Kiss those Math Headaches GOODBYE!

Posts tagged ‘math’

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!

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Fun Math Problem #2


Here is the second in my series of “Fun Math Problems.”

Feel free to try these problems. Share them with friends and colleagues. Use them however you see fit! I will post the answer to the problems two days later, after people have had time to respond.

To provide your response, simply send an email to me @ info@SingingTurtle.com
and make your Subject: Fun Problem.
Please show how you worked the problem. Thanks. I will post the names of the first three people who get this right.

The Problem:  Before you go out to lunch, you glance at the clock above your desk. When you come back from lunch, you glance at the clock again, and you notice something strange. The minute and the hour hand have exchanged places from the positions they had just before you went to lunch.

The question is:  how long were you away?

Rusting clock face

Image by The Hidaway (Simon) via Flickr

Origins of the Coordinate Plane


A fly …

Who would think that a mere fly could play a major role in the history of human thought?

But when it comes to the development of Algebra, that’s the story. I’ll explain how this works just a bit later in this blog. But it is all related to what is happening now in algebra classes all around the world.

For it’s spring, that time of year again when we get out the graph paper and the ruler. Kids are working on the Cartesian coordinate plane.

One about I like about the coordinate plane is that there’s an interesting story about how it was discovered, or should I say, invented. [Hard to know the right word for an intellectual Invention like the coordinate plane.]
(more…)

How to tell if a Number is Divisible by 8


I’ve explained a number of divisibility rules lately, offering tricks to tell if numbers are divisible by 2, 3, 4, 5, 6 and 7.

There is also a trick for divisibility by 8, and that’s what I’d like to explain in this post.

Essentially the trick for 8 is a lot like the trick for 4. If you’d like to refresh your memory on how that trick works, just go here. (more…)

Recent insight on the GCF (and GPGCF)


A while back I wrote a post about the GCF, and mentioned that there’s a number  related to it — a number that I call the GPGCF. “GPGCF” stands for the “Greatest Possible Greatest Common Factor.”

In short, the GPGCF is a number that sets an upper limit for the size of the GCF. I’ve seen many students struggle when searching for the GCF, seeking hither and yon for it. I had a sense that students were checking numbers that were too large. That’s what led me to try to figure out what must the the upper limit for the GCF.

If you check out that post (10/25/10), you’ll see that, for any two numbers, I said that the difference between those numbers has to be the GPGCF.

And I was correct, to a degree.

But I recently realized that my little theory needs modifying.

For while the difference between any two numbers can be the upper limit for the GCF, that difference is not the only quantity that can set an upper limit for the GCF. There’s another quantity that plays a role.

That other quantity, I recently realized, is the size of the smaller of the two numbers.

Take the numbers 8 and 24, for example.

The difference between these two numbers is 16, so I would have said that 16 is the upper limit for the GCF. But there’s actually another quantity that limits the size of the GCF, and that quantity is 8. For since the GCF of 8 and 24 must by definition fit into both 8 and 24, it must fit into 8. And common sense tells us that there’s no number larger than 8 that can fit into 8! So the size of this number — the smaller of the two numbers — also sets an upper limit for the size of the GCF.

So my revised theory about the GPGCF is this:  when you need to find the GCF for any two numbers, look at two quantities:  1) the smaller of the two numbers, and 2) the difference between the two numbers. Both of these quantities constrains the size of the GPGCF. So therefore, whichever of these is smaller IS the GPGCF. Once you’ve found the GPGCF, that makes it easier to find the actual GCF.

I know this sounds very abstract, so let’s look at a few examples to see what I’m blabbering on about.

Example 1:  What’s the GPGCF for 6 and 16?

Smaller number is 6; difference is 10.
6 and 10 both limit the size of the GCF, but
6 is less than 10, so 6 is the GPGCF.

Example 2:  What’s the GPGCF for 8 and 12?

Smaller number is 8; difference is 4.
4 is less than 8, so 4 is the GPGCF.

Example 3:  What’s the GPGCF for 30 and 75?

Smaller number is 30; difference is 45.
30 is less than 45, so 30 is the GPGCF.

Example 4:  What’s the GPGCF for 28 and 42?

Smaller number is 28; difference is 14.
14 is less than 28, so 14 is the GPGCF.

Now, let’s go one step further. From here, how do we figure out the GCF? I’ve done a bit more thinking about this, too, and I’ll share those ideas in my next post.

Not all variables are created equal


Are all variables the same?

Does every variable serve the same purpose?

When you think about it, you’ll see that the answer is “no.” Variables serve different purposes. When we explain this to students, we help them understand how variables work. Explaining this helps students understand how algebra “works.” You’ll see what I mean in a moment.

Consider the famous slope-intercept equation:  y = mx + b

A student recently asked me:  Are the  x and y variables the same as the m and b variables? What a great opportunity to explain something important!

I explained that the x and y variables serve completely different purposes than  the m and b variables. Here’s how.

The variables m and b are what I call “identifier” variables. By which I mean that they help us identify a specific line. To explain that, I asked the student a set of questions about something everyone understands — home addresses.

What would happen, I asked, if someone wanted to know where I live, and I told him that I live at 942? The student replied that this would not be enough info.

Then I asked, what if I told this person only that I live on Vuelta del Sur (a street name where I live in Santa Fe, NM)? Again the student said that this would not be enough info.

But what if I told this person that I live at 942 Vuelta del Sur. This, the student realized, would be enough information to enable someone to find my house. (All they have to do is Google me, and they’ll have my house AND directions!)

I pointed out that a similar situation applies to lines.

If I have a specific line in mind, and I want someone else to know the line I’m thinking of, is it enough to give this person just the line’s slope? No, for it could be any line with this slope, of which there are infinitely many parallel lines. What if I don’t give the slope but I do give the line’s y-intercept? Still not enough, as there are infinitely many lines that run through this y-intercept. But what if I tell the person both the slope and the y-intercept. Aha! The student could see — through drawings I made of this situation on a coordinate plane — that when you provide both slope and the y-intercept, there is one and only one line that could be indicated.

 

Three lines — the red and blue lines have the ...

Red & blue lines have same slope, so slope alone does not indicate a specific line; Red and green lines have same y-intercept, so y-intercept alone does not identify a specific line.

 

I explained that variables like m and b, which help identify a specific line, are “identifier” variables; their job is to identify a specific line. If your students are more advanced, you can explain that there are other identifier variables in different kinds of equations. For example, in the equation of a parabola:   y – k  = a(x – h)^2, the identifier variables would be the variables a, h, and k.

But what about variables like x and y? What do they do? What is their purpose?

These variables, I explained, have a completely different purpose. I call variables like x and y “ordered-pair generators.”

To explain this, I show students a simple linear equation like  y = 2x, and demonstrate how, using a “T-table,” you can use this equation to generate as many ordered pairs as you’d like, ordered pairs like (0,0), (1,2), (2,4), (3,6), etc. Point out that you can keep going and going. And then explain that the purpose of the x and y variables is to generate the infinitely many points that make up the line.

So the m and b variables tell us where the line is, and the x and y variables allow us to find the infinitely many actual points on the line. The two sets of variables, while different in purpose, work together toward a common goal:  to give us the equation of a line.

There are other purposes that variables serve, of course. And I’ll probably describe some of the other purposes in future posts. But the main point is that it helps students to recognize that variables do serve different purposes. Armed with that understanding, they can make much more sense of algebra’s formulas and equations.

Multiplication Trick — Multiplying by 5


Time for a math trick …

Q:  How do you multiply an even number by 5 in lightning speed?

A:  Divide the number by 2, then tack on a “0.”

Example:   5 x 24

Divide 24 by 2 to get 12.

Tack a “0” onto 12 to get 120. Presto, nothing up your sleeve. It’s that easy.

Why does it work? Hint: Think about how we multiply by 10. Then think about how multiplying by 5 compares to multiplying by 10.

Rotated version of File:Symbol support2 vote.svg.

Image via Wikipedia


Try these for fun (answers at bottom of post):

a)  5  x  16

b)  5  x  8

c)  5  x  28

d)  5  x  64

e)  5  x  142

f)  5  x  2,468

g)  5  x  6,042

h)  5  x  86,432

j)  5  x  888,888


Answers:

a)  5  x  16 = 80

b)  5  x  8 = 40

c)  5  x  28 = 140

d)  5  x  64 = 320

e)  5  x  142 = 710

f)  5  x  2,468 = 12,340

g)  5  x  6,042 = 30,210

h)  5  x  86,432 = 432,160

j)  5 x 888,888 = 4,444,440

k)  5  x  2,486,248 = 12,431,240