Inverse Operations and Fact Families: a week of ready-to-teach maths
Five days of lessons for Year 5 Algebra. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet, plus twelve fact triangles, four family mats with eighteen fact 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 triangles and the fact cards once; the triangles come back on Day 3.
- 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. Each plan explains the idea in plain words, lists the misconceptions children bring, and gives the answers. The big idea of the week: multiplication and division undo each other. Six rows of four dots make 24, and the same dots shared into rows of four make 6 rows. So three numbers, such as 6, 4 and 24, make a family of four facts, and a missing number is found with the inverse: in 6 × □ = 24, divide 24 by 6.
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 | One array, two ways | Read arrays as a multiplication and a division, and use one fact to undo another | One array, two operations |
| 2 | Fact families | Build the four facts of a family on fact triangles, and find a covered corner | A family from one array |
| 3 | Find the missing factor | Find a missing factor by dividing and check by multiplying, in sentences and stories | Find the missing factor |
| 4 | Does it belong? | Sort fact cards onto family mats, explain the cards that do not belong, and meet square families | Does it belong? |
| 5 | Multiply or divide? | Decide whether the box is the whole or a factor, then multiply or divide; play the unit quiz | Which inverse? and One product, a whole family |
How the week builds
Days 1 and 2 see the partnership in arrays and families; Days 3 and 4 use it to find missing numbers and to judge which facts belong; and Day 5 chooses the inverse wherever the box sits. It builds on Year 3’s inverse of addition and subtraction (AC9M3A01) and Year 4’s multiplication facts and related division facts (AC9M4A02), and it leads to the next unit, where children find unknown values in multiplication and division equations (AC9M5A02).
Materials for the week (one trip)
- From the classroom: scissors, envelopes for the cards, counters or cubes for arrays, mini whiteboards, this pack printed.
- Cut out once: the twelve fact triangles (Days 2 and 3) and the eighteen fact cards (Day 4); keep the four family mats whole. No maths equipment to buy.
Words for the word wall
array, row, column, product, factor, multiply, divide, inverse, fact family, unknown, whole
Dear families
This week in maths, Year 5 learns that multiplication and division undo each other. Three numbers such as 7, 8 and 56 make a family of four facts: 7 × 8 = 56, 8 × 7 = 56, 56 ÷ 7 = 8 and 56 ÷ 8 = 7. If one fact is known, the other three come free.
Try this at home
- Set out a muffin tray or an egg carton. Say the multiplication and the division it shows.
- Say a times-table fact; your child says the two division facts in its family.
- Ask: what times 9 makes 63? How can dividing help?
- Ask your child why 5 + 7 = 12 is not in the family of 5, 7 and 35.
My fact families this week
Fill one row a day. Choose three numbers and write their family.
| Day | The three numbers | The facts |
|---|---|---|
| Monday | ||
| Tuesday | ||
| Wednesday | ||
| Thursday | ||
| Friday |
Printed from the free seegongsik Inverse Operations and Fact Families teaching pack · seegongsik.com/au/y5/algebra/AC9M5A01/pack
One array, two ways
Children read an array both ways. Rows of dots build a multiplication: 3 rows of 7 is 3 × 7 = 21. The same dots shared back into rows of 7 make 3 rows: 21 ÷ 7 = 3. Only the question changes, so each operation undoes the other.
We are learning to
- read an array as a multiplication and as a division,
- explain that division undoes multiplication,
- use a known fact to write its inverse.
Success criteria
- I can draw an array for 6 × 7 and write a division it shows.
- I can explain why the same dots show both facts.
You need
Counters or cubes, about 40 per pair. Mini whiteboards. The worksheet, one per child.
Lesson flow (about 50 minutes)
| 10 min | Six rows of four Show the board (see the box). Ask for the multiplication, then the division, before pressing. Ask: “The dots did not move. How can the same picture be a division?” |
| 30 min | Arrays both ways Model 3 rows of 7 counters: 3 × 7 = 21; 21 ÷ 7 = 3 or 21 ÷ 3 = 7. Children do Parts A and B. Ask: “Which division does your array show: rows of 7, or 3 rows? Can it show both?” |
| 10 min | Undo it Children do Part C. Ask: “4 × 9 = 36. Without working it out, what is 36 ÷ 9?” |
Two half-sessions instead? Split after Part A; start Session B by building 8 × 3 with counters.
Watch for these ideas
- Writing a division with other numbers, such as 24 ÷ 2 for the 6 by 4 array. The division uses the same three numbers.
- Writing the division backwards: 4 ÷ 24. The total comes first.
- Counting every dot one by one. Rows times the number in each row gives the total.
Answers
- Part A: 3 × 9 = 27; 27 ÷ 9 = 3 or 27 ÷ 3 = 9. 7 × 4 = 28; 28 ÷ 4 = 7 or 28 ÷ 7 = 4. 5 × 8 = 40; 40 ÷ 8 = 5 or 40 ÷ 5 = 8.
- Part B: 6 × 7 = 42; 42 ÷ 7 = 6 or 42 ÷ 6 = 7. 8 × 3 = 24; 24 ÷ 3 = 8 or 24 ÷ 8 = 3. Part C: 54 ÷ 6 = 9; 6 × 8 = 48; 84 ÷ 12 = 7; 7 × 9 = 63.
One array, two ways
A. Write a multiplication and a division for each array
___ ÷ ___ = ___
___ ÷ ___ = ___
___ ÷ ___ = ___
B. Draw the array, then write both facts
C. Undo it
9 × 6 = 54, so 54 ÷ 6 = □48 ÷ 8 = 6, so 6 × 8 = □7 × 12 = 84, so 84 ÷ 12 = □63 ÷ 9 = 7, so 7 × 9 = □
Fact families
Children build fact families on fact triangles: the product at the top, the two factors below. Any two numbers give the third, and the three numbers give four facts. A square number such as 36 = 6 × 6 makes a family of only two.
We are learning to
- write the four facts in a family from three numbers,
- find a covered number on a fact triangle, and say the fact that finds it,
- explain why some families have only two facts.
Success criteria
- I can write all four facts for 7, 8 and 56.
- I can say a covered number and the fact I used to find it.
You need
The fact triangles (cut-out sheet 1), one set per pair. The worksheet, one per child.
Lesson flow (about 50 minutes)
| 10 min | Reveal the family Show the board (see the box). Children predict the facts before the reveal. Ask: “From 24, 6 and 4, how many facts can we make? Which ones?” |
| 30 min | Families and corners Model 4, 7 and 28: 4 × 7 = 28, 7 × 4 = 28, 28 ÷ 4 = 7, 28 ÷ 7 = 4. Pairs play cover a corner with the triangles for ten minutes, then do Parts A and B. Ask: “When the top is covered, do you multiply or divide? And for a bottom corner?” |
| 10 min | A family of two Children do Part C. Ask: “Which triangle in your set also makes a family of only two facts?” |
Two half-sessions instead? Split after the game; start Session B with Part A.
Watch for these ideas
- Mixing numbers from two families: 56 ÷ 7 = 9. Every fact uses the same three numbers.
- Stopping at two facts. Turn each one around: 7 × 8 and 8 × 7, then 56 ÷ 7 and 56 ÷ 8.
- Dividing a factor by the product: 7 ÷ 56. The product, the top number, is always the one divided.
Answers
- Part A: 7 × 8 = 56, 8 × 7 = 56, 56 ÷ 7 = 8, 56 ÷ 8 = 7. 9 × 4 = 36, 4 × 9 = 36, 36 ÷ 9 = 4, 36 ÷ 4 = 9. 6 × 12 = 72, 12 × 6 = 72, 72 ÷ 6 = 12, 72 ÷ 12 = 6. 3 × 11 = 33, 11 × 3 = 33, 33 ÷ 3 = 11, 33 ÷ 11 = 3.
- Part B: 9 (63 ÷ 7 = 9); 8 (48 ÷ 6 = 8); 45 (5 × 9 = 45); 11 (44 ÷ 4 = 11). Part C: 6 × 6 = 36 and 36 ÷ 6 = 6; the two factors are the same number, so turning a fact around gives the same fact. The 9, 9, 81 triangle is the other family of two.
Fact families
A. Write the four facts in each family
________________
________________
________________
________________
________________
________________
________________
________________
________________
________________
________________
________________
B. Cover a corner: find the missing number
Fact: ____________
Fact: ____________
Fact: ____________
Fact: ____________
C. A family of two
Write the family of 6, 6 and 36. Why does it have only two facts?
Find the missing factor
Children find a missing factor by dividing: in 5 × □ = 65, the box is the factor that makes 65 with 5, so it is 65 ÷ 5 = 13. They check by multiplying back, and use the fact triangles to see why dividing works.
We are learning to
- find a missing factor by dividing the product by the known factor,
- check the answer by multiplying,
- write a missing-factor sentence for a story.
Success criteria
- I can solve 7 × □ = 63 and check it.
- I can explain why dividing finds a missing factor.
You need
The fact triangles from Day 2. Mini whiteboards. The worksheet, one per child.
Lesson flow (about 50 minutes)
| 10 min | Which button? Show the board (see the box). Children write the factor on whiteboards before anyone presses. Ask: “18 is 24 − 6. Why is subtracting the wrong move for 6 × □ = 24?” |
| 30 min | Divide to find it Model 5 × □ = 65 on a triangle: 65 at the top and 5 below, so 65 ÷ 5 = 13; check 5 × 13 = 65. Children do Parts A and B. Ask: “Cover the box on a triangle. Which number do you divide, and by which?” |
| 10 min | Bigger numbers Children do Part C. Ask: “25 × □ = 175: how many 25s make 100? So how many make 175?” |
Two half-sessions instead? Split after Part A; start Session B with 5 × □ = 65 again.
Watch for these ideas
- Subtracting instead of dividing: 6 × □ = 24, so 18. Ask what times 6 makes 24.
- Dividing the wrong way round: 6 ÷ 24. The product is the number shared.
- Skipping the check. Multiply back to see whether the sentence is true.
Answers
- Part A: 63 ÷ 7 = 9; 42 ÷ 6 = 7; 72 ÷ 9 = 8; 60 ÷ 12 = 5; 52 ÷ 4 = 13; 96 ÷ 8 = 12; 51 ÷ 3 = 17; 90 ÷ 15 = 6.
- Part B: □ × 6 = 54, 9 bags; 7 × □ = 56, 8 cm; 8 × □ = 88, 11 players. Part C: 175 ÷ 25 = 7 (check 25 × 7 = 175); 240 ÷ 30 = 8 (check 8 × 30 = 240).
Find the missing factor
A. Divide to find the missing factor, then check by multiplying
| Sentence | Divide | Check |
|---|---|---|
| 7 × □ = 63 | ____ ÷ ____ = ____ | ____ × ____ = ____ |
| □ × 6 = 42 | ____ ÷ ____ = ____ | ____ × ____ = ____ |
| 9 × □ = 72 | ____ ÷ ____ = ____ | ____ × ____ = ____ |
| □ × 12 = 60 | ____ ÷ ____ = ____ | ____ × ____ = ____ |
| 4 × □ = 52 | ____ ÷ ____ = ____ | ____ × ____ = ____ |
| □ × 8 = 96 | ____ ÷ ____ = ____ | ____ × ____ = ____ |
| 3 × □ = 51 | ____ ÷ ____ = ____ | ____ × ____ = ____ |
| □ × 15 = 90 | ____ ÷ ____ = ____ | ____ × ____ = ____ |
B. Stories with a missing factor
Some bags hold 6 apples each, 54 apples in all. How many bags are there?
Sentence with a box: ____________________ Answer: ______________
A rectangle has an area of 56 square centimetres and is 7 cm wide. How long is it?
Sentence with a box: ____________________ Answer: ______________
8 teams, each with the same number of players, make 88 players. How many players are in each team?
Sentence with a box: ____________________ Answer: ______________
C. Bigger numbers
25 × □ = 175 □ = ______□ × 30 = 240 □ = ______
Does it belong?
Children sort fact cards onto family mats. A fact belongs to a family when it uses the same three numbers with multiplication or division. A card that adds or subtracts belongs to no family here, and a true fact with other numbers belongs to another family.
We are learning to
- sort facts into the families they belong to,
- explain why a fact does not belong,
- notice that a square family has only two facts.
Success criteria
- I can sort the fact cards and set aside the cards that belong to no mat.
- I can explain why 48 ÷ 6 = 8 is true but not in any family on our mats.
You need
The family mats and fact cards (cut-out sheet 2), one set per group. The worksheet, one per child.
Lesson flow (about 50 minutes)
| 10 min | Belongs or not? Show the board (see the box). Children show thumbs up or down before anyone presses. Ask: “5 + 7 = 12 uses 5 and 7. Why is it still not in the family of 5, 7 and 35?” |
| 30 min | Sort the cards Groups sort the eighteen cards onto the four mats, then do Parts A and B. Ask: “One mat ends up with only two cards. Is a card missing?” |
| 10 min | Make a family Children do Part C. Ask: “48 ÷ 6 = 8 is true. Which family does it belong to?” |
Two half-sessions instead? Split after the sort; start Session B with Part B.
Watch for these ideas
- Putting a card on a mat because one number matches. All three numbers must match.
- Thinking a true fact always belongs. It belongs only to the family of its own three numbers.
- Looking for four cards on the square mat. Turning 7 × 7 around gives the same fact.
Answers
- Part A: 6, 7, 42: A, F, K, P; 8, 9, 72: D, H, L, Q; 4, 12, 48: C, I, N, R; 7, 7, 49: E, M.
- Part B: B (72 − 9 = 63): it subtracts; G (48 ÷ 6 = 8): it is a true fact of another family; J (6 + 7 = 13): it adds; O (7 × 8 = 56): it is a true fact of another family. Part C: 9 × 11 = 99, 11 × 9 = 99, 99 ÷ 9 = 11, 99 ÷ 11 = 9. 8 × 8 = 64, 64 ÷ 8 = 8.
Does it belong?
A. Sort the fact cards onto the mats. Write the card letters.
| Family mat | Card letters | How many facts? |
|---|---|---|
| 6, 7, 42 | ||
| 8, 9, 72 | ||
| 4, 12, 48 | ||
| 7, 7, 49 |
B. The cards that belong to no mat
| Card | Why it does not belong |
|---|---|
C. Make a family
The family of 9, 11 and 99:
The family of 8, 8 and 64:
Multiply or divide?
The box can sit anywhere in a sentence, so children first ask where it would sit on a fact triangle. At the top, it is the whole: multiply the other two numbers. At a bottom corner, it is a factor: divide the whole by the factor they know. The sign does not decide: □ ÷ 4 = 5 is a division, but its box is found by multiplying.
We are learning to
- say whether the box is the whole or a factor,
- multiply to find a missing whole and divide to find a missing factor,
- write a sentence with a box for a story, and solve it.
Success criteria
- I can say whether the box in □ ÷ 4 = 8 is the whole or a factor, and find it.
- I can write a sentence with a box for a story and solve it.
You need
The fact triangles from Day 2. Mini whiteboards. The worksheet, one per child.
Lesson flow (about 50 minutes)
| 10 min | Whole or factor? Show the board (see the box). Children write M or D on whiteboards before anyone presses. Ask: “□ ÷ 4 = 7 has a divide sign. Why do we multiply to find the box?” |
| 30 min | Find the box Model □ ÷ 4 = 5 and 20 ÷ □ = 5 on one triangle, 20 at the top: in the first the box is the top, so 4 × 5 = 20; in the second it is a bottom corner, so 20 ÷ 5 = 4. Children do Parts A and B. Ask: “Where would the box sit on the triangle: at the top or at a bottom corner?” |
| 10 min | Quiz game Show “One product, a whole family”; the class calls each value before “Reveal next”. Then run the unit’s self-check quiz as a class game: pairs agree on an answer before it is pressed. Early finishers do Part C. |
Two half-sessions instead? Split after Part A; start Session B with the modelled pair again.
Watch for these ideas
- Dividing because the sentence shows ÷. In □ ÷ 4 = 7 the box is the whole, so multiply: 7 × 4.
- Working with the two numbers shown: □ ÷ 2 = 18, so 18 ÷ 2 = 9. Check: 9 ÷ 2 is not 18.
- Skipping the check. Put the answer in the box and read the sentence: is it true?
Answers
- Part A: whole, 4 × 8 = 32; factor, 77 ÷ 7 = 11; factor, 54 ÷ 6 = 9; whole, 9 × 12 = 108; factor, 24 ÷ 4 = 6; whole, 3 × 15 = 45.
- Part B: □ ÷ 6 = 8, 48 muffins; □ × 5 = 75, 15 rows; 96 ÷ □ = 12, 8 children (any true sentence with a box is right). Part C, for example: □ ÷ 7 = 8 (7 × 8 = 56) and 56 ÷ □ = 8 (56 ÷ 8 = 7).
Multiply or divide?
A. Is the box the whole or a factor? Then find it.
| Sentence | Whole or factor? | Multiply or divide? | □ = |
|---|---|---|---|
| □ ÷ 4 = 8 | |||
| 7 × □ = 77 | |||
| 54 ÷ □ = 6 | |||
| □ ÷ 9 = 12 | |||
| □ × 4 = 24 | |||
| □ ÷ 3 = 15 |
B. Write a sentence with a box, then solve it
Some muffins were shared equally onto 6 plates, with 8 muffins on each plate. How many muffins were there?
Sentence with a box: ____________________ Answer: ______________
75 chairs are set out in rows of 5. How many rows are there?
Sentence with a box: ____________________ Answer: ______________
Some children share 96 marbles equally, and each child gets 12. How many children are there?
Sentence with a box: ____________________ Answer: ______________
C. Your turn
Use 7, 8 and 56. Write a sentence with a box that you find by multiplying, and one that you find by dividing.
Find it by multiplying: ______________________________
Find it by dividing: ________________________________
Fact triangles
Cut out the twelve triangles (Days 2 and 3), one set per pair. The product sits at the top and its two factors below. Play cover a corner: cover one number with your thumb; your partner names it and the fact that finds it.
Teacher note: the 9, 9, 81 triangle makes a family of only two facts; it comes up at the end of Day 2.
Family mats and fact cards
Keep the four mats whole and cut out the eighteen fact cards (Day 4), one set per group. Sort each card onto the mat of its family. Some cards belong to no mat here.
Teacher note: the sort is on the Day 4 plan. Four cards belong to no mat: two use the wrong operation, and two are true facts of other families.
What we know: Inverse Operations and Fact Families
Work on your own. Show your thinking if you can.
- Write a multiplication and a division for this array.___ × ___ = ___ ___ ÷ ___ = ___
- Write the four facts in the family of 6, 11 and 66.
- 7 × □ = 84 □ = __________
- □ × 9 = 63 □ = __________
- □ ÷ 5 = 11 □ = __________
- 99 ÷ □ = 11 □ = __________
- Circle the two facts that do not belong to the family of 3, 12 and 36.12 × 3 = 3636 − 12 = 2436 ÷ 3 = 1236 ÷ 12 = 34 × 9 = 36
- Some stickers were shared equally among 3 friends, and each friend got 14. How many stickers were there? Write a sentence with a box, then solve it.
How did this week feel? Circle one: I can teach it · I am getting there · I need more help
Answers and marking guide
Answers
- 3 × 6 = 18; 18 ÷ 6 = 3 or 18 ÷ 3 = 6
- 6 × 11 = 66, 11 × 6 = 66, 66 ÷ 6 = 11, 66 ÷ 11 = 6
- 12 (84 ÷ 7 = 12)
- 7 (63 ÷ 9 = 7)
- 55 (the box is the whole: 5 × 11 = 55)
- 9 (99 ÷ 11 = 9)
- 36 − 12 = 24 (it subtracts) and 4 × 9 = 36 (true, but not made from 3, 12 and 36)
- □ ÷ 3 = 14, so 3 × 14 = 42: 42 stickers
A quick three-level guide
| Idea | Working towards | At standard | Beyond |
|---|---|---|---|
| Arrays and inverses (Q1) | writes only the multiplication | writes a multiplication and a division for the array | explains why the same dots show both facts |
| Fact families (Q2, Q7) | writes two facts, or keeps a true fact of another family | writes all four facts and finds both outsiders | explains why each outsider does not belong |
| Missing factors (Q3, Q4, Q6) | subtracts, or divides the factor by the whole | divides the whole by the known factor and checks | explains why dividing finds a missing factor |
| Missing wholes and stories (Q5, Q8) | divides when the box is the whole | multiplies to find a missing whole | writes the sentence for a story and explains the choice |
Eight questions, four ideas. A child at standard answers most questions and can say why, by naming the whole and the factors before choosing to multiply or divide.
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 | Reads an array both ways | Writes a family | Finds a missing factor | Sorts facts | Chooses the inverse |
|---|---|---|---|---|---|
The five columns are the five days: reading an array both ways, writing a family, finding a missing factor, sorting facts, and choosing the inverse.