I created the "Which One Doesn't Belong" discussion prompt for a professional development session with Mathematics 10 teachers. It has quickly become one of my favorite versions of this prompt. Participants were quick to notice that there was a triangle that was impossible. I could watch the room and see their puzzlement slowly turn to a realization that there were several impossible triangles in this collection. This generated a lot of valuable discussions about a variety of different characteristics of triangles. The conversation typically started with participants noticing that triangle C was the only triangle that "worked." In fact, with each group of teachers that I worked with, triangle C was the most commonly picked as the one that didn't belong. Then we went around the room and talked about what made each of the other triangles impossible. What makes it impossible?Triangle A - The angles of triangle A say "I'm an equilateral triangle" but the sides of triangle A say, "I'm an isosceles triangle." This disagreement is what makes the triangle impossible. Triangle B - The 90 degree angle says that this triangle is a right triangle but the sides are not a Pythagorean Triple. Most participants, remembering the 3-4-5 Pythagorean triple, said that the side that said 12 would have to be 10 to make this a right triangle. That would make this an x2 enlargement (similar triangle) to the 3-4-5 triangle. Triangle D - Participants noticed quickly that the angles in this triangle did not add up to 180. Less people noticed that this triangle also fails the Triangle Inequality Theorem which states that the sum of the lengths of two sides of the triangle is always greater than the third side. In this triangle 4 + 7 is not greater than 11 so this triangle would actually be a line. (Note that some versions of the Triangle Inequality Theorem allow the degenerate case where the sum of two sides is equal to the third side). What else makes it not belong?We can also talk about other reasons which each triangle doesn't belong. Triangle A is the only non-right triangle. Triangle B is the only triangle that doesn't have all three angles given. Triangle C is an vertex on the bottom instead of a horizontal side. Triangle D is the only triangle with odd side lengths and without a side of length 12. Effective FacilitationI've changed how I facilitate discussions using the "Which One Doesn't Belong" prompt. As described in Geoff Krall's book Necessary Conditions, how a teacher facilitates a task is just as important as the quality of the task. The region that I work in has the benefit of having a Diversity Team with several Culturally Relevant Pedagogy Specialists. Working with them has shown me that we often don't need a different lesson to be culturally responsive. We instead need be intentional about how we might facilitate a task through the stance of being a culturally responsive practitioner. I started this discussion by asking each participant to individually take a sticky note and decide which one they think doesn't belong and their justification. This let each individual have some time to consider their own reasoning. After recording their thoughts, I asked them to discuss their choice and their justification with others at their table. Did they choose the same triangle? If so, did they have the same justification or a different one? This allowed each participant to communicate their mathematical ideas and reasoning. Often, participants were sharing something that other people hadn't noticed. This allowed them to be positioned as someone with worthwhile mathematical ideas. After sharing, I asked each participant to walk to the screen to place their sticky note on the triangle that they had chosen. This gave everyone a chance to move, even if just a little bit. This gave me, as the facilitator, insight into what ideas the participants had come up with. A sign of a good WODB prompt is that at least a couple of people select each different triangle. Creating your own WODB promptCreating a WODB prompt can take some time. If you're looking for some ideas to help you get started, there are lots of examples at https://wodb.ca/. This website was created and is maintained by Mary Bourassa (@MaryBourassa). Christopher Danielson, who wrote a children's book featuring WODB images, titled Which One Doesn't Belong? A Better Shapes Book, gave this advice in a recent "My Favorite Theorem" podcast: "If you ever try to design a “which one doesn't belong” set, what you want to do is think about whatever your domain is, so say it's shapes, you want to think about four properties of shapes, and then cover up the first one, and design one that has these three, but doesn't have the first one. And then cover up the next one, design one that has those three, but doesn't have this one. And by the time you're done, you'll either realize that your set of four properties is more intertwined than you had originally thought, and now you’ve got to go back and revise, or you'll have a set where you know for sure that there's at least one reason for each not to belong. But then extra, an important key to this is that you have to be open to the possibility that some kid will see a reason for a shape to not belong that wasn't the reason you'd intended." EL
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