Reflection: High Quality Task Introducing Inductive and Deductive Reasoning - Section 3: Small Group Investigation: Bisecting Obtuse Angles

 

First of all, this particular activity gave me the chance to really break down the structure of a conditional statement.  What is special about everything that follows “if”?  (this is the given information, what we are starting with, the thing we are trying to investigate).  What is special about everything that follows the “then” (the conclusion we are trying to draw, the thing we are trying to prove after we have noticed some patterns)? 

 

How is filling in the blank with “congruent” different (and not as good) as filling in the blank with “acute”?  (All angles that are bisected create two angles that are congruent; we know this because that is the definition of bisect.  The thing we care about here is the fact that when OBTUSE angles are bisected, we get acute angles.) 

 

Additionally, this problem really helped me to drive home the idea of convincing a skeptic (essentially, the way I get students to “prove” their conjectures to one another).  I decided to actually play the role of the skeptic during class today; most of my students wanted to explain through the three small examples we investigated in the warm-up.  I asked, “well what if it’s only true for these three examples?”  In EVERY class, someone drew a 179 degree angle and bisected it, showing me that the resulting angles were both acute.  I decided to keep playing the skeptic, asking, “Well what about an angle measuring 179.99999 degrees?” This annoyed them, but helped me to make my point clear.

 

The best argument today was made by a student who had originally wanted to bisect a 179 degree angle.  This student finally said, “Let’s take a straight angle, which measures 180 degrees and bisect it.  We get two right angles.  Since an obtuse angle’s measure is bigger than 0 but less than 90 degrees, half of that must be less than 90, which means the angles are acute.”  Not everyone in the class seemed to understand the distinction between taking half of 179 and half of 180, so I made it a point to have students discuss in their groups, then share out again to the whole class.  I called on at least two other students to try to reframe the idea in their own words and explain why this argument was more convincing than putting forth the example of bisecting a 179 degree angle, which really drove home the idea of the function of a proof.

  Using Inductive and Deductive Reasoning to Understand the Concept of "Proof"
  High Quality Task: Using Inductive and Deductive Reasoning to Understand the Concept of "Proof"
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Introducing Inductive and Deductive Reasoning

Unit 4: Discovering and Proving Angle Relationships
Lesson 2 of 6

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.

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