Algebra | | DJS


Algebra is a fundamental skill in mathematics. It is the most abstract of skills but it develops your 'ability to think' more than anything else you study at school. And like school, the rules can change.... One of the best uses for this year is to use algebra to see why and how some of the (apparently clever) things we do with numbers work.

At the level of Lower School, the first jump is to use letters to represent unknowns. From there, it is usual to explore the meaning of the operators (+, -, X, /) and then to move to collection of like terms, to factorising and to equations in one variable. Life is not made easy when the variable is x and the sign for multiply is x too, so it makes a lot of sense to use a range of letters for tht variable AND to write the variable letter as 𝔁, even explaining how to draw it  - and why.

I was reminded, when reading a lot of Quora answers to a simple question in 2018, that the meaning of the equals sign needs explanation. For two reasons: (i) to move the understanding of those that think they already know how to do algebra and (ii) to undo the mistakes of those who never deal with the problems they generate, especially those who say things like ‘change the side, change the sign’. It is very much better for the sake of understanding to explore what the offending component is doing on one side and to ‘undo’ that.

Here is a simple example:    4𝒙+5 = 13

First see that there is an 𝒙, just one of them, and that this is what we want to find. So we need to separate it from the 4 and 5 on the left side of the equation.

Removing the 5 is the simplest first move. It is added to the left, so we remove it by subtracting 5 from both sides. You must say to yourself, if stuck, “What can I do to both sides that will make this simpler?” (and move towards the 𝒙 to being the only thing left). So let’s do that:

 4𝒙+5 - 5 = 13 -5                we can do the arithmetic, 13-5=8  and 5-5=0, so now

 4𝒙 = 8                     and we ask “What is the 4 doing to the 𝒙?”  Clearly multiplying, which will be undone by dividing both sides by four, which we can write down

4x / 4 = 8 / 4            whcih then become x = 2, as we do the arithmetic. Don’t be too quick with the arithmetic here. It needs to be seen that 4𝒙/4 did the trick and gave just 𝒙. Trusting that the choice of operation works will come later. possibly very much later.

Check by putting 2 into the original problem. Don’t miss out this checking process.

Notice that we have separated the manipulation from the arithmetic for the time being. But whenever people make mistakes, what they have wrong is, most often, forgetting to say internally what it is they are doing to both sides and, far less often, actually made a mistake with the arithmetic.

Here’s a harder example, useful for straightening out Y8 and Y9:

25 - 6𝒙 = 7                 One variable, but the big problem is that -6 multiplying 𝒙.

Most students will choose to remove the 25 first, so ‘take 25 from both sides’

gives us 25 -25 -6𝒙 = 7 -25      and therefore   -6𝒙 = -18.

At this point the class might be persuaded to divide both sides by -6 to reach 𝒙=3, but some willl declare the minus sign to be the big problem to deal with and, of those, some will say that what needs to be done is to ‘add 6x to both sides’. If so, they also need to ‘add 18 to both sides’ —as a separate move—and then have  18=6𝒙, which we know they can solve. It is important that it is seen that no correctly executed routes to solution are ‘wrong’. Instead, the lesson is for each student to identify what they think is the bit of the difficulty they can deal with. [And that it is a technique to apply outside maths lessons, too.]  The check to see if the answer works confirms that 𝒙 is 3. 

Often (too often, for me), someone will say that they can find (have found) the answer quicker by saying 25 - <what> = 7, so 18 = 6𝒙 so 𝒙= 3. Fine, but they have not moved towards being able to solve ‘harder’ problems until they have worked through the routine with easy ones. Their method, the sophisticated guess, works well right now, but won’t work with the levels of difficulty that are expected to be manageable (like at GCSE). A demonstration in class is called for, but it has to be done in a way that doesn’t put the offender down as much as allow everyone to be able to see why this (my seemingly plodding method,  not the sophisticated guess) apparently baby technique is useful.

25 (𝒙+2) - 6𝒙 (𝒙+2) = 7 (𝒙+2)    for example. There is an (𝒙+2) term everywhere (“Oooh, it’s already been factorised, isn’t that helpful?”)  Or move to the expanded position with Y9, which will look truly horrible, and would cause so many students to want to solve the quadratic -6𝒙2 + 6𝒙 + 36 = 0  (though too many will have wrong coefficients) and then, after more juggling, hopefully (3-𝒙)(𝒙+2)=0, so 𝒙=3 and 𝒙=-2 are solutions to be checked.

But the way that makes more sense at this point,  25 (𝒙+2) - 6𝒙 (𝒙+2) = 7 (𝒙+2),  is to see the common factor and ‘divide both sides by (𝒙+2)’, which will work provided (𝒙+2) is not zero (a side issue for now or later). The argument might be to ask “What is the leaast work we can do that makes this problem easier?” Some will see that 𝒙=-2 is an answer immediately, including the kid who likes the “What answer will work?” method.  Having done that division by (𝒙+2), we have the previous problem, 25 - 6𝒙 = 7    and so 𝒙=3.   One way to do this harder problem is to push for the 𝒙= 3 solution and many will see that the check is unnecessary from the previous problem. However, asking if the class can expand the brackets and rearrange the algebra offers an oppportunity to yet again check that 𝒙=3 contiinues to work, but also offers a chance to suggest that 𝒙=-2 might also work (best if one of the kids notices). At this point the class can do some ‘real’ maths and find out why, therefore discover (rediscover) when division fails (you can’t divide by zero and have a sensible answer) and so see improvemtents to their technique. A Y9 class might even be persuaded that seeing the factorising early not only produces the answer very quickly (fewer lines of writing), but also produces two very different routes to solution. It also gives the teacher a chance to point out that quadratics will have (must have?) two working answers, so finding just one is a ‘fail’, which, as part of the check that everyone always does (no they don’t) should cause them to yet again see the catch, or the clever move, is dividing through by (𝒙+2).   Which turns what could have been a boring lesson into a harder one with content for everyone.   A little play-acting by Teacher turns the whole thing into discovery so that everyone is more involved.

If using this with a Y9 or Y10 class, you might even replace (𝒙+2) with tan𝒙 or even (tan𝒙+2).  The point being that the factorising is very useful.

“When will we ever use this?”. Well it is algebra, so in an awful lot of maths lessons. The problems that the algebra solves will be found in real life all too easily, for those that can see them. Algebra is a mathematical tool, not unlike grammar in langauge.

DJS 20180518

One route with this problem,  25 (𝒙+2) - 6𝒙 (𝒙+2) = 7 (𝒙+2), would be to offer it at the start of the lesson as something to solve and see what happens around the class.   Taking the very different solutions as something for discussion allows several points to be raised, such as checking answers (boring but necessary), all sorts of issues about the state of their algebra, solving quadratics,  factorisation and so on. It might even make a good revision question for May. It would be instructive to see how many different routes to solution were used across the class. Able students need to see that there are many routes to solution; less able students need to see that there are things they can do (say to themselves) that will help, such as “What is the thing that is making this hard for me? So what can I do about that?” and “Is there anything that reduces the apparent difficulty that I can do?”. For example, some students might start by multiplying through by -1, saying that the -6x is the thing to ‘fi𝒙’ first. It might well be valuable to discuss the problem from several different startiing points. Some classes will need the problem replaced with a similar one (i.e. different coefficients) and to be pushed into starting in whatever way is worth trying next, perhaps by asking a student to provide the first move.

Some students discover, as a takeaway from the lesson, that the best bit of the whole 'but this is algebra and I hate it' thing is that the trick lies in asking the right question. Now there's a skill that applies to a much wider set of problems and issues. To me, the best bit is that they ask themselves what the best question is, by asking themselves, internally, what it is that is their immediate problem.

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