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Teaching CCSS.Math.Content.HSA-APR.B.2

Destroy polynomials, and leave no remainders.

  • Activities: 4
  • Quiz Questions: 0

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The Remainder Theorem. Just saying the words can make students shudder with fear and anxiety. Sorting through the mess of factors, variables, and zeros can seem like a daunting task for students, and explaining what everything means is just as daunting for teachers.

That is, until now.

Shmoop's A-APR.2 Teaching Guide has got your back. With activities and handouts galore, we'll help you teach students everything they need to know about the Remainder Theorem, how to apply it, and what it all means.

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.2?

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Instructions for You

Objective: Remember Deal or No Deal? We sure do—Howie Mandel's awesome soul patch will forever haunt us. So we're about to take a page from old Howie's book and run through a rousing round of Factor or No Factor.

In this game, students will get some solid practice in using the Remainder Theorem to identify different factors of a polynomial. They'll be working in groups in a gameshow-type format, earning points by figuring out whether a random binomial is one of the given polynomial's factors.

We couldn't afford Howie to host, but we're sure you'll do nicely. Sound like a deal?

Activity Length: 1 class period
Activity Type: Groups of 3-4
Materials Needed: Whiteboard, 16pieces of paper or cardstock, which will be taped to the whiteboard in a grid pattern to provide the choices for the game contestants (in the real game they're briefcases held by models, but in our game, cardstock and tape will have to do!), scratch paper, prizes if you want to include any

Preparation: For this game, you'll need four gnarly-looking fourth-degree polynomials, along with all of their factors. We'll give you four to use if you wanna (aww, ain't we sweet?), but you can totes use your own. Just make sure all their factors are in the form x – a, so the Remainder Theorem will actually apply.

x4 + 3x3 – 57x2 – 115x + 168 = (x – 1)(x + 3)(x – 7)(x + 8)
x4 – 5x3 – 28x2 + 188x – 240 = (x – 2)(x – 4)(x – 5)(x + 6)
x4 + 13x3 + 35x2 – 109x – 420 = (x – 3)(x + 4)(x + 5)(x + 7)
x4 – 11x3 + 8x2 + 116x + 96 = (x + 1)(x + 2)(x – 6)(x – 8)

Write one of the polynomials' factors on one side of each of the 16 cards (that's 4 cards per polynomial, with a binomial written on each card). Then write different point values on the other side of each card, ranging from 100 to 400. Tape the cards to the wall in a 4-by-4 grid pattern, in vertical columns from 100 to 400, with the point values showing and the factors themselves hidden.

Step 1: Separate the class into teams of 3 or 4 students, then have each team choose a spokesperson and a team name. Write each team's name on the board so you can keep score.

Begin by putting one of the polynomials on the board. Have your students copy this polynomial onto a blank sheet of scratch paper.

Step 2: Beginning with the first team, have each spokesperson choose one of the cards by reading off the point value, Jeopardy!-style. Then turn that card over, showing the binomial to the class.

Step 3: Have everyone in the entire class (not just the team that's playing this round) determine whether the factor on the card (x – a) is a factor of the polynomial by plugging in a for x. Have them work this up on a piece of scratch paper individually, showing all their work. Then each team should discuss quietly to make sure their team members are all in agreement before sharing their answer—we don't want a riot on our hands.

Step 4: Beginning with the team who chose the number, the spokesperson should say whether the binomial on the card is a factor of the polynomial or not. But don’t tell them if they're right just yet! Go around to each of the other teams and have each spokesperson say whether that team agrees or disagrees with the initial answer. Record responses on your own sheet of paper until all teams have given an answer.

Step 5: Now comes the big reveal: let 'em know if the binomial is indeed a factor, and give each team that was correct the indicated amount of points. (If you want to spice things up, each team that got the answer wrong can have the point value of the card deducted from their total.) Keep a running tally of each team's points on the board, so everyone can see 'em.

Step 6: Move to the next team and have them choose a card. Repeat the process until all of the factors for the polynomial have been discovered.

Then take all the cards down, re-shuffle them, and tape them back up on the board in a different random order (again, with the point values facing out and the factors hidden). Write your second polynomial up on the board, and repeat steps 2 through 5.

After you've gone through all four polynomials, tally up the points on the board and declare a winner. Now it's prize time!

Instructions for Your Students

Gameshows are an American institution. Who among us hasn't whiled away a whole afternoon on Jeopardy! or The Price Is Right reruns? And don't even get us started on Cash Cab.

In this activity, we're going to play a game similar to Deal or No Deal (without the models, the briefcases, and the thousands of dollars…sorry!) where your group will choose random cards and determine whether the binomial on that card forms a factor of the given polynomial. With a little practice, you'll be crowned the factor champ—it's not quite as prestigious as taking home a pile of prize money, but at least you'll get bragging rights.

Step 1: Get into a team of 3 or 4 people and come up with a team name. Be creative! No groan-worthy names like the Mathletes or Mathstronauts.

Pick a spokesperson for your team, too. When your teacher writes the first polynomial on the board, jot it down on a piece of scratch paper.

Step 2: If it's your team's turn, have your spokesperson pick one of the cards on the board by its point value—"We'll take the third column for 400, please." Try to sound as snooty as possible, like you're on a fancy game show.

If it's not your team's turn, just chill for a minute.

Step 3: When your teacher turns the card over, grab your scratch paper again. Write down the binomial (x – a), the point value (100 pts), and show all your work to show whether it is or isn't a factor of the given polynomial.

Do all this on your own paper (even if it's not your team's turn right now), then compare your answer to your teammates' answers when they're all finished. Make sure everyone agrees on what your team's answer is gonna be. Is the thing on the card a factor of the polynomial, or nah?

Step 4: If your team's up, have your spokesperson announce your team's answer when your teacher asks. If it's not your team's turn, have your spokesperson say whether your team agrees with the answer.

Step 5: Your teacher will holler the right answer, then award points based on if your team got it right (or possibly steal your points if you got it wrong). How'd you do?

Step 6: Now it's on to the next team, who'll choose another card and go through everything again, blah blah blah. Once the class has figured out every factor of the polynomial on the board, your teacher will bust out a new polynomial for a new round.

Repeat steps 2 through 5 until the game's over. If your team has the most points at the end, do a sweet, sweet victory dance—you've won! Collect your prize and enjoy the bragging rights.