Archive for: August 2007

If you’re new here, you may want to subscribe to my RSS feed. Thanks for visiting!

I am listening, for the second time today, to Jonathan Kozol’s absolutely brilliant speech on Alternative Radio on National Public Radio (NPR).

If you care about education in America at all, you should know about the work of this exceptional man.

The thrust of the speech is that apartheid is more alive and well than any time since the death of Dr. Martin Luther King Jr. As absurd as it sounds, it is true, and shamefully so.

Another great point he makes is how high-stakes testing is corrupting and distorting curricula. Of course you know this, but he is so eloquent about it that it makes you think harder than you probably ever have about it.

I cannot do the speech justice. You can order a copy of it at the above link. Or you can call 1-444-1977.

Here are two other links, to a sites where you can learn more about Kozol (and you should):

Jonathan Kozol’s Homepage
Rethinking Schools

When you read or listen to Mr. Kozol, you will thank your lucky stars if you are a homeschool family.

Don’t expect him to bash teachers, though. He knows they usually trying to do an almost impossible job under terrible odds. He doesn’t bash administrators, either (that’s where we part company). But he sticks it to the real evildoers - the politicians who back the “No Child Left Unstressed” (”No Politician Left Unenriched”) debacle. Those hypocritical gasbags who talk about “school pride.”

I’m going back to listen some more. This man really knows what it’s like to speak truth to power.

Please learn more about him.

A few posts ago, I offered some tips about how to check large division problems without having to multiply huge divisors and quotients to get even huger dividends.

One of the drawbacks to using the “crunch” method, which I described, is that it is not 100% accurate.

Often, people who need to defend the status quo (you know who they are, they work in the principal’s office) insist that checking by crunching is not acceptable, because it is not foolproof.

Let me give an example:

You could divide 1206 by 18 and get 64

It looks like it works. But it doesn’t. The real answer is 67.

Sometimes you can transpose digits, or make a mistake, the crunch of which will work out to the crunch of the real answer. After all, there are only 10 digits which all integers can crunch to.

It is very seldom, though, that you will crunch mistaken digits and do the multiplication, and have the answer come out to a crunch number that still has the same crunch number as the real answer.

The reason that mistakes are so seldom, is that it is easy to add numbers like 1+8 and 6+4 and multiply the results.

Although mistakes still can be made, much less mistakes are made with this method than with the cumbersome method you probably learned in school. Consider this: What is easier to do,

(1+8)*(6+4)
=9*1
=0, (all of which you can do in your head, with no training, in seconds)
or
18*64? (Do you really want to multiply that mentally if you don’t have to?)

There are many small-minded people in education, who insist that methods other than theirs must be foolproof, when their own methods are even less foolproof.

The mission of MathMojo is not only to teach easier, more effective methods, but also to make math more meaningful to you. And one way to do that is to sharpen your critical thinking skills.

Here is a perfect opportunity to do just that. Can you see the flaw in the small-minded person’s argument? They set up conditions that new things must fulfill in order for them to consider using them. But they don’t put their own things under the same conditions.

That phenomenon is one of the most prevalent flaws of society. Catch it when you see it, and call them on it.

Hotcha!

This has nothing to do with math, per se, but I think some readers will still find it useful.

As a member of the Macintosh Users Group of Oneonta, NY (MUGONE.com), I notice that even some savvy computer users are a little shaky on some of the basics. So once in awhile I’d like to put up a basic video tutorial on simple subjects just to solidify some people’s computer skills.

This week, I cover the difference between a web-browser and a search-engine. In the future I’ll cover some slightly more advanced skills, like how to sign up for and make a Squidoo lens.

Let me know if you notice anything on MathMojo or the Chronicles that you wonder about, or want to know how I did. I’m not exactly and expert, but I’ll help where I can.

Hotcha!

Brian

We’ve been talking about using factors to make long-division problems easier, sometimes being able to turn them into a manageable sequence of short-division problems, in which no paper and pencil (and certainly no calculators!) are needed.

Want to try another one? How about

962/52 ?

Well, they’re both even, so that’s going to be a piece of cake to start. Divide both by 2 and turn the problem into:

481/26

Can you factor them further? You can tell that 2 won’t be a factor. And a quick look at 26 tells you that 3,4,5,6,7,8,9 and 10 won’t factor into it. It’s only factor, other than itself and 1 is 13. When you factor 13 into 21, you get 2.

Now all you have to do is test if 481 is divisible by 13. If you know the trick to test for divisibility by 13, you could try that, but let’s just assume you don’t, and go ahead and divide it in our heads.

13 goes into 48 three times, with 9 left over. Carry the 9 to the front of the 1 in 481, and get 91. Divide that by 13, and whaddyaknow, it goes in exactly seven times. That gives us 37.

We have reduced the problem from

without much trouble, and no writing. I think you can handle 37/2 on your own from here.

Right, it’s 18, r. 1.

Remember, that’ the answer to 37/2. But if you want to check it as the answer to 962/52, you’re going to have to re-factor in the 13 and the 2 to the remainder. When you multiply the remainder (1) by the factors (2 and 13) you get 1 x 2 x 13, which is 26.

Checking 962/52 = 18, r. 26

962/52 = 18 r. 26

So the crunch to the problem is 0, remainder 26.

So the crunch to the answer is 0, remainder 26.

The crunch to the problem matches the crunch to the answer, so the answer is very probably right.

In the last post we looked at the problem of 926/18, and we simplified it to 463/9, so we could make it a short division problem.

What if the problem had been 927/18?
Both numbers are not even this time, so it is not readily apparent if they have common factors.

If you know how to factor (if you don’t, you can get a lesson at The Pretty Good Guide to Prime Factorization at MathMojo.com.) then you factor both of these numbers by 9.

Here’s a hint: If a number can be crunched to 9 or 0, then nine is a factor of that number. If you want to know more about crunching, I refer you to “The See-Say-Write Method of Speed Addition“.

There are also many hints you can find about how to determine if numbers are divisible by other numbers. MathMojo will eventually cover this in depth, but I’m sure you can find info if you google “divisibility rules.”

Ok, so let’s factor 927/18.

Using short division by 9, we get 103/2. How easy is the problem now? Just cut 103 in half in your mind and get 51 remainder 1. But remember, like in the last post,
If you factor a division problem before you solve it, you must multiply the remainder by that factor after you are done.

So the answer to 927/18 is 51 remainder 9, (not 1).

Go ahead and check it. Remember how? If not, check out this post.

(Is that title an oxymoron?)

Imagine you have to do this division:

926/18

How would you do it? Would you rewrite it with that funny division symbol (“division bracket,” or “right parenthesis followed by a vinculum over the dividend”)? Would you use a calculator? (Please say “no” to that!)

After you rewrote it, would you start by trying to figure out how many times 18 would go into 92? If you did, you would be doing it the way most people learned in school, and you would be wasting a lot of time and effort.
(more…)

They way we are usually taught to check division problems in school is unnecessarily complex. There is a better way. I always wondered why, after thousands of years of mathematics, schools generally haven’t figured that out. But I’d rather try solving the Riemann zeta-hypothesis than figure out why schools teach the way they do.

An astute reader in Iceland (yes, we get readers from the coolest places!) asked the following question:

I have a minor problem regarding crunching division problem. How would you crunch a problem like 275 divided by 11 = 25?
(more…)

Someone wrote in to ask:

4⁰ * 5³ is 125. Why isn’t it 0?

On the Math.Com website, problems such as 4 to the zero power times 5 to the third power have an answer of 125 as correct. Shouldn’t the answer be zero. If not, why? Thank you!

Professor Homunculus’ response:

The answer actually should not be zero, and here’s why:

Because 4 to the 0 power is 1, not 0.

So 4⁰ * 5³ would be 1 x 5³ which is 125.

Any integer raised to the zero power equals 1.

That is hard for most people to believe, so I wrote a little piece to explain why it makes sense. Here it is:

(more…)

“Why do I need to know fractions? Square roots? Algebra or geometry? I mean, why do we ever even need them in real life? I am never going to be a mathematician, and I hate math. So why do I have to learn this?”
(more…)

I was just fooling around with some numbers, and realized that 13^2 (which gives you 169) is the reverse of 31^2 (which gives you 961 - which is the reverse of 169).

Is there any name for a number, the reverse of which, when squared, will also yield the reverse of the original number’s square?

Here’s another one:

12^2=144
21^2=441

Do you know any others? Possibly with other powers?
Anyone know any use for it? Please leave a comment if you do.