Addition and subtraction strategies
Reordering and grouping numbers to add more easily, linking subtraction to addition, and adding three numbers
About four lessons of 35 to 45 minutes
Some sums are easier if you add them in a different order
Adding 8 + 5 + 2 in order is fine, but there is a faster way. Notice that 8 and 2 make a friendly 10. Add those first, then add the 5: 10 + 5 = 15. You get the same answer either way, but grouping the friendly pair first is quicker and less likely to go wrong.
Today you learn two tricks like this: reordering and regrouping numbers to make addition easier, and using addition facts you already know to solve subtraction, because 13 - 8 is really asking the same question as 8 + ? = 13.
- 8 + 5 + 2, reordered to 8 + 2 + 58 and 2 make a friendly 10, then + 5 = 15
- 3 birds, then 5 more, then 2 more land3 + 5 + 2 = 10 birds altogether
- 13 - 8, thought of as 8 + ? = 138 + 5 = 13, so 13 - 8 = 5
- 6 + 4 gives the same total as 4 + 6addition order does not change the sum
What students will be able to do
Students will use the commutative and associative properties as strategies for adding and subtracting, understand subtraction as finding an unknown addend by relating it to a known addition fact, and solve word problems that add three whole numbers with a total within 20.
- I can add two numbers in either order and get the same total.
- I can group three numbers in a helpful order to make adding easier.
- I can solve a subtraction by thinking of it as a missing addition partner.
- I can solve a word problem that adds three numbers together.
Standards this unit teaches
- 1.OA.B.3Common Core (US)Properties of addition and subtraction
Apply properties such as commutativity and associativity as strategies to add and subtract.
- 1.OA.B.4Common Core (US)Subtraction as an unknown addend
Understand subtraction as finding an unknown addend, linking it to a related addition fact.
- 1.OA.A.2Common Core (US)Add three numbers
Solve word problems that add three whole numbers with a total within 20.
- AC9M1N03Australian Curriculum v9 (ACARA)Add and subtract within 20 (Year 1)
Add and subtract within twenty using materials and part-part-whole facts to ten, with a range of strategies.
- AC9M1N04Australian Curriculum v9 (ACARA)Number sentences to solve problems (Year 1)
Solve one- and two-digit adding and subtracting problems by writing number sentences and using part-part-whole reasoning.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
Words to teach and display
- Commutative property
- adding two numbers in either order gives the same total, such as 6 + 4 = 4 + 6
- Associative property
- when adding three or more numbers, which pair you add first does not change the total
- Unknown addend
- the missing number in an addition sentence, such as the ? in 8 + ? = 13
- Related fact
- an addition and a subtraction built from the same three numbers, such as 8 + 5 = 13 and 13 - 8 = 5
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Adding in either order gives the same total
ConcreteCount 6 counters, then 4 more: 6 + 4 = 10. Now count 4 counters, then 6 more: 4 + 6 = 10. Same two groups, same total, just added in a different order. This is the commutative property: swapping the order of the two numbers you add never changes the sum.
This is genuinely useful, not just a fact to memorise. When one number is much easier to start from, such as a number close to 10, put it first: for 3 + 9, it is quicker to think 9 + 3 and count on 3 from 9 than to count on 9 from 3.
Check that 7 + 5 and 5 + 7 give the same total.
- 7 + 5 = 12, counting on 5 from 7.
- 5 + 7 = 12, counting on 7 from 5.
- Both orders give the same total, 12.
Answer: 7 + 5 = 5 + 7 = 12.
- If 8 + 3 = 11, what is 3 + 8?
- Why might it be easier to add 2 + 9 by thinking of it as 9 + 2 instead?
2. Grouping numbers to make adding easier
PictorialWhen adding three numbers, you get to choose which two to add first, and choosing wisely makes the whole sum easier. For 8 + 5 + 2, spot the friendly pair 8 and 2, which make a clean 10. Reorder using the commutative property to 8 + 2 + 5, add the friendly pair first, 8 + 2 = 10, then add the last number, 10 + 5 = 15.
This combines both properties: the commutative property lets you reorder the three numbers, and the associative property says that once reordered, it does not matter which adjacent pair you add first, the total comes out the same. Always look for a pair that makes 10 first; it turns three-number addition into two easy steps.
Add 2 + 9 + 8 by grouping helpfully.
- Spot the friendly pair: 2 and 8 make 10.
- Reorder to 2 + 8 + 9, then add the friendly pair first: 2 + 8 = 10.
- Add the last number: 10 + 9 = 19.
Answer: 2 + 9 + 8 = 19.
- In 4 + 6 + 7, which two numbers make a friendly pair to add first?
- Does it matter which pair you add first, as long as you add all three numbers?
3. Subtraction as a missing addition partner
AbstractEvery subtraction can be rethought as an addition with a missing number. 13 - 8 asks how many are left when 8 is taken from 13, but it is the exact same question as 8 + ? = 13, what number joins 8 to make 13. If you already know 8 + 5 = 13, you instantly know 13 - 8 = 5, with no counting back needed.
This is a genuinely powerful shortcut once your addition facts are secure: instead of counting backward, which is slow and easy to lose track of, you search your memory for the addition fact that matches.
Work out 15 - 9 by thinking of it as a missing addition partner.
- Rewrite as an addition: 9 + ? = 15.
- What joins 9 to make 15? Count on from 9: 10, 11, 12, 13, 14, 15, that is 6 hops.
- So 9 + 6 = 15, which means 15 - 9 = 6.
Answer: 15 - 9 = 6.
- Rewrite 12 - 7 as a missing addition partner question.
- If 6 + 8 = 14, what is 14 - 8 straight away?
4. Word problems that add three numbers
AbstractReal problems do not always involve just two numbers. 3 birds are on a branch, 5 more land, then 2 more land. How many birds altogether? Add all three: 3 + 5 + 2. Use the grouping strategy from section 2 to choose an easy order if one pair makes the sum simpler.
The most common slip in a three-number word problem is missing one of the numbers, especially the middle one. Read the problem twice and list all three numbers before adding, so none gets left out.
6 apples are in a basket, 7 more apples are picked, then 4 more are picked. How many apples in total?
- List all three numbers: 6, 7, and 4.
- Group a helpful pair if one exists, or add in order: 6 + 7 = 13.
- Add the last number: 13 + 4 = 17.
Answer: 17 apples altogether.
- A word problem mentions three numbers: 4, 9, and 6. Which should you add first, and why?
- Why is it important to list all three numbers in a word problem before adding?
Common misconceptions and how to address them
MisconceptionThe child believes subtraction also has the commutative property, so 12 - 5 is thought to give the same result as 5 - 12.
Why it happens: The commutative property genuinely does work for addition, so it feels natural to assume every operation works the same way.
How to address it: Compute both directly: 12 - 5 = 7, but 5 - 12 cannot be done with only 5 objects to start with. Show that order matters for subtraction, unlike for addition.
MisconceptionWhen adding three numbers, the child always adds strictly left to right and never looks for a friendlier pair to group first.
Why it happens: Left-to-right is the first method taught, and switching the order can feel like it is against the rules rather than a legitimate strategy.
How to address it: Explicitly hunt for a pair that makes 10 before adding anything, circling that pair first. Show with the bar model that the total comes out the same whichever pair is added first.
MisconceptionFacing a subtraction, the child counts back one by one from the larger number instead of using a known addition fact, even when the matching fact is already memorised.
Why it happens: Counting back is the first subtraction method taught, and switching to 'think addition' takes deliberate practice to feel natural.
How to address it: Before counting, ask 'what do I already know that adds up to this?' For 14 - 6, prompt: what plus 6 makes 14? If 6 + 8 = 14 is known, the subtraction is solved instantly.
MisconceptionIn a three-number word problem, the child adds only two of the three numbers mentioned, most often missing the middle amount.
Why it happens: Holding three separate quantities in mind while reading a story problem is harder than holding two, so one is easily dropped.
How to address it: Before calculating, list all three numbers separately on paper first, then add them. Checking the list against the story catches a missing number before any adding starts.
Guided practice (with answers)
1. If 6 + 9 = 15, what is 9 + 6?
Answer: 15. Addition gives the same total in either order.
2. Add 5 + 8 + 5 by grouping a friendly pair first.
Answer: 18. Group 5 + 5 = 10 first, then 10 + 8 = 18.
3. Work out 14 - 6 by thinking 6 + ? = 14.
Answer: 8, because 6 + 8 = 14, so 14 - 6 = 8.
4. 4 + 9 + 6, in a word problem. What is the total?
Answer: 19, because 4 + 9 = 13 and 13 + 6 = 19 (or group 4 + 6 = 10 first, then + 9 = 19).
5. Work out 11 - 8 by thinking 8 + ? = 11.
Answer: 3, because 8 + 3 = 11, so 11 - 8 = 3.
6. Add 2 + 9 + 8 by grouping a friendly pair first.
Answer: 19. Group 2 + 8 = 10 first, then 10 + 9 = 19.
Independent practice worksheets
Set the matching ChalkBee worksheets for independent work. The answer keys are computed in code, so they are never wrong. Addition and subtraction sheets build the fact fluency this unit's strategies rely on, and word problems practise adding three numbers in context.
Differentiation
- Keep all three-addend problems within 10 first, before extending to totals up to 20.
- Always circle the friendly pair (the two numbers that make 10) before adding anything, as a fixed first step.
- Use physical counters to act out 'think addition' for subtraction: start with the smaller pile, add counters until reaching the larger total, then count how many were added.
- Give one number of a three-number word problem already circled to model listing all three before adding.
- Add four numbers by grouping more than one friendly pair, such as 3 + 7 + 4 + 6 (two pairs that each make 10).
- Write a subtraction as a missing-addend equation with the blank in different positions, such as ? + 7 = 15.
- Write your own three-number word problem with a total within 20 for a partner to solve.
- Explore whether the commutative property works for three numbers added in any order, not just two, by testing several orderings of the same three numbers.
Assessment: exit ticket
A three-question exit ticket for the last five minutes, sampling grouping, subtraction as unknown addend, and a three-number problem.
1. Add 7 + 8 + 3 by grouping a friendly pair first.
Answer: 18. Group 7 + 3 = 10 first, then 10 + 8 = 18.
2. Work out 16 - 9 by thinking 9 + ? = 16.
Answer: 7, because 9 + 7 = 16, so 16 - 9 = 7.
3. 5 fish, then 6 more, then 7 more join the tank. How many fish in total?
Answer: 18, because 5 + 6 + 7 = 18.
Teacher notes and timings
- Rough timing across four lessons: Lesson 1 the commutative property (section 1), Lesson 2 grouping three numbers (section 2), Lesson 3 subtraction as an unknown addend (section 3), Lesson 4 three-number word problems (section 4) plus the exit ticket.
- These three standards cluster naturally because they are all reasoning strategies built on top of the Kindergarten number-bond and add/subtract-within-10 foundation, rather than new arithmetic content: reorder, regroup, and relate subtraction back to addition.
- Language to keep saying: does the order matter, which pair is friendly, and what plus this number makes that number. These phrases pre-empt the misconceptions above.
- Curriculum note: US Grade 1 states the commutative and associative properties (1.OA.B.3), subtraction as an unknown addend (1.OA.B.4) and three-addend word problems (1.OA.A.2) as three standards. ACARA Year 1 folds strategy-based adding and subtracting within 20 into one broad descriptor (AC9M1N03) and writing number sentences to solve problems into another (AC9M1N04), so the AU curriculum groups this content more broadly than the three separate US standards.
- Present mode and print both work: use the Print button for a student worksheet, or project the page and build the bar models together as a class.