Math 10C

Chapter 4 Exponents and Radicals

Lesson 44 Irrational Numbers

Getting Started

This lesson touches on some very deep mathematics,  some ideas that are the foundation of a LOT of concepts.

The textbook assumes that students already KNOW a lot of this, and they mostly skim over the details.

If that works for you, great!

I have created a set of several SHORT video tutorials to cover the background and BIG IDEAS of this lesson to help with either learning or being reminded of the concepts.

Totally optional, but if you find yourself not really understanding what an irrational number IS, these will hopefully help!

Number sets
Natural, Whole, Integer      LINK  (Start A)

Number sets
Rational Numbers      LINK    (Start B)

Rational Numbers  ⟶ Decimal numbers
Terminating Decimals     LINK    (Start C)

Rational Numbers  ⟶ Decimal numbers
Repeating Decimals     LINK    (Start D)

Rational Numbers  ⟶ Decimal numbers
THE BIG IDEA      LINK    (Start E)

Rational Numbers  ⟷ Decimal numbers
Using your calculator      LINK    (Start F)

Stupid numbers,  aka,  irrational numbers
 LINK    (Start G)

Number sets
REAL Numbers      LINK    (Start H)

Focus

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

TEXT Page 184

Read/Work through the  Focus  section at the top of page 184 in  your textbook.

If you are interested in art and mathematics,  an Internet search for the Golden Ratio will produce many hits that you can explore.
(but optional,  this isn’t information you need for this lesson, it’s simply a “real life” example of an irrational number)

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 185

Read/Work through the  Investigate  section on Page 185 in your textbook.

A list of materials  you need for the Investigation is given in the side bar.
(ruler and compass)

Investigating the Golden Ratio is (IMHO) super interesting!   But...
Your teacher may or may not want you to do this investigation, and, really,  you don’t need this to be able to tackle the ideas of this lesson.

You can watch the video tutorials to get what you need.....

Watch

I am going to use a drawing program to work through the steps.
You can do the same thing on paper with a compass and a ruler.

Video Tutorial  (drawing the rectangle)

You can measure the sides as outlined in the Investigation.  Your “calculations” should be close to your measurements, but no measurement will be perfect.

In this video,  I’ll calculate the Golden Ratio twice,  once using a square of size 10 cm, and once using a square of ANY size.

Video Tutorial, calculations

Demonstration

Totally optional, but the demonstration below creates a Golden Spiral.  You start with a square, do the steps you used in the Investigation, then repeat.  (draw another square, etc.)

You can see this idea illustrated below, with a Golden Spiral superimposed on a reasonably “famous” piece of artwork called The Great Wave off Kanagawa
LINK

Link the Ideas

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

TEXT Pages 186 - 187

Read/Work through the  Link the Ideas  section on Pages 186 - 187 in your textbook.

Watch

Video Tutorial   (fractional exponents with a numerator different than 1)

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  Convert From a Power to a Radical

TEXT Page 187

Work through Example 1 on Page 187 in your textbook.

Watch

Video Tutorial

Your Turn

YOUR TURN gives YOU the chance to try to do a problem ON YOUR OWN,  using the example as  a model.

Use the Video Tutorial for help, or to check your work, BUT... you should do your best to do this question ON YOUR OWN FIRST!

Video Tutorial

Example 2  Convert From a Radical to a Power

TEXT Page 188

Work through Example 2 on Page 188  in your textbook.

Watch

Video Tutorial

Your Turn

YOUR TURN gives YOU the chance to try to do a problem ON YOUR OWN,  using the example as  a model.

Use the Video Tutorial for help, or to check your work, BUT... you should do your best to do this question ON YOUR OWN FIRST!

Video Tutorial

Example 3  Convert Mixed Radicals to Entire Radicals

TEXT Pages 188 - 189

Work through Example 3 on Pages 188 - 189 in your textbook.

Watch

Video Tutorial

Your Turn

YOUR TURN gives YOU the chance to try to do a problem ON YOUR OWN,  using the example as  a model.

Use the Video Tutorial for help, or to check your work, BUT... you should do your best to do this question ON YOUR OWN FIRST!

Video Tutorial

Example 4  Convert Entire Radicals to Mixed Radicals

TEXT Page 189

Work through Example 4 on Page 189 in your textbook.

Watch

Video Tutorial

Your Turn

YOUR TURN gives YOU the chance to try to do a problem ON YOUR OWN,  using the example as  a model.

Use the Video Tutorial for help, or to check your work, BUT... you should do your best to do this question ON YOUR OWN FIRST!

Video Tutorial

Example 5  Order Irrational Numbers

TEXT Page 190

Work through Example 5 on Page 190 in your textbook.

Watch

Video Tutorial

Your Turn

YOUR TURN gives YOU the chance to try to do a problem ON YOUR OWN,  using the example as  a model.

Use the Video Tutorial for help, or to check your work, BUT... you should do your best to do this question ON YOUR OWN FIRST!

Video Tutorial

Example 6  Solve Problems Involving Irrational Numbers

TEXT Page 191

Work through Example 6 on Page 191 in your textbook.

Watch

Video Tutorial

Your Turn

YOUR TURN gives YOU the chance to try to do a problem ON YOUR OWN,  using the example as  a model.

Use the Video Tutorial for help, or to check your work, BUT... you should do your best to do this question ON YOUR OWN FIRST!

Video Tutorial

Key Ideas

Key Ideas summarizes the main new concepts of the lesson.

•	Carefully read through each of the bullet points in this section.

•	If you are unclear about any of these ideas,  ASK YOUR TEACHER FOR HELP!

Check Your Understanding    (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

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

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

Extend questions offer enrichment and extension.

Create Connections questions connect the math to everyday contexts.

Mini Lab (optional)  your teacher may or may not have you do these.

Notes

POWERS   I found myself using these and talking about them several times.   
I figured you might want to have them to print or at least to refer to.   
You can download the file here.    
A short video explaining the file here.   (powers)

Question 14,    At the end of the video I hesitated and figured I had a mistake, but turns out my answer WAS correct.   Didn’t realize the distance needed to be so great for a geo-synchronous orbit.

Question 23,    great question, but I didn’t make a video for this one.  
A PROJECT,  not sure if your teacher will want you to do this or not.

Question 24,    another great question, but I didn’t make a video for this one either.
A PROJECT,  not sure if your teacher will want you to do this or not.

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", "Question 19", "Question 20"}, {"Question 21", "Question 22",  ,  , }})