Defining Reasoning and Proof
 The Reasoning and Proof Standard | Inductive and Deductive Reasoning | Thinking About Reasoning in the Classroom | Questions and Answers | Connecting to the Other Process Standards | Summary | Your Journal
"For surface area, I would ask them, 'How did you find the surface area of this one rod?', and they would tell me, 'We took the length, multiplied by 4 to get all four sides, and added the two ends.' 'Now let's add another rod; how would you do that?' And then that got them started thinking about how you could generalize that into a formula."

## (Michelle Mullin, Grade 7-8 Teacher)

In both examples of the Building Rafts with Rods Student Work that were introduced earlier in this session, you used and observed inductive reasoning. Inductive reasoning involves generating data from specific cases of a problem, keeping an organized list of results, looking for a pattern in the data, and then making a generalization. In other words, we are using inductive reasoning when we make and justify a conjecture based on what we have observed.

Deductive reasoning, on the other hand, uses general rules to prove something about a particular case. In other words, deductive reasoning uses statements that we already know are true to prove other statements. In the Building Rafts with Rods problem, we first used inductive reasoning to develop a generalization from patterns, but in order to prove that the generalization always holds, we needed to use deductive reasoning. As students worked with the models to generalize a rule for finding the surface area, they were beginning to use deductive reasoning.

Another example of using inductive reasoning follows an activity in which students make observations from exploring angles in a triangle. After examining many triangles, students may come to the conclusion that the sum of the angles in a triangle will always be 180°. The students' conjecture is based on what they have observed from testing many triangles, so they are using inductive reasoning. As students relate this conjecture to other properties of triangles that we know to be true, they would move toward using deductive reasoning to justify their thinking. Middle-grades students will usually begin to justify their reasoning through the use of inductive reasoning. Our goal is to help them move toward more formal justification and proof by reasoning deductively.

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