Teaching CCSS.Math.Content.HSA-APR.C.5

Instructions for You

Objective: The objective of this activity is for students to practice calculating binomial expansion coefficients with an energy-packed, highly competitive, and chaotic game. Students can sometimes see binomial expansion as dull, tedious work, but when they're competing shoulder to shoulder with their peers, trying to make split-second decisions about binomials they've expanded, hopefully those boring associations will start to dissipate. And even if they're not huge fans of binomial expansion by the end of the game, most kids like slamming down cards, so they should have fun either way.

Activity Length: 1 class period
Activity Type: Groups (6 total)
Materials Needed: One piece of construction paper per group, markers, six index cards per person (or, three index cards cut in half), calculators, scratch paper, marker board or document camera to assist with the game
Attachments: Chaos With Coefficients Key

Step 1: Split your class into six groups (preferably four people per group). Get the groups to move their desks together. Each group should have their desks pointed inwards so they have a shared space. Have tables? Even better. Want to move the desks out of the way and let students sit in circles on the floor? More power to ya.

Step 2: Provide a piece of construction paper to each group and display four binomials to the entire class (we'll get to those binomials in a sec). Ask the students to expand each binomial on that sheet using one of the methods they've learned to expand polynomials (Pascal's Triangle? Binomial Theorem? Their favorite shortcut?). Each group will use these expanded forms to help answer questions. Encourage the students to help each other out when making these.

(x + y)²

(x + y)³

(x + y)⁴

(x + y)⁵

Here are the expansions to make sure they got them right:

(x + y)² = x² + 2xy + y²

(x + y)³ = x³ + 3x²y + 3xy² + y³

(x + y)⁴ = x⁴ + 4x³y + 6x²y² + 4xy³ + y⁴

(x + y)⁵ = x⁵ + 5x⁴y + 10x³y² + 10x²y³ + 5y⁴ + y⁵

Step 3: Give each student six index cards. For each card, have them write their name on one side, and one of the numbers 1 through 6 on the other side, making big letters/numbers. By the end, each student should have one card with the number 1 on one side and their name on the other, one card with the number 2 on one side and their name on the other, and so on, all the way up to 6.

Also, make sure each group designates a scorekeeper.

Step 4: Here's how this game's going to work:

• Pick a binomial to expand from the Chaos With Coefficients Key worksheet. Or if you're feeling particularly inventive, you can make your own up. Show the class the binomial, and give about a minute to expand it (you can adjust this as you need).
• Each student will work on their own scratch paper and expand the binomial using the four binomials they expanded before to help them. It's not a cheat sheet if they're allowed to use it, right?
• Take a look at the answer key. It lists the expanded form for whatever expression you chose. Pick one term and write its coefficient out for everyone to see. For instance, for the binomial (2x + 3y)², its expansion has the coefficient 4 in its first term, so you could put the number 4 up there.
• Students should keep their index cards face down or in their hand (like they're holding a royal flush) so that other people in their group can't see what card they've chosen.
• Students must decide which term number that coefficient came from (the first term, the second, etc.) and take that number card and put it face down in the middle of their group's table. They'll want to be quick about it, though.
• After time is up, tell the students to put their card on the pile if they haven't already. They must put a card on the pile even if they're just guessing.
• Reveal what term number that coefficient came from. You might need to console anyone who didn't get it right, and going over the binomial expansion might not be a bad idea, too.
• Ask the scorekeeper to take the pile in the middle and flip it over, so that the bottom card from the pile is now the top card facing up. The scorekeeper will award points as follows:
  • 1st correct card: +5
  • 2nd correct card: +4
  • 3rd correct card: +3
  • 4th correct card: +2
  • 5th or more correct card: +1
  • Incorrect card: -2

• The scorekeeper will give the cards back to their owners.
• Have one member of each group go up to the board, write everyone's name up there, and put tally marks next to their name for each point they got. If this isn't their first time to the board, they'll put another tally mark next to their name.
• For the last couple of rounds, move into the Chaos Finals. The game's the same, just using the Chaos Finals problems. Well, that, and all the point values are shifted up by 2. Whoever gets the most points at the end of the final round wins.

Instructions for Your Students

Student intro: Learning how to expand binomials using the binomial theorem or Pascal's Triangle can be a bit boring (to say the least). But what if it were a bit less "looking around a triangle created by a dead French guy to find coefficients" and more of a break-neck race to beat all your friends in a contest of speed and skill?

In this activity, we'll try the second approach. And if it's a bit too intense for you, don't worry; Pascal's Triangle will still be lying around when we're done if slower, more methodical, triangle-based math is more your speed.

Step 1: Your teacher will assign you to a group. Find the other members of your group, and arrange yourselves in a circle.

Step 2: Your group will be given a piece of construction paper and a marker. Your mission, if you choose to accept it, is to expand the following binomials using one of the methods you've already learned (Pascal's Triangle? Binomial Theorem? Shortcut? All good.). Put your heads together and get these done.

(x + y)²

(x + y)³

(x + y)⁴

(x + y)⁵

Step 3: You'll be handed six cards to work with. On the first card, write your name on one side, and the number 1 on the other. On the second card, write your name on one side, and the number 2 on the other. We hope you see where this is going. By the end of this step, all of your cards should have your name on one side, and one of the numbers 1 through 6 on the other. (It takes less time to do than to explain.)

You'll also need to pick a person in your group to be the scorekeeper. They'll keep score on a separate piece of paper.

Step 4: Here are the rules of the game:

• Your teacher will present a binomial expression to the entire class. Work on your own scratch paper and expand the binomial using the general models you already wrote down. (Remember, you're trying to win, so don't help the other members of your group. It's like you're playing poker. But with no money, and the same cards in your hand every round. So not like poker at all.)
• After a little bit, your teacher will write a number for everyone to see. That number is one of the coefficients from the expanded binomial, and it's your job to figure out which one. Is it in the first term? Then you'll be playing the card with a 1 on it. The second term? Then you'll need your 2 card. And so on. Put the correct card face down in the middle of your group's table. The key is to get your card there as quickly as you can, because the faster you get it there, the more points you get. (If you're right, that is.)
• After your teacher reveals the answer, the scorekeeper will take the pile in the middle and flip it over; the bottom card from the pile will now be the top card facing up. The scorekeeper will award points as follows:
  • 1st correct card: +5
  • 2nd correct card: +4
  • 3rd correct card: +3
  • 4th correct card: +2
  • 5th or more correct card: +1
  • Incorrect card: -2

• • The scorekeeper will give the cards back to their owners. It's just like they're working the Lost and Found counter at the local library.
• Have one member of your group go up to the board, write everyone's name up there, and put tally marks next to their name for each point they got. Or, if someone's name is already there, just put the marks down, please and thank you.
• Keep playing a few more rounds, until the game is over and the Supreme Master of Chaos is crowned. (Unfortunately, there's no cash reward; just the title.)