I spend two days with my students discovering volume. I do this because the volume formulas LOOK horrendous, however they make perfect sense. I like for my students to discover these formulas, because then they do not think of them as formulas. It just makes sense.
The first day I do volume of prisms, rectangular, triangular, and a cylinder using play-doh. The next day I do cones and pyramids. A few years ago I had a project where students made cones that fit into cylinders perfectly. I still use these models as they are such a great visual for students.
I start by showing them a cylinder, and having a student write the volume formula on the board. I have not asked them to memorize the volume formula of a cylinder, but after day 1, they all know it. It’s magical. I then pull a cone out of the cylinder and ask them what they notice and wonder. They notice that the circle has the same circumference and the height is the same. Excellent. I then ask them what they wonder about the volume of the two objects. Most groups decide that it is about 1/2, except for this group.
I then let them pour cheerios (or marshmallows) from the cone into the cylinder. They notice it takes THREE cones to fill up the cylinder. I have them talk with their table to adjust the volume of a cylinder. Today, ALL of my students said, “Divide the volume of a cylinder by 3!” and one group even told me that was the same as multiplying by 1/3.
After I write their observations on the board, I show them the actual formulas from their Geometry Booklets and ask them to write down what they learned today on that page. I think this is the most important part because I don’t want them to forget HOW they discovered the formula. Today a student said, “We just invented already invented math!” It was so awesome I had to film it, of course.
Now that is a magical day in math. Please, please, please do discovery learning with your students. It does take more time but will actually will save you more time in the end because you won’t have to re-teach it!
I Speak Math 
Can I get an amen ? 🙂 I need to find a way to make this happen with our students who don’t even get that volume is 3D because they’re so disconnected.
I personally hate all volume equations. I only see 3 different ones, volume of any prism or cylinder which is area of base times the height. It’s just stacking cubes that are 1 unit by 1 unit by 1 unit. The next is anything coming to a point like pyramid or cone that is one third of the prism with the same base as these students found out. The last is the volume of a sphere which is 4/3rds the volume of a cylinder with the same base and the height being the radius.
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You said that years ago, you had students make cones that fit perfectly into the cylinders and that you used these as models. How did you make these models? I have tried several times but they are never perfect enough to hold 1/3 the volume of the cylinder.
Hi Chris, That is a great question but I don’t remember. I believe we traced the circumference of the top and measured the height. Then used card stock. I remember wrapping it around in a cone and taping it. We even made “lids” for the tops of the cans. We also covered the paper in foil to make them more durable. They were not perfect either, but close enough for kids to realize it was 1/3.