As the students enter the room, I hand out the Do Now. On it, the students are asked to identify whether the triangles are congruent and why, and to name the triangles with their corresponding parts in the correct order. One set of the triangles (#4) cannot be proven congruent, and we will be using constructions to examine why this is the case later in the class period. When all students appear to have finished, we discuss their answers as a group.
The class began these Isosceles Problems during yesterday's class and finished the problem set for homework. To review, I ask the students to compare their answers within their groups. Then, we will discuss any problems on which several groups were unable to agree upon a common answer.
For this lesson's Triangle Congruence and Constructions activity, I ask the students to work in pairs. I make copies of these (1 page, back to back) such that one student in the pair has directions on the back page asking for an acute angle, while the other student has directions asking for an obtuse angle.
Starting with Problem #1 I ask each student to construct a triangle having the given side lengths using a compass and straightedge. Then, I ask the students to compare their triangle to their partner's considering the following questions:
I ask that the students do the same with Problem #2. Again everyone's triangles should be congruent if constructed correctly. In Problem #3, the pair's triangles will not be congruent, even though they have a pair of angles and two pairs of sides congruent. I take this opportunity to introduce the words "ambiguous" and "ambiguity", giving the students some exposure to the ambiguous case before they encounter it in trigonometry.
When we get to Problem #4, I ask the students in each group to compare triangles. In this case, their triangles should be similar because of the Angle-Angle Similarity Postulate, but not necessarily congruent.
Through this exercise I hope to convince the students that our reasons for proving triangles congruent (ASA, SAS, SSS, and AAS) do indeed guarantee congruence, while SSA and AAA are not sufficient to prove that two triangles are congruent.
Each year, at least one student asks, "Will our proofs will always work?" I am not sure why this occurs, but I have learned to expect it.
Last year, I gave this question a lot of thought, and came up with an activity in which the prove statement can't always be proven. As they complete the task, I want my students to recognize when this happens to be the case (i.e., the conditions when the proof will not work).
For the Mystery Proofs activity, students to work in pairs. The activity actually consists of 5 proofs. All of the possible "givens" for each proof are cut out, turned face-down on the students' desks, and each member of the pair randomly chooses 2 givens. In the case of the final proof, it will be 3 givens for each. Each student in the pair enters these givens in his or her proof, and then attempts to complete the proof.
Sometimes the students will be able to prove that their triangles are congruent. However, sometimes they will end up with SSA, in which case they need to state that congruence cannot be proven. When each member of the pair has completed a proof, they check each other's work, and then move on to the next proof.
Adding this small element of suspense - will my proof end up with SSA? - makes the proofs exciting and fun for the students.
Teacher's Note: I have left the format of the proofs up to the students. They can choose to do either flow chart or three column proofs. They were introduced to paragraph proofs earlier in the year, but I encourage the students to focus on these two particular types of proofs in this unit.
After the Mystery Proofs Activity, I will ask my students to solve some more problems with Isosceles Triangles. I will allow my students work in their groups, discussing the problems and comparing answers on this problem set, which consists primarily of numeric and algebraic problems involving the angle measures of isosceles triangles. The algebraic skills required to solve these problems include solving first degree equations and systems of equations.
I let my students know that any problems that they do not finish in class will be assigned as homework.
Today's Ticket Out the Door is a brief student self-assessment. I ask each student to give me feedback on how he or she feels they are doing with respect to both the proofs and the numerical problems that we are completing in this unit. This provides me with an opportunity to identify problems individual students might be having, and to get a feel for the overall class's progress as well.