'Adding It Up' Part 3: Breaking it Down

'Do not skip the steps! Do not skip the steps. Go step by step.'

The students in Professor Jorge Perez's remedial algebra class, at LaGuardia Community College, hear these words a lot.

For most of these students, math is challenging -- even frightening. They all failed an assessment used at LaGuardia and CUNY's other community colleges to determine whether a student is ready for college-level math. (Only the CUNY Community colleges offer remedial math, not the four-year institutions).

Professor Perez has been teaching regular and remedial math classes at LaGuardia since 1982. He forces his students to confront their fear of math by encouraging them to speak up about their feelings in class and having them keep a journal for credit. He also teaches them to break down each equation into discrete components.

Math is 'a language,' says Perez. 'But as any language, it has a grammar and it has syntax. And the students wants to do mathematics without paying attention to the grammar and the syntax.'

This means they often hurry to get the answer without thinking about how they got there, and whether the answer is right. Perez says they forget that another process in learning is 'reflection.'

It's not easy to explain this stuff on the radio so here are a few examples.  These equations are at about a 9th or 10th grade level. See if you can remember how they're solved.

Common mistakes

Professor Perez says most of his students get this question wrong:

(3x + 4) ²

The typical mistake is:

9x ² + 16

That's because students skip the steps. When you square an equation, it's really like this:

(3x +4)  (3x +4)

And that translates to:

9x + 12x + 12x + 16, which is then reduced to 9x + 24x + 16


Square roots and squares

Be Rational

Remember the phrase 'rationalize the denominator'? You're not alone.

Here's the kind of equation Professor Perez's class was tackling just before the final exam.


To solve for this, you have to rationalize it. That means getting rid of the irrational number, the square root of 5.

Any number times one is still the same number. And there are many ways of expressing the number one. Here's one way of doing it:


Why pick this? Because √5 x √5 = 5. And 5 is a rational number.

The answer is now (3√5)/5, giving us a rational denominator.

The trick is not to get distracted when there are multiple numbers in the denominator. So in a case where we have 7√15 in the denominator, the equation gets multiplied by √15/√15. Forget about the number 7 because it's not the problem. The irrational number is what we have to eliminate.

'Remember, this is like life,' says Perez. 'We worry only about the criminals. Not the good citizens.'

This report was compiled with assistance from the Hechinger Institute on Education and the Media at Columbia University.

Read more of Beth Fertig's Adding It Up series.