Last week my Math 7 students spent several days exploring ratios and unit rates (including scenarios with complex fractions in the mix). We haven’t used the word “proportion” yet… we’ve just been “playing” with numbers, really. Encouraging students to think flexibly (informally using ratio tables with recipes and other contexts, and just exploring ideas of equivalence in general) was comfortable for some, but made others extremely UNcomfortable.

Some students viewed having options as “fun”… a game… a puzzle… they found joy in exploring many different, correct ways to represent equivalence…

…others seemed to feel overwhelmed, yearning for ONE WAY to DO the THINGS that would ALWAYS work… but if I give ‘em time, I know they’ll come around!

These flexible-thinking days are an investment into students’ understandings in the future… when we start using vocabulary like proportional, direct variation, and constant of variation, but for now, let’s play a little longer… even if it means taking a bit more time than initially planned.

I experienced the first return-on-investment at the start of class Friday. Several students told me they’d started considering the date each day, and had begun representing the date as a complex fraction… all on their own. {*Note: They’re keeping track of the patterns by hand – I’m using Desmos here so it looks “pretty”.*}

Here was how they represented the date on their “Day 1”, which happened to be 10/9/19:

Here’s their “Day 2”:

Before they stacked Friday’s date in similar fashion as a new complex fraction, the students decided to guess whether the date was going to be “better” or “worse” than yesterday’s…

“I think it’s going to get ‘worse’ every day!”

Sure enough…

“We got even WORSE today! I think we’re going to get worse every day through Halloween!”

“Halloween will probably be the WORST!”

“I wonder how much better we’ll get on November 1!!!”

IT’S HAPPENING…!!!!!!

This exchange and student-driven “game” was the perfect way to end a week for me. Embracing these moments, even when they’re not exactly on topic with *my* lesson plans for the day, is critical and important! As we try to help our students think and understand versus apply tricks and quick-fixes… be patient and persevere.

Enjoy your precious mathematicians as you listen to them and guide them on their journey in the coming weeks and months.