Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
INTERPRETATION OF OBJECTIVE - G.GMD.A.1
This objective is all about developing and deriving the formulas for circumference, area, and volume. We want students to understand the formulas and not just simply use them.
(1) The student will be able to use informal and formal arguments to explain where the circumference, area, and volume formulas come from.
(2) The student will be able to use Cavalieri's Principle to relate volumes of different solids.
(3) The student will be able to use informal limit arguments to discuss the formula for the area of a circle and volume of a pyramid.
THE BIG IDEA
Formulas don't just appear... they come from a mathematical foundation. In this objective we build the conceptual understanding of where these formulas come from. Sometimes the development is very clear and in other case our arguments will be a little more informal due to the complexity of its 'proof'.
TRAPS & PITFALLS
Regular polygons have always tripped students up. The use of special right triangles and possibly trigonometry are road blocks for those that didn't understand them earlier. Break volume problems into two parts: calculating the base area and then calculating the volume.
Carelessness is paramount in this unit. Too many students get working on prisms and cylinders and become comfortable with the simple formula and then too often forget to take 1/3 of that volume for pyramids and cones.
The trapezoid is the single most confusing area formula that students learn. Take time to derive it and discuss where the variables come from. There is a great activity to help you with this.
The student needs to have had experience with perimeter and area from earlier years. The foundations of area definitely provide essential concepts to build on when looking at volume.
Area and volume are topics that continue at the next levels. Optimization questions, area under the curve, volume through rotation all come in later math and this objective sets the stage for them.
MY REFLECTIONS (over line l)
I have always loved this unit. I'm not sure what it is... maybe all the diagrams. Over the years I have learned that if a student is able to derive the formula then they are more likely to remember it correctly and USE IT CORRECTLY. I always spend time look at dissection and shearing as ways to visualize area.
While objective spoke only specifically to a few formulas, I have decided to expand that to others because they are essential foundational skills. I'm sure that the objective expects students to know how to work with a regular polygon but it has been my experience that they won't and our earlier work on special right triangles gets a nice review.
I have always used the classic pouring visual to demonstrate the 1/3 relationship between prism and pyramid and between cylinder and cone. It is a very clear and powerful way to informally determining the formula.