This is a simple and engaging problem for students. It involves making cylinders out of two 8 by 11 sheets of paper. The first cylinder is constructed by bringing together the two 8 inch sides of the paper, the second is constructed by bringing together the two 11 inch sides of the paper. Which one will hold the most popcorn?

The first web above will take you to an NCTM Illuminations version of the problem with a full lesson plan, student handouts, extension and reflection questions. It also has students working with both rectangular prisms and cylinders.

Web link2 link above will take you to a three-act math task version of the problem, created by Dan Meyer.

Both versions are engaging in different ways and worth trying. The Illuminations version is nice because students are actually making the different cylinders. They also have a series of questions that follow the “experiment” that bring in the formula for finding the volume of a cylinder to understand what happened. I like their progression of questions, but I would probably break up the sheet they offer. Some of those questions would be better as “push” or extension questions and I would either ask them verbally, or prepare questions 5, 6 and 7 as part two, three and four.  The Dan Meyer version is great because it adds an element of drama and excitement and the questions come from the students.

Here is a possible progression for using elements of both in one lesson:

1. Launch: In order to engage and assess prior knowledge, give students a picture of a cylinder and ask them to “Take a minute or so and write down everything you can about this figure”. Ask for volunteers to share their ideas and record them on newsprint.
2. Show the same figures with numbers added and ask the same question.
3. Say, “I saw something that’s got me thinking about cylinders and I’d like to share it with you”. 
4. Show the Act One from the Dan Meyer Popcorn Picker video – http://www.101qs.com/868-popcorn-picker
5. After watching the video, say, “Write down one question you are wondering right now. We’ll come back to it in a moment”.   
6. Rather than jumping right into a problem, let learners engage in some sense-making. Ask, “What are some things you saw/noticed?” and record their responses.
7. Then ask, “Let’s go back to what you were wondering. Turn and tell a partner what you were wondering and see if they were wondering the same thing”. Then have them share their questions with the whole group and record them on newsprint.
8. Say, “These are all great questions. The one that I’ve been thinking about that I’d like your help on is this one – which one will hold more popcorn? Take a few minutes and discuss some predictions at your table.”
9. Take a vote:
   • How many people think the taller one will hold more?
   • How many people think the wider one will hold more?
   • How many people think they will hold the same amount?
   • How many people have no idea?
10. Rather then just give the relevant information, we want students to think about what information they would need and what they would do with that information. Ask “What information do you need from me in order to answer our question?”
11. Inevitably, students will ask to know the dimensions of the paper. Rather than tell them, tell them that on their tables they will find identical pieces of paper and tape. They should make the two cylinders.
12. As each group completes their cylinder construction, give them a bag of popcorn. Put the wider cylinder on the table and but the taller one inside of it. Ask them to fill the taller cylinder with popcorn.
13. Ask “What do you notice?” and see if anyone wants to change their vote. If anyone does, ask a few volunteers for their rationale.
14. Invite students to carefully lift the taller cylinder so the popcorn falls into the wider one.
15. Ask, “What happened? Is the wider cylinder full, not full or overflowing? Did every group find the same results?”
16. Ask if anyone wants to see if Dan Meyer got the same results as we did and play the Act Three video – http://bit.ly/2fXnCCV
17. Give out the handout “Find the volume of our cylinders” and give at least two rulers to each table.
18. To support students in figuring out what happen, you might share this chart, looking at Which Measurement Impacts the Volume More.
19. Push/Extension questions
20. Give students 10 minutes to put their findings on newsprint with the instruction that someone should be able to look at what they write and understand their thinking.
21. Give students another 10 minutes for a gallery walk, where they should write comments on the posters of their colleagues. The two prompts are “What do you like/appreciate about this work?” And “What questions do you have?”