Factor Puzzles: Helping Math Make Sense

Guest blog post by Susan Smith and Kim Montague, Math is Figureoutable team

Feeling Puzzle-y?

Take a look at the puzzle on the left.

What do you notice?

What do you wonder?

If you noticed that 3 x 4 gives you the product of 12, which is the number on top…

And you noticed that 3 + 4 sums to 7, which is the number on the bottom… 

Congrats! You have figured out a Factor Puzzle. 

And I bet when you glance at the puzzle on the right, your brain is already thinking about what number each “?” must be to maintain those same relationships. 

Factor Puzzles can be addicting. And add great value to your classroom!

Anatomy of Factor Puzzles

Factor Puzzles are a routine we absolutely love.  They are a fun and engaging way to build additive and multiplicative reasoning for any student grades three and up.

The puzzle is a set of 4 squares, arranged with one centered on top, two side-by-side in the middle, and one centered on the bottom. Imagine placing numbers within those squares, so that the top number is 30, the two side numbers are 5 and 6, and the bottom number is 11. Can you see any relationships between those numbers?

That’s the challenge we pose to students: What do you notice about the puzzle you see? What relationships pop out to you?

Once students understand the relationships, identifying the top number as a product, the bottom number as a sum, and the middle numbers as factors or addends, you can challenge students by providing some of the numbers and leaving others unknown for them to solve.

For example, you could give students:

• the product and sum, leaving them to grapple with determining the two middle numbers.
• the two middle numbers, asking them to calculate the top and bottom numbers. 
• the product, sum, and one middle number, leaving students to find the one unknown middle number. 

With each unknown to solve, students are asked to consider different relationships.

Let’s Talk Strategy

Imagine a Factor Puzzle with 24 at the top, 11 at the bottom, and both middle numbers unknown. A student could first think about all the factor pairs of 24; 1, 24 ; 2, 12 ; 3, 8 ; 4, 6, and then consider which two pairs add up to 11, arriving at the conclusion that the numbers must be 3 and 8 or 8 and 3.

Another student might think about all the numbers that add to 11 and which pairs have a product of 24. The preferred strategy to cultivate with students would be to find the factor pairs first, as there are always going to be fewer factor pairs to filter through than pairs of addends for a given sum. Rather than tell students, we prefer to give them multiple experiences and ask guiding questions to help them draw their own conclusions about efficiency.

Through solving Factor Puzzles, students connect addition and multiplication. They apply reasoning instead of guessing. The routine also builds structure awareness and quietly prepares students for algebra. If students experience this kind of thinking early, expressions later do not feel like a brand new language. 

Another power of this routine is that it can be used with whole numbers, but also in higher grades integers and variables, which we will visit in the coming weeks!

Whole Number Factor Puzzle Problem String

On episode 300 of the Math is Figureoutable podcast, we shared the following Problem String of Factor Puzzles. As with all Problem Strings, remember to present and discuss each problem one at a time building connections and strategies as you go. Try this string with your class and let us know how it goes!

Present the first completed factor puzzle and ask, “What do you notice? What relationships pop out at you?” Once students have identified that the top number is the product of the middle two numbers, and the bottom number is the sum, present the next puzzle and ask, “Do the patterns still hold in this puzzle? How do you know?” Then present the first problem with unknowns.

With each puzzle, possible questions include:

• “What do the given numbers make you think about?”
• “What relationship are you using to solve for the unknown number?”
• “Are you thinking about factors or addends first? Which do you think is more efficient?”

Because Great Teaching Deserves Great Problems

First, thank you. If you are a listener of the Math is FigureOutAble podcast, we are so grateful you spend your time thinking deeply about math with us. And if you are not a listener yet, this is your invitation to start. It's where we dig into these ideas in real time, share what we're learning, and bring you the kinds of problems worth bringing back to your classroom.

If you're nodding along to this post, you already know something important: students are capable of deep thinking. The question is whether the math experiences we give them consistently build relationships, structure, and flexibility over time.

That kind of growth doesn't come from isolated lessons or random worksheets. It comes from intentionally sequenced problems, the kind where ideas build on each other and students have repeated opportunities to notice patterns, test strategies, and sharpen their thinking.

Factor Puzzles are a great place to start. Try the problem string with your class this week, use the resources below, and come back and tell us how it went. We'd love to hear what your students noticed.

Helpful Resources:

• Factor Puzzle Collection
• Blank Factor Puzzles