Factors and Multiples: a week of ready-to-teach maths
Five days of lessons for Year 5 Number. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet, plus cut-out square tiles and number 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 two card sheets once; the tiles and cards 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. Each plan explains the idea in plain words, lists the misconceptions children bring, and gives the answers. The big idea of the week: one multiplication fact says three things. 4 × 6 = 24 means 4 and 6 are factors of 24, 24 is a multiple of 4 and of 6, and 24 is divisible by 4 and by 6, because each division leaves no remainder.
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.
Two words that are easy to swap
A factor is the number that divides; a multiple is the number it divides into. 4 is a factor of 24, and 24 is a multiple of 4: say both sentences aloud whenever the words come up. In this pack, factors and multiples are counting numbers, 1, 2, 3 and on.
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 | Rectangles hunt factors | Build every rectangle for 12, 18 and 16 with square tiles, and write each one as a factor pair | A factor pair is a rectangle |
| 2 | Factors come in pairs | List every factor of a number in pairs, from the outside in, and find the factor that pairs with itself | Factors come in pairs |
| 3 | Multiples run the other way | Count in steps to list multiples, mark them on a grid, and sort numbers into multiples of 3, of 4, of both and of neither | Multiples run the other way |
| 4 | Divisible means no remainder | Divide and read the remainder to decide divisibility, and use divisible, factor and multiple in one sentence | Divisible means no remainder |
| 5 | Write it as a product | Write numbers as products of two and three factors, find missing factors, and play the unit quiz | Writing a number as a product |
How the week builds
Day 1 finds factors as the sides of rectangles; Day 2 lists them in pairs so none is missed; Day 3 turns round to the multiples a number makes; Day 4 joins the two with divisibility and remainders; and Day 5 writes numbers as products. It builds on the multiplication and division facts of Years 3 and 4, and it is the ground for comparing fractions with related denominators later this year and for prime and composite numbers in Year 6.
Materials for the week (one trip)
- From the classroom: scissors, pencils, coloured pencils, envelopes for the cards, this pack printed.
- For Day 4: 30 counters or the square tiles, to share out (optional).
- Cut out once, use all week: the 36 square tiles and the 24 number cards in this pack. No maths equipment to buy.
Words for the word wall
factor, factor pair, multiple, product, divisible, remainder, rectangle, array, square number, skip count, common multiple
Dear families
This week in maths, Year 5 looks inside whole numbers. A factor divides a number exactly: 3 is a factor of 12, because 12 ÷ 3 = 4 with nothing left over. A multiple is what you reach by counting in steps: 24 is a multiple of 6. We write numbers as products, such as 36 = 4 × 9, and test whether one number is divisible by another.
Try this at home
- Arrange 12 pieces of pasta, or 12 coins, into every rectangle you can. Each rectangle gives a factor pair: 1 and 12, 2 and 6, 3 and 4.
- Count in sixes while you walk, all the way to 60.
- Share a packet of biscuits fairly. Is the number divisible by the number of people, or are some left over?
- Ask: is 4 a factor of 20? Is 20 a multiple of 4? Both answers should match.
My factor hunt this week
Fill one row a day. Choose a number between 10 and 60 that you see at home.
| Day | My number | Its factor pairs | One multiple of it |
|---|---|---|---|
| Monday | |||
| Tuesday | |||
| Wednesday | |||
| Thursday | |||
| Friday |
Printed from the free seegongsik Factors and Multiples teaching pack · seegongsik.com/au/y5/number/AC9M5N02/pack
Rectangles hunt factors
Children arrange square tiles into every rectangle they can for 12, 18 and 16. The two side lengths of each rectangle are a factor pair, so a hunt for rectangles is a hunt for factors.
We are learning to
- explain that a factor divides a number exactly, with nothing left over,
- find every rectangle for a number of tiles and write each one as a factor pair,
- notice numbers of tiles that make only one rectangle, a single row.
Success criteria
- I can build all the rectangles for 18 and write their factor pairs.
- I can tell whether a number is a factor of 18 by trying to build the rectangle.
You need
The square tiles (cut-out sheet 1), one set of 36 per pair. The worksheet, one per child.
Lesson flow (about 50 minutes)
| 10 min | Twelve in a row Lay 12 tiles in one long row, then ask for another rectangle. Show the board (see the box). Ask: “Twelve tiles in one row is a rectangle 1 by 12. Can you make a fatter one?” |
| 30 min | Every rectangle Pairs build every rectangle for 12, then for 18 and 16, and draw each one on the squared paper (Parts A and B). A rectangle turned on its side is the same rectangle. Ask: “You made 2 rows of 9. Is 9 rows of 2 a new rectangle, or the same one turned?” |
| 10 min | Only one rectangle Pairs try 7 tiles and 13 tiles, then do Parts C and D. Ask: “Why can 13 tiles make only one rectangle?” |
Two half-sessions instead? Split after the rectangles for 12; start Session B with 18 tiles.
Watch for these ideas
- Counting 3 by 4 and 4 by 3 as two different rectangles. A turned rectangle is the same factor pair.
- Leaving out the one-row rectangle. Every number has the factor pair 1 and itself.
- Thinking a factor can be bigger than the number. A factor fits into the number exactly, so it is never bigger.
Answers
- Part A: 1 × 12, 2 × 6, 3 × 4. Part B: 18: 1 × 18, 2 × 9, 3 × 6, so the factors are 1, 2, 3, 6, 9, 18; 16: 1 × 16, 2 × 8, 4 × 4, so the factors are 1, 2, 4, 8, 16 (16 makes a square, 4 by 4).
- Part C: 7: 1 × 7 only; 13: 1 × 13 only; each has just two factors. Part D: yes, 3 by 6; no; no; yes, 8 by 2.
Rectangles hunt factors
A. Draw every rectangle you can make with 12 tiles
The factor pairs of 12: __________ __________ __________
B. Every rectangle for 18 and for 16
| Tiles | Factor pairs (one for each rectangle) | All the factors, smallest first |
|---|---|---|
| 18 | ||
| 16 |
C. Try 7 tiles and 13 tiles
How many rectangles can you make with each? ______ What do you notice?
___________________________________________________________
D. Can it be done? Circle yes or no.
Can 18 tiles make a rectangle 3 tiles wide? yes / no If yes, how long is it? ______
Can 18 tiles make a rectangle 4 tiles wide? yes / no If yes, how long is it? ______
Can 16 tiles make a rectangle 5 tiles wide? yes / no If yes, how long is it? ______
Can 16 tiles make a rectangle 8 tiles wide? yes / no If yes, how long is it? ______
Factors come in pairs
Children list every factor of a number by working in pairs from the outside in: 1 with the number, 2 with its partner, and so on, stopping when the pairs meet. A square number has one factor that pairs with itself.
We are learning to
- list all the factors of a number in pairs, so none is missed,
- know when to stop: when the next factor to try would meet a partner already found,
- explain why a square number has an odd number of factors.
Success criteria
- I can list every factor of 36 in pairs.
- I can say whether a number is a factor of another, and give its partner.
You need
The square tiles from Day 1, for checking (optional). The worksheet, one per child.
Lesson flow (about 50 minutes)
| 10 min | Pair them up Recall the three rectangles for 12. Show the board (see the box) and reveal the partners of 24. Ask: “1 pairs with 24 and 2 pairs with 12. Is 5 a factor of 24? How can we tell?” |
| 30 min | Outside in Model 30 on the board: write 1 and 30 at the two ends, 2 and 15 inside them, then 3 and 10, then 5 and 6, joining each pair with an arc. Children do Parts A and B. Ask: “We have tried 1 to 6. Why is there no need to try 7, 8 or 9?” |
| 10 min | The odd one out Children count the factors of 36, 45 and 20, then do Parts C and D. Ask: “36 has an odd number of factors. Which factor has no partner of its own?” |
Two half-sessions instead? Split after Part A; start Session B by modelling 30 from the outside in.
Watch for these ideas
- Finding 1, 2, 3 and 4 and forgetting their partners. Every factor brings its partner with it.
- Testing every number up to 36. Once the pairs meet in the middle, every factor has been found.
- Writing 6 twice in the list for 36. A factor that pairs with itself is listed once.
Answers
- Part A: 1 × 40, 2 × 20, 4 × 10, 5 × 8. Part B: 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 (9 factors); 45: 1, 3, 5, 9, 15, 45 (6 factors); 20: 1, 2, 4, 5, 10, 20 (6 factors).
- Part C: 36, because 6 pairs with itself (6 × 6 = 36). Part D: no; yes, 7 × 6; yes, 9 × 7; no; yes, 4 × 13.
Factors come in pairs
A. Every factor pair of 40, from the outside in
| Factor | Its partner |
|---|---|
B. All the factors, smallest first
| Number | Its factors | How many? |
|---|---|---|
| 36 | ||
| 45 | ||
| 20 |
C. The odd one out
Which number in Part B has an odd number of factors? ______ Why?
___________________________________________________________
D. Is it a factor? Circle yes or no, and give the partner.
Is 6 a factor of 40? yes / no If yes: 6 × ______ = 40
Is 7 a factor of 42? yes / no If yes: 7 × ______ = 42
Is 9 a factor of 63? yes / no If yes: 9 × ______ = 63
Is 8 a factor of 60? yes / no If yes: 8 × ______ = 60
Is 4 a factor of 52? yes / no If yes: 4 × ______ = 52
Multiples run the other way
Children count in steps to list multiples, mark the multiples of 4 and 6 on a number grid, and sort number cards into multiples of 3, of 4, of both and of neither. Factors and multiples are two ends of one fact: 4 is a factor of 24, so 24 is a multiple of 4.
We are learning to
- list the multiples of a number by counting in steps of it,
- use factor and multiple the right way round,
- find numbers that are multiples of two numbers at once.
Success criteria
- I can write the first ten multiples of 6 and of 8.
- I can sort numbers into multiples of 3, multiples of 4, both and neither.
You need
The number cards (cut-out sheet 2), one set per pair. Two colours of pencil. The worksheet, one per child.
Lesson flow (about 50 minutes)
| 10 min | Count in steps Count in sixes to 60 as a class, clapping on each multiple. Show the board (see the box). Ask: “Will we ever run out of multiples of 6?” |
| 30 min | Shade and circle Children do Part A, then shade the multiples of 4 and circle the multiples of 6 on the grid in Part B. Ask: “12, 24, 36, 48 wear both marks. What do they have in common?” |
| 10 min | Sort the cards Pairs sort the 24 number cards into the four labelled piles, write each number in its part of the rings (Part D), then do Part C. Ask: “Where does 24 go? Where does 10 go, and why?” |
Two half-sessions instead? Split after Part B; start Session B by counting in fours to 48.
Watch for these ideas
- Swapping the words: saying 24 is a factor of 4. The factor divides; the multiple is the number you reach.
- Stopping at the tenth multiple, as a times table does. Multiples go on without end.
- Thinking a multiple of 4 must end in 4. 12, 20 and 36 are all multiples of 4.
Answers
- Part A: 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60; 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80. Part B: both marks on 12, 24, 36, 48. Part C: true, false, false, true, true, false.
- Part D: both 12, 24, 36; only multiples of 3: 3, 6, 9, 15, 18, 21, 27, 30, 33; only multiples of 4: 4, 8, 16, 20, 28, 32, 40; neither 10, 14, 22, 25, 35. The overlap holds multiples of 12.
Multiples run the other way
A. The first ten multiples
Multiples of 6: ____, ____, ____, ____, ____, ____, ____, ____, ____, ____
Multiples of 8: ____, ____, ____, ____, ____, ____, ____, ____, ____, ____
B. Shade the multiples of 4. Circle the multiples of 6.
C. True or false? Circle one.
24 is a multiple of 8.true / false
8 is a multiple of 24.true / false
36 is a multiple of 5.true / false
5 is a factor of 35.true / false
7 is a factor of 56.true / false
9 is a factor of 30.true / false
D. Sort the number cards, then write each number in its part of the rings
What do the numbers in the overlap have in common? ___________________________
Divisible means no remainder
Children decide whether one number is divisible by another by dividing and looking at the remainder: a remainder of zero means divisible. Then they join the three words of the week: if 54 is divisible by 6, then 6 is a factor of 54 and 54 is a multiple of 6.
We are learning to
- divide and use the remainder to decide whether a number is divisible by another,
- explain how divisible, factor and multiple describe the same fact,
- use the last digit to test for 2, 5 and 10.
Success criteria
- I can show a division and say whether it leaves a remainder.
- I can use divisible, factor and multiple in one true sentence.
You need
30 counters or the square tiles, for sharing (optional). The worksheet, one per child.
Lesson flow (about 50 minutes)
| 10 min | Share it out Share 30 counters into 4 groups: 7 each, and 2 left over. Show the board (see the box). Ask: “Two are left over. Is 30 divisible by 4?” |
| 30 min | Divide and decide Children do Parts A and B, writing the division and its remainder each time. Ask: “48 ÷ 8 leaves nothing over. What does that tell us about 8 and 48?” |
| 10 min | Last-digit tests On the Day 3 grid, multiples of 2 end in 0, 2, 4, 6 or 8, multiples of 5 in 0 or 5, and multiples of 10 in 0. Children do Parts C and D. Ask: “Which last digits do multiples of 5 have? Why can a multiple of 10 only end in 0?” |
Two half-sessions instead? Split after Part A; start Session B with the sharing stories.
Watch for these ideas
- Thinking a remainder means the division went wrong. The remainder is the evidence that the number is not divisible.
- Turning it round: 24 is divisible by 6, but 6 is not divisible by 24.
- Thinking every even number is divisible by 4. 30 is even, and 30 ÷ 4 leaves 2.
Answers
- Part A: 42 ÷ 7 = 6, yes; 38 ÷ 5 = 7 r 3, no; 45 ÷ 4 = 11 r 1, no; 54 ÷ 9 = 6, yes; 72 ÷ 6 = 12, yes; 60 ÷ 7 = 8 r 4, no. Part B: yes, 9 stickers each; no, 8 full rows and 2 over; no, 9 teams and 1 over; yes, 16 packs.
- Part C: by 2: 70, 92, 136, 240; by 5: 70, 85, 105, 240; by 10: 70, 240. Part D: multiple, divisible; factor, multiple.
Divisible means no remainder
A. Divide, then decide
| Division | Answer and remainder | Divisible? |
|---|---|---|
| 42 ÷ 7 | ______ r ______ | yes / no |
| 38 ÷ 5 | ______ r ______ | yes / no |
| 45 ÷ 4 | ______ r ______ | yes / no |
| 54 ÷ 9 | ______ r ______ | yes / no |
| 72 ÷ 6 | ______ r ______ | yes / no |
| 60 ÷ 7 | ______ r ______ | yes / no |
B. Sharing stories
72 stickers are shared among 8 friends. Does everyone get the same number, with none left over? yes / no Show the division: ______________
50 chairs are set out in rows of 6. Are all the rows full? yes / no Show the division: ______________
A class of 28 makes teams of 3. Can every child be in a team of 3? yes / no Show the division: ______________
64 pencils go into packs of 4. Do the pencils fill the packs exactly? yes / no Show the division: ______________
C. Is it divisible? Use the last digit, and tick each box that is true.
| Number | by 2 | by 5 | by 10 |
|---|---|---|---|
| 70 | |||
| 85 | |||
| 92 | |||
| 105 | |||
| 136 | |||
| 240 |
D. One fact, three words
6 is a factor of 54, so 54 is a __________ of 6, and 54 is __________ by 6.
72 ÷ 8 = 9 with no remainder, so 8 is a __________ of 72, and 72 is a __________ of 8.
Write it as a product
Children write a number as a product of its factors in different ways, find missing factors, and split a factor again to write a number as a product of three factors. The week closes with the unit quiz as a class game.
We are learning to
- write a number as a product of two factors in more than one way,
- find a missing factor in a multiplication fact,
- write a number as a product of three factors (24 = 2 × 3 × 4).
Success criteria
- I can write 36 as a product in two different ways.
- I can find the missing factor in 56 = ___ × 8.
You need
The worksheet, one per child. The interactive unit open for the quiz game.
Lesson flow (about 50 minutes)
| 10 min | Many products Show the board (see the box). Children suggest products that make 36 before anything is revealed. Ask: “36 = 4 × 9. Who has a different product that makes 36?” |
| 30 min | Products and missing factors Children do Parts A, B and C. For Part C, model 24 = 4 × 6 = 2 × 2 × 6. Ask: “24 = 4 × 6. If we split the 4 into 2 × 2, what three factors make 24?” |
| 10 min | Quiz game Run the unit’s self-check quiz as a class game: pairs hold up their answer before you reveal it. Fast finishers do Part D. Ask: “Which question tricked us most? What would you tell a friend about factors and multiples?” |
Two half-sessions instead? Split after Part B; start Session B by splitting 24 into three factors.
Watch for these ideas
- Thinking a number has only one product. 36 is 4 × 9, and also 6 × 6 and 3 × 12.
- Writing a sum instead of a product, such as 36 = 30 + 6.
- Splitting a factor by adding: 4 = 2 + 2 does not give a product. Split by multiplying, 4 = 2 × 2.
Answers
- Part A (any two): 36 = 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6; 24 = 1 × 24, 2 × 12, 3 × 8, 4 × 6; 45 = 1 × 45, 3 × 15, 5 × 9; 30 = 1 × 30, 2 × 15, 3 × 10, 5 × 6.
- Part B: 8; 7; 9; 25; 6. Part C (any one): 24 = 2 × 2 × 6 or 2 × 3 × 4; 30 = 2 × 3 × 5; 60 = 2 × 2 × 15 or 2 × 3 × 10 or 2 × 5 × 6 or 3 × 4 × 5. Part D: 1 × 48, 2 × 24, 3 × 16, 4 × 12, 6 × 8.
Write it as a product
A. Two different products for each number
36 = ______ × ______ 36 = ______ × ______
24 = ______ × ______ 24 = ______ × ______
45 = ______ × ______ 45 = ______ × ______
30 = ______ × ______ 30 = ______ × ______
B. Find the missing factor
48 = 6 × ______56 = ______ × 863 = 7 × ______100 = 4 × ______90 = ______ × 15
C. Three factors, each bigger than 1
Write the number as a product of two factors, then split one of them again by multiplying.
24 = ______ × ______ = ______ × ______ × ______
30 = ______ × ______ = ______ × ______ × ______
60 = ______ × ______ = ______ × ______ × ______
D. Every product for 48
Write 48 as a product of two factors in every way you can.
___________________________________________________________
Square tiles
Cut along the dashed lines to make 36 square tiles, one set per pair. Lay the tiles edge to edge to build rectangles: every rectangle you can make with a number of tiles gives one of its factor pairs.
Teacher note: keep each set in an envelope. 36 tiles build every number this week asks for: 12, 16, 18, 24, 30 and 36 itself, which makes a square, 6 by 6.
Number cards and pile labels
Cut out the four labels and the 24 number cards. Lay the labels in a row, then put each card on the pile where it belongs (Day 3).
Teacher note: the four piles and the numbers on each are in the answers on the Day 3 plan.
What we know: Factors and Multiples
Work on your own. Show your thinking if you can.
- Write every factor pair of 20. ______________________________
- Write all the factors of 18, smallest first. ______________________________
- Write the first five multiples of 7. ______________________________
- Circle the multiples of 6: 16, 18, 26, 30, 42, 45
- Is 52 divisible by 4? Show the division. ______________________________
- Is 45 divisible by 6? Show the division. ______________________________
- Write 40 as a product of two factors in two different ways. 40 = ______________ 40 = ______________
- Fill the gaps: 9 is a ______________ of 27, and 27 is a ______________ of 9.
How did this week feel? Circle one: I can teach it · I am getting there · I need more help
Answers and marking guide
Answers
- 1 × 20, 2 × 10, 4 × 5.
- 1, 2, 3, 6, 9, 18.
- 7, 14, 21, 28, 35.
- 18, 30, 42.
- 52 ÷ 4 = 13 with no remainder, so yes.
- 45 ÷ 6 = 7 r 3, so no.
- Any two of 1 × 40, 2 × 20, 4 × 10, 5 × 8.
- factor; multiple (27 = 9 × 3, so 27 is divisible by 9).
A quick three-level guide
| Idea | Working towards | At standard | Beyond |
|---|---|---|---|
| Factors and factor pairs (Q1, Q2) | finds some factors, often without their partners | lists every factor pair and every factor, in order | explains how they know no factor is missing |
| Multiples (Q3, Q4) | lists the first few multiples by counting on | lists multiples and picks them out of a set | finds numbers that are multiples of two numbers at once |
| Divisibility (Q5, Q6) | divides but is unsure what the remainder means | decides divisibility from the remainder and shows the division | uses last-digit tests and explains why they work |
| Products and the three words (Q7, Q8) | writes one product, or swaps factor and multiple | writes a number as a product in two ways and uses the words correctly | writes a number as a product of three factors |
Eight questions, four ideas. A child at standard answers most questions and can say why, using the words factor, multiple and divisible the right way round.
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 | Builds rectangles | Lists factor pairs | Lists multiples | Tests divisibility | Writes products |
|---|---|---|---|---|---|
The five columns are the five days: rectangles, factor pairs, multiples, divisibility and products.