Math 8

Unit 1 Square Roots and the Pythagorean Theorem

Math 815  The Pythagorean Theorem

Getting Started

There are a few things to cover BEFORE we actually get into the lesson.

What is an angle?

An angle measures the spread between two rays

Video Tutorial

The sum of the angle measures in a triangle

Introduction

This section gives you a visual "proof"  for  the sum of the angles in a triangle.

Background

If you are standing up and someone asks you to turn 360˚, you know that is a complete circle, so you would spin and end up facing the direction you started.

If someone asks you to "do a 180....", we take this to mean a turn where we end up facing the opposite direction we started from.

So 180˚ forms a straight line as indicated in the diagram below.

The experiment

You should try this yourself in order to get the full benefit from the information

First, find a piece of scrap paper and draw a large triangle on it.

The size or shape doesn't matter, in fact, the more radical looking the triangle, the more convinced you may be.

Label each of the angles of the triangle with a letter or a spot of color or something....

Please, do it!!

Reading is not the same as doing, and it won't stick in your brain the same way as if you actually do it!

Once you make your triangle,  follow the steps below.

The steps

Below is a triangle with the three angles labeled.

The next step is to rip off each corner of the triangle as indicated below.

Now, take the pieces and arrange them so the angles are all joined together.

What is your conclusion!??

Don't read any further until you try this yourself!!

The picture worth 1000 words.... don't look till you try it yourself!!!

You should see that the angles, when they are placed together, form a line.

Final conclusion

The sum of the angles in a triangle is 180 degrees, since a line is 180 degrees.

The sum of the three angles in any triangle is 180 degrees.

Watch

I hope you did this yourself!!  but.... this is me doing the experiment with my drawing program....

Video tutorial

Interactive

An interactive demonstration of this important idea.

LINK
https://www.mathopenref.com/triangleinternalangles.html

Video tutorial

The connection between angle measure and side length in a triangle

Some ideas on using this demonstration,  LINK

A BETTER demonstration,  online    LINK
https://www.mathopenref.com/trianglesideangle.html

Conclusion....?   Video Tutorial

Focus

The FOCUS section introduces the main concept that will be covered in the lesson.

TEXT Page 31

Read/Work through the  Focus  section at the top of page 31 in  your textbook.

Watch

Video Tutorial

Investigate

Investigate an idea or problem, with a partner if possible.
The investigation provides the background for the ideas presented in the lesson.

TEXT Page 31

Read/Work through the  Investigate  section on Page 31 in your textbook.

You can do this activity on grid paper.
A protractor would help, but watch this HINTS video for some ideas of HOW to do the investigation.
(and I will also DO the first example....)

Need some grid paper to work with?
This folder has a few different varieties of grid paper you can use for this Investigation.
(I would use the 1 cm paper )

Three examples

You are welcome to make up three examples of your own!
But... since you probably won’t have a way to check,  you might want to use the examples below.
Start with the line segments shown, and go through the procedure I went through to introduce the investigation.
Once you have done this, you can check your work with the video tutorial that follows.

Example 1

•	copy the line segment onto a larger piece of grid paper
•	create a right triangle with the given line segment as the hypotenuse
•	draw squares on each side of the triangle
•	find the area of each square, and the side length of each square

Video tutorial (my solution)

Example 2

•	copy the line segment onto a larger piece of grid paper
•	create a right triangle with the given line segment as the hypotenuse
•	draw squares on each side of the triangle
•	find the area of each square, and the side length of each square

Video tutorial (my solution)

Example 3

•	copy the line segment onto a larger piece of grid paper
•	create a right triangle with the given line segment as the hypotenuse
•	draw squares on each side of the triangle
•	find the area of each square, and the side length of each square

Video tutorial (my solution)

Connect

Connect summarizes the math in the lesson.
A presentation of ideas connected to the lesson focus.

TEXT Page 32

Read/Work through the  Connect  section on Page 32 in your textbook.

I created another example of a right triangle, that you could print if you wanted to have it to refer to.
You can open the document HERE

Examples

Examples will give you TOOLS for solving problems, and also show you that there may be different ways to approach a question.

Example 1

TEXT Pages 32 - 33

Work through Example 1 on Pages 32 - 33  in your textbook.

Watch

Video Tutorial

Example 2

TEXT Page 33

Work through Example 2 on Page 33 in your textbook.

Watch

Video Tutorial

Practice Problems

Doing Practice Problems

Practice problems reinforce the math presented in this lesson.

•	Work through the practice problems for this lesson.
	These problems are for practice, you don’t need to hand them in!
	(your teacher may assign specific questions for you to complete)

•	Use the answers in the back of the book to CHECK YOUR WORK!

•	If you make a mistake, TRY TO FIX IT!

•	If you need help or want more information, view the video solutions provided.

•	If you still need more help,  CONTACT YOUR TEACHER!

Types of Practice Problems

Discuss the Ideas invites you to talk about the math.
If possible share your thoughts/ideas/answers with another person.
If you can help them understand, then you likely have good understanding yourself.

Check questions require basic understanding of the ideas of the  lesson.

Apply questions will require you to dig deeper for a solution.

Take It Further questions offer enrichment and extension.

Notes

Question 18,  HERE is a link to the diagram in the question, if you want to print it to work on the problem.
(I left it blank, no dimensions, you can add those yourself)

Video Solution Links

Check your work for every problem you do!
If you make a mistake,  try to fix it!
Use the video solutions for more information.
If a problem still doesn’t make sense, ASK YOUR TEACHER FOR HELP!

({{"Question 1", "Question 2", "Question 3", "Question 4", "Question 5"}, {"Question 6", "Question 7", "Question 8", "Question 9", "Question 10"}, {"Question 11", "Question 12", "Question 13", "Question 14", "Question 15"}, {"Question 16", "Question 17", "Question 18", , }})