## The straight lines debate

For a long time I’ve been confused about straight lines. Not because they confuse me, or that I don’t understand them, but the prolific use of the formula *y-yo = m(x-xo). *I personally prefer the y = mx +c version, and I couldn’t see a benefit to y-yo at all. I certainly couldn’t fathom why at A Level is was so heavily favoured by teachers and markschemes alike. I had asked around a few times, and the only coherent point I had got that made any sense to me was that when the gradient is fractional students who weren’t any good with fractions could multiply up by the denominator and deal with integers more easily. But my experience showed me that students used y-yo in a mechanical manner, and weren’t building any deep understanding of the geometry of straight lines.

I then noticed, while reading his blog at flying colours maths, that Colin Beveridge (@icecolbeveridge) was very heavily on the side if y-yo, and called y=mx+c “the baby formula”! I thought, perhaps “Colin can explain why there’s such a preference.” I asked, and sparked another “blog off”.

Colin wrote this excellent piece on his blog setting the case for y-yo. He argued that y = mx + c is a special case of the other, also mentioned the aforementioned fractional gradient argument, and made a few other (occasionally compelling occasionally contradictory) points. I felt that it did give me a better understanding of why people preferred it, but still did nothing to persuade me that it was the better version to use. (If you haven’t read Colin’s post, you should!)

I then responded with a guest post on Colin’s blog setting out my counter arguments. Since the post was written I’ve discussed these formulae with teachers and students alike. I have noticed something about the students views, those who I taught in year 12 ALL prefer y=mx+c, but those my colleague taught in year 12 mostly prefer y-yo. So I wonder if I had somehow imparted my preference to them. A friend of mine who is also a maths teacher summed it up pretty well for me when he said:* *

*“Ok, well so full honesty then- y=mx + c is probably better, however, when working with weaker students I find it easier to use the mechanical form, rather than the one which provides the most information about the line itself!” *(Steve Atkinson 2014)

Which version do you prefer? Do you have any strong views? I’d love to hear them.

I wondered whether to have the Ninja respond to your points in your excellent guest post, but he said it’d look petulant 😉 I do have some responses, though.

One is that the y-y0 version is MUCH quicker for finding, say, where a line crosses the x-axis. With y=mx+c, given the gradient and a point on the line, you’d need to substitute in your point, rearrange to work out c, then substitute in y = 0 and rearrange again to solve for x – most of which usually involves fiddly fractions; with y – y0 = m(x-x0), you sub in the given point and y=0, rearrange and you’re done.

I suspect we’ll need to agree to disagree on the ‘more chance for error’ point – I find students much happier multiplying out brackets than fussing with fractions. Even if they (or I ) can do it, they don’t like it, especially when it’s not necessary.

The Ninja would also like to clarify his ‘two constants’ remark: he claims he was misquoted, and would actually integrate dummy variables between the given values and x and y. He’s a law unto himself, I tell you.

Haha! I will have to conceed the X axis point, but don’t think it sways my thinking overall.

This is really interesting to read, as I never realised there was so much debate to be had! I must admit I’m in the y-y0 camp for A level teaching but reassure my students that they are equally welcome to use mx+c. (One does.) I think (y-y0)/(x-x0)=m is the most satisfying mathematical way to see that such a graph would be straight line. (Explaining why y=mx+c is actually a straight line would take more effort?) Also, at A level the two biggest uses for straight lines are tangents/normals at a point and linear regression. Why create the issue of solving for c when you can write the equation directly?! More interestingly, noone’s brought up ax+by=c which is what I was taught for Additional Maths!

Some good points, thanks for sharing.

I think a x+by+c=0 is more a form used to express the line, as opposed to calculate an equation.

I have found it depends what you are likely to be doing with the line. Younger students who are simply plotting benefit from being able to check their calculated coordinates with a general understanding of where the line is going to go through y=mx + c. Older pupils who are going to be finding normals at a point and similar processes will prefer to use the y-y0 version.

Realise this is an old post! Been debating this myself recently. Personally I prefer y=mx+c because I teach it in terms of connections between input-output functions, and the meaning of co-ordinates. That is (3, 5) is one particular input-output of y=2x-1, and a graph is simply one way of displaying this relationship (as is a table of values, a co-ordinate pair, and equation etc. etc.)

I have had a-level students who did not understand the link between co-ordinates and functions, or that gradient represented rate of increase or anything like this because they had been shown y-y1, and followed the “give me a formula and i can use it” school. Deep structure of mathematical knowledge was absent, and as a consequence as-level co-ordinate geometry was a complete nightmare to teach.

There is power and beauty in y-y1, if students understand what it means and how its got to. You can derive other things from it, or thing about the nature of the whole concept of change from IF its taught like that.

If its taught “chuck in the numbers and you’ll get an equation” then frankly, there’s no structural point but a few marks on a test and its potentially harmful to students who might take maths at A-level.

Aye, salient points well made. That is the issue with so many formulae. I was doing Normal Distribution revision with a class recently and they’d been taught it formulaically and had no idea what was going on. When I referred to a z value as “the number of standard deviations it is above the mean” they were all like “what? Is that what that number means?”