One short lesson that came from the Exeter sessions was how they teach lines and what formats they use to teach writing equations of lines. It is amazing how often we, as math teachers, fail to build connections between different elements of mathematics simply because we feel like we have to hold to some form of “history” or “tradition” that is involved in math.
As an example, let’s examine the old standby to writing equations of lines, y = mx + b. Of course, it is the math teachers go to. Here in Washoe County several years ago, we spent a huge amount of time (and money) learning how to teach lines by using y = mx + b through the “Algebraic Thinking” company. We had foldables for learning y = mx + b, we had lessons using it, we had just about everything we needed to teach y = mx + b except the most important thing; learners who understood why we were memorizing such an arcane idea, why did we use m, why did we use b, and why should we or they care?
After all, look at one lesson from Algebraic Thinking. It uses “formula” 4 times, “Hopefully your students have not forgotten” once, “Your students are going to have to be meticulous, careful, and exact on when simplifying today” is an actual statement in the lesson, and they actually say in their handout:
There is a point-slope formula, but the last thing most students need is another formula to remember. It’s a good idea to just stick to y = mx + b. They have to know it anyway, so they can use it for everything.
Every calculus teacher reading this just threw up a little in their mouth. Right, let’s not use the easier formula. Let’s go for the formula that takes the MOST amount of effort, and ram that down their throats through memorization. Glencoe, McDougal-Littel, and all the other textbooks do the same thing.
Let’s examine some ideas based on this y = mx + b from a pedagogical point of view, so let’s start with a problem. “Write the equation of a line with slope 3 and goes through the point (–2, 5).” Straightforward enough, right?
- First off, the learner has to memorize that the slope, 3, corresponds to the “m” in the equation y = mx+b. Why is it “m”? No one really knows. Seriously. Follow that link. There are several explanations there, and none of them are definitive.
- So the learner now substitutes the 3 in for the arbitrary “m” and gets y = 3x + b. Great.
- Next the learner has to substitute in the x and y co-ordinate of the (-2, 5) to get 5 = 3(-2) + b.
- Now some algebra, multiply and add to get 11 = b. (notice there are 2 steps there)
- Next the learner must substitute ONLY the “b” value back into the original step 2, but not the values of the (-2, 5) that they just substituted. Why? because the foldable says, just do it. (not really the answer I give in class. An equation has to have a “y” an “=” and an “x”, so we can’t substitute those again.)
- Then write the final equation, y = 3x + 11.
Summarizing, that is 7 steps, 2 of which are substitution, and one hidden step that is a NON-substitution that confuse the heck out of learners. All of this was done because we don’t have a really good reason to use y = mx + b except the textbook publishers have asked us to.
Also notice, THIS WILL NEVER COME UP AGAIN IN ALGEBRA 1 OR 2! At no point will a learner use this method for quadratics, absolute values, cubics, or anything else. It is an island unto itself that is only applicable in linear equations.
If we then ask the next question, “Take the equation written above, and write the equation of the line perpendicular that goes through (6, –2) and then also parallel to the original through (3, 7).” You have a learner using y = mx + b for about 5 minutes substituting and resubstituting over and over again. 5 minutes that is, if they are really good at it. Most learners give up because it is difficult and confusing.
I want to shake this up and look at how an Exeter teacher would ask their learners to write the same exact series of equations.
Write the equation of a line with slope 3 and goes through the point (–2, 5), then write the equation of the line perpendicular that goes through (6, –2) and then also parallel to the original through (3, 7).
This should take us about 45 seconds.
Why the difference? Because we are not going to start from y = mx + b.
Let’s start from the vertex form; y = a(x-h) + k.*
The first equation is y = 3(x–2) + 5 so therefore: y = 3(x+2) + 5
The perpendicular is y = -1/3(x-6) + –2 so therefore: y = –1/3(x-6) – 2
The parallel is y = 3(x-3) + 7
Okay, I exaggerated. It took less than 45 seconds. Notice that the learned and memorized information is the same for both situations (the y=mx+b and the vertex form); perpendicular slope is the opposite reciprocal, and parallel slope is identical.
But look how little effort goes into actually writing the equations. It is ONE substitution from the vertex form to the equation.
Want to graph it? Sure, start at (-2, 5) and go up 3, over 1. Okay, if you really want, I will let you distribute and add. 2 steps, piece of cake.
Wait, let’s look again at that vertex form of the line. y = a(x-h) + k. It looks surprisingly familiar. It looks like the vertex form of the quadratic, or the absolute value. Let’s line them up and see:
- y = a(x – h) + k : Linear vertex form
- y = a(x – h)2 + k : Quadratic vertex form
- y = a|x – h| + k : Absolute Value vertex form
I could go on, but I think the point is made. Using this form to write the equation of a line is faster, less to remember, makes connections to other units in math, and overall allows a learner to understand what they are doing instead of memorizing steps.
There are still some things that have to be memorized, for instance: What does the “a” do in front of the parenthesis? It makes it steeper. What if it is negative? it flips it over the x axis. What does the h do? What does the k do?
But notice, the answers to those questions are the same in the linear unit as in the quadratic unit as in the absolute value unit, or cubics, or … you get the point.
In the end, I have to ask the question, Why do we torture our learners with y = mx + b. It is arbitrary, and doesn’t make sense. It is hard and requires far more effort, and it is stupid.
Let’s start teaching math better. Throw away the textbooks. Seriously. Take the Algebra 1 textbooks and cut out the problems, throw everything else away. Now reorganize the problems so they make sense and build off one another.
Great, now photocopy them, scan them in, and post online for everyone to use.
That is essentially what Exeter has done. Why don’t we just start using their materials instead of buying the textbooks to begin with? The textbooks suck. We know it. Start acting on that knowledge.
*Why the vertex form and not the point slope form? I ask what is special about y – y1 = m(x – x1) the answer is nothing. Notice the subscripts? They confuse the heck out of learners. Where does this come from? The definition of slope. So why use the arbitrary “m”? Why not call it slope and therefore is “s”? There isn’t a good answer, but using “a” for it at least makes sense given the other vertex forms.
After all, if we are going to be arbitrary, at least let us be arbitrary consistently.