Welcome to the

Welcome to the

This 6-Hour Module focuses twoConcepts from the Next Generation

Sunshine State Standards

http://www.floridastandards.org/index.aspx

8

Big Idea #2Geometry

MA. 8 .G. 2. 4

Next Generation Sunshine State Standards

Subject Area:Grade Level:

Supporting Idea/Big Ideas:

Benchmark:

Mathematics

Benchmark

MA. 8. G. 2. 4

Validate and apply Pythagorean theorem to find

distances in real world situations or between points in the coordinate plane .  

8

Supporting Idea #6Number and Operations

MA. 8 .A. 6. 2

Next Generation Sunshine State Standards

Subject Area:Grade Level:

Supporting Idea/Big Ideas:

Benchmark:

Mathematics

Benchmark

MA. 8. A. 6. 2

Make reasonable approximations of square roots and mathematical expressions that include square roots, and use them to estimate solutions to problems and to compare mathematical expressions involving real number s and radical expressions.  

3 – Two Hour Parts

This lesson will be divided into 3 parts

Pythagorean’s Theorem and Square Roots by Hand (2 Hours)

Calculators in the Middle School Classroom (2 Hours)

Validate, Explore, Practice (2 Hours)

Before continuing with this Power Point…

Please take the

Content PretestContent Pretest for Block 13.docx

Because triangles are seen everywhere!

Why Do I Have to Learn This?

The Flatiron Building

• Located in New York

• One of the first skyscraper (1902)

• Sits on a triangular island

http://www.prometheanplanet.com/server.php?show=ConResource.20665

The Golden Gate Bridge

•San Francisco Bay

•Originally the longest suspension bridge in the world when it was completed during the year of 1937

•Since its completion, the span length has been surpassed by eight other bridgeshttp://www.prometheanplanet.com/server.php?show=ConResource.20665

The Louvre Pyramid

•Paris, France

•Glass and Metal Pyramid

•Main entrance to the museum

•Completed in 1989

•Landmark for the city of Parishttp://www.prometheanplanet.com/server.php?show=ConResource.20665

The Epcot

A theme park at Walt Disney World

Second part built

In 2007, Epcot was ranked the third-most visited theme part in the United States, and the sixth-most visited in the world.

http://www.prometheanplanet.com/server.php?show=ConResource.20665

One Way to Validate Pythagorean’s

Theorem

http://users.ucom.net/~vegan/images/Pythagoras_6.jpg

Things You Will Need

You will need a

•Partner

•3 Sheets of Graph/Grid Paper

•Scissors

•Ruler

•Crayons

Start with any right triangle

Draw any size right triangle in the middle of the page. You and your partner should have the same triangle.

½ab

a

bc

-b

a

Constructing Three Squares

Draw three squares so that each square corresponds to each side of the triangle.

The squares must be connected to the triangle.

½ab

a

b

a

a²

c²

b b²

a

-b

c

Cut and Color

Cut out the three squares and the triangle.

½ab

a

b

a

a²

c

c²

b b²

a

-b

Construct, Cut and Color

Take out another sheet of graph paper and construct three more triangles with area ½ab. Write ½ab on the three triangles.

Cut them out.

½ab

a

bc

½ab

a

bc

½ab

a

bc

Construct, Cut and Color

Finally construct a square whose length is (a-b).

To do this, put you a² and b² on the graph paper.

Then cut it out and color it.

a²b²

a-b

(a-b)²

Fit the 4 triangles into c².

½ab

½ab

½ab

½ab

c²

Validating Pythagorean’s Theorem Geometrically

There is a square left in the middle to fill?

Is it a²?

Is it b²?

Or is it (a-b)²

a

a²

b²

(a-b)²

Yes (a-b)² fits!

Validating Pythagorean’s Theorem Geometrically

Now take a² + b².Check to see if the same five objects fit into this configuration.

a

b²

(a-b)²

½ab

½ab

½ab

½ab

a²

They do!!

What does this meanGeometrically?

c² a²b²

a

b²a²

What does this meanGeometrically?

c²a²

b²

a

b²

It means that c² has the same area as a² + b².

We have now validated that a² + b² = c² !!

What does this mean algebraically?

It means that c² = (a – b)² + ½(ab)• 4Simplifying we get,

c² = a² + b²We have now validated Pythagorean’s

Theorem algebraically.

Another Way to Validate Pythagorean’s

Theorem

http://users.ucom.net/~vegan/images/Pythagoras_6.jpg

Cut and Color

Again begin with a², b², and c². ½ab

a

b

a

a²

c

c²

b b²

a

-b

Validating Pythagorean’s Theorem Geometrically

Construct a square with your four triangles and c².

This time the triangles don’t go inside.

Can you do it?

½ab

½ab

½ab

½ab

c²

Validating Pythagorean’s Theorem Geometrically

c²

Validating Pythagorean’s Theorem Geometrically

How about with your four triangles,a² and b²?

Can you do it?

½ab

½ab

½ab

½ab

a² b²

Validating Pythagorean’s Theorem Geometrically

a²

b²

Validating Pythagorean’s Theorem Geometrically

a²

b²

c²

What does this mean geometrically?

Validating Pythagorean’s Theorem Geometrically

a²

b²

c²

It means that without the triangles,

c² = a² + b², and geometrically validates Pythagorean’s Theorem.

What does it mean algebraically?

c²

a b

b

a

b

a

a b

It means that

(a + b)² = ½ab•4 + c²

a² + 2ab + b² = 2ab + c²

Subtracting 2ab from each side, we get

a² + b² = c² This validates Pythagorean’s Theorem algebraically.

Benchmark

MA. 8. G. 2. 4

Validate and apply Pythagorean theorem to find

distances in real world situations or between points in the coordinate plane .  

How will you explainValidating Pythagorean’s Theorem

to your eighth grade students?

Will you do it algebraically or geometrically or both?

Discuss this with your partner.

http://eppsnet.com/images/math-problems.gif

The Next Benchmark states

MA. 8. A. 6. 2

Make reasonable approximations of square roots and mathematical expressions that include square roots, and use them to estimate solutions to problems and to compare mathematical expressions involving real number s and radical expressions.  

Making Reasonable Approximations of Square

Roots

http://www.coverbrowser.com/image/bestselling-comics-2007/2502-1.jpg

Approximating the Square Root of a Number by Hand

There are several ways that you can approximating the square root by hand.

1. Using the number line to round to the

nearest integer

2. Using a Percentage Approximation

to round to the nearest tenths

place.

How do you approximate to the nearest integer using the number

line?

52

0 7 8

52

is between and . 52 749 864

Which integer is it closer to? 7

752

49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

7852

How do you approximate to the nearest integer using the number

line?

33

0 5 6

33

is between and . 33 525 636

Which integer is it closer to? 6

633

25 26 27 28 29 30 31 32 33 34 35 36

5 633

Using percentage approximations, how do you approximate to the

nearest tenths place.52

You know now that is closest to 7. Count how many spaces it is away from 7.

49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

7852

52

Now count how many integer square roots there are between 7 and 8.

3

15

Your approximation is 3/15 = 1/5 = 0.2. So your percentage approximation to the nearest tenth is 7.2.

Using percentage approximations, how do you approximate to the

nearest tenths place.33

25 26 27 28 29 30 31 32 33 34 35 36

5 633

You know now that is closest to 6. Count how many spaces it is away from 5.

33 8

Now count how many integer square roots there are between 5 and 6.

11Your approximation is 8/11 ≈ 0.72… So your percentage approximation to the nearest tenth is 5.7.

Now you try!!

Approximate to the nearest tenths place.

146Approximately 12.1

23Approximate to the nearest tenths place.

Approximately 4.8

How will you teach your eighth students to approximate the

square root of a number?1.Using the number line

to round to the nearest integer

2. Using a percentage approximation to round to the nearest tenths place.

http://eppsnet.com/images/math-problems.gif

There is still another method called the Classical Approach that you will learn in the next unit.