Going over Simultaneous Equations today with Year 11, we all agreed that the thing that is most confusing about solving equations like these are the negative numbers:
4 - 2x = 9 + 6y 6 - 2x = 7 - 2y
We also agreed that we much prefer these types of questions when we heave a sigh of relief realizing that we can add instead of subtract.
4 + 2x = 9 + 6y 6 - 2x = 7 - 2y
This is subtle. I would always start teaching simultaneous equations using some real-life examples (the ideas in this post).
I think real-life examples work in this case, because it is apparent what “extra” you are getting for the extra money. These problems create the need for the algebra – it becomes a way to represent what is going on by writing less. With these problems, everything is positive so it makes sense to think of the difference between the two situations or the two equations.
From there we can introduce the need to “multiply up” to get the same co-efficient for one of the terms. In my experience, most students don’t struggle too much with this concept.
What gets tricky is subtracting one equation from the other when negative signs are involved. We might identify the problem as being one of a lack of mastery of a fundamental concept – in this case negative numbers. So before teaching these types of simultaneous equations we could do lots of practice and drills on negative numbers. But I’m not sure that is always a helpful approach. Some things are just more confusing (present a higher degree of cognitive load if you like). There is a lot going on, and lots of it needs to be done mentally. So, maybe there is a case here for explicitly teaching an easier technique which is less prone to error.
A simple way to avoid subtraction is by always ending up with equations with opposite coefficients of one of the variables.
So, for example when solving the following:
4x - 3y + 1 = 0 (A) 3x - 7y + 15 = 0 (B)
multiply (A) by 3 and (B) by -4. A valid shortcut / rule to remember here is that multiplying by a negative simply “flips the sign” of any term it is being applied to. So we end up with:
12x - 9y + 1 = 0 -12x + 28y - 60 = 0
It’s easy to check that the sign of each term has been flipped before then proceeding to add the equations.
As any Maths teacher would, I believe strongly in teaching for deep understanding of concepts, not blind memorisation of rules (Nix the Tricks is a frequent reference point). But there is also a place for remembering certain procedures to reduce cognitive load (times tables being the most obvious example). Dani Quinn has written a great post on this in relation to “moving the decimal point” here. Manipulating Simultaneous Equations like these is another example of when there is a case for a bit of explicit teaching of a method.