Lesson+Plans

1. Topic- The triangle sum theorem. Target grade level- first year geometry students 8th, 9th, 10th graders. This lesson plan will be coming right after a quiz.
 * Section 3.5 **

2. Clearly stated objectives, standards, essential questions-Standards

a. Students will be able to identify and use the parallel postulate and the triangle sum theorem. b. Students will be able to measure angles in degrees and determine relations of angles in finding the missing angles. They will also use the properties of complementary, supplementary and vertical angles to find these missing angles. c. Students will be able to use the proof of the triangle sum theorem to prove that it works. d. Students will be able to construct parallel lines, draw a transversal and measure and compare angles formed.

3. PA content standards a. 2.3.8.8C- Measure angles in degrees and determine relations of angles. b. 2.4.11.11A- Use direct proofs, indirect proofs or proof by contradiction to validate conjectures. c. 2.9.8.8B- draw, label, measure and list the properties of complementary, supplementary and vertical angles. d. 2.9.8.8c-construct parallel lines, draw a transversal and measure and compare angles formed.

4. Accommodations for students- a. Students with ADD- There will not have to be as much individual assistance because this class is mainly a whole class discussion. I will make sure I pay attention to how much the student is paying attention and if I feel they are not participating enough that I ask them direct questions. When we are doing the proof in class, I will take a break throughout the proof and walk around to make sure all students are on task especially the students with ADD. I will have at least one of the ADD students be a volunteer during the parallel postulate demonstration. I will make sure the student understands what expected of them with the exit ticket as I am handing it out to all the students. I will individually go over and see how the students with ADD are faring. b. Students that are ahead- since this is largely a class discussion and demonstration class I can regulate how fast these students are going. They will not be able to get that ahead. I will also make sure that I regulate all questions by who is answering them. I don’t want my students who are ahead answering all the questions and the other students get behind. Also I don’t want to not have them answer questions cause then they will get bored and not pay attention. Its ok if the students who are ahead finish the exit ticket early, this just means they will really be on time for the next class. They will get the same homework as the rest of the class yet they will have expectations to finish it all and to have a higher success rate than average.

5. Active teaching, modeling, demonstrations-

a. Active teaching we will start with showing why the parallel postulate is true since we will need it to prove the triangle sum theory. 1. Parallel Postulate- Given a line a and a point not on the line there is one and only one line that contains the given point and is parallel to the given line. 2. Here I will start the class with a class discussion about parallel lines, a review of this. Next I will draw a line across the board and a dot above it. I will read the postulate and ask the class what they think about it whether it is true or not. I will say using this line here and this dot is there only one parallel line we can make? I will field some responses and reasons why to this. Next I will use three student volunteers to come up and help me demonstrate this principle. Here one student will hold the string down where the dot is and the other two will hold the string from either end. We will have the students try to find more than one parallel line that contains this point. Once the class has seen this they the students will sit down and we will go over what a transversal is and how it is related to this postulate. Next the students will write this in their math notebook and I will be setting up for the next demonstration while they do this. b. Active teaching through the visual of taking a straight line and forming a triangle and vice versa using the enlarger. Here I will take a straight line (180 degrees) and wrap it around three points to make a triangle and then connect the ends to show the 180 degrees will form a triangle. Now I will show this vice versa. Here I will take a paper triangle (any type of triangle will work) and place it on the documenting camera. I will rip off two of the corners and place them in line with the third corner and then draw a straight line (180) degrees and show the three corners or triangles angles equal the same as a line, which is 180 degrees. There will be questioning throughout this part to keep the students engaged. This will be listed under the questioning section. c. As a class we will prove that this idea on the chalk board, that we witness is in fact true, and the students will write down the proof in their math notebooks and the triangle sum theorem. 1. The sum of the measures of the angles of a triangle is 180 degrees. d. There will also be a demonstration section for homework on the computer through the website hotmath.com- here the students will explore the triangle sum theorem through an online program e. Class discussion under the questioning section

6. Tasks and activities- a. Warm up activity (6 minutes)- this will be one of those puzzles the class has to figure out by the information given about who went where and did what, 5 mins to do the puzzle and 1 min to go over the puzzle if they did not finish they can finish for homework, this will not be handed in yet it will be graded for class participation of working on the assignment. b. Parallel postulate class discussion and activity (10 minutes)- see active teaching section to see what we will do for this topic, this topic will go on before the triangle sum theorem since we need this idea for proving the triangle one c. Triangle sum theorem- (20 mins)- 10 mins of demonstration with time for discussion and questions included, then 10 mins of proof done as a class with definition and time for discussion and questions built in. d. Homework (1 min) is to explore the website and to complete the following problems to go over in class the next day. This assignment will be written on the board. There will be a link to this website on my website so the students don’t have to copy this down. e. Closure- exit ticket (5 mins)- students will complete this exit ticket for grading by themselves before they leave the classroom for the day.

7. Questioning- a. Parallel postulate section- today we will be talking about a new theorem but first there is some extra information we need before we can get there. 1. So who remembers what a parallel line is? 1. Field answers to this and then reveal on the board the complete definition along with a picture of what works and what doesn’t. Share with the class which picture works and why. 2. Lets say I have a straight line and a dot above that line (show the picture to this) how many sets of parallel lines can I make with drawing a line through this dot. 1. Field responses from this question and reason for the certain answers then say ok well we have a couple of answers lets see if we can come up with the correct answer. 3. Ask for three volunteers then hand out the roles and string. Ok so what happens when the line looks like this? Have the three create this line and the class will answer when called upon what happens, they are not sure what will happen to the line it looks like it is parallel but it looks like we may have to extend the string, we do this and find out the lines are not parallel, we will do this again but to the other side and find out the same thing. Maybe we can’t make any parallel lines you say. Lastly we will move the string and find a place where it is parallel. Now ask the class what we found. 1. Take answers until you get that we found only one parallel line. The say there must be a postulate for this and have the class record this in their math journals. b. Triangle sum theorem- now we will move onto the triangle sum theorem, which states that, the sum of the measures of the angles of a triangle is 180 degrees. We will try to show the triangle sum theorem is true in a couple different ways. What do we know that is equal to 180 degrees? · Field answers to this, until you get a straight line, if they care not coming up with this answer say what about a straight line, how many degrees does that have? · Now we take our straight line and move the ends to form a triangle. What happened here? o Did we take our big angle and make three angles? o If so then what do the three angles, a, b, c add up to? § Ask the class this question o That’s right the angles add to 180 degrees because we took the angles we had and moved them about. · Now lets start with this triangle here, if the theorem is correct then these two angles when connected to the third angle should equal the straight line or 180 degrees · Ok so we take angle a and angle c and place the corners at the end of b, what happened here? o Did we make a straight line? Does that mean the three angles added equals 180 degrees? o (Go to the board for this question) Since we moved angles a and c, from the place they were at and to a new place then that means these angles are equivalent to one another, lets remember this idea for homework. · Now we can see this visually, next lets write this mathematically through a proof, lets record this proof and the definition of the triangle sum theorem in your math journal. · Perform the two column proof on the board while asking questions periodically about it. o Now class can you please write this down in your math notebooks as we are doing it. Along with the definition.

8. Assessments- formal, informal, objective, subjective-

a. Informal- 1. Watch and record to see which students are participating in the proof and class work for class participation grade 2. There will be an exit ticket that will be graded as a informal assessment, which has higher weight than just class participation 1. This informal assessment will be fair and will make sure the students are on the right track yet it will be a little more weight than just class participation for the fact that the students are completing them by themselves. This is also a gage to help me see how the students feel after the first day. b. Formal- the students will have a formal quiz and test later on in the chapter that covers this section.
 * The exit ticket is under the assessment section of the wikispace.

9. Technology and tools- materials needed- a. Paper, pencil b. Chalk board c. Computers and website- for homework d. Documenting camera - here I will show that taking a straight line we can form a triangle e. Exit ticket f. Warm up sheet/ question g. String

10. Homework- a. Students will complete the following homework for the next class period, and any class work they did not finish in class b. Students will explore the following website to gather more information and to get more comfortable with the triangle sum theory. Website for homework
 * http://hotmath.com/util/hm_flash_movie.html?movie=/learning_activities/interactivities/triangleSum.swf&return_to=Geometry%20Activities&title=Angle%20Sum%20Theorem





//Geometry, Triangle Congruency Grade 7 or 8// Students will observe the relationship between the properties needed to describe a polygon that is congruent compared to congruent triangles Students will discover the SSS postulate about triangle congruency Students will explore the meaning of triangle rigidity. How do geometric relationships help us solve problems in the real world, for all different professions? How are geometric shapes related to each other, specifically triangles? Why is learning about triangles important when trying to find properties in other polynomials? How can measurements be used to solve problems? Warm-up at the beginning having to do with polygon congruency Student’s responses to the questions in the two different discussion questions. The student’s ability to realize which triangles are the same and which are not, by how they get into their groups. Homework worksheet Straws String Scissors Homework Congruency with polygons. || The students will come in and get out their homework, which will be collected for grading, therefore the homework will go into the homework bin which has already been established. The grade from the homework will have more to do with participation then the actual grade “correctness” grade. This is also a way for the teacher to figure out where her students started in the chapter because the homework is being collected would have been the first or second homework of the chapter. After they hand in this homework the students should get their warm up book which will be a book that they can take with them, with warm up questions from each class, each page will have a new date at the top of the page. This will try and help organize students that may not be so organized. On the board there will already be these following problems,   1) Are the following polygons congruent? Why or why not, be prepared to explain your answer to the class so write your answer down now as well. a) b) 2) The following triangles are congruent; list all the parts that are congruent? ΔABC≅ΔNPM
 * Section 4.2 **
 * **__Date:__** Day 3+4, this lesson cannot be taught in one day, but can be taught either in a long block period or over two days.
 * __Math emphasis/objective:__**
 * __Learning Objective:__**
 * __Essential Questions:__**
 * __Assessment (formative and summative):__**
 * __Materials/Handouts:__**
 * __Standards:__** NCTM “In grades 6-8 all students should…create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship, and examine the congruence, similarity, and line or rotational symmetry of objects using transformations”
 * __Students with Disabilities:__** Students who have visual impairments might have a difficult time with the project. One way in which the class can help this person is to give them more time to use his/her sense of touch to figure out the length of the straws when comparing the straw lengths that he/she has to the ones that other students have. I think that it’s important to have other students in the classroom help the visual impaired student as well as having the teacher help because it give the students more of an understanding of the impairments. Therefore, I think that the students around the visually impaired students can help him/her string the straws together to help make the triangle. If the students was blind there could be another teacher in the classroom to help, if this is the case the teacher should use ideas of co-teaching to help include the other teacher into the lesson.
 * __They should already have an understanding of:__**
 * Time || Activity ||
 * 10 minutes || **__Content Review (ten minute math, warm up problem/question, test prep review, interactive homework review)__**

This warm up will have the students think about what they have learned yesterday about polygons and it will help them in the lesson today about triangle congruence. While the students are doing this, the homework will be handed in and the teacher will go around and see if the students understand the warm up. Having the homework being handed in is a good way of letting the teacher be free to walk around while the students are doing the warm up because they don’t have to deal with looking at the homework answers. After the students seem to be finishing up with the warm up, the teacher should go over the answers and have a class discussion on why the students think the polygons are congruent, as in 1a and not as in 1b. ||  || After looking at what was learned the day before about polygons and congruency, we know that if all the sides and angles are the same then the polygons are congruent. But, in real life we don’t always have all this information. But, some great mathematicians of their time have come up with theorems, which help to prove that we do not need the measurements of all the, sides and the angles. The following activity will help us discover this new theorem about triangles and their congruence to other triangles. ||
 * 15 min || **__Active Teaching:__ (include activities, questions to be asked, work to be reviewed, materials to be demonstrated):**

Students will come up to the front of the classroom where the straws and sting will be, and get three straws of their choice as long as a piece of string. There will be 4 different colored straws 4 different length respectively. The students will take three different straws of their choice and a piece of string. They will then have to go back to their desk and create a triangle of straws, they will then have to try and look for another person with the same set of straws. This will take some time and with this the students will have to figure out what this means. On the board there will be questions for the activity. These questions include “What do we know about the triangles that you have with the people you are now in a group with?” “What do we know about the triangles you have compared to the ones you are not in a group with?” “Which, if any are congruent to each other, remember that you need two shapes to show congruency because it is a relative term?” Triangle Other Polygon ||
 * 10 min || **__Explore and Apply:__ (include how students will be grouped, how materials will be distributed, how students will be assessed during work time, how students at different levels or experiences will be accommodated):**

The teacher now will introduce the terminology of SSS, there being three sides on a triangle that are the same indicates that the triangles are congruent. And how the S’s are used to describe a side.
 * 10 min || **Whole class work review/discussion** __**(remember transition time, include questions to be asked, summary of lesson/activity, make connections to math emphasis/objectives):**__
 * After the students talk about this for a bit, the teacher should be going around to help facilitate the discussion and the construction of triangles. Then as a class, the teacher will ask the students to look at the triangles and as them the questions they were asked before hand. The students should then be guided into discovering that with just the knowledge of three sides do we know that triangles are congruent.The teacher will then also tell that the class that that means that the triangles are rigid, because if you were to string the straw together, you can’t move any of the straws to change the shape. The students will work with other shapes for homework to help demonstrate this idea. Rigidity is the reason why there are so many structures that use triangles to support the buildings, because the triangles cannot move like other shapes can.

The next class, or if the class were 90 min long, then the teacher could include this, will be about the ASA, SAS postulates using graph paper to show that as long as there is an including angle or including side there is triangle congruence ** || The homework tomorrow night will be more with the comparison to the three different postulates and how to use the different postulates in proving triangle congruence. Tonight’s homework will be more conceptual to help them with the lesson tomorrow. Students will have to take straws and sting home to work with other shapes and their rigidity. ||
 * || **__Homework (reflective and thoughtful; related to work of the day or work to come):__**

Name ___ 1. Using more then 3 straws, build to shapes with the same straws. Indicate below what the shape looks like, a drawing will be the best way to describe this. Are the shapes congruent to each other? If not, why not? Are the shapes rigid?

2. When are triangles used in the real world? How does this have to do with the fact that the triangles are rigid?

3. If we know that the sides are two triangles are the same, what else can we say about the triangles? (Hint: think about the other parts of triangles)