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Archive for the ‘Using stories to make math clear’ Category

Algebra Mistake #4: Combining Negative Numbers

Here’s a common mistake, and a very understandable one, too. Students need to combine two negative numbers, and they, of course, wind up with an answer that’s positive. Why? Because, they’ll say — pointing out that you yourself have told them this —  “Two negatives make a positive!”

This video gets to the root of this common misunderstanding by helping students understand exactly when two negatives make a positive, and when they don’t.


Make sure you watch the whole video, as there are practice problems at the end, along with their answers.











“Unpacking” Terms from Parentheses

How do you get math terms out of parentheses? And what happens to those terms when you remove the parentheses?

It seems like the process should be simple. But this issue often plagues students; they keep getting points off on tests, quizzes, homework assignments.  What’s the deal?

The deal is that there’s a specific process you need to follow when taking terms out of parentheses, and what you do hinges on whether there’s a positive sign (+) or a negative sign (–) in front of the parentheses.

But not to worry. This video on this page settles the question once and for all. Not only that, but the video provides a story-based approach that you can teach (if you’re an instructor) or learn (if you’re a student) and remember (no matter who you are). Why? Because stories are FUN and MEMORABLE.

So kick back and relax (yes, it’s math, but you have a right to relax) and let the video show you how this process is done.

And in customary style, I present practice problems (along with the answers, too) at the end of the video so you can be sure you understand what you believe you understand.






Algebra Mistake #3: (x + y)^2 = what?

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.



Algebra Mistake #2: Does a x a = 2 x a?

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 …


Reader Input on Slope Post

A longtime reader of Turtle Talk, Jeff LeMieux, of Oak Harbor, WA, sent in a suggestion based on today’s post on positive and negative slope. Jeff found a way to help students remember not only positive and negative slope, but also the infinite slope of vertical lines, and the 0 slope of horizontal lines … all using the letter “N.”

This is clearly a situation where the picture speaks more loudly than words, so I’ll just let Jeff’s submitted picture do the talking. By the way, to see this image even better, just double click it!


Slope Memory Trick

Thanks for putting this together and sharing it, Jeff!

How to Find the LCM for Three Numbers

Several readers have said they like my trick for finding the LCM described in the post “How to Find the LCM — FAST!” but wonder how to use the trick for finding the LCM for THREE numbers. Here is how you do that.

Essentially it involves using the same LCM trick three separate times. Here’s how it’s done.

Suppose the numbers for which you need to find the LCM are 6, 8, and 14.

Step 1)  Find the LCM for the any two of those. Using 6 and 8, we find that their LCM = 24.

Step 2)  Find the LCM for another pair from the three numbers. Using 8 and 14, we find that their LCM = 56.

Step 3)  Find the LCM of the two LCMs, meaning that we find the LCM for 24 and 56. The LCM for those two numbers = 168.

And that, my good friends, is the LCM for the three original numbers.

So, to summarize. Find the LCM for two different pairs. Then find the LCM of the two LCMs. The answer you get is the LCM for the three numbers.

Here are a few problems that give you a chance to practice this technique.

Find the LCM for each trio of numbers.

a)  10, 25, 30

b)  16, 28, 40

c)  14, 32, 40


The LCMs for each trio are:

a)  150

b)  560

c)  1,120

Josh Rappaport is the author of the Algebra Survival Guide and Workbook, which comprise an award-winning program that makes algebra do-able! The books break algebraic concepts down into manageable chunks and provide instruction through a captivating Q&A format. Josh also is the author of PreAlgebra Blastoff!, which presents an engaging, hands-on approach (plus 16-page color comic book) for learning the rules of integers. Josh’s line of unique, student-centered math-help books is published by Singing Turtle Press and can be found on

How to factor out the GCF with stories

At various times when I tutor, I find myself explaining the same concept repeatedly over several weeks.

Recently it has been that way with — drumroll please … factoring out the GCF from polynomials.

One reason I’m getting so much “experience” with this is that many kids find this process very difficult. It’s not hard to see why. First of all, the process of finding the GCF is, in itself, somewhat tricky. Then too, factoring out the GCF from all terms in a polynomial is a multi-step process; students need to get each step right, and then they need to perform the steps in the correct order. If that alone were not enough to tax children’s minds, students also get confused by the difference between how to multiply pure numbers (constants and coefficients), and how to multiply variables. (more…)