*(this follows on from my last post about the order of operations and the advantages of the acronym GEMS)*

I had a revelation (or, blinding flash of the bleeding obvious…) in my third or fourth year of teaching: when solving equations*, we are inverting the order of operations.

If that is crushingly obvious to you, please enjoy a smirk at my expense, but be aware this might be an issue for your colleagues and they could benefit from your expertise!

If not, I’m glad it’s not just me. Let me explain:

To evaluate 10+3 x 5², our order would be:

Exponents: 10+3 x 25

Multiplication and Division: 10+75

Subtraction and Addition: 85

If it were the other way around, as an equation (10+3 x *x*²=85) we would ‘know’ (more on this later…) to follow the following order:

Subtract 10: 3 x *x*²=75

Divide by 3: *x*²=25

(Eliminate) Exponents: *x* = 5, *x* = -5

In other words, we ‘know’ to reverse the order of operations. But we don’t say this to the children!

This implicit knowledge (or unknown known, for the Rumsfeld revivalists at the back) underpins the function machines approach to solving equations.

I have tried to conceal the origin of the image below– the teacher who shared this image from their work is generous to have done so! – and have included it to illustrate what I’m describing.

Using a function machine is a common approach for the early stages of teaching children to solve linear equations, rewriting

3*x* + 5 = 11

as

*x* -> x3 -> +5 -> 11

and then teaching the children how to ‘go backwards’ (i.e. use the inverse).

There are some problems with this.

- At worst, it collapses for harder questions (setting up a machine for unknown on both sides or questions of the type (
*x*+3)/(3*x*-4)=7 or even 13 –*x*= 7 is a tortuous process and teachers have to furrow their brows to prepare it. Pity the novice learners!). - At best, it is awkward when showing operations such as squaring.
- Eventually they will still have to learn to balance to solve (‘change side change signs’ and *shudder* ‘magic bridge’ are not defensible, in my opinion, and you should drop them*.)
- More fundamentally, this structure still doesn’t explain how to set it up successfully on their own. It doesn’t explain or teach anything, it is only a model for visualising. Structures are lovely, and can be illuminating once you know what is going on, but unless they can ‘see’ the order in which the equation was ‘built’ in, they will be trapped with random guessing when putting operations into the machine. Without explicit instruction about interpreting the order for setting up the equation, I think most children would make the reasonable inference that, if

3x + 5 = 11 becomes x -> x3 -> +5 -> 11

Then

5 + 3x = 11 becomes x -> +5 -> x3 -> 11

Because ‘5 happened first.’

Of course, you might have already spotted (or have already explicitly known) that the order of ‘building’ follows the order of operations: in 5 + 3x = 11 we can see that happened first because multiplication takes precedence over addition. From this we can conclude that, to solve the equation we must do the inverse of the order of operations: invert the addition before inverting the multiplication.

My bet is that most teachers, particularly those in their first few years of teaching, haven’t realised this and certainly haven’t communicated it to the children.

**Tried and Tested: SMEG **

Our approach has been:

- Teaching the order of operations in Y7 using GEMS, and getting children to be confident and fluent with it for several months, including with negatives and fractions and forms such as 3(4+5×2-20)/2. We leave algebra until the end of Y7/start of Y8 (once their number work and mental work is secure, especially around fractions, factors and directed numbers).
- Spending a lot of time on expressions and collecting like terms and manipulation
- Teaching children to form expressions, using GEMS to set it up correctly (e.g. rigorously teaching and checking they can differentiate between “I think of a number, I add 4 and then multiply the result by 3” and “I think of a number, multiply it by 3 and then add 4” or “I think of a number, square it, add 5 and divide by 2” and “I think of a number, divide by 2, square it and then add 5” and so on)
- Introducing one-step equations for 2-4 lessons (!), depending on the group, with a big focus on balancing and
**no decision-making about ‘what to do first**.’ The numbers used are deliberately NOT ones that facilitate mental calculation; the aim is to make it essential to learn to balance. We also teach checking methods at this stage and illustrate the idea of balancing with visuals (usually just images; an actual balance seems to be unfamiliar to this generation of children and hasn’t made any difference to their understanding, in my experience). We would also include examples where they have to simplify first, such as 2a + a -6a + 7a = 11 – 4 – 2, and so on. - Teach technical terms, such as eliminate, unknown, isolate, balance and inverse. This allows them to compress more ideas into an explanation and reduces the chance of them making incorrect leaps in logic or imagining patterns that aren’t there.
- Introducing a range of 2-steps and explicitly teach “We use GEMS to calculate, we use SMEG*** to solve (i.e. undo calculations)”. We then show them how to use SMEG to undo the calculation.

e.g.

31 = 3x – 5

Would be explained as:

*[Can we simplify either side? – this twist wouldn’t be in the first few examples and checks, but introduced soon]*

*What are we trying to isolate? The unknown, x. *

*What do we need to eliminate to do this? The operations on x’s side, which are x3 and +5. *

*S is first in SMEG: is there a subtraction or addition to eliminate? Yes, the +5. Eliminate that from both sides. *

*We have eliminated everything in the ‘S’ group. Is there anything in the M group, a multiplication or a division? Yes, the x3. Eliminate that from both sides. *

*The unknown is isolated; we’re done. *

(As I mentioned before, we try to avoid easy ‘in your head’ questions when teaching a process or a decision – although some will be shown after the first few examples so that they can see their intuition is correct – and try to avoid the most obvious layouts (e.g. not having most examples in the form ax + c = b) ).

This works even as the question becomes more complex:

(3-2a)/5 = -6

S: there is an addition (the +3 is added to -2a; we would have taught that idea already), but it is in a Group, so it needs to be dealt with at the end.

M: there is a division, we will eliminate that. Multiply both sides by 5.

3 – 2a = -30

The group is gone, we can start SMEG again.

S: There is an addition, not in a group. Eliminate that:

M: There a multiplication left, so eliminate that by dividing both sides by -2 and simplifying:

a = -33/-2 = 33/2

Unfortunately, we still have to teach that there is a special ‘first step’ with an unknown on both sides (that is, eliminate the smaller unknown). We explicitly teach them to distinguish between cases of one unknown and unknown on both sides, such as

3a +2 +a = 5

and

3a + 2 = a + 5

This might sound like a very intense level of detail! It’s been the product of several years of refining and trying to find out where the children come up against difficulties or their own reasoning and inferences can lead to errors or frustration. We’re pretty happy now that this approach is leading to success for close to 100% of children (most of our pupils would say maths is one of their favourite subjects, and equations are often given as an example of something they find easy to do).

(This photo is from 2015, when this pupil was in Y8 and much less confident with maths. It’s such a pleasure to see her now, as she is so much more assured – although still hilarious and expressive! – and has some really creative approaches. That said, I think a clear structure allowed her to achieve early success and self-belief, and her more methods are still underpinned by reliable processes such as balancing with SMEG).

*this is assuming the equation has a single unknown (i.e. no quadratics with terms). I’m focusing here on the foundational phases of solving an equation.

** 1) It conceals why anything works. 2) You aren’t changing signs, you are inverting the operation! 3) it collapses as soon as you aren’t eliminating addition or subtraction.

*** So, obviously SMEG makes it sound gross. I usually mention the brand of fridges as an offhand comment so that they do unwanted wondering. I was tickled to learn that SMEG (fridges) stands for Smalterie Metallurgiche Emiliane Guastalla (“*Emilian metallurgical enamel works of Guastalla*“)