Puzzle Proofs

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Objective

SWBAT complete congruent triangle proofs in flow chart form. SWBAT solve missing angles of triangles problems.

Big Idea

A hands-on approach to proofs helps students to build confidence and conceptual understanding!

Do Now

5 minutes

When the students walk in the room, there is a paper copy of the Do Now on their desks and the images are also displayed on my white board. The intent of the Do Now is to two-fold:

1. To give students practice with isosceles triangles and parallel lines

2. To practice solving quadratic equations and systems of equations

After giving the students sufficient time to work on the problems, we will discuss:

1. How do you start numerical problems like these? This is a discussion that we seem to have frequently, but some of my students still need to be reminded to fill angle measures into the diagram before even attempting to set up an equation or arrive at an answer.

2. What about the negative answers? Do they "work"? I don't know if this is something particular to just my district, but my students seem to have the phrase "reject the negative" stuck in their heads. In both of these problems, we check all of the values that we found to make sure that the values make sense in the problem. (MP1) The negative x and y values do check in the problems, and this provides a great opportunity to revisit the order of operations, as students often struggle with the notion that squaring a negative number results in a positive number.

Going Over the Homework

10 minutes

My students worked on Isosceles Triangle Problems during the previous lesson, and, they were asked to finish it for homework. As I ask individual students for the answers and as the students raise questions, I focus my discussion once again on:

  • reading the given information
  • understanding the vocabulary
  • marking and filling in the diagrams

These three things were a major emphasis yesterday, so we will continue to work on internalizing these strategies today. 

The Puzzle Proofs Activity

30 minutes

For this activity, my students work in pairs. Their desks are arranged in sets of two because they will be managing a set of loose materials. There are a total of 10 proofs for students to explore in the Puzzle Proofs activity. I have made enough copies of the proofs for each student in the class to have his or her own packet. However, during class, I hand out only one set for each pair. 

Teacher's Note: Using colored paper,  I have made two copies of the Statements and Reasons for each proof, and cut out each statement and reason. It takes a lot of time. I'd suggest laminating the statements and reasons so that you can use and reuse them for several years. Each set of statements and reasons goes in an envelope on which I have written the number of the proof.

Numerical Problems

20 minutes

After a brief discussion in which I listen to my students' reactions to the Puzzle Proofs activity, I ask my students to return to their normal groups of four. As they rearrange their desks, I will hand out the Numerical Problems packet. These problems are intended to provide more experience with angle relationships in triangles. The students work in their groups, discussing the problems and comparing answers as they go.

Any problems that are not completed during class will be part of the homework.

I found this problem set available on the Kuta Software website.

One More Proof and Homework

10 minutes

With 10 minutes remaining in the period, I hand out packets of the Puzzle Proofs to those students who do not yet have them. Recall that each pair of students was given only one of each proof in their packet.

I ask each student to write out and complete Proof #1 on their own. As the students work, I will walk around the room assessing each student's progress. In particular, I watch for students who are not marking their diagrams. Once again, I emphasize that this is one of the most important parts of completing a proof. So often I have students tell me they don't know how to finish a proof, but as soon as I have them mark their diagram, they immediately say, "Oh, now I get it!" With my students, learning to mark diagrams consistently is a huge step in the right direction!

Before students depart, we will discuss the proof as a class. Then, I will assign Proofs #2 and #3 in the packet for homework, as well as any unfinished Numerical Problems.