Geometry (A): READ: Inductive Reasoning Reading

Inductive Reasoning

Learning Objectives

Recognize visual patterns and number patterns.
Extend and generalize patterns.
Write a counterexample to a pattern rule.

Introduction

You learned about some of the basic building blocks of geometry in Chapter 1. Some of these are points, lines, planes, rays, and angles. In this section we will begin to study ways we can reason about these building blocks.

One method of reasoning is called inductive reasoning. This means drawing conclusions based on examples.

Visual Patterns

Some people say that mathematics is the study of patterns. Let’s look at some visual patterns. These are patterns made up of shapes.

Example 1

A dot pattern is shown below.

A. How many dots would there be in the bottom row of a fourth pattern?

There will be $4$ dots. There is one more dot in the bottom row of each figure than in the previous figure. Also, the number of dots in the bottom row is the same as the figure number.

B. What would the total number of dots be in the bottom row if there were $6$ patterns?

There would be a total of $21$ dots. The rows would contain $1, 2, 3, 4, 5,$ and $6$ dots.

The total number of dots is $1 + 2 + 3 + 4 + 5 + 6 = 21$ .

Example 2

Now look at a pattern of points and line segments.

For two points, there is one line segment with those points as endpoints.

For three noncollinear points (points that do not lie on a single line), there are three line segments with those points as endpoints.

A. For four points, no three points being collinear, how many line segments with those points as endpoints are there?

$6$ . The segments are shown below.

B. For five points, no three points being collinear, how many line segments with those points as endpoints are there?

$10$ . When we add a 5th point, there is a new segment from that point to each of the other four points. We can draw the four new dashed segments shown below. Together with the six segments for the four points in part A, this makes $6 + 4 = 10$ segments.

Number Patterns

You are already familiar with many number patterns. Here are a few examples.

Example 3 – Positive Even Numbers

The positive even numbers form the pattern $2, 4, 6, 8, 10, 12, \ldots.$

What is the $19^{th}$ positive even number?

The answer is $38$ . Each positive even number is $2$ more than the preceding one. You could start with $2$ , then add $2$ , $18\;\mathrm{times}$ , to get the $19^{th}$ number. But there is an easier way, using more advanced mathematical thinking. Notice that the $3^{rd}$ even number is $2 \times 3$ , the $4^{th}$ even number is $2 \times 4$ , and so on. So the $19^{th}$ even number is $2 \times 19 = 38.$

Example 4 – Odd Numbers

Odd numbers form the pattern $1, 3, 5, 7, 9, 11, \ldots.$

A. What is the $34^{th}$ odd number?

The answer is $67$ . We can start with $1$ and add $2, 33\;\mathrm{times}$ . $1 + 2 \times 33 = 1 + 66 = 67$ . Or, we notice that each odd number is $1$ less than the corresponding even number. The $34^{th}$ even number is $2 \times 34 = 68$ (example 4), so the $34^{th}$ odd number is $68 - 1 = 67$ .

B. What is the $n^{th}$ odd number?

$2n-1$ . The $n^{th}$ even number is $2n$ (example 4), so the $n^{th}$ odd number is $2n-1.$

Example 5 – Square Numbers

Square numbers form the pattern $1, 4, 9, 16, 25, \ldots.$

These are called square numbers because $1 = 1^2, 4 = 2^2, 9 = 3^2, 16 = 4^2, 25 = 5^2, \ldots.$

A. What is the $10^{th}$ square number?

The answer is $100$ . The $10^{th}$ square number is $10^2 = 100$ .

B. The $n^{th}$ square number is $441$ . What is the value of $n$ ?

The answer is $21$ . The $21^{st}$ square number is $21^2 = 441$ .

Conjectures and Counterexamples

A conjecture is an “educated guess” that is often based on examples in a pattern. Examples suggest a relationship, which can be stated as a possible rule, or conjecture, for the pattern.

Numerous examples may make you strongly believe the conjecture. However, no number of examples can prove the conjecture. It is always possible that the next example would show that the conjecture does not work.

Example 7

Here’s an algebraic equation.

$t = (n-1)(n-2)(n-3)$

Let’s evaluate this expression for some values of $n$ .

$n & =1;t=(n-1)(n-2)(n-3)=0 \times (-1) \times (-2)=0\\ n & = 2; t = (n-1)(n-2)(n-3) = 1 \times 0 \times (-1) = 0\\ n & = 3; t = (n-1)(n-2)(n-3) = 2 \times 1 \times 0 = 0$

These results can be put into a table.

$&n& &1& &2& &3\\ &t& &0& &0& &0&$

After looking at the table, we might make this conjecture:

The value of $(n-1)(n-2)(n-3)$ is $0$ for any whole number value of $n$ .

However, if we try other values of $n$ , such as $n = 4$ , we have

$(n-1)(n-2)(n-3) = 3 \times 2 \times 1 = 6$

Obviously, our conjecture is wrong. For this conjecture, $n = 4$ is called a counterexample, meaning that this value makes the conjecture false. (Of course, it was a pretty poor conjecture to begin with!)

Example 8

Ramona studied positive even numbers. She broke some positive even numbers down as follows:

$8 = 3 + 5& &14 = 5 + 9& &36 = 17 + 19& &82 = 39 + 43&$

What conjecture might be suggested by Ramona’s results?

Ramona made this conjecture:

“Every positive even number is the sum of two different positive odd numbers.”

Is Ramona’s conjecture correct? Can you find a counterexample to the conjecture?

The conjecture is not correct. A counterexample is $2$ . The only way to make a sum of two odd numbers that is equal to $2$ is: $2 = 1 + 1$ , which is not the sum of different odd numbers.

Example 9

Artur is making figures for a graphic art project. He drew polygons and some of their diagonals.

Based on these examples, Artur made this conjecture:

If a convex polygon has $n$ sides, then there are $n-3$ diagonals from any given vertex of the polygon.

Is Artur’s conjecture correct? Can you find a counterexample to the conjecture?

The conjecture appears to be correct. If Artur draws other polygons, in every case he will be able to draw $n-3$ diagonals if the polygon has $n$ sides.

Notice that we have not proved Artur’s conjecture. Many examples have (almost) convinced us that it is true.

Lesson Summary

In this lesson you worked with visual and number patterns. You extended patterns to beyond the given items and used rules for patterns. You also learned to make conjectures and to test them by looking for counterexamples, which is how inductive reasoning works.

Last modified: Monday, May 10, 2010, 2:08 PM