How Does a Parabola Grow?

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SWBAT identify the key features of a parabola, and to use the structure of the perfect squares to see a novel shortcut for plotting successive points on the graph.

Big Idea

It's not so much about the shortcut as it is about the structure it reveals!

Getting Started: Identifying the Features of a Graph

10 minutes

We're in the computer lab today.  Students will work to complete a series of seven exercises on Delta Math, the first four of which are about the graphs of quadratic functions.

The first module is simply about identifying the features of a parabola:

Given the graph of a parabola, students are asked to give the coordinates of the roots, the vertex, or the y-intercept, or to give the equation for the axis of symmetry.  A drop down menu allows them to choose whether their answer will consist of one or two coordinate pairs, the equation for a line that starts with x= or y=:

Students have seen this task before, at the start of the "Different Forms, Different Parameters" lesson.  It serves the same purpose here as it did there.  I want students to be able to identify the roots, y-intercept, axis of symmetry, and vertex of a parabola, and the best way to learn these definitions is through using those words in context.  

I buzz around and encourage everyone to try to get five in a row as quickly as possible.  "If you find that you don't know one of the vocabulary words you see here, ask your neighbor if they can give you a hint."

Usually it turns out that the most common errors kids make are to mix up a +/- sign or to reverse the order of x and y in a coordinate pair.  These are little errors that are easy to overlook, but when students have to correctly get five in a row on an unforgiving computer, they're also the type of errors that are quickly overcome.

Mini-Lesson in Small Groups: Using Shortcuts to Plot Points

10 minutes

When they get five in a row on the first module, students move on to the second, which prompts them as follows:

I give each student a few minutes to try this on their own.  It's very similar to what they've just been doing on the Quadratic Functions Gallery Walk, which was to find the coordinates of a set of points on a parabola and then graph them.  At the same time, we've also spent the last few days discussing strategies for quickly sketching the graph of a parabola.  Over the last few days, the importance of the vertex has become pretty clear to kids.  They realize that if they can just find the center of the parabola, then a lot of the rest will fall into place.  In contrast to that, simply plugging in numbers and hoping for the best can yield y-coordinates that are way too high or low to fit on a graph, and students recognize the inefficiency in that approach.

With all of that in mind today, I share a novel approach with students.  Every time I see that five or more students have made it this far, I invite them to the front board for a standing small group mini-lesson, and I say quietly and a little conspiratorially that I'd like to show them a neat little shortcut.

This shortcut is based on the way a sequence of perfect squares (0, 1, 4, 9, ...) grows from one term to the next.  I show students that if we start by finding the axis of symmetry and the vertex, we can use the idea that perfect squares grow by successive odd integers to find more points on the graph.  Then, we use the axis of symmetry to reflect these points, and just like that, we have a parabola!  Finally, we can use the roots to make sure that everything works out.

The quadratic functions generated for this activity by Delta Math are particularly well-suited for this trick, and of course this shortcut changes (although not to the point of irrelevancy) when we change the coefficient of x^2 to anything other than 1.  Half of the exercises in this activity do have a lead coefficient of -1, which provides a terrific opportunity to see quadratic functions in a new light, to factor out a -1, and to connect these ideas to those of negative slope.

I teach this way because in many ways, the shortcut is besides the point.  We're thinking about and using structure here (MP 7).  To understand this is to lay foundations for all sorts of mathematical ideas that extend far beyond the scope of Algebra 1.  The pattern of how the square numbers grow, which was the focus of the opener right before we jumped into the Gallery Walk is one worth being familiar with.  If students really take to this strategy, they'll be even more likely to be amazed when they see how the pattern reveals itself for narrower and wider parabolas.

Work Time

23 minutes

Once students successfully graph five consecutive parabolas, the next task is a combination of the first two.  Students must plot seven points on a parabola, and then identify one of the key features.  It's great reinforcement of what they've just done, and the vibe that results from the contagious self-confidence of kids makes it all worth it.  We've reached the end of April, and to see kids celebrate their self-efficacy is more satisfying than teaching any particular algebra topic.

If everyone gets at least this far today, I'm very happy.  There will be a few stragglers, but an advantage of using Delta Math is that I can quickly view of a summary of who has done what, and use that to plan my next steps.

The remaining modules are not essential to today's lesson, but there here for anyone who moves quickly through the first three.  This is an example of how technology can cut down on the heavy-lifting of differentiation.