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Lesson 1.2: Do the angles of a d-triangle sum to 180 degrees? This lesson gives the students the opportunity for deeper exploration into the properties of d-triangles. It can be assumed that most students accept that the sum of the angles of a triangle is equal to 180 degrees. This, however, is not always the case. In fact, the sum of a geometric triangle can range from 0 to 540 degrees. In hyperbolic geometry, the sum of the angles of a triangle is less than 180 degrees. This lesson allows the students to explore the property of angle measure, as well as some of the ideas the students have come up with in groups. Strand: Non-Euclidean Geometry - Hyperbolic Learning Objectives: 1. Students will be able to construct planar and d-triangles and calculate the sum of each triangle’s angles. 2. Students will be able to show by construction that the sum of the angles of a d-triangle is less than 180 degrees. 3. Students will be able to gain a basic foundation of the term “limit” by relating it to the sum of the angles of a d-triangle. Materials: Computers equipped with Cabri Geometry II, a set of disks each having <hyperbol.men> on it, lesson lab sheets for each student. Procedure: (suggestions) 1. Set Induction/Attention-getter: Pose this question: What do the angle measures of a triangle sum to? It is assumed that the students will respond “180 degrees.” Tell them that this may not always be true. Allow the students to express ideas and thoughts about why this may not always be true. 2. Group students using method of choice. 3. Distribute lesson labs and provide plenty of time to complete them. If students finish early, encourage them to “explore more” by testing other ideas the students come up with on their own. 4. Conclude with a class discussion summarizing what the students observed during the lesson. Assessment: Authentic forms of assessment are usually best when conducting a lesson that relies heavily on discovery based learning. Keep this in mind when assessing the students. The possibilities for assessment are left up to the instructor. Do the angles of a d-triangle sum to 180 degrees? Planar Construction Step 1 Use Cabri to construct three triangles. and the one obtuse. One should be right, one acute, Step 2 Use the angle measure tool to find the measure of the angles of each triangle. Step 3 Use the calculate tool to sum the measure of the angles for each triangle. Investigate 1. Manipulate each of your triangles by pulling each vertex. What do the measure of the angles sum to? Is this true no matter how you pull the triangle? 2. Construct a table for each triangle to show what the angles sum to. Make a Hypothesis 3. Do you think that there is an instance where the sum of the angles of a triangle is less that 180 degree? Greater than 180 degrees? 4. Make a hypothesis about whether the sum of the angles is 180 degrees. Explain why you have made this hypothesis. Poincare Disk Construction Step 1 Open <hyperbol.men> while in Cabri. four new buttons. This will augment the tool bar with Step 2 Use the Unit Circle tool to construct a Poincare disk. Step 3 Draw a d-triangle on the Poincare disk with the tool called d-triangle. Step 4 Measure the angles of the d-triangle using the angle measure tool under the augmented buttons. Investigate 5. If you were to sum the measure of the angles of the d-triangle, what would be the result? 6. Construct a table to show your calculations. Does the table contradict the rule for Euclidean geometry that states: The sum of the angles of a triangle is equal to 180 degrees? 7. Summarize your findings. What do the angles of a d-triangle sum to? Do you think this is true for every d-triangle? The Plane vs. The Poincare Disk 8. Make a chart showing the theorems and rules that have held in the plane and on the Poincare disk as well as those theorems and rules that have worked in both. 9. Make a list of the things that we have not tested that you are interested in testing. Explore More 10. Exchange the list you created in (9) with another group. Look at some of their ideas and try testing them in the plane and on the Poincare disk.