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.

Derman has read 3 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.

Derman has read 50 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.

Derman has read p/2 pages.

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

I can solve the equation:

p/2 + 84 = 2/3(p)

to find a value of p that works.

Derman's book is 504 pages long.

Another advantage of this method is that students can choose their guesses strategically to make the calculations easier to do, which cuts down on errors. There's a built-in differentiation here, too. Students who need to try several guesses before they see the pattern are encouraged to do so, and they gain valuable practice in calculation. Students who feel more confident can generalize after a single guess, or can even begin with a generic guess as their first step.