# Honors Geometry Companion Book, Volume 1

2.1.1 Using Inductive Reasoning to Make Conjectures (continued) Example 3 Biology Application

In this example you are asked to make a conjecture based on the data. Since a conjecture is based on inductive reasoning, base a statement upon the results of specific examples in this table and that statement will be a conjecture. Look at the table and consider a statement that is true about the number of birds observed on each day. On Day 1 and Day 2, more cardinals were seen than either mockingbirds or blue jays. This pattern holds for Days 3, 4, and 5. So, there are equal numbers of each type of bird in the neighborhood and more cardinals were observed eating the food each day. The data suggests that the food is most desired by the cardinals.

Example 4 Finding a Counterexample Steps of Inductive Reasoning 1. Look for a pattern. 2. Make a conjecture. 3. Prove the conjecture or find a counterexample.

A conjecture cannot be proven true using specific examples. However, a conjecture can be proven false using only one specific example. An example that shows a conjecture is false is called a counterexample.

This conjecture states that for all positive numbers, a number’s square is greater than or equal to the number. Squaring a number means to multiply the number by itself. Consider specific examples to find a counterexample. If the example involves a number greater than 1, then the conjecture holds. However, the conjecture includes all positive numbers, so numbers between 0 and 1 are included and must be considered. When the example includes a fraction (a number between 0 and 1), the conjecture does not hold. Therefore, any example involving a fraction is a counterexample.

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