3
RELATIONSHIPS IN MATHEMATICS 271 in a little school at the edge of the Sand Dunes made a collec- tion of the interesting plants that are found growing in this re- gion. Letters were sent to a similar grade in a Florida school near the ocean beach suggesting that "we exchange collections." As a result beautiful shells were sent in exchange for the interest- ing plant collections. THE PROPERTIES OF RELATIONSHIPS IN ELEMENTARY MATHEMATICS. BY J. S. GEORGES, The University High School, Chicago. In a recent study1 an attempt was made to determine the nature of reading difficulties encountered in mathematics. The number of reading difficulties caused by a lack of understanding of the nature of mathematical relationship between the elements in a statement comprises eleven per cent of the total number of cases reported. When the ratio is interpreted in the light of the small number of relationships studied the findings are quite significant. An important observation in the study of this type of reading difficulties reveals the fact that the symmetric property of certain mathematical relationships such as "is equal to/’ "is complement of," offer both difficulties of expression and of interpretation. It is quite obvious that the statements "a equals b," and "a and b are equal" mean the same thing, but the student often finds it diffi- cult to put the second statement in the form of the first. The same thing is true of the reflexive and the transitive properties of relationships. Believing that a clear understanding of the nature of a mathe- matical relationship has considerable bearing upon its proper study, that is its symbolic expression and interpretation, this paper presents a brief resume of the properties of a few of the many relationships studied in elementary Mathematics. Relationships in mathematics may be classified according to their properties as follows: (1) reflexive relations, (2) symmetric relations, and (3) transitive relations. A given relationship between any two elements or entities a and b is reflexive when upon making b identical with a the rela- tion still holds. For example, the relation of equality is reflexive, for a == a’, while the relation "is greater than" is not reflexive, for a is not greater than a. iQeorges, J. S., "The Nature of Difficulties Encountered in Reading Mathematics," The School Review. XXXVII. March. 1929. 217-226.

THE PROPERTIES OF RELATIONSHIPS IN ELEMENTARY MATHEMATICS

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RELATIONSHIPS IN MATHEMATICS 271

in a little school at the edge of the Sand Dunes made a collec-tion of the interesting plants that are found growing in this re-gion. Letters were sent to a similar grade in a Florida schoolnear the ocean beach suggesting that "we exchange collections."As a result beautiful shells were sent in exchange for the interest-ing plant collections.

THE PROPERTIES OF RELATIONSHIPS IN ELEMENTARYMATHEMATICS.BY J. S. GEORGES,

The University High School, Chicago.In a recent study1 an attempt was made to determine the

nature of reading difficulties encountered in mathematics. Thenumber of reading difficulties caused by a lack of understandingof the nature of mathematical relationshipbetweenthe elements ina statement comprises eleven per cent of the total number ofcases reported. When the ratio is interpreted in the light of thesmall number of relationships studied the findings are quitesignificant.An important observation in the study of this type of reading

difficulties reveals the fact that the symmetric property of certainmathematical relationships such as "is equal to/’ "is complementof," offer both difficulties of expression and of interpretation. It isquite obvious that the statements "a equals b," and "a and b areequal" mean the same thing, but the student often finds it diffi-cult to put the second statement in the form of the first. Thesame thing is true of the reflexive and the transitive propertiesof relationships.

Believing that a clear understanding of the nature of a mathe-matical relationship has considerable bearing upon its properstudy, that is its symbolic expression and interpretation, thispaper presents a brief resume of the properties of a few of themany relationships studied in elementaryMathematics.

Relationships in mathematics may be classified according totheir properties as follows: (1) reflexive relations, (2) symmetricrelations, and (3) transitive relations.A given relationship between any two elements or entities a

and b is reflexive when upon making b identical with a the rela-tion still holds. For example, the relation of equality is reflexive,for a == a’, while the relation "is greater than" is not reflexive,for a is not greater than a.

iQeorges, J. S., "The Nature of Difficulties Encountered in Reading Mathematics," TheSchool Review. XXXVII. March. 1929. 217-226.

27^ SCHOOL SCIENCE AND MATHEMATICS

A. given relationship between any two elements or entitiesa and b is symmetric if the relation is conversely true,that is if ais related to & in any manner then b is related to a in the samemanner. Thus the relation "is equal to" is symmetric, for a.is equal to b implies that & is equal to a; while the relation "isbisector of" is not symmetric, for a is bisector of b does not implythat b is bisector of a.A given relationship between any three elements or entities

a, b, and c is transitive if it is true that the relation between aand b, and the same relation between b and c together implythat the same relation holds between a and c. Thus the relation"is equal to" is transitive, for a = &, and b = c, together implythat a == c; but the relation "is perpendicular to" is not transi-tive, for a is perpendicular to b, and b is perpendicular to c donot together imply that a is perpendicular to c; in fact a is paral-lel to c.

Table I presents a classification of some of the most common re-lationships in elementary mathematics. The sign (+) indicatesthat the relationship has the property at the head of the column,while the sign (�) means the relationship does not have thatproperty.

TABLE I�PROPERTIES or MATHEMATICAL RELATIONSHIPS.Relation Reflexive Symmetric Transitive

1. Equality, is equal to......-....................� 4-4-4-2. Inequality, is greater than...-..-...-.....-. � � 4-3. Inequality, is less than.�......-..�..-...- � � 4-4. Similarity, is similar to..-...-........��.. 4- 4- 4-5. Congruence, is congruent to.���� 4-4-4-6. Equivalence, is equivalent to���.- 4-4-4-7. Parallelism, is parallel to...������ 4-4-4-8. Perpendicularity, is perpendicular to.. � 4- �

9. Bisection, is bisector of�... ��.�.�- � �

10. Complementary, is complement of�� � 4- �

.

11. Supplementary, is supplement of.�-� � 4- �

12. Factor, is factor of...��..���..�� 4- � 4-13. Multiple, is multiple cf........�.��.� 4- � 4-The reader might find it interesting to construct a similar

table for such relations as concurrence, collinear, coplanar,concentric, prime factor, exponent, logarithm, function, tangent,asymptote, commensurable, etc.

Tests may be constructed based upon these three propertiesto reveal understanding of mathematical relationships. Such atest, rather simple, is presented below.In each blank in the following statements insert the word or words which

make the statement true:A. c, 6, and c any three numbers.

1. If a ==6. and 6>c. then o�.�.���.c.

INDUSTRIAL GEOGRAPHY 273

2. If a<b, and b=c, then CL......-............C.3. If a>b, then b....................a. .

4. If a<&, and &< c, then a........-..-......-.c.5. If a is a factor of &. arid 6 ==c^ then a-.-.-...............c.6. If a is a multiple ot b, arid b is a factor of c, then a...........-........c.7. If a is a square of &, and 6 is a cube of c, then CL...................C.8. If a is a square of 6, and b is a square root of c, then a�.................c.

B. Z, in, and n any three straight lines.9. If I is parallel to m, and m is perpendicular to n, then L-................n.

10. If ? is perpendicular to w, and w is perpendicular to n, thenL.....-...-.-......n.

11. If Z+wi>^» and l-}-n>mj then w4-^� -�..-�.-� -?.C. x, y, and z any three angles.

12. If x is supplement of y, and y is equal to -?, then a;....................z.13. If x is complement of y, and y is complement of z, then a;-................j?.

14. If x is equal to y, and y is complement of 2, then a;.....-......-.......z.D. P, Q, and R any three plane figures.

15. If P is similar to Q, and Q is congruent to R, then P....................R.16. If P is congruent to Q, and Q is equivalent to R, then P.-...........-...-R.17. If P is similar to Q, and Q is equivalent to R, then P...-........-......R.

INDUSTRIAL GEOGRAPHY.

BY BERTHA WILLIS,Iowa City High School, Iowa City, Iowa.

Geography is a subject of intense human interest as it inti-mately affects the life of every individual and is, also, of greatcultural value. It should hold a prominent place in the highschool curriculum, if not placed on the list of required subjects.Too often the class is composed of those unhappy individualswho have failed in the first half of a year course and must thentake up some half year course at the end of the first semester.It is very difficult to keep such a class interested. While thesubject presents a great variety of interests, it just as readilylends itself to monotony. I remember my course in high schoolgeography with a feeling of great weariness, induced by a longseries of map drawing and compilation of dry facts.The following are a few of the devices I have used to arouse

the interest of the class. The work is organized into units andthe exercises accompany the units. The units are not arrangedin the order taken up.

UNIT I. CAUSES AND EFFECTS OF INDUSTRIAL GEOGRAPHY.Aims�to create a personal interest and to determine how man is

engaged in satisfying human wants.

Study 1. Occupations.Exercises:

a. Make a list of commodities used during the last 24 hours.b. Group the 30 occupations given into six groups according to their

similarity and name each group. Be prepared to defend your group-ing in class.