"Um, I don't know..." one student replied as she continued building the model.
"How is this model going to help you solve our problem?"
"We're going to count up the cubes and figure out the area."
"Ah ha, we're trying to figure out the area. Okay. But is your model the same size as that table? Will they have the same area?"
At this point in the conversation I found myself guiding students to think about scaling concepts, in terms of multiplication and division. Not the original plan when I started teaching the lesson, but that is the beauty of a 3-Act math task-- the open problem solving and inquiry-based format means that the math instruction can often take spontaneous turns in whatever direction the students' work and conversation leads us.
|Act 1 "Mystery Picture"|
In this particular task ("Piles of Tiles" by Graham Fletcher), students were presented with a picture of a cross-shaped table on which a handful of 1 inch squared math tiles were placed along the edge of the table. The question-- are there enough tiles in the bag to cover the whole table?
The classroom teacher had asked me to come in and demo a 3-Act. Her goal is to help her students become better problem solvers in math, to help them think more creatively in math and to engage them in more meaningful math tasks.
This was the students' first 3-Act task, and most did have a hard time getting started. It's still not typical for a math teacher to ask students to dive into a problem with little guidance and without a textbook chapter title hint at the top of a worksheet cluing them into which operation(s) to use. When I asked students what they thought we should do first, I received quite a few blank stares. Fair enough. The one thing most students did know right away-- they wanted to use the manipulatives I had put out!
Linking cubes were everywhere as students started recreating the shape of the table. They weren't quite sure why they were building the table, but they did know that building that table and filling it in with cubes should help them. It was from this "playing" that the conversation above was spawned-- I asked students to think about the dimensions they were given (60 tiles long), and then think about a number that could be useful when building their models.
- One group said "6!" right away, seeing the 6 in the 60, but it took some prying for them to think about 6 as a factor of 60, and that being the reason why 6 could make a good size scale model.
- Another team, overhearing my conversation with the other, then suggested that 12 cubes as the length of their scale model might work because it is also a factor of 60-- nice!
One student in the class started solving the problem by drawing a model and breaking the cross-shape into five small squares. From there she confused the dimensions, but her thinking was on the right track. It took everything I had not to give her too much information and to instead ask the right questions to get her thinking about where her mistakes were.
We ended my demo lesson that day without closure. We did take time to reflect on our work up to that point, and shared some of the students' thinking so far, but we still had not solved the problem-- another jarring situation for the 4th graders. They expected me to give them the answer that they hadn't figured out yet-- not so much. Instead, their classroom teacher excitedly explained that she would be taking over the lesson the next day, and would let the students continue their work on the problem. The students' reactions? Cheers and applause.
That's right, 4th graders were cheering for math.