While at a PD session collaborating with other teachers in my state, one teacher shared a resource with me that can REALLY help your students ability to understand significant features of a graph for a wide variety of functions. Desmos Graphing Calculator is a tremendously user friendly online graphing utility that the kids really enjoy. In the opening few minutes of this class, I direct my kids to the Desmos link via our course webpage. After about 30 seconds of making sure that everyone has access to the site, I display the graph on the Lesson Image and challenge the students to recreate the function on their own Desmos graph by manipulating an equation. The students also love it if you make it really competitive.
About 90% of the students will feverishly guess and check with the graph (MP1) although other systematic approaches are better (especially since we explored this topic in the last lesson) it is GREAT critical thinking for the students to guess and check and in real-time see how the coefficeint changes influence their graph (MP7).
The remaining 10% of the students will take a much slower, systematic approach to finding the equation of the graph. Although these students aren't plugging values in at break-neck speed like the rest of the class, it does not mean that they are not processing the task at hand!
Once several students have the correct answer, I ask the class how they approached the task. We take time to comment on the pro's and con's of a variety of methods, and ulitimately make the connection to the previous lesson where we showed that factoring can be shown as breaking down a quadratic function into two linear functions.
Be prepared for the students to want to continue to play around with Desmos. You may need to ask them to turn their technology to "Courtesy Mode" so that you can continue on with the class.
Please see the attached video narrative that explains the rationale behind this lesson and the transition to the Common Core with the standard.
I have also attached the examples that I work through with the students help. In this lesson, I use a strategy called "Watch One - Try One - Teach One".
Watch One: The students watch me walk through an algebraic example, and copy it into their notes.
Try One: The students try a problem on their own using the first example as a guide.
Teach One: The students create, solve, and teach an example to a student. The "create" element is key, because if the students can build, solve, and explain the mathematics... then it is a great indication that they truly understand it inside and out.
An excellent "old school" tactic that I use to check for understanding is whiteboard practice. Early in my teaching career, a veteran teacher allowed me to to borrow her set of 16"x12" whiteboards for a lesson that I created. I immediately realized the power that they have as a teaching tool. For whatever reason, when a student is given a whiteboard they are more likely to participate without the fear of being wrong. It never fails that EVERY student raises a board with a response on it - even if it is wrong. With 30 other boards popping up in the air at the same time the student does not have to fear being wrong as much as other "check for understanding" teaching strategies. It is also instant feedback for the teacher! Any students to struggle I make sure to check on in in the next section of the lesson.
If you don't have the funds to go out and purchase a set of whiteboards, do what I did: Go to your local Home Improvement store and buy a couple sheets of shower board. Have your shop teacher (or do it yourself like I did) cut the shower board into mini whiteboards. Cut up an old towel into small pieces to use as erasers, and presto, your new instructional tool is complete! I use my whiteboards all of the time for a wide variety of applications.
At this time I circulate the homework assignment. I offer support to the students where needed and work with individuals who struggled with the Watch One - Try One - Teach One section of the lesson. The assignment is due at the start of class on the following day.