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Two and Three Dimensional Shapes

Bell Ringer: How many lines of symmetry does a circle have? A. one B. two C. infinite number D.none of the above

This Friday we will not have P.A.T. time. We must cover this lesson today. We may have time for 2 days of P.A.T. time next week. Today we will work on objective 3d some more. We have worked on three dimensional objects— prisms, pyramids, cylinders, cones, and spheres. Who would like to share their homework from last night. Today, we will discuss congruence and similarity.

Congruence and Similarity Distribute geoboards and rubber bands to each student or pair of students, depending on the number available. Make a rectangle that is two units in height and 4 units in width on the geoboard using the rubber band. Then, make a rectangle that is four units in height and 2 units in length. What can you observe about the rectangles you have made? You have made two congruent rectangles.

Congruent – having the same shape and size For figures to be congruent, they may be turned differently, but they must be the same shape and size. We are going to go through several examples of different polygons. Corresponding sides and corresponding angles. For figures to be congruent, all corresponding sides and all corresponding angles must be congruent (the same).

Corresponding sides – sides in two figures that are in the same position in relation to the other sides and the angles of the figure Corresponding angles – angles in two figures that are in the same position in relation to the other angles and the sides of the figure

Often polygons may be the same shape, but slightly different in size. Create the following example using the geoboards. Make the same 2 by 4 rectangle. Then create a rectangle that is 1 x 2. The two rectangles are the same shape. The angles on both rectangles are the same (right angles). The lengths of the two rectangles are not the same. The widths of the two rectangles are not the same. The lengths and widths are proportional to one another. Tell students that these rectangles are an example of a pair of polygons that are similar.

Similar – having the same shape but different size. Two polygons that are similar because they are the same shape but different sizes. Similar figures may have different length sides (they must be proportional), but corresponding angles must be the same. Figures may be similar if they are turned differently.

Congruent:

Similar:

Congruent or Similar?

Are these triangles similar? Just because they are both triangles, they are not similar. Because there are different types of triangles and the angles can be different measures, all triangles are not similar.

Construct one pair of similar scalene triangles. Construct one pair of similar isosceles triangles. Construct one pair of equilateral triangles. Now construct a pair of triangles that are not similar.

1) How would the perimeter of two congruent figures compare? (The perimeters would be the same.) 2) How would the area of two congruent figures compare? (The areas would be the same.) 3) How would the perimeter of two similar figures compare? (The perimeters would be different, but would be proportional to the differences in the length of the sides.) 4) How would the area of two similar figures compare? (The area would also be different.) 5) Are all squares similar? (Yes, because all have four ninety degree angles, and the sides are always the same length in a square. Therefore, all squares are similar. This is also true for rectangles and circles.)

Closure: What is the difference between congruent and similar?