Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
Lesson Plan: Pythagorean Theorem and Special Right Triangles
Lesson Summary
This lesson will provide students with the opportunity to explore concepts including
the Pythagorean Theorem, rational and irrational numbers, and the relationship of
side lengths of special right triangles.
Key Words
 Hypotenuse
 Irrational Numbers
 Legs
 Pythagorean Theorem
 Rational Numbers
 Right Triangles
 Simplest Radical Form
Background Knowledge
We are assuming the students are proficient at calculating square roots and squares,
solving multi-step equations as well as basic quadratic equations. We are also
assuming they have a strong background in basic geometry concepts including
finding area and perimeter and classifying angles. Lastly, we assume students have
basic graphing calculator proficiency in finding squares, finding square roots, and
using the table function.
NCTM Standards Addressed
 Geometry and Spatial Sense Standard: Students identify, classify, compare
and analyze characteristics, properties and relationships of one-, two- and
three-dimensional geometric figures and objects. Students use spatial
reasoning, properties of geometric objects and transformations to analyze
mathematical situations and solve problems.
o Grade 7: Characteristics and Properties
 Indicator #3: Use and demonstrate understanding of the
properties of triangles.
For example:
a. Use Pythagorean Theorem to solve problems involving right
triangles.
b. Use triangle angle sum relationships to solve problems.
o Grade 8: Characteristics and Properties
Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
Indicator #1: Make and test conjectures about characteristics and
properties (e.g., sides, angles, symmetry) of two-dimensional figures
and three-dimensional objects.
Number, Number Sense and Operations Standard: Students demonstrate
number sense, including an understanding of number systems and operations
and how they relate to one another. Students compute fluently and make
reasonable estimates using paper and pencil, technology-supported and mental
methods.
o Grade 7: Number and Number Systems
 Indicator #3: Describe differences between rational and irrational
numbers; e.g., use technology to show that some numbers
(rational) can be expressed as terminating or repeating decimals
and others (irrational) as non-terminating and non-repeating
decimals.


Learning Objectives
 Understand Pythagorean Theorem
 Apply Pythagorean Theorem to real-world scenario
 Understand the difference between rational versus irrational numbers
 Recognize the relationships of side lengths in special right triangles
 Apply knowledge of special right triangles to real-world scenarios
Materials
 Ziploc bags containing colored straws of different lengths. Use lengths so that
not all combinations will result in a triangle.
 Calculators
 Activity and extension activity sheets
 Pencils
Suggested Procedure
 The time frame for this lesson is 2 days.
 Day 1
o Group students into heterogeneous groups of 3.
o Attention Getter: Pass out Ziploc bags with straws to students and pose
the question “Can you get a right triangle from any 3 side lengths?” Ask
students to work individually on this question and then discuss their
findings with their group members. Facilitate class discussion after 2
minutes.
o Pass out Pythagorean Theorem activity sheet and have students work on
this activity sheet within their groups. Walk around and observe.
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o

Day
o
o
o
o
Dr. Antonio R. Quesada
Director, Project AMP
Remember to let students determine conjectures and solutions – do not
give answers! After completion of this activity sheet, facilitate a debrief
discussion of the activity.
Ask students to summarize individually in their math journals what they
learned today.
2
Pass out Extension Sheets on 30°, 60°, right triangles and 45°, 45°, right
triangles.
Have students work in same groups on these activities.
At conclusion of activities, debrief as a whole class.
Ask students to summarize individually in their math journals what they
learned today.
Assessments
 Observation
 Class Discussion
 Written solutions and explanations
 Math Journals
 Future quiz and/or test
The Pythagorean Theorem and Special Right Triangles
Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
Group Members ____________________________
______________________________
____________________________
______________________________
Before We Begin
Can a right triangle be formed using any three lengths of sides?
Select three straws from the bag. Use the corner of a piece of notebook paper for a
right angle. Use two of your straws placed along the edges of your notebook paper as the legs
of the right triangle. The third straw will form the hypotenuse of the right triangle. Do your
straws form a right triangle? Compare your results with those of your members. How many
of you were able to form a right triangle?
Lesson 1
We recall the area formula for a square is A = s2. Using the dimensions given, complete
the table by finding the area of the squares in the diagram below.
1.)
a
b
c
3
4
5
7
10
149
6
8
10
5
171
14
a2
2.) What relationship do you notice with the areas of the three squares?
b2
c2
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Dr. Antonio R. Quesada
Director, Project AMP
3.) Based on the patterns observed in the table, what conjecture can you make?
4.) Notice two of the dimensions in the table are expressed as radicals, why do you think
they are expressed as radicals and not in decimal form?
5.) When writing the decimal form, is it possible to give an exact answer? Why or why not?
Definition:
Irrational number – An irrational number is a number that cannot be expressed as a
fraction
for any integers and . Irrational numbers have decimal expansions that neither
terminate nor repeat.
For the remainder of this lesson, you may want to express irrational answers as decimal
approximations (for real world applications) or in simplest radical form (to express a more
precise value).
6.) Now, create your own dimensions. Does your conjecture still hold?
7.) Now, use that relationship to find the missing length in each of the right triangles below.
119 ft
60 ¾ in
57 ft
b
164.01 cm
c
100 cm
57 in
a. )
a
b.)
c.)
8.) Summarize, in your own words, what you have learned today.
Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
Real World Application: Play Ball!
2nd Base
90 ft.
90 ft.
X
3rd Base
1st Base
90 ft.
90 ft.
Home
Plate
The distance from each consecutive base is 90 feet and it can be necessary to
determine how far the catcher will have to throw to get the ball from home plate to
2nd plate or the distance of the throw from the 3rd baseman to 1st base. We will use
the Pythagorean Theorem to find answers to these questions.
1. What other shape is the baseball diamond?
2. Explain how you might use the Pythagorean Theorem to determine a value for x.
Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
3. Where is a right triangle located?
4. Determine what side of the right triangle is the hypotenuse.
5. Determine the legs of the triangle.
6. Now, set up the problem so that you can use the Pythagorean Theorem to find out how far the
catcher will have to throw the ball from home plate to 2nd base to the nearest foot.
7. How far will the third baseman have to throw to get to 2nd base. How can you determine this
answer from the work that you have already done. Explain.
Extension 1
1.) Complete the table for the special right triangle below. Express irrational values in
simplest radical form.
Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
b
30º
a
c
60º
a
1
b
c
a2
b2
c2
3
3
6
5 3
10
144
2.) Do you notice a relationship between the lengths of the sides of this special right
triangle?
3.) Express the above relationship in terms of x.
Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
Using the TI-Nspire
From the Home Menu, select Lists & Spreadsheets.
Label the first column “sl” (for short leg), the second column “ll” (for long leg), and the
third column “hyp” (for hypotenuse).
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Dr. Antonio R. Quesada
Type in the formula “=a( 3 )” in the formula line of column b.
Enter any values you wish into column a to test your conjecture.
Was your conjecture accurate? Explain.
4.) What conclusions can you make about 30-60 right triangles?
Director, Project AMP
Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
Real world application – Locked Out!
José has locked himself out his house. Fortunately, he did leave an
upstairs window open and does have access to an extension ladder. The
ladder, when fully extended safely, will extend to 24 feet. For optimal
safety reasons, he wants to maintain a 60-degree angle with the ground.
If he extends the ladder to 18 feet, (a) how far will the base have to be
away from the wall and (b)how high up on the wall will the ladder reach?
Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
Extension 2
1.) Complete the table for the special right triangle below. Express irrational values in
simplest radical form.
45º
c
a
45º
b
a
b
5
5
3
C
a2
b2
c2
3 2
49
1
2.) Do you notice a relationship between the lengths of the sides of this special right
triangle?
3.) What conclusion can you make about 45-45 right triangles?
Project AMP
Dr. Antonio R. Quesada
Director, Project AMP
Real World Application
6 feet
An Isosceles Right Triangle is the shape of the opening of a tent. What is the largest
sized object that can be put through the opening without touching the sides of the
opening?