Teaching Factors and Multiples in 4th Grade Small Groups

Factors and multiples are two related but different ideas built directly from multiplication facts students already know. This guide walks through distinguishing the two terms, finding factor pairs systematically, recognizing multiples, and connecting both to the fact fluency work already underway.

Direct Answer

A factor of a number is one of two (or more) whole numbers that multiply together to give that number — for example, 3 and 4 are factors of 12, since 3 × 4 = 12. A multiple of a number is the result of multiplying it by a whole number — 12 is a multiple of both 3 and 4. These are related but different ideas, and students commonly treat them as interchangeable or reverse them. A common 4th grade instructional focus is finding all factor pairs of a number up to 100 systematically — checking each whole number from 1 upward to see if it divides evenly, and recognizing when a number can stop being checked — along with recognizing multiples using skip-counting or known multiplication facts. Knowing multiplication facts well is what makes finding factors and multiples efficient, rather than a matter of guessing. Prime and composite numbers extend this naturally: a number with exactly two factors, 1 and itself, is prime, while a number with more than two factors is composite. Specific grade-level expectations and number ranges vary by state and curriculum.

Where This Skill Fits

Prerequisite: Multiplication Fact Fluency (3rd grade), since factors and multiples are built directly from known multiplication fact relationships → Current: Factors and multiples, including systematic factor pairs and prime versus composite numbers → Next: Multi-Digit Division (4th grade), where factor and multiple reasoning supports estimating and checking quotients.

Essential Prerequisite Skills

Solid recall of multiplication facts through at least 10 × 10
Understanding of the relationship between multiplication and division (a fact family)
Comfort with skip counting for common numbers

Helpful Prior Knowledge

Experience with arrays as a way to represent multiplication
Familiarity with organizing information systematically, such as in a table or list
Comfort recognizing when a number divides another evenly versus leaves a remainder

Common Student Thinking / Misconceptions

Student may think: "Factor" and "multiple" mean the same thing, or a student says "12 is a factor of 3" when 12 is actually a multiple of 3, and 3 is a factor of 12.

What this may reveal: This may indicate the student hasn't yet connected each term to a distinct direction — factors are the smaller numbers that build up to a product, while multiples are the results of multiplying up from a number.

Possible teacher response: Anchor both terms to the same fact family (3 × 4 = 12): "3 and 4 are factors of 12 — they multiply together to make it. 12 is a multiple of 3, and also a multiple of 4 — it's what you get when you multiply up."

Student may think: When listing factor pairs of a number, a student stops after finding one or two pairs instead of checking systematically.

What this may reveal: This may suggest the student is factoring by guessing rather than checking each whole number in order, so pairs get missed. See Memorizing Facts Without Using Related-Fact Relationships.

Possible teacher response: Model checking 1, then 2, then 3, and so on in order, recording each factor pair found and stopping once the factors start repeating in reverse order.

Student may think: A student lists multiples of a number that skip around or aren't the result of multiplying by a whole number, such as including a number's half.

What this may reveal: This may reveal that the student is thinking of "related numbers" broadly rather than specifically as the result of multiplying by 1, 2, 3, and so on.

Possible teacher response: Have the student build the list by skip counting aloud from the number itself, confirming each new value is the number times the next whole number.

Student may think: Every number has an unlimited or unclear number of factors, so there's no way to know when to stop looking.

What this may reveal: This may indicate the student hasn't yet noticed that factor pairs start repeating once the checking number passes the square root of the target number, or more simply, once pairs begin to reverse.

Possible teacher response: Show a full factor-pair table for a number like 24 and point out where the pairs start reversing (4 × 6, then 6 × 4), which signals that every pair has been found.

Student may think: A prime number is any number that "feels hard to divide," rather than a number with exactly two factors.

What this may reveal: This may suggest the student is relying on intuition rather than actually checking a number's factor pairs before deciding whether it's prime or composite.

Possible teacher response: Have the student list all factor pairs for a candidate number and count them, confirming prime status only when exactly one pair (1 and itself) is found.

Useful Visual Models

Arrays — useful for visualizing factor pairs as the two dimensions of an array, since a rectangle with whole-number rows and columns represents one factor pair of its total. A limitation is that arrays become impractical to build or draw once numbers grow past about 30 or 40, so students eventually rely on the multiplication fact itself rather than a drawn array.
Equal Groups — useful for visualizing multiples as repeated groups of the same size, making it clear that a multiple is what accumulates when a number is added to itself repeatedly. A limitation is that equal groups can get unwieldy for larger multiples, so skip counting or known multiplication facts become more efficient as the list grows.

Small-Group Teaching Sequence (about 15–30 minutes)

Activate Prior Knowledge (2–4 min): Review a multiplication fact family (such as 3 × 4 = 12 and 12 ÷ 4 = 3) and name the numbers involved.
I Do (4–6 min): Teacher models systematically finding all factor pairs of a number and listing several multiples of a different number.
We Do (5–8 min): Teacher and students find factor pairs and multiples together, checking whether specific numbers are factors or multiples of a target number.
You Do (4–8 min): Students find factor pairs and list multiples independently, including one prime-versus-composite check.
Quick Check (2–3 min): One problem asking the student to distinguish a factor from a multiple for the same target number.

I Do Example

Find all factor pairs of 18, then list the first five multiples of 4.

"To find all the factor pairs of 18, I'll check whole numbers in order, starting at 1, and ask: does this number divide 18 evenly?" "Does 1 work? 1 × 18 = 18, yes — 1 and 18 are a factor pair." "Does 2 work? 2 × 9 = 18, yes — 2 and 9 are a factor pair." "Does 3 work? 3 × 6 = 18, yes — 3 and 6 are a factor pair." "Does 4 work? 18 ÷ 4 leaves a remainder, so no." "Does 5 work? No." "Does 6 work? I already found 6 as part of a pair with 3 — once pairs start repeating in reverse, I know I've found them all." The factor pairs of 18 are: 1 × 18, 2 × 9, 3 × 6. "Now for multiples of 4, I multiply 4 by 1, 2, 3, 4, and 5: 4, 8, 12, 16, 20." Notation: Factors of 18 = {1, 2, 3, 6, 9, 18}. Multiples of 4 (first five) = 4, 8, 12, 16, 20. Emphasized language: "check in order," "does it divide evenly," "pairs start repeating," "multiply up" for multiples versus "multiply together to build" for factors.

We Do Example

Problem 1: Find all factor pairs of 24 together. "What's the first number we should check?" (1; 1 × 24 = 24.) "What's next?" (2; 2 × 12 = 24.) "Keep going — does 3 work? Does 4 work? Does 5 work?" (3 × 8 = 24; 4 × 6 = 24; 5 does not divide evenly.) "When can we stop checking?" (Once the next number we'd check, 6, already appeared as part of a pair.)

Problem 2: Is 5 a factor of 30? Is 30 a multiple of 5? "How could we check whether 5 is a factor of 30?" (30 ÷ 5 = 6, with no remainder, so yes.) "Now, is 30 a multiple of 5? How is that question different from the first one, or is it the same?" (It's really the same relationship, viewed from the other direction — 5 × 6 = 30, so 5 is a factor of 30, and 30 is a multiple of 5.)

You Do Examples

Find all factor pairs of 36, checking whole numbers in order.
List the first six multiples of 7.
Is 9 a factor of 63? Is 63 a multiple of 9? Explain how you know.
Find all factor pairs of 17. Based on what you find, is 17 prime or composite? Explain.

Quick Check

Ask the student to find all factor pairs of 20, list the first four multiples of 6, and explain in their own words the difference between a factor and a multiple.

A student who finds the factor pairs and lists the multiples correctly but cannot clearly explain the difference between the two terms is showing partial understanding — the computations are working, but the vocabulary distinction is not yet secure. A student who completes both tasks correctly and can explain that factors multiply together to build a number while multiples are what result from multiplying that number up is showing full understanding of this skill.

If the student demonstrates understanding → introduce prime and composite classification for numbers up to 50, using factor pairs as the evidence. If the student needs more support → return to a single, smaller target number (10 or fewer) and rebuild factor pairs using an array before reintroducing the checking-in-order strategy for larger numbers.

If Students Demonstrate Understanding

Move toward classifying numbers up to 50 or 100 as prime or composite using factor pairs as evidence, and begin connecting factor and multiple reasoning to estimating and checking quotients in multi-digit division, where recognizing a divisor's multiples speeds up finding a reasonable quotient.

If Students Need More Support

Return to a single target number of 10 or fewer and build its factor pairs using arrays, so the pairing is visible rather than abstract. Keep a multiplication chart or a list of known facts nearby as a concrete support, since checking for factors and generating multiples both depend on quick fact recall. A useful teacher prompt is, "Can you find two whole numbers that multiply together to make this number?" for factors, and "What do you get when you multiply this number by the next whole number?" for multiples.

Before → Current → Next Skill Relationships

Before: Multiplication Fact Fluency (3rd grade)
Current: Factors and multiples, including systematic factor pairs and prime versus composite numbers
Next: Multi-Digit Division (4th grade)

Related Misconceptions

Memorizing Facts Without Using Related-Fact Relationships
Treating "factor" and "multiple" as interchangeable terms (for example, calling 12 a "factor of 3" when 12 is actually a multiple of 3, and 3 is a factor of 12)

Related Visual Models

Relevant SMS Resources

Browse more 4th grade multiplication and division routines, task cards, and quick checks in the full catalog.

Teacher FAQ

What's the simplest way to explain the difference between a factor and a multiple?

Factors are the smaller numbers that multiply together to build a number, while multiples are the results of multiplying that number by a whole number. Anchoring both to the same fact family, such as 3 × 4 = 12, keeps the distinction concrete.

How do I know when a student has found all the factor pairs of a number?

Have the student check whole numbers in order starting at 1. Once the next number to check has already appeared as part of a found pair, every factor pair has been found.

Why does fact fluency matter so much for this skill?

Checking whether a number is a factor, or generating a list of multiples, depends on quick multiplication and division fact recall. Without that fluency, the process becomes slow guessing rather than efficient checking.

How much time should I spend on prime and composite numbers?

Treat it as a brief, natural extension once factor pairs are solid. Have students count the factor pairs they find for a number and classify it as prime (exactly one pair, 1 and itself) or composite (more than one pair).

My student can list multiples but struggles to find factor pairs. What should I do?

Return to arrays: have the student try building rectangles with whole-number rows and columns that equal the target number, which makes each factor pair visible as a physical arrangement.

How does this connect to the division work coming up?

Recognizing a divisor's multiples makes it faster to estimate and check a quotient during multi-digit division, since students can quickly judge which friendly multiple of the divisor fits into the dividend.