Knowing what tools to use and what those tools can do for you is an important part of any skill. If you know how to write a situation that would require someone to use induction or deduction, you will understand these two tools much better. Then, when you need to decide whether you should use induction or deduction to solve a problem, you'll know exactly what kinds of situations call for each type of reasoning.
Each of the tabs below demonstrates how to describe a situation that needs induction and deduction. Study these examples carefully before attempting to identify suitable situations on your own.
Induction
Deduction
Inductive reasoning requires many examples or observations--enough to identify a pattern, a regularly occurring relationship among the parts of the problem. Read this example of a situation that includes the factors needed for inductive reasoning:
Manuel has been collecting data for his geology class. He has been collecting rocks from the creek, and writing down what kinds he finds.
Question
Why might this example work for induction?
Manuel has lots of examples of rocks. He could use all this data to predict something about the other rocks he might find. To make his prediction, Manuel could make a conjecture that describes a relationship among the rocks he's found so far.
Let's look at another example.
Jessica has been volunteering as a teacher's aide at a local elementary school. After a few days helping out in several classrooms, Jessica wonders if there is a relationship between the ratio of boys to girls sitting at each table and how noisy that classroom is throughout the day.
Question
Why does this example work for induction?
Jessica thinks that there is a relationship--that a conjecture can be made about the situation. Instead of suggesting that teachers with noisy classrooms rework the groups based on her hunch, though, Jessica should collect data about boy/girl ratios and the noise level in each classroom. She believes there is a relationship--now the next step is to "prove" her conjecture is true, at least in the examples she sees.
To write a situation that requires inductive reasoning, you need to identify a set of data (examples) and also the need for a conjecture. What data does Hector have? What conjecture might he make?
Hector is currently enrolled in a geometry class, so he has some large data sets available related to geometric shapes. For instance, Hector has measured the interior angles of about twenty quadrilaterals that he has drawn for his various assignments.
Question
What could Hector use inductive reasoning to prove?
A conjecture that identifies and states a pattern that he sees in the measurements of interior angles of quadrilaterals.
If you encounter a scenario that involves (or could involve) rules, laws, facts, or data, you've found an opportunity to use deduction. In a geometry or science course, rules are typically called laws, properties, or theorems. Facts are simpler statements that describe an object or relationship, whereas data are facts that you gather through direct observation.
Logan knows the speed limit is 55 mph, and he is driving 65 mph.
Question
How might Logan (or his passenger) use deductive reasoning to make a prediction?
The speed limit is a law or rule that includes a consequence: getting a ticket. Given the situation, Logan or his passenger can draw a new conclusion--Logan is likely to get a ticket. If Logan knows anything about fatal car crash statistics, he could also deduce that he's more likely to die or kill someone else when he speeds.
In geometry, definitions, postulates, and theorems are the rules that allow for deductive reasoning.
How can deduction help Morgan out?
Morgan wants to prove the Pythagorean Theorem only works for right triangles.
Question
Why does this example work as an opportunity to use deduction?
Morgan wants to prove something new based on what is already known about the theorem. The Pythagorean Theorem only mentions right triangles, but it might work on other kinds of triangles as well. Morgan will have to use definitions, postulates, and theorems to show that her claim is true (or to disprove it).
To write a problem or scenario that requires deduction, you need to look for situations in which definitions, postulates, or theorems are needed to prove or justify something.
Angelina knows that vertical angles have the same measure and that angles with the same measure are congruent. Angelina wants to prove that vertical angles are congruent.
Question
Should Angelina focus on collecting examples of vertical angles that are congruent? Or should she look for laws that state known facts about the relationship between types of angles?
To use deductive reasoning, Angelina should look for definitions, postulates, and theorems that will help her "construct" a formal proof. If she looks for examples, Angelina will have to rely on inductive reasoning.
