Sunday, September 9, 2018

A mathematical beginning of the school year activity with low bar and a high ceiling

I'm teaching a really broad range of classes and subjects this year (6th grade math, middle school geometry, algebra 2, calculus, Japanese 1 and Japanese 3) so for my math intro activity I wanted to explore fraction concepts in a way that both 6th graders and algebra 2 students could enjoy and would reveal the struggles and confusions of both groups.

Voila: my summer in fractions.

This activity went so much better than I expected that I wanted to share it somewhere.  It took one and a half class periods to complete (really just one- half of the first day after other intro activities and half of the next day).  I thought it might be funish in that the kids got to share their summers with each other and color a little, but I expected to hear some grumbling about fraction hate.  Instead, it generated some of the most interesting conversations I've had with students to date.  First I'll talk about how I ran it, then at the bottom of the post I'll list the cool conversations/insights we had.

On day 1, I did some basic introduction stuff and with the 6th graders, my favorite intro to math class game where I put cards on their backs with different numbers on them and then they have to go around asking each other yes or no questions about their number and try to guess what it is.  With Algebra 2 we did an intro activity where they wrote on a note card a number that represented their summer (45 hrs playing zelda, 17 for the age they turned, 5 am for the latest they stayed up, etc.) and their general attitude about math on the back.  Then we guessed who was associated with each card.

Then after those intro activities I handed them the above worksheet and did one or two of the categories on the top (sleep and vacation) for myself on the document camera so they could see some of the thinking involved then I let them work the rest of the period.  Finishing it was homework.

Then the next day I showed their pie charts on the document camera anonymously and they tried to figure out who had made which pie chart.  I would then ask a mixture of math questions about the charts and personal questions (if 1/7 of the pie chart represented reading, how many hours a day on average is that?  What did you read?  Oh yeah? me too.  Did you binge watch the show also?  How on earth did you spend a quarter of your summer staring off into space?  What did you think about?)

Finally, I taped all their charts to the wall.  It was a beautiful sight:
Here are some of the cool conversations/insights I had with the 6th graders

  • How many days are in the summer?  How many weeks is that?  How many hours is that?  So if you spend one day a week on an activity does that represent 1/7 of your summer?  If you sleep while on vacation, how do we account for that in a pie chart?
  • If you spent 8 hrs a day sleeping and 8 hrs is 1/3 of a day, how much of a week do you spend sleeping?  How much of the whole summer do you spend sleeping?
  • Lots of arguments about units.  "I'm confused because the fraction that represents my sleep is in hours but the fraction that represents my vacation time is in weeks?  How can I put both of those on the same pie chart?  Do I use 84 days or 12 weeks for this fraction?"
  • How do we add up all the fractions and what do they equal?  What does it mean if they don't equal one or are more than one?
  • How do we break the pie chart up into enough sections that we can represent every fraction we generated?
  • So much reducing!  If they were on vacation for 16/84 days what's the simper fraction.  
Here are some of the cool conversations/insights I had with the high schoolers.
  • I showed them how to use their graphing calculators to add up all the fractions and output a fractional answer since their fractions were much more complicated than the 6th graders and they were more attuned to precision.  They fell in love with the calculators IMMEDIATELY because they can avoid doing fraction work with them.  But we still reviewed how to find the LCM of more complicated fractions.
  • We went over how to turn the fractions into degree measures.  How to use a protractor, and how to find the center of the circle.
  • When showing the pie charts on the screen I had them estimate how big different sections were (is the amount of time he spent playing video games a tenth or a twelfth) and that generated an amazing conversation when we decided one of the sections was between 1/4 and 1/5.  What fraction IS between 1/4 and 1/5?  One student suggest 1/4.5.  What IS that?  Is it halfway between?  Can we find a rule that gets us the half way point between any two fractions?
And all students loved the activity.  The lowest ability students could do it, and the higher ability students came up with lots of interesting questions.  The ones who liked geometry got to play with compasses and protractors and colors.  We all got to learn cool things about each other and tease each other a little. And I got to see everyone's comfort level with fractions and decimals and some long division.  A little geometry and algebra snuck in too.  WOOT!  Success.

With my algebra 2 and calc students I also did this answer key ethics worksheet that generated some interesting discussions.