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8th grade Instructional Unit

Unit 2 – Exponents & Square Roots Review
Unit 3 – Pythagorean Theorem Instructional Unit

by
Nancy Perzel

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Day 5 - Pythagorean Theorem Formula

Warm Up:

Begin by asking the students to find a partner to check their homework from last night. Then together they should write a short note to another friend who “missed class yesterday” giving a short summary of what they learned. Allow a few pairs to share; this will give you a good idea of where they are at, what they remembered and any misconceptions that might exist.

Activity 1:

At this point, it’s important to move towards a more concrete representation of the conclusions that were drawn in class yesterday… It’s time to bring it all together. The area of each square is the length of the side, squared. And the reverse is true. The length of the side of the square is equal to the square root of the area…

Moving towards the formula… replace the numbers with variables, calling the legs ‘a’ and ‘b’. As a class, label the areas accordingly… Make sure to remind the class of the difference between a (the length of the side), and a² (the area of the square of the side). This can be a point of confusion later on.

Now if we call the hypotenuse, c, the following holds true a² + b² = c²

Activity 2:

Proof of the Pythagorean Theorem and formula.

For a description of 98 (!) different proofs of the Pythagorean Theorem click here.

There are many Pythagorean Theorem Proof videos to choose from on Teacher Tube

Click here for a great electronic resource of interactive animations to explore the different proofs of the Pythagorean Theorem. Moving figures and using manipulatives are a great way to help students grasp the meaning of mathematical ideas intuitively.   

Activity 3:

Now it’s time to begin finding the length of the hypotenuse without relying on manipulatives.

What is the length of the hypotenuse for a right triangle with a base equal to (7) and a height equal to (8)?

7² + 8² = c²

49 + 64 = 113

c² = 113 -> Clarify that this number represents the area of the square on the side of the hypotenuse… to find the length of the hypotenuse, you must take the square root of this value.

1) An estimate of the √113 is between √100, 10 and √121, 11. A good estimate would be 10.5.

2) Students can use calculators to compute the exact answer. For this example, the exact value is 10.6301458127… rounded to the nearest tenth is 10.6

Additional Examples:

Provide an estimate of the value for the length of the hypotenuse before using a calculator!

1) a = 12, b = 13

2) a = 11, b = 4

3) a = 12, b = 12

4) a = 16, b = 10

5) a = 9, b = 11

6) a = 12, b = 21

Additional Technology Resources:

- http://www.flocabulary.com/pythagorean-theorem/

- http://www.brainpop.com/math/geometryandmeasurement/pythagoreantheorem/preview.weml

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