School started this past week and just like most teachers, I have been working hard to learn who my students are and what they already know. I put a problem up on the board during an Algebra class:

x/3 = 5

I then asked a pretty simple question: how do we solve this problem? Every student who raised their hand was able to tell me that they had to multiply by 3 or that x was 15 because 15 divided by 3 was 5. My next question, however, seemed to stump them: why are you multiplying by 3? I received responses from my students such as, “Division is in the problem, and that is the opposite of division,” or, “multiplication cancels division out so x will be alone.” As happy as I was that they were able to tell me these things, I was still asking them to dig a little deeper. So then I wrote the following on the board:

1/3 x = 5

I asked them to look at that equation and think about whether it was the same or different from the first equation I wrote on the board. Some thought it was the same, but many thought it was different (others didn’t raise their hand). We talked a little more about the second equation:

me: The second equation says 1/3 x. What operation is this?

student a: It is multiplication.

me: It absolutely is! So, let’s look further. We know this is multiplication. What happens when we multiply 1/3

times x?

student b: Well, we have to change x to a fraction by putting it over 1.

student c: Yes, then we multiply fractions. We multiply the top together, and the bottom together.

student d: OH! I see it! 1 * x = 1x, or just x; and 3 * 1 = 3. So, it’s 1x/3, or just x/3.

student b: Which is the same as the top equation.

FANTASTIC! I didn’t say much because I really wanted them to converse with each other. Then we continued.

me: Yes! So, now knowing that, why are you multiplying by 3?

student e: Well, if 1/3 x is the same as x/3, 1/3 x is multiplication and the opposite is division.

student a: Yes, but to divide fractions, you have to multiply by the…

me: …the…? Starts with an ‘r.’

student f: Reciprocal! And the reciprocal of 1/3 is 3.

student c: Yes! So that’s why we multiply by 3!

This seems like something so simple, but I challenged my students’ thinking that day. I made them look a little deeper into the problem that they had learned how to solve last year. We had a long discussion about how knowing the procedures and steps to take to solve a problem are great, but understanding the why behind what they are doing is amazing. We talked about how understanding these whys help us to apply our understanding to more challenging problems.

These conversations are incredibly powerful and important to have with our students. However, sometimes they are overlooked simply because there is not enough time to teach every topic that is covered like that. I like to have these types of questions as my openers, or a part of a station activity. Students will learn a flipped lesson, and then I will have them look a little deeper into these types of problems. By teaching them to understand the “why” first, they will figure out the “how.” To end, I challenge you to challenge your students’ thinking. Ask them the “why” every single day.