Download Do the angles of a d-triangle sum to 180 degrees?

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.