How is that possible? Tell me the answer?
Some of my students haven’t picked up on my sneaky side yet. There are no free answers in my class, including this visual which greeted them today:
Some students had seen this before, but few could figure out the mystery of the infinite chocolate. In my afternoon class, one student took charge, showing the subtle differences in the sizes of the pieces as they are reconnect…a future math teacher in the making. Today’s opener wasn’t intended to connect to anything course-related; it’s just a fascinating geometric mind trick, and great for generating math conversation right away. You can Google this problem and find a number of versions, many which explain the illusion, but we ended this opener with a video which shows some potential geometric shenannigans.
Today I desired a short and snappy opening hook, as my goal was to get students to the boards right away to work on binomial theorem problems. This was the second day students viewed videos and took notes for homework, and the response has been outstanding. Classes the last two days have been energetic, as the group doesn’t need to hear me drone on….they heard that at home. The focus today was terms in a binomial sequence – enjoy the video notes here. Also, pay attention for the rough edit at the end due to my mistake….was more fun to leave that in than to edit it out.
A unique sculpture greeted students as they entered class today:
There’s a lot of math goodness happening in this picture, but I don’t want to steer conversation in any particular direction right off. Time for some Noticing and Wondering! Students shared their thoughts on the back board:
Most of our class time today will be spent completing a jigsaw activity which guides students through many of the rich connections between Pascal’s Triangle, Combinations and the Binomial Theorem. Knowing that I would eventually talk about Pascal’s Triangle (one of my favorite shares of the year!), I was hoping to see if we could generate ideas on triangular and tetrahedral numbers organically. This visual opener did the trick. And while I ran out of time today for my Triangle chat, it’s in my pocket for tomorrow!
After sharing this experience on Twitter, Annie Fetter (the queen of noticing and wondering) chimed in with her ideas:
So many great ideas for packaging to be had here, but thinking I share it and leave it to my geometry colleagues to explore.
I opened today with what I had hoped would be a rich discussion concerning a past contest problem, but turned into something more substantial. Here is the problem:
I wasn’t too surprised when many, many students reached for their graphing calculator…this is what freshmen do. I was however surprised when I asked the class to volunteer their ideas on how they could simplify the expression – placing ideas on the board. Some appear below, and we discussed why or why not the procedures were valid.
Many misconceptions regarding what it means to “cancel” in a fraction were revealed; in fact, the very nature of what it means to reduce was the star of our discussion.
Later in class, I provided a hint which I hoped would provide some clarity with our factorial challenge. Some students immediately saw the link to the original problem and simplified. But how much closer are wo to finding the largest prime divisor? After simplifying, it was back to the calculators….which doesn’t really help much here. Listing the remaining factors of 16! after dividing common factors of 8! leaves us with the clear answer : 13.
The moral of the story – those who never touch a calculator discover the mechanics of this problem much more quickly than those who take a sledgehammer to it with a Nspire CX.