How to teach factors and multiples

Why this matters

In one elementary class, most students copy the factors of 24 from the board and can't say why 13 has only two factors. A student who thinks 3 can't be a factor of 24 "because 24 is even" keeps that idea, because nobody asked her to test it. The teacher has to wait for the quiz to find out who memorized the list and who understands it.

Factors and multiples

A factor of a number divides it with nothing left over, and the number is a multiple of each of its factors. 3 and 8 are a factor pair of 24, so 24 is a multiple of both 3 and 8. A prime number has exactly two factors, 1 and itself, and a composite number has more. Arranging objects in equal rows turns each factor pair into a picture: 4 rows of 6 chairs, turned on its side, is 6 rows of 4, so the two arrangements are one factor pair. The Common Core asks students to find all factor pairs for a whole number from 1 to 100, recognize that a whole number is a multiple of each of its factors, and tell whether a number is prime or composite (CCSS 4.OA.B.4).

In the Flint activity, students find factor pairs by arranging chairs in equal rows, and Sparky draws every try. A list of the factors of 24 copied from the board never makes a student test an idea like "3 can't work because 24 is even". In the chat, Sparky asks that student to skip count by 3s, and the student finds 3 rows of 8 on their own.

A teacher with 25 students can't draw every arrangement each student suggests, or stop each wrong idea with the right question. Sparky, Flint's AI tutor, does both in each student's chat.

How the chair-arranging activity works

Sparky plays an event planner setting up chairs for the class play. Every student starts with the same message: 24 chairs, in equal rows, with no chairs left over. The activity's guidelines tell Sparky to take each student through four steps in order:

1. Find every way to arrange 24 chairs: 1 by 24, 2 by 12, 3 by 8 and 4 by 6. Sparky asks how the student knows there are no more.
2. Try 2, 3 and 4 rows of 13 chairs for a smaller show. Each try leaves a chair over, so only 1 row of 13 works, and Sparky calls 13 a prime number.
3. List the chair totals that work when every row holds 4 chairs, until the list passes 24 and the student spots it.
4. Write a three-line summary of what they found.

Each time a student names an arrangement, Sparky draws it as a grid with one dot per chair and shows leftover chairs in orange. Sparky never lists an arrangement the student hasn't found. It asks one question instead, such as "Could you make 3 rows?" A student who finds two arrangements in a row without help moves faster, and after the multiples step gets 36 chairs (6 rows of 6 makes a square) and then 29. A student who gets stuck is told to skip count by the number of rows, or tries a smaller number like 12 and then comes back.

One student finds the factor pairs of 24, then discovers a prime

Lena skip counts and hasn't learned factors yet. Her chat below runs from the first message through 13 chairs and multiples.

Sparky asks a question instead of giving the answerSparky builds on what Lena already found

L

Lena

Starting point: Skip counts, new to factors

Sparky

I'm setting up chairs for the class play, and I have 24 chairs. I need them in equal rows with no chairs left over. How could I set them up?

Lena

2 rows of 12?

Sparky

That works, with no chairs left over. Can you find another way?

Lena

I counted by 4s: 4, 8, 12, 16, 20, 24. So 6 rows of 4.

Sparky

Nice skip counting. That works! Could you make 3 rows?

Lena

No, 3 doesn't work because 24 is even.

Sparky

Let's check that. Skip count by 3s. Do you land on 24?

Lena

3, 6, 9, 12, 15, 18, 21, 24. It works! 3 rows of 8. And 4 rows of 6 is a new one.

Sparky

Look at your 4 rows of 6 and your 6 rows of 4. Which two numbers does each one use?

Lena

Both use 4 and 6. It's the same one, just turned.

Sparky

Exactly. 4 and 6 are a factor pair of 24. So are 1 and 24, 2 and 12, and 3 and 8. Now the show got smaller, and I have 13 chairs. Can you try 2 rows?

Lena

2 rows of 6 is 12, so there's 1 left over.

Sparky

That orange chair doesn't fit in a row. What happens with 3 rows?

Lena

3 rows of 4 is 12, 1 left over again. 4 rows of 3, still 1 left. Only 1 row of 13 works!

Sparky

You found it. 13 is a prime number: its only factors are 1 and 13. 24 is a composite number, because it has more factors. If every row holds 4 chairs, which totals work?

Lena

4, 8, 12, 16, 20, 24, 28. Hey, 24 is there!

Sparky

Those are multiples of 4, and 24 is one of them. Is 24 also a multiple of 3?

Lena

Yes, because 3 rows of 8 worked. So it's a multiple of all its factors?

Sparky

Exactly right.

Found all four factor pairs of 24, explained why 13 is prime, and saw that 24 is a multiple of its factors

Lena found every arrangement herself. When she said 3 rows can't work because 24 is even, Sparky asked her to skip count instead of telling her she was wrong. Both drawings are Sparky's own. Lena is an example student, and her chat is shortened.

See who found every factor pair

Sparky grades each chat against the activity's rubric. The activity page shows who needs help now, what the class understands, and what to practice next.

Set up the activity for free

The chair-arranging activity is in Flint's public library, with Sparky's role, guidelines, greeting and a rubric already written. Open it, select Use activity to copy it into a free Flint account, and move it to a class. In the activity's Preview tab, select Try session myself to test it as a student before you assign it.

The guidelines are written for a 4th grader, the grade where the Common Core puts factor pairs and multiples (CCSS 4.OA.B.4). To use the activity with another grade or other numbers, change Guidelines for Sparky. For example:

For more math activities, see Flint for math.