The Algebra Onion


I have been teaching algebra for what seems like an eternity with my top set Year 10 at the moment. We have covered solving one, two, multi-step equations (with success) and moved more recently onto brackets and factorisation. We are almost at the end of the unit, and as such are at the most difficult parts that rely heavily on deeper understanding of more basic concepts. Students struggled at first with the ‘why’ of factorising (as we weren’t using it to solve anything at that point) but soon saw how it was useful for solving quadratics once we got there. A few couldn’t quite see how the following worked:

(x -2)(x +3) = 0 therefore x = 2 or x = -3

They couldn’t quite make the connection until I showed them that if ab = 0, then a, or b (or indeed both) must equal zero, and that in our case, a was (x – 2) and b was (x + 3).

Job done. At least for all but one student.

This particular student has been fine with getting answers correct the whole time, but she was becoming increasingly frustrated with the ‘whys’ and ‘hows’ of what was going on. I was really pleased in some way, as it demonstrated not only a willingness to really figure out what was happening, but also a determination on her part.

After sitting with her a few minutes I realised this would take some time. I was confident enough that the rest of the class could get on, so we cracked on. Her fundamental issue was that she couldn’t for love nor money see where the middle part of a quadratic equation was coming from, or more importantly, WHY is was coming from the method when multiplying out brackets. In other words:

(x + 3) (x + 1) = x^2 + 3x + x + 3 = x^2 + 4x + 3

The 3x + x part she could calculate, but just didn’t get it. I could easily have just said ‘because it is’ or reassured her that it didn’t matter, she was getting the right answer. But she was *so* confused and her confidence in maths was draining in front of me.

So we started peeling the onion. We went back to a single bracket, using algebra, and soon discovered that the problem was still there at this level. “Why can’t you just add together what is in the bracket first? That’s what you’re supposed to do”. Yes, but the bracket contains an ‘unknown’ number. If I asked you to add 6 and ‘question mark’ together, you can’t, so we have to leave them separate for now. Again, this was not about getting incorrect answers, she WAS keeping them separate, she just didn’t know WHY it worked.


So we peeled back another layer, and looked at  p(1 + 3). Again she correctly identified that it would become 4p, but disputed that p(1+3) = p(4)  – “why is there a 4 in the bracket now, why is there even a bracket??). Perhaps I shouldn’t have left the sum in the bracket for one of the steps, I’m not sure it helped.

But then the root of the problem started to present itself. We went back even further, and focused on number, with no algebra whatsoever. I asked her what 4(2 + 3) would be, and she again correctly said 20 (“Because it’s BIDMAS”). A lightbulb moment … for me. She was getting totally bogged down with the idea that she *HAD* to do the bracket first, and that was sending her brain into overdrive. Furthermore, after a little discussion about 4(2 + 3) , she had never really noticed or accepted that 4(2 + 3) is exactly the same as 4×2 + 4×3 . In her mind it was only ever 4 x 5. Hence when we switch  to algebra: a(b + c) = ab + ac never made sense to her. But now she (hopefully) can see that not only is it TRUE, but more importantly, it’s LOGICAL. She had always accepted it worked, but never grasped why.


I’d love to say all this worked out over five minutes or so, but in truth it took a long time. I have no doubt it was therefore not a good lesson for anyone except this one student (fortunately the rest of the class miraculously got on with their work the entire time this played out), but I *think* we cracked it. We’ll see today…


5 thoughts on “The Algebra Onion

      • Lets use a+b for the vertical side and c+d for the horizontal side
        Draw rectangle
        vert side – a lower part, b upper part, draw line across between the parts
        horiz side – same for c and d, well, c is the left part, d the right,draw vert line
        then total area is (a+b)(c+d)
        and the four small rectangles have area ac+ad+bc+bd
        But it’s the same rectangle !
        This is like explaining calculus over the telephone !

  1. Sometimes you have to invest a little time in individual students for the greater good of the class. I wouldn’t worry about that, first of all.

    Secondly this situation is testament to the fact that a child’s understanding of maths is a house of cards, and it doesn’t take much for that to be under threat if the basic principles aren’t strong.

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