This lesson helps students start talking about the two dimensional faces of a three dimensional object. This is the second day of a 2 day lesson (Exploring Flat Patterns) and it gives students an opportunity to practice visualizing and talking about individual faces of the "box". It not only helps support student learning of surface area but also the area of composite figures because as students discuss the problems and the necessary modifications they are referring to each face separately.
It is also really important for students to develop a positive attitude about mistakes, so I like to give them opportunities to investigate, learn from, and correct mistakes. Whenever we work with errors I like students notice what is correct about them as well and why a person might have started out with this mistaken idea.
In this warm up students are presented with two flat patterns that will not fold perfectly into a box without gaps or overlaps. Student are asked why they would not work and how they might be modified. Initially students are not given a copy of the patterns, but are asked to write silently for two minutes about what they think. They are told they will be given time to discuss and explore their ideas to help their group decide. This gives them the chance to think about how they might investigate their modifications. Some might want to cut out the patterns and try to fold them, others might want to draw them on grid paper and visualize the folding, etc.
After the silent write time I ask students to share with their "math family" group and discuss their ideas. I remind them in their discussions to ask each other for and provide evidence or clarification and to disagree politely. Sentence frames and starters taped to their desks still help with this.Table top sentence starters As I circulate I may need to probe with questions to get the conversation moving by asking questions like "what do you think of her idea?", "Would it help if you asked her to show you what she means?", etc. Prompting a group discussion This type of questioning helps students to listen to, critique, and develop an argument.
It is important for me as the teacher to remain impartial while listening to their ideas. While I am using the information to assess and guide my own teaching I don't want to give students any indication as to the "correctness" of their responses. If the teacher interjects judgement of right or wrong it may stunt the argumentation and investigation process. We get to the heart of the practices by not telling students what they can figure out for themselves.
Once students have had a chance to discuss and explore their ideas with the group I ask them to share with the whole class. I may call for a volunteer to explain or show why the patterns would not fold perfectly into a box or I may call on specific students depending on what I heard from the small group discussions. My job then becomes one of facilitating feedback by asking "what do you think about this new idea?", or "discuss with your group whether Dale's idea will work."
The main ideas I want students to come away with are that the "tabs" need to close both ends of the box and there needs to be no overlapping. In the last lesson (exploring flat patterns) students looked at a given box with specific dimensions, but in this lesson I wanted them to focus on the more general parameters of a flat pattern. This is another experience that helps them visualize and transition between 2nd and 3rd dimension.
At this point I display a second Activity prompt asking students to draw a flat pattern that will fold into a new box with given dimensions. I instruct each member of the "math family" group to come up with a different flat pattern. Students will have a conversation about what constitutes same or different patterns. I leave that up to them and say they just have to convince the group that theirs is different. This is a nice way of having students practice providing evidence.
As I circulate I focus more on questions like "how can you tell the flat pattern will fold into the given box?", or "how do you know that each of these patterns will work?" This focuses the attention on the number of faces, the dimensions of the faces, and the orientation of the faces.
For their exit ticket I ask students to do a 2 minute silent write to answer the following question: Do you think everyone's flat pattern for this box has the same area? I keep the visual model displayed and remind them to explain their reasoning. I prompt them with the idea that while everyone's pattern may look slightly different there are many things that are the same about them and I want to know if area is one of them.
After they share their explanation with their "math family" groups I give them more time to modify their ticket and explain why they made the changes. I expect some students will need reminding about the definition of area and say the patterns are not the same because they look different or that they are the same because they fold into the same box. I am hoping to surface the idea that they have the same area because each pattern contains identical faces each with the same dimensions.
As I listen to their "math family" group discussions I may need to guide the conversation back to area. "Remember, I'm not asking if they fold into the same box here, we know that they do, I'm asking if the patterns have the same area." "Can anyone argue that they do have the same area?", "How might you measure the area?", etc.
They may be left in some confusion but that is okay. I want them to enter class in the following lesson (Making Sense of Surface Area) with the idea of area in mind.