As we start today's class I invite students to discuss a problem from the previous lesson or present on a different of problem that students created in the last lesson. I take the opportunity to revisit key ideas that came out in the previous lesson. My goal is to help the class rediscover key vertical line concepts.
These next steps are typical for many of the mini projects we do in class. We have students work in partners or groups to create a problem to share with the class. This can be done on chart paper or in their notebook and then shared via document camera or on the board. For example, if my students are working on chart paper, I discuss the layout of their work and request that the diagram is clear and that their algebra is hidden until the class has had a chance to solve their problem.
Watch Students Create Right and Straight Angle Problems for an overview of my strategy for these types of lesson.
The main idea is to encourage divergent thinking and have students create their own algebraic supplementary and complementary angle problem. I remain at the ready to help student wrestle with the idea of linear solutions, decomposition, substitution and combining like terms.
I might have them write the problem only on the white board and then show their chart when the class is ready. As they work, I will stop the class to highlight examples and save student work to share with other classes. As this part of class unfolds, I go into facilitation mode and help groups create a problem that they find interesting enough to share. I help them play with line layouts to find a diagram that peaks their interest.
For example, if a student creates a pair of vertical angles and one angle is 30, they have to be comfortable with the idea of representing this angle as x. If x = 30 then we can decompose x into 100x – 99x (or any other combination) and then rewrite this as 3000 – 99x by substituting 30 into the term 100x. Then the other angle would also be x but needs to be composed in a different way. They could create a expression such as 5x – 4x and then substituting 30 into 4x and getting 5x – 120.
The great questions surround the ideas of this decomposed equality. Essentially students want to know how these expressions could ever be equal. Since we have covered slope and linear systems, I look for opportunities to spiral back and discuss linear equations. This can get complex and requires strong algebraic intuition on my part. Fortunately students love to control the parameters of their problem. They love to take ownership of their problem, so I usually don't have to work too hard.
At the end of today's lesson, I have students present their solutions to the problems they solved and discuss ideas around vertical angles. I like to take a moment and return to the original proof around why vertical angles should always be equal. I ask students to write their own version of the proof and include illustrations to match.