## Kiss those Math Headaches GOODBYE!

### Everyday Life Sparks Mathematical Puzzles

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.

“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

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.

### 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.

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)

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!

### How to Combine Positive & Negative Numbers — Quickly and Easily

If you or someone you know struggles when combining numbers with opposite signs — one positive, the other negative — this post is for you!

To be clear, I’m referring to problems like these:

– 2 + 7 [first number negative, second number positive], or

+ 13 – 20 [first number positive, second number negative]

To work out the answers, turn each problem into a math-story. In this case, turn it into the story of a tug-of-war battle. Here’s how.

In the first problem, – 2 + 7, view the – 2 as meaning there are 2 people on the “negative” team; similarly, view the + 7 as meaning there are 7 people on the “positive” team.

There are just three things to keep in mind for this math-story:

1)  Every “person” participating in the tug-of-war is equally strong.

2)  The team with more people always wins; the team with fewer people always loses.

3)  In the story we figure out by how many people the winning team “outnumbers” the other team. That’s simple; it just means how many more people are on that team than are on the other team. Example: if the negative team has 2 people and the positive team has 7 people, we say the positive team “outnumbers” the negative team by 5 people, since 7 is 5 more than 2.

Now to simplify such a problem, just answer three simple questions:

1)  How many people are on each team?
In our first problem, – 2 + 7, there are 2 people on the negative team and 7 people on the positive team.

2)  Which team WINS?
Since there are more people on the positive team, the positive team wins.

3) By how many people does the winning team OUTNUMBER the losing team?
Since the positives have 7 while the negatives have only 2, the positives outnumber the negatives by 5.

Now ignore the answer to the intro question, Question 1, but put together your answers to Questions 2 and 3.

All in all, this tells us that:  – 2 + 7 = + 5

For those of you who’ve torn your hair out over such problems, I have good news …

… THEY REALLY ARE THIS SIMPLE!

But to believe this, it will help to work out one more problem:  + 13 – 20.

Here, again, are the common-sense questions, along with their answers.

1)  How many people are on each team?
In this problem, + 13 – 20, there are 13 people on the positive team and 20 people on the negative team.

2)  Which team WINS?
Since there are more people on the negative team in this problem, the negative team wins.

3) By how many people does the winning team OUTNUMBER the losing team?
Since the negatives have 20 while the positives have only 13, the negatives outnumber the positives by 7.

Just as you did in the first problem, put together your answers to Questions 2 and 3.

All in all, this tells us that:  + 13 – 20  = – 7

Now try these for practice:

a)  – 3 + 9

b) + 1 – 4

c)  –  9 + 23

d)  – 37 + 19

e) + 49 – 82

a)  – 3 + 9 = + 6

b) + 1 – 4 = – 3

c)  –  9 + 23 = + 14

d)  – 37 + 19 = – 18

e) + 49 – 82 = – 33

Josh Rappaport is the author of five books on math, including the Parents Choice-award winning Algebra Survival Guide. If you like the way Josh explains these problems, you will very likely 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!

### Let STUDENTS make the math Problems, for a change

If you want students to look at you like you’re crazy — and have fun because you know you’re doing a good thing — try this.

Tell students it’s their turn to make up a math problem.

Math Meeting Board and Lesson (Photo credit: Old Shoe Woman

Yes, they’ll give you that look like, what are you talking about? But it’s o.k. Persist. Not only that … tell them to make up a word problem just like one in the textbook or on the worksheet. And tell them to make it relevant to their own lives.

For example, if you’re doing problems on rate, time and distance, suggest that students make up a skateboarding problem. One of my students came up with this:

You want to skate over to Ted&Tom’s (a local hangout), and you need to get there by 2:15 pm. If you’re 3 miles away and you leave at 1:30, going 4 mph, will you get there in time? [Answer:  You’ll get there right on time, not a minute too soon or too late.]

See how easy it is? Not really hard.

Or, let’s say that you’re doing ratio problems. Suggest that students do a problem on price comparisons. Another one of my tutees came up with this:

Lip gloss is on sale, 4 tubes for \$7. At that rate, can you buy 12 tubes if you have exactly \$20? [Answer:  No, since you won’t get the special if you have only \$6 for the last set of lip gloss tubes.]

The benefits for students are many.

1)  Students start to see that math problems are “all around them.” i.e., They start to see math in their everyday situations. And they start to realize that they can actually use the math you’ve been teaching them to figure out  real-life problems.

2)  By developing their own problems, students grasp the concepts in the problems more deeply. In the same way that we teachers learn by teaching, students learn by making (and solving) their own problems.

3)  Making problems is a creative activity, and once students see they can pull their problems from real life, they start to enjoy the activity. And because this involves creativity, this exercise engages the “creative types” who often feel like math does not “speak to them.”

4)  If you take the activity one step further, you can help students build their critical thinking skills. The one step further is: require that students get a whole number answer for their problem. This requirement forces students to think about how the numbers in the problem affect the value of the answer. And when they need to fine-tune those problem numbers to get out a particular kind of numerical result (like a whole number answer), they learn about the “innards” of the problem. They learn how the problem works more deeply than they would if they only were solving a problem someone else gave them.

5)  If you make the solving process cooperative, you can add even more fun to the process. By this I am suggesting that after students make the problems, they give them to other students to solve them. This way two students can exchange problems. I’ve seen students really get into this. They start making problems harder until they are just at the level that makes their partner “sweat.” But they enjoy this process, and it helps them get to know each other. I’ve found that this is a good way to get some fun socializing into a math class.

One last nice thing:  I’ve found that students cannot actually make up problems if they don’t know how to solve the problems. That means that this exercise tells you, the teacher, which of your students do understand the problem. And if they don’t get it, you can help them get it by helping them make the problem. It’s a nice, indirect way to teach.

So give it a try in your class or teaching situation, whatever that may be. I have a hunch you’ll find it as helpful and enjoyable as I have found it to be.

### Reader Input on Slope Post

A longtime reader of Turtle Talk, Jeff LeMieux, of Oak Harbor, WA, sent in a suggestion based on today’s post on positive and negative slope. Jeff found a way to help students remember not only positive and negative slope, but also the infinite slope of vertical lines, and the 0 slope of horizontal lines … all using the letter “N.”

This is clearly a situation where the picture speaks more loudly than words, so I’ll just let Jeff’s submitted picture do the talking. By the way, to see this image even better, just double click it!

Slope Memory Trick

Thanks for putting this together and sharing it, Jeff!