G-GMD.B. Visualize relationships between two-dimensional and three-dimensional objects
We used the Mathematics Assessment Project formative assessment lesson on 2D Representations of 3D Objects.
As with other MAP lessons, students start with a pre-assessment (and they are given an opportunity to revise their work at the end of the lesson).
Next we moved to a class problem: The cylinder is full of water. This water flows out through a pipe at the bottom of the cylinder. Imagine looking down on the cylinder as the water flows out of it. Draw the shape of the surface of the water at five different levels. Will the radius of the shape change?
(The student who answered yes was including the pipe in his answer.)
And another: Draw the shape of the surface of the water at five different levels. Again, imagine looking down on the bottom cylinder. As the bottom cylinder fills with water, what is the shape of the surface of the water? Draw the shape of surface of the water in the bottom cylinder, at five different levels.
Student teams gets sets of cards and match eat set of shapes of the surface of the water with a top (water flowing out) or bottom (water flowing in) container. They draw in the shapes that are missing.
The MAP lessons include deliberate instructions to students on how to work together:
Some students used held models of 3D objects to help them visualize how the water level affected the size and shape of the cross section.
Some students didn’t need the models.
Teams worked well together, ensuring that everyone was participating and agreeing on the solutions.
The MAP lessons also include deliberate instructions for how to share work with other teams when it is time to do so:
I am realizing more and more that we have to deliberately teach our students how to work together. Speaking of which, have you read Elizabeth’s blog post on Rooting Out Blind Spots in the Language of Group Roles in Complex Instruction-based Group Work? I am excited to learn alongside Elizabeth and incorporate some of her ideas for team work this coming year with my students.
A few students commented on this lesson in their reflection at the end of the unit.
- The lesson on two dimensional representations of three dimensional objects was helpful because it gave me different visualizations in my head of different objects, which helped me be able to answer questions more precisely.
- Although I enjoyed the popcorn activity, I feel like the most helpful activity was the one where we cut out the cards and thought through the gradual shapes of the water as it emptied the top figure and filled the bottom figure. It allowed me to work with a team to reason out our arguments and working with my peers in a collective effort enlightened me with thoughts and ideas that I had not previously thought of or would have otherwise ventured to discuss.
And so the journey to not only provide opportunities for students to work together but to deliberately teach students how to work together continues …