I start with a simple quadratic function [f(x) = -4.0x^2 +10x + 6] on the board and ask my students to brainstorm and then pair-share what it might model and why they think so. Appropriate responses might include the path of a projectile, the area of rectangle, the shape of a lenses or mirror. The intent is to help students connect functions models expressed algebraically to real-world problems. (MP4) I discuss the importance of this in my video.
I randomly select students to share out possible models, then ask them to individually consider, by giving each student a notecard to record, what, at least two, terms of the equation might represent if modeling the path of a projectile. (MP2) This gives me a chance to see which students are on track while having all students practice writing communication, an task that is still a challenge for many. (MP3)
After a moment or two I collect the notecards, shuffle them, and read them aloud to the class, telling my students I to listen for those descriptions that make sense, those that don't and why. I have a small class, so students may recognize and acknowledge their notecards even as they're being critiqued. In a larger class the notecard strategy allows every student to contribute to the discussion without being singled out.
At least a few students should recall that a negative sign with the quadratic term means the parabola is "flipped upside down", to use student language. When they say that I restate it more accurately like this, "Yes, the parabola is upside down which means that it has a maximum point for its vertex and the y-values become increasingly negative as you move from that vertex." (MP6) This leads to the next key observation, the constant term indicates that function crosses the y-axis at 6. Seldom does anyone connect the y-intercept to the starting height of the projectile, nor that the linear terms gives information about the speed of the projectile. Since that's what this lesson is working towards I lead students to this discovery in the next step.
I challenge students to work with a partner to rewrite the equation to show different information. They usually ask for clarification and I respond with a question to lead them to rewriting in factored form, something along the lines of "How could you rewrite this function to show where the projectile touches the ground?"
As students work I walk around offering assistance and redirection as needed. (MP1) I chose this problem since it should be easy enough for students to factor. The factors are (4x+2) and (-1x+3) or -2(x-3)(2x+1) which students can find the zeros and know the projectile touches down at either (-0.5, 0) or (3,0). I ask what the negative sign represents for the x value of the first point and again, most students can plot it on a graph but few if any can say what it actually represents in terms of the projectile. To help them visualize this and the earlier key points we found, I hold a softball in my hand and physically walk it on a path from my hand to a point in the back of the room as though I had thrown a slow pitch. They can now more easily recognize that the ball doesn't start on the ground behind me (-0.5, 0). I ask for a volunteer to sketch a graph of path of the softball on the board, then ask for another volunteer to label the axes. (MP4) This is tough for some of my students because they still don't connect graphs to actual events and don't see equations as models for real-world happenings.
key features of original equation: height at start is 6m, speed upwards is 10m/s, gravity effects = -4.0m/s^2
I tell my students they will be working with their left shoulder partner for this section but that I expect each student to write out their solutions and reasoning in their own words. I distribute the Keys work and give them about 20 minutes to complete the worksheet and prepare for class discussion. (MP1, MP2)
While they're working I walk around giving encouragement and assistance as needed. For example, some students will struggle with the last question because it doesn't give a specific function but instead asks them to really think about what a function would look like for each given context. For those students I might suggest "Have you considered what an average salary for an attorney might be?" or "How would the population of India change from 1993 to 2013, increase, decrease or stay constant?"
After 20 minutes or when all the teams are done, I tell my students they have about five minutes to prepare to share their solutions and reasoning with the class. I then randomly select teams to present each problem, with the rest of the class critiquing both the solution and the reasoning. (MP3)
To wrap up this lesson I ask my students to write notes about the new terms we've discussed today and to include examples and sketches. (MP6) I post the list of the terms (see below) I would like them to have but encourage them to add any others they think they might need to remember.
TERMS: intercept, intervals where increasing, decreasing, maximum and minimumvalues, symmetries, end behavior
There are always a few students who are uncomfortable with the idea of writing things out in their own words so I explain that this is not about what I want or need to see it's about them having these terms in their notes in a form that makes sense to them.