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Archive for the ‘Positive & Negative Numbers’ Category

Algebra Mistake #5: How to Combine a Positive and a Negative Number without Confusion

So, you’d think that combining a positive number and a negative number would be a fairly straightforward thing, huh?

Well, unfortunately, a lot of students think it’s easy. They think it’s too easy. They think there’s one simple rule that guides them to the very same kind of answer every time. And that’s exactly where they get into trouble.

The truth is that combining a positive and a negative number is a fairly complicated operation, and the sign of the answer is dependent on a nmber of factors.

This video reveals a common mistake students make when tackling these problems. it also shows the correct way to approach these problems, using the analogy of having money and owing money to make everything make sense.

So take a look and see if this explanation doesn’t end the confusion once and for all.

And don’t forget: there are practice problems at the end of the video. Do those to make sure you’ve grasped the concept.















Algebra Mistake #4: How to Combine Negative Numbers without Confusion

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.










“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.








How to Remove Math 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: 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 …




Making Sense of Inequalities

OK, teachers, homeschooling parents and tutors … raise your hand if you’ve ever felt uncomfortable when students pose that question about inequalities?

That question being:  why do we flip the inequality symbol when we multiply or divide by a negative number?

I’d have to, sheepishly, raise my hand.

So when I got asked that question once again last week, I decided to figure it out and come up with an answer that would help students understand this point.

What I came up with is that it’s easiest to explain this through a combination of examples and logic. First, the examples.

Let’s break the situation up into three cases. We could have inequalities in which the numbers on the two sides are A) both positive, B)  both negative, or C) one number positive, the other negative.

Let’s start with Case A. Suppose we start with the statement, 2 < 4

Now, multiply both sides by a positive number (let’s use 3), and we get:  6 < 12. Still true, right?

But take the original inequality and multiply it by a negative number (let’s use – 3), and we get: – 6 < – 12. Not true, right? But if we flip the sign, we do get a true statement:  – 6 > – 12

Case B. Now let’s start with two negative numbers in our true inequality:  – 4 < – 2 If we multiply both sides by a positive number (3 again), we get:  – 12 < – 6, which is again true.

But if we multiply this inequality by a negative number (– 3 again), we get: 12 < 6, which is obviously false. However if we once again flip the sign, we get a true statement:  12 > 6.

Finally, Case C. Now we start with an inequality that has both a positive and a negative number:  – 2 < 4. If we multiply both sides by  positive 3, we get:  – 6 < 12, which is still true.

But if we multiply both sides by our – 3 again, we get:  6 < – 12, which is once again false. And again we need to flip the sign to make it true:  6 > – 12.

So far so good, but this lacks the logic of an explanation. How can we bring in some logic and reasoning, to help students see why all of this stuff happens?

Here’s my — granted, informal — way of explaining this. When we multiply or divide a number by positive numbers, we don’t change its sign; if the number was positive, it stays positive, and if it was negative, it stays negative. But when we multiply or divide a number by a negative number, we do change its sign … either from positive to negative, or from negative to positive.

So the reason that we flip the inequality symbol must be related to the fact that — by multiplying or dividing both sides of the inequality by a negative number — we are changing the signs of both numbers in the inequality. But how exactly does this work?

The answer, it turns out, is rooted in the relationship between the absolute value of numbers and their relative sizes. For numbers that are positive, there’s one way to tell which number is larger … the number with the larger absolute value is the larger number. For example, comparing 4 and 12, we know that 12 is larger than 4 because the absolute value of 12 is larger than the absolute value of 4. But for numbers that are negative, the exact opposite is true. For two negative numbers, the number with the larger absolute value is actually the smaller number. For example, compare – 4 and – 12. Their absolute values are 4 and 12, respectively, but the number with the larger absolute value is in fact the smaller number, not the larger number. In this example, – 12 (with the bigger absolute value of 12), is in fact smaller than  – 4 (with the smaller absolute value of 4).

So the point to remember here is that there are two different relationships between the absolute values of numbers and the relative sizes of numbers. For positive numbers, the greater the absolute value, the greater the number. But for negative numbers, the greater the absolute value, the smaller the number.

This fact has an impact on inequalities where we change the signs of the numbers. Before changing the signs of the numbers, the numbers on the two sides of the inequality had one size relationship; one number was larger than the other (let’s say that Number A is, at this stage, larger than Number B). But when we multiply or divide both of these numbers by a negative, we flip the signs of both numbers. And by flipping the signs of both numbers, we change the size relationship of the numbers to each other. The one that was the larger one ends up being the smaller one, and vice-versa. So in our abstract example, if Number A was larger than Number B before their signs were changed, after both signs are changed, Number A will be smaller than Number B.

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  Just click the links in the sidebar for more information! 

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.


“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)


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  Just click the links in the sidebar for more information!