This lesson opener builds off of yesterday's lesson. For a brief overview of the idea of a Daily Desmos bell ringer, click HERE.
To open class, I display the first slide of the PowerPoint. The students then begin work on their iPads (on Desmos) trying to re-create the graph of the of the function on the screen. If you do not have access to this technology, you can find an alternative or use graphing calculator technology. Also you might have students come to your computer and enter their guesses as the whole class watches.
When students first tried this type of activity, yesterday, most of them were feverishly entering guesses trying to get close close to the graph. Today, however, the students should have a more systematic approach connecting a factored form to the x-intercepts. In essence, they build the function up from the linear factors. What is novel about the graph presented today is the double root" which causing the graph to hit the x-axis and rebound, or "bounce." This will temporarily stump students, who will likely look at the graph and assume only a single root as a factor which results in causing it to cross the axis, rather than "bounce" off). However, they will soon realize that something different must be happening at this particular point on the graph. Many of the students will eventually catch on to this idea of a "double root" - allowing everyone to make the discovery independently is key. If students are still struggling after 4-5 minutes, prompt them by suggesting they looking at the graph of something like x^2+2x+1 and what they know about the factored form (x+1)^2, this should help them make the connection to a "double" root.
The important part of this lesson for the teacher is PATIENCE. In my experience as a math teacher, it is very exciting to show how concepts overlap, or illustrated this to them on a graph. HOWEVER, it is much MORE powerful to let the students play the role of the mathematician and have them try to piece together what is going on. Sure, it takes a little longer than if you just told them, but it also creates a culture of inquiry in the classroom (MP2 - Reason abstractly, MP4 - Model, MP5 - Use Tools Strategically). I do not worry about keeping students engaged who finish the task a couple of minutes early. I simply challenge them to create other functions that have double roots and to see how "cool" of a graph they can create!
After everyone has the first graph created, we move on to the remaining three in the presentation. These should now take only a fraction of the time that it took to construct the first graph.
In this lesson, I allow students ample opportunity to try to graph the function on the opening slide and as a part of the conversation I asked my students to share their successes and frustrations with the activity. Many students will respond that they were "Soooo close to finding the proper equation, but they just couldn't get the graph to turn back upwards instead of crossing the x-axis".
In all of this discussion, I made careful note to let the students know that I appreciate their willingness to share, especially if they didn't quite get the correct solution. It is common for me to say the phrase "I LOVE wrong answers!" once or twice every period. I do this because I want the students to know that we can learn so much more from wrong answers as opposed to only looking at correct ones. Most of the time, I students organically to provide me with these, but sometimes I might lead them down the wrong road myself just to make a point!
To wrap up the class session, I ask the student to write a few sentences summarizing what they learned in class today. I begin the journal by saying Today, we learned that __(and I have my students fill in the blank with their explanation in their own words)_____. In my experience, this is often difficult for students to put into writing. It is good practice for them to reflect on their learning, and begin to become accustomed to communicating it in a variety of forms.
When I do a journal of this nature, I do not make it anything fancy. I "push out" the journal to my class through our online learning environment and make it due on the following day. I make sure that my students know of this requirement before class begins (I post it on our course agenda so that the students see it right at the start). This ensures that the students know that they will be held accountable for their learning for the day. Another way that this could be done is to give you students a blank half sheet of paper to write their responses on.
A small portion of my students' overall grades are connected to these and other small writing assignments. Usually they are reflective in nature, as in this lesson. Embedding writing in your mathematics curriculum is a great way to show student growth over the course of a school year.