Introducing Inductive and Deductive Reasoning

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Objective

Students will be able to use inductive reasoning to look for patterns and generalize the patterns; then, they will use deductive reasoning to prove the generalization they made.

Big Idea

Students will use inductive and deductive reasoning, making connections to how this kind of mathematical thinking fits in with the notion of proof.

Warm-Up

15 minutes

I give the following Warm-Up problems today because they offer my students two familiar mathematical situations in which they can apply inductive and deductive reasoning:

1. Consider the sequence: 2, 4, 7, 11, …
(Find the next three terms and explain how you know)

2. Solve the following equation and give a reason for each part of your process: 

3(2x-5)+20=5(2x-5)

The first problem asks students to explain how they make sense of a pattern and extend it. A great followup question that I like to ask my students is, "How would you determine the three terms preceding the given starting point in the sequence?"

The second problem highlights the idea that when we apply the order of operations we are making a logical decision. In other words, solving an equation is an example of following agreed-upon rules to come to a conclusion. This allows students to see that they always need a reason to back up every statement they make, which is essential for good proof writing.

Bisecting Obtuse Angles Video Narrative

2 minutes

Small Group Investigation: Bisecting Obtuse Angles

10 minutes

Next, I facilitate a small group investigation by drawing three examples of an obtuse angle that has been bisected on the whiteboard and asking students to conjecture about the two newly formed angles in their notebooks.  When I ask students to examine small examples, identify the patterns that occur within these examples, and form conjectures about the phenomena occurring within the examples, they are essentially expressing regularity in repeated reasoning (MP8).

I put a sentence frame on the whiteboard to get students to write their conjectures in the format I want (conditional statement):

If an obtuse angle is bisected, then the two newly formed congruent angles are _________.

By discussing the structure of conditional statements, students can look for and make sense of the "if...then..." statements they are given and better understand how they can use them in their own proof writing (MP7).

I have found that this whole-class discussion is a great way to introduce the language of proof with respect to arguing from given information towards a conclusion.  As a result, I like to push the conversation further, asking students how they would convince a skeptic that their conclusion will ALWAYS be true--this helps us to further develop our understanding that proof requires showing something is true in not just a few examples, but in call cases.

Ideally, students say something to the effect of "An obtuse angle's measure is less than 180 degrees, so bisecting an obtuse angle means taking half of something that is less than 180 degrees; this will always result in something less than 90 degrees, which, by definition means that the newly formed angles are acute." We'll see.

Check for Understanding: Finding the nth Term

15 minutes

I draw the following table on the whiteboard and give students 2-3 minutes to find the rule for the nth term.

x

2

3

4

5

6

7

 

y

7

4

1

-2

-5

-8

 

When students are reviewing homework in their groups, I quickly sort students' work into two groups: one group of tutors, one group of tutees.  After the homework review, I create new, small groups of students, with the tutors teaching the tutees about how they came up with the equation and coming up with more examples for the tutees to practice.

 

Inductive and Deductive Reasoning Investigations

20 minutes

I give students a set of shapes and ask them to sketch the 25th shape and describe the 30th shape.  To do this, students have to examine the pattern closely and explain what they see changing in the even-numbered shapes as well as the odd-numbered shapes so they can find a way to generalize the pattern. 

After students work with those shapes, they investigate examples of adding congruent segments to a given segment and determine each sum.  Essentially, students apply the idea of the addition property of equality and conclude that the lengths of the summed segments are equal.  As the teacher, my main goals are to begin teaching students some of the language needed for proof writing and to show them that when writing a proof, they must show the conclusion is true in all cases, not just in a few examples.

While students work on these investigations, I circulate the room, listening in on how students see and explain the patterns they observe.  I make notes to myself about which groups of students might be best to call on for debriefing the lesson, which is important since I want to send students the message that there will always be multiple ways to get to the correct solution and that it is important for us to be able to make connections, which deepens our understanding.

Debrief Inductive and Deductive Reasoning Investigation

15 minutes

I debrief the investigations with the whole-class, formalizing their discoveries by writing them in our Notes.  I call on students to share out what they found for the 25th and 30th shapes in the first investigation--these students come up to the whiteboard or document camera to draw their shapes and explain how the shapes fit the patterns their groups saw.  After taking questions from the audience, asking questions like, "how do you know?" and "did anyone else come to the same conclusion but see the pattern in a different way?" we move onto debriefing the second investigation.  

Again, I call on students to share out the conjecture they made based on looking at a few examples, and then we algebraically prove the conjecture is true as a whole class.