Math 20-2

Chapter 1 Inductive and Deductive Reasoning

Lesson 1.3  Using Reasoning to Find a Counterexample to a Conjecture

GOAL

The goal of this lesson is stated at the top of page 18 in your textbook, and is also written below.
Keep this in mind as you work through the lesson material.

Invalidate a conjecture by finding a contradiction

Explore...

Try to answer the question in the sidebar on page 18.

Watch

Video Tutorial

Learn About the Math

Information and Examples that present the ideas of this lesson

TEXT Page 18

Read/Work through the Learn About the Math section on Page 18 in your textbook.

Go through this material,  AS BEST YOU CAN,  before you view the video tutorials that follow.

Introduction

Video Tutorial  (introduction to the problem)

Demonstration

This question is connected to a figure that is sometimes called the “Mystic Rose”.
You can put points on a circle, and connect them up with chords,  to form quite amazing figures.
Imagine if you had to actually COUNT the different regions formed to support the conjecture being explored here!!

Example 1  Testing a conjecture

Video Tutorial

Reflecting

I didn’t make a video for this section

Apply the Math  (Examples)

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

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

Prime      Composite      Divisors  (vocabulary, optional)

A few math terms that will keep coming up, over and over and over...
If you already KNOW them,  no problem, nothing to see here!
But...  if you want a short presentation on these important words,  watch the video 
(and do a bit of exploration with the demonstration below)

Video Tutorial

Example 2  Connecting to previous conjectures

TEXT Pages 19 - 20

Work through Example 2 on Pages 19 - 20 in your textbook.

Watch

Video Tutorial

Your Turn

I didn’t make a video here.
For Francesca,  I presented several counter-examples to her conjecture in the video from the previous section.
For Steffan,  you would NOT be able to find a counter-example since his conjecture IS true!
But... how would you know for sure!?
(it was a very compelling way to explain that it was true...)

Example 3  Using Reasoning to find a counterexample to a conjecture

TEXT Page 21

Work through Example 3 on Page 21 in your textbook.

Watch

Video Tutorial

Your Turn

I answered this question when I did Example 3

In Summary

Key Ideas and Need to Know 
Presents and 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!

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

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

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

Closing  A summary question that often asks YOU to explain ideas in your own words.

Extending questions offer enrichment and extension.

Notes

Question 9,    I didn’t do a video for this one.
Check the answer in the back for one possible solution (which I shared in the video for question 5)

Question 12,    I didn’t do a video for this one.
Check the answer in the back for one possible solution

Question 13,    I didn’t do a video for this one.
Check the answer in the back for one possible solution
If you come up with an interesting solution , share it with your teacher!

Question 18,    I didn’t do a video for this one.
Check the answer in the back for a possible solution

Question 20,    I didn’t do a video for this one.
Check the answer in the back for a possible solution

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 10"}, {"Question 11",  ,  , "Question 14", "Question 15"}, {"Question 16", "Question 17",  , "Question 19",  }, {"Question 21", , , , }})