LESSON 1: Similarity in Right Triangles

Description: On lesson one of our unit, we will introduce students to similar right triangles via the Geometer’s Sketch Pad. First we will have students use GSP so that they can have a visual understanding of the properties that are needed in order for two right triangles to be similar. Namely, that two right triangles need to have one other angle congruent (besides the right angle). Then, in our GSP activity, we will have them use the distance feature so they can see the ratios that exist between corresponding sides of similar right triangles. After this (possibly the next day), we will go over what students have done using GSP.

Objectives: 1. Students understand the relationship(s) that exist(s) between a right triangle inscribed inside another right triangle as shown in the picture below.

2. Students understand geometric mean using similar right triangles. Namely that (segments) AD/BD = (segments) BD/DC in the picture above.

3. Students can do general problems where side length values are given. An example would be. Given that segment AD = 3m and that segment DC = 12m; what is the length of segment DB?

4. Students can do application problems and/or real life problems that incorporate similarity between right triangles.

Standards: This lesson relates to both the NCTM and the ISBE Standards in many ways.

NCTM:

Geometry:

1. Students will be expected to show that they understand the mathematical relationships that exist between similar right triangles. For example, they will be expected to use the Angle-Angle Postulate in order to show why two similar right triangles are in fact similar and that ratios exist between side lengths in similar right triangles.

Reasoning and Proof:

1. Students will be expected to prove that, if given a right triangle as shown below. The two right triangles that are created when the altitude from point B (the intersection point of the two sides) is drawn to the hypotenuse (which is segment AC) are similar to one another and are similar to the triangle that they are inscribed in (in this case, triangle ACB).

Connections:

1. Students will learn that similarity in right triangles interconnects in that similar triangles have all the same angle measure and that their sides have related lengths.

ISBE:

STATE GOAL 9: Use geometric methods to analyze, categorize and draw conclusions about points, lines, planes and space:

  1. Students will recognize and apply relationships within and among geometric figures by seeing that the two inscribed triangles and outside triangle are all similar to one another.
  2. Students will construct logical arguments for geometric situations using technology by using GSP in order to prove similar right triangles.

Materials: Geometer’s Sketch Pad, Computer Lab, Computer with Projection Capabilities, Student Worksheets

TEACHER’S ACTIVITIES / STUDENTS’ ACTIVITIES
1. Take Attendance. / 1. Meet in computer lab.
2. Preview of lesson to be done. / 2. Listen attentively.
3.Pass out assignment. / 3. Open GSP and look at assignment.
4. Help students through worksheet. / 4. Work on worksheet and ask questions.
5.Pass out homework at the end of the day. / 5. Wrap up computer work and get homework assignment.

Accommodations: For students with special needs, I will have the Special Education Teacher translate all of the written material into Braille for those students that have visual disabilities. Also, for these students, special computers will be set up so that if needed, they can work by themselves at a computer. Lastly, I will set up groups where students with troubling typing will be paired up with students that work hard and that will explain things.

Assessment: In order to assess my students, I will look at what they have done for their homework and see what knowledge they gained as a result of my lesson. Their grade will be out of three points. Zero points if they did not hand it in. One point if they handed it in and tried most of the problems. Two points if they handed in the assignment fully completed with a few major errors. Three points if they handed in the assignment fully completed with (at most) a few minor errors.