Objectives:

• Review some properties of angles and lines
• Explore some properties of triangles
• Prove Triangle Sum Theorem

Introduction: (7-15 minutes) (Teacher: use an overhead when presenting the material, and ask questions to engage the students. Students: taking notes when necessary and listen attentively , also, if there are any uncertainties they should ask questions.)
We will be exploring triangles today, more specifically; we will prove the Triangle Sum Theorem, which states that the sum of the measures of the angles of a triangle is 180º.  What does this mean exactly?  We will be using Geometer’s Sketch Pad in a few moments, but we must first review some important properties of angles and lines that we will be using today.  First of all, how many degrees is a straight line?  Now this is an important property of straight lines that we will be using today to help us prove the Triangle Sum Theorem.  Now we are going to do a quick review of vertical angles and corresponding angles.

In this picture which angles are vertical angles and which ones are corresponding?
The vertical angle theorem says that vertical angles are equal in measure and they make a shape that looks like two V’s with their bottoms attached.  This is an important property that we will use to prove our theorem.  Now, what do corresponding angles look like?  They make an F shape.  The next picture shows us the two different angles and the points that make them.

So angle FCE and angle IAB are corresponding (in the yellow), and so is DCE and ECF, if we were to flip our F around we would see that angle DCE and FHA are corresponding, as well as angle GCF and HAI.  The vertical angles are in the blue.  So DCG and FCE are vertical angles.  Where else are there vertical angles?
Now we are ready for the lab.

Activity:(30-35 min.) (Teacher: walk around and ask and answer any questions that my arise, keep the students on task. Students: work with group to construct proof and better understand triangles, should be communicating and asking questions of each other.)
While following the steps students should answer questions along the way and reflect.  Students will be broken up into groups of 3 or 4 depending on how big the class is, and will work together to understand the activity.
Steps for GSP (once it is opened make sure the labels setting is on, it is in the display menu):
1) Click on the point icon in the upper left hand side, and put 3 dots on the screen.  They should be labeled A, B, C, so they form a triangle shape.

2) Next connect the points by first making sure the segment icon is on, upper left side (not the line, or ray icon).  Then click on the arrow icon on the upper left side.  After that, hold down the shift key and click on each of the points.  Once they are all highlighted go to the construct menu and click on segments.

Questions: Now that we have our triangle, what do you notice about it, what type of triangle is it (scalene acute, isosceles right, equilateral acute)?  What is the angle measure?  (To find this out, highlight, in this order, B then A then C and go to the measure menu and click on angle, do the same thing except starting with A, then again with C)  What do they add up do?  We could do this in our heads, but it is nice to have it on the screen, so go to the measure menu and click on calculate.  Then click on each angle with a plus sign in between each one.  Once they are all entered click on okay.
Still not convinced…

What is the angle of a straight line?  Let’s construct our proof:
1) Highlight points B, A, and line AB and point C (holding the shift key down the whole time) Then go to the construct menu and click on parallel line.  We should have a parallel line with line AB.

2) Now, go to the upper left corner and click on the segment icon, this time drag it to the ray icon and then press the pointer icon.  Highlight points A and B (in that order) and go to the construct menu and click on ray.  Do the same thing for C and B (in that order)

3) In order to measure angles we need to make some points on these new lines.  First click on the dot icon, and put a point on ray AB above point B (this should be point D).  Do this again for a point on the parallel line to the right (point E) and left (point F) of B.  Then again on ray CB above point B (point G).

Questions: What do you notice about angle DBF and BAC, also GBF and BCA?  How are they related, what is their measure?  How are angles GBF and ABC related?  What is their measure? (Let’s measure these angles.  We must first click on the pointer icon in the upper left corner, then we can start high lighting again.  First click on points DBE and measure by going to the measure menu and clicking on angle just as before.  Just repeat with angles GBD and GBF to find out their angle measure.)  Once you have the angle measures then calculate to find the total.  Are you surprised?  Does this work for every triangle?  If time allows try this again with different triangles to verify your answer.  These questions along with a reflection of what we did in class today are your homework.

 
 

Wrap-up: (last 4-5 minutes) (Teacher: let students know there is a journal assignment, answer any last minute questions.Students:listening to the conclusion of the activity and logging out of computer.)
We have just proven that the sum of the measure of the angles of a triangle is 180º, this was done using properties of angles such as vertical angles, corresponding angles, and also properties of straight lines, all straight lines have measure of 180º.  So using all that we see that the sum of the angles of triangle ABC is 180º.

Accommodations:
This lesson involves a lecture component and a hands-on/technology component.  Ideally it addresses the needs of three types of learners: kinesthetic, auditory, and visual. The lecture and the group communication should help the auditory learner understand the material well.  The kinesthetic and visual learners will be able to use the technology component of the lesson plan to construct their understanding of the material.  By having the students in groups, they should be able to learn from each other and use each other’s diverse learning styles to better understand the material.

Students with special needs will not have any difficulties with the lecture component, but the technology component may pose a problem.  By having the students in groups, if the student with special needs is unable to use a computer, one of his/her group members will handle that part and they can communicate with the other group members to understand the material.

NCTM and ISBE standards:
Students will develop their sense of Number and Operations by having them do some mental calculations and judge the reasonableness of the computational results of the sum of the angles (ISBE goal 6). By using visualization, and geometric representation to prove the theorem, students are expanding their Geometry knowledge (ISBE goal 9).  Students will be using Measurement in order to make sure the angles add to 180º(ISBE goal 7).  By constructing knowledge through exploration and observation, students will develop their sense of Problem Solving.  This lesson is designed for the students to make and investigate mathematical conjectures and have them build their Reasoning and Proof skills.  Throughout the activity students will need to use Communication with their peers, by thinking coherently and clearly when discussing the proof with their peers.

Assessment:
Students will be assessed based on their journal assignment.  Length will not be as important as quality of journal.  Students should think and reflect about what we did in class for at least an hour before writing it.