Congruence Conjectures Video Analysis Emily Nelson 12/06/2011

Video Analysis Watch intro of lesson

Objectives Students will be able to… Chosen benchmarks state the definition of congruency. state the SSS, SAS and ASA Postulates and use them to prove or disprove if two triangles are congruent. give examples of real-world applications of congruent triangles. Chosen benchmarks MA.912.G.4.3: Construct triangles congruent to given triangles. MA.912.G.4.6: Prove that triangles are congruent or similar and use the concept of corresponding parts of congruent triangles.

Evaluations Question 1: recall of the postulates from the lesson Question 2: provoke justification of congruency by applying the learned postulates Question 3: addressed the last objective involving identifying real-world examples of congruent triangles Inherent understanding of congruency necessary for all three questions

The Data

Evaluation Scores - Without 2 (c), 10 (55%) students receive a score above 60% - With 2 (c), 9 (50%) students received a score above 60%.

How does my questioning facilitate learning through problem solving? Effective questioning is very important in order to “elicit responses from students about material that otherwise would have been presented by the teacher” (Posamentier, pg. 62). Higher-order/Open-ended vs. Lower-level/closed Subdivided LL into… Yes/no Factual Leading

Percentage of HL vs. LL questions During classroom discussion, I asked about 100 questions, which I then analyzed.

HL Examples How do we know that these triangles are congruent? Okay, they are asymmetrical, but how do we really know they are congruent? No? Why not? How, why? What determines that congruence? Why, why…well, let’s ask someone to explain why they think they are congruent? Okay Brad, tell me why they are not congruent? Yes? Using what postulate?

Leading Questions Is it just three? Okay, so we have the base and the two sides. What about (points to each angle in a triangle)? So, everyone got the same triangle, right, if one side is five inches? No? Okay, so each line when they meet, they create a vertex, right? Okay, so the vertices match up, but we want to make sure the angle is between them, right? If we match up all the parts of the triangle, they would be the same, right? Yes? They are the same, right? At least one, right, because you can rotate that angle between the 5-inch long side and the 7-inch long side to create whatever angle, right?

Yes/No Questions Are any of these triangles congruent? Did you compare your triangles? Did everyone get the same one? No? Did everyone have the same triangle? And the last group, did you all get the same triangles? Umm, did they look the same? Did they look pretty much the same? They are the same, right? Did you get two that were different sizes? No? Do they look congruent or close to it? Tim, it looks like you’re thinking about it. Do you have an idea? Tim, can I have you finish reading this one?”

Sequencing (17 seconds of 2nd video) Does anyone have two triangles that aren’t the same? (LL = Lower Level – Quick yes/no)  Okay, about 5 3/8 or 5 ½ so are they congruent or not congruent? (LL – Yes/no) Do they have all the same sides? (LL – Factual, yes/no) Why do you think no, student Q? (compared two congruent triangles) (HL – requires explanation) Do they look congruent or close to it? (LL – Yes/no) What can we decide about SAS? (HL – generalizing SAS postulate)

Faults Only six instances in which I did not repeat a student’s answer word-for-word. Did not call on non-volunteers very much called on 5 different non-volunteers to read a definition or problem on a slide called on 4 different non-volunteers to explain their thinking. Other students probably did not feel accountable for having answers of their own.

Group Work Paired, face-to-face 6:54 – students working 17: 48 - “They are unique and therefore not congruent.”

The 4 F’s Forming Skills: ought to have emphasized the importance of working quietly and efficiently. Functioning Skills: should have modeled how students would move through each conjecture with their partner Formulating Skills: set the expectation of each partner communicating their understanding to the other. Fermenting Skills: hold students accountable for the justification of their conclusions.

Participation Classroom demographics 12 females and 12 males 13 of which are white, 3 of which are black, and 8 of which are either Hispanic, Asian or a different ethnicity. 13 of the 22 students present in class that day contributed to the whole-group discussions. about half females and half males – representative of the class. 10 are white, one is Hispanic, one is Black, and one is Asian. Calling on mostly volunteers.

Video Student Understanding Order confusion Teacher: “Okay Student E, why do you think yes?” Student E: “Because it has the same angle and two of the same sides.” Teacher: “Okay, but what order are they in? We have Side-Side-Angle, right? And like Student K was saying before, these sides look the same (pointing to the two unmarked sides) Segment from 23:15 Again specified the order of the Side-Side-Angle conjecture because I noticed many students were constructing a triangle using the SAS order instead. “oh man, I have to do this again?” and “wait, I’m confused now.”

Formative Assessment SSS Postulate segment (30:10) SAS Postulate segment (2nd video) Both instances, I focused on the conjecture before moving on.

The Great Debate (8 min) Student S: “They’re not congruent.” Teacher: “Okay, so we have one for not congruent. Why, why…well, let’s ask someone to explain why they think they are congruent? Student K, do you have an idea?” Student K: “Umm…” Teacher: “Okay so we know that these two are the same (pointing to two sides), right? – because of the hash marks. And then, they both have a…what kind of angle?” Student E: “Right Angle.” Student K: “Umm, I think I’ve changed my mind…” Teacher: “You’ve changed your mind, what do you think about them?” Student K: “Well, I mean…that side could theoretically be longer. Just because it looks congruent doesn’t mean it is.”

The Great Debate Teacher: “You’ve changed your mind, what do you think about them?” Student K: “Well, I mean…that side could theoretically be longer. Just because it looks congruent doesn’t mean it is.” Teacher: “ That’s very important to note just because it looks congruent, but in this case, they share the same side. So, could the length really differ for each one? Students: “Yes, no.” Teacher: “But that is really good to be observant and notice because a diagram could be deceiving. So, who thinks it is congruent, the two triangles are congruent? Raise your hand if you think so. Raise your hand if you think they’re not congruent. Okay Brad, tell me why they are not congruent.” Student B: “I still think they’re not congruent because the angles on the two sides, it doesn’t say that they are equal.” Teacher: “Okay, but what was the postulate we were just talking about?” Hailey: “If two sides and the included angle of one angle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.” Teacher: “Okay, so we have Side-Angle-Side, right?” Student in the back: “Not necessarily.” (Didn’t hear this comment until the tape!)

Revisions Extend over two days Include a worksheet Incorporate more higher-order questions aimed at the whole group rather than just willing volunteers. Avoid repeating student answers and elaborating for them Reorganize presentation Each conjecture, then definition. Post student examples on the board