I’ve been refining the way I teach equations in science for a while now, and think I’ve found a way that is reaping good results. The key principles are as follows:

- No matter what the content or equation, I always get pupils to follow one method. It works for every equation in Biology, Chemistry and Physics.
- Drill pupils on equations, symbols for each variable and units.
- Always interleave equations during practice to ensure pupils have the opportunity to select the right equation.
- Model how to think, lay out the working and check the answer using a visualiser.
- Pupils learn only one version of a formula (and NO formula triangles).

**Method**

Here are the steps:

- Box the value to be calculated & underline given values
- List all values with units
- Write equation which unites values
- Check units – convert if necessary
- Substitue
- Combine number if possible, then rearrange
- Calculate final answer & write units

Here is what this looks like in practice:

The rationale (written from the perspective of a model-thinking pupil):

- Boxing the unknown value makes the objective of the calculation clear to me: the purpose of this calculation is to work out what the value of this boxed variable is. Underlining all the known values ensures that I read the question properly and capture all of the information.
- I then list all of the values using the letter that represents each variable. I will write the number and units for all given values, put a question mark next to the letter of the variable that I am trying to find out the value of. I have all of the information I need to succeed in this calculation now.
- I now look at all of my variables and think to myself: which formula do I know that uses these variables? I am hoping there is a formula which includes the unknown variable (‘?’) and at least some of the others have a value. I write this formula down.
- I know that every formula will only work if each variable has the correct units. So I will check my list of values, and convert any units that do not match the units that suit the formula.
- I will re-write the formula underneath the first, but this time I will substitute any letters for numbers. For the unknown variable, I will keep the letter that is in the formula. I will get one mark for correct substitution.
- Having substituted, if there is more than one number on one side of the equation, I can combine these. This normally leaves three values (two known, one unknown), which I can easily re-arrange, to make the unknown the subject.
- I calculate the answer and make sure I write the units (checking against the formula).

**Pedagogy Rooted in Cognitive Science**

**Drill**

Pupils lay their working out in exactly the same way for all equation calculations. This consistency helps them to automate the process and free up their working memory to focus on the specifics of the problem at hand. I continuously drill pupils on the formula, the symbols for each variable and the units to ensure these facts transfer into their long-term memory. This means most of the steps discussed above are simple acts of retrieval.

I only get my pupils to remember one version of a formula. This is because, firstly I do not want pupils to rely on formula triangles – I want them to be able to re-arrange. Besides, my method means re-arranging will only ever include three variables, one of which is unknown, and all pupils are capable of doing this. Secondly, the exam (certainly AQA) award a mark for correct substitution. Pupils are far more likely to substitute correctly into one formula than to risk re-arranging the formula incorrectly and losing all hope for any marks subsequently.

**Interleaving**

Once I have taught a few equations, with limited amounts of practice, I stop giving practice questions which require pupils to use the same equation. This is because pupils fall into habits mindlessly answering questions, since they can copy the format of the question that have answered previously. Instead, I vary the type of equation they must use. This has two benefits. First, it stops pupils answering questions mindlessly. Secondly, it provides them with the opportunity to practice selecting the correct equation, in addition to actually performing the calculation. This is the aspect of practice that most pupils who are proficient at the fundamentals in maths require. Therefore this aspect requires deliberate practice.

**Modelling**

For every new equation, I model for pupils exactly how to approach the question using the seven step method using a visualiser (live video which projects onto the board). I think out loud. I write in A4 lined paper for pupils to copy down and follow exactly how they should lay their working out. I make every step very explicit. After the first example, I ask pupils to do the next one on their own, and then model it again on the visualuser, this time, asking random samples of students to tell me what to do and why.

I then proceed to model a more complex example such as one where they are required to convert units, or one where rearranging is necessary. This builds up their confidence, but also prevents their working memories from becoming overloaded. If I expect them to attempt a calculation which requires unit conversion, re-arrangement, a standard form answer to a certain number of significant figures all in one go – they will be over-burdened.

Deliberate practice is all about breaking down a complex skill such as solving equations, into constituent steps, which are then practiced individually.

Before teaching this seven step method, I will make sure my pupils understand and practice substitution, units and unit conversions, standard form, significant figures etc. All separately, as necessary. Lots of this will not require much time since they will have covered it in maths (our science department is in talks with maths about when concepts are introduced). But never assume this!

**Pitfalls**

Pupils might not actually read the question in its entirety – rather they may just box the thing they are trying to find out, and then underline any number they see. This might mean pupils do a calculation without actually thinking about the context of the question. This might get them all the marks for most questions, but it is important for conceptual understanding that I model the thinking behind the calculation and remind pupils to do the same. If not, pupils might miss important clues about what the variable actually is. For example, I once wrote a question which said: ‘2 x 10^{4 }Joules is required to heat one kilogram of lead by 1^{o}C. 12000 g of lead was heated by 4^{ o}C. How much energy was required? Give your answer in standard form, including units.’ This stumped lots of pupils because they failed to recognise that the first value was not change in energy, but rather, specific heat capacity.

If a question has more variables than required in a formula, will pupils confidently pick the right formula? Practice of these types of questions can help overcome this pitfall.

Hope this quick guide to how I teach equations was useful – please comment/critique as this will help us all!

I like most of this article. But your example problem left out units, which is UNFORGIVABLE! I would take off 1/3 of the points for this problem. Not only that, but that technique does not allow you to properly solve some problems, because you don’t know which quantities you can combine. I highly recommend changing the picture…

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When I first started using this method to teach equations, I did include units, but found that a) with the step of converting units already tackled (step 4) and b) in the context of GCSE I haven’t yet found missing out units to be a hindrance, it wasn’t problematic. I realise the danger of limiting the application to this context…

But I do see its essential value in helping pupils gain a deeper understanding of units and their relationships. It also is an excellent way to check calculations. But I do feel adding units at this stage would require lots of explicit teaching of units and relationships, which is something we are yet to add to our schemes of work at my current school. It is definitely work in progress and on our agenda. Thanks for pointing this out though!

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Units are essential. Take a look at how equations are handled WITH units in the book

“Calculations In Chemistry — An Introduction” by Dahm and Nelson from WWNorton.

See the chapter on Gas Laws. I think those guys have it figured out!

— Nelson

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I echo comments above. Stage 5 should be to substitute a value not a number (i.e. number plus unit) This allows an instant check to be made by the user to see that the units tally. If not, there was possibly an error in rearranging or in unit conversion (stage 4)

I am aware that AQA awards marks for correct substitution even if there is an error in calculation. However, I would suggest rearranging the equation symbols before substituting values helps students recognise the relationships between variables rather than ‘flying blind’. In AQA physics there are a number (19?) of equations that candidates need to recall as these are not provided within the examination materials. Helping them to understand, rather than memorise these equations could contribute to successful outcomes.

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I think this is excellent! One thing I would recommend is to not plug in values for variables until the very end, and then to do so with units. There’s a lot of value in being able to solve for a variable in terms of other variables. More importantly, doing so sets students up for having higher level conversations about how different physical realities depend on each other.

For example, solving the combined gas law for final pressure: P2 = (P1*T2)/(V2*T1). If we solve for variables first, I can ask a question like, what happens to the final pressure when the final temperature is decreased? We can have conversations about direct and inverse relationships.

The same is true for more complicated scenarios. Like what is the coefficient of friction for a block on a slope of angle theta? Using a variety of equations, we end up with mu_k = cos(theta)*a/g. Again we can ask questions about the relationships between the variables or analyze the units – how come mu_k ends up unit-less? Or introduce an experimental problem: design an experiment to find mu_k between a surface and a block using a wide range of angles.

When students plug in values for variables (even if they plug in with units) before isolating, the *meaning* of their answer is lost.

I had a lot of success this year with giving my students a process much like the one you describe here. I added the step “solve in terms of variables” before plugging in: https://needtoknowscience.wordpress.com/2018/04/10/the-journey-begins/

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Thanks for your well-thought our comments! I completely agree with you on this – but i’ve come to the conclusion this works best beyond GCSE. At gcse, there are lots of pupils for whom this would make it harder. To get around this, I try to use the version of the formula where the relationships make most sense e.g. V = E/Q and have lots of discussions around units and the relationships.

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I suppose that’s true. I’ve seen the same in my introductory vs. advanced courses. I wonder if we could do better in our early algebra classes and give greater importance to manipulating variable expressions over numbers so students can be comfortable with it from the beginning.

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Yes that is true. But I don’t think it matters with the level of expected maths at gcse science. Pupils who substitute first are not at a disadvantage.

Another factor is the exam boards favouring substitution first – there is a mark awarded for correct substitution, that is not awarded after re-arranging if the rearranging is incorrect.

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Ah I see. We’re set up a little differently in the US

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