Why slope is important

It has the same slope ratio. And we're going to get in a second what I mean by ratio. And here's what I'm talking about. When we talk about slope or steepness, the way it's defined is by change of Y over change of X. Like in a fraction. Change in Y on top of change in X. That's why we call it a ratio. Remember ratio is like a fraction.

So if I were to draw a little triangle here that represents how steep my line is, this would be my change in Y piece because Y is up and down. This would be my change in X piece, because X is horizontal. And whatever those numbers were on the graph, I would write as a fraction. That's one thing you want to keep in mind. Sometimes we write it using this little triangle. This triangle is the Greek letter delta, which is tricky. Not only do you have to learn math but now you have to learn Greek.

This means change in Y on top of change in X. That delta just represents the word change. And a third way we write this is using the letter M. M stands for slope, and if I had two points, I'll use them up here. Let's say I had this point I'm going to call it X from my first point and then Y from my first point. Here's my second point. X from my second point I'm going to use that little 2 to show it's my second point.

Y for my second point. Then there's a formula I could use using those X and Y numbers to find M, or the slope. This is the same thing just written in a different way. I'm finding out how much did my Y values change and putting that on top of how much did my X values change in a fraction.

So this formula is really important anytime you have two points like this. This is a huge change in the meaning of the letters x and y, and many students are justifiably confused by this. Suddenly, formulas and equations serve a different purpose than before.

It is important to discuss explicitly the rules of mathematical grammar. The discussion is about the slope of the graph of a linear equation. And you get the definition of a derivative for free along the way.

By the way, I tested this on my family during dinner a while back. But they did seem to like it. In other words, if you start with a specific input and start changing it a little bit, how much does the output change? If the output changes a lot, then we say that the function is very sensitive to small changes in the input. If the output changes only a little, we say that the function is not so sensitive.

But we want to find a quantitative measure of sensitivity. A natural thing to do is to the measure the ratio of the change in output over the change in input. So what are the simplest cases?

First, there is the function where the output is always the same, no matter what input you feed it. Second, there is the function whose sensitivity is the same, no matter what the input is. We note that how much and which way the graph is tilted corresponds to the sensitivity of the function. There it is, the slope. I also want to thank the Harvard Calculus Consortium for showing me such a down-to-earth approaches to teaching mathematical abstraction.

Deane…I agree with you, especially with drawing a box. I am an engineer by training and often sketch a box to mimic a physical system when teaching linear or non-leaner equations.

However, I do think we start teaching kids math from day one by underestimating their ability to grasp abstract concepts. We are forever looking for some trick or shortcut to get the concept across.

I feel that we are handicapping them as they move into more difficult math courses in elementary, middle school, high school, college, etc. Different people learn in different ways and have different skill areas and might even learn different things best in different ways — for directions I might do better with a map and written instruction for a recipe.

Some kids might do better with physically manipulating an object and others watching it on the computer. No Kid Left Behind should mean we work with everyone to achieve to the best of their ability and we celebrate their strength areas. Not every kid is going to be a math wiz — and the math wiz kids should be allowed to excel in that area. You are just torturing them with the same material year after year after year. I feel our job as teachers is not to just teach slope but to make connections to concepts that will help the kids store and build a logical framework.

I have my students design roof lines for passive solar housing projects, this can go as deep or surface as you please and it helps connect slope with steepness and practical design issues like snow load and sun angle through the year. If we can meaningfully teach more topics students at different levels may find more inspiration and connections that stimulate their creativity and curiosity then we did in junior high and high school. One practical dilemma is the time it takes to develop more connected meaningful concepts and the time we have to fit the complete content of a class into their brains.

Your email address will not be published. Notify me of follow-up comments by email. Notify me of new posts by email. Notify me of followup comments via e-mail. The unemployment-rate graph in Figure 4, below, illustrates a common pattern of many line graphs: some segments where the slope is positive, other segments where the slope is negative, and still other segments where the slope is close to zero. The slope of a straight line between two points can be calculated in numerical terms. As an example, consider the slope of the air-density graph, above, between the points representing an altitude of 4, meters and an altitude of 6, meters:.

Thus, the slope of a straight line between these two points would be the following: from the altitude of 4, meters up to 6, meters, the density of the air decreases by approximately 0. Suppose the slope of a line were to increase. Graphically, that means it would get steeper. Suppose the slope of a line were to decrease. Then it would get flatter.

These conditions are true whether or not the slope was positive or negative to begin with. A higher positive slope means a steeper upward tilt to the line, while a smaller positive slope means a flatter upward tilt to the line.

A negative slope that is larger in absolute value that is, more negative means a steeper downward tilt to the line. A slope of zero is a horizontal flat line. A vertical line has an infinite slope.