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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! 

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

The Clouds Part, and a Log Rule MAKES SENSE!


Have you ever been befuddled by the rules for logs?

More specifically, have you ever looked at this rule:

log (v w) = log v + log w

and thought: Now why in the world is that true?! What exactly is this saying? I know that I, myself, have had that thought. And for me the desire to understand this rule never went away. Till I got it some time ago.

[By the way, keep in mind that the v and the w in the parentheses are multiplying each other, so that v w actually means: v times w]

And the good news is: I think I can explain this rule in a way so that pretty much everyone who knows basic algebra can grasp it.

O.K., first, I knew that this log rule was related to another rule, the  exponent rule that says:

(a^b) x (a^c) = a^(b + c)

Remember: this is the rule that says if you have two exponential terms  with the same base, and those two terms are multiplying each other, you just keep that base and add the exponents. For example:
(3^2)  x  (3^5) =  3^(2 + 5) = 3^7

But how exactly does this exponent rule relate to the more confusing-looking log rule?

To get ready to see this, one preliminary concept must be clear. The concept is that whenever you see a log term, you’re basically seeing an exponent. Why? Because every log represents an exponent. For example:  log 2 of 8 is the exponent of 3 since 2^3 = 8. 

Put another way, the term log 2 of 8 is asking a question. It’s asking: what exponent would you plunk on the right shoulder of the smaller number, 2, to get the much bigger number 8? The answer is 3, since 2^3 = 8.

Now you try this.

What question is log 3 of 81 asking? Answer: What exponent would we put on 3 to get 81?
What is the answer to this question? Answer:  4, since 3^4 = 81.
So based on all of that, log 3 of 81 = 4.

Now that we’ve got this concept straight, let’s look at the log rule again.

log (v w) = log v + log w

If we substitute in some numbers, this rule will be easier to think about. So let’s substitute 4 for v and 8 for w. After doing that we get:

log (4 x 8) = log 4 + log 8

Next, keep in mind that we can insert a base, and we can actually use any base we wish, as long as we use the same base for all three terms. A handy base would be 2 since 4 and 8 are both powers of 2. So when we use 2 as our base, the equation now reads:

log 2 of (4 x 8) = log 2 of 4 + log 2 of 8

One more thing before we tackle this sucker. Let’s  express the product inside parentheses as 32, which is ok since 4 x 8 equals 32, right? So now the equation reads:

log 2 of (32) = log 2 of 4 + log 2 of 8

Now, after all of that work, let’s finally have some fun. “Having fun,” of course, is relative, but if you’re a math person, “having fun” probably means: let’s  figure out what this crazy equation is saying. So here goes …

Based on what we’ve been saying, the left side of the equation asks the question: what exponent would we put on 2 to get the number 32. So what about that … ? What exponent would we stick on the left shoulder of 2 to get 32? The answer, of course, is 5, since 2^5 = 32. O.K., so far so good: the left side of this equation is clearly equal to 5.

Now how about the right side? While the left side asked one question, the right side asks two questions because it has two log terms. First, the term, log 2 of 4, asks: what exponent do we put on 2 to get the number 4? That, of course, is 2, since 2^2 = 4. And the next term, log 2 of 8, asks: what exponent do we put on 2 to get the number 8? That, of course, is 3, since 2^3 = 8.

So the two log terms on the right side are 2 and 3. And we are supposed to add those terms because the equation says to add them. And what is 2 + 3? It is 5, the same number we just got for the left side of the equation. So that is that. The rule works. We can see it working!

And all it is really saying (for this example) is this:

The exponent you put on 2 to get 32 [which is 5] is the sum of the exponents you put on 2 to get the factors of 32, 4 and 8. Or, stated more succinctly and more generally:  the exponent you put on a base to get a certain number is the sum of the exponents you put on that same base to get the factors of that certain number.

That is all that this formula is saying; nothing more, nothing less. So if you understand what I’ve explained here, you understand this rule more deeply. And that is a cool thing. So pat yourself on the back, and go  enjoy the rest of your day!

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…)

How to Divide Fractions: from annoying to FUN!


O.K., I’m ready to share my amazing approach to dividing a fraction by another fraction. Well, maybe not breathtaking … like Andrew Wiles’ proof of Fermat’s Last Theorem … but at least interesting. And best of all, fun and student-friendly!

Last week I asked if anyone had any tricks up their sleeves that make it easier for students to divide fractions. And I said that I would share a trick after I heard from you.

I got a nice response from Michelle, who said that she has used the mnemonic “KFC” (like the fried chicken), which in her class stands for Keep-Change-Flip. The idea being that you KEEP the first fraction, and next you CHANGE the sign from multiplication to division. Finally you FLIP the second fraction, the fraction on the right. We have similar mnemonic where I live, which goes by the phrase: Copy-Dot-Flip, with the “dot” meaning the dot of multiplication.

But what I want to share with you is a completely different approach to dividing one fraction by another, an approach that saves time, and makes it both easier and more fun — in my humble opinion — than the standard approach.

The approach I’m going to show you works for any complex fraction situation you might encounter, such as these:

fraction-images2
For this blog post, I’m going to limit my chat to complex fractions of the arithmetic type, meaning those with numbers only, and no variables. And if it seems important, I’ll do another post later on using this very same process for algebraic fractions.

So what is this amazing approach, anyway? Well, it’s based on something I discovered on day when I was just messing around with fractions divided by fractions. I realized that after you do the KFC or the Copy-Dot-Flip, what you get — in general — is actually something really easy to grasp, as this next image will show you, along with a Quick Proof:

fraction-c3b7-1

If you take a moment to think about it, the terms in the numerator of the result — terms a and d — have something in common; they were on the outside of the original complex fraction, so I call these terms the “outers.” In the same way, the terms in the denominator of the result — terms b and c — were both on the inside of the complex fraction, so I call them the “inners.”

So when you divide fractions in this vertical format, the answer is simply the outers, multiplying each other divided by the inners, multiplying each other.

I find that students find this easy to remember and a cinch to do. This next sheet summarizes the idea, and also provides a fun way of remembering the concept, thinking about the stack of terms as a fraction “sandwich.”

fraction-c3b7-2

So, to put this in words, the four-level complex fraction that you start out with can be thought of as a sandwich, with two pieces of bread at top and bottom, and slices of bologna and cheese in the middle.

The main point is that to simplify the fraction sandwich, all you need to do is put the two slices of bread together in the numerator and multiply them, And then put the bologna and cheese together in the denominator, and multiply them.

Using this idea it becomes a lot easier to simplify these complex fractions. Here’s an image that shows how it is done, and how this approach saves time over the way we were taught to do it, using reciprocals.

fraction-c3b7-a2

And there’s more good news. This new way of looking at complex fractions also gives students a cool, new way to simplify the fractions before they get the answer. And when you do simplify fully, the answer you get will be a fraction that’s already completely reduced, so you won’t have to stress about that part.

The next two pages show you this fun and easy new way to simplify:

fraction-c3b7-31

or, or what? …  Here’s what …

fraction-c3b7-4

So now you might like to see the whole process from start to finish, so you can decide for yourself if this technique is for you. Well that’s exactly what we’re showing next. As you can see I consistently highlight the outers with pink, the inners with yellow.

fraction-c3b7-51

And finally, a “harder” problem, you might say. But check it out. Is it really any harder than the one we’ve just done? You decide.

fraction-c3b7-6

In my next blog I’ll give you a few problems like these, so you can get used to this trick, and start shaving precious seconds and nano-seconds off the time it take you to do your homework, so you spend more time doing all of those things that you want to do more:  texting, watching You-Tube, taking hikes, skating (roller and ice), etc. etc. , etc. You know better than me.

Happy Teaching and Learning!

—  Josh

 

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! 

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Place Value Metaphor


During the summer I get to tutor a lot of elementary age students, remediating them on the basics.

Almost invariably I find that these students are confused about PLACE VALUE, and considering how critical this concept is to all of math, I decided to write this post.

Whenever I have the least suspicion that a student might be confused about place value, I check with a simple test.

I have them write down the number 22, then I ask them if they can tell me the difference between the two 2s. Often they cannot.

Tutoring a girl this past week I came up with a way of understanding place value that really resonated with the student. I want to share it because you may be able to use it, or a modification of it, with your students. First it’s important to know that this student’s mom teaches ballet, and the girl dances at her mom’s studio. 

I asked the girl if she has ever been to a ballet performance, and of course she said yes.

Then I drew a quick diagram of the stage and first few audience rows. I pointed to two seats, one in the front row, another seat several rows back. I asked her if the two seats would cost the same amount. This girl knew that the close seat costs more money because it is closer to the action on stage.

Then I used that idea to explain place value. I showed this girl that just as seats can be more or less valuable because of where they are, so too digits can be more or less value based on where they are in a number. 

She got this idea very quickly, and now she understands place value.

For children with different interests, use whatever makes sense. For example if you’re teaching a boy who loves baseball, make the rows of seats those at a baseball game, and so on.

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