## Where math comes alive

### How to factor out the GCF with stories

At various times when I tutor, I find myself explaining the same concept repeatedly over several weeks.

Recently it has been that way with — drumroll please … factoring out the GCF from polynomials.

One reason I’m getting so much “experience” with this is that many kids find this process very difficult. It’s not hard to see why. First of all, the process of finding the GCF is, in itself, somewhat tricky. Then too, factoring out the GCF from all terms in a polynomial is a multi-step process; students need to get each step right, and then they need to perform the steps in the correct order. If that alone were not enough to tax children’s minds, students also get confused by the difference between how to multiply pure numbers (constants and coefficients), and how to multiply variables. (more…)

### Dividing Fractions: from annoying to FUN!

O.K., I’m ready to share my amazing approach to dividing a fraction by another fraction. Well, maybe not breathtaking … like Andrew Wiles’ proof of Fermat’s Last Theorem … but at least interesting. And best of all, fun and student-friendly!

Last week I asked if anyone had any tricks up their sleeves that make it easier for students to divide fractions. And I said that I would share a trick after I heard from you.

I got a nice response from Michelle, who said that she has used the mnemonic “KFC” (like the fried chicken), which in her class stands for Keep-Change-Flip. The idea being that you KEEP the first fraction, and next you CHANGE the sign from multiplication to division. Finally you FLIP the second fraction, the fraction on the right. We have similar mnemonic where I live, which goes by the phrase: Copy-Dot-Flip, with the “dot” meaning the dot of multiplication.

But what I want to share with you is a completely different approach to dividing one fraction by another, an approach that saves time, and makes it both easier and more fun — in my humble opinion — than the standard approach.

The approach I’m going to show you works for any complex fraction situation you might encounter, such as these:

For this blog post, I’m going to limit my chat to complex fractions of the arithmetic type, meaning those with numbers only, and no variables. And if it seems important, I’ll do another post later on using this very same process for algebraic fractions.

So what is this amazing approach, anyway? Well, it’s based on something I discovered on day when I was just messing around with fractions divided by fractions. I realized that after you do the KFC or the Copy-Dot-Flip, what you get — in general — is actually something really easy to grasp, as this next image will show you, along with a Quick Proof:

If you take a moment to think about it, the terms in the numerator of the result — terms a and d — have something in common; they were on the outside of the original complex fraction, so I call these terms the “outers.” In the same way, the terms in the denominator of the result — terms b and c — were both on the inside of the complex fraction, so I call them the “inners.”

So when you divide fractions in this vertical format, the answer is simply the outers, multiplying each other divided by the inners, multiplying each other.

I find that students find this easy to remember and a cinch to do. This next sheet summarizes the idea, and also provides a fun way of remembering the concept, thinking about the stack of terms as a fraction “sandwich.”

So, to put this in words, the four-level complex fraction that you start out with can be thought of as a sandwich, with two pieces of bread at top and bottom, and slices of bologna and cheese in the middle.

The main point is that to simplify the fraction sandwich, all you need to do is put the two slices of bread together in the numerator and multiply them, And then put the bologna and cheese together in the denominator, and multiply them.

Using this idea it becomes a lot easier to simplify these complex fractions. Here’s an image that shows how it is done, and how this approach saves time over the way we were taught to do it, using reciprocals.

And there’s more good news. This new way of looking at complex fractions also gives students a cool, new way to simplify the fractions before they get the answer. And when you do simplify fully, the answer you get will be a fraction that’s already completely reduced, so you won’t have to stress about that part.

The next two pages show you this fun and easy new way to simplify:

or, or what? …  Here’s what …

So now you might like to see the whole process from start to finish, so you can decide for yourself if this technique is for you. Well that’s exactly what we’re showing next. As you can see I consistently highlight the outers with pink, the inners with yellow.

And finally, a “harder” problem, you might say. But check it out. Is it really any harder than the one we’ve just done? You decide.

In my next blog I’ll give you a few problems like these, so you can get used to this trick, and start shaving precious seconds and nano-seconds off the time it take you to do your homework, so you spend more time doing all of those things that you want to do more:  texting, watching You-Tube, taking hikes, skating (roller and ice), etc. etc. , etc. You know better than me.

Happy Teaching and Learning!

—  Josh

### Place Value Metaphor

During the summer I get to tutor a lot of elementary age students, remediating them on the basics.

Almost invariably I find that these students are confused about PLACE VALUE, and considering how critical this concept is to all of math, I decided to write this post.

Whenever I have the least suspicion that a student might be confused about place value, I check with a simple test.

I have them write down the number 22, then I ask them if they can tell me the difference between the two 2s. Often they cannot.

Tutoring a girl this past week I came up with a way of understanding place value that really resonated with the student. I want to share it because you may be able to use it, or a modification of it, with your students. First it’s important to know that this student’s mom teaches ballet, and the girl dances at her mom’s studio.

I asked the girl if she has ever been to a ballet performance, and of course she said yes.

Then I drew a quick diagram of the stage and first few audience rows. I pointed to two seats, one in the front row, another seat several rows back. I asked her if the two seats would cost the same amount. This girl knew that the close seat costs more money because it is closer to the action on stage.

Then I used that idea to explain place value. I showed this girl that just as seats can be more or less valuable because of where they are, so too digits can be more or less value based on where they are in a number.

She got this idea very quickly, and now she understands place value.

For children with different interests, use whatever makes sense. For example if you’re teaching a boy who loves baseball, make the rows of seats those at a baseball game, and so on.