Algebra and Angle Pairs

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Objective

SWBAT draw appropriate diagrams given verbal descriptions, and identify and solve problems involving adjacent, supplementary, and complementary angles.

Big Idea

More basics of Geometry: Drawing diagrams, identifying and naming angles and angle pairs, and solving algebraic problems involving angle pairs. And an answer to that eternal question - why can't we just measure instead of doing constructions?

What About This Construction Stuff?

25 minutes

For the "Do Now" I hand out a reading entitled, "An Explanation of Constructions."  This is a brief discussion of what constructions are, their origins, and why their rules are what they are. I took this reading from Math Open Reference, made a few slight adjustments, and really like it, though the article contains at least one inaccuracy.  (It states that a midpoint divides a line into two equal segments.)  I therefore ask the students to keep an eye out for errors as they read. The reading is short and fairly easy, but helps to give students some context with regard to the constructions they have been doing, and watching for errors keeps them focused and interested!  

In the reading, I used bold font on a few key vocabulary words.  After the students have finished the reading, we briefly discuss the words in bold, the contents of the article, and any errors they might have found.

We then review the two constructions we have learned (construct a perpendicular bisector and an angle bisector) by completing the Construction Review (MP5).  When the students have completed the constructions, I pepper them with questions to help review the concepts we have been learning: 

  • What point did you find when you bisected the line segment? 
  • What segment relationships were created?
  • What angle relationships were created by the angle bisector? Why? 
  • What are our definitions?

Practice with Drawing Diagrams

20 minutes

Two key components to success in geometry are:

  • understanding symbols 
  • being able to draw diagrams

Toward these ends, I hand out a practice sheet that hits both of these concepts and colored pencils:

Practice with Symbols and Diagrams

I ask a student to read aloud Number 1, asking his classmates to listen carefully for his interpretation of the symbols.  I have the students sketch an appropriate diagram, using the colored pencils to mark congruent segments AM and MB.  I walk around the room, checking the students' work and stressing that this is how we will always proceed whenever we encounter a problem with midpoints.  

We repeat this process with the remaining problems.  

More on Angle Pairs

40 minutes

I hand out the resource entitled Angle Pairs.  The first few pages of this serve as review of the previously learned angle concepts and as guided notes, introducing the students to the topics of adjacent angles, complementary, and supplementary angles.  Problem Number 5, a table, enables the students to use repeated reasoning (MP8) in order develop expressions for the Supplements and Complements angles. 

After completing and discussing Number 5, I work with the class on Numbers 6 and 7.  I emphasize four steps:

  1. Define the variables and expressions
  2. Read and translate the problem from English to "math"
  3. Solve the problem
  4. Check to make sure that the answer makes sense.  

I spend a lot of time on step Number 2, because I find this is often the hardest part for students. I really try to hammer home the importance of the word "is" and the necessity of reading slowly and carefully, translating one phrase at a time.

When it appears that everyone understands these problems, I ask that the students work together in their groups to complete remaining problems. I move about the room, from group to group, checking the students' progress and answering questions.

Lesson Closing and Homework

5 minutes

With 5 minutes left in the period, I hand out the homework and highlight with the students several of the items that we worked on today:

  1. The importance of drawing diagrams when trying to understand and solve geometric problems
  2. The four steps that we used when solving "word" problems