Every student has a gut feeling when it comes to probability, and I feel like I have been too quick to theorize their gut instincts in the past. This year to introduce Grade 9 probability, I decided to exploit gut feelings to introduce the topic.
To do this, I needed a semi-familiar situation, some friendly competition, and a time pressure to make decisions.
To fit these criteria, I invented the Dice Auction.
You are invited to an auction, and given a budget of $10. Everyone at this auction has an identical budget. You all are bidding on possible events when two 6-sided dice are rolled. After all the spots are auctioned off to the highest bidders, the two dice will be rolled 20 times. Each time the event you purchased occurs, you collect a single prize.
Bidding always begins at $1 and goes up in increments of $1. You cannot bid against yourself. The order of the events up for auction will be known beforehand. If you choose not to spend your money, the auctioneer will sell you a prize for $2 after all the bidding is completed.
Your task: optimize the number of prizes you receive.
To begin class, I informed the students that they were attending the auction described above. I had a class of 22 students so I came to school armed with about 150 total "prizes" (an assortment of candies). I counted out groups of 10 pattern blocks which became our currency. Each student also received a single handout with this description of the events up for auction on the front and this set of assessment questions on the back.
I gave them five minutes to decide which events they wanted to target. I also warned them that sharing their reasoning at this point might result in auction sabotage. I don't usually discourage the sharing of reasoning, but I wanted to delay it in this instance. I wanted to accentuate each student's individual gut feelings.
After the five minute preparation time, the auction began. I opened each item at $1 and two pre-service teachers (@Mr_Harms_ & @HeidiLNeufeld) collected the funds after each event closed. It took about 30 seconds per auction event, and the presence of more hands made collection much easier.
Once all the spots were taken, I gave the students time to answer the assessment questions. After that, the dice were rolled 20 times and the prizes were distributed. Students kept track of how many times their purchased events were rolled (in the tally section of the handout) and later we combined these results so everyone had the data for their reflection questions.
Bidding began rather timidly, and that enabled some to pick up spots for very cheap. I quickly realized that their perception of probability was very skewed. Certain events (like both number greater than or equal to 5) were selling for higher prices than others (like a single one is rolled). I didn't bother mentioning this in the heat of the action, but as the finds were being collected, I made comments like, "wow, that went fast" or "I thought more people would want that". These were more or less done randomly with no attention to the chances of them actually being rolled. I wanted students to think and re-assess on the fly.
I could see student marking certain events from further down the list as they lost out on ones they wanted. I assumed they were trying to find events that had similar likelihoods of occurring. The patterns of bidding became recurrent, with the same students jumping in early, and others braving the high amounts. The $5 plateau seemed to slow down many bidders.
I allowed students to buy prizes at the rate of $2 per prize because I wanted to offer a way for the very reserved students to take in the action, and still receive some candy. As it turns out, many used this strategy to guarantee at least one candy. I had two different students mention they were going to spend $8 and leave $2 as a guarantee. This struck me as interesting.
I began by asking who felt they "won" the auction. Naturally, those with the largest stack of prizes asserted that they had. Others then claimed that they just got lucky. This was exactly what I was hoping for. The pressure of the auction format had forced students to think on the fly about the value of certain events. Naturally, missteps were made. We decided that it may be unfair to call the winner lucky until we determined how likely his events were to be rolled.
One student had won the event "A single one is rolled" and it occurred 11 times. I began by asking students how many possibilities for rolling two dice exist and this led us to the inevitable conversation about a (2, 1) being unique from a (1, 2). I anticipated (and even prompted them toward) this.
We ended up listing all 36 possibilities in an ordered pair structure (no one suggested we use a table). After we had that, students were quickly testing to see if they overpaid. I purposely chose 20 rolls because it necessitated students to compare fractions with denominators of 20 and 36.
I gave the student four assessment questions; they were briefly introduced before the period ended. I essentially wanted them to reflect on the events, and their likelihood. I got an incredibly high percentage of assignments back the next day, and they included very insightful comments:
" 'Numbers multiply to a prime number' was underpaid for because I think many thought that you can't multiply to a prime. We forgot about 1"
"I knew I was guaranteed 6 because of 'missed three in a row'. I am happy with this, so I didn't overpay"
"I've used dice a lot, and they have bigger chance of landing a 6 or a 1"
"Sum to 8 occurred less often because I mean it's a number that's in the middle with many chances"
" 'missed three in a row' was overpaid for because you can only win 6 prizes. You should invest somewhere with more potential"
" 'No one else collects' was rolled less often than I thought. It surprised me that someone collected every round."
It was clear to me that students continued to think about their actions, and some even provided how they would approach the auction differently. It might be a nice extension to ask them to "set fair prices" for all 30 events.
I love the conversations surrounding this task. Students were upbeat, blaming each other for overpaying, and then justifying why they weren't. I used the happenings to introduce the idea of favourable events, total events, probability, odds, and even got into the fundamental counting principle. For my Grade 9s, that is almost the entirety of the curricular outcomes (although we play with them a lot longer because... it is fun).
This task slowed the theorization of their curiosity. By allowing them to follow their gut instincts, the decisions became personal and the task became vibrant. Notation arose out of the necessity to communicate how they conceptualize chance. That is the starting point for any unit on probability.