A collection of Lesson Plans for the Unit on Area.

- Education > General
- Mathematics > General
- Mathematics > Geometry
- Mathematics > General
- Mathematics > Geometry

- Grade 9
- Grade 10
- Grade 11
- Grade 12

**Topic**: Area of Parallelograms and Triangles

**Learning Objectives:**

*The Student Will*derive area formulas for parallelograms and triangles*The Student Will*apply and extend basic area formulas.

**Motivational Problem**: Draw a rectangle and a series of "knocked-over rectangles" (parallelograms). Compare and contrast the areas? On what does the area of the parallelogram depend?

**Overview**

The key techniques for this topic are applying the basic postulates of area to finding the areas of parallelograms and triangles. By "cutting and pasting" the parallelogram, and applying the Area Congruence Postulate and the Area Addition Postulate, you can see that the area depends on base and height. Then, by noting that any triangle is really half a parallelogram, the formula for the area of a triangle follows immediately. These proofs offer an excellent opportunity to integrate simple, narrative proofs from the early part of curriculum into a new topic that students should already have familiarity with.

Immediate applications include finding the formula for the area of a rhombus, finding the areas of right triangles and equilateral triangles, and developing strategies for finding the areas of isosceles triangles.

Once you have the formula for the area of a triangle, you can find the review the techniques of finding the area of of arbitrary polygons in the plane by "completing the rectangle" and subtracting off right triangles.

**Key Questions**: How does the area of parallelogram relate to the area of rectangle? Why is the area of a triangle half base times height? What are the strategies for find the areas of various triangles?

**Looking Ahead:** Decomposing figures into triangles will be key in finding areas of trapezoids and general polygons.

**Topic**: Area of Trapezoids and Regular Polygons

**Learning Objectives:**

*The Student Will*apply techniques of area to find formulas for the area of polygons

**Motivational Problem**:Find the area of the trapezoid whose vertices are (0,0), (10,0), (8,3), and (2,3).

**Overview**

Use the motivational problem to review the techniques of "completing the rectangle" in the xy-plane, and decomposing the trapezoid into a rectangle and two triangles by dropping altitudes. Highlight the trapezoid as a sum of two triangles by connecting opposite vertices with a diagonal, which leads to the elegant result.

Highlight the same decomposition techniques with regular polygons. Have students verify that decomposition from the center yields congruent, isosceles triangles, thanks to the consequence that regular polygons can be inscribed in circles. Pay special attention to the hexagon (decomposed into 6 equilateral triangles) and the square (decomposed into 4 isosceles right triangles).

**Key Questions**: How do we find the area of a trapezoid? Give a geometric interpretation of the average of the bases of a trapezoid. Why can a regular polygon be decomposed into congruent, isosceles triangles? How might we estimate the area of a circle?

**Looking Ahead:** As the number of sides in a regular polygon increases, the figure becomes more and more like a circle, and the apothem becomes more and more like a radius.

**Topic**: Area and Arclength of a Circle

**Learning Objectives:**

*The Student Will*apply formulas for area and perimeter of a circle

**Motivational Problem**: Have the students find the area of several regular n-sided polygons inscribed in a circle of radius 1. If the students know trigonometry, they should be able to do this without assistance; otherwise provide them with the appropriate apothem lengths.

**Overview**

Have students find areas for several values of n: say, 4, 5, 6, 7, 8. Have them try a very large value--say, n=50, and then n=1000. Ask them to draw this polygon--it is virtually indistinguishable from a circle, and its area (when rounded) is approximately 3.14. **Note** that this is *not *a proof that the area of a circle is pi times the radius squared, just an investigation. A proof would have to come much later, requiring much more advanced mathematics.

For sectors and arc-length, the key is to remember that in each case, we are merely looking at some *portion* of the full circle. A proportion will always be used to compute sector area or arc-length--the question to be asked is always "What percentage of the full circle am I using?" The answer to that question, along with the area or circumference of the original circle, will be sufficient to find sector area and arc-length.

**Key Questions**: How can we approximate the area of a circle? How can we approximate the perimeter of a circle? What is the strategy for finding arc-length and sector area?

**Looking Ahead:** Arc length and sector area are important ideas in trigonometry, especially when it comes to the concept of *radian measure* for trigonometric functions.