# 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:

 $x=8$ $x-8=0$ $x=-1/2$ $2x=-1$ $2x+1=0$

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.

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