Unit
Title: Geometry – Right Triangle
Properties from
Pythagorean
Theorem to Trigonometry
8^{th}
Grade Algebra/Geometry
Description:
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.
Goalsgeneral outcome(s)
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
measurement.
Objectives
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.3.3.1.8.2: The student represents and applies geometric
properties and relationships to solve realworld and mathematical
problems. The students apply the
Pythagorean theorem in realworld problems and find the relationship among
sides in a 4545 and 3060 right triangles.
Analyze
characteristics and properties of two and threedimensional geometric
shapes and develop mathematical arguments about geometric relationships 

Materials
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 Projector
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
Measuring Tape
Agenda Notebooks
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
Do Now:
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:
Math Agenda
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 a^{2}. 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 a^{2 }+ b^{2}
= c^{2}. 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.
Lesson procedure
The students will return to
nongroup 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.
Closure
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!
Assessment
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.
Homework
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.
Accommodations
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 stepbystep 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.
Reflections
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: a^{2 }+ b^{2}
= c^{2}
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.