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Math 106: Contemporary Math9 chapters | 106 lessons

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Lesson Transcript

Instructor:
*Maria Airth*

Maria has a Doctorate of Education and over 15 years of experience teaching psychology and math related courses at the university level.

Solving problems is not just a simple, straightforward process. There are a few principles that can help you as you approach any problem solving scenarios. This lesson covers those principles with examples.

Problem solving in math - and in all other areas of your life, really - is one of the trickiest things to do. Many problems are similar, but must be solved in different ways. Some look very different from each other, but should be solved in similar ways. How can you know how to solve a problem, what to do to get an answer, where to even start?

Well, there are five principles that you can keep in mind to help you attack any problem solving situation you may find:

- The Always Principle
- The Counterexample Principle
- The Order Principle
- The Splitting Hairs Principle
- The Analogies Principle

In this lesson, I will discuss these principles as well as give examples for each.

**The always principle** is fairly simple. Basically it states that something is true, 100% of the time, with no exceptions.

The sun always rises in the East; always. No exceptions. If it is morning and you are looking at the sun, you are looking to the East. Likewise, if it is late afternoon and you are facing the sun, you are facing west. No exceptions.

In math, there are a few examples of the always principle:

- The product of two negative numbers is always a positive: -2 * -3 = +6
- Negative numbers are always less than positive numbers: -3 < 2
- Dividing by 0 always ends in an 'undefined' result

So, when you approach a problem, look for any 'always scenarios' that might be present.

Do you have a friend or relationship who is always late? Have you given up on them ever being on time? If that person ever showed up somewhere on time, then it would be a **counterexample** to your concept that they are always late. A counterexample is an example that disproves a statement or theory. So, one act of showing up on time would forever disprove that your friend was 'always' late.

Again, in math we have some statements that we think are 'always' statements, but if even a single counterexample can be found, then we have disproved the 'always.' Take prime numbers for example. We know that prime numbers are numbers which can only be divided by themselves and 1. Some examples are 3, 5, 11, 13 and 17. What do you notice about each of these? I notice that they are all odd numbers. It would be easy to state that 'all prime numbers are odd.'

However, there is a counterexample: 2 is a prime number but is not odd. When solving problems, watch out for counterexamples to commonly held ideas.

A great example of how order matters is the old riddle about having to get a chicken, a fox and a bag of grain across a river one at a time. You cannot leave the fox with the chicken (the fox would eat it). You cannot leave the chicken with the grain (the chicken would eat the grain). So, how do you do it? First you take the chicken across. Then go back for the grain. When you get to the other side, leave the grain and pick up the chicken again for the return. Next, swap the chicken and the fox, leaving the chicken on the original side. Drop off the fox with the grain and return for the chicken. There, you've done it! But, it is clear that the order really did matter a great deal here.

**The order principle** states that order usually does matter. The most well-known order principle in math is the order of operations, which gives the order in which to conduct mathematical operations: PIMDAS, parenthesis, indices, multiplication, division, addition, subtraction, which is the order in which mathematical problems should be solved.

2 + 6 * (-2 +1) - 6^2 /2 equals 220.5 from left to right, but equals -22 when using the correct order of operations. When solving problems, make sure you are following the proper order.

**The splitting hairs principle** relates to things that seem to be the same, but are not truly identical. Consider two popular cola drinks. I have personally had the experience of asking for one brand and being brought the other accompanied by the phrase, 'They're really the same thing.'

No, they aren't. The two drinks may look the same, but many people do truly have a preference for one over the other and can tell the difference. No matter how similar they seem, they are not identical.

In math, a great example of the splitting hairs principle is the use of the terms 'equal' and 'equivalent.' Many people treat these terms as if they are the same. They aren't. Equal refers to two things being exactly the same, like these triangles:

Equivalent refers to two things being similar in effect and results, like these two triangles, which are proportionately equivalent to each other:

When problem solving, make sure that you understand the definitions of the terms and use them properly.

**The analogy principle** uses comparisons and relationships to illustrate concepts that are unknown. If you met a person that did not know what a zebra was, you might compare it to a horse with stripes. In this way, you use the analogy of a horse to illustrate the concept of a zebra.

The best example of using analogies in math is this lesson. For each new principle, I gave a real world analogy to assist you in understanding the concept before moving on to discuss the mathematical application of the principle. So, in problem solving, if you can draw a relationship or compare one thing to another to get a better understanding of it, do so.

There you have it, the principles of problem solving are:

**The always principle**

When you approach a problem, look for any always scenarios that might be present.**The counterexample principle**

Watch out for counterexamples to commonly held ideas.**The order principle**

Make sure you are following the proper order.**The splitting hairs principle**

Understand the definitions of terms and use them properly.**The analogies principle**

Draw a relationship or compare one thing to another to get a better understanding of it.

Thank you for watching, and I wish you good problem solving!

You should be able to explain the five principles of problem solving after watching this video lesson.

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Math 106: Contemporary Math9 chapters | 106 lessons

- Critical Thinking and Logic in Mathematics 4:27
- Logical Fallacies: Hasty Generalization, Circular Reasoning, False Cause & Limited Choice 4:47
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