Tuesday, 16 May 2017

Approaches to teaching simultaneous equations

My esteemed colleague Mark Horley (@mhorley) wrote an excellent blog recently about the balance between the need for understanding when teaching simultaneous equations balanced against ensuring procedures are straightforward enough to support pupils ability to follow (read it here). Reading his reflections led me to reflect on my own approach to simultaneous equations, as well as others I have previously seen, and one that occurred to me literally as I was thinking about them. This blog is designed to act as a summary and chart my journey through the teaching of this topic.

Elimination: This is probably the first method I used, and is definitely the sort of approach I was taught at school. Very much a process driven method, I can't remember understanding much about the algebra beyond the idea that I was trying to get rid of one variable so that I could find the other. I find that the subtraction often causes problems (which is partly why Mark's idea of multiplying by -2 instead of 2 is very interesting) and of course the method doesn't generalise well to non-linear equations. I can see this being a popular approach for those people teaching simultaneous equations in Foundation tier.

                    Substitution: Another one from school,
                    this was the alternative I was taught to 
                    elimination, which was mainly because it
                    was necessary to solve non-linear 
                    simultaneous equations. I can't remember
                    it being the method of choice for myself 
                    or any of my classmates, and that is 
                    certainly borne out with my experience of
                    using it with any other than the highest 
                    attaining pupils.

       Comparison: Similar to elimination, but for me less 
       process driven and more focused on understanding the
       relationship between the two different equations. This 
       removes the difficulty around dealing with subtracting 
       negatives, and allows for the exploration of which
       comparisons are useful and which aren't, so it is a little
       less 'all or nothing' than the process drive elimination
       approach. It also copes nicely with having variables with
       coefficients that are the additive inverse of each other, for 
       example in the pair of equations above if instead of the
       approach outlined we multiply the second equation by 3 
       and get:

       4x - 3y = 9     and       6x + 3y = 21

       then the comparison would be "the left hand sides have a 
       total of 10x, and the right hand sides have a total of 30, so
       10x = 30."

       This is the approach I used when recapping simultaneous 
       equations with my pupils in Year 11 and they certainly 
       took to it a lot better than the elimination or substitution 
       that had used with them the previous year.

                                               Transformation: This approach is the
                                               one I have very recently considered, but
                                               not yet tried. The general idea is that you 
                                               isolate one of the variables, and then look
                                               at how you can transform that variable in
                                               one of the equations into the other. The
                                               same transformation applied to the other
                                               side of the equation then gives a solvable 
                                               equation. Although the equation may be 
                                               slightly harder to solve at first, I do believe
                                               this approach has merit. I would suggest 
                                               that this approach develops pupils'
                                               appreciation of the algebra and the
                                               relationships between the different 
                                               equations in a similar way to the
                                               comparison approach above. I can also see 
                                               this approach working for non-linear
                                               equations, like the one below:



I will almost certainly give this approach a try when I next teach simultaneous equations - when I do I will try and blog the results!         


  1. The transformation approach is interesting. I think this topic is one where exposing students to many different "methods" should hopefully lead to deeper understanding. There is a risk that they get overwhelmed, thinking that they need to "learn all the methods". But I'd hope that as they work through them (generally I don't think I even call them anything) with well-chosen examples, they see the logic in it all and generally build confidence in algebra. Of course, neither of us talked about representing the equations graphically. Is that something that comes later, or should we aim to build that from the start?

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