Prime factorisation with exponents: a week of ready-to-teach maths
Five lessons for Year 7 Number. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet with practice and a stretch section, plus prime and composite sorting cards, prime tiles, factorisation match cards, a mini-check and every answer with its working.
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.
- Print the two cut-out sheets, one copy per pair, and cut them out before Day 1: the sorting cards are for Day 1, the prime tiles for Day 3 and the match cards for Day 4.
- Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.
- Teach straight from the plan. Timings, talk prompts, misconceptions and worked answers are all on the one page.
The idea in plain words
A prime number has exactly two factors, 1 and itself: 2, 3, 5, 7, 11 and so on. Every other whole number above 1 is composite and can be split into smaller factors, and if you keep splitting, you always end at primes. Whichever split you start with, you finish with the same primes: 60 is always 2, 2, 3 and 5. Repeated primes are written once with an exponent, so 60 = 2² × 3 × 5. This prime factorisation works like a fingerprint: no other number has it, and it tells you at a glance what divides the number.
One day, one lesson
The five lessons fill a week of maths, one lesson of about 60 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point for two shorter sessions, and every worksheet ends with a stretch section for students who finish early.
How we write a prime factorisation
Write the primes smallest first, each one once, with its exponent: 360 = 2³ × 3² × 5. Read 2³ as “two cubed” and 2⁴ as “two to the power of four”. An exponent of 1 is not written. The number 1 is neither prime nor composite: it has only one factor. No calculators this week; the arithmetic is the practice.
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 | Students learn and do | On screen |
|---|---|---|---|
| 1 | Primes and composites | List factor pairs, sort numbers into prime and composite, sieve the primes to 50, and say why 1 is neither | Prime or composite |
| 2 | Factor trees | Split a number into factors again and again until every branch ends in a prime, and see two different trees reach the same primes | A factor tree finds the primes; The first split does not matter |
| 3 | Repeats become powers | Gather repeated primes with exponents, write the primes smallest first, and expand a power back out | Exponents gather repeats |
| 4 | The division ladder | Divide by the smallest prime again and again, check by multiplying back, and find the mistakes in four claimed factorisations | Rebuilding from the powers |
| 5 | The fingerprint at work | Use a prime factorisation to spot squares, find common factors and count factors, and solve problems with it | Self-check quiz as a class game |
How the week builds
Day 1 sorts the building blocks from the numbers built from them; Day 2 takes numbers apart with factor trees; Day 3 records the primes compactly with exponents; Day 4 adds a second method and the habit of checking by multiplying back; and Day 5 uses the factorisation to answer questions about the number. It builds on Year 6 work with prime, composite and square numbers and on last week’s squares and square roots, and it prepares the common factors and multiples of the fraction work later in Year 7 and the exponent laws of Year 8.
Materials for the week (one trip)
- From the classroom: pencils, coloured pencils, scissors, small envelopes or paper clips for the cards, this pack printed.
- Cut-out sheet 1, one per pair: 24 number cards to sort (Day 1) and 20 prime tiles (Day 3).
- Cut-out sheet 2, one per pair: 24 factorisation match cards (Day 4).
- No calculators and no other equipment to buy.
Words for the word wall
factor, factor pair, multiple, prime, composite, prime factor, factor tree, branch, leaf, division ladder, prime factorisation, index, exponent, power, base, squared, cubed, index form, expanded form, highest common factor
Dear families
This week in maths, Year 7 is breaking numbers down into prime numbers, the numbers with exactly two factors such as 2, 3, 5 and 7. We draw factor trees, write the result with exponents (60 = 2² × 3 × 5), and use it to answer questions about the number.
Try this at home
- Pick a house number, a year or a price and ask: is it prime? (Try dividing by 2, 3, 5 and 7.)
- Ask your child to draw a factor tree for a number of their choice, then start again with a different first split. Do the branches end in the same primes?
- Ask: what number is 2³ × 5? (40.) Why is it not 2 × 3 × 5?
- Ask: why is 1 not a prime number?
Our prime finds this week
Fill one row a day with a number you met, its prime factorisation, and a way to check it.
| Day | The number | Its prime factorisation | Checked by |
|---|---|---|---|
| Monday | |||
| Tuesday | |||
| Wednesday | |||
| Thursday | |||
| Friday |
Printed from the free seegongsik Prime factorisation with exponents teaching pack · seegongsik.com/au/y7/number/AC9M7N02/pack
Primes and composites
Students list factor pairs to decide whether a number is prime or composite, sieve the primes to 50, and settle the special cases: 1 is neither, and 2 is the only even prime.
We are learning to
- list every factor pair of a number,
- decide from the factor pairs whether a number is prime or composite,
- explain why 1 is neither prime nor composite.
Success criteria
- I can show that 69 is composite with a factor pair.
- I can find every prime up to 50 with a sieve and say why 1 is not one of them.
You need
The 24 number cards from cut-out sheet 1 (one set per pair), coloured pencils, the worksheet. No calculators.
Lesson flow (about 60 minutes)
| 10 min | Factor pairs decide Show the board (see the box). Then list the factor pairs of 15 together: 1 × 15 and 3 × 5, so 15 is composite. Ask: “7 has one factor pair and 12 has three. What does the number of pairs tell you?” |
| 15 min | Card sort Pairs sort the cards into prime, composite and neither. For every composite they say a factor pair that is not 1 × the number. Ask: “Which card nearly went in the wrong pile? What gave it away?” |
| 35 min | Pairs and the sieve Students complete Parts A to C; early finishers do Part D. Share the sieve: why does crossing out the multiples of 2, 3, 5 and 7 catch every composite up to 50? |
Two shorter sessions instead? Split after the card sort; start the second with Part A.
Watch for these ideas
- Thinking every odd number is prime. 9, 15, 21 and 57 are odd and composite.
- Calling 1 prime. A prime has exactly two factors, and 1 has only one.
- Not knowing when to stop looking for pairs. Once the smaller number of a pair passes the larger, the pairs repeat, so every pair has been found.
Answers
- Part A: 18: 1 × 18, 2 × 9, 3 × 6, composite; 23: 1 × 23, prime; 36: 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6, composite; 29: 1 × 29, prime; 45: 1 × 45, 3 × 15, 5 × 9, composite; 51: 1 × 51, 3 × 17, composite.
- Part B: the 15 primes up to 50 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.
- Part C: true (every other even number has 2 as a factor); false (9 = 3 × 3 is odd and composite); false (1 has only one factor); true (5 is a factor of it).
- Part D: 97 is prime: none of 2, 3, 5, 7 divides it, and a factor pair of 97 would need a number below 10, as 10 × 10 = 100. 91 = 7 × 13, so it is composite.
Primes and composites
A. Every factor pair
| Number | All its factor pairs | Prime or composite? |
|---|---|---|
| 18 | ||
| 23 | ||
| 36 | ||
| 29 | ||
| 45 | ||
| 51 |
B. Sieve the primes to 50
Cross out 1. Circle 2 and cross out every other multiple of 2. Do the same for 3, 5 and 7. Circle every number left.
C. True or false? Give a reason.
2 is the only even prime number. ________ because ____________________________
Every odd number is prime. ________ because ____________________________
1 is a prime number. ________ because ____________________________
Every number above 5 that ends in 5 is composite. ________ because ____________________________
D. Stretch
To test whether 97 is prime, which primes must you try dividing by, and why can you stop there? ________________________
Is 91 prime? Show how you know. ______________________________
Factor trees
A factor tree takes a composite number apart: split it into two factors, split again, and stop when every branch ends in a prime. Students fill in trees, grow their own, and find that two different trees for 72 end at the same primes.
We are learning to
- split a composite number again and again until every branch ends in a prime,
- check a finished tree by multiplying the leaves,
- explain why two different first splits end at the same primes.
Success criteria
- I can draw a finished factor tree for 44 and check it by multiplying the leaves.
- I can say how I know a tree is finished.
You need
The worksheet, coloured pencils (one colour to ring each prime leaf). No calculators.
Lesson flow (about 60 minutes)
| 10 min | Split until prime Show the first picture on the board (see the box). Then grow a tree for 40 together: 4 × 10, then 2 × 2 and 2 × 5. Multiply the leaves to check: 2 × 2 × 2 × 5 = 40. Ask: “How do you know the tree is finished?” |
| 15 min | Two trees, one answer Show the second picture. Partners do Part C, each starting 72 with a different split. Ask: “Your trees look different. Compare the leaves. What do you notice?” |
| 35 min | Fill and grow Students complete Parts A and B, ringing each prime leaf; early finishers do Part D. |
Two shorter sessions instead? Split after Part C; start the second with Part A.
Watch for these ideas
- Stopping at a composite leaf such as 4 or 9. Check every leaf: can it still be split?
- Splitting off 1, as in 7 = 1 × 7. That split never ends; a prime leaf stays as it is.
- Losing a leaf when writing the answer. Multiply the leaves back to check.
Answers
- Part A: 84 = 4 × 21, 4 = 2 × 2, 21 = 3 × 7 (leaves 2, 2, 3, 7); 90 = 9 × 10, 9 = 3 × 3, 10 = 2 × 5 (leaves 3, 3, 2, 5); 168 = 8 × 21, 8 = 2 × 4, 4 = 2 × 2, 21 = 3 × 7 (leaves 2, 2, 2, 3, 7).
- Part B: 48: 2, 2, 2, 2, 3; 100: 2, 2, 5, 5; 126: 2, 3, 3, 7; 75: 3, 5, 5. Trees vary; the leaves do not.
- Part C: both trees end at 2, 2, 2, 3, 3, for example 8 = 2 × 4, 4 = 2 × 2, 9 = 3 × 3, and 6 = 2 × 3, 12 = 2 × 6, 6 = 2 × 3.
- Part D: 1000 has 6 prime leaves (2, 2, 2, 5, 5, 5). A tree with one split belongs to a product of two primes, such as 15 = 3 × 5 or 49 = 7 × 7.
Factor trees
A. Fill in the empty circles
Ring every prime leaf. Then multiply the leaves to check.
B. Grow a tree
C. One number, two trees
What do you notice about the two sets of leaves? ________________________________________
D. Stretch
A tree for 1000 starts 10 × 100. How many prime leaves will it have? ______ Find a number whose tree has only one split. What must be true about it? ________________________
Repeats become powers
A factor tree often ends with the same prime more than once. Students gather each repeat into a power, write the primes smallest first, and run the other way: from index form back to the number.
We are learning to
- write repeated primes as powers, such as 2 × 2 × 2 = 2³,
- write a prime factorisation in index form, primes smallest first,
- expand a number in index form and find its value.
Success criteria
- I can write 2 × 3 × 3 × 3 × 7 as 2 × 3³ × 7.
- I can say why 2³ is 8 and not 6.
You need
The prime tiles from cut-out sheet 1 (one set per pair), the worksheet, scrap paper for trees. No calculators.
Lesson flow (about 60 minutes)
| 10 min | Count the repeats Show the board (see the box). Then write 2 × 3 × 3 × 3 × 7 and gather it together: 2 × 3³ × 7 = 378. Ask: “2³ looks like a 2 and a 3. What does it mean, and what is it worth?” |
| 15 min | Build with tiles Pairs build 98 and 150 from prime tiles and write each in index form, then lay out tiles for a partner to name. Ask: “Your partner laid out 3, 3, 5 and 5. How do you write it, and what number is it?” |
| 35 min | Index form both ways Students complete Parts A to C, using a factor tree on scrap paper for Part C; early finishers do Part D. |
Two shorter sessions instead? Split after the tiles; start the second with Part A.
Watch for these ideas
- Reading 2³ as 2 × 3. The exponent counts the twos: 2³ = 2 × 2 × 2 = 8.
- Gathering by adding: 3 + 3 + 3 + 3 is 12, but 3 × 3 × 3 × 3 = 3⁴ = 81.
- Leaving a prime twice, as in 2² × 3 × 2. Gather it: 2³ × 3.
Answers
- Part A: 2³ × 5 = 40; 3⁴ = 81; 2² × 3² × 7 = 252; 2 × 5³ = 250.
- Part B: 2⁴ = 2 × 2 × 2 × 2 = 16; 3³ = 3 × 3 × 3 = 27; 2 × 5² = 2 × 5 × 5 = 50; 2³ × 7 = 2 × 2 × 2 × 7 = 56; 2² × 3 × 11 = 2 × 2 × 3 × 11 = 132.
- Part C: 80 = 2⁴ × 5; 54 = 2 × 3³; 196 = 2² × 7²; 360 = 2³ × 3² × 5.
- Part D: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024; 1024 = 2¹⁰ and 2000 = 2⁴ × 5³; 2⁵ = 32 is bigger than 5² = 25.
Repeats become powers
A. Gather the repeats
Write each product in index form, primes smallest first.
2 × 2 × 2 × 5 = ______________________
3 × 3 × 3 × 3 = ______________________
2 × 3 × 2 × 7 × 3 = ______________________
5 × 5 × 2 × 5 = ______________________
B. Spread it out
| Index form | Written out | Value |
|---|---|---|
| 2⁴ | ||
| 3³ | ||
| 2 × 5² | ||
| 2³ × 7 | ||
| 2² × 3 × 11 |
C. From the number
Grow a factor tree on scrap paper, then write the number in index form.
80 = ______________________
54 = ______________________
196 = ______________________
360 = ______________________
D. Stretch: powers of 2
Write 1024 and 2000 in index form. ______________________ Which is bigger, 2⁵ or 5²? ________
The division ladder
A second way to factorise: divide by the smallest prime that goes in, again and again, until 1 is left; the primes down the side are the factorisation. Students check every answer by multiplying back and find what is wrong with claimed factorisations.
We are learning to
- factorise a number with a division ladder,
- check a prime factorisation by multiplying it back,
- spot a claimed factorisation that is wrong and say why.
Success criteria
- I can factorise 280 with a ladder and write it in index form.
- I can say what is wrong with 60 = 2 × 30, even though it multiplies back.
You need
The 24 factorisation match cards from cut-out sheet 2 (one set per pair), the worksheet. No calculators.
Lesson flow (about 60 minutes)
| 10 min | Down the ladder Model 280: 2 leaves 140, 2 leaves 70, 2 leaves 35, 5 leaves 7, 7 leaves 1. So 280 = 2³ × 5 × 7. Ask: “Why start with the smallest prime? What do you do when 2 no longer divides?” |
| 15 min | Rebuild and match Show the board (see the box). Pairs match the 12 numbers to the 12 factorisations, multiplying back to check each match. Ask: “Four cards start with 2³. How did you tell them apart?” |
| 35 min | Ladders and errors Students complete Parts A to C; early finishers do Part D. Share Part C: which claims multiply back and are still wrong? |
Two shorter sessions instead? Split after the match; start the second with Part A.
Watch for these ideas
- Dividing by a composite number such as 4 or 6. Every number down the side must be prime.
- Stopping before 1. The ladder ends when 1 is left at the bottom.
- Trusting a claim because it multiplies back: 2 × 3 × 15 = 90, but 15 is not prime.
Answers
- Part A: 180: divide by 2, 2, 3, 3, 5, so 180 = 2² × 3² × 5; 252: divide by 2, 2, 3, 3, 7, so 252 = 2² × 3² × 7; 330: divide by 2, 3, 5, 11, so 330 = 2 × 3 × 5 × 11; 441: divide by 3, 3, 7, 7, so 441 = 3² × 7².
- Part B: 2² × 3 × 5² = 2 × 2 × 3 × 5 × 5 = 300; 2⁵ × 3 = 2 × 2 × 2 × 2 × 2 × 3 = 96; 3² × 5 × 7 = 3 × 3 × 5 × 7 = 315; 2 × 3³ × 5 = 2 × 3 × 3 × 3 × 5 = 270.
- Part C: 84 right. 72 wrong, as a prime appears twice, so it is not gathered (2³ × 3²). 90 wrong, as 15 is not prime (2 × 3² × 5). 150 right. 64 wrong, as it makes 32, not 64 (2⁶).
- Part D: 504 = 2³ × 3² × 7. 12 = 2² × 3: yes, those primes are all there; 18 = 2 × 3²: yes, those primes are all there; 25 = 5²: no, 2³ × 3² × 7 has not got 5²; 14 = 2 × 7: yes, those primes are all there; 16 = 2⁴: no, 2³ × 3² × 7 has not got 2⁴.
The division ladder
A. Climb down the ladder
Divide by the smallest prime that goes in. Stop when 1 is left. Then write the number in index form.
B. Rebuild the number
| Index form | Written out | Number |
|---|---|---|
| 2² × 3 × 5² | ||
| 2⁵ × 3 | ||
| 3² × 5 × 7 | ||
| 2 × 3³ × 5 |
C. Right or wrong?
Each claim says it is the prime factorisation in index form. Tick the right ones; for the others, say what is wrong.
| 84 = 2² × 3 × 7 | |
| 72 = 2² × 3² × 2 | |
| 90 = 2 × 3 × 15 | |
| 150 = 2 × 3 × 5² | |
| 64 = 2⁵ |
D. Stretch
504 = 2³ × 3² × 7. Without dividing, decide whether 504 is divisible by each of 12, 18, 25, 14, 16, and say why.
______________________________________________________________________________
The fingerprint at work
A prime factorisation shows at a glance what a number is made of. Students use it to spot squares and cubes, find common factors and common multiples, and count factors.
We are learning to
- decide from a factorisation whether a number is a perfect square or a cube,
- find the highest common factor of two numbers from the primes they share,
- use prime factorisations to solve problems.
Success criteria
- I can say why 2² × 5² is a perfect square and find its square root.
- I can find the highest common factor of 2² × 3 and 2 × 3².
You need
The worksheet and the unit’s self-check quiz on the board. No calculators.
Lesson flow (about 60 minutes)
| 10 min | What the fingerprint tells Write 2² × 5² = (2 × 5)² = 10² = 100, then 2³ × 5, which is not a square. Ask: “Why do even exponents make a perfect square?” |
| 30 min | Problems Students work through problems 1 to 6, writing each step; 7 and 8 are the stretch. |
| 20 min | Class quiz game Run the self-check quiz as a class game (see the box), then wrap up. Ask: “What can you tell about a number from its prime factorisation that you cannot tell from the number?” |
Two shorter sessions instead? Split after problem 3; start the second with problem 4.
Watch for these ideas
- Calling a number a square because one exponent is 2. In 2³ × 5² the 2³ spoils it.
- Multiplying every prime from both numbers for the highest common factor. That gives a common multiple; the factor uses only the shared primes.
- Forgetting 1 and the number itself when counting factors.
Answers with working
- 2³ × 5² no (an odd exponent); 2² × 3² = 6²; 3⁴ × 7² = 63²; 2 × 3 × 5 no (an odd exponent).
- odd exponents on 2 and 3, so multiply by 6: 2⁴ × 3² = 144 = 12².
- both have 2, 3 and 5 at least once, so the highest common factor is 2 × 3 × 5 = 30.
- 2 × 3 × 7 × 11 = 462; composite factors include 6, 14, 21, 22, 33.
- yes: 2³ × 3³ = (2 × 3)³ = 6³ = 216.
- the smallest number using the primes of both: 2² × 3² = 36, so after 36 seconds.
- 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72, which is 12 factors.
- a factor takes 0 to 4 twos (5 choices) and 0 to 2 fives (3 choices): 5 × 3 = 15 factors of 400.
The fingerprint at work
Show each step. Check by multiplying back where you can.
Problems (7 and 8 are the stretch)
- Which of these are perfect squares? Give the square root of each one that is: 2³ × 5²; 2² × 3²; 3⁴ × 7²; 2 × 3 × 5.
- What is the smallest whole number you can multiply 2³ × 3 by to make a perfect square?
- 60 = 2² × 3 × 5 and 90 = 2 × 3² × 5. Use the primes they share to find the highest common factor.
- A number is 2 × 3 × 7 × 11. What is the number? Name three of its factors that are not prime.
- Is 2³ × 3³ a cube number? If so, the cube of what?
- Two lights flash together, then one flashes every 12 seconds and the other every 18 seconds. Use 12 = 2² × 3 and 18 = 2 × 3² to find when they next flash together.
- Stretch. List every factor of 2³ × 3². How many are there?
- Stretch. Without listing them, how many factors does 2⁴ × 5² have? Explain your method.
Sorting cards and prime tiles
Prime or composite? (Day 1)
Cut out the 24 cards. Pairs sort them into three piles: prime, composite and neither.
Teacher note: prime 2, 17, 31, 37, 41, 53, 59, 61, 67, 89 (10); composite 9, 15, 21, 27, 33, 39, 49, 51, 57, 63, 77, 87, 91 (13); neither 1.
Prime tiles (Day 3)
Cut out the 20 tiles. Lay tiles side by side to multiply them: 2, 2 and 3 make 12.
Factorisation match cards
Twelve numbers (Day 4)
Twelve prime factorisations (Day 4)
Pairs lay out all 24 cards and match each number to its prime factorisation, multiplying back to check. For a second round, turn the cards face down and play memory.
Teacher note: the matches are 24 = 2³ × 3, 40 = 2³ × 5, 45 = 3² × 5, 54 = 2 × 3³, 56 = 2³ × 7, 63 = 3² × 7, 75 = 3 × 5², 88 = 2³ × 11, 98 = 2 × 7², 99 = 3² × 11, 108 = 2² × 3³, 117 = 3² × 13.
What we know: Prime factorisation with exponents
Work on your own, without a calculator. Show your working.
- Is 39 prime or composite? Show how you know. ______________________
- List the prime numbers between 20 and 40. ______________________
- Use a factor tree to write 120 as a product of prime powers. ______________________
- Write 2 × 2 × 2 × 7 × 7 in index form. ______________________
- What number is 2⁴ × 3 × 5? ______________________
- Use a division ladder to write 198 in index form. ______________________
- Which of these is the prime factorisation of 100: 10², 2² × 5², 4 × 25? Say why the other two are not. ______________________
- Is 2⁴ × 3² a perfect square? If it is, what is its square root? ______________________
How did this week feel? Circle one: I can teach it · I am getting there · I need more help
Answers and marking guide
Answers with working
- Composite: 39 = 3 × 13.
- 23, 29, 31, 37.
- For example 120 = 10 × 12, 10 = 2 × 5, 12 = 2 × 2 × 3, so 120 = 2 × 2 × 2 × 3 × 5 = 2³ × 3 × 5.
- 2³ × 7².
- 2⁴ × 3 × 5 = 2 × 2 × 2 × 2 × 3 × 5 = 240.
- Divide by 2, 3, 3, 11, so 198 = 2 × 3² × 11.
- 2² × 5². 10² uses 10, which is not prime; 4 × 25 uses 4 and 25, which are not prime.
- Yes: every exponent is even, so 2⁴ × 3² = (2² × 3)² = 12² = 144. Its square root is 12.
A quick three-level guide
| Idea | Working towards | At standard | Beyond |
|---|---|---|---|
| Primes and composites (Q1, Q2) | calls every odd number prime, or calls 1 prime | decides prime or composite with a factor pair and lists the primes in a range | knows which primes to test, stopping where the square passes the number |
| Taking numbers apart (Q3, Q6) | stops at a composite leaf, or divides by a composite on a ladder | finds the prime factors with a tree or a ladder and checks by multiplying back | picks a first split or method that keeps the work short |
| Index form (Q4, Q5, Q7) | reads 2³ as 2 × 3, or leaves a prime twice | writes and reads index form, primes smallest first, and says why 10² and 4 × 25 are not prime factorisations | moves between written out, index form and the number without slips |
| Using the fingerprint (Q8) | calls a number square because one exponent is 2 | says a number is a perfect square when every exponent is even, and finds its root | uses the factorisation for common factors, multiples and factor counts |
Eight questions, four ideas. A student at standard answers most questions with working and checks each factorisation by multiplying it 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 | Prime or composite | Factor trees | Index form | Ladder and rebuild | Using the fingerprint |
|---|---|---|---|---|---|
The five columns are the five days: primes and composites, factor trees, index form, the ladder, and using the factorisation.