Proofs about Angles

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Objective

SWBAT write simple formal proofs involving angle pairs

Big Idea

Students will apply postulates to write two column proofs using vertical angles, complementary angles and supplementary angles.

Do Now

7 minutes

 As students walk in the room, I hand them a sheet of paper with a list of postulates and properties on one side and examples on the other. Students are instructed to match the postulate with its example. Each of these postulates was explored in the previous two lessons.

There are several ways this Do Now could be used. Sometimes I use it as a quick quiz, to check for student understanding and retention. In this case, I do not allow students to use their notes. After about 4 minutes, they hand in their sheet and I grade it after the lesson.

It can also be used as a regular Do Now activity; however, there are variations in the way we go over it. At times, I have had students check their own work using a colored pencil to make corrections. Another way to grade the activity is to have students swap papers and check their peers’ work. After going over the answers, I ask the students how many questions they got correct. For example, “Raise your hand if you got 6 or more correct. Raise your hand if you got 7 or more correct…etc.”

Students will continue to use these postulate and properties in writing formal proofs. It is important to know early on in the unit that students understand the concept.

Mini-Lesson

15 minutes

In the Proofs about Angles Mini-Lesson, we review precise definitions of previously studied terms:

  • right angles
  • vertical angles
  • supplementary angles
  • complementary angles
  • a linear pair of angles

I hand students a sheet which has a chart on it with the definitions already filled in. Students are instructed to draw an example to illustrate each term (MP4, MP6). We then use the presentation to go over theorems associated with the terms. These theorems are very intuitive and students may already be familiar with some of them. We will use these theorems and definitions in the lesson activity. 

After students complete the chart, we write a formal proof showing that vertical angles are congruent using the definitions and theorems from their chart, in addition to the postulates studied in previous lesson. This is the first proof of a theorem students see in the unit and it may cause them difficulty. We brainstorm ideas together first and then write the statements and reasons.

Activity

18 minutes

In the Proofs about Angles Activity, students work independently to write two column proofs using the Side-Side-Side postulate (G.CO.9). It is important to have the students take the first few minutes to look at the given statements and label the information on the diagrams. Then they brainstorm about what information they will need to write the proof. I'll encourage them to use information explored in previous lessons to write the proofs. If students have difficulty on their own, I pair up two students to help each other.

After about 10-12 minutes, I'll call on a student to present his or her proof to the whole class. We critique the student’s reasoning (MP3) and fix any misconceptions or errors. For these presentations and classroom conversations it is important to determine your level of flexibility and differentiation. For example sometimes a student may combine or skip steps when writing their proof which, depending on the level of the class or the students, I accept different answers when necessary.

Summary

5 minutes

At the end of the lesson, I have the students add the new theorems to their list of postulates and properties, which they created in the previous lesson. Even though they took notes during the lesson, rewriting the theorem and keeping a list on one page makes it easier for them to find useful postulates, theorems and properties.