The Next Step in Math Problems is Up to You

This is Day 18 of 31 Days of Math Learning Success. Follow all the days here and check out others that are writing for 31 days here.

Your scientists were so preoccupied with whether or not they could that they didn’t stop to think if they should.
~Dr. Ian Malcolm, Jurassic Park

At this point in the 31 days, you’ve read how much you should get out there and try things. Have faith in you. And don’t just sit with a blank piece of paper – write!

It’s been a bootcamp in DO!, for sure.

And now I’m going to reign you in.

What goes up, must come down.

Everything in math (just about) has an opposite. Subtraction is the “undo” of addition. Division is the undo of multiplication. There’s even “undo” stuff for some matrix and function operations!

If you have a chance, look at a pre-algebra or algebra textbook. One section will be factoring, and the next will be multiplication. These are opposites (officially called “inverses,” but who’s counting. :D)

When you learn to hammer a nail, you also need to know how to pull it out.

And that’s what early textbooks try to teach – doing and undoing. Back and forth.

You must make the decision to do or undo.

Consider this expression: $\frac{2}{14}$ . Are you itching to reduce it?

What if it was part of a bigger expression like this: $\frac{2}{14}+\frac{3}{14}$

Still so hip on reducing it?

You might be. But there’s a good chance you just worked hard to get that $\frac{2}{14}$ . Like this:

$\frac{1}{7}+\frac{3}{14}$
$\frac{1}{7}\left[\frac{2}{2}\right]+\frac{3}{14}$
$\frac{2}{14}+\frac{3}{14}$

So before you reduce a fraction, make sure you didn’t just un-reduce it to make it handier for the next step.

How about multiplication?

When you encounter $2 \times 3 \times 15 \times 7$ , you’ll probably want to multiply it out.

But what if it’s part of this problem: $\frac{2 \times 3 \times 15 \times 7}{35}$ ?

You’d now be left with $\frac{430}{35}$ , and have to figure out how to reduce it.

Consider doing this:

${\frac{2 \cdot 3 \cdot 15 \cdot 7}{35}}$ (what we started with, using dots)

$\frac{2 \cdot 3 \cdot (3 \cdot 5) \cdot 7}{35}$ (factor the 15 into $3\cdot 5$ )

$\frac{2 \cdot 3 \cdot 3 \cdot (5 \cdot 7)}{35}$ (rearrange parenthesis)

$\frac{2 \cdot 3 \cdot 3 \cdot (35)}{1 \cdot 35}$ (this is an extra step for clarity)

$\frac{2 \cdot 3 \cdot 3 \cdot}{1} \cdot \frac{35}{35}$ (I see a 1 coming on!)

$\frac{2 \cdot 3 \cdot 3 \cdot}{1} \cdot 1$

$2 \cdot 3 \cdot 3$

$18$

Rearranging the factors in the top of that fraction allowed you to “cancel” the 35’s fairly quickly and easily. Once you’re used to doing this (and especially if you’re not using a calculator), it’ll be far easier than handing $\frac{430}{35}$ .

But that’s up to you.