Math 20-2

Chapter 1 Inductive and Deductive Reasoning

Lesson 1.6  Reasoning to Solve Problems

GOAL

The goal of this lesson is stated at the top of page 45 in your textbook, and is also written below.
Keep this in mind as you work through the lesson material.

Solve problems using inductive or deductive reasoning

Explore...

Try to answer the question in the sidebar on page 45.

Answer

A “famous” trick question.... you would light the match first
(and then the other items have pros and cons,  probably the candle...?)

Investigate the Math

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

TEXT Pages 45 - 46

Read/Work through the Investigate section on Pages 45-46 in your textbook.

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

?  How can Emma’s communication about her reasoning be improved?

Video Tutorial

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.

Example 1  Use reasoning to solve a problem

TEXT Page 46

Work through Example 1 on Page 46 in your textbook.

Here is a slightly different way of asking the question  (setting up the rules).
Also note that when you ask the question this way,  you would likely start with a smaller number of people, look for patterns, and then use those patterns to (hopefully) find a general rule or formula to help answer the question.

Imagine you are at a very polite party, where everyone shakes hands with everyone else.  
No one was rude enough to shake hands with the same person twice and no one was silly enough to shake hands with her/himself.  
There were 100 people at this party.  
You need to find out exactly how many handshakes there were.

Demonstration

Here is a demonstration of the problem,  with a visual explanation of Kim’s solution.
See if you can make sense of it.  (check it out FIRST, before you watch the video that follows)
..and note,  this is just like the “Mystic Rose” diagram we saw in a previous lesson,  check HERE for an example

Video Tutorial

Your Turn

You should do your best to do this question ON YOUR OWN FIRST!
(I didn’t do a video here)

This is inductive reasoning, looking at patterns and specific examples.
If we used deductive  reasoning, we would start with a general case,  like “n” people, and try to come up with a rule.
(the rule is   ((n (n+1))/2)   but you don’t need to know that!   It’s actually not hard to prove that rule)

Example 2  Use reasoning to solve a problem

TEXT Pages 47 - 48

Work through Example 2 on Pages 47 - 48 in your textbook.

Watch

Video Tutorial

Your Turn

I answered this when I did Example 2
This should be deductive reasoning.
I will admit that I don’t always have a clear idea of which is which.  
THIS website had a really good overview and helped explain the differences.

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

NOTE!   Many of these problems are trick questions.  
Once you see the answer you go “of course....”
However, if you don’t see it,  they can be frustrating.
If I was the TEACHER of this course, I would encourage my students to DO THEIR BEST,  try, give it some time, 
but..... don’t get too encouraged if you can’t get an answer.
This is NOT like the other math you will study in this course where you learn a METHOD and then practice that method.
For many of these problems, there is no set method, you just need to use logic and “perspiration” to try different things to come up with an answer.
I know for sure that if you put me on the spot and asked me to solve some of these problems without having the answers to help me,  I would NOT be able to do lots of them!

Do your best,  and use the solutions to help you develop background and skills that will help for solving problems that involve logical thinking.

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 11,    here is a link to a page that goes through, step by step, and uses logic to solve the problem.
LINK

Question 12,    I didn’t do a video for this one.
Check the answer in the back.
I know some people would immediately be thinking of the different possible arrangements and what would work.
Other people (like me!) would have a much harder time visualizing the possibilities.

Question 14,    here is the demonstration I used in the Video

Question 17,    I didn’t do a video for this one.
Here is the sample answer from the back of the book
e.g., A problem can be solved using inductive reasoning if it has a pattern that can be continued. 
A problem can be solved using deductive reasoning if general rules can be applied to obtain the solution. 
It is not always possible to tell which kind of reasoning is needed to solve a problem.

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 13", Question 14 , "Question 15"}, {"Question 16",  , "Question 18", "Question 19", "Question 20"}})