# Contradictions and Identities

After students have been solving equations in algebra for a while, they may get lulled into thinking that all equations have a single solution. Of course, this is not the case.

Begin this activity by asking them a simple question. Do they think it’s possible to write an equation that has no solution? After they think about that for a little while and offer up some answers to you, then ask them this question: Is it possible to write an equation that would have all real numbers as its solution set?

Then offer this example:

5t + 8 = 5t – 9

If students subtract 5t from both sides, they will get 8 = -9 which is, of course, false. Whenever the variable drops out AND they end up with a false statement, they have encountered a contradiction.

Next, offer this example:

3x + 21 = 3 (x + 7)

This statement is an identity. Any value from the set of real numbers that we substitute for x will be a valid solution.

Now give students index cards and have them create 5 contradictions and 5 identities, each on a separate card. These will be used for Part 2 of this activity.

From the previous activity, you should now have 200 or more flashcards with contradictions and identities if you have a class of 20 students. Take those flashcards and mix in about 20 equation flashcards that DO have solutions. These should be handwritten too so that they don’t look different than the other cards!

Write a number on the back of each card so you can keep the presentation order straight and also designate the teacher answer key by writing a “C” for contradiction, an “I” for identity, and an “S” for solution.

Break the class up into teams with 3-4 students in each. Give them a worksheet to write their answers.

Now the class is ready for a game of speed. Tell them they will have about ½ minute to come up with each answer. Set up a timer so you can clock them.

For this activity, they can’t solve the equation. They can only look at it, discuss it briefly, and then designate on the worksheet whether it is a “C,” “I,” or “S.”

You may want to break this activity into several class sessions. Grade their final worksheets and be ready to discuss and solve any equations that students didn’t categorize properly.