Where We've Been: Students have just finished an introduction to the fundamentals of Geometry. They've learned basic figures, notation, and terminology.
Where We're Going: Students will now begin to experience the reasoning aspect of geometry. An overarching goal is for students to understand the similarities and differences between inductive and deductive reasoning and to recognize how mathematicians use each of these types of reasoning to support the other.
So in this section of the lesson, we want to make sure students can identify pairs of angles that are vertical , linear, supplementary, congruent, or complementary. Students have already learned to do these things. This is just a refresher.
Using student whiteboard strategies, I run through the following prompts:
1. Create a diagram that shows Angle 1 vertical to Angle 2. [Think, Pair, Share]
2. Create a digram that shows Angle 1 and Angle 2 forming a linear pair. [Think, Pair, Share]
3. Given that the measure of angle ABC is 42 degrees, sketch and label a diagram of angle PQR, the complement of angle ABC. [Think, Pair, Share]
4. Given that the measure of angle JKL is 65 degrees, sketch and label a diagram of angle HTM, the supplement of angle JKL. [Check Boards]
5. If the measure of Angle 1 is 52 degrees, what is the measure of an angle that is congruent to Angle 1? [Check Boards]
In this lesson, I want students to get a feel for how inductive reasoning works.
For starters, I provide some definitions:
Inductive reasoning: Reasoning that uses observations of specific details in order to make conjectures about general truths.
Conjecture: An educated guess about how things are generally
While I don't stop to define every term, I am very intentional about using academic vocabulary terms repeatedly in context so that students will understand their meaning and learn to use them. For example, I describe inductive reasoning as bottom-up reasoning. I explain that inductive reasoning is all about making generalizations from specific cases.
I drive home the point that the greater the number of examples that support a particular conjecture (in the absence of a counterexample, of course) the stronger the evidence for that conjecture.
Next, I provide some examples of inductive reasoning starting with a non-math example like: John recalls that every time he has seen dark clouds in the morning sky, it has rained at some time during the day. He makes a conjecture that whenever there are dark clouds in the morning sky, it is going to rain. Then I move on to a math example. For instance I might use dynamic geometry software (e.g., Geometer's Sketchpad) to demonstrate how we might come to a conjecture about the angle measures in a triangle. This serves as important modeling for what students will have to do, themselves, later in the lesson.
In this section of the lesson, each student will receive a protractor and calculator. Students will work individually to complete the Vertical Angles and Linear Pair Conjectures activity.
It is important that students attend to precision (MP6) when making their measurements so that they have reliable data upon which to base their conjectures.
If students are having trouble making conjectures, here are some probing questions that might move them forward:
1. As you analyze the angle measurements, are there any relationships that are consistent through the four cases we have before us?
2. Do you get any consistent results when you add two angle measures together?
3. How did you arrive at this measurement (assuming they've made an error)?
When students have made their best attempts at writing conjectures, I try to find exemplars to share with the class. If there are no exemplars, then I model the language I would like to see for conjectures. Usually something like..."If two angles are vertical angles, then they are congruent"...or..."If two angles form a linear pair, then they are supplementary".
In this section, I reinforce the major take-home lessons for the day using a "think-out-loud" to model the thought process that goes into making a conjecture. I start by creating intersecting lines on Geometer's Sketchpad. This, of course, forms vertical angles and linear pairs.
Being very deliberate and using sufficient theatrics to hook students attention, my think-out-loud goes something like this:
By just looking, it seems like the two acute angles are about the same and the two obtuse angles are about the same...
How about I do some measuring to see if that's actually the case...
(After measuring) In fact, it is true for this case.
But I wonder if I just got lucky with this case. Let's try some other cases...
(Varying the angles widely and sometimes wildly) It seems that I can generalize and say that the two acute angles are congruent no matter what, and the two obtuse angles are congruent no matter what.
Oh wait, those are vertical angles. OK...revision... it seems that if two angles are a vertical pair, then they have to be congruent. That's my conjecture! And I have strong evidence to support it.
I also notice that when I vary the angles, and the acute angles gets bigger, the obtuse angles get smaller...and vice versa. It's like their balancing each other out. I wonder if they always have to add up to the same number and that's why they have to balance each other out.
Let's measure and see. Look at that, the sum is 180 degrees for these two angles that are next to each other. Is that generalizable?
In fact, it seems that if two angles form a linear pair, then they must be supplementary, no matter what.
Finally, I have students write a reflection summarizing what they have learned for the day.