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3 Acts - Pop Box Design

2/4/2012

29 Comments

 
Ever wonder why companies make the decisions that they do? My wife and I drink more pop than I am willing to admit, and one thing I noticed while at the store is that the twelve packs of Coke and Pepsi do not have the same design.  Let's look at them (warning I do not know if this works in the States). This is also a precursor to this lesson.

Act 1 - The Boxes

I asked my students which one uses the least amount of cardboard, and in relation, which company made the best choice for the environment?

This was a good starter.  I had kids talking about which looks bigger, and a trend over at least two classes (I'll see later on in the future), is that the majority of students say that Pepsi is the clearer waster, or they have equal amounts of carboard.

Act 2 - The Measurements

I ask students what do we need to know to solve this?  Students came up with volume, area, dimensions etc.  at which point I introduce them to our cm cubes.

Picture
We talk only a little bit about the difference between volume and surface area.  In fact I do not use the words in particular, we talk only about "How many cubes are necessary to build this prism?" and "How many squares can we count on the outside?" an idea I lifted from Christopher Danielson.  I gave students the following sheet and basically said, GO!
surface_area_investigation.docx
File Size: 19 kb
File Type: docx
Download File

surface_area_investigation.pdf
File Size: 198 kb
File Type: pdf
Download File

Here I talked to students a lot about short cuts.  It was great when they said they had no short cuts, and I ask, "You just counted?"

"Yes," replies the student.  I then proceed to count one-by-one each outside square.  Before I can even get to three, the student interrupts and says, "No, no, no, you times the length by width for one side, then times it by two.  You do this for three of the sides, since they each have an equal side" (paraphrased).  Bwa ha ha! I laugh to myself, they have it, and I didn't have to write a single formula on the board!

Bringing it back to the pop box now only involves asking, what do we need to know to solve the pop box, problem.  Students tell me they need the dimensions of the box so I give them these...

Act 3 - The Reveal

This is also a good time to have students talk about anything we are missing.  My wife after a few days of seeing me make this asked about the overlap, and if that makes a difference, I haven't looked into it yet, let your students do it too!

Sequel

  • The boxes  are designed with the following can configuriations...
How many different configurations can you find, and which one of these uses the least amount of cardboard? Do any of them use less cardboard than Pepsi and Coke's configurations?
  • Write a letter to one of the companies that explains your mathematics.  Suggest for them why you think they should switch or keep their design.

Reflections and Moving Further

This was extremely engaging for one of my two classes.  I think for me the pay out of the reveal is not as great as most of these 3act stories, so it can lose some students.  I want to work on that out for next year.

What else I need to do is to prepare more guiding questions to start off the investigation, so students can be invested in the problem earlier.  They have not seen surface area before and so are not acquainted with it.  Students knee jerk reaction at this age is to find volume (well actually they just say lxwxh without knowing what that means), so I want to guide them away from that. If they focus on volume to early on they count the problem off to early.

What I really liked about this investigation was the next day when I wanted to teach them surface area of triangular prisms I handed them the nets, and asked what do we have to do now. Students quickly came up with the idea that we needed to find the area of each separate shape and them together.   They knew the only formula they needed was to find the area of the triangle. We talked about the relation between rectangles and triangles.  This was a fairly easy process, because students can see if you cut a rectangle down the diagonal it makes two triangles (therefore bh/2).  This was a fairly simple review of the area of a triangle and extension of their learning of Surface Area.  It was fantastic to see how quickly they adapted to the new information! Students were able to further adapt this to other nets that used basic shapes, and it was a very simple natural extension of the same logic (Thanks to Kate Nowak for the idea of nets and surface area).

My only hiccup came with cylinders.  I was ill prepared to lead students into a discovery of the area of a circle (technically they learned it last year... but they're only in grade 8), so I fell back onto lameness. I hope to alleviate the lameness, any suggestions? I have learned that great teaching comes from thinking through all aspects of the lesson and leading them through the inquiry.  My cylinders lesson bombed, because I was lazy and had little prepared!  I was humbly reminded how easily it has to slip into bad teaching, let's change that Timon!
29 Comments
Damian Watson link
2/4/2012 07:52:58 am

Another classic, I really enjoyed this approach to surface area. I generally use a multi-link cube exercise where students make a cube of there choice and work out the surface area - this is way more exciting. I think Act 3 is brilliant - I particularly like the way you have cut of the excess and put it together. Well done.

On the cylinder - I use a toilet roll or kitchen roll centre (cardboard) and cut it apart.I can't see at the moment how you could make a 3-act out of it but at least it lets then see it and allows both visual and kinaesthetic learners an opportunity to make some connections.

I ask my students to put together some digital photps (of there shapes) and present how to work out the area & surface area of them. I am trying to get them to use higher order thinking skills by doing this - hopefully I will see some good results?

Great work.

Reply
Blair Miller
2/4/2012 11:24:59 am

Brilliant stuff! Full disclosure, I'm "stealing" this for use in Math 8 in about a month. Act 3 is the icing on the cake, then you kick in a stellar sequel. This got me wanting to produce something again, it's been too long for me. Thanks for being an inspiration and look forward to collaborating again sometime soon.

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Timon Piccini link
2/4/2012 11:37:30 am

Thanks Blair, and Damian!

I appreciate the encouragement. Just let me know how it goes for you guys, and if you think of anything we can add.

Reply
Brian Miller
2/5/2012 02:35:11 am

This is really nice stuff - exactly what I was looking for.

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Brian Miller
2/5/2012 02:37:54 am

Can you post a .pdf of the handout? My .docx converter is not transferring this document correctly.

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Timon Piccini link
2/5/2012 04:40:45 am

Done and done.

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Peter (@polarisdotca) link
2/6/2012 01:44:51 am

In the Sequel, you give us the volume and ask for surface area. After the finding an answer, it's pretty interesting (and probes much deeper into mathematics, I think) that they may be more than one answer! There are clearly (at least) 2 ways to package 12 cans. Hmm, what about 13 cans? I wonder if you could get the students thinking backwards like this on the Worksheet by strategically filling in the volume column and/or the surface area column and getting them to find the dimensions. Nice opportunity to check and discover with their neighbours.

Great activity! I have 2x12 pop boxes at home. Next time we buy a 4x6 case, I'll be keeping the boxes and setting my kids (my actual kids, not students) loose on them!

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Timon Piccini link
2/6/2012 04:42:30 am

I'm glad you like it. My favourite moment in this whole things was one of my ESL students came up to me and showed me that he had figured out the volume, and he was blown away that they were the same. It was a pretty awesome moment, that I hadn't even asked him to solve.

I like the idea of giving the volume and finding the surface area. That's a pretty big jump for these kids, but it could be super interesting. I would also get them to make a table of different dimensions (maybe get them to graph it on an ipod or some other three dimensional graphing program).

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Brian
2/18/2012 03:16:05 am

Do you recommend groups of 2, 4, or having students work alone?

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Timon Piccini link
2/18/2012 04:37:11 am

I would say anything more than two, and you have someone who is riding on the coattails of the other. The investigation portion students usually do on their own, but I encourage partner work.

Reply
susan russo
6/29/2012 12:45:37 am

I'm a little late to the party here, using summer vacation to read some great math blogs. I did a related activity with a sophomore Geometry class a couple of years ago, but instead of giving them all the same product, I gave each group different things with different numbers and told them THEY were to design the packaging. One group, for example, had 24 golf balls. Another group had 10 baseballs. One group had 64 crayons (definitely the hardest), another had 12 rolls of toilet paper, another had 6 rolls of paper towels, etc. I gave them all the dimensions of a single item, and then told them their parameters: They were going to stock a shelf with their item at a store. The shelf was 36" long and 18" high. The goal was to design a package that would adequately display as many of the packages as possible, but the design should also be pleasing to the eye. They were to look at surface area but also volume (I had them calculate how much empty space was inside their package and how much empty space was on the shelf.) Then I made up prices for packaging material: Shrink-wrap plastic vs. cardboard. They had to incorporate the prices into their designs, too: would the costs be restrictive? Would they/could they/should they change from one material to another? We figured on no overlap with a cardboard box (no flaps) and that the plastic would fit perfectly. It was a great project for them - they immediately went to work. Should we do 4 x 3 x 2 or 6 x 2 x 2 or 1 x 12 x 2 or.... One group didn't write any of their calculations down on paper but instead filled up my entire board with work and discussion and then took a picture of it for posterity! I *almost wish I was teaching geometry again to refine the project.

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Laura
6/20/2014 04:28:17 am

Susan, this is such a cool project! I teach Alg II, and I'm teaching Geometry for the first time during summer school. Is there any chance that you'd be able to share those materials? I'd love to see them. Many thanks, and have a great summer! My email is lcutrona(at)es-cat(dot)org.

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kathryn labiuk
9/17/2019 05:16:23 am

Hi Susan - Would you be willing to share a word doc for this activity? Sounds really great and a perfect extension for what we are doing right now.

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Kathryn
2/25/2020 10:14:42 am

Hi Susan! Do you have this as a word document? Would really like to see how it was all laid out. This is an awesome extension!

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Allison Cope
4/7/2021 07:00:26 pm

Can anyone share the resources with me on this project? I teach PreAP Geometry and think this is a great idea!

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Morgan
4/22/2022 08:35:39 am

This sounds like a great project. My students are just now getting into surface area and volume and I want to do a hands on activity. Would you be willing to share this lesson plan with me?

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Rachel Tabak link
8/8/2012 09:14:42 am

I'm also late to the game! I wanted to share a resource in response to your plea for help re: surface area. The teacher in this video (http://goo.gl/KufSM) does some things that I'm not so into (he needs to learn to be less helpful!), but I do think there's a small stroke of genius around 15:30. Check it out. And--thanks for sharing this fantastic write-up and set of resources.

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Mike Wiernicki link
3/18/2014 11:44:19 am

Excellent 3 Act! I've thought of trying to compare boxes, but the coke/Pepsi 12 pack comparison is extremely cool. Nicely done! I like how the open ended ness of the follow-up leads to other configurations of cans

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Lisa Bejarano link
3/19/2015 11:12:36 pm

This is a great activity for introducing Surface Area with meaning rather than formulas! I wrote a blog post about it here:
https://crazymathteacherlady.wordpress.com/2015/03/18/introducing-surface-area-with-pop-box-design/

Reply
Allyson
2/8/2017 09:08:44 am

I really like this. I am going to adapt it for my grade 7 math class. I am a teacher in Ontario, not too much where your posting from (my apologies). You mentioned cylinders... in a previous year I have brought in Pringles containers and had the student take them apart/ investigate. We discuss whether the container makes the most sense and what other connections do we notice.
Thanks
Allyson.

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John Golden link
10/23/2017 05:42:25 pm

We did this as an investigation. First: surface area & volume of cube buildings. 2nd: approximate V & SA of 1 can with cubes. 3rd: look at the pop cases. Some did by making cube models. My challenge question (since they were done more quickly) was about how they were using cubes to represent cans, but the actual cans are taller than they are wide. Does that make a difference? Others measured what would be the actual box with cubes. 4th: measurements in cm to figure out the surface area precisely. This was way more challenging than I thought with all the experiences to that point. Pictures: https://twitter.com/mathhombre/status/922602707390730242

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Catherine Tuczek link
10/27/2017 10:12:25 am

Hello! My name is Catherine Tuczek, and I am Curator of School and Public Learning at The Henry Ford, a museum in Dearborn, MI. We have been working with teachers on resources to accompany our new exhibit Mathematica. They found your great lesson plan on pop box design. If possible, we would love to include it in our resources, giving you full attribution. Can you please get in touch with me to discuss whether you'd permit this? Thanks, and looking forward to hearing from you!

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Kristen
3/23/2018 01:27:21 pm

Silly question but is there any way you can post the video on YouTube? Our school has Vimeo banned. Thanks!!

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Timon Piccini link
3/23/2018 01:51:45 pm

There should be a link on the Vimeo page to download it when you are home. I don’t really have a YouTube presence.

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Kara Rehbein
4/29/2020 06:49:37 am

I did not see the option to download the video. Would you be willing to enable that? And, thanks for sharing your ideas. I'm excited to try this with my students.

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Laura
5/29/2020 03:41:38 pm

I tried as many ways as I could, and there was no direct link from Vemeo. I copied a picture and gave it a title. I could copy paste the URL from the Vemeo page where the video was played, and link that to the title in my powerpoint slide. When you click it, you go to the Vemeo page, advertising and all.

For the dimensions piece, I used a snipping tool to put each pic on a separate slide.

I would love to embed the video in my slideshow. It such an amazing activity and my kids would learn so much!! I wish there were easier access to show the videos. The kids stay more engaged when I can simply hit the play arrow instead of being rerouted.

But please don't loose the main idea. I am so excited to do this with my kids. It will be really interesting and a great, hands on, relevant way to explore surface area and nets.

The activity is culturally relevant as well. Nicely done.

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Tonia Kuehn
3/3/2021 08:00:58 am

Didn't have cubes. This interactive was a life saver.
https://www.nctm.org/Classroom-Resources/Illuminations/Interactives/Isometric-Drawing-Tool/

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Mary (Cassie) Winchell
5/4/2021 08:54:01 am

Thank you SO much for putting this out for your colleagues' use! The first time I used all your components as-is, I was bummed that my students' "wonders" weren't about smart packaging, or surface area, or cardboard, or efficiency, or wasted space. They were wondering "Why are we watching this? and "What's with the polar bear?" So with later classes I did the asking myself, like you did: "Which company's packaging choices are better for the environment? For their own company from a cost standpoint?"

I also couldn't get my first student groups to use the info from Act 2 to attempt to answer the question. But they DID once I changed Act 2 from a video to a static page that contained information snipped from Walmart's website. I Googled "dimensions of coke 12 pack" and one of the options was a link to Walmart where they gave specifications that included dimensions. Did the same for Pepsi. I have kids drawing pictures of the boxes, labeling them with dimensions, etc.

Another thing that I'm glad I did is show them using cans of soda how the cans of soda are oriented in Act 1.

It helps, I think, that we have Schoology, which allows us to make it so that students have to get past checkpoints to get to other materials. So my students had to make a guess at which box was designed "better" before they could see Act 2. They have to answer some questions about "How many square inches of cardboard are needed to make the Pepsi box?" etc. before they can see Act 3.
After Act 3 they'll have a checkpoint called "Grand Finale" where I ask them to reflect on whether their prediction was correct, and THEN I show them an article about Coke changing their box design and we talk about that decision:
https://medium.com/@ryansheffer/technology-is-a-12-pack-of-coke-6e50f3232e9e

I've had a hard time getting a lot of my students to do asynchronous work (We have 50% of our students in attendance each day, on alternating days) but I can tell that some students who haven't gotten to Act 3 are going to keep progressing on their own because they are excited.

THANK YOU!

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anonymous link
3/2/2022 10:43:58 am

hi

Reply



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