Q&A #226

Student ahead of the class

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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

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