Basic Understandings


 Joan Cannon
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1 Activity: TEKS: Exploring Transformations Basic understandings. (5) Tools for geometric thinking. Techniques for working with spatial figures and their properties are essential to understanding underlying relationships. Students use a variety of representations (concrete, pictorial, numerical, symbolic, graphical and verbal), tools, and technology (including, but not limited to, calculators with graphing capabilities, data collection devices, and computers) to solve meaningful problems by representing and transforming figures and analyzing relationships. (G.2) Geometric structure. The student analyzes geometric relationships in order to make and verify conjectures. The student is expected to: (B) make conjectures about angles, lines, polygons, circles, and threedimensional figures and determine the validity of conjectures, choosing from a variety of approaches such as coordinate, transformational, or axiomatic. (G.5) Geometric patterns. The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to: (C) use properties of transformations and their compositions to make connections between mathematics and the real world, such as tessellations; and (G.11) Similarity and the geometry of shape. The student applies the concepts of similarity to justify properties of figures and solve problems. The student is expected to: (A) use and extend similarity properties and transformations to explore and justify conjectures about geometric figures; Note: The following middle school TEKS provide the foundation for this lesson. (7.7) and spatial reasoning. The student uses coordinate geometry to describe location on a plane. The student is expected to: (B) graph reflections across the horizontal and vertical axis and graph translations on a coordinate plane. (8.6) and spatial reasoning. The student uses transformational geometry to develop spatial sense. Exploring Transformations Page 1
2 The student is expected to: (A) generate similar figures using dilations including enlargements and reductions; and (B) graph dilations, reflections, and translations on a coordinate plane. Overview: Materials: This activity allows students to explore transformations using graphs and to discover the algebraic rules to do transformations using a TI graphing calculator. Exploring Transformations: Reflections Exploring Transformations: Rotations Exploring Transformations: Translations Exploring Transformations: Dilations Coordinate grid paper Xlist/Ylist tables Calculator Procedures Solutions for LISTS Quarter inch or centimeter grid paper (for dilation not in lesson) Straight edges Map (color) pencils T83+/TI84+ Grouping: Pairs or groups of 4 Time: 1 or 2 classes Lesson: Procedures 1. Give students the following scenario: Notes Crystal and Tania are planning to use a flag theme on posters for a school dance. They want to use the transformations they learned in geometry to arrange the flags in different patterns. The flag will have vertices at the following points: A(0,9) B(5,9) C(5, 7) D(1,5), E(1,1) F(0,1) G(0,9) Provide students with coordinate grid paper and the Xlist;Ylist tables. The teacher may want to use a Exploring Transformations Page 2
3 Procedures Each transformation will be plotted on a coordinate grid. The coordinates of the transformation will be recorded in lists and the transformations will then be done using a TI83/84 graphing calculator. 2. Distribute Exploring Transformations: Reflections and the map colors. Direct students to complete the activity by creating the tables and graphs and answering the questions. Notes transparency to describe the scenario and plot the points on a transparency with the students. Allow about 10 to 15 minutes for completion of this first part. Suggest that students use the different color map pencils to distinguish between the different transformations. Circulate among students to answer questions and ensure that students are completing the work properly. Calculator Procedures give directions for using the graphing calculator for reflections. You may want to have students stop after Problem 7 to make sure students understand how to enter and graph the information in the calculator. You might want to make a transparency of the first page of the Calculator Procedures to help students. 3. Discuss the results as a class and identify some procedures to create reflections; have students write these procedures in their own words. It is important for the students to put the procedures in their own words to ensure that they truly understand how to form each of these transformations. Students need opportunities to communicate both verbally and in written form. 4. Distribute Exploring Transformations: Rotations and repeat steps 2 and 3 above. Exploring Transformations Page 3
4 Procedures 5. Distribute Exploring Transformations: Translations and repeat steps 2 and 3 above. Notes 6. Distribute Exploring Transformations: Dilations and repeat steps 2 and 3 above. Homework: It is important for the students to put the procedures in their own words to ensure that they truly understand how to form each of these transformations. Assign the writing for homework, and grade them for clarity and accuracy. Students need opportunities to communicate both verbally and in written form. Exploring Transformations Page 4
5 Coordinates for vertices of the flag are: Exploring Transformations Reflections A(0,9), B(5,9), C(5, 7), D(1,5), E(1,1), F(0,1), G(0,9) 1. Use the 1st grid on the attached grid sheet to plot and label the given points. Connect the points to create the original flag. 2. Fill in the x and y lists with the given coordinates. 3. Graph the reflection of the original flag across the yaxis. Label the points. 4. Fill in the x and y lists with the coordinates of the reflection. A. Which values changed? B. How did they change? 5. Before graphing the reflection of the original flag across the xaxis, predict how the coordinates will change. Fill in the x and y lists with the coordinates you predicted. 6. Now graph the reflection of the original flag across the xaxis. Label the points. A. Which values changed? B. How did they change? C. Did this change match your prediction? Exploring Transformations Page 5
6 7. Now use the information you have gathered to graph each reflection on the graphing calculator. Use List 1 for xvalues and List 2 for yvalues for the original flag. Use List 3, List 4, List 5 and List 6 (as needed) for the x and yvalues of the reflections over the yaxis and the xaxis. Is there a short cut for inputting the coordinates of the reflections in the calculator? Explain your answer. 8. On the next grid, plot the original flag and graph the line y = x. Now graph the reflection of the original flag across this line. Label the points. 9. Again, fill in the x and y lists with the coordinates of the reflection A. Which values changed? B. How did they change? 10. On the next grid, plot the original flag and graph the line y = x. Now graph the reflection of the original flag across this line. Label the points. Predict how the coordinates will change. 11. Fill in the x and y lists with the coordinates of the reflection A. Which values changed? B. How did they change? C. Did this change match your prediction? 12. Now use the information you have gathered to graph each reflection on the graphing calculator. Use List 1 and List 2 for the original flag. Use List 3, List 4, List 5 and List 6 (as needed) for the reflection over the line y = x and over the line y = x. Exploring Transformations Page 6
7 Exploring Transformations Rotations 1. On another grid, begin with the original flag. Then graph a rotation 90 clockwise and label the points. Predict the new coordinates. 2. Fill in the x and y lists with the coordinates of the rotation. A. Which values changed? B. How did they change? C. Did this change match your prediction? 3. Begin with the original flag on another grid. Then graph a rotation 180 clockwise and label the points. Predict the new coordinates. 4. Fill in the x and y lists with the coordinates of the rotation. A. Which values changed? B. How did they change? C. Did this change match your prediction? 5. Now use the information you have gathered to graph each rotation on the graphing calculator. 6. Describe what you have noticed about the coordinates of the original flag and the coordinates of the reflections and rotations. Exploring Transformations Page 7
8 Exploring Transformations Translations 1. On another grid, begin with the original flag and then translate that flag by the vector <6,4>. 2. How do you think the coordinates will change? Fill in the x and y lists with the coordinates of the translation. A. Did the new coordinates match your prediction? B.. How can you use the calculator to complete display the translation? Exploring Transformations Page 8
9 The scale factor for the dilation is 2. Exploring Transformations Dilations 1. Graph the original flag on the grid paper. (Be sure to make the x and y axes large enough for the dilation.) Graph the translation with the scale factor of Fill in the x and y lists with the coordinates for the dilation. 3. How can the calculator be used to complete this dilation? 4. Was your procedure for using the calculator to graph the dilation work? Explain your answer. 5. Repeat the dilation of the original image on your calculator using a scale factor of 0.5. Exploring Transformations Page 9
10 Calculator Procedures 1. Enter the values for x and y in L1 and L2 by pressing Stat and choosing 1 EDIT. Enter the data into the appropriate lists. 2. Press 2nd, y= and ENTER to select Plot 1. Press ENTER again to turn the plot on and then select the continuous plot. The default X List and Y List are L1 and L2. 3. Press GRAPH to view the graph in Plot 1. Reflect across the yaxis 1. Press Stat and choose 1 EDIT. Now select L3. Enter , 2 nd, L1, and ENTER. The negation of L1 is now L3. 2. Press 2 nd, y=. Press 2 and Enter to select Plot 2 and turn it on. Turn on the continuous plot. The x values for the reflection are in L3, so change the X list to L3. 3. Press GRAPH to view the graph in Plot 2. Reflect across the xaxis 1. Press Stat and choose 1 Edit. Now select L4. Enter , 2nd, L2 and ENTER. The negation of L2 is now L4. Exploring Transformations Page 10
11 2. Press 2 nd, y=, Press 3 and ENTER to select Plot 3 and turn it on. Turn on the continuous plot. The x values for the reflection are in L3, so change the Y list to L4. 3. Press GRAPH to view the graph in Plot 3. (Plot 2 can be turned off, if desired.) Reflect across y = x. 1. All values to graph this reflection have already been entered. The required values for X are in L2, the original Y values, and the Y values (original X values) are in L1. 2. Turn off PLOT Press GRAPH. Exploring Transformations Page 11
12 Rotation 1. Select Plot 2. Now change the X list to L2 and the Y list to L3. 2. Press GRAPH. 3. Select Plot 3. Now change the X list to L3 and the Y list to L4. 4. Press GRAPH. Translation 1. Press Stat and choose 1 Edit. Now select L5. Press 2 nd, L1, and +6. Now select L6. Press 2 nd, L2, and Turn off Plot 3. Select Plot 2. Change the X list to L5 and the Y list to L6. 3. Press GRAPH. Exploring Transformations Page 12
13 Dilation 1. Press Stat and choose 1 Edit. Now select L5. Press 2 nd, L1, and *2. Now select L6. Press 2nd, L2, and *2. 2. Reset the window to an appropriate size. 3. Press GRAPH. 4. Press Stat and choose 1 Edit. Now select L5. Press 2 nd, L1, and *.5. Now select L6. Press 2nd, L2, and * Reset the window to an appropriate size. 6. Press GRAPH. Exploring Transformations Page 13
14 Exploring Transformations Page 14
15 Exploring Transformations Page 15
16 Solutions for Lists: L1 L2 L3 L4 L1+6 L24 L1* 2 L2* 2 L1 *.5 L2 *.5 Exploring Transformations Page 16
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