Wednesday, April 23, 2008

There is Hope - I Think

Today I read an article in the New York Times concerning education in the USA. As you might expect it does not present a lot of great news. Our local situation is probably better than what is described in this article but we will all have to live with some of these problems if the author is correct.

Here is the link.

Monday, April 21, 2008

Earth Day

Significant and Insignificant

April 22nd is Earth Day. Since there is no practical way to determine otherwise, let's call it Earth's birthday. The best scientific estimate for the age of the earth is about 4.55 billion years, plus or minus 70 million years. This estimate was calculated from the work of an Iowan, Clair Patterson, in 1953. See the Bill Bryson book, “A Short History of Nearly Everything” (p.149-160) for more information.

From what I can tell, what we would call human beings (homo sapiens) have existed on the earth for about the last 250 000 years. Of course this is a rough estimate but it gives the opportunity to do some interesting math. Large numbers are not easily understood by anyone. In one of my earliest posts (October 30th, 2007), I mentioned that one million seconds is about 11.5 days while one billion seconds is about 31.7 years.

Using the above numbers, 4.55 billion years for the age of the earth and 250 000 years for the time human beings (homo sapiens) have been on earth, there are a few ways to show the relative insignificance of man's time on earth compared to the time that earth has been in existence. One way to do this is to compress the entire age of earth into one 24 hour day. Based on that, how long have human beings existed on earth?

The easiest way to solve this would be to use a proportion:

4 550 000 000 years : 250 000 years = 24 hours : X hours

Solving for X we get about .001318681319 hours

Multiply that by 60 min./1 hour and by 60 sec./ 1 min. (which are forms of one)
and we get about 4.747252747 seconds.

Therefore in the 24 hour history of earth, human beings have been on the scene about the last five seconds. Most people would find that pretty stunning.

Another interesting thing to do with the information is more appealing to students on a visual level. Assume that the average sized man has a wing span (arms stretched out wide), from the tip of the fingernail on the middle finger of one hand to the tip of the fingernail on the middle finger of the other hand, of six feet or 72 inches. Assume that length of 72 inches is all of earth's history. What part of that length would represent the length of human beings time on earth? Again set up a proportion:

4 550 000 000 years : 250 000 years = 72 in. : X in.

Solving for X we get about .003956043956 inches which is very close to .004 inches, or about 1/250 of an inch. In the Bryson book (p.337), one of the authors he cites says, “... a single stroke with a medium-grained nail file [would] eradicate all of human history.”
That is a pretty good visual for students to see that the 1/250th of an inch scraped off one of those middle finger nails would represent all of human beings' time on earth.

Another visual way to look at that would be to imagine that the 4.55 billion years of earth be represented by a 100 yard football field. What would the 250 000 years of human beings be on that field?

4 550 000 000 years : 250 000 years = 100 yards : X yards

Solving for X we get about .005494505495 yards. Multiply by 36 in./1 yard you get about .197802197 inches or about 1/5th of an inch. Imagine looking at a football field from one goal line to the other goal line and realizing that the time of human beings would only be about 1/5th of an inch. That is pretty stunning too.

The very bright student may realize that some of these numerical answers can be obtained from the other numerical answers. Time could be spent on that for those interested. For example, the football field (100 yards) is fifty times as long as the wing span of 6 feet. Multiply 1/250th of an inch times fifty and you would get 1/5th of an inch.

I love math!

Wednesday, April 16, 2008

Number Tricks

I have been gone from posting anything for awhile. One of my former students sent me an e-mail saying, "patiently waiting for another blog" so I thought I better get back to posting and that maybe somebody was actually reading them. If you want to comment on a posting directly to my e-mail, send it to me at crxos@yahoo.com.


NUMBER TRICKS

All students enjoy what many people call "Number Tricks." Number tricks are
in two basic types. The first type involves the student starting with their own
chosen number, performing several operations on it, and then ending up with
the same number they started with. The second type involves the student,
again, starting with their own number, performing several operations, but
then everyone in the group ending up with the same number, instead of
their starting number. The students find the first type pretty interesting but
are more fascinated by the second type. At first they are somewhat mystified
that these work. That opens the door for the Algebra teacher to demonstrate
algebraically why they do work and then challenge the students to develop
their own number tricks. I eventually show them why each trick works by showing
three columns that have (1) the direction written out, (2) a numerical example,
and (3) the algebraic expression for that step. I require students to develop an
example of both types of tricks and each trick must have at least six steps.
They show their number tricks just like I show mine to them. The first column is
where they write the direction for that step. In the second column, they write a
numerical example of that step. In the third column they write an algebraic
expression for that step. I encourage them to be as creative as possible. I give
them an example of "dressing up" their directions to allow more creativity. Of
course you cannot do project this until they possess the algebraic skills needed.
This has proven to be an enjoyable project for my students involving creativity
and algebraic knowledge.

(Please excuse the fact that the three columns I wanted to show, separating the direction, the number, and the algebraic expression didn't come out well. The auto editor crams them together so they are a little hard to read. If needed I can e-mail you my document copy which is easier to read.)


Here is an example of the first type of number trick in which everyone ends up
with the same number they started with:

Pick any number: 32, X
Multiply by 20: 640, 20X
Add 1000: 1640, 20X + 1000
Divide by 2: 820, 10X + 500
Subtract 300: 520, 10X + 200
Divide by 5: 104, 2X + 40
Subtract 40: 64, 2X
Divide by 2: 32, X

The teacher should do this trick at least twice (before showing the algebraic expressions)
so the students can see that it works regardless of what number they choose at the beginning.
You might ask them if it matters if they had started with a negative number or a decimal.
If they understand the algebraic expressions they will be able to answer that question.

Here is an example of a number trick in which no matter what number the student chooses
to begin with, they will all end up with the same number - in this case the current year.

Pick any number: 3.14, X
Multiply by 100: 314, 100X
Add 250: 564, 100X + 250
Multiply by 4: 2256, 400X + 1000
Add 1000: 3256, 400X + 2000
Subtract 400 times your
original number: 2000, 2000
Add 8: 2008, 2008

I would have several number tricks of each type ready on that first day in order
to heighten their interest. Make them as fun and interesting as possible. Many times
students would ask how old I was. This was my way of telling them.

An example of "dressing up" the above trick would change the directions to:

Pick any number
Multiply by the number of years in a century
Add the number of pennies in $2.50
Multiply by the number of sides in a quadrilateral.
Add the number of meters in a kilometer.
Subtract twenty squared times your orginal number
Add the value of two to the third power

The steps to any number trick can be "dressed up" in a variety of ways depending
on the age, maturity level, and mathematical knowledge of the class. They could
involve other school subjects also. (Add the number of milliliters on a liter or subtract
the year in which the Declaration of Independence was signed.)

The students may very well surprise you with their creativity in this situation.