I don’t have a very long drive home every day, but even in my 10-15 minute ride, I’m always thinking about something. This change is weather is causing quite the migraine for me tonight, and while some of my thinking this evening was along the lines of, “Please don’t throw up!,” 🙂 I was also thinking about math. Why?
Today, my student teacher, Ashley, introduced the students to our Teapot Box Challenge (evaluation here). When she asked the students about calculating the perimeter of the net — one of the expectations — she noticed that some students had questions. She said that she’d do a mini-lesson tomorrow on this as the class was off to music.
During our prep time today, we spoke about this perimeter expectation. We thought that students need to realize that the perimeter is found by adding up the lengths of the sides along the unfolded net. Easy, right? This option definitely makes sense if we think of the net in the “unfolded” sense, with the prism or pyramid being the “folded” option.
Could the “net” also be referring to the prism or pyramid? If so, I have more questions:
- Would the perimeter refer to just the perimeter of the base as the “area around the outside” is really just impacted by the size of the base?
- Would the perimeter change depending on how the prism or pyramid is placed (e.g., in a rectangular prism, would the perimeter change if the prism is laid flat versus standing tall)?
I really don’t know the answers to these questions, and I’m hoping that one of my blog readers might be able to help me out. What do you think? How would you get students to uncover this learning? Is there just one right answer here or could there be many? Inquiry doesn’t just get my students thinking more, but it has me thinking more as well! Thanks, in advance, for sharing any thoughts (and maybe helping lessen at least some of my migraine). 🙂
Aviva
Hi Aviva,
To answer you question it all depends on what you are looking at. Is the net fully made? Then yes the perimeter is just the base. However, I think for your project the students will need to know the total covering perimeter. I can’t remember but I thought the student were to make nets that cover a teapot. In which case the practical application would be like a cardboard box and there you would need to know the total perimeter used because it takes up that amount of space on the sheet of paper (I could be wrong in this).
And yes I believe the perimeter could change depending on the type of net that you have created. This is a cool expectation to look at as periemeter doesn’t always rely on the area (big area doesn’t mean big perimeter).
Hope this helps.
Thanks Jonathan! Your comment has me thinking even more. Eventually, the group that has the winning teapot design (we’re having a vote) will need to enlarge their net to make the 3-D figure to use as the packaging for the teapot. Right now, they’re making models using pizza boxes. So, I guess if transferred to the big idea of where this project is going, then the net should really be the unfolded one (our first thought). Would there be any reason (given this project outline) that it would be better to look at the perimeter in terms of the base? Okay, I think that my “math migraine” just got worse … but in a good way! 🙂
Aviva
If it is just the base then you could think of shipping. No much space does it take up on the pallet. In this case it would be a perimeter and in a way area. How many can fit in this space? Do we have enough room? Etc.
Thanks Jonathan! What does your second sentence mean? I’m thinking there’s a typo somewhere, but I just can’t seem to figure it out tonight — sorry!
This latest comment is now getting me to think about perimeter versus area. Do students need to find out both? I guess so much depends on the context, and what we really want students to figure out and why. Who knew that this one little requirement would be so complicated?! I think that my new blog post title needs to be, “My Marvellous Math Migraine!” 🙂
Aviva
Sorry yeah small typo. It was how much space does it take up on a pallet (or skid). Sorry. I honestly don’t know how auto correct works sometimes. I would think that in grade five students should start to explore the relationships between area and perimeter. How do they differ? What is dependent on eachother? Are they the same? What makes the greatest area? What makes the greatest perimeter? Why do I need to know this? I think this teapot challenge can do a lot of this thinking has it has great context for the learning. Marylin burns has some great lessons on this. One is the perimeter of your foo, with string, then find the area. Then take the string and make a square. compare the new perimeter and new area, why isn’t it the same?
Sorry if this is creating more of a migraine. I love math a little too much.
Thanks for clarifying, Jonathan! This would be great, as we looked at many of these concepts earlier in the year when we focused on area and perimeter. (Our Board has a Math Delivery Plan that we have to follow, and area and perimeter is in Term 1.) That being said, this provides the perfect context to re-explore this topic and in a very meaningful way. I wonder what parts could be explored earlier on in the project, what could be explored throughout the process, and what could be looked at near the end. I emailed my student teacher the link to this blog (and the great conversation here) with the hope that she can also do some thinking about where she might want to go next. I think that your ideas will help her out as much as me! Thanks for getting both of us thinking! (A “math migraine” is the good kind of migraine. 🙂 )
Aviva
Aviva,
Like you say, the “perimeter” isn’t well defined when you take a 2D and make it into a 3D shape. It can be quite complicated, but opens some interesting avenues for exploration.
The first is the projected 2D area of the 3D shape. Which 2D area does the student want “projected”? Is it the covered floor area (which, like Jonathan said, makes the most sense if you talk about shipping)? Or is it the projected area on one of the sides, which might be important if you think about putting many teapots (in their boxes) into a larger 3D box? It’s also interesting to think about how the project area of a side for a rectangular prism compares to pyramid for packing purposes.
The next is more complex and looks at the relationship between 2D perimeter versus edge lengths in a 3D shape. This is a great puzzle and in upper years could be used for a good algebra challenge. At the grade five evel it could still be used to ask, “What happened to the perimeter, why doesn’t the 2D perimeter and 3D total edge length equal each other?”
– With all of the sides of a 3D “box” laid out on a surface, then each side is a part of the perimeter. If you want the concept of starting with a 2D form that easily folds into a 3D shape (i.e., like a ready-made packing box or filing box) then the six sides of a rectangular prism are already attached along 5 of the edges (folding corners). In its 2D shape, you would count each exposed side to calculate the perimeter.
– When you fold it into the 3D shape every exposed edge will make contact with another exposed edge. If you measure all of the edge lengths of the 3D shape, then the total length will be less than the perimeter of the original 2D shape.
The mathematical relationship between the 2D perimeter and 3D total edge length can be well defined for a particular 3D shape (i.e., a rectangular prism, pyramid or a tetrahedron). At the more basic level, however, it might be a good exercise to simply ask why the 2D perimeter and the 3D edge length are different and to explore how the sides share an edge.
Cheers,
Greg
Thank you so much for the comment, Greg! I totally love hearing your thinking. I’m excited to share your comment with Jonathan too. When my student teacher and I were talking this morning about this, we thought that it would be neat to have students compare perimeters and see if they can figure out why the perimeters change (between the net and the prism). You’ve asked some wonderful questions here that I’m eager to ask the class as well. And just as a thought, if you ever want to come in and do a little “guest math mini-lesson,” the classroom’s always open. Please let me know! I’d love to have you!
Aviva
Love the comments that Greg left. Didn’t even think about the relationship between the 2d 3d. Great add on. It would be interesting to expose them to it as it does come into play in later yrs. love community learning.
I totally agree, Jonathan! My student teacher, Ashley, and I actually spoke about this today as well. I think this would be the perfect extension and really get students thinking. I love how blogging can lead to such a great discussion and so much learning for all!
Aviva
Thanks Aviva.
I think my “expertise” is more on the applied-math side of things – y’know: structures and bone mechanics.
Thanks for the reply, Greg! With your expertise in structures, any chance that you might be free for a Skype call next week if students had questions as they work on their “bridge task” for structures? Just a thought …
Thanks again!
Aviva
Sure, no problem.
Depending on the day I might be able to come in. I’m currently available Monday, Thursday and Friday.
Greg
Thanks for letting me know! I’d think maybe Thursday. Let me see. Depending on the number of questions, we could arrange a visit or students could maybe tweet you too. Let me email my student teacher and talk to her about this tomorrow, and I’ll be in touch! Thank you so much!
Aviva