L5+O'Halloran,Erin+Marissa


 * UNIVERSITY OF MAINE AT FARMINGTON**
 * COLLEGE OF EDUCATION, HEALTH AND REHABILITATION**
 * LESSON PLAN FORMAT**


 * Teacher’s Name: Ms. Erin O'Halloran** **Date of Lesson: 5 (Perspective)**
 * Grade Level: 10** **Topic: Pythagorean Theorem**

__**Objectives**__

 * Student will understand that** computing areas and volumes is based on the object's properties and dimensions.
 * Student will know** Pythagorean Theorem, right angles, right triangles, legs of a triangle, hypotenuse, proof Pythagorean triples.
 * Student will be able to** analyze the Pythagorean Theorem and how it can be used to solve problems.

__**Maine Learning Results Alignment**__
//** Maine Learning Results: Mathematics- C. Geometry Geometric Figures Grades 9 - Diploma Students justify statements about polygons and solve problems **//


 * Rationale: By analyzing Pythagorean Theorem and the importance of proofs in Geometry. Students will understand that, for this time period, the idea of the Pythagorean Theorem is one of the discoveries of all time.**

__**Assessment**__
Students will be pre-assessed on their intelligence on right triangles. They will need to identify the parts of a right triangle. Depending on how well students do, there may be a review session on parts of triangles and the importance of right triangles.
 * Formative (Assessment for Learning)**

Students will use several graphic organizers to organize their thoughts. They will have one dedicated to solving problems dealing with Pythagorean Theorem and another dedicated to proving Pythagorean Theorem. Students will complete these graphic organizers together in a Think-Pair-Share grouping and the teacher will come around to collect the organizers, making sure the students easily organized the Theorem.

Once students have understand the Theorem, they will need to create a rap about Pythagoras and his famous theorem. Students will use Garage Band or Audacity to create the sound files. Students are allowed to use the information they learned in class, or they can research to find more information on Pythagoras. Students can mention the Theorem itself, other theorems Pythagoras discovered, and/or the life of Pythagoras. The raps/music files will be posted on the class wiki. The rap/music piece should between 1:30-2:00 minutes long. This project is worth 10% of the students' final grades and will be scored out of 100 points with a checklist.
 * Summative (Assessment of Learning)**

__**Integration**__
Technology: The rap or music piece will be created using Garage Band or Audacity. The groups will need a microphone, either built into the computers or and external microphone, so they can record their voices. Students can add music and effects to their voices using the modes in Garage Band and Audacity. A good site to find special effects is http://www.freesound.org where students can download free files of effects rather than record themselves.

History: Students have the option of including facts about Pythagoras in their raps. There seems to be some controversy whether or not the famous Pythagorean Theorem was actually discovered by Pythagoras. This controversy and other interesting facts could be mentioned in the rap.

__Groupings__
Students will be split into Think-Share-Pair groupings during class time. Students will be given problems on the board dealing with Pythagorean Theorem and right triangles. When the two problems are presented in class, students will have time to solve one of the math problems themselves. Once a certain amount of time has passed, students will pair with another partner who did the same problem to discuss it and how they solved it. Then finally the pairs will discuss the problem with another pair who did a different problem and vice versa.

__**Differentiated Instruction**__
**Linguistic**: Have students make a script of what they are going to say during the rap and have them pass it in for evaluation. Students will have the option to create a story board so they can organize their thoughts about what is going to be said. **Intrapersonal**: Have each student make a rap on their own rather than in groups. Students have the option of creating the rap on their own if they want but it is suggested that they work in groups. **Interpersonal**: It is suggested that students work in groups of two or three. Students can work together to determine what topics they want to discuss in the rap. **Bodily**/**Kinesthetics**: Have students make a dance or video along with the rap and have the whole class learn it. Students who do this can perform it in front of the class or record it and present the video in class. **Musical**: Students can have their rap rhyme so that it is catchy and easy to remember. They could pass out the lyrics of their rap to the rest of the class and have the rest of the class sing along to the rap. **Natural**: Students have the option of including facts and the time period of Pythagoras in the rap. This way, students can take a more historical approach to the rap.
 * Strategies**

//**I will review student’s IEP, 504 or ELLIDEP and make appropriate modifications and accommodations.**//
 * Modifications/Accommodations**


 * Absent:** If students are absent, then they will need to get the notes of Pythagorean Theorem and the proofs of the Theorem from another student in the class. Should the absent student have a question, they can email the teacher and set up an appointment. If student is absent when the project is assigned, he/she would need to contact the teacher and they will receive the assignment. Because the raps will not be presented in class, an extension could be made pending the reason for the student's absence.


 * Extensions:** Students will use Garage Band or Audacity to create a rap using the effects included in the software. Sound clips can be created by the students themselves or from sites such as freesound.org. The teacher would challenge students by having them apply their intelligence to create situations based on how they learn: linguistic, logical, spatial, kinesthetic, musical, interpersonal, intrapersonal, or naturalist. The students can also create a music video if they so desire using iMovie. The music video may include video from the students themselves or clips from online.

> ==__Materials, Resources and Technology__ == >
 * LCD Projector
 * Computer to show software and slides
 * Text book
 * Hand outs
 * Graphic organizer
 * Checklist
 * Story board template
 * Garage Band tutorial

__Source for Lesson Plan and Research__
[|Pythagorean Theorem] ; includes information on what Pythagorean Theorem is, three different proofs, Pythagorean triples, Pythagoras and Euclid. Properties of[| right triangles] ; legs, hypotenuse, right angle

__**Maine Standards for Initial Teacher Certification and Rationale**__

 * //Standard 3 - Demonstrates a knowledge of the diverse ways in which students learn and develop by providing learning opportunities that support their intellectual, physical, emotional, social, and cultural development.//**
 * Rationale:** **Beach Ball:** Students get to find their own resources in order to find out important or interesting facts about Pythagoras. The project can be taken several ways including describing Pythagorean Theorem, describing Pythagorean's life, or describing the setting of Geometry in Pythagoras' time. Students do not have to make a rap, but they can represent the information in another musical way. They can create a rhyme with sound effects- as long as Garage Band and effects were used.
 * Clipboard:** Because students will have the chance to fill out graphic organizers, the information provided in class will be organized and correct. Students will have time in class to work on their Garage Band product during class. This way, students have a clear procedure of how to work with the program.
 * Microscope:** Students will find their own resources if they chose to cover more on Pythagoras' life. They can use many different resources, as long as the information if backed up. Students will get to work on their projects together or by themselves during class. This way, they can discuss their procedure with other students or the teacher.
 * Puppy:** There will be class time for the students to work on their project. Students will be working together, if they choose, on their project during class time. They can bounce ideas off each other.


 * //Standard 4 - Plans instruction based upon knowledge of subject matter, students, curriculum goals, and learning and development theory.//**
 * Rationale:** Refer to content notes for what students will know. // Students will be able to analyze the Pythagorean Theorem and how it can be used to solve problems. // The facet of learning for this lesson is perspective. In mathematics, there is really no room for opinions; the information is either correct or incorrect. However, when discussing proofs, opinions can sometimes form. Depending on certain people, points of view can be very important when discussing who discovered what proof. Students can choose to further research Pythagoras and they may come across a theory that Pythagoras stole a theorem from another philosopher. Students can express their opinions in their final product.


 * //Standard 5 - Understands and uses a variety of instructional strategies and appropriate technology to meet students’ needs.//**
 * Rationale:**

**Linguistic**: Have students make a script of what they are going to say during the rap and have them pass it in for evaluation. Students will have the option to create a story board so they can organize their thoughts about what is going to be said. **Intrapersonal**: Have each student make a rap on their own rather than in groups. Students have the option of creating the rap on their own if they want but it is suggested that they work in groups. **Interpersonal**: It is suggested that students work in groups of two or three. Students can work together to determine what topics they want to discuss in the rap. **Bodily/Kinesthetics**: Have students make a dance or video along with the rap and have the whole class learn it. Students who do this can perform it in front of the class or record it and present the video in class. **Musical**: Students can have their rap rhyme so that it is catchy and easy to remember. They could pass out the lyrics of their rap to the rest of the class and have the rest of the class sing along to the rap. **Natural**: Students have the option of including facts and the time period of Pythagoras in the rap. This way, students can take a more historical approach to the rap.


 * //Standard 8 - Understands and uses a variety of formal and informal assessment strategies to evaluate and support the development of the learner.//**
 * Rationale:** Using both the formative and summative assessments assigned for this lesson, the teacher can determine if the students fully understand the different polygons and their properties. Because the formative and summative assessments are using two different sides of the brain, students with different learning styles will be able to learn the way they want to.


 * Formative (Assessment for Learning)**

Students will be pre-assessed on their intelligence on right triangles. They will need to identify the parts of a right triangle. Depending on how well students do, there may be a review session on parts of triangles and the importance of right triangles.

Students will use several graphic organizers to organize their thoughts. They will have one dedicated to solving problems dealing with Pythagorean Theorem and another dedicated to proving Pythagorean Theorem. Students will complete these graphic organizers together in a Think-Pair-Share grouping and the teacher will come around to collect the organizers, making sure the students easily organized the Theorem.

**Summative (Assessment of Learning)** Once students have understand the Theorem, they will need to create a rap about Pythagoras and his famous theorem. Students will use Garage Band or Audacity to create the sound files. Students are allowed to use the information they learned in class, or they can research to find more information on Pythagoras. Students can mention the Theorem itself, other theorems Pythagoras discovered, and/or the life of Pythagoras. The raps/music files will be posted on the class wiki. The rap/music piece should between 1:30-2:00 minutes long. This project is worth 10% of the students' final grades and will be scored out of 100 points with a checklist.

__Teaching and Learning Sequence__
For classroom arrangement, I will have students sit in a perimeter so they everyone can see the screen without other peoples' heads in the way. It is vital to see which polygon is being discussed during the lecture to minimize confusion. During the Think Pair Share exercise, the desks will be arranged in groups depending on how many are in the group. The notes for my class will be displayed from a computer using an LCD projector.

Pythagorean Theorem, right angles, right triangles, legs of a triangle, hypotenuse, proof Pythagorean triples.

Day 1: Pre-assess students on right triangles' parts (15 minutes) Review parts of triangles if necessary: legs, hypotenuse, angles (10 minutes) Introduce Pythagorean Theorem (15 minutes) Proof 1 of Pythagorean Theorem (10 minutes) Proof 2 of Pythagorean Theorem (10 minutes) Think Pair Share exercise (20 minutes)

Day 2: Introduce Garage Band assignment and assigned groups (5 minutes) Show students how to use Garage Band (15 minutes) Have students work on the rap/music file (60 minutes)


 * Content Notes**

The Pythagorean Theorem is a statement about triangles containing a right angle. The Pythagorean Theorem states that:

**//"The area of the square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides."//**



**Figure 1** According to the Pythagorean Theorem, the sum of the areas of the two red squares, squares A and B, is equal to the area of the blue square, square C.

Thus, the Pythagorean Theorem stated algebraically is:

for a right triangle with sides of lengths a, b, and c, where c is the length of the hypotenuse.

Although Pythagoras is credited with the famous theorem, it is likely that the Babylonians knew the result for certain specific triangles at least a millennium earlier than Pythagoras. It is not known how the Greeks originally demonstrated the proof of the Pythagorean Theorem. If the methods of Book II of Euclid's **//Elements//** were used, it is likely that it was a dissection type of proof similar to the following:

"A large square of side a+b is divided into two smaller squares of sides a and b respectively, and two equal rectangles with sides a and b; each of these two rectangles can be split into two equal right triangles by drawing the diagonal c. The four triangles can be arranged within another square of side a+b as shown in the figures.

The area of the square can be shown in two different ways: >>
 * 1.** As the sum of the area of the two rectangles and the squares:

>> Now, setting the two right hand side expressions in these equations equal, gives >> >>
 * 2.** As the sum of the areas of a square and the four triangles:

Therefore, the square on c is equal to the sum of the squares on a and b. (Burton 1991)

Find all right triangles whose sides are of integral length, thus finding all solutions in the positive integers of the Pythagorean equation:



The three integers (x, y, z) that satisfy this equation is called a Pythagorean triple. >> **x y z** >> >> >> **3 4 5** >> >> >> **5 12 13** >> >> >> **7 24 25** >> >> >> **9 40 41** >> >> >> **11 60 61** >> >> **Proposition:** In right-angled triangles the square on the hypotenuse is equal to the sum of the squares on the legs. >> >> >> **Figure 2** >> Euclid began with the Pythagorean configuration shown above in Figure 2. Then, he constructed a perpendicular line from C to the segment DJ on the square on the hypotenuse. The points H and G are the intersections of this perpendicular with the sides of the square on the hypotenuse. It lies along the altitude to the right triangle ABC. See Figure 3. >> >> >> **Figure 3** >> Next, Euclid showed that the area of rectangle HBDG is equal to the area of square on BC and that the are of the rectangle HAJG is equal to the area of the square on AC. He proved these equalities using the concept of similarity. Triangles ABC, AHC, and CHB are similar. The area of rectangle HAJG is (HA)(AJ) and since AJ = AB, the area is also (HA)(AB). The similarity of triangles ABC and AHC means >> >> >> >> and therefore >> >> >> >> or, as to be proved, the area of the rectangle HAJG is the same as the areaof the square on side AC. In the same way, triangles ABC and CHG are similar. So >> >> >> >> and >> >> >> >> Since the sum of the areas of the two rectangles is the area of the square on the hypotenuse, this completes the proof. >> >> Euclid was anxious to place this result in his work as soon as possible. However, since his work on similarity was not to be until Books V and VI, it was necessary for him to come up with another way to prove the Pythagorean Theorem. Thus, he used the result that parallelograms are double the triangles with the same base and between the same parallels. Draw CJ and BE. >> >> >> >> The area of the rectangle AHGJ is double the area of triangle JAC, and the area of square ACLE is double triangle BAE. The two triangles are congruent by SAS. The same result follows in a similar manner for the other rectangle and square. (Katz, 1993) >>
 * __Some Pythagorean Triples__:**


 * Handouts**