How do you help your students make sense of volume formulas?
We start with a prism and think about slicing the prism into cross sections that are all congruent to the base. While we can think about an infinite number of these horizontal slices that fit into the prism, they are limited by the height of the prism. Using the formula V=b*h, where b represents the area of the base, seems to make sense to my students.
What about a cylinder? It doesn’t take long for them to recognize the similarities between a prism and a cylinder. We can still use V=b*h, it’s just that since the base is always a circle, b will always equal πr2.
What about a pyramid?
We compare the Power Solids for the square pyramid and the square prism. How are they alike? They have congruent bases. They have equal heights.
How many times will the pyramid fill the prism?
Three students have it right. All of these students used the formula for the volume of a pyramid to calculate volume when they were in middle school. Three of them might understand that formula.
I have a bucket of water for the demonstration. I fill the pyramid with water and pour it into the prism. (I do realize this would be better letting the students do the work … there’s always next year, right?)
Students look and wonder. Will they be correct? I fill the pyramid with water a second time and pour it into the prism.
Is it full? Not yet.
I fill the pyramid with water a third time and pour it into the prism.
The volume of the prism is b*h. How many times will the pyramid fill the prism? What is the volume of the pyramid? (1/3)b*h.
We look at the cylinder and the cone. How are they alike? They have congruent bases. They have equal heights. How many times will the cone fill the cylinder?
What is the relationship between the formulas for the volume of the cone and the volume of the prism? Again, we don’t have a new formula to memorize.
The cone works like a pyramid, it’s just that since the base is always a circle, b will always equal πr2.
And so the journey continues … and maybe eventually my students will have made sense of these formulas before they get to my high school geometry class?
Easing the Hurry Syndrome 
Pat Flynn
April 26, 2014 at 8:20 pm
Nice to see others do this too. Many teachers I know simply hand out the formulas and go. It is SO important that students make seance of the formulas. A quick test of student understanding is asking a student to find the surface area of a compound shape (like a cone on top of a cylinder.). Students who do not understand the formulas will have the area of two extra circles.
jwilson828
April 26, 2014 at 10:43 pm
Great idea. Thanks, Pat. I think it is so interesting that all of my students can tell me the formula for the volume of the square pyramid, and yet only three students can tell me that the pyramid will fill the corresponding prism (congruent height, congruent base) 3 times.