Negative Exponents

I’m pretty pleased with the sequence that we did for introducing negative exponents today. We’ve been doing patterning for ages, so students are used to the routine of modelling visual patterns.

This one was a bit different than the usual linear and quadratic patterns. We were modelling the number of dots or terminal branches.

Students had great success with making tables and spotting that the number doubled and doubled and doubled as we got higher figure numbers. It took a minute for some to connect that with writing repeated (2)(2)(2)(2) and making it 2^x. Students graphed for a positive domain quite readily, but needed a push to think about if the domain of this would be positive x only. The visual patterning breaks down, but the equation continues, and we got there by using the tables. As you go down the table (for larger x values) we multiplied by 2 each time. To go up the table (for smaller values of x) we divide by 2 each time.

We had light bulb moments as we saw the numbers getting smaller and smaller and smaller, as we divided by 2 each time. They made connections to graphical features saying there’s an asymptote, and telling that the range is y>0 but not equal to 0.

We debriefed about how 2^-3 would be 1/(2^3) etc. It’ll take some practice to get the skills solid, but it was a nice introduction. We worked on expressing different bases to different negative powers.