Unit Title: Geometry – Right Triangle Properties from
Pythagorean Theorem to Trigonometry
8th Grade Algebra/Geometry
The students will first begin with a lesson initiating activity that will spark their interest in the Pythagorean theorem. It is a hands on activity that will introduce the concept. In this lesson the students will be introduced via power point presentation about the history of the Pythagorean theorem and how and who discovered it. They will complete an outdoor activity in which they will measure student heights and the lengths of their shadows and other right triangle objects such as trees, light posts, et. The students will collect the data and then come back into the classroom to find the distance from the shadows to the top of the objects. Students will discuss methods and strategies used to solve the problems. A review will be completed and then homework assigned.
Know and understand the
Pythagorean theorem and its converse and use it to find the length of the
missing side of a right triangle and the lengths of other line segments and, in
some situations, empirically verifies the Pythagorean theorem by direct
The student demonstrates the Pythagorean relationship in right triangles using models or diagrams (for example, manipulatives, dot, graph, or isometric paper).
The student will apply the Pythagorean theorem in real world problems through hands on activities and projects.
The students will represent and apply geometric properties and relationships to solve real world and mathematical problems.
Sunshine State Standards (or your State Standards) addressed in the lesson.
MA.C.3.3.1 The student represents and applies geometric properties and relationships to solve real world and mathematical problems.
MA.C.18.104.22.168.2: The student represents and applies geometric properties and relationships to solve real-world and mathematical problems. The students apply the Pythagorean theorem in real-world problems and find the relationship among sides in a 45-45 and 30-60 right triangles.
Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships
Right Triangle in Graph/Grid Paper reproduced one for each group so at least 15 copies (always good to have extra).
Graph or Grid Paper for each student (30 copies)
Overhead Transparency of the example used for the Initiating Activity
A second Overhead Transparency with the shapes cut out for the students to demonstrate the activity.
Scissors, Pencils, Graph Paper, Ruler
Students need paper, pencils, and portfolios
Dry Erase Board/Markers or Chalk Board/Chalk
Computer/Lap Top with Power point software for Power Point Presentation
Internet Access for the links from within the power point presentation
Grouping of students
When the students first come
in the classroom they are in rows all facing forward working independently
copying the agenda for the day. As the
transition to the initiating lesson activity begins the students will move their
desks into groups of three or four (depending on how large the class and what
works best). After the activity they
are back in rows working independently during instruction of the lesson. The final activity outside which will
reinforce the concept requires two partners to work together. Once back in the classroom students are in
rows working individually through the example homework problems during the
ending review of the concept.
Time Block Schedule 90 minutes
I always have an Agenda on the board so that the students can copy the activities for the class period in their Agenda notebooks. They complete any activity we didn’t finish at home. This also serves the purpose so that they will all know what activities will be taking place during the class period. As soon as the students come into the classroom this is the first thing they copy down and complete until first directions are given by the teacher to begin the lesson. It generally looks something like this:
Pythagorean Theorem Demonstration Group Activity
Power Point Presentation / Questions & Answers
Outside Activity Reinforcing Pythagorean Theorem Lesson
Review the Pythagorean Theorem Homework
Motivation/Lesson Initiating Activity
The students will complete a lesson initiating hands on group activity in which they will cut out a triangle on graph/grid paper (1 copy given to each group). Showing a transparency example, explain that the group will make three squares with sides that are equal to each side of the triangle with given grid paper. Begin with side a. Measure the length of side a with a ruler. On the blank piece of grid paper, draw a square with sides that are the same length as side a. Label this square a2. Repeat these steps to create squares for sides b and c. Cut out the squares. Place each square next to the corresponding sides of the triangle. Now show that a2 + b2 = c2. Place the squares made from sides a and b on top of square c. You will have to cut one of the squares to get a perfect fit. Have a group come up to the overhead projector and show how they completed the activity for the entire class to see. As a teacher, continuously circulate and assist students to make sure each one has completed the activity and comprehended the concept.
The students will return to non-group setting all facing forward for an introduction to the Pythagorean Theorem through the use of a power point presentation. The students will take notes and write five questions on a separate piece of paper as they learn about the mathematician Pythagoras and the proof of the theorem and how it works. (See information that will be put onto power point slides enclosed). Have students also read the information orally. After the presentation explain to the students that the next step will be that they will pass their questions to the student sitting beside them as many times as needed so that no one has they’re own questions. Using a timer allow the students three minutes to answer the questions. Instruct the students that once the time is up you will ask students to volunteer questions and answers they have. Write the questions and answers on an overhead transparency. Make sure specific questions about Pythagoras and the theorem are answered such as “Give two examples of Pythagoras beliefs that lead to his theorem.” “Was the Pythagorean theorem first discovered by Pythagoras?”, “How do we prove the theorem?”. Instruct students to turn in questions and answers with their names on the papers and collect for a grade.
Next, Explain that the students will be going outside and completing a group activity (in groups of two/three). They will need the following materials: a measuring tape, paper, and pencil. Have the students write the instructions and steps of the activity down in their notes, which they will take with them outside. Step one: Measure the height of your partner and then measure the length of their shadow. Using the Pythagorean theorem you will find the diagonal distance from the end of the shadow to the top of your partner’s head. If time permits allow students to do the same with other objects around them such as posts, signs, or other objects that are measurable forming right triangles (remember two sides must be measured). Ringing a bell that they are accustomed to the students know to stop, look, and listen as the teacher explains that they will go back into the classroom.
Once the students are in the classroom ask students to volunteer to go up to the board and draw a diagram of what they measured outside of their partner’s height and shadow. Next call upon a student to come up and solve the solution using the Pythagorean Theorem. Hand out to the students a teacher created homework packet for students to complete. Go over one problem from the homework with the students as a review from the lesson. Ask a student to volunteer to read the problem. Use the overhead transparency to complete the problem with the class. As students are following the steps ask questions such as “How will we solve this problem?” “What formula needs to be used?”, “What values will need to be substituted into the formula?”, “What algebra concepts/steps will need to be used in order to find the solution?”. If time permits allow students to begin homework problems in class, although I usually never have time left for the students to begin their homework in class!
Throughout all the activities I always have a class roster in which I can check off that a student has shown to me they have comprehended the concept and proved so by successfully completing the given activity and/or when they demonstrate and answer any posed questions relating to the concept. Also students will be assessed through their individual homework and results of any post test given.
FCAT Practice Problems Packet in which students will be required to answer multiple choice questions where they must bubble in answers, record their answers in FCAT style grids, and answer short response questions in which they must show their work and explain in words how they arrived at their solutions to the problems. A problem will also be given in which they must explain why the situation is not possible - explaining what is wrong with the problem and/or what part of the problem was an error made in the steps leading up to the solution.
Some sample problems: Given diagrams of a baseball field and marked with different dimensions ask students how far would it be from home plate to second base. Given the diagram of the ladder and house with dimensions labeled ask the students how they would determine the missing length. A student who is five feet tall standing outside has a shadow of four feet in length, what is the distance from the end of the shadow to the top of the person’s head. Anne lives 12 miles west and 5 miles south of her friend Mary. There is a trail between their homes (looking at the drawing given with the dimensions) find the shortest distance between their houses. A tree, which is 15 feet tall, casts a shadow. The distance from the top of the tree to the end of the shadow creates a diagonal, which measures 25 feet. What is the measure of the tree’s shadow? Again the answers will be in multiple choice, grid response, and think/solve/explain short response. This gives them practice for the F.C.A.T. at the same time practice learning the concept.
What modifications could be made to the lesson for students with learning disabilities or L.E.P. students? For ESE/LEP students some suggestions would be to actually give them a copy of the notes/slides of the power point presentation, pair the students up with students who are strong in the subject area, make sure those students are seated at the front of the classroom/nearest to the teacher. Give the students extra assistance and support during group/individual activities. Pair up students during individual activities. Also have step-by-step directions printed for them in large print if needed. Allow students more time to complete assignments, shorten assignments, and give students pictures and diagrams of each activity problems. Assess students through work completed in portfolios or during assigned activities and/or during verbal responses.
To be completed after the lesson is taught. What went well? What needs to be changed for the next time?
I will let you know when I actually teach the lesson and get the results!!!
Overhead Transparency of Activity and A second overhead with shapes cut out so those students can demonstrate the activity.
Power Point Presentation Information
Pythagoras of Samos (560 - 480 BC)
Pythagorean Theorem is what you have just learned by completing the previous group activity. The way to find the measure of the length of a right triangle when only given the value for two sides is to use this the Pythagorean theorem: a2 + b2 = c2
Let’s find out more information about the man who proved this theory and how he did so. In the next slides will link to the following websites about Pythagoras and the theorem. http://virtualscholar.com/geom/geom1.htm. Where you will take notes and come up with five questions relating to the lesson that you will write on a separate piece of paper.