Lesson 2: Similarity and congruence
By: Tim Lehman
The lesson 2 is based on an exercise from an Addison-Wesley 1996 workbook, and one of the most famous cats in history......
For this lesson, students learn about similarity by constructing figures that are and are not similar. Even though students might not have been exposed to similarity before, most already know what it means. They realize figures that are similar have the same shape and congruent figures are "twins." The goal behind this lesson is students will get a better understanding of the mathematical definition of these terms. Three of the figures, Morris I, Morris II, and Boris, will be discussed here. Two others, Morris III and Doris, can be added (being (3x, 3y) and (x,3y), respectively). The figures can be easily plotted on GSP (as well as graph paper.) First, we will graph Morris I. Each figure has points broken up into 5 sets. The points in each set are to be connected. On GSP, simply add all the points of a set, press plot, and then construct segment. Repeat the process for each set. After graphing Morris II and Boris, compare and contrast them with Morris I.
In the chart below, the points are given for Morris I. Connect the points after plotting them, by constructing a line from each one to the next point on the list and connecting the last point of each set to the first one. The points of Morris I are considered (x,y). To find Morris II, multiply each x and y-coordinate of Morris I by 2. For Boris, multiply the x-coordinate of Morris I by 3 while the y-coordinate remains the same.The first couple have been done for you.
| Morris I | Morris II | Boris | |
| (x,y) | (2x,2y) | (3x,y) | |
| SET I | (5,0) | (10,0) | (15,0) |
| (7,2) | (14,4) | (21,2) | |
| (7,7) | |||
| (6,10) | |||
| (5,7) | |||
| (2,7) | |||
| (1,10) | |||
| (0,7) | |||
| (0,2) | |||
| (2,0) | |||
| SET II | (1,3) | ||
| (2,2) | |||
| (5,2) | |||
| (6,3) | |||
| SET III | (3,3) | ||
| (4,3) | |||
| (4,5) | |||
| (3,5) | |||
| SET IV | (5,6) | ||
| SET V | (2,6) |
Click on Morris I, Morris II, and Boris for GSP sketches of each.
Now, the students should compare their three drawings. Which one is similar to Morris I? Which one is not? How do the angles compare for similar shapes? What about sides? The angle measure is most evident to me at the point of the ears of the three figures. The nose of each drawing is a good place to compare side lengths and also shows equal angles do not mean shapes are similar.
