Accumulation 5: Lines
If you have a function y(x), that has a constant derivative, m, and contains the point 
This is why I need your help!
I want to ban all use of the slope-intercept form, y = mx + b, as a method for writing the equation of a line!
The reason is that using the point-slope form to write the equation of a line is much more efficient and quicker. Given a point 
Algebra 1 books, for some reason that is beyond my understanding, insist using the slope-intercept method. You begin by substituting the slope into 
Where else do you learn the special case (slope-intercept) before, long before, you learn the general case (point-slope)?
Even if you are given the slope and y-intercept, you can write 
If for some reason you need the equation in slope-intercept form, you can always “simplify” the point-slope form.
But don’t you need slope-intercept to graph? No, you don’t. Given the point-slope form you can easily identify a point on the line,
Help me. Please talk to your colleagues who teach pre-algebra, Algebra 1, Geometry, Algebra 2 and pre-calculus. Help them get the kids off on the right foot.
Whenever I mention this to AP Calculus teachers they all agree with me. Whenever you grade the AP Calculus exams you see kids starting with y = mx + b and making algebra mistakes finding b.
Teaching Calculus 
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I know this is really old, but I think the reason that slope-intercept is so popular early on is that it is initially easier to teach. Early math is all about concrete examples, so if Suzie has $5 and wants to save $3 per day, it’s not difficult to write a function for how much money Suzie will have after d days (y = 3d +5 or y = 5 + 3d which actually makes a lot more sense in that context). So many of the “real world” linear problems before and after Algebra 1 are basically like that (sort of like start plus accumulation but without the integrals) so it makes perfect sense to teach y-intercept and slope as THE form of a line.
When we get into systems, Ax +By = C makes sense for the types of word problems textbooks tend to have.
It’s not until we get into function notation (af(x-b)+c) that point slope makes sense, because it puts linear function transformations into the same “language” as all other functions. For years I’ve been teaching point-slope for that reason (and I like it more), but I had a kid tell me this year that he was really impressed that you didn’t need the y-intercept to graph a line.
This is my first year teaching AP Calculus and I think one thing that is really different about the lines we deal with is that they very rarely happen around x=0, making point-slope much more appealing, and slope intercept less useful.
I think even in lower level math, starting at point-slope and moving to other forms is so much easier than all the substitution and solving and substituting again, but that is my personal preference.
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I agree with you!
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Excellent blog post. I definitely love this site. Stick with it!|
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We public school teachers are forced to teach to the “common core curriculum” and are rated on our students’ ability to demonstrate understanding of such. The common core curriculum for grades 8 and 9 (at least for New York) lists slope-intercept form as a performance indicator. Our hands are tied; we must teach it, or risk a lower evaluation by our districts. I agree 100% with your post however. Can you influence the common core curriculum writers?
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I have no influence over the Common Core – wish I did.
I do not know just exactly what the Common Core will require. I certainly hope they will NOT require students to do things or solve problems by some specified method. A correct answer found by any correct method, with supporting work, should receive full credit. Otherwise they will be penalizing students for doing correct mathematics.
As I hope I made clear, I am not in favor of using the slope-intercept method as a way to write the equation of a line. Students should certainly know what the slope-intercept form looks like, know what the m and b stand for, and know how to graph directly from slope-intercept form. They should be able to change their answer into slope-intercept form (which I suspect will be necessary to recognize the correct multiple-choice choice), and also other forms such as Ax + By = C. But to force them to use such a clumsy method for writing the equation in the first place is ridiculous.
I’m happy that the AP Calculus exams always ask for “an” equation of a line and take any correct form.
Of course if the CCSS asks only multiple-choice questions, or if they accept any correct method, then using the point-slope method will work just fine. Let’s hope they do that.
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It isn’t Common Core’s fault; this is a part of almost all state standards, and predates the Common Core Initiative by several decades. If you want to blame an organization, blame the National Council of Teachers of Mathematics – but remember they are just trying to help us teach the way we’ve been teaching math for centuries. For that matter, remember that “Common Core” does not write curriculum; the textbook writers do that. Get with NCTM & the publishers if you don’t like “Common Core problems.” As for the S-I Method, I rather agree, but now you’re fighting with the graphing calculator manufacturers as well!
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You can teach slope intercept form without solving for b! Teach point slope form first–then have students simply to get y alone and boom-slope intercept form! Then you have taught the standard without teaching the pointless “solve for b”
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