Where math comes alive

Algebra Survival Guide — Second Edition
by Josh Rappaport
illustrated by Sally Blakemore

The Algebra Survival Guide — Second Edition is here, but not yet released to the general public. Now’s your chance to order it at a 25% discount through April 10th. Just go to SingingTurtle.com.

But first, let me tell you about what’s new … a massive, 62-page chapter on advanced story problems.

It’s no secret that algebra gives students the ‘jitters,’ and word problems give them the ‘shakes.’ As a dastardly duo, the word problems of algebra are just about as nerve-wracking as anything in the teenage years.

The Algebra Survival Guide — Second Edition takes a hard look at algebra’s word problems and offers time-tested advice for cracking them. With a new 62-page chapter devoted to these word problems, the new edition tackles the ultimate math nightmares of the puberty years: problems involving rate, time and distance, work performed, and mixture formulas, among others. Added to the pre-existing 20-page introduction-to-story-problems chapter (in the Algebra Survival Guide — 1st Edition), it’s like having a book within a book.

The Algebra Survival Guide — Second Edition also includes:
  • 12 additional content chapters that explain fundamental and advanced areas of algebra
  • a unique question/answer format so students hear their own questions echoed in the text
  • conversational style written in the voice of a friendly tutor
  • step-by-step instructions
  • practice problems after each new concept
  • chapter tests
  • an expanded glossay and index
  • lively illustrations by award-winning artist Sally Blakemore
Finished Spit Fire
The many cartoons not only provide well-deserved comic relief for math learners, they also offer a visual way to grasp algebra’s challenging abstractions. Example: The above cartoon illustrates a real-world mixture problem by showing different percent concentrations of paint.

With all of these features, the Second Edition Algebra Survival Guide is ideal for homeschoolers, tutors and students striving for algebra excellence.

The Second Edition also aligns with the current Common Core State Standards for Math, so it’s ideal for today’s teachers, as well. Its content chapters tackle the trickiest topics of algebra:

Properties, Sets of Numbers, Order of Operations, Absolute Value, Exponents, Radicals, Factoring, Cancelling, Solving Equations, and the Coordinate Plane.

So, have some fun learning algebra!

• Updated version of Josh and snake

So here’s the situation: you’re at the breakfast table, enjoying a bowl of steaming-hot steel-cut oats and maple syrup, and you just poured yourself a mug of black coffee. But then you realize you want to pour some milk in the coffee (sorry, purists). But the milk is in the frig, six feet away. So of course you walk to the frig, grab the milk, bring it to the table, pour some in your coffee, return the milk to the frig and sit back down. Question: could you have done this more efficiently?

Yes, of course. You could have brought your cup of coffee with you as you walked to the frig, poured the milk right there at the frig, returned the milk, and then walked back to the table.

Being Smart?

“Morning Joe”

When I realized this this morning, I thought … hmmm. Had I used a bit of forethought, I would save myself an entire round trip from the table to the frig. And while I have no problem making that extra trip (hey, just burned 1.3 calories, right?), the experience made me wonder if anyone has ever developed a mathematics of efficiency for running errands.

I could imagine someone taking initial steps for this. One would create symbols for the various aspects of errands. There would be a general symbol for an errand, and there would be a special ways of denoting: 1) an errand station (like the frig), 2)  an errand that requires transporting an item (like carrying the mug), 3) an errand that requires doing an activity (pouring milk) with two items (mug and milk) at an errand station, 4) an errand that involves picking something up (picking up the mug), and so on. Then one could schematize the process and use it to code various kinds of errands. Eventually, perhaps, one could use such a system to analyze the most efficient way to, say, carry out 15 errands of which 3 involve transporting items, 7 involve picking things up, and 5 involve doing tasks at errand stations. Don’t get me wrong! I have not even begun to try this, but I’ve studied enough math that I can imagine it being done, and that’s one thing I love about math; it allows us to create general systems for analyzing real-world situations and thereby to do those activities more intelligently.

Of course, one reason I’m bringing this up is to encourage people to think more deeply about things that occur in their everyday lives. Activities that appear mundane can become mathematically intriguing when investigated. A wonderful example is the famous problem of the “Bridges of Konigsberg,” explored by the prolific mathematician Leonhard Euler nearly 300 years ago.

Euler in 1736 was living in the town of Konigsberg, now part of Russia. The Pregel River, which flows through Konigsberg, weaves around two islands that are part of the town, and a set of seven lovely bridges connect the islands to each other and to the town’s two river banks. For centuries Konigsberg’s residents wondered if there was a way to take a walk, starting at Point A, crossing each bridge exactly once, and return to Point A. But no one had found a way to do this.

One of the famous Seven Bridges of Konigsberg

One of the famous Seven Bridges of Konigsberg

Enter Euler. The great mathematician sat down and simplified the problem, turning the bridges into abstract line segments and transforming the bridge entrance and exits into points. Eventually Euler rigorously proved that there is no way to take the walk that people had wondered about. This would be just an interesting little tale, but it has a remarkable offshoot. After Euler published his proof, mathematicians took his way of simplifying the situation and, by exploring it, developed two new branches of math:  topology and graph theory. The graph theory ideas that Euler first explored when thinking about the seven bridges sparked a branch of math that’s used today to determine the most efficient ways of connecting servers that form the backbone of the internet!

Of course, there’s also the classic example of Archimedes shouting “Eureka!” and running through the streets naked after seeing water rise in his bathtub. In that moment, Archimedes, who had been trying to help his king figure out if the crown that was just made for him had been created with pure gold, or with an alloy, saw that the water displacement would help him solve the problem. In the end, Archimedes determined that the crown was not pure gold, and the king rewarded the great thinker for his efforts.

As I write this, I find myself wondering if any of you readers can think of other situations in which everyday life experiences led mathematicians or scientists to major discoveries. It would be enlightening to hear more of these stories.

And, if no such stories spring to mind, check out this site, which lists several such stories.  http://www.sciencechannel.com/famous-scientists-discoveries/10-eureka-moments.htm

In any case, the way that such discoveries occur shows that you never know where a seemingly trivial idea might lead … so it’s good to keep your eyes and mind open.

Everyone who teaches math knows it’s easy for students to make mistakes. After all, there’s often just one way to get a math problem right, but many ways to get it wrong.

And on top of that, the more steps a problem has, the greater the chances for making a mistake. When you consider all of this, doesn’t it seem miraculous that anyone ever gets a math problem right?!

As easy as it is to make math mistakes —and despite the fact that children know that mistakes happen — kids often get quite upset when math mistakes “happen” to them. 

So what’s the solution? Through my work tutoring students, I’ve found that the way we talk to students about mistakes can have a big impact on their “math esteem,” the way they view themselves as young math students. To bolster students’ “math esteem.”

This month I’d like to share a few helpful approaches, in hopes that they’ll inspire and help parents.

STRATEGY #1: NORMALIZING. One approach is to assure students that their mistakes are actually normal. “Oh, that’s a very common mistake,” I’ll say. Or even better: “That’s a really understandable mistake.” And then I’ll explain why it is that people might make this mistake.

EXAMPLE: Take the problem, 82 – 35. When students start work on this problem, they subtract in the 1s place. If students think about this correctly, they see 2 – 5 and realize they need to “borrow,” or “re-group.” But some students never get that far. Interestingly, their minds turn this part of the problem around, and they see it instead as 5 – 2. As a result, they get an answer of 3 at this stage. I tell students that it’s understandable to look at 2 – 5, re-cast it as 5 – 2, and mistakenly get 3 as the partial answer. And then I suggest a reason for this mental flip-flop. Students get flummoxed by the idea of subtracting a larger number from a smaller one (2 – 5), so their minds turn it around, changing it to the more familiar problem of subtracting a smaller number from a larger one (5 – 2). Once I tell my student this is a possible cause of his mistake, he usually relaxes and either agrees that this is what he did, or he offers a different explanation for how he made the mistake. The key is to make it clear that this kind of mistake is understandable and therefore nothing to be ashamed of.

Helping students relax around their math mistakes also gives parent and child the opportunity to brainstorm ways to avoid such mistakes. Brief example: I tutored a boy who would often say “Yikes!” whenever he noticed he was supposed to subtract a larger number from a smaller one. So together, we agreed he should continue saying “Yikes!” In fact, saying “Yikes!” served as a “mathematical alarm clock,” alerting the boy that he needs to “borrow” or (to use the more modern term) “re-group.” So together, we turned this boy’s natural reaction into a useful tool for avoiding mistakes.

To broaden the discussion a bit, a pre-algebraic mistake that falls into this “typical mistake” category involves simplifying terms like 2^3 (2 to the 3rd power). 2^3 means 2 x 2 x 2, so it equals 8. However, when students first learn this idea, they often fall back to thinking of multiplication, and as a result, they often view 2^3 as

2 x 3. As a result, they often get 6 when they simplify this term. A parent can reassure her child that this is a typical mistake, and point out that she made it because she is not yet used to working with exponents. To follow up, then, it would be helpful to give such a student a set of focused practice problems — perhaps 10 to 20 problems just like 2^3, so she can immediately grasp the concept of exponents and get practice simplifying these terms correctly.

STRATEGY #2: EXPRESSING INTEREST. Another response to mistakes that I’ve found useful involves telling students that that I find their mistakes interesting. I’ll say things like, “Hmmm, that’s a fascinating mistake!” Or, even better: “Wow, that’s an amazingly interesting mistake!” At first kids often look at me as if I ‘have a screw loose.’ That’s because they’ve developed an attitude that mistakes are simply “bad,” something to be thrown into the mental garbage can — not pondered with interest. Also there’s the prejudiced idea of: “Who but a nutcase could possibly find math fascinating?”

No matter, though. If you do take an interest in mistakes, your ability to reach your child and help her with math will improve in several ways.

EXAMPLE: A child is just starting pre-algebra, so she is just beginning to solve simple equations. She needs to solve the mini-equation: 11c = 44. Her answer:
c = 33. Your response, “That is so interesting. I’m dying to know how you got that answer.” Your child tells you that she simply subtracted 11 from both sides. Aha! You have just learned something very important. Your daughter had not realized that 11c means 11 times c. She had been viewing 11c as 11 + c, and that explains why she subtracted 11 from both sides of the equation. She thought the 11 was connected by addition, and she was (correctly, in her mind) doing the opposite operation — subtraction — to eliminate this term. If you, the parent, had not taken an interest in her mistake, you possibly would have never realized that she held this misunderstanding. But now that you do know this, you can help her realize that in general, whenever a number and a variable stand side by side with nothing in between, those two terms are linked by multiplication, not by addition. Progress!

This leads to a point I like to make when I talk with students about mistakes and our approach toward them. Mistakes, I say, are like weeds. If you simply snip away at a weed, it will pop right up again in a few days. If you really want a weed to disappear, you need to rip it out by the roots. Likewise, if you tell a child that she simply made a math mistake, and that she simply needs to do things a different way, that very same mistake will reappear soon. Just as we need to get weeds out by the root, we need to uproot mistakes as well. And generally, the only way to do that is to trace the student’s incorrect thinking back till we find its conceptual “roots.” In the last example, the “root” of the girl’s problem was her misunderstanding about the meaning of a term like 11c. Without uprooting that misunderstanding and replacing it with the truth — that 11c means 11 times c — not 11 plus c — this mistake would have recurred.

STRATEGY #3: DOCUMENTING. It’s all well and good to use Strategies 1 and 2, but to give your work with mistakes that extra “oomph,” it’s critical that you also teach children how to document their mistakes. I do this by setting up and using what I call a “Mistake Journal.”

The “Mistake Journal” is quite simply, a bundle of identical sheets of paper on which students document their mistakes, over time. Each sheet has empty lines on which children respond to the following writing prompts:

1. What mistake did I make? [Here children write down the actual math mistake they made. They don’t need to copy down their entire answer if the mistake is just a small part of the answer, as it usually is. They need only copy down the line just before the mistake, and the line with the mistake.]

2. What’s the right way to work this problem? [Here children write down — always with your guidance — how they are supposed work the problem correctly, the part with the mistake.]

3. What was my misunderstanding? [In this section children track their thinking back and explain what lies at the root of their mistake. Students often need help wording their understanding of the root of the mistake, so stand by, ready to help, at this stage.]

4. How can I avoid this mistake? [Here children come up with creative ideas on how to avoid making this mistake again. At this stage, you help your child learn how to: a) spot the tricky situation when it comes up in a problem, and b) work the problem without making this same mistake.]

To make Mistake Journal templates, I simply write out these four question-prompts on a blank sheet of paper, and I leave lines for the students’ answers. Then I photocopy this sheet, and keep a sizeable (o.k., huge) stack of these empty Mistake Journal sheets close at hand. Then, whenever we come upon a mistake that warrants an entry, we grab one of the sheets, and the student answers the questions. I three-hole-punch the sheets and have students to keep their completed Mistake Journal entries in a three-ring binder, since that keeps the sheets in good shape.

But don’t stop there. Make sure that your students use their Mistake Journal sheets! Sometimes, while tutoring, I’ll just ask my young charge to read a recent Mistake Journal entry. To check whether or not my student has grasped this Mistake Journal lesson, I’ll give my student a few problems that require her to put her new understanding to use. For example, if she’s the student who was confused by the problem: 11c = 44, I’ll give her five problems just like that one, to see if she has really mastered the concept. I suggest that you do the same thing with your child.

Also, if you give your children tests, quizzes or math evaluations of any kind, encourage your kids to look over their Mistake Journal entries while studying for the evaluation. Reviewing those potential mistakes (and notes on how to avoid them) helps children feel more confident when they take the test.

So while children’s responses to math mistakes can be fraught with anxiety, the three techniques described in this article can help lower that level of anxiety. I encourage you to try out these approaches, to see if they make your math learning environment more relaxed, less frustrating, and more successful overall.

Josh Rappaport lives and works in Santa Fe, New Mexico, along with his wife and two children, now teens. Josh is the author of the briskly-selling Algebra Survival Guide, and companion Algebra Survival Guide Workbook. Josh is also co-author of the Card Game Roundup books, and author of PreAlgebra Blastoff!, a playful approach to positive and negative numbers. 

At his blog, http://www.mathchat.wordpress.com, Josh writes about the “nuts-and-bolts” of teaching math. Josh also leads workshops on math education at school and homeschooling conferences., and he tutors homeschoolers nationwide using SKYPE. You can reach Josh by email at: josh@SingingTurtle.com

To find the GCF for three or more numbers,  follow these steps:

1)  Determine which of the given numbers is smallest, then find the smallest difference between any pair of numbers.

2)  See what is smaller:  the smallest number, or the smallest difference. Whichever one  is smallest, that number is the GPGCF (Greatest Possible GCF). That means that this is the biggest number that the GCF could possibly be. Or, more formally we would say:  The GCF, if it exists, must be less than or equal to the GPGCF.

3)  Check if the GPGCF itself goes into all of the given numbers. If so, then it is the GCF. If not, list the factors of the GPGCF from  largest to the smallest and test them until you find the largest one that does divide evenly into the given numbers. The first factor (i.e., the largest factor) that divides evenly into the given numbers is, by definition, the GCF.


Problem:  Find the GCF for 18, 30,  54.

1)  Note that the smallest number is 18, and  the smallest difference between the pairs is 12 [54 – 30 = 24;  54 – 18 = 36;  30 – 18 = 12] .

2)  Of those four quantities (the smallest number and the three differences), 12 is the least. This means that the
GPGCF = 12.

3) Check if 12 divides evenly into the three given numbers: 18, 30 and 54. In fact, 12 doesn’t divide evenly into ANY of these  numbers. Next we check the factors of 12, in order from largest to smallest. Those factors are: 6, 4, 3 and 2. The first of those that divides evenly into all three numbers is 6. [18 ÷ 6 = 3;  30 ÷ 6 = 5;  54 ÷ 6 = 9]. So the GCF = 6. And we are done.

Find the GCF for 24, 148, 200.

1)  Note that the smallest number is 24, and that the smallest difference between the pairs is 52 [200 – 148 = 52;  200 – 24 = 176;  148 – 24 = 124] .

2)  Of those four quantities (the smallest number and the three differences), 24 is the least. This means that for this problem, the GPGCF = 24.

3) Check if 24 divides evenly into the three given numbers: 24, 148 and 200. While 24 does divide evenly into 24, it does not divide evenly into 148 or 200. So next we check the factors of 24, in order from largest to smallest. Those factors are: 12, 8, 6, 4, 3 and 2. The first of those that divides evenly into the three given numbers is 4. [24 ÷ 4 = 6;  148 ÷ 4 = 37;  200 ÷ 4 = 50]. So the GCF = 4. And, once again, we are done.

The process may seem a bit long, but once you get used to it and start doing it in your mind, not on paper, you should find that it actually is quite fast. And you’ll find yourself figuring out the GCF for three or more numbers all in your mind — with no need for pencil and paper — while everyone around you will be making prime factor trees or using calculators. And surely that is a good feeling.

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! 

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 Amazon.com  Just click the links in the sidebar for more information! 

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 Amazon.com  Just click the links in the sidebar for more information! 

Hi everyone,

I have some exciting news.

I will be conducting a workshop this Tuesday morning, and the workshop is with a school in Nigeria.

Thanks to my good friend Ibraheem Dooba (aka Professor Brainy), and also thanks to the marvels of modern technology, this Skype workshop has been all set up and is ready to go.

The school whose teachers are receiving the workshop (not sure if that’s the right way to phrase it, or if there is a way to phrase that!) is the Esteem International School, in Nigeria’s capital city, Abuja. This is a wonderful school for elementary age children. Like students in the United States, students at the Esteem School are learning aspects of algebraic thinking in elementary school. And since teaching algebra to young students is a challenge, that will be the focus of my workshop.

If anyone here or anywhere in the world is interested in receiving a workshop on this topic, please let me know, and we’ll set one up.

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