Ever thought this after you got back a math test … ?
“Why did I do that? I used a rule where it doesn’t apply!”
Yep, that’s exactly what we’re looking at in Algebra Mistake #3, a case of “overgeneralizing.”
The situation we’re dealing with involves over-generalizing everyone’s “favorite” property, the distributive property!
How’s that? Well, you’re supposed to use the distributive property when a number multiplies terms inside parentheses.
But sometimes students get a little bit — shall we say — “carried away” — and use the distributive property principle in other situations, too. The results are a tad bit comic, if you’re the teacher, but not so funny if you’re the student and you’ve made the mistake 19 times on a test with 20 problems.
Anyhow, after you watch the following video you shouldn’t have to worry about this again because we’ll get the two wires in your mind untangled so you never make this mistake again. So just relax, watch and learn.
And oh yes, don’t forget that we’ve provided some practice problems at the end of the video to help you make sure you’ve got the concept nailed down.
Now that you’ve gotten a taste for the benefits of analyzing algebraic mistakes, it’s time to explore a second common mistake. This one is so common that nearly every student commits it at least once on the road to algebra success.
As you watch the video, notice how by thinking hard about two expressions, we can think this mistake through to its very root, thus discovering the core difference between two similar-looking algebraic expressions.
And along the road, we’ll learn a general strategy for decoding the meaning of algebraic expressions. What I like about this strategy is that you can use it to understand the meaning of pretty much any algebraic expression, and you’ll see that it’s not a hard thing to do. In fact, it just involves using numbers in a nifty way.
Best of all, students usually find this approach interesting, convincing and even a bit fun. So here goes, Common Algebra Mistake #2 …
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.