Finishing Five Point Graphs

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SWBAT sketch quick graphs of quadratic functions, identifying and labeling the key points on each: the roots, the vertex, and the y-intercept.

Big Idea

A focus on special cases can reveal a lot about the fascinating behavior of quadratic functions.

Opener: Can We Make a Five Point Graph?

10 minutes

The purpose of today's opener (on the first slide of these lesson notes) is to review the idea of "Five Point Graphs" from yesterday's lesson, and to show one special case of such a graph.  A big part of today's lesson will be to examine some of the special cases where the five points laid out yesterday might overlap, or not exist at all.  If you're unfamiliar with the "Five Point Graphs" activity, please take a look at yesterday's lesson to get up to speed.

I give students a few minutes to get started on this problem, and it soon becomes clear that we're not going to be able to find the five unique points that we did yesterday.  That's because this function has one root, and that root is also the vertex of the parabola.  The best we'll be able to do, if we follow the steps laid out yesterday, is to make a "three point graph."

When we left of yesterday, students had a few choices for how to go about creating a five point graph.  Depending on the route students take to complete this problem, there are different ways that the uniqueness of this function might present itself.  I watch to see how each student approaches the problem, and then I ask questions accordingly:

  • If a student starts by factoring to find the roots, they'll see that this function has two equal roots, which, for our purposes, means there's just one root to plot on the graph.  I'll ask, "So if the roots are equal, and we know that they axis of symmetry is halfway in-between the roots, then where is the axis of symmetry on this function?"
  • If they start by using the formula to find the axis of symmetry and then evaluating the function to get the vertex, they'll see that the vertex is on the x-axis.  I'll ask, "Wait, doesn't that mean that the vertex is a root?  So where is the other root?"
  • If they determine the axis of symmetry and then the discriminant of the function, they'll see that the discriminant is zero.  I'll ask, "But if the discriminant is zero, then its square root is also zero, which means that the distance from the axis of symmetry to the root is zero.  What will that look like?"

I take a few laps around the classroom, watching students work, encouraging them to help each other out, and asking these questions.  When it seems like the time is right, I ask for a volunteer to show us what they've got on the board, which leads to a conversation in which we synthesize some of the points that are hinted at in my questions above.  This won't be the last special case we look at today.  There's more of that to come as we move into work time.

Work Time: Finishing Up Five Point Graphs

28 minutes

You can see on the second slide of today's lesson notes that I give students a few options of what to do today.  I expect them to finish the Five Point Graphs assignment first, however, and for most kids, the time they get today will be exactly what they need.  The primary purpose of this work list is to ensure that anyone who finishes early can quickly make plan for what to do next.  The middle two work options are the same as they were at the end of last week.

If you haven't already, take a look at yesterday's lesson to see how the Five Point Graphs assignment works.  Following the opener, I tell students to find their work and to pick up where they left off.  I circulate to make sure that everyone has what they need, and soon everyone is cruising.

I have two roles today.  One is simply to keep moving around and providing help as needed, to individual students or small groups.  The other is to look for opportunities to lead a whole-class discussion about a problem or two, just like I did at the end of the opener, in order to help everyone learn something new.  Such opportunities will come at different times in any given lesson.  It's all about paying attention to kids, waiting for a quorum to have a similar question, and then guiding everyone to see what's going on.

Whenever it's impossible to find five unique points on the graph of a quadratic function, there's something to learn.  Each of the special cases on this assignment represents an opportunity to do that.  Rather than trying to record every detail of what might happen here, I've shared some pictures of the board.  Take a look at these photos and try to reconstruct the kinds of conversations that would lead to notes like these:

One thing that makes the "Five Point Graph" idea particularly useful is that we can't always get five unique points on a the graph of a quadratic function.  We already saw this in today's opener.  We might not have time to look at functions #5 and #7 during whole-class discussions, but each of these problems allows students to develop a deeper understanding of quadratic functions, their graphs, and their key points.


For students who make short work of this assignment, it may be worthwhile to have them take another look at the first four problems.  I point out that the y-coordinate of the vertex is the same for functions #1 through #4.  We might also observe that the discriminant is the same for each function.  Why is this happening?  Is there any relationship between the discriminant and the y-coordinate of the vertex?  If you have a conjecture about this, does it hold for other quadratic functions?

The relationships seen here are quite evident if we rewrite all four of these functions in vertex form.  Slides #3 through #7 of the lesson notes can also be used to transition into a conversation about vertex form, and functions d, e, and f have a similar relationship to those on the front of the Five Point Graphs assignment.

Exit Task: Sketch a Quick Graph

5 minutes

Today's exit task (on the last slide of the lesson notes) is the same as it was a week ago, at the end of the Features of a Parabola lesson.  I expect students to notice their progress when they complete this task.  As they submit their work and leave class, I'll check with as many students as I can, to ask if they feel like they've gotten better at quickly sketching parabolas.

No matter what kids have done the last few days, I want to see how well students can graph and label the key points on a quadratic function in a short amount of time.  In addition to demonstrating to individual students how much they've learned, these exit slips will give a quick snapshot of where everyone stands, and I'll be able to plan scaffolds and extensions accordingly for the coming days.  Everyone is going to master each learning target at different times, and it's a given that everyone will have a different depth of understanding at the end of this unit.  This exit slip is a base level that I hope everyone reaches by now, and it provides a clear picture of whether or not we're reaching that goal.