonds within 20, and solve simple one-step problems using concrete objects and pictorial representations.
National Curriculum context
In Year 1, addition and subtraction are introduced as related operations grounded in practical experience and the manipulation of concrete objects. Pupils memorise and reason with number bonds to 10 and 20 in multiple forms — for example, 9 + 7 = 16; 16 – 7 = 9; 7 = 16 – 9 — realising the effect of adding or subtracting zero and understanding that addition and subtraction are inverse operations. They combine and increase numbers by counting forwards and backwards, and discuss and solve problems in familiar practical contexts using the language: put together, add, altogether, total, take away, distance between, difference between, more than and less than. This rich vocabulary enables pupils to develop the concept of addition and subtraction flexibly, so they can apply these operations in varied contexts. One-step missing number problems — such as 7 = ? – 9 — begin to develop algebraic thinking from the earliest stage, challenging pupils to work backwards and reason about unknowns.
4
Concepts
2
Clusters
3
Prerequisites
4
With difficulty levels
Lesson Clusters
Understand addition and subtraction as operations
introduction CuratedEstablishing the conceptual meaning of both operations (combining/augmenting; taking away/difference) is the essential first move. Both concepts are mutually listed in co_teach_hints.
Know number bonds and solve missing number problems within 20
practice CuratedFluent number bonds are the knowledge base; missing number problems are the application. Both co-teach with the operation concepts and with each other.
Teaching Suggestions (1)
Study units and activities that deliver concepts in this domain.
Addition and Subtraction Facts Within 20
Mathematics Worked Example SetPedagogical rationale
Number bonds to 10 and 20 are the essential facts that underpin all later arithmetic. Pupils must move from counting all (combining two groups and counting from 1) to counting on (starting from the larger number) and eventually to known facts recalled from memory. The part-whole model is the key representation: it shows that addition and subtraction are inverse operations on the same structure, not two separate operations. Missing number problems develop early algebraic thinking.
Access and Inclusion
4 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)
Number bonds within 20
Keystone knowledge AI FacilitatedMA-Y1-C008
Number bonds are pairs of numbers that add together to make a given total. Knowing number bonds to 10 and to 20 fluently — both the addition facts and the related subtraction facts — is one of the most important early arithmetic skills. Mastery means pupils can instantly recall any number bond to 20 (e.g. 7 + 13 = 20, 20 – 13 = 7, 13 = 20 – 7) without counting, and can use these known facts flexibly to derive related calculations.
Teaching guidance
Use ten frames and double ten frames as primary concrete tools because their structure (5+5 per frame) makes complements visible. Part-whole models (bar model) help pupils see the relationship between the parts and the whole. Practise bonds to 10 first until secure, then extend to 20. Use 'bond collector' games, missing number activities and rapid-fire oral practice. Ensure pupils see all three forms of each bond: a + b = c, c – b = a, b = c – a. The curriculum explicitly states pupils should reason with bonds in 'several forms'.
Common misconceptions
Pupils memorise addition bonds but do not automatically recognise the related subtraction fact (e.g. knowing 7 + 6 = 13 but not knowing 13 – 6 = 7). They may know bonds to 10 but not extend them: knowing 3 + 7 = 10 but not recognising 13 + 7 = 20. Pupils often rely on counting on rather than instant recall, which is appropriate initially but must give way to automatic recall by end of KS1.
Difficulty levels
Finding pairs that make 5 and 10 using a ten frame and counters.
Example task
Put 3 red counters on the ten frame. How many yellow counters do you need to fill the frame to 10?
Model response: 7 yellow counters. 3 + 7 = 10.
Recalling number bonds to 10 without concrete support and beginning to learn bonds to 20.
Example task
What goes with 4 to make 10? What goes with 13 to make 20?
Model response: 6 goes with 4 to make 10. 7 goes with 13 to make 20.
Instant recall of all addition and subtraction bonds within 20, including relating addition bonds to their subtraction inverses.
Example task
What is 8 + 6? What is 14 – 6?
Model response: 8 + 6 = 14. 14 – 6 = 8.
Using known number bonds to solve related problems and explain relationships between facts.
Example task
If you know 7 + 8 = 15, what other facts do you know? List as many as you can.
Model response: 8 + 7 = 15, 15 – 7 = 8, 15 – 8 = 7. I also know that 17 + 8 = 25 because it is 10 more.
CPA Stages
concrete
Children use ten frames with two-colour counters to discover number bonds to 10: placing red counters in some cells and yellow in the rest makes the complement visible. Numicon plates show bonds through interlocking pairs that fill a 10-plate. Double ten frames extend the approach to bonds within 20.
Transition: Child builds any bond to 10 on a ten frame without counting the empty spaces one by one — they see the complement as a known fact and announce it immediately.
pictorial
Children draw ten frame diagrams and use part-whole models (bar models) to represent number bonds. Bond collector record sheets encourage systematic recording of all bonds to 10 and 20 in all three forms (addition and both subtractions).
Transition: Child completes part-whole models for any bond within 20 without counting on, and writes all four facts in the fact family from a single drawn model.
abstract
Children recall all addition and subtraction bonds within 20 instantly, in any form, without visual support. They respond to rapid-fire oral and written questions covering all three representations of each bond.
Transition: Child recalls any bond within 20 in any form (addition, subtraction, missing number) within 3 seconds without counting on or using visual support.
Delivery rationale
Primary maths (Y1) with concrete stage requiring physical manipulatives (Ten frames, Double ten frames). AI delivers instruction; facilitator sets up materials.
Access barriers (2)
Number bond fluency is often assessed through timed activities. Children with processing speed difficulties may know the bonds but be unable to produce them within the expected time window.
Number bonds within 20 require holding a target number and a part simultaneously while computing the missing part. The fluency expectation (instant recall) demands automaticity that takes longer to develop for children with working memory difficulties.
Addition as combining and augmenting
knowledge AI FacilitatedMA-Y1-C009
Addition describes two distinct situations: combining two separate groups into one (aggregation: 3 apples + 4 apples = 7 apples) and increasing a quantity by adding more (augmentation: I had 3 apples and got 4 more). Pupils need to recognise and work with both structures. Mastery means pupils can represent addition in multiple ways (concrete objects, number line, number sentence) and select appropriate strategies for different problems.
Teaching guidance
Use rich problem-solving contexts to expose both structures. For combining: use two groups of objects and physically push them together. For augmenting: begin with a set and add objects one at a time, counting on. Connect the '+' symbol to the action of combining or adding. Teach pupils to 'count on from the larger number' (e.g. for 3 + 7, start from 7 and count on 3) rather than always starting from the first addend. Connect to number bonds for quick recall without counting.
Common misconceptions
Pupils often treat addition as only 'putting two groups together', not recognising the augmentation (adding to) structure. They may count all objects from the beginning rather than counting on, which is inefficient and error-prone. Pupils frequently forget that addition is commutative (3 + 7 = 7 + 3), restarting a count rather than choosing the more efficient order.
Difficulty levels
Combining two groups of objects by pushing them together and counting all to find the total.
Example task
Here are 3 red cubes and 4 blue cubes. Push them together. How many altogether?
Model response: 7. There are 7 cubes altogether. 3 and 4 makes 7.
Representing addition using a number line — counting on from the larger number.
Example task
Use the number line to work out 5 + 3. Start at 5 and count on 3 jumps.
Model response: Start at 5, jump to 6, 7, 8. The answer is 8. 5 + 3 = 8.
Solving addition problems within 20 using recall or efficient strategies, recording as a number sentence.
Example task
There were 9 children on the bus. 7 more got on. How many children are on the bus now? Write a number sentence.
Model response: 9 + 7 = 16. There are 16 children on the bus.
CPA Stages
concrete
Children physically push two groups of objects together (combining) or start with a group and add more objects one at a time (augmenting). They count the total by touching each object. Both structures of addition are practised with real objects before any recording.
Transition: Child combines or augments two groups of objects and finds the correct total by counting on from the larger group rather than recounting all objects from the beginning.
pictorial
Children use number lines with drawn jumps to represent addition as counting on. Simple pictures of objects being joined together illustrate combining problems. Bar models (part-part-whole diagrams) begin to represent addition structure.
Transition: Child draws jumps on a number line to solve addition within 20, starting from the larger number and counting on the smaller number, landing on the correct answer consistently.
abstract
Children write addition number sentences using the + and = symbols, solve addition problems mentally using recalled number bonds, and interpret addition word problems selecting the correct operation.
Transition: Child writes correct addition number sentences for word problems involving both combining and augmenting contexts, using recalled bonds rather than counting on.
Delivery rationale
Primary maths (Y1) with concrete stage requiring physical manipulatives (Counters, Interlocking cubes). AI delivers instruction; facilitator sets up materials.
Access barriers (2)
Understanding addition as both combining (aggregation) and augmenting requires grasping two distinct conceptual structures. Without concrete demonstrations of both, children may develop a narrow understanding limited to one structure.
Addition word problems require parsing sentence structures to identify the operation. Children with receptive language needs may understand addition conceptually but be blocked by the linguistic framing of word problems.
Subtraction as taking away and finding the difference
knowledge AI FacilitatedMA-Y1-C010
Subtraction also describes two distinct situations: taking away (removing objects from a group: 7 – 3 = 4, removing 3 from a group of 7) and finding the difference (how many more is 7 than 3?). Both structures result in the same calculation but require different reasoning and contexts. Mastery means pupils can interpret subtraction problems correctly and choose appropriate strategies.
Teaching guidance
Use concrete objects for 'take away' problems — physically removing objects from a group. For 'difference', place two rows of objects side by side and count the gap. Use the number line: count back for take away, count the distance between two numbers for difference. The curriculum requires problems to use the language 'take away', 'distance between' and 'difference between', so expose pupils to all these terms explicitly. Connect to missing number problems (7 = ? – 9) to develop flexible thinking.
Common misconceptions
Pupils often interpret all subtraction as 'taking away' and cannot solve 'difference' problems by subtraction (they count instead). They may always compute the larger minus the smaller, even when the context requires the smaller from the larger (e.g. computing 9 – 7 = 2 correctly but not knowing how to interpret 7 – 9 in context). Subtracting zero also causes confusion — some pupils think 'taking away nothing means you get zero'.
Difficulty levels
Physically removing objects from a group and counting the remainder for 'take away' problems.
Example task
Here are 8 cubes. Take away 3. How many are left?
Model response: 5. There are 5 cubes left.
Using a number line to count back for take-away problems and count the gap for difference problems.
Example task
Use the number line. Start at 12. Count back 5. What number do you land on?
Model response: Start at 12: 11, 10, 9, 8, 7. I land on 7. 12 – 5 = 7.
Solving both take-away and difference problems within 20, choosing an appropriate strategy and recording as a number sentence.
Example task
Ali has 15 stickers. Ben has 9 stickers. How many more stickers does Ali have than Ben?
Model response: 15 – 9 = 6. Ali has 6 more stickers than Ben.
CPA Stages
concrete
For 'take away': children remove objects from a group and count what remains. For 'difference': children line up two rows of objects side by side and count the gap between them. Both structures are practised physically so children understand subtraction has two distinct meanings.
Transition: Child solves both take-away and difference problems with objects, correctly distinguishing between removing objects (take away) and comparing two groups (difference).
pictorial
Children use number lines to count back for take-away problems and count the distance between two numbers for difference problems. Crossing-out pictures show removal; comparison bar diagrams show the gap between two quantities.
Transition: Child uses a number line to solve subtraction within 20 by counting back (take away) or counting the gap (difference), selecting the appropriate strategy for the problem type.
abstract
Children write subtraction number sentences using the – and = symbols, use subtraction vocabulary (take away, difference, how many more) in context, and solve word problems by selecting the correct operation.
Transition: Child writes correct subtraction number sentences for word problems involving both take-away and difference contexts, and explains which structure the problem uses.
Delivery rationale
Primary maths (Y1) with concrete stage requiring physical manipulatives (Counters, Interlocking cubes for comparison rows). AI delivers instruction; facilitator sets up materials.
Access barriers (1)
Subtraction as both taking away and finding the difference requires understanding two distinct word problem structures. The language of difference ('How many more?', 'What is the difference?') is particularly challenging for children with SLCN.
Missing number problems
knowledge AI FacilitatedMA-Y1-C011
Missing number problems, such as 7 = ? – 9, require pupils to reason about the relationship between numbers in an equation rather than simply computing an answer. These problems lay the foundations of algebraic thinking, requiring pupils to understand the equals sign as expressing equivalence (not just 'the answer comes after') and to use inverse operations to find unknowns. Mastery means pupils can work flexibly with the equals sign in all positions and use known facts to deduce missing values.
Teaching guidance
Use bar models (part-whole diagrams) to make the structure of the problem visible before presenting it as an equation. Start with missing addends (? + 3 = 7) as these are more intuitive, then progress to missing subtrahends and to problems where the missing number is at the start or in the middle of the equation. Connect explicitly to known number bonds — if pupils know 7 + 9 = 16, they can solve 7 = 16 – 9. Avoid always writing equations as a + b = c; mix the position of the equals sign.
Common misconceptions
Pupils often interpret the equals sign as meaning 'write the answer here' (operational understanding) rather than as expressing a balanced relationship (relational understanding). This means they are confused by equations where the equals sign is in the middle or at the left (e.g. 7 = 3 + 4). They may write expressions like 3 + 4 = 7 + 2 = 9 (running the equals sign on) which represents a fundamental misconception about equality.
Difficulty levels
Solving missing number problems with a missing addend using concrete objects: ? + 3 = 7.
Example task
I have some cubes and 3 more cubes. Altogether there are 7. How many cubes did I start with? Use cubes to find out.
Model response: 4. I started with 4 cubes because 4 + 3 = 7.
Solving missing number problems in different positions using a part-whole model: 6 + ? = 11, ? – 4 = 5.
Example task
Fill in the missing number: 6 + ? = 11
Model response: 6 + 5 = 11. The missing number is 5.
Solving missing number problems where the unknown is in any position, understanding = as equivalence.
Example task
Find the missing number: 7 = ? – 9
Model response: 7 = 16 – 9. The missing number is 16 because 16 – 9 = 7.
CPA Stages
concrete
Children use part-whole models with physical objects: cubes in a tray divided into three sections (whole, part, part), with one section covered by a cup or cloth. They work out the hidden part using their knowledge of the visible parts and the whole.
Transition: Child finds the hidden quantity in a part-whole model without needing to peek, using subtraction or known bonds, and explains: 'I know 3 and 4 make 7, so 4 must be hiding.'
pictorial
Children solve missing number problems using drawn part-whole diagrams with a question mark in one section. They also work with number line diagrams showing a gap and equation cards with blanks in different positions.
Transition: Child solves missing number problems from a drawn part-whole model where the unknown is in any position (missing part or missing whole), writing the corresponding equation correctly.
abstract
Children solve written missing number equations where the unknown can be at the start, middle or end, and where the equals sign can be on either side. They understand = as expressing equivalence, not as 'the answer comes after'.
Transition: Child solves missing number equations with the unknown in any position and the equals sign on either side, explaining their reasoning using the inverse relationship between addition and subtraction.
Delivery rationale
Primary maths (Y1) with concrete stage requiring physical manipulatives (Part-whole trays (three sections), Cups or cloths for covering). AI delivers instruction; facilitator sets up materials.
Access barriers (1)
Missing number problems (__ + 3 = 7) require algebraic thinking — reasoning about an unknown quantity. This is highly abstract for Year 1 and particularly challenging for children who need concrete anchoring. The empty box IS the abstraction.