gifted students || manipulating numbers

MANIPULATING NUMBERSAuthor(s): John FirkinsSource: The Mathematics Teacher, Vol. 76, No. 4, Gifted Students (April 1983), pp. 256-260Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27963462 .

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activities -0 3

Edited by Evan M. Maletsky, Montclair State College, Upper Montclair, NJ 07043 Christian R. Hirsch, Western Michigan University, Kalamazoo, MI 49008 Daniel S. Yates, Mathematics and Science Center, Richmond, VA 23223

MANIPULATING NUMBERS By John Firkins, Gonzaga University, Spokane, WA 99258

Teacher's Guide

Grade Level: 7-10

Materials: Cardboard disks, bottle caps, or centimeter squares cut from graph paper

Objectives: To investigate the difference between the consecutive sums of the num bers along the three sides of a triangle. Stu dents must collect and organize data, develop problem-solving strategies, and extend and generalize the pattern pre sented.

Procedure: Duplicate a set of work sheets for each student. Give each student cardboard disks or other items on which to write each of the digits 1 through 9.

On the board or overhead draw three circles at the vertices of an equilateral trian

gle. Place the digits 1, 2, and 3 in the circles.

Compute the sum of the numbers on each side of the triangle as in the example that follows.

Notice that if this arrangement is select

ed, the consecutive sums are 3,4, and 5. No matter how the numbers 1, 2, and 3 are

placed inside the circles, the consecutive sums are always 3, 4, and 5. Therefore, we can conclude that the difference between the consecutive sums is always 1.

The teacher should work through the ac

tivity sheets before attempting this activity with students. Some guidance may be nec

essary. The answer key gives only some of the possibilities. The case where the sums differ by zero, or are equal, is a special case that has been used by many teachers as a

problem-solving exercise (1978). For exam

ple, the following are solutions for consecu tive sums that differ by zero :

3 54 1 0 2

All aums9

2 5 3

4 1 6 All sums 11

5 2 4

3 6 1 All Sums 10

4 2 3

615 All sums 12

This section is designed to provide mathematical activities suitable for reproduction in worksheet and

transparency form for classroom use. This material may be photoreproduced by classroom teachers for use in their own classes without requesting permission from the National Council of Teachers of Mathematics.

Laboratory experiences, discovery activities, and model constructions drawn from the topics of seventh, eighth, and ninth grades are most welcome for review.

256 Mathematics Teacher



Answers:

Sheet 1

4 2 3

6 1 5 Sums differ by 0;

1

4

It may be of interest to show that if one

solution is found, another can be obtained

by subtracting each number from 7 and re

placing the number by the result.

Sheet 2

7 9 ; .'.''.? . - --'> -:,;?--7

Sums differ by 2. Sums differ by 3.

4 4

Sheet 3

'?''':'.;;;4?.';'^,?:1:-;'

't':--*'.'.'iL:--:3?L.:'- ? '

For circles per side, 3(n ?

1) integers are needed.

The largest possible difference between

consecutive sums on the sides is ( ?

l)2.

BIBLIOGRAPHY

Caldwell, Janet. "Magic Triangles." Mathematics

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g & ADKINS MATH SERIES.

^ These "no-nonsense" books are

V pedigogically sound, and they have ^ been widely used for over 20 YEARS.

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BATES PUBLISHING COMPANY 277 Nashoba Road Concord, Massachusetts 01742

April 1983 257



SHEET 1

Place each of the digits 1 through 6 in the circles on the sides of the triangle. Arrange them so that the consecutive sums of the three numbers along each side differ by the indicated amounts.

in which the consecutive sums differ by more than 4? If not, can you explain why it is impossible?

From the Mathematics Teacher, April 1983



SHEET 2

Place each of the digits 1 through 9 in the circles on the sides of the triangle. Arrange them so that the consecutive sums of the four numbers along each side differ by the indicated amounts.

Can you find a solution in which the consecutive sums differ by more than 9? If not, can you explain why it is impossible?

Sums differ by 9. From the Mathematics Teacher, April 1983



SHEET 3

Arrange the numbers so that the consecutive sums along the sides differ by the largest amount possible. Hint: Begin by placing consecutive integers starting with 1 across the bottom.

Use the numbers 1 through 12. Use the numbers 1 through 15.

xVx'7'--r^^

Ufi11

Use the results from worksheets to complete the table.

Number of circles drawn on each side

of the triangle

Number of integers needed to fill

the circles

Largest possible difference between consecutive sums

2

3 4 5 6

3 6

1

Look for patterns in the numbers in each column of the table. How would you extend them for the next set of entries?

Suppose a triangle has circles on each side. How many consecutive integers starting with 1 would be needed to fill all the circles?_

What do you think would be the largest possible difference between consecutive sums on the sides ?_

From the Mathematics Teacher, April 1983



gifted students || manipulating numbers

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