## Kiss those Math Headaches GOODBYE!

### Reader Input on Slope Post

A longtime reader of Turtle Talk, Jeff LeMieux, of Oak Harbor, WA, sent in a suggestion based on today’s post on positive and negative slope. Jeff found a way to help students remember not only positive and negative slope, but also the infinite slope of vertical lines, and the 0 slope of horizontal lines … all using the letter “N.”

This is clearly a situation where the picture speaks more loudly than words, so I’ll just let Jeff’s submitted picture do the talking. By the way, to see this image even better, just double click it!

Slope Memory Trick

Thanks for putting this together and sharing it, Jeff!

### Remember the Difference in LOOK between Positive and Negative Slope

Some ideas just slap you in the face.

I got slapped this morning as I was flying home from LA to Albuquerque. Those little cocktail napkins they hand out with “beverage service” often give me the urge to write. So this morning, nerdily enough, as I sipped my orange juice at 30,000 feet above the Salton Sea, I worked on figuring out a better way to help students grasp the difference in look between positive and negative slope.

That’s when I got “slapped.”

First, you must realize that I use the three-letter abbreviations of POS and NEG for positive and negative. Do some of you use these as well? I mention this because those abbreviations hold the key. You have to use the first letter of the NEG abbreviation and the last letter of the POS abbreviation.

The first letter of NEG is, of course, “N.” But look what I noticed …

Visual Clue for Negative Slope

The trick for POS is a tad more complicated. But I’m hopeful it will work.

Visual Clue for Positive Slope

So what do you think? Will this work for your students?

If you test it out, please let me know what you find. I’m interested to know. Thanks!

### 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.

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.