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Archive for the ‘Understanding Variables’ Category

Algebra Mistake #2: How to Understand the Difference between A x A and 2 x A without Confusion


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 …

 

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How to Decrease Algebraic Mistakes – Part 1


Anyone who has worked with students learning algebra knows the truth to the maxim:  MISTAKES HAPPEN.

This is the first in a series of posts offering PRACTICAL SUGGESTIONS for decreasing the number of algebraic mistakes students make.

No Mistakes

Let's Reduce Mistakes in Algebra!

First, it’s useful to recognize a key fact:  we can’t help students with mistakes if we don’t know what causes those mistakes.

Years of tutoring have taught me a lot about why students make mistakes. And one major cause of mistakes in algebra is that students combine terms that should not be combined. Not all their fault, though. Students are often confused about what they may and may not combine. And it is tricky!

Take a problem like this:  8 – 2(3x – 7)

Certainly some kids can simplify this expression with no trouble. But in my experience, many struggle with a problem like this (when first learning it), and quite a few stay befuddled for quite some time.

The biggest mistake is that students think they can and should combine the 8 and the 2 through subtraction, proceeding like this:

8 –  2(3x – 7)

=            6(3x – 7)

=            18x – 42

Q:  How can we help students avoid this mistake?

A:  Use a mark that show students what gets combined and what stays separate.

I will start to elaborate on how I do this in tomorrow’s post.

Extra, extra!   I thought it would be interesting for you readers to send in comments on the kinds of algebraic mistakes that “drive you up the wall” the most. When I get a number of comments in, I will conduct a poll to see which mistakes people find most vexing. Should be “fun.”

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…)

Not all variables are created equal


Are all variables the same?

Does every variable serve the same purpose?

When you think about it, you’ll see that the answer is “no.” Variables serve different purposes. When we explain this to students, we help them understand how variables work. Explaining this helps students understand how algebra “works.” You’ll see what I mean in a moment.

Consider the famous slope-intercept equation:  y = mx + b

A student recently asked me:  Are the  x and y variables the same as the m and b variables? What a great opportunity to explain something important!

I explained that the x and y variables serve completely different purposes than  the m and b variables. Here’s how.

The variables m and b are what I call “identifier” variables. By which I mean that they help us identify a specific line. To explain that, I asked the student a set of questions about something everyone understands — home addresses.

What would happen, I asked, if someone wanted to know where I live, and I told him that I live at 942? The student replied that this would not be enough info.

Then I asked, what if I told this person only that I live on Vuelta del Sur (a street name where I live in Santa Fe, NM)? Again the student said that this would not be enough info.

But what if I told this person that I live at 942 Vuelta del Sur. This, the student realized, would be enough information to enable someone to find my house. (All they have to do is Google me, and they’ll have my house AND directions!)

I pointed out that a similar situation applies to lines.

If I have a specific line in mind, and I want someone else to know the line I’m thinking of, is it enough to give this person just the line’s slope? No, for it could be any line with this slope, of which there are infinitely many parallel lines. What if I don’t give the slope but I do give the line’s y-intercept? Still not enough, as there are infinitely many lines that run through this y-intercept. But what if I tell the person both the slope and the y-intercept. Aha! The student could see — through drawings I made of this situation on a coordinate plane — that when you provide both slope and the y-intercept, there is one and only one line that could be indicated.

 

Three lines — the red and blue lines have the ...

Red & blue lines have same slope, so slope alone does not indicate a specific line; Red and green lines have same y-intercept, so y-intercept alone does not identify a specific line.

 

I explained that variables like m and b, which help identify a specific line, are “identifier” variables; their job is to identify a specific line. If your students are more advanced, you can explain that there are other identifier variables in different kinds of equations. For example, in the equation of a parabola:   y – k  = a(x – h)^2, the identifier variables would be the variables a, h, and k.

But what about variables like x and y? What do they do? What is their purpose?

These variables, I explained, have a completely different purpose. I call variables like x and y “ordered-pair generators.”

To explain this, I show students a simple linear equation like  y = 2x, and demonstrate how, using a “T-table,” you can use this equation to generate as many ordered pairs as you’d like, ordered pairs like (0,0), (1,2), (2,4), (3,6), etc. Point out that you can keep going and going. And then explain that the purpose of the x and y variables is to generate the infinitely many points that make up the line.

So the m and b variables tell us where the line is, and the x and y variables allow us to find the infinitely many actual points on the line. The two sets of variables, while different in purpose, work together toward a common goal:  to give us the equation of a line.

There are other purposes that variables serve, of course. And I’ll probably describe some of the other purposes in future posts. But the main point is that it helps students to recognize that variables do serve different purposes. Armed with that understanding, they can make much more sense of algebra’s formulas and equations.