Wednesday, December 15, 2010

Rates of Change

Sometimes we just stumble onto stuff. I've taught slope intercept form for a lot of years, but the more I try to stay true to the concept, the more my students demonstrate innovative ways of seeing things. We just finished up linear equations and their graphs. The equations part of things usually culminates with a student taking two points and writing the standard form of the equation by way of this process:

Step 1: Find slope
Step 2: Write in point slope form
Step 3: Solve for y to get slope-intercept form
Step 4: Rewrite in Standard Form

I've usually encouraged this process as it seemed to be the efficient way to cover the majority of these skills. But this year, I kept it much more loose. We all agreed that if the two points given define a line with an integer for the y-intercept, then it's pretty easy to write in slope intercept form.

I mean, c'mon, we've got the slope and we see the y-intercept plain as day.

But what happens here?

We know the slope is 1/7 and we can estimate the y-intercept, but how can we know the y-intercept?

Rate of Change

Most of the time when we discuss Slope we talk about the path from one lattice point to another. For example, the slope between A and B is 1/7 because we go up 1 and over 7 to get from A to B.

But we have been doing more with rate of change as a unit rate which means that for every 1 unit of horizontal change, we increase 1/7. Yeah, I know that sounds elementary, but it makes a difference in how your kids see these things.

If each step increases 1/7, we increase a total of 4/7 to get to the y axis. Therefore, y-intercept is 4 4/7. We still talk about point-slope form, but this beats the heck out of the plug and chug that usually goes along with it.


CalcDave said...

This is good for your visual types.

I'm of the mind that there are multiple intelligences in math: visual, numerical, and analytical.

Some people like to see the pictures like this and understand better that way. Others like to make t-charts and see how the numbers make patterns and play out. Others still understand the symbolic manipulation best.

I guess I'm just saying not to discount the previous ways you were doing it.

David Cox said...

I'm not trying to pit one against the other. I just think that the visual gives context to the symbolic manipulation. We are really focusing on multiple representations this year and I think that this helped. A lot.

Julia Tsygan said...

This seems useful. Overall, graphing lines is too often done too algorithmically. Let kids figure it out for themselves, given a variety of concepts and connections like the ones you mention here.

What about lines where you can't see them cross any lattice points? Lines with irrational slopes - Can you graph such a line? Including questions such as this one really clarifies the whole topic.