Sydney Sherrieb Angle Bisectors & Centers of Triangles Class: Geometry Date: 4/15/14 Topic: Angle Bisectors & Centers of Triangles Period: N/A I. Estimated Time: 1 hour Standards: CC.2.3.7.A.2 – Visualize and represent geometric figures and describe the relationships between them. CC.2.3.HS.A.3 – Verify and apply geometric theorems as they relate to geometric figures. CC.2.3.HS.A.14 – Apply geometric concepts to model and solve real world problems. Goal: All students will explore and understand angle bisectors and how they effect the triangle’s incircle. II. Objectives: Students will be able to find/construct the incenter of a triangle using Geometers Sketchpad, and other classroom tools (compass, protractor, and ruler). Students will be able to define the incenter of a triangle. Students will be able to define the incircle of a triangle. Students will be able to construct the incircle of a triangle using Geometers Sketchpad and other classroom tools (compass, protractor, and ruler). Students will be able to explain the relationship between the incenter and incircle of a triangle. III. Materials/Equipment: IV. Laptops Pencil Compass Ruler Calculators Procedures: A. Engagement Activity/Anticipatory Set: Sydney Sherrieb Students will do a worksheet reviewing angle bisectors from last class. The teacher will go over the answers as a class with the students. B. Body/Instructional Plan: The teacher will explain the Geometer Sketchpad activity to the students. The teacher will show the students how to open Geometers Sketchpad on the laptops and get students started on the activity. The in-class activity will be completed individually by students but they are able to communicate with their classmates if they have questions or comments on the activity. The teacher will be around to answer student questions and make sure students stay on task. Once students have completed the activity the teacher will generate a student led discussion. The teacher will probe students to think about the least amount of information required to find the incenter of a triangle. Students will be asked to justify their response. C. Closure and Extending Activities: After students complete the activity students will do an extending worksheet on incircles. Students will present their work on the document camera and discuss how they found the incircle within the triangle. Students will then do a closing problem that will be turned in on the way out of class. V. Adaptations: Students with IEPs will be given an opportunity to explain rather than write their responses to worksheets. If students seem to finish the activity quickly without struggling the teacher can ask students to discuss differences between the incenter and centroid of a triangle. VI. Evaluation of Students: Students will compare their responses to the in-class activity questions with their classmates to determine whether or not they found the same results. The teacher will also evaluate students work when they present The students will turn in their closing activity for the teacher to see. Sydney Sherrieb References: Math Open Reference http://www.mathopenref.com/wsincircle.html Math Open Reference http://www.mathopenref.com/triangleorthocenter.html Sydney Sherrieb Name _____________________________ Date ______________________ We are going to investigate what happens when we draw the three angle bisectors of a triangle using Geometer’s Sketchpad. First, open up Geometer’s Sketchpad, and open up a new sketch. In order to draw three angle bisectors of a triangle, we need to first draw a triangle: 1) Click on the line segment tool in the tool bar on the left side of the screen. Click anywhere in the white space and drag the pointer, stretching the line segment. Click to end the segment. Your sketch should look similar to this: 2) Stay on the line segment tool. Put the mouse over one of the endpoints of the segment and you will see a blue halo around the point. Click the point and drag your mouse to where you want the end of that segment, and then click to end the segment. It should look similar to this: Sydney Sherrieb 3) Now, connect the two segments. Put the mouse over one of the points and when you have a blue halo around the point, click and drag until you get a blue halo around the remaining point. Click to end the segment. Click on the arrow selection tool, and then click somewhere in the white space to deselect everything. It should look similar to this: 4) Now your triangle is made. Before proceeding to drawing the bisectors, we must label your triangle. In order to do this, click on the arrow tool in your tool bar on the left side of the screen points. . Then, click on all three points until they all have a pink halo around the Sydney Sherrieb When all three points are highlighted, click on Display at the top, then click on Show Labels. Your sketch should look similar to the following: 5) Before proceeding, click somewhere in the white space to deselect everything. Now we are going to make our angle bisectors. In order to do this, we need to highlight the points in a specific order, let’s say ∠ABC. Make sure the arrow tool is still selected and click on the points in the order of A, B, C so that all the points have a pink halo around them. Click on Construct and then click on Angle Bisector. Sydney Sherrieb Your sketch should look similar to the following: 6) Repeat the previous step to get angle bisectors for all three angles. Remember, the order in which the points are selected is important. Your sketch should look like the following: Now you have a triangle with three angle bisectors! Save your work and answer the following questions. Sydney Sherrieb QUESTIONS 1. Notice that the three angle bisectors intersect at a point. There is a name for this point. If you don’t know it, open up an Internet browser and search for the name of the point of intersection. Point of intersection of the three angle bisectors - ________________________ ** This point is the center of the triangle’s incircle, which is the largest circle that will fit inside the triangle and touch all three sides. Label this point on your sketch. This can be done the same way that the points of the triangle were labeled. Make sure the arrow tool is selected, then click on the point and make sure it has a pink halo around it. Then click on Display and Show Label. Let’s construct the incircle! Step 1: From Geometer’s Sketchpad, click on the arrow tool on the left and then click in any of the white space to deselect everything. Then click on the incenter so it has a pink halo around it, and then click on one of the sides of the triangle. Your sketch should look like the following: Sydney Sherrieb Step 2: Click on the Construct menu and click on Perpendicular line. Your drawing should look like the following: Sydney Sherrieb Step 3: Click on the circle selection tool in the left tool bar. Then, put your mouse over the incenter until it is highlighted pink, and then click and drag the mouse until the point on the circle reaches the intersection of the selected side of the triangle and the perpendicular line. When both the selected side of the triangle and the perpendicular line are highlighted in blue, click again. Your circle should be highlighted in pink. Sydney Sherrieb Step 4: Click on the selection arrow tool and then click on any of the white space to deselect everything. Then, click on any of the vertices. Hold the button down and move the mouse around. Try this with the other two points. Write your response below. a) What happens to the triangle as the mouse is moved around? b) What do you notice about the point of intersection? c) What about the incircle? 2. Share your findings with the person sitting next to you. Does their sketch look similar to yours? Does their sketch do the same thing as yours when they move it around? Write your responses below. Sydney Sherrieb Name:_______________________ Each figure shows a triangle with ones of it’s angles bisected. The bisected angles are marked as 1 and 2. ANSWER KEY 1) Find m∠1 if m∠2 = 25° m∠1=25° 1 2 2) m∠CED = 96°. Find m∠1. C D 2 m∠1=48° 1 E 3) Find m∠ ABC. m∠1 = 26°. A m∠ABC=52° 1 C B 2 Name:_______________________ Sydney Sherrieb Each figure shows a triangle with ones of it’s angles bisected. The bisected angles are marked as 1 and 2. 4) Find m∠1 if m∠2 = 25° 1 2 5) m∠CED = 96°. Find m∠1. C D 2 1 E 6) Find m∠ ABC. m∠1 = 26°. A 1 C B 2 Name:_________________________ Sydney Sherrieb Closing Activity Given three points corresponding to Lazytown, Crazytown, & Busytown construct a triangle and find the incenter of the three towns. Label the incenter as C. Label the segment from Lazytown to Crazytown, Booty Street. Label the segment from Crazytown to Busytown Sassafras Road. Label the segment from Busytown to Lazytown WallabeyWay. Lazytown Crazytown Busytown Construct the largest circle between the three segments in the above triangle. 1) Justify below why C is the incenter of the three towns. 2) What is the largest circle inscribed in the triangle called? Analysis of Technology Used in the Lesson Geometer’s Sketchpad Sydney Sherrieb The use of technology in the lesson is purposeful and deliberate. Using sketchpad students can see the relationship between the incenter and incircle within a triangle. I believe seeing visual representations of any mathematical topic are helpful, especially when dealing with mathematical relationships. The directions throughout this lesson are clear. If students haven’t used Geometer’s Sketchpad before it may seem daunting but I tried to make the directions as clear and concise as possible. I think that using Sketchpad first before using the compass and protractor is helpful and beneficial. This is because students are simply able to drag any point on the triangle and see how the rest of the figure is effected. However, drawing a picture with a pencil and tools in the classroom we aren’t able to do that. We would have to do multiple drawings/representation to reach the same conclusion. The technology saves time and I believe is more visually appealing to students. Students are also less likely to make any mathematical error using sketchpad because the angles and segments are measured by the computer not the student. This allows for more exploration from the students, given they don’t have to worry about calculating or measuring themselves. When completing this activity students should be familiar with angle bisectors, properties of triangles, and properties of circles. From this activity students understand how angle bisectors effect the triangle’s incircle.