With the Unit 6 Exam coming up in two days, I want to hit the ground running today. As students arrive, I post the opener (on the first slide of the lesson notes). When the late bell rings, I say, "In two minutes, I'll show the solutions to this opener." There's a brief chorus of "What? Just two minutes?" from some students, but then everyone realizes that they're familiar with this task, and that it can be done quickly. Everyone scrambles to get going, and I take pride in seeing how students urge each other to get to work.
At precisely the two-minute mark, I post the solutions, which are on the second slide of the lesson notes. I ask everyone how they did, and if they have any questions. For many kids, this is a great confidence builder, and for others, it will help them to identify what they know and what they don't about this this learning target.
I'll take questions from the class, and I might spend a minute or two sketching area models for a couple examples. Four weeks ago, students saw area models like these for the first time, and I urge everyone to acknowledge how far they've come. After all we've done, this feels comfortably approachable to students, and that's a big deal. Some kids may still need to demonstrate their mastery of SLT 6.1, and they'll have that opportunity today.
Following the fast-paced opener is another brief review, this time of the relationship between the equation and the graph of a quadratic function. First, I frame the work of the next two days by posting the third slide of the lesson notes and reminding everyone that the two-day exam for Unit 6 is coming up at the end of the week. I explain that on the first day there will be 20 short open response questions, that the second part will consist of 25 multiple-choice questions, and that as usual, each part of the exam will be graded on the learning targets for Unit 6. "For the opener, we just reviewed SLT 6.1," I say. "Now we'll take a quick look at the graphs of quadratic functions, which is what SLT 6.4 is about."
I flip to slide #4, which shows the graph of f(x) = x^2 + 8x - 20. The roots, the vertex, and the y-intercept are labeled. I ask a few questions, starting with, "What is this graph called?" After we correctly name it a parabola, I ask, "Where is the vertex of this parabola?" Students simply have to read the coordinates from the front board. Again: my purpose today is to host a confidence-building review session. Then, "How many roots does this parabola have?" and "What are they?" Finally, I ask students to identify the y-intercept, and I ask if everyone can see how each of these features is related to the equation for the function.
I check the time, because I want to ensure that students have at least 25 minutes of individual work time today. As long we've got time to fit it in, I then pose two tasks on the next two slides (#5 and #6). First, students are instructed to sketch a graph of f(x) = 2x^2 + 16x - 40, which is simply the first function multiplied through by 2. I leave the original graph on slide #5, so we can add to it and compare the two functions. Whether students start by finding the roots (which are the same) or the vertex (which is on the same axis of symmetry, but lower), there are insights that will come naturally here. These two graphs are certainly related. With slide #6, we repeat the process for the function f(x) = (1/2)x^2 + 4x - 10, which is the first function scaled by one-half, and which has the same roots as the first two functions, but a higher vertex and y-intercept. I allow students to lead me through the steps they'd take - whether we start by finding the vertex or the roots, either way is fine - and when we're done, we can consider the three related graphs on the front board.
This task gives serves to summarize some of what students have learned about the effect of changing the lead coefficient of a quadratic function, but it also gives us the chance to run through some background skills, like plugging different lead coefficients into the axis of symmetry formula and factoring for a values other than 1. It also provides background for using the quadratic formula, which some students will do today.
The rest of today's class is a work period, and students should use the time to complete any assignments they wish. On the last slide of the lesson notes, I provide some suggestions. My message to teachers reading this lesson is similar: you should view the seven lessons preceding this one as a suggested sequence that will be adapted to the experience of your students. For practical purposes, I've shared one new assignment in each lesson, but real work time is much more flexible than that. For example, even though I wrote about the Five Point Graphs assignment more than a week ago, if that's what a student is able to do today, then that's ideal. With that in mind, please take a look at the last few lessons to see the assignments that are in play today.
Also with that in mind, of course my ideal would be that all students mastered SLT 6.1 (I can find the product of two polynomials.) weeks ago. But kids need space to learn at their own pace, and maybe some students are finally really getting it today. When I see that happening, I let them at it!
I've prepared a Mastery Assignment for SLT 6.1, which gives a set of exercises at increasing levels of complexity. As the instructions on that assignment indicate, each subset of exercises represents a greater level of mastery than the one preceding. Students can do as much of this assignment as needed to achieve the mastery grade they want, or they can simply use it to review for the exam. Some students will use this assignment to get a 2 on this SLT for the first time. Others will grapple with the end of the assignment to see if they can raise their grade from a 3 to a 4.
Still others will ignore this assignment entirely and are ready to work with the quadratic formula - that's great too. Part of what distinguishes this unit from the others is that it's an rigorous study of pretty much pure algebra. Students bring different amounts of experience and expertise to the table, so it's fully expected for there to be a wide range of results here.
Today isn't the first chance that students have had to see the quadratic formula. Some used the formula on Delta Math and others received today's Mastery Assignment for SLT 6.5 during a previous work session.
The quadratic formula isn't essential to hit in an Algebra 1 course, but students who want to learn to use it and who have some understanding of the axis of symmetry and the discriminant are going to dig it. A key purpose of the Five Point Graphs activity is to help students build a conceptual understanding of how and why the quadratic formula works, and for any students who get to this point, it's an example of beautiful, elegant mathematics.
You'll see in this image that I use an example from the What Does a Do? assignment to show how the formula works. The functions on the back of that assignment (which I introduced a few days ago) are great examples for use in this context. For the strongest students, the best options are to finish the five point graphs on the back of What Does a Do? and get as far as they can on the Mastery Assignment for SLT 6.5.
There's also a connection to completing the square, which like the quadratic formula is non-essential to Algebra 1, but which is so cool that we shouldn't deny kids the chance to learn a little about it. If kids have that down, then the quadratic formula is more of a shortcut than anything. In this course, I don't attempt a formal derivation of the formula by completing the square, but to do so would require little more than connecting some dots that are already in place. Around this time, I ask my brightest students whether they'd prefer to use the formula or the algebraic steps of completing the square to solve equations, and somewhat surprisingly, they're unanimous in preferring to complete the square. I'm thrilled about that, because to me it indicates that they expect to understand how and why things work. In terms of being better prepared for their future study of math, facility with completing the square is going to come in handy.