Unknowns in Multiplication and Division: 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 twenty balance cards, four property mats with sixteen equation 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 balance cards and the equation cards once, and keep each set in an envelope.
- 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: an equation is a balance, and the unknown is the number that keeps it true. The inverse finds it (in 6 × □ = 42, divide 42 by 6), and the properties of multiplication often find it without any working at all (in 5 × 13 = 13 × □, the box is 5). The board writes the unknown as a question mark and the sheets use a box; both mean the same number.
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 equals sign balances | Sort balance cards into piles of equal value, judge equations with a calculation on each side, and find the number that balances | Finding an unknown in an equation |
| 2 | Undo it | Run number machines backwards: divide to undo multiplying, multiply to undo dividing | Multiplication and division undo each other |
| 3 | Where the unknown sits | Name the box as a factor, the divisor or the dividend, and choose the step that frees it | Where the unknown can sit |
| 4 | Use the properties | Sort equation cards by the property that finds the box: turn it around, group it, split it, or double and halve | Using fact families to find unknowns |
| 5 | Solve and check | Free the box, check by putting the number back, and write equations for stories; play the unit quiz | Solving with the inverse operation |
How the week builds
Day 1 makes the equals sign a balance; Days 2 and 3 undo multiplying and dividing wherever the unknown sits; Day 4 finds unknowns with the properties of multiplication; and Day 5 puts it together and checks every answer. It builds on Year 4’s unknowns in addition and subtraction equations (AC9M4A01) and this year’s inverse operations and fact families (AC9M5A01), and it leads to Year 6, where equations bring brackets and mixed operations (AC9M6A02).
Materials for the week (one trip)
- From the classroom: scissors, envelopes for the cards, mini whiteboards, this pack printed.
- Cut out once: the twenty balance cards (Day 1) and the sixteen equation cards (Day 4); keep the four property mats whole. No maths equipment to buy.
Words for the word wall
equation, equals, balance, unknown, factor, product, dividend, divisor, quotient, inverse, property
Dear families
This week in maths, Year 5 finds unknown numbers in equations such as 6 × □ = 42 and □ ÷ 5 = 6. Dividing undoes multiplying and multiplying undoes dividing, so 42 ÷ 6 = 7 fills the first box and 6 × 5 = 30 fills the second.
Try this at home
- Play a number machine: your child secretly multiplies a number by 4 and tells you the answer. You find the start, then swap.
- Ask: what times 8 makes 56? How can dividing help?
- Ask: is 4 × 6 the same as 8 × 3? Check both sides.
- Ask: if 7 × 8 = 56, what is 8 × 7? How do you know without working it out?
My equations this week
Fill one row a day. Write an equation with a box, then the number in the box.
| Day | My equation | The box is |
|---|---|---|
| Monday | ||
| Tuesday | ||
| Wednesday | ||
| Thursday | ||
| Friday |
Printed from the free seegongsik Unknowns in Multiplication and Division teaching pack · seegongsik.com/au/y5/algebra/AC9M5A02/pack
The equals sign balances
Children meet the equals sign as a balance: both sides of an equation have the same value. 3 × 5 = 30 ÷ 2 is true because both sides are 15, even though neither side is a single number. A box in an equation stands for the number that keeps the balance true.
We are learning to
- read the equals sign as ‘has the same value as’,
- decide whether an equation with a calculation on each side is true,
- find the number that balances an equation.
Success criteria
- I can say whether 2 × 9 = 3 × 6 is true, and show the value of each side.
- I can find the number in the box that keeps an equation balanced.
You need
The balance cards (cut-out sheet 1), one set per group. Mini whiteboards. The worksheet, one per child.
Lesson flow (about 50 minutes)
| 10 min | Find the missing number Show the board (see the box). Children write their choice on whiteboards before anyone presses. Ask: “How could you find the missing number without trying every number?” |
| 30 min | Piles of equal value Model 3 × 5 = 30 ÷ 2: each side is 15, so it is true. Groups sort the balance cards into piles, then do Parts A and B. Ask: “Two cards sit in the same pile. Which equation do they make?” |
| 10 min | Balance it Model 4 × 4 = 2 × □: the left side is 16, so 2 × □ must be 16 too, and □ = 8. Children do Part C. Ask: “Which side do you work out first, and why that one?” |
Two half-sessions instead? Split after the sort; start Session B with Part B.
Watch for these ideas
- Reading = as ‘the answer is’, so an equation with a calculation on the right looks wrong. Ask what each side is worth.
- Writing the value of one side in the box: 4 × 4 = 2 × □, so 16. The box is one factor of that value.
- Judging by eye, such as calling 6 × 8 = 7 × 7 true because 48 and 49 are close. Work out both sides.
Answers
- Part A: 24: 4 × 6, 8 × 3, 72 ÷ 3, 2 × 12; 36: 2 × 18, 6 × 6, 4 × 9, 12 × 3; 30: 90 ÷ 3, 10 × 3, 15 × 2, 5 × 6; 48: 12 × 4, 3 × 16, 6 × 8, 96 ÷ 2; 20: 4 × 5, 80 ÷ 4, 40 ÷ 2, 2 × 10. Any two cards in a pile make a true equation.
- Part B: true (18 and 18); false (35 and 36); true (8 and 8); true (40 and 40); false (28 and 42); false (56 and 54); true (8 and 8); true (5 and 5). Part C: 7 (both sides 42); 10 (both sides 40); 3 (both sides 9); 2 (both sides 33).
The equals sign balances
A. Sort the balance cards into piles of equal value
Write the value of each pile and two true equations made from its cards.
| Value | A true equation | Another true equation |
|---|---|---|
B. True or false? Work out both sides.
C. Find the box that balances
Undo it
Children run number machines forwards and backwards. A machine that multiplies by 3 turns 5 into 15; when 36 comes out, run it backwards and divide by 3: 12 went in. Dividing undoes multiplying and multiplying undoes dividing, so an equation with one operation is undone in one step.
We are learning to
- run a number machine forwards and backwards,
- undo multiplying by dividing, and dividing by multiplying,
- free the box in an equation with the inverse.
Success criteria
- I can find what went into a machine from what came out.
- I can write □ = … to free the box, and find it.
You need
Mini whiteboards. The worksheet, one per child.
Lesson flow (about 50 minutes)
| 10 min | Show the inverse Show the board (see the box). Children say the division before anyone presses. Ask: “8 × 5 = 40. Which two divisions undo it?” |
| 30 min | Machines backwards Model the × 3 machine on the board: 5 in, then 36 out. Pairs play guess my start for five minutes: one says a rule and an output, the other finds the start. Children do Parts A and B. Ask: “The machine divides by 4. What do you do to run it backwards?” |
| 10 min | Two steps Children do Part C. Ask: “The machine multiplied, then divided. Which step do you undo first?” |
Two half-sessions instead? Split after the game; start Session B with Part A.
Watch for these ideas
- Running the machine forwards when the output is given: × 3 with 36 out, so 108. Ask what went in.
- Undoing with the same operation: a ÷ 5 machine with 6 out, so 6 ÷ 5. Dividing is undone by multiplying.
- Undoing a two-step machine in the machine’s order. Undo the last step first.
Answers
- Part A: out 42; in 9; in 7; out 8; in 36; in 56.
- Part B: □ = 64 ÷ 8 = 8; □ = 11 × 3 = 33; □ = 84 ÷ 12 = 7; □ = 5 × 9 = 45. Part C: 18 × 2 = 36, 36 ÷ 4 = 9, so 9 went in; 35 ÷ 5 = 7, 7 × 3 = 21, so 21 went in.
Undo it
A. Fill the empty box on each machine
B. Free the box, then find it
| Equation | Free the box | □ = |
|---|---|---|
| □ × 8 = 64 | □ = _____ ___ _____ | |
| □ ÷ 3 = 11 | □ = _____ ___ _____ | |
| 12 × □ = 84 | □ = _____ ___ _____ | |
| □ ÷ 9 = 5 | □ = _____ ___ _____ |
C. Two-step machines: what went in?
Where the unknown sits
The box can be a factor, the divisor, or the dividend (the number being divided). Children name its job before choosing a step: a factor is the product divided by the other factor; the divisor is the dividend divided by the quotient; the dividend is the quotient times the divisor. One family, such as 8, 3 and 24, can hide its unknown in each place.
We are learning to
- name the box as a factor, the divisor or the dividend,
- choose the step that frees the box for each job,
- write equations with the box in each place.
Success criteria
- I can say what the box stands for in an equation, and find it.
- I can write three equations from one family, with the box in a different place each time.
You need
Mini whiteboards. The worksheet, one per child.
Lesson flow (about 50 minutes)
| 10 min | Where is it? Show the board (see the box). Children write the number before anyone presses. Ask: “In 36 ÷ ? = 9, is the unknown the number shared or the number of shares?” |
| 30 min | Name its job Model □ × 3 = 24 (a factor: □ = 24 ÷ 3), 24 ÷ □ = 3 (the divisor: □ = 24 ÷ 3), □ ÷ 3 = 8 (the dividend: □ = 8 × 3). Children do Parts A and B. Ask: “The box is the divisor. Do you divide or multiply to find it? Why?” |
| 10 min | Stories Children do Part C. Ask: “Is the unknown how many altogether, how many groups, or how many in each group?” |
Two half-sessions instead? Split after Part A; start Session B with the modelled family again.
Watch for these ideas
- Multiplying whenever there is a ÷ sign: 24 ÷ □ = 3, so 72. The divisor is found by dividing.
- Mixing up the dividend and the divisor. In □ ÷ 3 = 8 the box is the number being divided, so it is the biggest.
- Thinking the box can only sit on the left of the equals sign.
Answers
- Part A: a factor, □ = 48 ÷ 6 = 8; the divisor, □ = 63 ÷ 7 = 9; the dividend, □ = 11 × 4 = 44; the dividend, □ = 6 × 8 = 48; a factor, □ = 60 ÷ 5 = 12; the divisor, □ = 84 ÷ 12 = 7.
- Part B, for example: □ × 12 = 48, 48 ÷ □ = 12, □ ÷ 12 = 4; □ × 11 = 66, 66 ÷ □ = 11, □ ÷ 11 = 6. Part C: 72 ÷ □ = 12, 6 cartons; □ ÷ 6 = 9, 54 cm; 8 × □ = 96, 12 cards (any true equation is right).
Where the unknown sits
A. Is the box a factor, the divisor or the dividend? Free it and find it.
| Equation | The box is the | Free the box | □ = |
|---|---|---|---|
| □ × 6 = 48 | □ = _____ ___ _____ | ||
| 63 ÷ □ = 7 | □ = _____ ___ _____ | ||
| □ ÷ 4 = 11 | □ = _____ ___ _____ | ||
| □ ÷ 8 = 6 | □ = _____ ___ _____ | ||
| 5 × □ = 60 | □ = _____ ___ _____ | ||
| 84 ÷ □ = 12 | □ = _____ ___ _____ |
B. One family, the box in three places
The family 4, 12 and 48:
The family 6, 11 and 66:
C. What is the unknown in each story?
72 eggs are packed in cartons of 12. How many cartons are there?
Equation: ____________________ Answer: ______________
A ribbon was cut into 6 equal pieces, each 9 cm long. How long was the ribbon?
Equation: ____________________ Answer: ______________
8 friends share 96 cards equally. How many cards does each friend get?
Equation: ____________________ Answer: ______________
Use the properties
Some boxes can be found without working out either side, because of how multiplication behaves. Turn it around: 5 × 13 = 13 × □, so □ = 5. Group it: 3 × 2 × 5 = 3 × □, so □ = 10. Split it: 5 × 17 = 5 × 10 + 5 × □, so □ = 7. Double and halve: 18 × 5 = 9 × □, so □ = 10.
We are learning to
- find an unknown with a property of multiplication instead of the long way,
- name the property we used,
- explain with an array why a property keeps an equation balanced.
Success criteria
- I can sort an equation card onto the property that finds its box.
- I can show with an array why splitting a factor keeps the balance.
You need
The property mats and equation cards (cut-out sheet 2), one set per group. The worksheet, one per child.
Lesson flow (about 50 minutes)
| 10 min | Show the family Show the board (see the box). Children say the four facts before anyone presses. Ask: “Which two facts in the family are the same factors turned around?” |
| 30 min | Sort by property Model the four examples above on the board. Groups sort the sixteen cards onto the mats and find each box, then do Parts A and B. Ask: “Could you find this box without working out the left side? Which property lets you?” |
| 10 min | Why splitting works Children do Part C. Ask: “Where are the 4 × 10 and the 4 × 3 in your array?” |
Two half-sessions instead? Split after the sort; start Session B with Part B.
Watch for these ideas
- Working out both sides the long way when a property would do. Ask what is the same on both sides.
- Halving both factors. When one factor is halved, the other must double to keep the balance.
- Adding the ones without multiplying them: 5 × 17 = 5 × 10 + 7. Both parts are multiplied by 5.
Answers
- Part A: Turn it around: B (9), E (23), J (6), P (12); Group it: D (8), G (20), L (10), M (12); Split it: A (3), F (4), K (8), O (2); Double and halve: C (30), H (50), I (14), N (10).
- Part B: split it, 6; double and halve, 10; group it, 20; turn it around, 11. Part C: 4 × 13 = 4 × 10 + 4 × 3 = 40 + 12 = 52.
Use the properties
A. Sort the equation cards. Write the card letters and their boxes.
| Property | Card letters | The boxes |
|---|---|---|
| Turn it around | ||
| Group it | ||
| Split it | ||
| Double and halve |
B. Name the property, then find the box
C. Why splitting works
Draw an array of 4 rows of 13. Split it into 4 × 10 and 4 × 3, then write the equation it shows.
Equation: ______________________________________________
Solve and check
Children put the week together as a routine: see where the box sits, free it with the inverse, and check by putting the number back. The check is the habit that matters: □ ÷ 7 = 4 gives □ = 4 × 7 = 28, and 28 ÷ 7 = 4 shows the balance holds.
We are learning to
- free the box with the inverse operation, wherever it sits,
- check an answer by putting it back into the equation,
- write and solve an equation for a story.
Success criteria
- I can solve an equation and show my check.
- I can find and fix a wrong answer by checking it.
You need
Mini whiteboards. The worksheet, one per child.
Lesson flow (about 50 minutes)
| 10 min | Solve it Show the board (see the box). Children write the step that frees the unknown before anyone presses. Ask: “? ÷ 3 = 8. Why does multiplying free the unknown?” |
| 30 min | Free it, check it Model □ ÷ 7 = 4: □ = 4 × 7 = 28; check 28 ÷ 7 = 4. Children do Parts A and B. Ask: “Read your check aloud. Is the equation true?” |
| 10 min | Quiz game 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 equation again.
Watch for these ideas
- Dividing the two numbers shown when the box is the dividend: □ ÷ 7 = 4, so 4 ÷ 7. The box is the number being divided.
- Checking with the freeing step instead of the equation. Put the number back into the equation itself.
- Writing the step and stopping. Work it out, then check.
Answers
- Part A: □ = 72 ÷ 9 = 8 (8 × 9 = 72); □ = 12 × 6 = 72 (72 ÷ 6 = 12); □ = 56 ÷ 8 = 7 (56 ÷ 7 = 8); □ = 99 ÷ 11 = 9 (11 × 9 = 99); □ = 15 × 5 = 75 (75 ÷ 5 = 15); □ = 108 ÷ 9 = 12 (108 ÷ 12 = 9).
- Part B: □ ÷ 4 = 13, 52 oranges; 8 × □ = 88, 11 chairs. Part C: 3 ÷ 4 is not 12, so the answer is wrong; □ = 12 × 4 = 48, and 48 ÷ 4 = 12.
Solve and check
A. Free the box, find it, then check
| Equation | Free the box | □ = | Check |
|---|---|---|---|
| □ × 9 = 72 | □ = _____ ___ _____ | ||
| □ ÷ 6 = 12 | □ = _____ ___ _____ | ||
| 56 ÷ □ = 8 | □ = _____ ___ _____ | ||
| 11 × □ = 99 | □ = _____ ___ _____ | ||
| □ ÷ 5 = 15 | □ = _____ ___ _____ | ||
| 108 ÷ □ = 9 | □ = _____ ___ _____ |
B. Write an equation with a box, solve it and check it
Some oranges are packed equally into 4 bags, with 13 oranges in each bag. How many oranges are there?
Equation: __________________ Answer: ____________ Check: __________________
88 chairs are set out in 8 equal rows. How many chairs are in each row?
Equation: __________________ Answer: ____________ Check: __________________
C. Check Sam’s answer
Sam solved □ ÷ 4 = 12 by writing □ = 12 ÷ 4 = 3. Check the answer. If it is wrong, fix it and check again.
Balance cards
Cut out the twenty cards (Day 1), one set per group. Work out each card and sort the cards into piles of equal value. Any two cards in a pile make a true equation: one card, an equals sign, then the other.
Teacher note: there are five piles of four. The piles are on the Day 1 plan.
Property mats and equation cards
Keep the four mats whole and cut out the sixteen equation cards (Day 4), one set per group. Put each card on the mat of the property that finds the box, then find the box.
Teacher note: the sort is on the Day 4 plan. A card that two properties both explain is fine on either mat.
What we know: Unknowns in Multiplication and Division
Work on your own. Show your thinking if you can.
- Is 6 × 8 = 4 × 12 true? Show how you know. __________
- 3 × 8 = 6 × □ □ = __________
- □ × 7 = 63 □ = __________
- 48 ÷ □ = 6 □ = __________
- □ ÷ 9 = 4 □ = __________
- What went into this machine?
- Find the box without working out the left side. Name the property you used.8 × 15 = 4 × □□ = ______ Property: ______________________
- A baker puts 72 rolls into bags of 6. How many bags are there? Write an equation 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
- True: both sides are 48
- 4 (both sides 24)
- 9 (□ = 63 ÷ 7)
- 8 (the box is the divisor: □ = 48 ÷ 6)
- 36 (the box is the dividend: □ = 4 × 9)
- 11 (66 ÷ 6 = 11)
- 30, double and halve
- □ × 6 = 72 (or 72 ÷ 6 = □), 12 bags
A quick three-level guide
| Idea | Working towards | At standard | Beyond |
|---|---|---|---|
| Balance (Q1, Q2) | works out one side only | works out both sides and finds the number that balances | balances without working out both sides |
| Undoing (Q3, Q6) | uses the same operation again | undoes with the inverse and checks | explains why the inverse undoes the step |
| Where the unknown sits (Q4, Q5) | divides wherever the box is | names the box and chooses the step that frees it | explains why the divisor is found by dividing |
| Properties and stories (Q7, Q8) | works everything out the long way | uses a property, or an equation for the story | names the property and explains why it keeps the balance |
Eight questions, four ideas. A child at standard answers most questions and can say why, by naming where the box sits and the step or property that frees it.
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 | Balances equations | Undoes a machine | Names the unknown | Uses a property | Solves and checks |
|---|---|---|---|---|---|
The five columns are the five days: balancing equations, undoing a machine, naming the unknown, using a property, and solving and checking.