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.
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.
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.
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.
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