From: Mary Lou Derwent (for Teacher2Teacher Service)
Date: Apr 21, 1998 at 09:07:18
Subject: Re: Student ahead of the class

Hi,

Now I know that the student is in the eighth grade. However, I am not sure 
what academic level in mathematics we are talking about. Has this student 
finished an elementary algebra course? This is something that can be done 
independently. It is not the ideal situation but it is certainly an 
alternative.  

If algebra is not taught until the ninth grade in your school district, ask 
an algebra teacher in the high school for a text. I would suggest the D.C. 
Heath text by Larson, Kanold and Stief.  The student will need a graphing 
calculator or at least a scientific one to work with the text.

Has your student ever taken the National Junior High Mathematics Exam? This 
is a test that is given every November to students in grades 6-8. You can 
write to Dr. Walter E. Mientka and ask him to send you copies of the last 
five year's tests and an answer key.

Dr. Walter E. Mientka, Executive Director    e-mail: walter@amc.unl.edu
American Mathematics Competitions
1740 Vine Street                             FAX: 402-472-6087
University of Nebraska                       Tel: 800-527-3690
Lincoln, NE 68588-0658 U.S.A.                Tel: 402-472-5114 (off hours)

               World Wide Web Page: http://www.unl.edu/amc/

Have the student do these tests writing out all of the work and explaining 
how the problem is solved. When a test is completed, give the student the 
answer key and have the student look at the different ways to answer 
questions. There are about 250,000 students who take this test every year and 
only a few, less than 10, get perfect papers. I find the problems 
interesting.

There is a problem that I have given to this age student and the student has 
done many experiments with it.

Consider non-terminating repeating decimals. All of these decimals can be 
represented by rational numbers. What is interesting to look at is the 
sequence of digits in the repeating decimal.

Suppose the decimal is 0.25252525252.....  This can be graphed on a piece of 
graph paper. Choose any point as a starting point. From this point draw a 
segment two squares long going north. Then draw a segment five squares long 
going east. Then draw the third digit, a segment two squares long going south 
and the fourth digit, a segment five squares long going west. You end up back 
at the starting point and have constructed a rectangle with height 2 and 
width 5.

The decimal 0.3333333.....   will generate a square.

What does 0.124124124124124...  generate? one square north, two squares east, 
four squares south, one square west, two squares north, etc. This starts to 
meander around the grid and tends to cover all of the squares. Do the digits 
in sequence, north, east, south, west. north, east, etc.

What is true about the repeating decimals representing 1/7 = 
0.142856142856.....or 2/7 or 3/7, etc.? These eventually close.

The question is:  what must be true about the sequence of numbers so that 
they form a closed figure returning back to the place where it started?

You can help this student by having him/her make careful graphs, labeling the 
graphs with the original decimal, and trying to categorize those which close 
and those which don't.

Another good fraction is the 13ths.

Have you a set of pentominoes?   Ask the student how many different
rectangles can be made using all of the tiles.

I hope that I have given you some ideas to try.

- Mary Lou, for the Teacher2Teacher service

Post a public discussion message
Ask Teacher2Teacher a new question