Open GeoGebra & Format Worksheet

Transcription

1 Open GeoGebra & Format Worksheet 1. Close the Algebra view tab by clicking on the in the top right corner. 2. Show the grid by clicking on the show grid icon, located under the toolbar. 3. Select the Move Graphics View tool from the toolbar. Move the grid until the origin appears near the center of the screen. Make sure both axes show points from -6 to From the Options menu, select Point Capturing and Fixed to Grid. This ensures that students will only work with integer coordinates. Plotting Polygons Page 1

2 Begin Example 1 1. Create a rectangle that has vertices A(-4,2), B(-4,-6), C(2,-6), D(2,2). Select the polygon tool from the toolbar. Click on the coordinates of the points A-D, one at a time. Click on the point A again to close the polygon. 2. Ask students to determine the lengths of sides of the rectangle. Allow students to share their methods of finding side lengths (counting, 4+2, etc.). Relate this to distance between two points. 3. Measure the side lengths using the distance tool. Click on the triangle to open the menu under the angle tool. Select the Distance or Length tool. Click on the line segment that corresponds to the side you are measuring. The length will appear near the midpoint of the segment. Do this for segments and Find the distance between points by clicking on the points individually. Use this process for and. Plotting Polygons Page 2

3 Begin Example 2 1. Create a trapezoid that has vertices E(1,6), F(1,2), G(7,2), H(5,6). Select the polygon tool from the toolbar. Use the same process that was used to create a rectangle in Example 1. Change the color of the trapezoid by selecting the Move item tool. Then click on the trapezoid. You can either right-click and select Object Properties and click on the color tab. Or you can use the Set Color and Transparency shortcut from the toolbar. 2. Ask students to determine the lengths of sides of the trapezoid. Allow students to share their methods of finding side lengths. There may be discussion about the length of, as it is not a vertical or horizontal segment. Allow students to share their thoughts and to provide reasoning for their thinking. Do not evaluate their ideas yet, but record their guesses on the board. 3. Measure the side lengths using the distance tool. Select the Distance or Length tool. Measure the lengths of the four line segments using either method from Example 1, part Students will notice that the length of is Ask students why this length is not an integer value. Allow students to discuss their ideas with a partner and to make a conjecture about the lengths of diagonal sides. After several minutes, pairs will share their thoughts with the class. Lead the discussion so that students agree that diagonal side lengths will not always be integer values. Explain that for this lesson, we will only find the lengths of horizontal and vertical line segments. Plotting Polygons Page 3

4 Horizontal & Vertical Lines: Discussion Have students work in pairs on computers using GeoGebra software. Students will graph the following polygon on the coordinate plane using the coordinates as the vertices. A(-3,-2), B(4,-2), C(4, -5), D(-3,-5) Ask students to identify the shape using their prior knowledge of properties of polygons. Ask students to provide their reasoning. They should determine that the shape is a rectangle because the opposite sides are congruent and all four angles are right angles. Next, guide a discussion about the line segments of the rectangle. An example script is given below. T: Are there any horizontal or vertical line segments? S: Yes T: Which line segments are horizontal? S: and T: How do you know? S1: It s a straight line. T: (Provide a counterexample, such as.) This is a straight line. So it is also horizontal. S1: No. S2: It goes across. T: [Provide a counterexample, such as a segment with coordinates (-3,3) and (5,4).] Okay, like this? S3: It has to be on one of the grid lines. T: Well, as long as it is parallel to the x-axis it will be horizontal. In this case, they will be on the grid lines because we are only dealing with integers. Is there a clue in the ordered pairs that might tell us these points will make a horizontal line segment? Let s look at first. S: Oh! They both have the -2! T: What does the -2 represent? S: The y-value? T: The y-coordinate. Yes! Can you make a conjecture about ordered pairs of horizontal line segments? Work with your partner and write a conjecture. Plotting Polygons Page 4

5 Allow students a minute or two to discuss this with their partner. Circulate the room to listen to their ideas. Select pairs of students to share their conjectures, beginning with the least formally written conjecture. (Anticipated response: Horizontal line segments will have the same y-coordinate.) Tell students that they will now test this conjecture on. The conjecture will hold true for this segment. Now ask students to write a conjecture about vertical line segments. Allow them two minutes to work with their partners before sharing their ideas with the class. Students should decide that two points on a vertical line will have the same x-coordinates. Test this conjecture using and. (Use a similar line of questioning in the script above if students have trouble making this conjecture.) Now follow the script below to extend students thinking and understanding. This portion of the lesson is designed to get students to consider the properties of shapes based solely on the coordinates of its vertices. T: Now we will move on to a new polygon. Without plotting the vertices, let s take a look at the ordered pairs to see what we can discover about the shape. This polygon has vertices at E(-1,-1), F(-1,4), G(4,-1). What can you tell me about this shape? S: It s a triangle. T: How do you know? S: It has 3 points. T: Okay so since it has 3 vertices that are not on the same line, what does that tell us? S: That it will have 3 lines. T: So when we connect the 3 vertices, we will end up with a 3-sided figure? S: Yes. T: That s correct! What else do we know? Students should now discover that, based on their conjectures, will be a vertical line and will be a vertical line. Students may confuse points F and G because they both contain a 4. Allow students to correct one another if this comes up in discussion. (Note: Be sure to correct students vocabulary when they use terms improperly. Examples are provided within the scripts above.) Now have students graph the polygon with vertices EFG and check their responses. Plotting Polygons Page 5