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Teaching CCSS.Math.Content.HSA-APR.B.3

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  • Quiz Questions: 0

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Nothing. Nada. Goose egg. Zero. It's one of the most important and influential concepts in math—especially when it comes to factoring polynomials in order to graph them on the coordinate plane. Who knew that nothing could be so helpful? (Well, you did, but there's no need to rub it in.)

In Shmoop's A-APR.3 Teaching Guide, we'll give you all the activities you need to get your students psyched about identifying the zeros of polynomials using factoring, and using those zeros to construct rough graphs of those polynomials. What could be better than that? Nothing.

What's Inside Shmoop's Math Teaching Guides

Shmoop is a labor of love from folks who love to teach. Our teaching guides will help you supplement in-classroom learning with fun, engaging, and relatable learning materials that bring math to life.

Inside each guide, you'll find handouts, activity ideas, and more—all written by experts and designed to save you time. Here are the deets on what you get with your teaching guide:

  • 3-5 in-class activities specifically designed with the Common Core in mind.
  • 4 handouts (with separate answer keys!) that'll get your students thinking deeply about the concepts and calculations.
  • Additional resources that'll help make any math topic hip, hot, and happening.
  • A note from Shmoop's teachers to you, telling you what to expect from teaching the standard and how you can overcome the hurdles.

Want more help teaching Teaching CCSS.Math.Content.HSA-APR.B.3?

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Instructions for You

Teacher Text
Objective: "A city," says Plato, "is man writ large against the sky." We're not sure what he's talking about exactly, but we can get behind the whole large-writ thing. In this activity, your students will practice their polynomial factoring and graphing chops on a giant posterboard. Enough of that sissy regular-sized paper. "Go big or go home"—that was Plato too, right?

Teaching something is one of the surest bets to really internalizing a concept, so we'll also have students give a short group presentation on how they pieced together their graphs and found their zeros. Plato would be proud.

Activity Length: 1 class period
Activity Type: Groups of 3 or 4
Materials Needed: 4 large blank posters per group, lined paper, markers, rulers, whiteboard or document camera

Step 1: Split your students up into groups, then pass out 4 posters to each group. Have students come up with a team name and jot it down in marker at the top of each poster.

Step 2: Write the following function on your whiteboard or display it with your document camera:

y = x2 − 4x − 12

Then have students take a minute or two to factor the equation (each student should show their work on their own lined paper).

Once the group members have put their heads together and all agreed on an answer, have them write the factored form of the equation in marker at the top of their poster, leaving plenty of space underneath for the awesome graph they're about to make.

Step 3: Tell students to graph the equation, paying special attention to the x-intercepts (zeros), y-intercepts, and the general shape of the graph (spoiler alert: it's a parabola).

Have them use most of the poster for the graph—we want these things giant-sized. Don't let 'em use calculators, though; this should be part of the discovery process, and half the fun is making the graph. It's a good idea to first have them graph it in pencil but then overlay with marker so when they display their poster to the class, it can be seen across the classroom.

Take a stroll around the room, answering questions and making sure each group member is participating. We don't want a Hermione Granger situation here.

At the bottom of their posters, have students write down the zeros of the function and the coordinates for each x-intercept.

Step 4: Have students tape up their posters on one wall of your classroom. Ask a group to come up to their poster hanging on the wall and explain to the class how they approached this problem, and why their graph looks the way it does.

What shortcuts did they come up with, if any? How did they graph the problem? Are there any shortcuts for being able to tell how a graph looks without the use of a graphing calculator? You want to allow the groups to teach each other.
If you feel like any important stuff got skipped over, have a little class discussion after the group is done with their presentations, focusing on how the factored form relates to the graph.

Step 5: Give students the following problem for their second poster.

y = 2x2 − 7x − 4

Repeat all instructions from Steps 1 through 4. Make sure they tape these posters on a separate wall of the classroom. Your goal is to have them taped on all 4 walls of your room. Symmetry!

Step 6: The third poster will have this problem:

y = (x – 3)(x2 – 4)

Repeat Steps 1 through 4 again. If you want to make this problem a little trickier (and if you've already gone over factoring cubic functions with your class), you can display it in its completely un-factored form as y = x3 – 3x2 – 4x + 12 instead.

Step 7: The last problem for the groups to work on is:

y = (x + 2)(x2 + x – 30)

Again, you can show this instead as y = x3 + 3x2 – 28x – 60 if you want to make it more challenging. You're the boss!

Instructions for Your Students

Everything's better on the big screen. You see details you would've never noticed, the acting is more intense, and the explosions literally sear your eyes. That's what we're talkin' about.

So why not practice factoring and graphing on a giant canvas? None of that small-screen, home-video nonsense. In this activity, you'll epically demolish polynomial functions into their essential factors, track down their zeros, and build a huge graph on a poster-sized piece of paper. It's the silver screen of algebra.

Step 1: Split off into groups of 3 or 4, then come up with an awesome team name. Once your teacher passes out the posters, write your group's name in marker at the top of all 4 posters (preferably by whichever group member has the coolest handwriting—we don't want it to look like a bunch of scribbled gibberish).

Step 2: Your teacher will show the first polynomial on the board. Grab a pencil and factor that thing (do this on your lined paper, not on the poster yet). Feel free to discuss this with the group while you're factoring to make sure you're all on the same page. Pun intended!

When you're all done and in agreement, write the factored form of the equation using a marker on the top of your poster, underneath your team name.

Step 3: Draw a huge set of x-y axes and graph your function on the poster. No calculators allowed! First, graph in pencil (a little trial and error is totally fine) but then overlay that with marker and make sure it's big enough to be seen across the room.

Identify the zeros of the function (aka the x-intercepts) and the y-intercepts, then write the coordinates of each one at the bottom of your poster. Once you're done and everything's lookin' good, take a minute to talk with your group about how you solved this problem—you might be called on to give a (very short) presentation in a second.

Step 4: Tape your poster up on one wall of the classroom that your teacher tells you. Now your teacher will ask a group to explain to the class how they approached this problem. When your group is called on, it's cool to elect a main spokesperson here, but make sure all your group members say something in the presentation.

How did you relate the factored form of function to the graph? What shortcuts did you come up with, if any? How did you graph the problem? It'll be a short talk, so put away those PowerPoint slides.

Step 5: Your teacher will put up another polynomial on the board. Grab your second blank poster and repeat Steps 1 through 4.

Step 6: Now your teacher will bust out your third problem—this one'll be a bit trickier.Repeat all instructions from Steps 1 through 4 again. Déjà vu, right?
Step 7: One more poster problem, this time a tiny bit more complicated. You've got this! It may look more gnarly, but just keep at it and you'll figure it out. Repeat Steps 1 through 4, blah blah blah. You know the drill.