# Habits of Mind: Abstracting Regularity

November, 2010

Abstracting regularity is a process of noticing repetition or pattern in a series of calculations or figures and describing that regularity, for example, by writing an equation. In the CME Project, this idea leads to a technique for solving word problems which we call guess-check-generalize. This is not the same as the familiar "guess-check-revise" technique in which students estimate an answer and then change their guess up or down to zero in on the correct answer.

Instead, guess-check-generalize gives students a way to find an equation that models the situation. Even students who are very good at solving equations can have a difficult time figuring out what equation to solve, so generating the equation is where they need the help.

Here's an example of a word problem from CME Project-Algebra 1:

Derman says, "I'm halfway done with this book. If I read another 84 pages, I'll be two thirds of the way done." How many pages are in Derman's book?

Students using the guess-check-generalize method would begin by choosing any convenient number to use as a guess. They're not trying to accidentally hit on the answer, or even to estimate it. They're just picking a good number to calculate with.

Since students are not limited to numbers that would be a reasonable number of pages in a book, they might guess any number for which it's easy to find half or two-thirds.

Guess: 6 pages.

Is 3 + 84 equal to 2/3(6)?

No, 87 is not equal to 4.

Some students like to choose a number like 100, because they feel confident making calculations with it.

Guess: 100 pages.

Is 50 + 84 equal to 2/3(100)?

No, 134 is not equal to 66 2/3.

Eventually, after several checks like this, students see the regularity in the process of checking a number. They see where the number of pages they guess comes up in each step of their check. This allows them to develop an algorithm to check a "generic" guess.

Guess: p pages.