Writing and factoring quadratics

Another “game”. This one hits the following skills:

  • The relationship between roots and factors of a quadratic or polynomial function.
  • Factoring quadratics.
  • Multiplying binomials, the distributive property, and simplifying polynomials.

It is a cooperative game, vaguely modeled on War. Each pair of students is given a deck of cards.

Each student draws two cards, keeping them hidden from their partner. These cards represent the roots of their quadratic function, using the following key:

  • Black cards are positive and red cards are negative.
  • Cards 2–10 represent the value shown.
  • Jack = \frac{1}{2}
  • Queen = \frac{1}{3}
  • King = \frac{2}{3}
  • Ace = \frac{3}{4}
  • Joker = \frac{1}{4}

For example, if Student A draws a red jack and a black 8, then the roots of their quadratic function are -\frac{1}{2} and +8.

Student A turns these two roots into factors:


Student A then multiplies these factors together to give a quadratic function in standard form: (x-8)(2x+1) = 2x^2-15x-8.

Meanwhile, Student B has been doing the same thing with their two cards. When both students have finished writing their quadratic functions they share only their functions. They should not share the original roots, because each student is now going to factor their partner’s quadratic function.

Assuming everything goes as planned, both students correctly factor their partner’s quadratic and discover the original roots. If they are correct, those cards go into the discard pile. If they are incorrect, the cards go back into the deck. Play continues until all cards are in the discard pile.

A popular source of confusion was students incorrectly multiplying their binomials leading to a quadratic that didn’t factor. I encouraged students to help each other find mistakes and work through the process (it’s a cooperative practice game, after all).

Extension: Draw more than two cards and make cubic, quartic, etc., polynomials to factor with the rational root theorem.


The “Balloon Pop” review game

This is my students’ favorite review game. I’m pretty sure I originally got the idea from Elissa Miller, but I’ve been working on the rules with my classes for the last couple years. I think that last year we developed the definitive version.

Teams start with four balloons. Not pictured here are the fun and usually clever names they come up with.
balloonpop2A question is projected on the board for each team to answer. I usually give a time limit of around two minutes, but go longer or shorter depending on the question and my goals.
balloonpop3After time is up, I call on a random student from each team to come and show me their answer. While the chosen students come to the front of the room, the answer is displayed for the other students.
balloonpop4Each team that gets the question correct must take another team’s balloon. I usually limit students to about 20 seconds to decide whose balloon to steal, otherwise they’ll spend all day making their choice. I also randomize the order that groups steal balloons to make it a little more fair and interesting. In the example, Groups 3, 7, and 10 appear to have answered incorrectly. Group 7 appears to have made some enemies. And Group 1 took a balloon from Group 3, while Group 2 took one from Group 5, etc. A student once told me that this game ends friendships. What more could you ask for?


Play continues until the period ends or we run out of questions. The winner is the team with the most balloons.

Unlike the physical version I was inspired by, this version takes place entirely in a PowerPoint (the blank template is at the end of the post). Some notes about the file:

  • I use Extended Desktop and project the PowerPoint to my projector, leaving the actual file open on my computer monitor. This allows me to move the balloons around without restarting the slideshow.
  • I always edit the file at 50% magnification, because…
  • balloonpop5There are four balloons on the sides of the score slide that you can quickly copy and paste into each group’s box. Hold down CTRL and drag while you have something selected to quickly make a copy of it.
  • balloonpop6When you’re making your questions and answers, the answer is on a text box to the right of the question. Again, editing at 50% magnification will make it easier to navigate without scrolling so much.
  • balloonpop8Click the “Score” button on a question slide to go back  to the score slide.
  • balloonpop7Click the numbered buttons on the bottom of the score slide to go to each question. They turn green after you’ve clicked them once because I got tired of forgetting which question we were on.
  • Click anywhere on a question slide to show the answer.

Click here to download my blank balloon pop template (.ppt).

Simplifying radicals war

We played this game in, like, October. It’s based off of someone else’s hard work—probably this post, but, you know, October.

I’m presenting my version of it though because I did a little work to find what I think is a better rule-set. One of the problems that you run into if you use the “normal” values for cards (jack = 11, queen = 12, king = 13) is that the majority of the numbers you get (around 60%) aren’t able to be simplified. In my version, 61% of the possible numbers can be simplified, while also having enough variety to keep it interesting. (It’s possible to make almost all of the possibilities simplifiable, but you have to use only 2, 4, 8 and that’s a little boring…)

Here are the rules. It’s based on War and involves two players:

  • Remove all 3s, 7s, 10s, and jokers from the deck.
  • Jacks count as 4. Queens count as 8. Kings count as 12. Aces count as 1.
  • Each player starts with half the deck.
  • Each player draws a card. Use the two cards to form the radicand by concatenation. So if the two cards drawn are 5 and 2, the square root is \sqrt{52}. A king and a queen makes \sqrt{128}. Students will need to agree on a method for determining which number comes first.
  • Both players try to simplify the radical as quickly as possible. I’m not usually into speed-math, but sometimes it’s okay, I suppose.
  • Whoever simplifies the square root first gets to keep both cards. The goal is to collect the whole deck.
  • If the square root isn’t able to be simplified, players take their cards back and try again. You can also play by switching the order of the cards if the first order didn’t work and only taking the cards back if neither order works.

Click here for a copy of the slide I used in the classroom. It’s the same as above, but with some pictures. It’s kind of ugly…