Addition and Subtraction Undo Each Other: a week of ready-to-teach maths
Five days of lessons for Year 3 Algebra. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet, plus fact-family triangles, box cards, a mini-check and every answer.
Start here: five minutes to Monday
- Skim the week at a glance on the next page.
- Print the five days. Each day is two A4 sheets: a plan and a worksheet.
- Cut out the fact-family triangles and the box cards once; they are reused all week.
- Open the free interactive unit on your board. Every plan tells you which picture to show and which buttons to press.
- Teach straight from the plan. Timings, talk prompts, misconceptions and answers are all on the one page.
No maths background needed
This pack is written for the busy generalist teacher. Addition and subtraction are one relationship read two ways: 8 and 5 make 13, and 13 take away 5 leaves 8. Each operation undoes the other, which is what inverse means. A part-part-whole bar shows it, and the same idea finds the number in a box (7 + □ = 16) and checks a subtraction by adding back.
One day, one lesson
The five lessons fill a week of maths, one lesson of about 50 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point if your timetable runs two shorter sessions.
The week at a glance
One lesson a day for a week. Each day stands on the day before, so run them in order.
| Day | Lesson | Children learn and do | On screen |
|---|---|---|---|
| 1 | The fact family | Show three numbers on a part-part-whole bar and write the four facts of their family | The fact family |
| 2 | Build and undo | Undo an addition with a subtraction and a subtraction with an addition, and think addition to subtract | Build and undo |
| 3 | Partition partners | Split a whole into two parts in different ways, and find a missing part by subtracting | Partition partner |
| 4 | Find the unknown | Decide whether the box is a part or the whole, then use the inverse to find it | Find the unknown; Which operation? |
| 5 | Check by inverse | Check a subtraction by adding back, and an addition by taking a part away; fix what is wrong | Check by inverse |
How the week builds
Day 1 shows the relationship on a bar, and Day 2 feels it as an undo. Day 3 uses it to split numbers, and Day 4 to find a missing number in a box. Day 5 turns it into a check that children run on their own work. It leans on Add and Subtract to Three Digits (AC9M3N03) and the Year 2 number sentences unit (AC9M2A01).
Materials for the week (one trip)
- From the classroom: scissors, pencils, this pack printed.
- Connecting cubes in two colours, counters, and a few paper cups or small bags.
- A 0 to 30 number line on the floor or the board, for walking forward and back.
- Cut out once, use all week: the fact-family triangles and the box cards in this pack.
Words for the word wall
fact family, part, whole, add, take away, inverse, undo, partition, split, missing number, unknown, number sentence, check
Dear families
This week in maths, Year 3 is learning that addition and subtraction undo each other. Knowing 8 + 5 = 13 also gives 13 − 5 = 8, and the same idea finds missing numbers and checks answers.
Try this at home
- Give three numbers that belong together (6, 9, 15) and ask for all four facts.
- Hide some pegs or coins in your hand: there were 12, now 5 are showing. How many are hidden?
- Ask your child to check a subtraction by adding back: 14 − 6 = 8, so 8 + 6 should be 14.
- Play “undo”: I added 7 and got 15. What did I start with?
Our fact families this week
Fill one row a day: three numbers that belong together, and one fact from their family.
| Day | Our three numbers | A fact from the family |
|---|---|---|
| Monday | ||
| Tuesday | ||
| Wednesday | ||
| Thursday | ||
| Friday |
Printed from the free seegongsik Addition and Subtraction Undo Each Other teaching pack · seegongsik.com/au/y3/algebra/AC9M3A01/pack
The fact family
Three numbers that make an addition make a family of four facts: 8 + 5 = 13, 5 + 8 = 13, 13 − 8 = 5 and 13 − 5 = 8. A part-part-whole bar shows why: the two parts join to make the whole, and the whole splits back into the parts.
We are learning to
- show three related numbers on a part-part-whole bar,
- write the four facts of a fact family,
- explain why each take-away fact belongs: the whole take away one part leaves the other.
Success criteria
- I can write the four facts in a family.
- I can show the family on a bar.
You need
Connecting cubes in two colours. The fact-family triangles (cut-out sheet 1). The worksheet, one per child.
Lesson flow (about 50 minutes)
| 10 min | Cube trains Make a train of 8 red and 5 blue cubes: 13 altogether. Turn the train around, then snap off the blue part. Ask: “We snapped off 5 and 8 were left. What number sentence is that?” |
| 30 min | Four facts Reveal the family on the board (see the box). Children write the four facts for each worksheet bar, then fill in their own bar for 20. Ask: “Why can’t 5 − 13 be in the family?” |
| 10 min | Triangle cards Pairs take a complete fact-family triangle, say its four facts aloud, then swap. Ask: “Which number is the whole? How do you know?” |
Two half-sessions instead? Split after the board; start Session B with the worksheet.
Watch for these ideas
- Writing a take-away with a part first (5 − 13). Subtraction starts from the whole.
- Bringing in a new number (13 + 5 = 18). A family uses only its own three numbers.
- Thinking 8 + 5 and 5 + 8 have different answers. Turning the bar round keeps the whole.
Answers
- 9 + 6 = 15, 6 + 9 = 15, 15 − 9 = 6, 15 − 6 = 9.
- 7 + 4 = 11, 4 + 7 = 11, 11 − 7 = 4, 11 − 4 = 7.
- 12 + 5 = 17, 5 + 12 = 17, 17 − 12 = 5, 17 − 5 = 12.
- 8 + 8 = 16 and 16 − 8 = 8: only two different facts, because the two parts are the same.
- Own bar: 13 + 7 = 20, 7 + 13 = 20, 20 − 13 = 7, 20 − 7 = 13.
Fact families
Each bar shows a whole and its two parts. Write the four facts of its family.
____ − ____ = ____ ____ − ____ = ____
____ − ____ = ____ ____ − ____ = ____
____ − ____ = ____ ____ − ____ = ____
____ − ____ = ____ ____ − ____ = ____
Number 4 is a doubles family. How many different facts does it have? Why? ______________________________
Your own bar
The whole is 20 and one part is 13. Fill in the bar, then write the family.
____ − ____ = ____ ____ − ____ = ____
Build and undo
Every addition has an opposite that puts things back. Start at 0, add 5 and add 8 to reach 13; take away 8 and take away 5 and you are back at 0. Addition and subtraction are inverse operations: each one undoes the other.
We are learning to
- undo an addition with the matching subtraction, and a subtraction with the matching addition,
- use the undo to get back to the start number,
- think of the addition to work out a subtraction (13 − 5: 5 and what make 13?).
Success criteria
- I can undo an addition or a subtraction.
- I can use addition to work out a subtraction.
You need
A 0 to 30 number line on the floor or the board. Counters. The worksheet, one per child.
Lesson flow (about 50 minutes)
| 10 min | Walk and walk back A child stands on 4 on the floor number line, walks forward 6 to 10, then walks back 6. Ask: “Where did they end up? Which walk undid the first one?” |
| 30 min | Build and undo Build and undo on the board (see the box). Children do the worksheet undo chains, then the think-addition take-aways and the mystery start. Ask: “To undo + 8, what do you press? Why not − 5?” |
| 10 min | Undo it In pairs, one child secretly adds or takes away from a start and says the new number; the other says how to get back. Ask: “How did you know which way to go back?” |
Two half-sessions instead? Split after the undo chains; start Session B with the think-addition take-aways.
Watch for these ideas
- Undoing with the same operation (+ 8 undone by + 8). The undo is the opposite operation.
- Undoing with a different number (+ 8 undone by − 5). The undo uses the same number.
- Counting back one at a time for every take-away. Think of the addition fact instead.
Answers
- Undo: 13, take away 7, back at 6; 6, add 9, back at 15; 32, take away 12, back at 20; 17, take away 8 then 5, back at 4.
- Think addition: 13 − 5 = 8; 16 − 9 = 7; 14 − 6 = 8; 17 − 8 = 9; 12 − 7 = 5; 20 − 13 = 7.
- Mystery start: 9, because 15 − 6 = 9 undoes the add 6.
Build and undo
Undo it
1. Start at 6. Add 7. You are at ______. Undo it: ______________. You are back at ______.
2. Start at 15. Take away 9. You are at ______. Undo it: ______________. You are back at ______.
3. Start at 20. Add 12. You are at ______. Undo it: ______________. You are back at ______.
4. Start at 4. Add 5, then add 8. You are at ______. Undo both: ______________. You are back at ______.
Think addition
Use the addition fact to find each answer.
13 − 5 = ____ because 5 + ____ = 13
16 − 9 = ____ because 9 + ____ = 16
14 − 6 = ____ because 6 + ____ = 14
17 − 8 = ____ because 8 + ____ = 17
12 − 7 = ____ because 7 + ____ = 12
20 − 13 = ____ because 13 + ____ = 20
Mystery start
I started at a number, added 6 and got 15. What was my start? How did you undo it?
Partition partners
A whole can be split into two parts in many ways: 13 is 6 and 7, or 5 and 8, or 10 and 3. Fix one part and the other is decided: if one part is 4, the other must be 13 − 4 = 9. The whole and one part find the other by subtraction.
We are learning to
- partition a number into two parts in different ways,
- find the missing part when the whole and one part are known,
- write a partition as an addition and as a take-away (13 = 4 + 9, 13 − 4 = 9).
Success criteria
- I can split a whole into two parts in different ways.
- I can find a missing part by subtracting.
You need
Counters and a paper cup for each pair. The worksheet, one per child.
Lesson flow (about 50 minutes)
| 10 min | Under the cup Show 13 counters, then hide some under a cup so that 4 are showing. Ask: “How many are under the cup? How do you know without lifting it?” |
| 30 min | Fix one part Move the split on the board (see the box). Children find every way to split 10, then the missing partners and the story on the worksheet. Ask: “Have we found every way to split 10? How can we be sure?” |
| 10 min | Cup game Pairs play Under the cup with 15 counters, writing each round as a take-away. Ask: “Which take-away did you write? Which number was the whole?” |
Two half-sessions instead? Split after the splits of 10; start Session B with the partners and the cup game.
Watch for these ideas
- Adding the whole and the part (13 + 4) to find a missing part. A part is smaller than the whole.
- Missing splits when listing them. Work in order: 0 and 10, 1 and 9, 2 and 8, and so on.
- Counting 4 + 6 and 6 + 4 as different splits. They are the same two parts, turned round.
Answers
- Splits of 10: 0 + 10, 1 + 9, 2 + 8, 3 + 7, 4 + 6 and 5 + 5 (six ways, or eleven if turned-round splits count).
- Partners: 13 − 4 = 9; 16 − 9 = 7; 20 − 7 = 13; 18 − 11 = 7; 25 − 10 = 15; 30 − 14 = 16.
- Story: 17 − 9 = 8 children in the classroom.
Partition partners
Every way to split 10
Count 4 + 6 and 6 + 4 as the same split. You may use 0.
Find the partner
Example: the whole is 13 and one part is 4, so the other part is 13 − 4 = 9.
| Whole | One part | Other part | Take-away sentence |
|---|---|---|---|
| 16 | 9 | ||
| 20 | 7 | ||
| 18 | 11 | ||
| 25 | 10 | ||
| 30 | 14 |
Story
There are 17 children in the class. 9 are at the library. How many are in the classroom? Write the take-away sentence. ______________________
Find the unknown
A box in a number sentence stands for a missing number. Ask: is the box a part or the whole? If it is a part, take the other part from the whole (7 + □ = 16, so 16 − 7 = 9). If it is the whole, put the parts back together (□ − 5 = 9, so 9 + 5 = 14).
We are learning to
- decide whether the box in a number sentence is a part or the whole,
- use the inverse operation to find the missing number,
- check the answer by putting it back in the sentence.
Success criteria
- I can find the number in the box.
- I can say whether I added or subtracted, and why.
You need
Counters and a small bag. The box cards (cut-out sheet 2), one set per pair. The worksheet.
Lesson flow (about 50 minutes)
| 10 min | Mystery bag Put 7 counters in a bag, secretly add more, and say there are now 16. Write 7 + □ = 16 on the board. Ask: “Is the box a part or the whole? How will we find it?” |
| 30 min | Free the box Find the unknown, then choose the operation on the board (see the box). Pairs sort the box cards into a part and the whole, then children do the worksheet. Ask: “In 15 − □ = 6 the box comes after the minus. Is it a part or the whole?” |
| 10 min | Put it back Children check three of their answers by writing the number back into the sentence. Ask: “Does the sentence come out true? What would show it was wrong?” |
Two half-sessions instead? Split after Find the unknown; start Session B with Which operation? and the card sort.
Watch for these ideas
- Doing the operation that is shown (7 + □ = 16, so 23). Undo it instead: 16 − 7.
- Taking a box after a minus as the whole (15 − □ = 6, so 21). The 15 is the whole.
- Not checking. Put the answer in the box: 7 + 9 = 16 is true.
Answers
- 9 + □ = 17: part, 17 − 9 = 8; □ + 7 = 15: part, 15 − 7 = 8; 18 − □ = 11: part, 18 − 11 = 7; □ − 6 = 8: whole, 8 + 6 = 14.
- 12 + □ = 20: part, 20 − 12 = 8; □ − 9 = 9: whole, 9 + 9 = 18; 25 − □ = 10: part, 25 − 10 = 15; □ + 13 = 19: part, 19 − 13 = 6.
- Story: □ − 6 = 9, the box is the whole, so 9 + 6 = 15 stickers at first.
Find the unknown
Is the box a part or the whole? Use the inverse to find it, then put it back to check.
| Number sentence | Part or whole? | I worked out | □ = |
|---|---|---|---|
| 9 + □ = 17 | part / whole | ||
| □ + 7 = 15 | part / whole | ||
| 18 − □ = 11 | part / whole | ||
| □ − 6 = 8 | part / whole | ||
| 12 + □ = 20 | part / whole | ||
| □ − 9 = 9 | part / whole | ||
| 25 − □ = 10 | part / whole | ||
| □ + 13 = 19 | part / whole |
Story
Mia had some stickers. She gave away 6 and has 9 left. How many did she have at first? Write a number sentence with a box, then find the box.
Number sentence: ______________________ □ = ______
Check by inverse
Addition checks subtraction. To check 16 − 9 = 7, add the answer to the part taken away: 7 + 9 = 16 rebuilds the whole, so the answer is right. If the check does not rebuild the whole, the answer is wrong. Subtraction checks addition in the same way.
We are learning to
- check a take-away by adding the answer and the part taken away,
- check an addition by taking one part away from the total,
- explain how the check shows that an answer is right or wrong, and fix a wrong one.
Success criteria
- I can check a take-away with an addition.
- I can find and fix a wrong answer.
You need
The worksheet, one per child. The unit’s self-check quiz.
Lesson flow (about 50 minutes)
| 10 min | Spot my mistake Write 17 − 8 = 9, 15 − 7 = 9 and 12 − 5 = 7. Ask the class to find the wrong one without counting back. Ask: “Which addition proves that one is wrong?” |
| 25 min | Check it Check claims on the board (see the box). Children check every worksheet answer with the inverse and fix the wrong ones. Ask: “The check gave 14, not 15. Was the answer wrong, or the question?” |
| 15 min | Class quiz Run the self-check quiz as a team game: teams hold up one, two or three fingers for their choice. Ask: “Which picture from this week would prove that answer?” |
Two half-sessions instead? Split after the take-away checks; start Session B with the addition checks and the quiz.
Watch for these ideas
- Checking with the two numbers in the question (16 + 9). Add the answer and the part: 7 + 9.
- Trusting a check that does not match. If the check is not the whole, the answer is wrong.
- Fixing a wrong answer by changing the question. Keep the question; find the right answer.
Answers
- Opening: 15 − 7 = 9 is wrong, because 9 + 7 = 16; it is 8.
- 18 − 9 = 9 right (9 + 9 = 18); 15 − 8 = 6 wrong (6 + 8 = 14; it is 7); 20 − 6 = 14 right (14 + 6 = 20).
- 17 − 9 = 7 wrong (7 + 9 = 16; it is 8); 23 − 10 = 13 right (13 + 10 = 23); 14 − 5 = 8 wrong (8 + 5 = 13; it is 9).
- 7 + 8 = 15 right (15 − 8 = 7); 9 + 6 = 14 wrong (14 − 6 = 8, not 9; it is 15); 12 + 7 = 19 right (19 − 7 = 12). Ella: 18 + 12 = 30, so she is right.
Check by inverse
Check each take-away
Add the answer and the part taken away. Does it rebuild the whole? Fix any wrong answer.
| Someone wrote | My check | Right or wrong? | Fixed answer |
|---|---|---|---|
| 18 − 9 = 9 | |||
| 15 − 8 = 6 | |||
| 20 − 6 = 14 | |||
| 17 − 9 = 7 | |||
| 23 − 10 = 13 | |||
| 14 − 5 = 8 |
Check each addition
Take one part away from the total. Do you get the other part?
| 7 + 8 = 15 | |||
| 9 + 6 = 14 | |||
| 12 + 7 = 19 |
Write a check
Ella says 30 − 12 = 18. Write the addition that checks it. Is she right? ______________________
Fact-family triangles
Cut out the twelve triangles. The whole sits at the top; the two parts sit at the bottom. Cards 1 to 6 are complete (Day 1: say the four facts). Cards 7 to 12 hide one number (Days 3 and 4: find it).
Teacher note, the hidden numbers: card 7 is 8, card 8 is 5, card 9 is 8, card 10 is 18, card 11 is 7, card 12 is 12.
Box cards
Cut out the twelve cards. On Day 4, pairs sort them into two piles, “the box is a part” and “the box is the whole”, then find the number in each box.
Teacher note. The box is a part: 6 + □ = 13 (7), □ + 9 = 17 (8), 14 − □ = 6 (8), 11 + □ = 19 (8), 16 − □ = 9 (7), □ + 5 = 12 (7), 20 − □ = 14 (6). The box is the whole: □ − 4 = 9 (13), □ − 8 = 7 (15), □ − 10 = 5 (15), □ − 3 = 12 (15), □ − 7 = 7 (14).
What we know: Addition and Subtraction Undo Each Other
Work on your own. Draw a bar if it helps.
- Write the four facts for 6, 9 and 15. ______________ ______________ ______________ ______________
- Start at 11 and add 7. How do you get back to 11? ______________________
- The whole is 18 and one part is 11. The other part is ______.
- 8 + □ = 15 □ = ______
- □ − 7 = 6 □ = ______
- 20 − □ = 12 □ = ______
- Someone wrote 16 − 7 = 8. Write the addition that checks it: ______________ Right or wrong? ______
- Sam had some marbles. He won 5 more and now has 14. Write a number sentence with a box: ______________ □ = ______
Answers and marking guide
Answers
- 6 + 9 = 15, 9 + 6 = 15, 15 − 6 = 9, 15 − 9 = 6.
- Take away 7 (18 − 7 = 11).
- 7 (18 − 11 = 7).
- 7 (the box is a part: 15 − 8).
- 13 (the box is the whole: 6 + 7).
- 8 (the box is a part: 20 − 12).
- 8 + 7 = 15, which is not 16, so it is wrong. 16 − 7 = 9.
- □ + 5 = 14, so □ = 9 marbles (14 − 5).
A quick three-level guide
| Idea | Working towards | At standard | Beyond |
|---|---|---|---|
| Fact families (Q1) | writes the additions but not the take-aways | writes all four facts and shows them on a part-part-whole bar | explains why a doubles family has only two different facts |
| Inverse and parts (Q2, Q3, Q7) | undoes with the same operation | undoes with the inverse, finds a missing part, and checks a take-away by adding | chooses whether to check by adding or by taking away |
| Unknowns (Q4, Q5, Q6, Q8) | does the operation shown (7 + □ = 16, so 23) | decides whether the box is a part or the whole and uses the inverse | writes a number sentence with a box for a story and solves it |
Eight questions, three ideas. A child at standard can say for each box whether it is a part or the whole, and check a take-away by adding back.
Weekly class record
Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.
| Name | Fact families | Undoes | Partitions | Finds the box | Checks |
|---|---|---|---|---|---|
The five columns are the five days: fact families, undoing, partitioning, finding the box, and checking by the inverse.