G.GMD.A.1 -- 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.
G.GMD.A.3 -- Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
G.GMD.B.4 -- Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
(1) This week is all about building foundations in area. There are two facets to the concept development learning about the origin of the formulas for area and then applying them to solve problems. While the objective speaks very specifically to circumference and area of a circle we will look at many others as well. It is essential to have a strong grasp on all area formulas before starting into volume.
(2) The deep foundations of area will make the calculation and understanding of volume much easier so this week we will continue to look at area relationships. In this week we develop formulas for trapezoids and regular polygons. Regular polygons have been historically a difficult topic for students. We need to spend this time so that when we attempt to find the volume of a hexagonal prism we will have the tools to do it.
(3) With the foundation of area well laid, we are ready to head into volume. In this week we begin working with prisms and Cavalieri’s Principle. Prisms and cylinders are the easiest for students to work with. Their congruent cross sections help students to understand the formula as well as calculate the volume. We will elevate the volume of prims questions by doing composite prism shapes.
(4) We continue as expected into the volume of pyramids and cones. We will use Cavalieri’s Principle and a spreadsheet activity to look at the 1/3 relationship. In an attempt to approach the 1/3 relationship differently (that using the pouring demonstration) we will use technology to help us see 1/3 through an informal limit arguement.
(5) This is a cool week!! We learn all about cross sections and visualizing 2-D forms in 3-D shapes. I have created a very fun activity using a computer applet to learn all about the cross sections of a cube. Finally we finish with rotating cross section to form volume.