Kiss those Math Headaches GOODBYE!

Archive for the ‘Simplifying Expressions’ 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.

 

 

 

 

 

 

 

 

 

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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.

 

 

 

 

 

 

 

Factor Quadratic Trinomials, Part 2


This video shows the fastest and easiest way I know of for factoring quadratic trinomials. Give it a watch and see if you agree.

“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 #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 …

 

 

 

How to Remove (“Unpack”) Algebraic Terms from Parentheses


As you’re probably aware, I’m a big believer in using stories to bring math to life. Especially when you’re teaching tricky concepts, using a story can be the “magic switch” that flicks on the light of understanding. Armed with story-based understanding, students can recall how to perform difficult math processes. And since people naturally like stories and tend to recall them, skills based on story-based understanding really stick in the mind. I’ve seen this over and over in my tutoring.

Stories from My Tutoring Work

The kind of story I’m talking about uses an extended-metaphor, and this way of teaching  is particularly helpful when you’re teaching algebra. Ask yourself: what would you rather have? Students scratching their heads (or tearing out their hair) to grasp a process taught as a collection of abstract steps? Or students grasping  a story and quickly seeing how it guides them in doing the math? I think the answer is probably pretty clear. So with this benefit in mind, let’s explore another story that teaches a critical algebraic skill: the skill of  “unpacking” terms locked inside parentheses.

To get the picture, first imagine that each set of parentheses, weirdly or not, represents a corrugated cardboard box, the kind that moving companies use to pack up your possessions. Extending this concept, the terms inside parentheses represent the items you pack when you move your goodies from one house to another.  Finally, for every set of parentheses (the box), imagine that you’ve hired either a good moving company or a bad moving company. (You can use a good company for one box and a bad company for a different “box” — it changes.) How can you tell whether the moving company is good or bad? Just look at the sign to the left of the parentheses. If the moving company is GOOD, you’ll see a positive sign to the left of the parentheses. If the moving company is BAD, you’ll spot a negative sign there.

Here’s how this idea looks:

+ (    )     The + sign here means you’ve hired a GOOD moving company for this box of stuff.

– (    )     This – sign means that you’ve hired a BAD moving company to pack up this box of things.

Now let’s put a few “possessions” inside the boxes.

+ (2x – 4)  This means a GOOD moving company has packed up your treasured items: the 2x and the – 4.

– (2x – 4)  Au contraire! This means that a BAD moving company has packed up the 2x and the – 4.

[Remember, of course, that the term 2x is actually a + 2x. No sign visible means there’s an invisible + sign before the term.]

What difference does it make if the moving company is GOOD or BAD? A big difference! If it’s a GOOD company, it packs your things up WELL.  Result: when you unpack your items, they come out exactly the same way in which they went into the box. So since a good moving company packed up your things in the expression:  + (2x – 4), when you go to unpack your things, everything will come out exactly as it went in. Here’s a representation of this unpacking process:

+ (2x – 4)

=      + 2x – 4

Note that when we take terms out of parentheses, we call this “unpacking” the terms. This works because algebra teachers fairly often describe the process of taking terms out of (   ) as “unpacking” the terms. So here’s a story whose rhetoric  matches the rhetoric of the algebraic process. Convenient, is it not?

Now let’s take a look at the opposite situation — what happens when you work with a BAD (boo, hiss!) moving company. In this case, the company does such a bad job that when you unpack your items, each and every item comes out  “broken.” In math, we indicate that terms are “broken” by showing that when they come out of the (  ), their signs,  + or – signs, are the EXACT OPPOSITE of what they should be. So if a term was packed up as a + term, it would come out as a – term.  Vice-versa, if it was packed up as a – term, it would come out as a + term. We show the process of unpacking terms packed by a BAD moving company, as follows:

– (2x – 4)

=      – 2x + 4

And that pretty much sums up the entire process. Understanding this story, students will be able to “unpack” terms from parentheses, over and over, with accuracy and understanding.

But since Practice Makes Perfect, here are a few problems to help your kiddos perfect this skill.

PROBLEMS:

“Unpack” these terms by removing the parentheses and writing the terms’ signs correctly:

a)  – (5a + 3)

b)  + (5a – 3)

c)  – (– 3a + 2b – 7)

d)  + (– 3a + 2b – 7)

e)  6 + (3a – 2)

f)  6 – (3a – 2)

g)  4a + 6 + (– 9a – 5)

h)  4a + 6 – (– 9a – 5)

ANSWERS:

a)  – (5a + 3)   =   – 5a – 3

b)  + (5a – 3)  =  + 5a – 3

c)  – (– 3a + 2b – 7)  =  + 3a – 2b + 7

d)  + (– 3a + 2b – 7) = – 3a + 2b – 7

e)  6 + (3a – 2)  =  + 3a + 4

f)  6 – (3a – 2)  =  – 3a + 8

g)  4a + 6 + (– 9a – 5)  =  – 5a + 1

h)  4a + 6 – (– 9a – 5)  =  + 13a + 11


Josh Rappaport is the author of five books on math, including the Parents Choice-award winning Algebra Survival Guide. If you like how Josh explains these problems, you’ll certainly  like the Algebra Survival Guide and companion Workbook, both of which are available on Amazon.com  Just click the links in the sidebar for more information! 

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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