August, 2010

Demonstrating the technique with a new problem.

Choose a nice number.

Now here's the problem:

Chiko says, "There's an amazing fact about the number I'm thinking of. I call this fact the four 4's. Using my number, you can get four different answers if you add 4, subtract 4, multiply by 4, or divide by 4. That's not impressive, but if you add all the answers you get 60." Is one of our nice numbers the answer to this problem?

## The Guess-Check-Generalize Method

Pick a number, any number.

Check to see if it works.

Check more numbers until you get a feel for the calculation

involved in checking a guess.

When you’re ready, use a generic guess, like n.

The guess-checker for n gives you an equation that you can solve.

An equation is a point-tester for its graph

Al Cuoco, the lead developer on CME Project, found in his decades of classroom experience that students often failed to make the critical connection that a point is on the graph of an equation if, and only if, its coordinates made the equation true.

For example, students might be asked to graph this equation:

The students knew how to transform this equation:

by completing the squares into this form:

The students were even able to use this new equation:

to sketch an accurate graph:

But then, when asked whether the point (7:5; 3:75) was on the graph, all the students knew how to do was to graph the point and see if it looked as though it were on the graph.

With a real understanding of the concept that an equation is a point-tester for its graph, students would answer the question of whether the point (7:5; 3:75) was on the ellipse by substituting the coordinates into the equation:

Since the coordinates of the point do not create a true

statement when they’re substituted into the equation, the point

is not on the graph.