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.