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Athena Matherly, Barbara Perez, Guy Barmoha, Ed Knote, Lewis Prisco GeoGebra Geometry Workshop Instructions for Presenter Overview This workshop is designed to introduce Middle School Math teachers to the wonders of GeoGebra. GeoGebra is a dynamic, interactive software that will bring the Mathematics classroom alive. With the help of GeoGebra, students will become involved, interested, and excited to learn the many concepts of Algebra and Geometry. The activities done in this workshop are designed for 6th, 7th, and 8th grade students. Presenters Materials The presenter should be prepared with this document “Instructions for Presenter” to guide him/herself through the workshop, copies of the “GeoGebra Geometry Handout” for the participants (to be handed out at the end of the workshop), laptop, projector, and the following files: o Dynamic Worksheet – Isometries: Reflection (html), Rotation (html) (Located in folder called GeoGebra Files) o Dynamic Worksheet – Angles: Parallel Lines (html) (Located in folder called GeoGebra Files) o Power Point Presentation – “GeoGebra_Workshop_Geometry” (Located in the GeoGebra Workshop Folder) Each participant should have a laptop computer with GeoGebra downloaded before the workshop. Table of Contents (Total Time: 90 minutes) Discussion of teaching strategies (10-15 min) – page 2 Presentation of Dynamic Worksheet – Reflection and Rotation (10 min) – page 2 Introduction and Overview of Geogebra (10 min) – pages 2,3 Recreate Dynamic Worksheet – Reflection (10 min) – pages 3,4 Recreate Dynamic Worksheet – Rotation (15 min) – page 4 Presentation of Dynamic Worksheet – Angles (5 min) – page 4 Recreate Dynamic Worksheet – Angles (20 min) – pages 4,5 Bonus Material – pages 6-9 1 Athena Matherly, Barbara Perez, Guy Barmoha, Ed Knote, Lewis Prisco GeoGebra Geometry Workshop The GeoGebra Geometry workshop will begin by setting up the following scenario and asking the following question: You’re trying to get your students to understand the properties of isometries, (reflections, rotations, translations, and dilations), how do you usually show this to your students? Discussion: (Estimated time: 10 – 15 minutes) Participants may suggest showing the students a figure on a piece of paper and folding the paper to see the line of reflection and where the reflected image would be. For a rotation they may show a clock and investigate how the hands rotate, or they may draw a point of rotation on the board and show an object being rotated about that point. Keep a mental or physical list of suggestions and ask some probing questions such as: What happens to the reflected image when the original image moves closer or farther from the line of reflection? How would you go about showing reflection in a coordinate plane? What happens to the rotated image when the original is moved? Presentation of Dynamic Worksheet: (Estimated time: 10 min) Files: Reflection (html), Rotation (html) Participants will be exposed to the use of a pre-made dynamic worksheet that will show an example of a reflection and a rotation. The teachers will be able to see how dynamic geometry software is so helpful when we want to make a change in the original construction and observe the consequences of that change. With this example we can observe how the reflected image changes as the original image is manipulated. Dynamic geometry software allows the student to see the material in a more entertaining and engaging way than with paper and pencil. The student becomes more interested in the material because of this. Once we have their interest and attention learning flourishes. Now is a good time for the presenter and the participants to have a discussion about the benefits of using this software as opposed to an overhead, whiteboard, or chalkboard. Introduction and Overview of GeoGebra: (Estimated time: 10 minutes) There are a couple of basics that need to be shown before constructing anything in GeoGebra. We will go over the following to get the teachers acclimated to how to navigate through the software. View menu: show axis, grid, and algebra window Toolbar: Click the arrow on the bottom right side of the button to see a list of the options within that menu. 2 Athena Matherly, Barbara Perez, Guy Barmoha, Ed Knote, Lewis Prisco Pointer Button: Move mode, think of this as home base. If you want to get out of a mode click the pointer button. On the bottom left hand side of the screen you can always see what mode you‟re in. (This will be just an introduction to some of the options in GeoGebra, the other options will be discussed throughout the investigations in the workshop) Investigation of Isometries Goal 1: Create reflected image (Estimated time: 15 minutes) Open Blank GeoGebra File Step 1: Create Polygon Ask, “What would I need to construct first in order to show the students a reflection?” We need an image or a figure. Ask “Which button has the option Polygon?” After choosing the polygon tool, we will put down 5 points. Show and emphasize that you need to click on the first point to close the figure. Step 2: Construct a line of reflection Ask, “What will I need next to reflect this image?” The participants should tell you that you need a line of reflection (that will be a line of symmetry). Have the participants find the line tool and construct a line of reflection. Step 3: Reflect the original polygon Now to reflect the image over the given line we would need to find and select the reflection tool, the select the polygon (notice how it is bolded when the pointer moves over it) and then select the line of reflection. Step 4: Observations At this point the participants can play with their construction for a few minutes. Ask what they observe and to describe their observations. Can they move and change the reflected image, why or why not? Why are the points in the reflected image labeled like they are? Can you move the line of symmetry, how? Step 5: Use the properties menu Right click on the original image and select the properties. Demonstrate the many different properties that can be changed (color, thickness, show label, etc…). Show how you can stay in the properties menu to change the properties of any of the objects in the construction. 3 Athena Matherly, Barbara Perez, Guy Barmoha, Ed Knote, Lewis Prisco Step 6: Extension (If time permits) From here we can connect the corresponding points with segments. We could find the intersection points of this segment and the line of symmetry, then by showing the segment‟s lengths between corresponding points and showing the length from one of the points to the line of symmetry and observe the relationship between those distances. Goal 2: Create rotated image (Estimated time: 15 minutes) Open Blank GeoGebra File Step 1: Create polygon Step 2: Rotate polygon Ask the participants to find the rotation mode. Before using this mode read the description to see what we would need in order to rotate. We need a point to rotate around, we could use one of the vertex points of the polygon or another point we construct. Place a point anywhere on the drawing pad, then choose the rotate tool and click on the polygon and the point. At this moment GeoGebra will ask what angle of rotation you would like, you can type in a degree, name of an angle, or slider. Step 3: Create a slider for the angle of rotation Find and select the slider tool and click somewhere in the drawing pad. Select the angle option for the slider and we can change the name to anything we want, let‟s say „a‟. We can change the range of values and the incremental change in the angle. Now choose the rotation tool. Click on the polygon and the point of rotation, when the screen appears for the angle, type in „a‟ and click „ok‟. Go to the pointer mode and slide the point on the slider to change the angle of rotation. Observe what happens when the slider moves. Investigating Angle Relationships Goal 1: Two parallel lines cut by a transversal (Estimated time: 20 minutes) Open Blank GeoGebra File Step 1: Construct a line and a line parallel to that line Step 2: Construct a line that, transversal, which intersects the two parallel lines Step 3: Use the Angle tool to measure the angles, more points may have to be constructed on the lines so we have three points to pick for each angle. 4 Athena Matherly, Barbara Perez, Guy Barmoha, Ed Knote, Lewis Prisco Step 4: Move points and lines around and observe which angles will always stay congruent when two parallel lines are cut by a transversal. Make conjectures about your observations. Here is an example of two images from a GeoGebra file that followed the steps above. Goal 2: Investigating Triangle Sum (Estimated time: 25 minutes) Open Blank GeoGebra File Step 1: Construct triangle ABC, using the segment tools Step 2: Measure the three interior angles of the triangle Step 3: Calculate the sum of the angles by inserting text and typing in “sum =” + This is assuming that the three interior angles are named , , and . Step 4: Drag the vertices around and observe the individual angle measures change while the sum remains constant, 180 degrees. Proving the observation Step 4: Create a polygon for Triangle ABC Step 5: The participants will prove this using a visual model. A line parallel to one side of the triangle through the opposite vertex will be constructed. Then the triangle will be rotated about the midpoints of the other sides. Step 6: Find the midpoints of the other two sides, and then set up two angle sliders to rotate the original triangle about the midpoints. Move the sliders to see if you can get the three angles to form a straight angle (180 degrees), proving the observation we saw earlier. 5 Athena Matherly, Barbara Perez, Guy Barmoha, Ed Knote, Lewis Prisco Here is an example of two images from a GeoGebra file that follows steps 5 and 6 above. We can see in the picture on the right that the three interior angles of the triangle form a straight angle, 180 degrees. This should be a sufficient visual proof of the triangle sum theorem. Bonus Material (If time permits) If time allows we will like to present some worksheets to show some of the other options that can be done using GeoGebra. Some of the worksheets will include but not be limited to: Median/Centroid Investigation After the construction we will look at how the two segments on each median relate to each other. The sum, difference, and product always seem to change. However, the quotient seems to be the same. 6 Athena Matherly, Barbara Perez, Guy Barmoha, Ed Knote, Lewis Prisco Let‟s see what happens when we change the orientation of the triangle. Even with the different orientation of the triangle we notice that the quotient is still constant, and it‟s still 2! What conjecture can we make about the lengths of the segments on each median? ASS Triangle Congruency Test Investigation The triangle was constructed using LK , KI , and , making sure the angle is not included in the given segments. We can see that there are two triangles that can be formed, however, is there a time when one triangle is formed? If so, what condition should be met? Can we use GeoGebra to prove or disprove any other possible triangle congruency tests (AAA, ASA, SAS, SSS, SAA)? 7 Athena Matherly, Barbara Perez, Guy Barmoha, Ed Knote, Lewis Prisco Investigating Symmetry in pictures/images Another great use of GeoGebra is to use it to check for line symmetry. A nice way to do that is to get a picture and import it into GeoGebra. Then attempt to construct what you think may be a line of symmetry, using GeoGebra we can test to see if it really is a line of symmetry. We will first do this by inserting a picture to the geometry window of GeoGebra, to do this we need to go to the following menu: After choosing the correct tool we need to click anywhere in the geometry window and browse our hard drive for a picture we would like to insert. We can then resize the picture by fixing its corner points. We can do this by constructing two points and making those new points the corner points of the image. Then by right clicking on the image we can choose the properties of the image. In the properties window we can change the placement of 3 of the corners, we are only changing the position of 2 corners and leaving the third one (also the fourth one) free. Here we change Corner 1 to point A and Corner 2 to point B. Then by moving point A or B we can resize the image. Once we are satisfied with the size of the image we want to set it as a Background image, this can also be done in the properties window. Now it is time to construct a line that we think may be a line of symmetry, then construct a point on one side of the line on the figure and reflect that point by about the line, then put the trace on both points. Then pick the original point and move it along the picture and see if the reflected point traces on or off the picture. For example: 8 Athena Matherly, Barbara Perez, Guy Barmoha, Ed Knote, Lewis Prisco Appears to be a line of symmetry Does not appear to be a line of symmetry Some of the Sunshine State Standards being addressed in this workshop are: Strand C Geometry and Spatial Sense Benchmark MA.C.1.3.1 The student understands the basic properties of, and relationships pertaining to, regular and irregular geometric shapes in two and three dimensions. Standard 1 The student describes, draws, identifies, and analyzes two- and three-dimensional shapes. Benchmark MA.C.2.3.1 The student understands the geometric concepts of symmetry, reflections, congruency, similarity, perpendicularity, parallelism, and transformations, including flips, slides, turns, and enlargements Standard 2 The student visualizes and illustrates ways in which shapes can be combined, subdivided, and changed. Benchmark MA.C.3.3.1 The student represents and applies geometric properties and relationships to solve real-world and mathematical problems. Standard 3 The student uses coordinate geometry to locate objects in both two and three dimensions and to describe objects. Note: The Geogebra Geometry Handout will be given to the participants at the end of the workshop as an overview of the steps that were taken throughout the activities. Note: The PowerPoint slide show (GeoGebra Workshop (Geometry)) will be used as an outline of the workshop with objectives, goals, tasks, and Sunshine State Standards. 9

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