## Kiss those Math Headaches GOODBYE!

### Algebra Mistake #2: Does a x a = 2 x a?

Now that you’ve gotten a taste for the benefits of analyzing algebraic mistakes, it’s time to explore a second common mistake. This one is so common that nearly every student commits it at least once on the road to algebra success.

As you watch the video, notice how by thinking hard about two expressions, we can think this mistake through to its very root, thus discovering the core difference between two similar-looking algebraic expressions.

And along the road, we’ll learn a general strategy for decoding the meaning of algebraic expressions. What I like about this strategy is that you can use it to understand the meaning of pretty much any algebraic expression, and you’ll see that it’s not a hard thing to do. In fact, it just involves using numbers in a nifty way.

Best of all, students usually find this approach interesting, convincing and even a bit fun. So here goes, Common Algebra Mistake #2 …

### Origins of the Coordinate Plane

A fly …

Who would think that a mere fly could play a major role in the history of human thought?

But when it comes to the development of Algebra, that’s the story. I’ll explain how this works just a bit later in this blog. But it is all related to what is happening now in algebra classes all around the world.

For it’s spring, that time of year again when we get out the graph paper and the ruler. Kids are working on the Cartesian coordinate plane.

One about I like about the coordinate plane is that there’s an interesting story about how it was discovered, or should I say, invented. [Hard to know the right word for an intellectual Invention like the coordinate plane.]
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### Finding domain and range — with color!

Have you ever noticed that a lot of students struggle with the idea of domain and range? This concept, taught mostly in Algebra 2,  often confuses students to the point where they cannot even identify the domain and range of a simple, continuous function.

I don’t really understand why students struggle with this concept, but I recently found a way of showing the idea that makes it considerably easier — using color to mark up a function.

Here’s an example of a problem where students need to figure out the domain and range by looking at a graph, like this:

What I have students do is use two colors to sort of “box in” the function. With one color, green in this case, students mark the left bound and right bound of the function by drawing vertical lines. And with another color, red, students mark the lower bound and upper bound by  drawing horizontal lines. I have students write in the phrases:  left bound, right bound, lower bound, and upper bound, like this:

Finally I ask students to figure out the domain and range by writing three-part inequalities for x and y, respectively, like this:

I’ve used this approach with a number of students, and so far no one has been unable to find the domain and range when using it. So it appears to be a winner. Try it yourself, either as you teach a concept, or as you re-teach it to those who are struggling.

### Using Color to Show Perimeter

Many things look better in color, right?

So why should that be any different  in math?

I’ve found that taking a “colorful approach” to math not only makes mathematical objects look more interesting and pleasurable, it can also make mathematical concepts more clear.

Here’s an example from something I did today — I used color to show a shortcut for finding the perimeter of rectangular-ish objects.

I was tutoring a boy who had to find the perimeter of this figure:

Right object, find perimeter

This student did not see that there is a short-cut that could help him find the perimeter. I wanted to make this clear, so I reached for my color pencils and colorized both the left vertical segment and the two right vertical segments. My goal was to help the student see that the sum of the two right vertical segments equals the long left vertical segment.

The student realized this after I colorized it. Then I used a different color, red, to show that the sum of the two horizontal segments on top equals the longer horizontal bottom segment, like this:

At this point I felt that the student was ready to see the math that relates to the whole figure, so I wrote the math, using color to relate the numbers to the colors of the sides of the figure, like this:

At this point the student was able to see the shortcut in this kind of problem, which together we wrote as follows:

This is a fairly basic example of how color can, quickly and effectively, illustrate math concepts. Feel free to share examples of how you use color in your math lessons. I’m curious to learn (and share) a variety of ways, for I see that color has great potential.

Image via Wikipedia

### Common Algebra Mistake: Interpreting Negative Signs in Front of Parentheses

Certain areas of algebra are like pebbles in your shoe: looked at closely, they  are very small — tiny in the scheme of things — yet they are “oh-so-bothersome!”

As a tutor, I’ve long felt this way about negative signs before parentheses. It’s a small thing really, and it seems simple to grasp for those of us who get it. Yet students make so many mistakes when facing this situation; to them it’s quite irritating.

So there I was again, trying to help a girl see how to simplify the expression: – (– 5x + 3y – 7)

But this time I thought of something different, the word “opposite.”

I talked for a moment with my tutee about the idea of opposites, and then I started out like this:

Q:  So, what’s the opposite of black?

She replied:  White (with the teenage “that’s-totally-obvious-what are you-doing?-insulting-my-intelligence? accent)

I told her not to worry, this would lead back to the problem. Next I gave her two terms for which she were to find the opposite, as in:

Q:  opp (tall, happy)

She wrote:   (short, sad), still wondering …

And I continued:

Q:  opp (heavy, up)

She wrote:  (light, down), sighing.

Then I explained that in math we express the idea of “opposite” with nothing more than the negative sign.

Then I gave her some problems with the negative sign:

Q:  –  (cold, left)

A:  (hot, right)

and

Q:  –  (under, near)

A:  (over, far)

She was still giving me that “this-is-so-easy-I-could-die” kind of look. When I thought about that, I realized it was good!

Next I  explained that in math, just as in real life, there are opposites. And we find mathematical opposites by examining signs. For example, the opposite of 5 is – 5; opposite of – 3/4 is 3/4; opposite of – 3x is 3x; opposite of y is – y, and so on.

Then I gave her these problems:

Q:  – (+ 2x, – 5)

Still she was with me:  – 2x, + 5

and

Q:  – (– 4y, + 3x, – 6)

A:  + 4y, – 3x, + 6

The sighing was slowing down, finally. Then I simply told her that we’re going to “lose” the comma (how’s that for modern slang!), both in the original expression and in their answer. Then I gave her a new problem:

Q:  – (5a – 3a – 9)

This puzzled her a bit. So I explained that she needs to mentally group the term with the sign that lies to the left. And that if no sign is showing, as for leading positive terms, she needs to mentally insert the invisible positive sign:  5 becomes + 5;  2a becomes + 2a. Once she got that, she was able to proceed:

Q:  – (5a – 3a – 9)

A:  – 5a + 3a + 9

And so on … one success after another. The concept was sticking. And best of all, she had a conceptual framework — the concept of opposite — that she could “lean against” any time she got stuck.

The longer I tutor the more I realize that this kind of conceptual framework — a story or concept we know from everyday life, which relates to the algebra in a direct way — is a big key to helping students grasp algebra. I use these kinds of stories in my book, the Algebra Survival Guide, providing stories we know from everyday life, which serve as analogies that show how the math works. For example, in the Guide I use a “tug-of-war” analogy to show how you solve problems like:  – 3 + 8.

Tur-of-War Teaches – 3 + 8

I’ve had so much success with this “story”-approach to algebra that I am working on an eBook that provides a whole litany of stories that work for algebra. It is fun to work on, and kids like this approach because it gives them a new way — an everyday way — to relate to the math.

So in any case, my suggestion is that when you teach or review the concept of negative signs before parentheses, you might just try the “opposites”  approach and see how it works with your students.