As I mentioned in my previous post, the start of the school year in my Grade 9 class is dedicated to reviewing some basic concepts from elementary school – in particular, integer operations, fractions, and fraction operations. I try to embed these skills as components of tasks that ask students to make decisions, generalize, problem solve, and/or engage with novel or less-familiar mathematical ideas – or, as Nat Banting (who does this so well) describes it, “embedding atomic skills into tasks so that the basic skills are developed and used as tools of mathematics, rather than the ultimate goal of mathematics.” At the beginning of the year, these kinds of activities allow me to simultaneously identify and assess the needs of students who are still struggling with basic concepts and to challenge students who are ready to learn something new.
As a way of reviewing integer operations, we spent some time playing Integer Bingo, which I wrote about here. A task involving Fraction Talks images served as a re-introduction to fractions, and was followed by a clothesline number line task to elicit reasoning about comparing and ordering fractions. When we moved on to fraction operations, I used Nat’s (Min + max)imize structure (see also 16 Boxes) and the Open Middle idea to design (I use the term loosely, because the general idea is not mine) the following task:
Students draw four boxes and an operation to create a fraction expression (determined prior to the game), as above (image credit to Open Middle). Before I roll four dice, we decide on whether we are aiming for the largest sum, the smallest sum, or the sum that is closest to 0. After each roll, students must make a decision about where to put the number, and cannot change their minds once the number has been written. (With my students, I use two dice that have the numbers 1, 2, 3, -4, -5, -6 and two that have the numbers -1, -2, -3, 4, 5, 6.) After four rolls, they determine the sum. I ask volunteers to share their expressions and their thinking, and we work together to determine the answer that is largest/smallest/closest to 0. Repeat, changing the goal and/or the operation after one or several rounds.
Although I had planned the task to last only about 20 minutes, it ended up sustaining productive discussion about fraction operations for an entire class period, with both of my Grade 9 classes (and for some time the next day, too). Both misconceptions and strong reasoning emerged, especially when the task was modified so that a subtraction sign separated the fractions and when a box was added before the first fraction to create a mixed number. The next day, I extended the task by rolling the 4 dice simultaneously and asking students to find (and provide justification for) the biggest/smallest/closest to 0 sum or difference (again, determined prior to rolling) that was possible by forming two fractions with the 4 numbers.
Because the previous activity took longer than I had expected, I didn’t get a chance to try out a new task that I had planned, also for reviewing fraction operations, inspired both by (Min + Max)imize and… The Price is Right. I am sharing it both to document it for the future and in the hopes that you will try it with your own students and report back with feedback!
Again, at each round, four dice are rolled (and again, I use the integer dice described above), giving students four numbers to create two fractions with (modify the number of dice and the numbers on the dice as desired). Each student uses these dice to make two fractions, which they secretly add; mini-whiteboards would work well here, so that students can hold them up for all to see at the end. Then, a final die is rolled (say, with 0,1,1,1,2,2 on the sides). This die is the target number (“price”). The students now compare their sums. Whoever is closest to the target number, without going over, is the winner. Points might be allocated based on whether or not the target number is reached exactly (e.g., 10 points for hitting the target number exactly; 5 points for being the closest, without going over).
I like the idea because it invites students to reason about fractions in a way that goes beyond drill (“& kill”), because they must think not only about how to perform the operations involved, but also about how to form fractions that will meet the constraints of the game (the sum must be less than or equal to 2) and about the risk of forming certain fractions. E.g., a larger sum is more likely to win if 2 is called, but this outcome is less likely than 1. 0 is an even less likely target number, but a student who chooses the strategy of always making a fraction below 0 (if possible) is guaranteed to never overshoot the target. What other strategies might emerge? And do they change as the game goes on? I anticipate the task to elicit interesting discussions about risk and probability, in addition to fraction operations. Once students have played a few rounds, the game could be played in partners.
I’m not sure that I’ll have time to try out this task during the current unit, but I do hope to try it sometime this semester. I’d love to hear about it, along with any feedback or extensions, if you give it a go with your own students! Finally, if you’re interested in more great fraction tasks for students, you will find a wealth of ideas here. (And have I mentioned www.fractiontalks.com?)