Math Trickery

Hello there dear reader. I have a request for you, if you wouldn't mind. Would you please pick a number between 10 and 20? Ah, you've picked the number 14, lovely. 

Now would you please pick a number between 20 and 30? 23 is an excellent choice. 

Now I shall perform a card trick for you. Please watch the video below: 

Nicole Horlings, September 18 2017

Now, it isn't a coincidence that I just pulled out two queens. So how did I do this? 

This past week, my math teacher presented this card trick to our class, then divided us into groups and handed us packs of cards while telling us to replicate the trick. We excitedly took our cards, picked a number between ten and twenty, and thought hard about the specific steps of performing the trick. We thought that it was odd that we would have to figure out the sum of the two digits of the number that we would need to pick for each round of the trick, and our investigation into the importance of this step led us to realizing the importance of the number 9. 

Nicole Horlings, September 15 2017

You choose a two digit number, then take the sum of the two digits and subtract it from the number. The result is always a multiple of 9. 

Once we had figured out the trick, we felt quite proud of ourselves. Our teacher smiled when we explained to her how we had figured out the trick, then asked, "Why?"

I was utterly confounded by her simple question. Hadn't we explained the trick? What more was there to say?

I learned that understanding "how" something works mathematically is different than understanding "why" it works mathematically. Once again I was reminded that I grew up with a strong focus on procedural understanding and very little focus on conceptual understanding.

My teacher proceeded to show us a much more complicated way to arrive at the number 9.

Nicole Horlings, September 15, 2017

You use the processes of factoring, removing redundancy, and combining numbers to arrive at 9

Our teacher then converted this equation into an algebraic expression, replacing each of the variables with a letter. 

Nicole Horlings, September 15 2017

The card trick works because the sum of the two digits of a number subtracted from the number always equal a multiple of 9. 

That night when I came home, I proudly proclaimed to my family that I could explain a card trick using algebra. Never before had I been excited about algebra, or thought of algebra as being "cool." Yet there I was, taking pictures of the blackboard, and thinking about when I could replicate this lesson in my own classroom.

Why?

Honestly, I'm still trying to answer that question. I know that the card trick was "how" I became interested in algebra, but I feel like I am making uncertain guesses at the "why." Beginning with the card trick, I was presented with something that didn't look like a typical math equation, so I allowed myself to make mistakes while I put effort into this challenge, and persisted in trying to find the answer. These three things - effort, persistence, and seeing mistakes as learning opportunities - are the basic components of having a growth mindset. I was pushed forward with high expectations from a teacher who believed that I was capable, and was guided to see math as something that you discover, not just something that mystically pops into your head.