I’ve arrived at that one frustrating point in teaching Algebra – teaching linear equations. I don’t know why the concepts are so difficult for students to grasp. I’ve tried teaching it a myriad of ways, and have yet to discover the best method for teaching this section. It takes me nearly a month to teach all the things that go with slope, slope-intercept and standard form equations. And those are just the basic concepts of linear equations. (After the tortuous month is over, something kicks in and they finally get it and claim it’s SO easy).
And why it is difficult for students to remember that horizontal lines have a slope of zero and vertical lines have undefined slopes is beyond me.
I always start by trying to show them how to arrive at that conclusion using the slope formula. Since vertical lines invariably end up having zero in the denominator – which is THE big mathematical no-no – then their slope is undefined. They insist on calling it zero.
I resort to making them actually divide by zero on the calculator (after having arrived at some vertical line ratio using their slope formula), because it gives them “ERROR” since dividing by zero is a big mathematical no-no. And even though it says ERROR on their calculator, they still want to call it zero. The few who can actually remember that it’s undefined don’t want to call it undefined – they call it unfined, unrefined, or unidentified …
We analyze how linear equations with positive slopes increase or rise (from left to right) and how equations with negative slopes decrease or fall. Then we contrast that with a horizontal line, which neither rises or falls – and since it does “nothing” then it is “nothing” (0). But with a vertical line, you can’t tell if it is rising or falling; therefore since you can’t DEFINE which way the line is going, it is UNdefined.
Then I get desperate and resort to telling them that hOrizOntal has O’s in it, which looks like zeroes, because horizontal lines have a slope of zero. And that Vertical starts with a V, which strongly resembles the U in Undefined.
…that horiZontal has the letter Z in it, just like Zero …
…that Zero starts with a letter Z which actually has not one, but two horizontal lines ….
