Here’s an idea I came up with today for helping students understand more deeply the mistakes they make in algebra.

One thing that makes algebra difficult is that students have, basically, no sense as to whether something is true — or not — when they look at algebra. They have virtually no intuition about this. However, they do have intuition as to whether or not things are correct in arithmetic.

But we can use this idea to help students understand algebra. For example, we can use this approach to help students understand what is “wrong” when they make mistakes in algebra.

For example, let’s say that a student makes the following mistake:

3x – 4 – 2x = 12

+ 2x + 2x

5x – 4 = 12

What is the student is doing wrong? The student is adding 2x to the same side of the equation two times, instead of adding it to both sides of the equation.

How can we help the student see that this is wrong?

Change it to an arithmetic situation. Ask them is the following makes sense:

9 + 3 = 12

– 3 – 3

6 = 12

They will see that this is wrong because they know that the addition is wrong. What is more, they will get the general idea that it makes no sense to subtract 3 twice from the same side of the equation.

This mistake — in the algebra — will make little sense unless you do something like this, something they can grasp.

I agree 100%. I usually suggest that students substitute small prime numbers greater than 1 for the variable(s) in the problem, then see if the operation they were contemplating will work or not. This will work with just about any algebraic manipulation: exponents, distributing, solving equations, solving inequalities, etc.

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