Teaching Equations big-picture style

I started Equations with my year 8 maths. I love equations – probably why I love Algebra – and have decided to buck the textbook trend.

Lesson

Instead of teaching technique by technique, I decided to go with the big picture. And really, students in year 8 already know this big picture. I started the lesson by asking them “What do you already know about equations?” and listed these 4 main “principles” :

1. There’s an = sign (expressions don’t; it’s a number sentence)
2. Left hand side is the same as the Right hand side (LHS = RHS) in value, i.e. size and direction
3. Solve for the unknown (x) by isolating it or making it the subject (sentences have subjects) AND
4. Use inverse or opposite operations

After this discussion of less than 10 minutes, the students doing worksheets that spanned at least 3 sections of their textbook. I pointed out that the answers were on the back of the worksheet. Why? They replied that the focus was on the working out. (Conditioning works) ðŸ™‚

Using these principles, you don’t really need to distinguish whether or not it’s one-step or two-step or “what do you do to x to get x +3?”. I had a student who said she couldn’t answer the latter (from the textbook) when she’s already solving problems like x / 3 = 8, or even 2x + 15 = 6. That’s because the reason we use inverse operations is because we want to isolate x ; that’s the context.

I also had a student ask how they know if they’re on the right track. The textbook says – and rightly so – check by substitution. Instead, I said well check whether the principles still hold, i.e. is LHS = RHS? If not, then you’re definitely off-track. So really, check correctness through checking by principles and by substitution.

I purposely did not teach move it to the other side and change its direction (e.g. +5 in the LHS becomes -5 in the RHS). Sure, that’s what eventually happens after applying inverse operations but I insist in not doing this shortcut….not until I’m convinced that the students know why.

Rationale

Those who’ve been following me in this learning (and teaching) blog would recognise my constant battle with keeping maths relevant and meaningful (even fun) for those who are not naturally good at it. Maths has a lot of techniques and short-hand ways of doing things which, though can be followed step-by-step, often detracts from the beauty of maths itself and its many applications.

In doing this approach I’m hoping to have more time for practice (or drill) and (equation) fun (so many interactive games out there). I hate the term drill and use ‘practice’ because maths is a language too and fluency requires practice, not just understanding. For example, I know and understand the rudiments of Spanish and French grammar but nowhere near as fluent as I am in English (I am ESL) because I don’t practice those two as much as English.

I am hoping this will work, i.e. I don’t know if it will but I sure hope so.

8 thoughts on “Teaching Equations big-picture style”

1. Bon Crowder says:

If they get the big pic – which 8th graders are still un-polluted enough to – that’s awesome!

Bon

2. Bon Crowder says:

Me again – revisiting. Teaching a dev math class in college this semester. I think I’m going to try this with them.

My teaching philosophy is much different now after publishing MathFour.com for a year and interacting with parents and teachers, like you, Malyn.

It’s so much harder teaching understanding instead of just content. Not only because it’s challenging, individualized, and takes serious thought, but also because college students at this point are so totally conditioned against it.

Wish me luck, pray for me and/or sacrifice a chicken. I’ll need it all.

• malyn says:

Thanks Bon. All the best.

I have this new (to me) theory that maths should be thought like other languages. Content is important (as in vocabulary, slang, nuances). Structure is also important (as in grammar). Fluency is important (as in reading, writing and speaking).

Deliberate practice is important. Less important are the ‘strands’ split out like so (Number, Algebra, Data, Geometry).

What I need to do is think more about this and see if I can flesh this out a little bit more….for my sake at least.

• Bon Crowder says:

I like the analogy. I would say, though, that instead of a language where you’re learning grammar/syntax, suppose you already know it and you’re studying the LITERATURE in the language.

It’s the metaphors, the rhymes and the alliterations, the onomatopoeia that make literature wonderful. And that allow the students to write their own – regardless of the crudity of it.

After you see that you express yourself in these wild and wonderful ways, making sure you have commas in the right place is just a matter of editing.

Now THAT’s math!

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