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Posts tagged ‘Algebraic Expressions’

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.

 

 

 

 

 

 

 

 

 

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“Algebra Survival” Program, v. 2.0, has just arrived!


The Second Edition of both the Algebra Survival Guide and its companion Workbook are officially here!

Check out this video for a full run-down on the new books, and see how — for a limited time — you can get them for a great discount at the Singing Turtle website.

 

Here’s the PDF with sample pages from the books: SAMPLER ASG2, ASW2.

And here’s the website where you can check out the books more fully and purchase the books.

 

 

 

 

 

 

 

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 #1: – 1^2 vs. (–1)^2


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 …

 

 

 

The “Unknown” Order of Operations


Talk about a major point that’s usually unspoken …

We make such a big deal out of the Order of Operations in Algebra, and yet there’s a second order of operations, equally important but seldom mentioned.

First, to clarify, the standard Order of Operations (caps on the two O’s to indicate this one) helps us simplify mathematical expressions. It tells us how to take a group of math terms and boil them down to a simpler expression. And it works great for that, as it should, as that’s what it’s designed for.

EXAMPLE:  this Order of Operations tells us that, given an expression like:  – 2 – 3(4 – 10), we’d first do the operations inside PARENTHESES to get – 6, then we’d MULTIPLY the 3 by that – 6 to get – 18. Then we would SUBTRACT the – 18 from the – 2, to get 16. You know, PEMDAS.

But it turns out that there’s another order of operations, the one used for solving equations. And students need to know this order as well.

In fact, a confusing thing is that the PEMDAS order is in a sense the very opposite of the order for solving equations. And yet, FEW people hear about this. In fact, I have yet to see any textbook make this critical point.  That’s why I’m making it here and now: so none of you  suffer the confusion.

In the Order of Operations, we learn that we work the operations of multiplication and division before the operations of addition and subtraction. But when solving equations we do the exact opposite: we work with terms connected by addition and subtraction before we work with the terms connected by multiplication and division.

Example: Suppose we need to solve the equation,
4x – 10 = 22

What to do first? Recalling that our goal is to get the ‘x’ term alone, we see that two numbers stand in the way: the 4 and the 10. We might  think of them as x’s bodyguards, and our job is to get x alone so we can have a private chat with him.

To do this, we need to ask how each of those numbers is connected to the equation’s left side. The 4 is connected by multiplication, and the 10 is connected by subtraction. A key rule comes into play here. To undo a number from an equation, we use the opposite operation to how it’s connected.

So to undo the 4 — connected by multiplication — we do division since division is the opposite of multiplication. And to undo the 10 — connected by subtraction — we do addition since addition is the  opposite of subtraction.

So far, so good. But here’s “the rub.” If we were relying on the PEMDAS Order of Operations, it would be logical to undo the 4 by division BEFORE we undo the 10 with addition … because that Order of Operations says you do division before addition.

But the polar opposite is the truth when solving equations!

WHEN SOLVING EQUATIONS, WE UNDO TERMS CONNECTED BY ADDITION AND SUBTRACTION BEFORE WE UNDO TERMS CONNECTED BY MULTIPLICATION OR DIVISION.

Just take a look at how crazy things would get if we followed PEMDAS here.

We have:  4x – 10 = 22

Undoing the 4 by division, we would have to divide all of the equation’s terms by 4, getting this:

x – 10/4 = 22/4

What a mess! In fact, now we can no longer even see the 10 we were going to deal with. The mess this creates impels us to undo the terms connected by addition or subtraction before we undo those connected by multiplication or division.

For many, the “Aunt Sally” memory trick works for PEMDAS. I suggest that for solving equations order of operations, we use a different memory trick.

I just remind students that in elementary school, they learned how to do addition and subtraction before multiplication and division. So I tell them that when solving equations, they go back to the elementary school order and UNDO terms connected by addition/subtraction BEFORE they UNDO terms connected by multiplication/division.

And this works quite well for most students. Try it and see if it works for you as well.

Josh Rappaport is the author of the Algebra Survival Guide and Workbook, which together comprise an award-winning program that makes algebra do-able! Josh also is the author of PreAlgebra Blastoff!, an engaging, hands-on approach to working with integers. All of Josh’s books, published by Singing Turtle Press, are available on Amazon.com

Visual Symbols Slay the Distributive Property


Progress …

I made some progress yesterday helping a boy understand the distributive property, and it was mostly due to the use of visual symbols.

I’d like to share the process of my tutoring, for it shows how important it is to break an algebraic procedure into its constituent “baby” steps. I’d also like to share the process because I used a fairly unusual technique — using visual, non-algebraic symbols to take the place of algebraic symbols. As you’ll see, this technique has some interesting advantages.

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How to Decrease Algebraic Mistakes – Part 5


This is the fifth in a series of posts on how to help students make fewer mistakes in algebra.

No Mistakes

Let's Reduce Mistakes in Algebra!

So far I have introduced a form of notation I have developed, the double-slash, which looks like this:

//

and I have described some of the ways that students can use it.

I’ll continue the conversation by showing how this notation can help students combine like terms with greater care.

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