Factors, multiples and prime numbers
Factor pairs as rectangles, multiples as skip counts, and telling primes from composites
About four lessons of 45 to 60 minutes
How many ways can you arrange the chairs?
Your class of 12 is putting on a play, and you have to set out 12 chairs in equal rows. You could do 3 rows of 4, or 2 rows of 6, or even 1 long row of 12. But could you do 5 equal rows? Try it: 12 does not split evenly into 5, someone is left over. The numbers that do work, 1, 2, 3, 4, 6 and 12, are exactly the factors of 12.
Factors and multiples are the hidden structure of every whole number. Today you will build factor pairs as neat rectangles, list multiples by skip counting, and discover why some numbers, the primes, can only ever be arranged in one single row.
- 12 chairs in 3 rows of 43 and 4 are a factor pair of 12, because 3 x 4 = 12
- 24 cookies shared onto plates24 splits evenly as 4 x 6 or 3 x 8, so 4, 6, 3 and 8 are all factors of 24
- Counting in 3s: 3, 6, 9, 12these are the multiples of 3, what you land on when you skip count
- 7 stars can only make 1 row of 77 is prime, its only factors are 1 and 7
What students will be able to do
Students will find all the factor pairs of a whole number by building arrays, recognise that a number is a multiple of each of its factors, list multiples by skip counting, and decide whether a number up to 100 is prime or composite by counting its factors.
- I can build the factor pairs of a number as rectangles.
- I can list all the factors of a number in order without missing any.
- I can list the first several multiples of a number.
- I can explain how factors and multiples are opposite directions of the same fact.
- I can decide whether a number is prime or composite and say why.
Standards this unit teaches
- 4.OA.B.4Common Core (US)Factor pairs, multiples, primes and composites
Find all factor pairs for a whole number in the range 1 to 100. Recognise that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1 to 100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1 to 100 is prime or composite.
- AC9M5N02Australian Curriculum v9 (ACARA)Prime, composite and square numbers (Year 5)
Identify and describe the properties of prime, composite and square numbers and use these properties to solve problems and simplify calculations.
- AC9M6N01Australian Curriculum v9 (ACARA)Factors, multiples and divisibility (Year 6)
Express natural numbers as products of their factors, recognise multiples and determine whether one number is divisible by another. Australia meets factor pairs and divisibility at Year 6, so this US Grade 4 unit runs about one to two years ahead of the ACARA placement.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
Words to teach and display
- Factor
- a whole number that divides another exactly, with no remainder
- Factor pair
- two factors that multiply to give the number, such as 3 and 4 for 12
- Multiple
- the result of multiplying a number by a whole number, so 12 is a multiple of 3
- Array
- objects in equal rows and columns, the rectangle a factor pair makes
- Prime number
- a number with exactly two factors, 1 and itself, such as 7
- Composite number
- a number with more than two factors, such as 12
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Factors are equal rows
ConcreteGive each pair 12 counters and ask them to make a rectangle with equal rows. Every rectangle that works reveals a factor pair. A 3 by 4 rectangle uses all 12 counters in 3 equal rows of 4, so 3 and 4 are factors of 12, because 3 x 4 = 12. A factor is a number that fits into another an exact number of times, with none left over.
Try to make a rectangle with 5 equal rows and it fails: 12 counters cannot fill 5 equal rows, some are left over. So 5 is not a factor of 12. Factors are exactly the numbers that make a full rectangle with nothing left over.
The word factor and the word divide are two sides of one coin. 3 is a factor of 12 because 12 divides by 3 exactly, giving 4. If there is a remainder, it is not a factor.
- Is 4 a factor of 12? Show the rectangle that proves it.
- Why is 5 not a factor of 12?
- What does it mean for a number to divide another exactly?
2. Finding every factor pair
PictorialTo find all the factors of a number, hunt for every rectangle it can make. For 12 there are three: 1 by 12, 2 by 6, and 3 by 4. Reading off both sides of each rectangle gives the full list of factors: 1, 2, 3, 4, 6 and 12. Notice they come in pairs that multiply to 12.
Work through the counting numbers in order and test each one: does 1 divide 12? Yes, and its partner is 12. Does 2? Yes, partner 6. Does 3? Yes, partner 4. Does 4? Yes, but its partner 3 is already listed, so you have met in the middle and can stop.
Listing in order from the outside in, 1 and 12, then 2 and 6, then 3 and 4, means you never miss a factor. Every number has at least two factors: 1 and itself.
Find all the factors of 24.
- Test each number in order. 1 x 24 works, so 1 and 24 are factors.
- 2 x 12 works, and 3 x 8 works, and 4 x 6 works.
- 5 does not divide 24 exactly, so skip it. After 4 and 6 the pairs would repeat, so stop.
Answer: The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.
- Which two factors of 12 multiply together to make it, other than 1 and 12?
- Why can you stop testing once the pairs start to repeat?
3. Multiples: the other direction
PictorialA multiple is what you get by multiplying a number by 1, 2, 3 and so on. The multiples of 3 are 3, 6, 9, 12, 15 and on forever, exactly the numbers you land on when you skip count in 3s. A number is a multiple of each of its factors, so because 3 is a factor of 12, 12 is a multiple of 3.
Factors and multiples are opposite directions of the same fact 3 x 4 = 12. Looking down, 3 and 4 are factors of 12. Looking up, 12 is a multiple of both 3 and 4. Factors of a number are that number or smaller and run out; multiples are that number or bigger and never end.
- List the first five multiples of 5.
- Is 20 a multiple of 4? How do you know?
- Which never runs out, the factors of a number or its multiples?
4. Prime and composite numbers
PictorialNow the special case. Some numbers can make only one rectangle, a single row. Try to build 7 counters into equal rows: the only rectangle is 1 by 7. That is because 7 has just two factors, 1 and 7. A number with exactly two factors is called a prime number.
A number that can make more than one rectangle has more than two factors, and is called composite. 6 is composite because it makes 1 by 6 and 2 by 3, giving four factors: 1, 2, 3 and 6. 12 is composite too, with six factors.
One special number, 1, is neither prime nor composite: it has only a single factor, itself, so it does not have the two factors a prime needs. The smallest prime is 2, which is also the only even prime.
- How many factors does a prime number have?
- Is 9 prime or composite? Draw the rectangle that decides it.
- Why is 2 a prime number?
5. Deciding prime or composite
AbstractTo decide whether a number is prime or composite, find its factors. If the only factors are 1 and the number itself, it is prime. If there is even one more factor, it is composite. You do not need the whole list, a single extra factor settles it.
Quick tests help. Every even number bigger than 2 is composite, because 2 divides it. If the digits add to a multiple of 3, then 3 is a factor, so the number is composite. Numbers ending in 0 or 5 (except 5 itself) have 5 as a factor.
Watch out for numbers that look prime but are not. 9 is odd, yet 9 = 3 x 3, so it is composite. Being odd does not make a number prime.
Decide whether 12 and 7 are prime or composite.
- 12 has factors 1, 2, 3, 4, 6 and 12. That is more than two factors, so 12 is composite.
- 7 has only the factors 1 and 7. Exactly two factors, so 7 is prime.
- As a check, 12 makes several rectangles while 7 makes only a single row.
Answer: 12 is composite, and 7 is prime.
- Is 15 prime or composite, and which extra factor tells you?
- Name a prime number between 10 and 20.
Common misconceptions and how to address them
MisconceptionFactors and multiples are mixed up, so a student lists 24 and 36 as factors of 12.
Why it happens: The two words describe the same fact family and are easy to swap before their directions are clear.
How to address it: Anchor the directions: factors are the number or smaller and run out, multiples are the number or bigger and never end. 24 and 36 are bigger than 12, so they are multiples, not factors.
Misconception1 and the number itself are left off the factor list.
Why it happens: They feel too obvious to count, so students jump straight to the interesting middle factors.
How to address it: Every rectangle includes the 1 by n row, so 1 and the number are always a factor pair. Start every list with 1 and its partner.
MisconceptionFactor pairs are missed because the search is random rather than in order.
Why it happens: Hunting for factors at random, a student easily skips one, such as forgetting 4 when listing the factors of 12.
How to address it: Test the counting numbers in order, 1, 2, 3, 4, and record each pair from the outside in until they meet in the middle. Order means nothing is skipped.
Misconception1 is called a prime number.
Why it happens: 1 seems to fit the pattern of only dividing by 1 and itself, but for 1 those are the same single factor.
How to address it: A prime needs exactly two different factors. 1 has only one factor, so it is neither prime nor composite. The smallest prime is 2.
MisconceptionEvery odd number is assumed to be prime.
Why it happens: The first few primes, 3, 5, 7, are all odd, so odd and prime get tangled together.
How to address it: Show 9 = 3 x 3 and 15 = 3 x 5: both are odd yet composite. Odd tells you 2 is not a factor, but other factors may still exist.
MisconceptionA bigger number is assumed to have more factors.
Why it happens: It feels as if size and number of factors should go together.
How to address it: Compare 61, a prime with just two factors, against 60, which has twelve. The count of factors depends on a number's structure, not its size.
Guided practice (with answers)
1. List all the factors of 18.
Answer: 1, 2, 3, 6, 9 and 18. The pairs are 1 x 18, 2 x 9 and 3 x 6.
2. Write the first five multiples of 4.
Answer: 4, 8, 12, 16 and 20.
3. Is 3 a factor of 12? Show why with a rectangle.
Answer: Yes. 12 makes 3 rows of 4, so 3 x 4 = 12 and 3 is a factor.
4. Is 13 prime or composite?
Answer: Prime. Its only factors are 1 and 13.
5. Is 21 prime or composite, and why?
Answer: Composite, because 3 x 7 = 21, so it has factors beyond 1 and 21.
6. Explain how you know 12 is a multiple of 3.
Answer: Because 3 is a factor of 12 (3 x 4 = 12), so counting in 3s reaches 12.
Independent practice worksheets
Set the matching ChalkBee worksheets for independent work. The answer keys are computed in code, so they are never wrong. Begin with listing factors, then move to prime and composite once factor lists are secure.
Differentiation
- Keep physical counters or square tiles out so students can build each rectangle before listing factors.
- Provide a factor-pair recording table (this by that) to organise the search from the outside in.
- Start with numbers that have clear rectangles, such as 6, 8, 10 and 12, before tackling awkward ones.
- Supply a multiplication grid so a student can scan a row to spot factor pairs.
- Find the common factors of two numbers, such as 12 and 18, as a first step toward greatest common factor.
- Investigate square numbers, where one factor pair is a number times itself, such as 9 = 3 x 3.
- List the primes up to 50 and look for patterns in where they fall.
- Explore numbers with many factors (12, 24, 36) and describe what they have in common.
Assessment: exit ticket
A three-question exit ticket for the last five minutes. It samples finding factors, identifying multiples, and deciding prime or composite.
1. List all the factors of 16.
Answer: 1, 2, 4, 8 and 16 (pairs 1 x 16, 2 x 8, 4 x 4).
2. Is 30 a multiple of 6? Explain.
Answer: Yes, because 6 x 5 = 30, so counting in 6s reaches 30.
3. Is 17 prime or composite?
Answer: Prime, because its only factors are 1 and 17.
Teacher notes and timings
- Rough timing across four lessons: Lesson 1 factors as arrays (section 1), Lesson 2 finding every factor pair (section 2), Lesson 3 multiples (section 3), Lesson 4 prime and composite (sections 4 and 5) plus the exit ticket.
- Language to keep saying: a factor divides exactly with nothing left over, factors run out and multiples never end, a prime has exactly two factors. These pre-empt most of the misconceptions.
- Keep counters or square tiles available through the pictorial sections. Building the rectangle is what makes a factor pair concrete before it becomes a list.
- The array diagrams show exact rows and columns, so a 3 by 4 array is genuinely 12 cells. Point out that turning a rectangle on its side (3 by 4 to 4 by 3) is the same factor pair, not a new one.
- Curriculum note and a US and AU alignment: the US places factor pairs, multiples and prime versus composite together in Grade 4 (4.OA.B.4). ACARA introduces prime and composite numbers at Year 5 (AC9M5N02) and factor products and divisibility at Year 6 (AC9M6N01), so this unit runs about one to two years ahead of the Australian sequence and suits an extension or early-introduction context there.
- Present mode and print both work: use the Print button for a clean teacher copy or a student handout, and project the page to teach straight from the diagrams.