New Strategies for Teaching Properties of Number Systems

New Strategies for Teaching Propertiesof Number Systems

John L. Creswell and Margaret WiscambCollege of Education, University of Houston^ Houston, Texas 77004

. Bruner has hypothesized that any subject can be taught to anygrade level in some intellectually honest form. Using this as a chal-lenge the authors have been teaching elements of group theory tostudents from the sixth grade through the fifth year mathematicslevel in high school.Many mathematics teachers have been having difficulty teaching

the structure of number systems, i.e., the natural numbers, thesystem of integers, etc. It occurred to the authors that demonstratingthe structure of some simple algebraic system using concrete objectsmight be worthwhile. Since the group is one of the simplest algebraicsystems, it appeared that this might be the one to use.

In the summer of 1969 the authors had occasion to teach a group ofchildren who had just completed the fifth grade. Each was talented inarithmetic and was attending summer school for enrichment ex-periences. After two weeks, their teacher called for help since she hadexhausted all her background material.

Since these were talented children and since this material had notbeen introduced to students below junior high school level previously,it was decided to see what reaction these pre-sixth graders wouldmanifest.The children were familiar with the properties of the whole num-

bers and could give examples and illustrations of the closure, com-mutative, associative, and other properties. It was decided to con-front them with the problem of finding a system which was not com-mutative. After having a twenty-minute discussion about the in-gredients of a number or mathematical system, they were instructedto go to the library to see if they could find examples of non-com-mutative systems.The next day they came back to class and each reported that he

could find no such system. All systems they could find or think ofwere commutative. It was here that the idea of the manipulations ofthe equilateral triangle was introduced.Each child was given an equilateral triangle such as pictured in

Figure II. As the authors demonstrated manipulations on the chalk-board, the children followed through at their desks.By numbering the vertices of the triangle 1, 2, and 3, we began to

form elements by reflecting the triangle on axes passing through eachvertex. After we did the first one through vertex 1, the children

635

636 School Science and Mathematics

formed the other two elements themselves. The same thing occurredwhen the elements formed from the rotations were constructed.Throughout this activity the children were very actively involvedand enthusiastic.Once the six elements were identified, the children were asked if

there were more than six elements which could be formed in thismanner. They were led into discovering that six and only six elementscould be formed. (The reader will recall that there is a theorem to thiseffect, i.e., from a set of n elements, n\ permutations may be formed.)

After forming the elements, the author led the students into con-structing the table of the elements (see Table 1) by operating on twoelements at a time, using the manipulations of the triangles. Forexample, if we wished to have A o D, we would take the triangle tostandard position and then reflect on the imaginary rod throughvertex 1, i.e., A. Having done this, we would then do D, i.e., rotate thetriangle 120° counter-clockwise (see Figure 1). This would then pro-duce the element 5. These children needed only one or two examplesin order to complete the table on their own. (It was here that somediscovered that A o D was not the same as D o A. We did not pressthe point at this time.)Once the table was completed the children were asked to investi-

gate the table to determine the properties of the system. The readercan imagine the tremendous enthusiasm of the students as they dis-covered that the system had closure, and the surprise as they foundthat the system was not commutative. Further investigation revealedthat the associative property held. Many were surprised that a sys-tem could be associative and not commutative.In investigating to determine if the system had an identity, the

children had to rely on their understanding of the concept of identity,rather than recognize that one (1) was the multiplying identity andzero (0) the adding identity. They had to know and understand thefunctions of an identity element, and it was found in this exercise thatalthough most knew about one (1) and zero (0), many had troubledeciding if this system contained an identity element. The method ofguided discovery was used here, and soon it was felt that all under-stood the concept of identity.

Proceeding one step further, the students were led to understandthe concept of inverse through a similar exercise as described above.The authors believe that teaching structure, i.e., properties of

number systems, can be accomplished much more effectively in thismanner than by the usual method of drill depending primarily uponnumbers, which can be easily memorized. For example, it doesn^ttake much understanding to recognize that a (�5) is the adding in-

Teaching Properties of Number Systems 637

verse of a (+5), or that 1 is the multiplying identity and that 0 is theadding identity.

After proceeding through the rotations and reflections of theequilateral triangle, the students, on their own, did the same thingfor the square. Later, it was learned that these same students haddeveloped a mathematical system from the rotations and reflectionsof the pentagon.

It is felt that talented fifth and sixth grade students can benefitmuch from the exercises described above. Following is described somesimilar activities for junior high school.The demonstration at the junior high school level begins with the

formation of the elements of our system. The usual approach is to leteach student have an equilateral triangle in his hands. (See illustra-tion below)

FIG. I

Using a larger triangle to represent standard position, we can letour first three elements be rotations of the triangle in a counter-clockwise direction. Thus a rotation through 120° will be defined to beelement

/I 2 3^D =(-C 2 3)\3 1 2/.\3 1 2/

That is, a rotation through 120° takes the vertex numbered 1 intostandard position 3, 2 into 1 and 3 into 2. Two 120° rotations, or arotation through 240° will be called element

/I 2 3\^=( )

\2 3 I/.

Likewise three 120° rotations, or a rotation through 360° will be theelement which we shall denote by

/I 2 3\

^M 2 3/.

638School Science and Mathematics

The remaining three elements will be reflections of the triangle ontoitself; see Figure II. The reader can visualize axis la as a rod. There-flection of the triangle across this rod will be our fourth element

/I 2 3\A = }

Continuing, we have the reflection of the triangle across the axis 26which we will call element

/I 2 3\B =( )

\3 2 I/

and the reflection across axis 3c, element

/I 2 3\

^^ 1 J.Thus we have the set of elements {A, B, C, D, £, 1}.

FIG.II

For any two elements x and y of the set, we may define the binaryoperation o on this set, x o y, to be first x followed by y. For example,in order to find A o C, one would have to reflect on rod la (see FigureII) and then reflect on rod 3c from standard position. On paper wehave

/I 2 3\ /I 2 3\ /I 2 3\AoC =( ) o ( )=( )

\1 3 2/ \2 1 3/ \2 3 I/ ^which is the rotation through 240°. This is in effect permutationmultiplication. However, for the junior high school level the oper-ation is performed by actually manipulating the triangles. Theteacher may, if he wishes and if his class is sufficiently sophisticatedin mathematics, explain permutation multiplication.

The. result of all possible combinations results in Table I.We have a set of elements and a defined operation on the set. Now

the teacher must lead his class to discover what properties exist byexamining Table I.


TABLE I

o

ABCDEI

A

IEDCBA

B

DIEACB

C

EDIBAC

D

BCAEID

E

CABIDE

I

ABCDEI

1. Closure. Yes, for the result of operating on two elements of theset produces a unique third element of the set.

2. Commutative. No, for it can be seen that A o Cy^C o A. (Thisshould be stressed, for it is probably the first time the studentshave encountered a non-commutative system.)

3. Associative. Yes, this can be verified by examining several dif-ferent combinations such as (A o D) o E==A o (-D o £). Sincethis is a finite system, it poses no serious problem.

4. Identity. Yes, the identity is J, for it can be verified that I o A =A o I==A for any element of the set. (Check column I androw 1)

5. Inverses. Yes, for it can be seen that for any element x of theset there exists an element x~1 such that x o x^Î. (each rowand column contains I)

The authors have used the foregoing demonstration with studentsin the 6th, 7th, 8th and 9th grades. (If skillfully used this procedurewill impress upon students the idea of what certain properties en-able them to do with operations on sets.) Contrast this with the rotelearning which still persists in far too many classrooms today.The group of the rigid rotations and reflections of the square may

be used in the same manner. Figure III indicates the rotations andFigure IV indicates the reflections.

FIG. Ill FIG. IV

One 90° clockwise rotation is element

/I 2 3 4^\/I 2 3 4\R"{4 1 2 ,)�Ri =(\4 1 2 3^

640School Science and Mathematics

two 90° rotations

/I 2 3 4\R. = ( ) ;

\3 4 1 2/

three

/I 2 3 4-^_ /I 2 3 4\

=\2 3 4 l)9^3 = (\2 3 4 \}

and four

_ /I 2 3 4\

M 2 3 4/L

/I 2 3 4^^4 =(

V 2 3 4^

The reflections become the following elements: the horizontal re-flection is

=f2 3 4’),\4 3 2 I/^

the vertical reflection

/I 2 3 4\’"(2 1 4 3)-reflection across the diagonal 1-3 is element

/I 2 3 4^_ /I 2 3 4\

\1 4 3 2/D == _

\1 4 3 1)

and across the diagonal 2-4, element

/I 2 3 4^/I234\Df =()

\3214/ �

Df =(\3 2 1 4,

Defining the operation o as for the equilateral triangle, we haveTable II as follows:

TABLE II

IRiR2RzHV D D’

IRiRiR^HVDD’

IRiR2RzHVDD’RiR^RsIDD1VHR2RzIRiVHD’DRzIRiR2D’DHVHD’VDIR2RsR^VDHD’RÎRi%DHD’VRiRzIR^D’VDHRsRiR;I


The properties of this system may be verified in the same manner aswas done with the equilateral triangle.The same demonstration outlined above has also been done for

4th and 5th year high school mathematics students. Their reactionwas much the same as that of the junior high school students. How-ever with the more mathematically mature students the idea of thegroup was introduced first.We define a group as follows:A set G forms a group with respect to the operation o if the fol-

lowing properties hold true:

1. For any a and b in G, a o b is in G". (closure)2. For any o, 6, c in G, (a o b) o c^==a o (& o c). (associative)3. There is an element i in G such that for any a in G, i o a=a o

i==a (Identity)4. For every a in G there exists an a~1 in G such that a o a~1

==a~1 o a==i (inverses)

Then a permutation is defined as a 1-1 correspondence of a set ontoitself. Using the set of elements {1, 2, 3}, it is seen that there are 3!or 6 permutations on three elements and that these permutationelements are the same as those derived concretely by using the ro-tations and reflections of the equilateral triangle.

It follows that there are n\ permutations on n elements. The prod-uct of two of these permutations, say P and R, symbolized by P oR, means first perform P and then perform R, Thus if

/I 2 3 n\P =( )

\bi 62 &3 � � � bn/

and

then

/bi &2 bs ’ � � bn\

\Ci C’2 Cz - � - Cn)

/I 2 3 n\PoR=( )\^1 Cs ^3 � � � Cn/ �

It is quite evident that this operation is the same as was defined pre-viously for the triangle.

Permutation multiplication is not, in general, commutative. Forexample, for

/I 2 3>D={-C 2 3)\3 1 2/\3 1 2/


/I 2 3\A =( - I

V 3 2/,

/I23\/I 2 3N/I23\/I23\= ()but A o D =(}\213/\32I/,

Do A = ()but -4 o D = (\213/\3 2 ^

thus jD o AÂ o D.This operation, on the other hand, is associative and may be proved

so very easily. Suppose we let P and R be the permutations on nsymbols denned above and let

T =/CiC2C3 ’ � ’ Cn\

\^1^2^3 � � � dn/ ,

then

/I 23 � � � n\(P o R) o T = ( )

Vid^s � � � dn/

Now

/bib2bs ’ � � bn\ /I 2 3 � � � n \R o T = ( ) and P o (R o T) = ( )

VlJ2^3 � � � dn/ \d^dz ’ � � Jn/.

Hence (P o R) o T=P o {R Q T) and permutation multiplication isassociative.

Since all permutations on n symbols are in the original set and theproduct of two permutations is a permutation, the operation isclosed.The identity is

/.f12 3---"),.(12 3---")U 2 3 � � � n)U 2 3 � � � n)

because I o R==R o I==R for any permutation jR.

P~1 (the inverse of P) would be

/^2&3 � � � bn\

V 2 3 � �

� n )because

_ /I 2 3 � � � n \ /b^bs � � - bn\"

\bib2bs ’ . � bn)° \1 2 3 � � � u )/I 2 3 � � � n \ /bib2bs ’ � - bn>p o p-i = ( ) o (Vl^3 � � � bn/ \1 2 3 � � � U

/123...,.^\1 2 3 � � � n)


Consequently, this is a group.It is hoped that some student will discover that the rotations of the

equilateral triangle form a subgroup of the group of rotations and re-flections of this triangle, as do the rotations of the square. It is alsohoped that they will discover that both of these subgroups are com-mutative.The brave, creative and innovative teacher will endeavor to lead

his students to discover that the two commutative subgroups aboveare isomorphic to the group of integers modulo three under addition,and to the group of integers modulo four for addition, respectively.Many interesting discoveries and discussions should be the result.

Since there seems to be a real shortage of elementary examples ofnon-commutative groups, let us introduce two such examples, oneinfinite and one finite.

GROUP ALet S be the set of linear functions of the form ax-\-b, where a and

b are real numbers, aÔ, and let the operation, o, be composition offunctions. Thus if/, g are elements of S

teo/)0.)=K/^))For example, if f(x)==2x+l, g(x)==5x-2, then g o f(x) == g(2x+1)=5(2a;+l)-2==10;r+3 (5, o) is a group.

1. S is closed under the operation o. If f(x)=ax-}-b\ g(x)==cx-\-d,ayÔ, cÔ, then g o f(x)=g(ax-\-b)=c{ax-\-b)Jrd=caxJr(bc-{-d),where cayÔ. Thus the composite of two elements of S is an element ofS.

2. The group identity is i{x)=x.3. Composition of functions is associative.4. Each element of S has an inverse. If f(x)==ax-}-b, aÔ, then

f~\x) = x/a� b/a since

y-i of(x) = f-\ax + b) == l/a{ax + b) - b/a = x,

and

f o/-1^) = /(l/o x - b/a) == a(l/a x - b/a) + b = x

However, (5, o) is not commutative. Consider f(x) = x+1; g(x) = 2x.Then g o (x) =g{x+1) = 2(x+1) = 2x+ 2, while

fog<ix) =/(2rv) == 2^+1.

GROUP BNow for our finite group let us take the set of linear functions with

coefficients from the set of integers modulo three, again with the


operation of composition of functions. There are only six elements inthe set: x, x+1, x+2, 2x, 2x+l, 2x+2. They form a noncommuta-tive group, as can be seen from Table III.

TABLE III

0

2x+22X+12xx+2x+1x

2x+2

scx+1x+22y

2x+l2x+2

2x+l

x+2x

x+12x+22x

2x+l

2x

x+1x+2x

2x+l2x+22x

x+2

2x+l2x

2x+2x+1x

x+2

x+1

2x2x+22x+lx

x+2x+1

x

2x+22x+l2xx+2x+1x

Amusingly enough, the first of these two examples is not reallydifferent from a group of matrices under matrix multiplication, andthe second is the same as the group of permutations on three objects.

For consider the linear function f(x) =ax+b, ayÔ. We assign tothis function the matrix

/a 0\Mf = ( )

\b I/

This is non-singular, since oÔ. Then if g(x) =cx+d, cXO,

(c 0\M, ==( )

\d I/

Corresponding to the function g of(x) we have the matrix

/a0\/c0\ fac0\fc U\ fac U\

lo=:\. . }\ . .== , , , J == M^MrM,={)}==(}=M,\bl/\dI/ \bc + dI/

The operation is thus preserved and the group A is isomorphic tothe group of 2X2 matrices of the form

/a 0\c i)°"0’under matrix multiplication.Group B can be shown to be isomorphic to the set of permutations

on three objects, say the numbers 1, 2, 3.To set up the one-to-one correspondence between function and

permutation, for each function we look at the functional values of 1,2, 3, reducing modulo 3 where necessary, except that we shall repre-sent 0 by the numeral 3.


Thus for the function f(x)=x+l, we have /(I) =2; /(2)== =3;(3) = 1. So to x-\-1 we assign the permutation

/I 2 3\( ) = ^\2 3 I/\2 3 1,

In the same way we find

x corresponds to (l 2 3)^M 2 3/

/I 2 3\x + 2 corresponds to ( } = D

\3 1 2/

/I 2 3\2» corresponds to ( ) == C"

\2 1 3/

/I 2 3\2x + 1 corresponds to ( j = B

\3 2 I/<321

(’23)-\132/ ^2x + 2 corresponds to \1 3 2^

Clearly if/ and g are two functions in Group B, g o/ will correspondto the permutation obtained by taking the permutation assigned to/ and following it by the permutation assigned to g. For example

(x + 1) o (2x) = 2x + 1

/I 2 3\/1 2 3\ /I 2 3\

\2 3 lA2 1 3/ \3 2 I/

where we perform the permutation on the right first.(Compare Table I and Table III)It is the creative, innovative teacher who looks beyond his textbook

to find ideas which wdll help explain mathematical concepts and en-rich their teaching. The authors have found the materials and tech-niques presented herein to be of great value and it is hoped that otherteachers also will find them to be useful.

SEE YOU IN CHICAGONOVEMBER 1970

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New Strategies for Teaching Properties of Number Systems