Reflecting on the Rules Teacher Notes
Topic Area: Reflections of Polygons
NCTM Standards: Understand and represent translations, reflections, rotations, and dilations of objects in the plane by using sketches, coordinates, vectors, function notation, and matrices. Use Cartesian coordinates and other coordinate systems, such as navigational, polar, or spherical systems, to analyze geometric situations.
Objective The student will be able to use the properties of reflections to use the matrix function of a calculator to represent the vertices of a polygon, write a reflection as a product of two matrices, find the reflection of a polygon across a given line, and write a translation as a series of reflections.
Getting Started As a class, discuss the definition of a reflection. Ask students to give examples of reflections in everyday life. Provide picture examples of reflections from various areas such as nature, quilting, corporate logos, and other areas. Discuss with them various career areas that use reflections such as architecture, photography, sports, and graphic design.
Prior to using this activity: Students should have an understanding of the coordinate plane. Students should have an understanding of operations with matrices. Students should know the matrix rules for reflections across the x-axis, y- axis, line y = x, and the origin point.
Ways students can provide evidence of learning: The student will be able to write a matrix equation for a reflection for a given problem. The student will be able to determine the line of reflection. The student will be able to relate reflections to other translations.
Common mistakes to be on the lookout for: Students may use the wrong line for a given reflection. Students may confuse the x and y values in the reflection. Students may use the incorrect matrix rule. Students may enter a matrix into the calculator incorrectly.
Definitions Reflection Matrix, Rotation Line of Reflection Translation Point Reflection Activity 4 • Geometry with the Casio fx-9750GII and fx-9860GII Reflecting on the Rules “How-To”
The following will demonstrate how to enter a set of coordinates into a matrix using the Run-Matrix mode on the Casio fx-9750GII and apply this to reflections. Using the vertices of (1, 1), (3, 4), and (6, 2), find the coordinates of the image under the given reflections. a. Create a reflection matrix for the x-axis and a matrix for the polygon. b. Reflect the points across the x-axis.
To create a reflection matrix and a matrix for the polygon:
1. From the Main Menu, highlight the RUN•MAT icon and press l or press 1.
2. Press q(MAT). With Mat A highlighted, press l and press 2l2l to set up the dimensions.
3. Press l and then enter the values of the matrix to be used for a reflection across the x-axis by pressing 1l0l0ln1l.
4. Press dNl to prepare to enter the matrix for the ordered pairs. Enter 3l2ll and enter the x-coordinates in the first column and the y-coordinates into the second column.
To reflect the figure across the x-axis:
1. Press d twice to return to the initial RUN•MAT screen. Enter iw(MAT)q(Mat)ag(B) mqaf(A)l.
2. To store this answer, press lq(Mat)Ln (Ans)bqaG(C)l.
Activity 4 • Geometry with the Casio fx-9750GII and fx-9860GII To draw the reflection:
1. Press l and enter w(Mat)q(ML)a G(C),1kbiq(List)1l to store the x-values in List 1. Repeat this process to store the y-values in List 2.
2. Press p and highlight the STAT icon and press l. Enter the first pair of coordinates at the bottom of the lists so that the graph will be a polygon.
3. Enter Le(V-Window)q(Init)l to set up the view window. Press q(GRPH)u(SET)N w(XY)Nq(List)1lNq(List)2l lq(GPH1) to graph the to graph the polygon.
4. Press Lw(Zoom)r(Out)l to see the graph as shown.
Activity 4 • Geometry with the Casio fx-9750GII and fx-9860GII Reflecting on the Rules Activity
Many designers use reflections to create patterns and obtain symmetry. It has been shown that symmetry is very pleasing to the eye and thus it is used in many architectural designs. Take a look at some buildings that you find pleasing and see if this is not true. In this activity, we will look into the relationship between reflections and other forms of transformation. We will also take two simple polygons and create a quilt block that could be incorporated into a bed cover.
Questions
1. What are the coordinates of the vertices for N M ABC? C ______
Enter these into a 3x2 matrix. Find coordinates for L the images for each of the following reflections using A B the proper reflection matrix. P
2. a. Across the x-axis ______
R Q b. Across the y-axis ______
c. Across the origin ______
d. Across the line y = x ______
3. Draw each image and label the vertices.
4. How many reflections of ABC are needed to equal a 90 rotation of the same triangle? ______
5. Reflections over which lines will give a 90 rotation? Graph this on the calculator. ______6. Which reflection is equal to a 180 rotation? ______
Activity 4 • Geometry with the Casio fx-9750GII and fx-9860GII 7. What is the maximum number of reflections needed to translate ABC to PQR? What is the minimum number? ______
8. What reflections are needed to create LMN? ______
A quilt block is made of two pattern pieces. The first is a triangle with vertices at (2, 0), (6, 2), and (6, 2). The second is a parallelogram with vertices at (0, 0), 4, 2), (6, 6), and(2, 4).
9. Reflect the triangle across the origin point, the y = x line, and the line y = x. What are the new coordinates? a. Origin Point: ______b. Line y = x: ______
c. Line y = x: ______
10. Plot all four triangles.
11. Reflect the parallelogram across the y-axis, x-axis, and the origin point. What are the new coordinates? a. Origin Point: ______b. x-axis: ______c. y-axis: ______
12. Plot all four parallelograms.
Activity 4 • Geometry with the Casio fx-9750GII and fx-9860GII Solutions
1. A(2, 1) B(6, 1) C(6, 5)
2. a. A’(2, 1); B’(6, 1); C’(6, 5)
b. A”( 2, 1); B”( 6, 1); C”( 6, 5)
c. A”’(2, 1); B”’(6, 1); C”’(6, 5)
d. A””(1, 2); B””(1, 6); C””(5, 6)
Activity 4 • Geometry with the Casio fx-9750GII and fx-9860GII 3. N M C d. b. L A B
P c. a.
R Q 4. Two reflections
5. Line y = x and y-axis
6. Reflection across the origin point.
7. three; two
8. Reflection across the y-axis and then a reflection across the line y = x. or Reflection across the line y = x and then a reflection across the y-axis.
9. a. Origin Point: (2, 0), (6, 2), (6, 2) b. Line y = x: (0, 2), (2, 6), (2, 6) c. Line y = -x: (0, 2), (2, 6), (2, 6)
11. a. Origin Point: (0, 0), (4, 2), (6, 6), (2, 4) b. x-axis: (0, 0), (2, 4), (6, 6), (4, 2) c. y-axis: (0, 0), (4, 2), (6, 6), (2, 4)
Activity 4 • Geometry with the Casio fx-9750GII and fx-9860GII 10. & 12.
Activity 4 • Geometry with the Casio fx-9750GII and fx-9860GII