The main topics we study in this unit are:
* various angle relationships
* drawing geometric figures, including basic geometric constructions
* pi and the area of a circle
* slicing 3-dimensional solids
* surface area and volume.
In the first lesson of the unit, we examine various angle relationships: angles that are formed when several rays originate from the same starting point, vertical angles (formed when two lines intersect), and corresponding angles (formed when a line intersects two parallel lines). Then, the lesson Angles in a Triangle presents and proves the well-known result that the angles in a triangle sum to 180 degrees. With this knowledge, students are now able to solve various problems that involve unknown angles.
Next, students practice drawing geometric figures. Basic geometric constructions are done just like in ancient times: with only a compass and straightedge (a ruler without measurement units). These constructions help students to think about the main defining features of a figure. Personally I have always enjoyed geometric constructions because they are like little puzzles to solve.
Students also draw figures using a normal ruler and compass in the lesson Drawing Problems. They especially determine whether the given information defines a unique figure (triangle or a parallelogram).
Then we turn our attention to pi. Students first learn the definition of pi as a ratio of a circle’s circumference to its diameter in the lesson Circumference of a Circle. Then they learn and practice how to calculate the area of a circle in a wide variety of word problems and applications. We also briefly study the proof for the formula for the area of a circle. I feel it is important that students encounter justifications for mathematical formulas and procedures and even read some proofs before high school. We don’t want students to think that mathematics is only a bag of magic tricks or formulas to memorize that seemingly came out of nowhere. Proofs and logical thinking are foundations to mathematics and school mathematics should not be left without them.
After this, we slice three-dimensional solids with a plane, and learn that the result is always a two-dimensional shape. Students see that in a concrete way by slicing cubes and pyramids made of modeling clay. Some Internet links (provided in the lesson) will also help students to visualize what happens when a solid is cut with a plane.
In this unit, students also solve a variety of problems concerning surface area and volume and practice converting between various units of area and volume. While these topics tend to involve lots of calculations and less possibilities for hands-on activities, they are very important in real life.