Consider x + y = 5
How do you know what this will look like plotted on equal aspect axes? How do your students know? Some possible ways include:
- using a graphical calculator display
- rearranging the equation and plotting three points (including one to ‘make sure’)
- using one intercept and the gradient of –1
- recognising a member of the ‘family’: x + y = k
- mentally covering y so you can see the intercept x = 5; then doing the same with x.
You only need two points to know about a straight line, so you might as well use the points on the axes. You can do this by substituting zero for each variable in turn, which is the fifth method described above. The only time this doesn’t work for linear graphs is when the line goes through the origin, which you would know from the form of the equation. The more I think about this the crosser I get about the hours spent using the second method above. Instead, students could be building up their recognition capabilities (the fourth method) and their mental capabilities. This can help them extend the ‘cover-up’ approach to further cases, e.g. 2x + 3y = 1, or even to other formats: –y = 3x – 1. This approach also makes a strong link to the visual representation.
I used to hope that my graduate teaching interviewees would be able to say something intelligent about the following collection of linear graphs, without reaching for a calculator, setting up tables of values or rearranging formulae:
x + y = 5
x – y = 5
y – x = 5
–x – y = 5
Sadly, a few had to plot points for each line, appearing to have no sense of how the equation reveals the visual features of the graph. I am not blaming them – rather I am sad for a system that prioritises methods over meaning.
The fifth method above could be dismissed as a trick, but actually demonstrates a powerful tool of mathematical enquiry – substituting zero to find out more about the structure.
There are other places in the school curriculum where cover-up can be used, and arguments develop about whether we should be teaching children ‘tricks’. If so-called tricks work, then they must be based on mathematical principles, so let’s look at the principle. Cover up methods work only when zero can be substituted for the term and the equation or identity still holds. So, teaching ‘cover up’ rather than teaching ‘substitute zero’ can lead to misapplication of the method. For example, 2⁄x = y cannot be plotted using cover-up.
I have been reading online tutorials about using cover-up with partial fractions and very few explain why it works (some do not even say when it works). Even some of them that say ‘why’ still talk about ‘making factors disappear because they might be zero’.
My reconstruction of a two-term case goes like this (and I hope it makes sense, now I have been so critical of some other expositions):
All we have to think about are the equivalent expressions for the numerator:
1 ≡ A(2x + 3) + B(x + 1)
Note that I have written this as an identity, because it has to be true for all values of x. This means it has to be true for x = –1 and x=
Values for A and B follow almost immediately. There has been no attempt to ‘hide’ offending factors of the denominator; instead, the power of subbing zero has been used.
To be exact, however, I do need to consider that some of these rationals are undefined at the values I have used; so is it legitimate to say that the numerator identity has to be true for all values of x? Well, yes, because whatever values of A and B emerge, the identity has to be true for all other possible x and it does not depend on the denominator. I could stick both sides of the identity over some other denominator and 1 would still equal (2x + 3) – 2(x + 1) for all x.
So, that is two places – linear graphs and partial fractions – where subbing zero simplifies things and also provides some insight. The Maclaurin series also uses subbing zero to good purpose, when a Taylor series at some other value might lead to heavy calculations. For example, Leibniz used Maclaurin when expanding arctan θ to get an approximation of π. The denominators of the derivatives were pretty much out of control for calculation purposes unless zero was subbed.
I think my favourite use of subbing zero is when polynomials are expressed as products of their factors and a zero for any factor makes the whole thing zero. We can generate any function we like that has the values 1, 2, 4, 8, … for x = 1, 2, 3, 4, …, by constructing something nasty like f(x) = (x 1)(x – 2)(x – 3)(x – 4) sin3 x + 2x – 1, and thus avoid students jumping to conclusions about the function that produced those values. A ‘nastiest function’ competition maybe?
Finally, a bit of fun – express this product without brackets:
(x – a)(x – b)(x – c)(x – d)…(x – z)
In my last blog about partial fractions, if I mentioned the cover-up method, I did not say much about it. Since then, I have been thinking about the many places in mathematics where we learn something about a mathematical structure by eliminating parts of it.
Anne Watson has two maths degrees and a DPhil in Mathematics Education, and is a Fellow of the Institute for Mathematics and its Applications. Before this, she taught maths in challenging schools for thirteen years. She has published numerous books and articles for teachers, and has led seminars and run workshops on every continent.