# Algebra: you use the letter ‘x’ more than you ever have done in your whole life!

I started my first algebra unit with Year 7 on Monday.  It’s the essentials of algebraic manipulation: adding, subtracting, multiplying and dividing terms, including those with indices, as well as expanding and factorising single brackets, all pretty standard beginnings in algebra.  In the past I think my first lesson on algebra would consist of a brief introduction then some simple collecting like terms.  From what I’ve seen, that’s generally what comes up first in most schemes of work.  This time, though, I’ve tried to be more deliberate and more pedantic over the details.  Really, really pedantic, because it’s insecurity with the small details that causes so many mistakes for the rest of our students’ experience.  I’m starting out by assuming they have done no algebra.  Many of them have done a little but I’m not prepared to risk that all their different primary experiences were the same, or solid.  (This is a not a criticism of primary teachers, more that I want to take sole responsibility for something so important).

So today was different.  We started with some context for algebra – where it’s used throughout the modern world and how powerful it is.  I asked the students what they already knew when they heard the term “algebra” and the overriding theme was “letters instead of numbers”.  One boy said “you use the letter x more than you ever have done in your whole life!“, which had to get quote of the day.  We talked about why we might need algebra: sometimes we don’t know a number so a symbol is used instead, with algebra we can write statements that are true for all numbers without writing infinitely many examples, that kind of thing.

Then we went into the pedantry.  Over two lessons we are practising the conventions, making these clear now so that we can refer back to them in the future and hopefully reduce those problems that come with a flimsy understanding of the algebraic language.  These are the kind of things we’re practising:

• Why we write a curvy $x$
• Writing $2 \times x$ and $x \times 2$ as $2x$, no multiplication sign, numbers before letters
• Identifying equivalent expressions such as $x + 2$ and $2 + x$ (and why similar statements with subtraction are not equivalent – we learnt about commutativity at the start of the year, they’re superb with it)
• Not writing a coefficient of 1, so $1x$ is just $x$
• Always using a fraction bar for division, so $x \div 2$ is written $\frac{x}{2}$
• If a symbol looks different, it may represent a different number, so $X$ is not the same as $x$ (this one bugs me no end!)

After talking through the conventions the students did an exercise with a number of expressions written in “ugly” ways that they had to rewrite using the conventions.

The other thing I wanted to do first was focus on the meaning of algebraic expressions.  After examples they did an exercise with worded statements that had to be written algebraically.  Things like “add two to a number”, “subtract three from a number”, “subtract a number from 3”, “treble a number and add 4”, building up to things like “add five to a number and then multiply by 7” or “subtract 10 from a number, multiply the result by 6 and then divide by 8”.

I like the way this gives what could be just “letters and numbers” a meaning in language.  I think it will help when we solve equations or rearrange formulae and want them to think about the inverse order of operations.  Being able to say “what’s happening” to the unknown number is a big step in understanding how to reverse the process and I am more convinced all the time that this tacit understanding that we teachers have needs to be made completely explicit to our students – they need to practise the things we take for granted more than we think they do, however obvious these things might seem to us.

[As a side point, I am also very keen not to introduce the idea of substitution for a long time yet: I don’t want anyone to think that a letter has to have a fixed value associated with it.  This unit will take us till the end of the year then at the start of Year 8 we have unit A2 (Solving Linear Equations) and substitution comes in A3 (Formulae).  By this time we can cover these topics quite comprehensively since our students will have had plenty of practice in the underlying skills of manipulation.  I have done this deliberately: of course simple substitution is easier than factorising into a bracket, but that doesn’t mean it needs to come first.]

What’s interesting are the misconceptions and misunderstandings that arise straight away.  We had “I need to write $2 + x$ rather than $x + 2$ because we put numbers before letters” (even though we wrote down, “When multiplying, write numbers before letters”).  We had to reinforce which operations are and are not commutative, so what the difference is between $\frac{x}{6}$ and $\frac{6}{x}$.  There were ideas that were raised that I normally encounter further down the line (into GCSE) and that I hadn’t even thought of in my planning.  Take “Add five to a number then divide by 2”.  Many students wrote $\frac{(x + 5)}{2}$.  When we did Order of Operations earlier in the year we looked at calculations like this and the implication of brackets with the fraction bar.  My colleague taught the first lessons on using brackets and frequently warned the students away from “overzealous brackets”, teaching them to be aware of when brackets are not needed as well as when they are.  As soon as I wrote $\frac{(x + 5)}{2}$ on the board there were hands shooting up to tell me about the overzealous brackets.

I’ve enjoyed these lessons particularly because I’m starting to see how things we made explicit earlier in the year with number, specifically so that we could come back to them when we started algebra, have had an immediate positive effect.  I’ve also enjoyed these lessons because, by forcing myself to think about a scheme of work from scratch, I’ve been able to structure the students’ learning incrementally, building in time for things I never would have done in such detail before, and I’m excited to see how it pays off.

1. Matt says:

“As a side point, I am also very keen not to introduce the idea of substitution for a long time yet: I don’t want anyone to think that a letter has to have a fixed value associated with it.”

I found this comment interesting, would you mind expanding in more detail? Personally I like to teach substitution as one of the first aspects of algebra since then pupils can always use it to check for themselves whether or not two expressions are equal. For example, they can convince themselves that 2a + 3b – 4a = 3b – 2a no matter what the values of a and b are.

I have found it helpful framing questions like: what is the value of 3x + 2 when x = 7? What about when x = 10? Or -1? Etc so students know that x could take any value and it’s not fixed.

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• I don’t know if there’s a right or wrong here. My thought process is this:
1. Many students struggle to work abstractly because they are so tied to numbers.
2. When we solve equations, ‘x’ always has a fixed value.
3. If algebra is about generalisations of number I want them to think about concepts that hold for number (such as commutativity) and apply them to algebraic expressions without the need to test with numbers in order to verify what they’re writing (this is rather fuzzily explained, I know).
4. We’re using the identity symbol from the start. If I use the equals and identity symbols together they’ll be huge confusion.
5. We can talk about testing still. If “+ 2 then x 3” is identical to “x 3 then + 6” then we can test with some numbers. What I’m not doing is lessons on substitution.

Hope that makes a little sense.

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2. Matt says:

Yep makes sense, thanks!

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3. […] Algebra: you use the letter ‘x’ more than you have ever done in your whole life!, by Jemma […]

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4. Nice piece; I just finished my first year of teaching algebra. In the US, we start the basics of algebra in Year 8 (what we call 7th grade), and those students who qualify then take algebra in year 9. Those who don’t qualify take a rather boring year of math (or maths as you say) that has some algebra in it as well, but only linear equations up to systems of linear equations. I taught both year 8 and 9 this year and it was the algebra class for year 9. There’s a definite need for “interleaving” so they can recognize that 1/4 of x = x/4 without having to resort to “It’s the same as 1/4 times x/1, you remember how to multiply fractions don’t you?” Also, when we get to quadratics, there is always someone who forgets that x^2 = 2x is solved by putting in standard form and factoring.

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• The x/4 = 1/4 of x (and similar) is a huge one that students often don’t realise. This is why I believe you have to practise these things explicitly. Many, many kids will never just “realise” or pick it up as they go along. I’m not prepared to leave these things to chance.

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• I know – your thoughts on maths teaching are very similar to mine.

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5. Have you seen the Dragon Box app? Amazingly clever puzzle game that teaches algebra without you realising you are doing algebra.

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• Hold on. Just realised you’re Richard Vadon from More or Less, so I need to stop and give kudos to you for just about *the best* programme on the radio!

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