Additional Lesson Summaries

Geometry Unit: Extending to Three Dimensions

Lesson#1: What Is Area?

Lesson Summary

Standards: 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.


The goals of this lesson are for the students:


  • Students review the area formula for rectangles with rational side lengths and prove the area formula for an arbitrary rectangle.
  • Students use a square grid to estimate the area of a curved region using lower approximations, upper approximations, and average approximations.


It shows how finding the area of a curved figure can be approximated by rectangles and triangles. By refining the size of the rectangles and triangles, the approximation of the area becomes closer to the actual area. Students experience a similar process of approximation in Grade 8 in order to estimate pi

Lesson#2 What are the Properties of Area

Lesson Summary

Standards: 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.


The goals of this lesson are for the students:

  • Students understand properties of area:

1. Students understand that the area of a set in the plane is a number greater than or equal to zero that measures the size of the set and not the shape.

2. The area of a rectangle is given by the formula length × width. The area of a triangle is given by the formula 1 2 × base × height. A polygonal region is the union of finitely many non-overlapping triangular regions; its area is the sum of the areas of the triangles.

3. Congruent regions have the same area.

4. The area of the union of two regions is the sum of the areas minus the area of the intersection.



In this lesson, we make precise what we mean about area and the properties of area. We already know the area formulas for rectangles and triangles; this will be our starting point. In fact, the basic definition of area and most of the area properties listed in the student outcomes above were first explored by students in third grade. Since their introduction, students have had continuous exposure to these properties in a variety of situations involving triangles, circles, etc. It is the goal of this lesson to state the properties learned in earlier grades explicitly. In that sense, this lesson is a summative experience for students rather than an introductory experience.

Lesson#3 What are the Scaling principle for Area

Lesson Summary

Standards: 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.


The goals of this lesson are for the students:


  • Students understand that a similarity transformation with scale factor r multiplies the area of a planar region by a factor of r squared.
  • Students understand that if a planar region is scaled by factors of a and b in two perpendicular directions, then its area is multiplied by a factor of ab.


In Lesson 3, students experiment with figures that have been dilated by different scale factors and observe the effect that the dilation has on the area of the figure (or pre-image) as compared to its image. The move will be

made from the scaling principle for area to the scaling principle for volume. This shows up in the use of the formula V=BH; more importantly, it is the way we establish the volume formula for pyramids and cones. The scaling principle for area helps us to develop the scaling principle for volume, which in turn helps us develop the volume formula for general cones