During the first week or so of the school year, I take time to review some basic concepts from previous years with my Grade 9 students – in particular, integer operations, fractions, and fraction operations. The challenge is designing tasks that embed review and practice within problems or activities that also call for generalization, problem solving, or engagement with novel or less-familiar mathematical ideas. I deem this necessary because, of course, not all students come to high school with the same understandings and strengths, and while some do need the opportunity to review basic concepts, others are ready to move on. For the latter group, review activities are at a high risk of being perceived as baby-ish and not worth the time, which means that I need to design these tasks very carefully – that is, in such a way that I can simultaneously assess the needs of students who are still struggling with basic concepts and challenge students who are ready to learn something new.
I’d like to share one task that, I think, had something to offer to all of my students: Integer Bingo, which is based on a task developed by and discussed in Serradó (2016). I chose this task for several reasons:
- It gives students an opportunity to review integer operations, and me an opportunity to assess their understanding and note any misconceptions in this area.
- The review is embedded within a task that challenges students to wrestle with the less-familiar concepts of randomness, relative frequency, theoretical probability, and elementary outcomes.
- It’s easily adaptable (to suit a variety of grade levels and to target a variety of concepts) and extendible. Not only does it offer practice with a targeted concept, it can serve as a launch into the study of probability.
- Who doesn’t like bingo?
Here’s how the task was enacted in my class.
Preparation: In preparation for the task, I marked two sets (one yellow and one white) of nine ping-pong balls with the integers from -4 to +4 and placed each set into a paper bag. (A random number generator such as this one can also be used, but I elected to use physical random generators for a particular reason, which I’ll discuss below). I also generated a 4×4 bingo card for each student in my class using Excel, with each cell containing a number that is the sum of two random integers between -4 and +4. Numbers may be repeated. (Click here to download a set of 38 unique cards.)
Game play – Round 1: Unlike in most bingo games, where the goal is to get a line, students’ goal in Integer Bingo is to get a “blackout” (to mark off all of the numbers on their card). During the game, I repeatedly draw, with replacement, a number from each bag. If the sum of those numbers is on a student’s card, they can cross it off; if it appears more than once, they cross it off only once per draw. Draw, add, repeat, until someone crosses off all of the numbers on their card and calls “bingo!”
The review, obviously, is in determining the sums (another skill can easily be targeted by changing the operation, or changing the numbers – of course, this changes the possibilities for numbers on the cards). At several points during the first round, I asked students to explain how they reasoned through the computation. The numbers are small, but the task still allowed misconceptions to surface – e.g., “Two negatives make a positive, right?” Important conversations to have at the start of the year.
Design: After the first round and the first “bingo!,” which took somewhere between 5 and 10 minutes (next time, I’d like to time it for comparison), I presented students with the following challenge: “In your groups, design a card that you think will be most likely to win. You must be able to justify your choices of numbers.” I handed out blank cards, whiteboards, and markers.
Circulating, I noticed that some groups filled out their cards in a matter of several minutes or less. Here’s an example of one of these:
When I asked them about their choices, many students stated that “these were the numbers that came up most often” – in other words, their arguments were based purely on the observed frequencies, rather than on an underlying theoretical probability distribution. This might explain why, for example, -1 appears more often on the above card than 0 and 1. Other students seemed to be looking beyond the observed frequencies to the underlying distribution, referring to “more possibilities” or “more ways to get” certain numbers, but couldn’t clearly articulate their reasoning beyond this when prodded.
I brought the class together again to discuss these issues. Through the conversation, we were able to establish that one way to reason about more likely outcomes is to list all of the ways that the integers -8 to 8 can be produced as a sum of integers between -4 and 4. This discussion served as a theoretical foothold, allowing students to articulate their intuitions through the mathematical process of determining possibilities. I offered students time to explore these possibilities, probing and challenging their thinking as I interacted with the groups. Because many of the groups had already hastily filled in the card I handed out at the outset of the construction phase, I eventually handed out a second card for them to fill out using what they had discovered. (Next time, I will hand out the blank cards near the end of this phase.)
Game play – Round 2: Once all of the groups had finalized their cards, we played the game again. I allowed student groups to play with both of their cards (mainly because I was worried about time; otherwise, I would have asked them to choose the one they thought would be more likely to win).
As most groups had filled their cards with 0, ±1, ±2, and ±3, there were audible groans around the room when sums such as 8 and -7 came up – sometimes, several times in a row. Again, we played until a group crossed off all of the numbers on their card. This round was noticeably shorter than the first (but I would like to time both rounds next time to compare).
Discussion: During the post-game discussion, I asked students to explain why, if 0 and ±1 were so likely and ±7 and ±8 were so rare, we still observed them several times during the drawing process, which led us to discussing the distinction between possibility and probability: “It’s rare, but it can still happen.” We also grappled with the question of whether -7 and -8 were equally likely outcomes, which hinged on the question of whether (-3,-4) was the same outcome as (-4,-3) (a question I had posed to several of the groups during the design phase, which sparked some debates among the students). “Yes, because it’s the same sum,” argued several students; “no, because you can get it in two ways,” argued others. Having physical random generators was helpful during this particular discussion, because the drawing process was transparent. As one student eventually explained, “to get -7, you can either draw a yellow -3 or -4 from the first bag, and then you need to draw a white -4 or a -3 from the second bag. But to get -8, you need a -4 from both bags. So -7 is more likely.”
Shortly after this point, the class drew to a close.
Next week, as we continue our review, I am going to extend this task by having students design cards most likely to win if the numbers drawn are multiplied. Since students will be familiar with game play, and because I would like them to consider not only the probabilities, but also the possibilities themselves, we won’t start with a practice round.
To reiterate, this task fulfilled my criteria for a back-to-school review activity, but also has other bonus features that made it stand out to me as a Good Task:
- It gave me an excuse to briefly delve into the topic of probability with my students, which is unfortunately not part of the Math A90 curriculum. Those who are lucky enough to be teaching probability may find this to be a fruitful space to explore several important concepts related to the topic; I could see this task being stretched over several days, as I suggest below.
- It gave my students an opportunity to review a basic concept (integer operations), and gave me an opportunity to assess their understanding and note any misconceptions in this area.
- The review was embedded within a task that challenged students to wrestle with less-familiar concepts (in this case, randomness, relative frequency, theoretical probability, and elementary outcomes). As a result, it was appropriate and engaging for most students in the room – and additionally so, because there were stakes attached to designing a card well: namely, winning the game. (Although I’m not into bribing students into learning, I will note that when suckers are on the line, students tend to pay attention.)
- It’s readily adaptable and extendible. Next week, we will consider possibilities when integers between -4 and 4 are multiplied; the task can be made even more interesting when the operation is randomized as well (e.g., use a die with multiplication, addition, and subtraction symbols). Imposing the winning arrangement to be a line, an X, or an L (for example) adds yet another layer to the task. If, on the other hand, your students are just starting to explore the concept of probability, you might wish to (as in Serradó, 2016) introduce another round between the introductory round and the construction phase, where students must choose between two pre-made bingo cards, play the game with their selection, and reflect on the outcome before designing their own card. In this context, I think the task could be easily stretched over several days, with several iterations of choose-play-reflect and/or construct-play-reflect that give students opportunities to develop increasingly sophisticated reasoning about randomness, uncertainty, and probability.
If you have other ideas for how this task can be adapted or extended, I’d love to hear them. I’m looking forward to trying out different variations with my students as we warm up for the new school year.
Serradó, A. (2016). Enhancing reasoning on risk management through a decision-making process on a game of chance task. Paper presented at the 13th International Congress on Mathematics Education, Hamburg, Germany, July 2016. Available at https://iase-web.org/documents/papers/icme13/ICME13_S13_Serrado.pdf