Teaching CCSS.Math.Content.HSA-APR.A.1

Instructions for You

Objective: The great thing about polynomials and algebra is that you can always use actual stuff to represent your variables. Jellybeans, piles of gold, live kittens, you name it—whatever it is, it's always massively helpful to have physical, visual objects to show how variables interact.

In this hands-on (literally) activity, your students will grab handfuls of colorful poker chips from a bag or hat, and then construct a polynomial equation to represent their pile of chips. Then they'll partner up with another student and combine their two equations using addition, subtraction, and multiplication.

But that's not all—after that, they'll take polynomial addition to bold new heights by combining and re-combining their group's equation with another group's, doubling their size over and over until the whole class is one giant supergroup with a giant super-polynomial.

That's a whole lotta poker chips, so we recommend buying a cheap set instead of using your actual casino winnings. Cash those things in.

Activity Length: 1 class period
Activity Type: Individual at first, then gradually increasing groups (1, 2, 4, 8, etc.)
Materials Needed: Lots of poker chips in 4 different colors (like white, red, blue, and black), scratch paper, whiteboard, markers, a big bucket or hat

Step 1: Explain to your class that each poker chip color represents a different variable, and that black chips represent constants. Maybe run a bunch of chips through your fingers, so your class knows how cool and with-it you are.

Then write the following on your whiteboard (and obviously you can adjust this if you've got different colors):

w = white
r = red
b = blue
constants = black

Grab a small handful of chips yourself and show them how to represent this as a polynomial. If you drew 5 white, 3 red, 2 blue, and 3 black chips, for instance, you'd write this on the board as 5w + 3r + 2b + 3 (the black chips don't need a variable, since they represent constants).

Off to the side of your whiteboard, also write down an equation for the total number of poker chips you're starting with. If you've got 100 of each, that's 100w + 100r + 100b + 100. We'll use your grand total at the very end, in a big dramatic reveal.

Step 2: Once everyone understands the setup, dump all the chips into a bucket or hat, and have your students come up one at a time to the front of the class. Tell each of them to grab a handfuls of chips using their left hand (this is important).

Then have them go back to their seats and carefully set their handful down on their desks. Each student should grab a sheet of paper, write their name at the top, and then express their handful of chips as a polynomial equation starting with L = …. This'll be their left-hand equation. They don't have to write it with their left hand, though (unless they're a southpaw). We want this stuff legible.

Step 3: After everyone has their L equation, bring them back up to the front again and (you guessed it) draw another handful of chips with their right hand. Then it's back to their seats again, and have them write out an equation for R right underneath their left-hand equation. They should have something like this:

L = 5w + 3r + 2b + 3
R = 3w + 6r + 4

Important note: try to make sure your entire stock of poker chips gets used up. If there are any left over, you can grab two handfuls and participate in the game yourself. We're gonna be finding the grand total number of chips eventually, so we want them all in play.

Step 4: Now have students add their two polynomials into a single equation for their total chips (T = …), showing all their work below their original equations. The idea is for their papers to look like so:

L = 5w + 3r + 2b + 3
R = 3w + 6r + 4

T = L + R
T = (5w + 3r + 2b + 3) + (3w + 6r + 4)
T = 8w + 9r + 2b + 7

Step 5: Time for the good stuff. Tell each student to grab their paper and (quickly) go pair up with someone else in the class (preferably someone who's not their mortal enemy). Have each pair combine their two T equations using addition, subtraction, and multiplication. Both partners should write the following items out on their papers (and you can jot it down on the board too if it'll help):

Partners: _________ (both their names go here)

Sum: T₁ + T₂ = _____________

Difference: T₁ – T₂ = ____________

Product: T₁T₂ = ______________

Make sure they leave enough space to show all their work. Have them both fill in all their own blanks, checking in with each other to make sure they've got the same answers. (You can be strolling around the room at this point, helping out anyone who needs it.)

Have each partner circle their answer for "Sum: T₁ + T₂"—don't circle the other guys.

Step 6: Now things start to get really wild. Tell each pair to go find another pair in the class, forming groups of 4. On the back of their papers, have each student write out the following:

Group A: ________________ (the names of the first set of partners)
Group B: ________________ (the second set)

Addition of A + B's sum: __________________________

Then tell the group to add their circled answers from the last step, giving them a brand-new super-polynomial. This bad boy should be the sum of all 4 of their poker-chip totals.

Step 7: Ready for this? Have each group of 4 go and combine their group with another group of 4. Repeat Step 6 again, combining each group's sum with the new group's. Have all 8 students write out the same stuff as last time:

Group A: ________________ (the names of the first set of 4 students)
Group B: ________________ (the second set)

Addition of A + B's sum: __________________________

Step 8: You see where we're going with this. Repeat Step 6 over and over again, doubling the group size every time.

Here's the idea: we want to eventually combine your entire class's polynomials into one mega-equation. Students might need more sheets of paper for this, but we want the same info from Step 6 filled out every time. By the end, everyone's blanks for Group A and Group B should include half the entire class, so tell 'em to leave more and more space as they combine groups.

The coefficients should be racking up pretty high by the end, too. And the final total, of course, should represent the total amount of poker chips you started with at the beginning of class (assuming all your chips are in circulation). Make sure the whole class's final equation matches the polynomial you wrote on the board earlier for the total chips.

If so, you're done! If not, see if the class can track down their error, looking back over all the steps on their paper(s) to see if they can't weed out the problem.

By the way, if you're short on time, you can always stop the activity after the partnering-up in Step 5. Do what you gotta do, you know?

At the end of the period, have all your students hand in their papers with all their work (and tell 'em to staple everything together if they used more than one sheet). That was a polynomial problem of epic proportions.

Instructions for Your Students

We're big fans of math here (obviously), but we'll be the first to admit that super-abstract problems can be even more painful than getting a root canal while riding the subway. If you're anything like us, you'd rather have something visual or physical, something you can hold in your hands, instead of just a bunch of x's and y's.

Don't worry, we've got you covered. This activity is all about turning something visual (a handful of colorful poker chips) into something mathematical (a polynomial equation), then combining your equation with other people's until your whole class makes one giant mega-equation.

You can't cash in your chips afterward for cold, hard cash though. Sorry.

Step 1: You ready to chip it up? First things first: your teacher will run you through the basics of the game. You'll be turning a handful of poker chips into a polynomial equation, with a different variable for each color. For instance, they could be something like this:

w = white
r = red
b = blue
constants = black

Colors may vary. No purchase necessary. See your teacher for rules and conditions.

Step 2: When the teacher give you the go-ahead, walk up to the front of the room and use your left hand to grab a small-ish handful of poker chips from the bag or hat your teacher offers. It's real important that you use your left hand! Hey, we don't make the rules. Wait, actually we do. Nevermind.

(By the way, it's okay if you don't get every color in your handful.)

Back at your seat, bust out a sheet of paper and write your name at the top. Oh, and you can set your chips down on the desk. Now write out a polynomial equation for your handful, starting with L = … (for "left"). For example, if you get 5 white, 3 red, 2 blue, and 3 black chips, you'd write this on your paper as 5w + 3r + 2b + 3 (the black chips don't get a variable, since they represent constants).

Step 3: When the whole class is done, everyone will go back up to the front again and draw another handful of chips, this time with your right hand.

Head back to your seat and write a new equation as R = … underneath the left-hand one on your paper. You should have something like this:

L = 5w + 3r + 2b + 3
R = 3w + 6r + 4

Step 4: Combine your two hands into a single equation for your total chips (T = …) by adding them together and combining like terms. Show all your steps, too. In our example above, your paper would look like this:

L = 5w + 3r + 2b + 3
R = 3w + 6r + 4

T = L + R
T = (5w + 3r + 2b + 3) + (3w + 6r + 4)
T = 8w + 9r + 2b + 7

Got it? You're basically making one big polynomial out of both your hands.

Step 5: Now it's time to get mobile. Take your paper and go find a partner in the class. Once you're paired up, both of you should write this down on your sheet of paper, underneath the stuff you've already written:

Partners: _________ (both your names go here)

Sum: T₁ + T₂ = ____________

Difference: T₁ – T₂ = ____________

Product: T₁T₂ = ____________

Yep, that's right—you'll be adding, subtracting, and multiplying your two T equations together. You can work together on this, but make sure each of you writes it down on your own sheet of paper, showing all work. You wanna get credit on this thing, right?

When you're done, circle your answer for "Sum: T₁ + T₂", but don't circle anything else.

Step 6: When the whole class is done and your teacher gives the signal, now you and your partner should go find another set of partners and join forces with 'em. Boom, now you're in a group of 4 people. Jot this down on the back of your paper:

Group A: ________________ (the names of the first set of partners)
Group B: ________________ (the second set)

Addition of A + B's sum: __________________________

Now your whole group should work on combing the sum from each set of partners into one super-sum. Take the "Sum: T₁ + T₂" that you circled and add it to the "Sum: T₁ + T₂" from the other set of partners. (You can ignore the difference and product from here on out (unless your teacher says otherwise, of course).) Everyone should be writing this down on their own paper.

Combine all like terms and simplify, blah blah. You know what to do.

Step 7: Now things are about to get wiiiiild. Your entire group of 4 should now go find another group of 4, forming a supergroup of 8. Guess what's next? Yep, write the following stuff down again on your paper (or on a new sheet if you're out of room):

Group A: ________________ (the names of the first set of 4 students)
Group B: ________________ (the second set)

Addition of A + B's sum: __________________________

Make sure you leave enough space for everyone's names. Now you'll do the same thing you did in Step 6—add the first group's total sum with the second group's, which should give you one giant polynomial. And we bet you can guess what's coming next.

Step 8: That's right—now combine your group of 8 with another group of 8, and repeat Step 6 again. Do this over and over, doubling your group size every time until your group is the entire class (and make sure you write down all the notes from Step 6 every time).

At the end, you should have one enormous polynomial that combines every single person's poker chips. Does your very last polynomial match up with the total that your teacher wrote on the board? If so, you guys are true champs. If not, you're still champs—you just need to figure out where things went wrong along the way.

At the end of the game, staple all your papers together, make sure your name is on 'em, and turn those babies in. You're a polynomial-adding expert now.