Beans, Bundles, and Base 4

Fifteen years into this profession and I still get humbled when I teach new content. This year I teach a new class for me, called Math in Modern Society. It's a concurrent class for mostly seniors, that offers them essentially freshman math college credit for most trade schools and 2-year community colleges. 

Our topic this week was change of base. Specifically, learning about other number systems other than base 10 and being able to convert back and forth. In all my schooling, I never learned this. Not once. 

I opened up the book and immediately shut it. 

I opened up the notes and work that another teacher shared with me and immediately shut down. 

In fact, I avoided it until the day before I had to teach it. Super responsible.

And when I finally sat down to tackle it, I realized that I needed to make this visual. I was doing calculations and algorithms without understanding what I was doing. And I knew my students would feel the same way. 

So, I busted out some of my son's dinosaurs and started trying to visualize what this process looked like. It led me to this lesson, where students used beans (not enough dinosaurs at my house) to build, bundle, and write numbers in bases other than 10. 

So we started by counting out beans, and "bundling" them into cups based on the base we were working in. This is really just place value, something students have been working with since elementary school, but in a different number system. In base 10, we bundle every 10 individual objects to make one group of 10. In base 4, we bundle every 4 individual objects to make one group of 4. When we get enough cups, we bundle those into a larger package. The value of each digit depends on its position, just like it does in our familiar base 10 system. The only thing changing is how many we need to make each new group.

For example, the following scenario involves 11 beans. In a base 4 number system, this number would be written as 23 base 4, representing 2 powers of 4 (cups of beans) and 3 extra beans. 

We then started transitioning to the algorithm, or process, for converting these numbers without the beans and cups.

This was an important step for me. I didn't want students to simply memorize a process for converting numbers without understanding why it worked. The beans gave us something concrete to refer back to. When we used the algorithm, we could connect each part of the calculation to the groups we had physically built. The remainders weren't just numbers appearing in a division problem. They represented the leftover beans, and the quotients told us how many larger groups we could make.

And that's really what I wanted students to walk away understanding. We aren't learning an entirely new concept. We're taking something we already know about place value and applying it to a different system.

Fifteen years in, and I'm still learning new math right alongside my students. This unit was a good reminder that sometimes I need to shut the book, pull out some dinosaurs, and figure out what the heck I'm doing before I can help my students understand it.