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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.

What methods are there to solve and understand mathematical problems? This lesson will review three methods to understand mathematical problems (verbal, graphical, and by example). Each will be illustrated with examples.

Hi. Today, we are going to be talking about the **three-way principle of mathematics**. Basically, there are three ways to solve a problem in math: verbally, graphically, or by example. In this lesson, we will discuss each of these principles by solving sample problems using each type.

To really illustrate how you can use each of the three principles to solve problems, I think it would be great to have two very different styles of math problems and work each out using each principle. So, here are the two samples we will focus on through the lesson:

- Mathematically speaking, what is a set?
- 2
*x*= 20, solve for*x*.

Let's see how each principle can help us solve these problems.

The **verbal principle** literally refers to working through a problem using words to talk or write out the solution. The first sample question was mathematically speaking, what is a set? The verbal solution to this is to write it out. A mathematical set is a group of numbers or items that are grouped together based on a stated standard of inclusion. It is indicated by brackets.

The second sample is similar: 2*x* = 20, solve for *x*. Verbally, we might say, 'This means 2 times what number equals 20. Knowing that 2 tens is 20 gives us our answer. *x* must equal 10 because 2 times 10 is 20.' Solving things verbally can be time consuming and takes a real understanding of the concept before you even get started. The other methods help when you don't necessarily know the subject matter as well.

Solving a problem **graphically** means to show the problem in a visual way. In math, we tend to think of drawing graphs, charts, or diagrams, but any visual representation of a problem is also a form of graphical illustration. Any type of picture or drawing out the problem is a form of solving the problem using the graphical principle.

Back to sample question one: mathematically speaking, what is a set? We could draw the concept of a set by imagining a set of cows and drawing a circle around just the cows in this picture:

Now we can visually see that a set is a grouping together of items based on a pre-stated condition (here, the pre-stated condition was that the animal had to be a cow).

The second example, 2*x* = 20, solve for *x*, is perfect for a graph. We will just need to graph the line *y* = 2*x*, and then find the value of *x* when *y* is 20. Draw your *x* and *y*-axis and label the vertices. Then, complete the graph of the line. Once you have drawn the line, find the point where *y* = 20; you will see that the value of *x* at that point is 10. The graph helps you see that you are doubling your *x* value each time, which could help you come to your answer before actually having to calculate each product.

To solve problems **by example**, you do just what it sounds like: create examples to illustrate the solution. Here are a few examples of a set:

- The set of positive even numbers: {2, 4, 6, 8, 10, 12, â€¦}
- The set of all numbers between -2 and 4: {-2 <
*x*< 4} - The set of odd integers greater than 1, but less than 8: {3, 5, 7}

By seeing these examples, it is clear that a set is a group of items (normally numbers in math) that comply with a pre-stated condition for inclusion.

For sample two, 2*x* = 20, solve for *x*, we can think of a word problem to get a less abstract example of what we are trying to solve. Consider this example of the same problem: John has twice as many books as Jane. John has 20 books. How many books does Jane have?

We can see that the original question is asking, twice what is 20? If you still aren't sure, then you can just try some numbers: twice 5 is 10; twice 12 is 24; twice 10 is 20. Many math problems can be solved in this example format, which is also called trial and error. You continue to try answers (make examples of the calculation) until you get the correct answer. As you can see, it could be a long process or you could hit on the pattern quickly through the use of the examples.

So, in this lesson, we reviewed three ways to solve and understand problems. To illustrate these methods, we walked through two sample problems for each method:

- The
**verbal method**required using words to explain the problem and solution. - The
**graphical method**used pictures, graphs, or charts to visually illustrate the problem and solution. - The
**by example method**led to showing many examples of the same concept in order to determine the pattern of the problem and solution.

Not all these methods work equally well with every type of problem, but, as you saw in this lesson, you can use any of the methods to help you find solutions to your mathematical problems. Which method you choose should take into account your specific learning style - you have a greater chance of succeeding if you play to your personal strengths. Thanks for watching and good solving!

This lesson on the three-principle of mathematics could prepare you to:

- Solve mathematical problems using the verbal, graphical, and by example methods
- Recognize the strengths and limitations of each method

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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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- Conditional Statements in Math 4:54
- Logic Laws: Converse, Inverse, Contrapositive & Counterexample 7:09
- Reasoning in Mathematics: Inductive and Deductive Reasoning 7:03
- Reasoning in Mathematics: Connective Reasoning 8:16
- Polya's Four-Step Problem-Solving Process 7:52
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- Using Mathematical Models to Solve Problems 6:35
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