Teaching CCSS.Math.Content.HSA-APR.C.4

Instructions for You

Objective: In this activity, students will act as math-lawyers to prove the innocence of several polynomial identities. Are they really who they say they are? The fate of an innocent polynomial hangs in the balance.

You'll be the courtroom judge (props if you can find a gavel and/or giant powdered wig to bring), so let 'em know that nothing less than full proof will satisfy you and exonerate these poor guys.

Activity Length: 1 class period
Activity Type: Teams of 3 or 4
Materials Needed: Small, portable whiteboards (one per team), markers, large regular whiteboard at the front of the class, gavel and wig (optional)

Step 1: Split your students off into teams of 3 or 4, and pass out a small whiteboard and markers to each team. Explain that each group is a legal defense team that has to convince you, the judge, that each polynomial isn't lying about their identity. The teams will be presenting their findings to the whole class (who'll be acting as the jury, except they don't get to vote), so have each team elect a spokesperson.

Have them also choose a team name and write it at the top of their whiteboard—preferably something really lawyer-y (like Peterman, Peterman, & Ellis, Attorneys at Law or something like that).

Step 2: Give each team one of the following polynomial identities to work on (you can write 'em on the board or just give each team a slip of paper with an identity written on it):

(x + y)² = x² + 2xy + y²
x² – y² = (x + y)(x – y)
x³ + y³ = (x + y)(x² – xy + y²)
x³ – y³ = (x – y)(x² + xy + y²)
(x + a)(x + b) = x² + (a + b)x + ab(x² + y²)² = (x² – y²)² + (2xy)²

You'll need a different polynomial identity for each group. Go ahead and come up with some extra identities if you have more groups than this—they can be as simple or as complicated as you like, depending on your class's level.

Have students work in their groups to develop a proof of their polynomial identity, then write it down on their team whiteboard, showing all their work.

Step 3: When everyone's finished, choose one group to present their arguments first. Have the whole group come up to the front of the class, and have their spokesperson slowly, step-by-step, present their argument/proof to you, writing it down on the big classroom whiteboard as they go. (They can consult their individual whiteboards for this, but we want the whole class to see every step.) You can also give them the option to have a separate writer and spokesperson, as long as the writer follows along with the spokesperson's arguments.

At the end of the proof, ask your jury (the rest of the students at their seats) if they think all the steps make sense. Give the defending counsel a chance to explain further if there's any confusion.

Step 4: From your lofty judge's bench, look over their proof and decide if it's correct and makes logical sense. If they did a good job, you can announce that the polynomial is "not guilty."

Tell the rest of the class, "Ladies and gentlemen of the jury, I find the defendant (guilty or not guilty)." If their proof wasn't quite right and you found the polynomial guilty, you can either give them a chance to talk amongst themselves for 2 minutes and fix it, or you can go the board and fix it yourself. Either way, make sure everyone sees the properly worked-out proof.

Step 5: Move on to the next group of lawyers, and repeat Steps 3 and 4. Hopefully the defendants all got off the hook.

When all the groups have presented, you can have a quick class discussion if you noticed any persistent issues everyone had. And you can always do another round if you come up with another handful of polynomial identities.

Another option for discussion is to have the whole class try to present some alibis to the court—real-world example of what each identity was doing when the crime was committed. Were they busy finding the area of a rectangle, or showing a relationship between every third odd number on a Tuesday?

With that, this court is adjourned.

Instructions for Your Students

The crime of the century has just been committed. There's panic in the streets, and a bunch of your polynomial friends have been labeled as suspects. It's up to you, the state's most powerful math-lawyer, to prove their innocence and get them off the hook.

You know that x² – y² is really who he says he is—his full name is (x + y)(x – y). But the judge is a tough one, and you've got to prove beyond a shadow of a doubt that the two are equivalent. Will you save a life today? Will you wow the paparazzi with your lush, lawyerly haircut? Only time will tell.

Step 1: Your teacher will separate everyone into groups of 3 or 4, then hand each group a mini-whiteboard and markers. Your fellow group members are your partners at a super-prestigious law firm, with headquarters in the tallest building in town.

Choose one person to be your group's spokesperson, and then decide on an awesome, lawyerly team name. Remember, you're a law firm, so try to make it sound as hoity-toity as possible (like "The McLatchy Group" or "Johnson, Johnson, & Johnson, Attorneys at Law").

Step 2: Once your lawyer team is fully assembled and named, your teacher will give you a specific polynomial identity for your group to work on. Using your whiteboard, come up with a solid, unassailable proof that your polynomial identity is true. Make sure you show every little step on your whiteboard. This is a court of law, for crying out loud.

Step 3: When everyone's done, court is officially in session: the judge (aka your teacher—just imagine them in a huge white powdery wig) will call the teams up one at a time. When it's your team's turn, go up to the front of the class with your whole group.

Now your spokesperson should carefully explain and write down every step of your proof on the big classroom whiteboard, so everyone can see. The rest of the class will be the jury (though they don't actually get to vote on whether you're right). It's the judge you need to convince, and we hear they're the toughest in the county.

The jury will get to ask questions at the end of your proof, though. If someone didn't understand a particular step, answer their questions and do any extra explainin' you need to.

Step 4: Once your spokesperson has run through the entire proof (with the rest of you solemnly nodding along and looking serious), the judge will decide whether your polynomial is guilty or not guilty. If the verdict is not guilty, congrats! You've successfully proved its identity. If the verdict is guilty, try to figure out where your group went wrong and fix it, if your teacher says there's time.

Step 5: Grab a seat when you're done, and watch as the next group goes to court. See if you can follow their arguments, and feel free to ask questions at the end if you missed something. They should be able to explain it super thoroughly, lawyer-style.

Your teacher might also do a quick cool-down discussion at the end, if there's time. Now get out there and celebrate your courtroom victory today.