Kiss those Math Headaches GOODBYE!

Posts tagged ‘Help Students Understand Mistakes’

Algebra Mistake #2: How to Understand the Difference between A x A and 2 x A without Confusion

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 …


Algebra Mistake #1: How to Understand the Difference Between -1^2 and (-1)^2 without Confusion

Welcome, welcome, welcome to my series on COMMON ALGEBRA MISTAKES!

We’re going to have some fun spotting, analyzing, dissecting, exploring, explaining and fixing those COMMON ALGEBRA MISTAKES, the ones that drive students and teachers UP THE WALL!

I’ve had so much experience tutoring that I find these mistakes fascinating, and I intend to share my (ok, bizarre) fascination in this series of videos.

Also, be aware that I’m very much OPEN to suggestions from you folks on mistakes that you’d like me to explore. I highly value the experience and wisdom of you students and educators, and I want to do all I can to work with you to un-earth the mistakes of algebra, and bring them to the light of day so we can find ways to stay out of their way!

Here’s the first video on these mesmerizing mistakes. Could any mistake be more classic than this very one? I doubt it. But watch the video and form your own opinion …




Turtle Talk – January 2010

Note: Below is a copy of my January 2010 ezine, Turtle Talk.

If you’d like to subscribe to Turtle Talk to get it as soon as it gets published, just go to this site:

There is a cute animation of moving turtles.

Turtle Talk
— a newsletter —
January 2010
Vol. XIII, Issue #1


“If you think dogs can’t count, try putting three dog biscuits
in your pocket and then giving Fido only two of them.”
— Phil Pastoret

MathChat Blog Update:

In case you did not know, I have been spending a fair amount of time
writing on my blog, and there are many articles and ideas over there.
I have articles on many topics. Here’s just a small sampler:

Multiplication tricks
Mental math shortcuts
Dividing a fraction by a fraction using the “bologna-cheese” sandwich
Using color to elucidate ideas in geometry
Using colors to clarify concepts in algebra
How to solve algebraic mixture problems
Using “master equations” to make word problems less scary

To check out the blog, just visit:

Feel free to leave comments, too.

Twitter presence too:

I send out tweets about blogs and other issues of interest
to math teachers and students. If you’d like to FOLLOW ME,
just go to twitter, type in joshmathguy and sign up to follow.

January Problem of the Month

Samantha gets 12 out of 14 problems correct on a test.
Then she gets half of the remaining problems correct.
If every problem is worth the same number of points,
and Samantha ends up getting a score of 60% on the test,
how many problems are on the test altogether?

To get credit you must show your work as well as get the problem right.

First person to send in the right answer gets a FREE COPY of any
Singing Turtle Press book.

Send your answers to:

In the next newsletter I’ll name the first 9 others who get this

DON’T FORGET: when you submit answers for the Problem
of the Month, please write
on the Subject line of your email’s header and, knowing that
it is optional, feel free to share:

— your age (if a child)
— your state or hometown
— status: student, teacher, tutor, etc.
— whether in school or homeschooled
— anything else about you of interest


Discussion: A Way to Help Students Notice
& Understand the Mistakes they make in Algebra

Have you ever noticed that students often have a hard time spotting and understanding what is actually wrong — when they make mistakes in algebra?
One reason is that — when students enter the “Land of Xs and Ys,” they have little intuition as to what should happen. It’s as if they know that “they’re not in Kansas anymore,” so as far as they’re concerned, there could be “munchkins” around the corner. With such a level of uncertainty, they have little idea what to expect.
And if students don’t know what SHOULD happen, they also don’t know what SHOULD NOT happen. However, it’s helpful to remember that there is still one way to reach such early algebra students — appeal to their understanding
of how numbers work.
Here’s an example of how you can use this to your advantage:
Suppose a student makes the following mistake when solving an equation:
3x – 4 – 2x = 12
+ 2x + 2x
5x – 4 = 12
What’s the student is doing wrong? Adding 2x to the SAME SIDE of the equation TWICE, instead of adding it to BOTH SIDES of the equation ONCE.
Of course, if you asked the student why s/he did this, the student would probably offer up some half-true, yet misguided phrase like, “you have to add 2x on both sides.”
What you do here is get a “change of venue.” Move the problem from the field of algebra to the field of arithmetic.
Simply substitute numbers for letters. Ask the student if the following steps would make sense:
10 – 3 = 7
 + 3 + 3
13 = 7
Generally, students will see right away that the answer is wrong. And that will alert them that something is wrong up above. And usually students can see that it is weird to add 3 twice on the left side.
Once they get that idea — either on their own, or with a little prodding — go back to the algebraic situation and ask them if they can NOW see what’s wrong. Very often they can.
When I was studying to become a teacher long ago, I learned a key idea about how people learn: to learn anything we need to relate new information to something we already know and understand. In essence, that’s what I am suggesting we do when students make procedural mistakes in algebra. Students know arithmetic (hopefully!). So use that. Put the algebra in an arithmetic context. Usually students will see the problem in the arithmetic setting. Then they’ll realize that algebra, even though it looks different than arithmetic, still works a lot like arithmetic. Getting that idea across will help students greatly as they continue in their study of algebra.
In any case, feel free to give this a shot, and see how it works for you. And feel free to share any feedback on how this works for you.

Online Algebra now Available from Author
of the Algebra Survival Guide

Josh is now offering services online through SKYPE:
algebra tutoring sessions, and entire Algebra 1 classes.

If interested, please send email to:,
telling what kind of help you are looking for.


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Well, that’s all for this month, though I will continue to blog and send out tweets on items of interest.

Have a great month, and I’ll be back in February.

— Josh

Tutor Tales — Algebra/Arithmetic Connections

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.