The Pythagorean Theorem and Special Right Triangles



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

▪ 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. 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.

o Ask students to summarize individually in their math journals what they learned today.

• Day 2

o Pass out Extension Sheets on 30°, 60°, right triangles and 45°, 45°, right triangles.

o Have students work in same groups on these activities.

o At conclusion of activities, debrief as a whole class.

o 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

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 |a2 |b2 |c2 |

|3 |4 |5 | | | |

|7 |10 |[pic] | | | |

|6 |8 |10 | | | |

|5 |[pic] |14 | | | |

[pic]

2.) What relationship do you notice with the areas of the three squares?

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 [pic]for any integers [pic]and [pic]. 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.

a. ) [pic]b.) [pic]c.) [pic]

8.) Summarize, in your own words, what you have learned today.

Real World Application: Play Ball!

[pic] [pic]

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.

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.

[pic]

|a |b |c |a2 |b2 |c2 |

|1 |[pic] | | | | |

|3 | |6 | | | |

| |5[pic] |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.

Using the TI-Nspire

From the Home Menu, select Lists & Spreadsheets. [pic]

[pic]

Label the first column “sl” (for short leg), the second column “ll” (for long leg), and the third column “hyp” (for hypotenuse).

[pic]

Type in the formula “=a([pic])” in the formula line of column b.

[pic]

Enter any values you wish into column a to test your conjecture.

[pic]

Was your conjecture accurate? Explain.

4.) What conclusions can you make about 30-60 right triangles?

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?

Extension 2

1.) Complete the table for the special right triangle below. Express irrational values in simplest radical form.

|a |b |C |a2 |b2 |c2 |

|5 |5 | | | | |

|3 | |3[pic] | | | |

| | | |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?

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?

-----------------------

60º

30º

c

a

b

100 cm

164.01 cm

a

119 ft

c

57 ft

60 ¾ in

57 in

b

45º

45º

a

b

c

90 ft.

90 ft.

2nd Base

X

3rd Base

1st Base

90 ft.

90 ft.

Home Plate

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