Exponential Innovation

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While I was on my vacation this summer in Sanibel Island, I read The Classroom Chef by John Stevens and Matt Vaudrey. What, you say?! You read a professional development book when you were supposed to be relaxing on vacation?! Well, yes, I'm a math geek, so while most people were reading trashy romance novels on the beach, I was reading about how to pump up my math lessons.

That reading, along with other conversations and books about innovative teaching, inspired me to take a fresh look at my lessons. I was remotivated to bring innovation into my classroom. I just love how educators can inspire each other! I was also lucky enough to hear Dave Burgess speak on my district's opening day that year—he was amazing and inspiring!

At the same time, I was participating in an online course inspired by The Innovator’s Mindset, a book by George Couros, with George Couros and Katie Martin. It was another opportunity to reflect on what innovation could look like in my classroom and to learn from other educators.

So I thought about one of my really boring lessons and tried to figure out how I could be as crazy and innovative as John, Matt, and Dave and make a really memorable lesson for my students. The first topic that came to mind was negative exponents. I looked around the internet to find some ideas that I could run with. One that caught my eye was from Yummy Math. I wanted to be able to relate negative exponents to something in real life that my students could understand. I liked the idea of the beanstalk story, but how could I make it memorable? My first idea, although I loved it, did not happen. I wanted to construct a giant beanstalk (out of paper mache maybe?) in the corner of the room that was so tall it went through a hole in the ceiling tiles. Instead I just made up an elaborate story about a former student named Jack who gave me some beans at the end of the year, and some slides in my Smart notebook to go along with it. (I made my beanstalks with Google Drawings—I just love that app!)

After I told my story and showed my graph, I asked my class to predict what the height of the beanstalk would be at month 6. I told them to look for patterns because math, and nature, are full of patterns. (That's my big push this year—look for patterns.) I had them all answer the question in Google Classroom. Only one student figured out the pattern and predicted correctly. We discussed how the height was changing—I had them figure out that it was changing multiplicatively, not additively. Then I showed them a graph of what the beanstalk would have looked like if it had grown additively.

I used this to introduce the concept of exponential vs. linear growth and we drew in the exponential curve and linear line. We also explored what this data would look like in a table and then we graphed with Desmos. I pointed out that one equation had an exponent and one didn't. I also noted that things that grow exponentially grow faster than things that grow at a linear rate—we compared the heights at month 5 in both graphs.

The next day, I had the class do a Desmos card sort with the concepts of exponential and linear. I was happy to see that most of them had remembered the concepts from the beanstalk lesson! Then I finally got around to introducing negative exponents. In Google Classroom, I asked them to predict what the beanstalk's height would have been at month 1 if I had thought to measure it. This time a few of them saw the pattern and figured it out. From here, I gave out my guided notes. I taught them how to use the calculator to find exponents with the ^ button. I had them complete the positive exponents in the Powers of 2 table on their own. I reiterated the first point in the notes that said negative exponents are reciprocals of positive exponents so we went through the negative exponents together. Then I had them complete the next two tables independently.

After the tables were complete, I showed them my method of circling the base and negative exponent so they are aware that the negative exponent only affects the base and not any coefficients. I showed them how to "swing it down" to the denominator and leave the coefficient in the numerator. They picked up on this procedure fairly quickly.

So the first day, the beanstalk story that captured my students' attention wasn't about negative exponents, but it set the stage for the discussion on negative exponents the next day and they had a real life situation (sort of) that they could relate it to. After class the first day, my classroom aide told me she was watching one student in particular and he was really into the whole story—this is a student who rarely can focus on a lesson. After school, the student I tutor asked me if that story was real. Even though I didn't have my life-sized beanstalk prop, I think the lesson went well and the students will definitely remember the difference between exponential and linear growth!