Taxicab geometry, rabbits, and Pascal's triangle—discoveries in a sixth-grade classroom

Taxicab geometry, rabbits, and Pascal's triangle—discoveries in a sixth-grade classroomAuthor(s): DAVID M. CLARKSONSource: The Arithmetic Teacher, Vol. 9, No. 6 (OCTOBER 1962), pp. 308-313Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41184644 .

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Taxicab geometry, rabbits, and Pascal's triangle- discoveries in a sixth-grade classroom

DAVID M. CLARKSON Poughkeepsie Public School, Poughkeepsie, New York

During the 1961-62 academic year Mr. Clarkson taught in Grades 1-9

at the Spackenkill Union Free School.

1 wo aspects of the current experimenta- tion with our elementary-grades mathematics instruction have been perhaps underemphasized. One is the opportunity a teacher has to take advantage of the imagination and initiative of her pupils re- leased by the freer approach to the curriculum, and the other is the effect on the teacher herself when the pursuit of a topic new to elementary school leads her to discover a rich vein of mathematics on her own. The following article is an ac- count of the discovery of triangular numbers by a sixth grader, and its consequence, the pursuit of an important mathematical idea by both the class and the teacher. The project took three forty-five minute class periods and the student remarks quoted were taken from a tape recording of the sessions.

One day last November I went into a sixth-grade classroom to teach a lesson from the "Madison Project" syllabus. In previous weekly meetings the class had been busy writing names for as many numbers as possible using four and only four "4V' At the beginning, the children had used only the signs for the four basic operations, but it soon became apparent that new symbols for more complicated operations would be necessary if the list of numbers was to be expanded. By the time of the November class we had introduced the use of the decimal point, parentheses, the square root sign, and the factorial

sign. Four factorial (4!) means4X3X2Xl or 24. Many of the children had been able to write the names for all whole numbers up to 30 and one boy had completed a list through 72 with most of the rest up to 100 as well. An example of the sophistication of the solutions is one little girl's name for 31:

_ V4 4Î+V4+- -

.4

The pupils had obviously enjoyed this game and my weekly class with them usually started with some additions to our class list or a question about the use of new symbols.

I was about to begin my "Madison Project" lesson when a boy in the back of the room asked if there were a symbol for 4+3+2 + 1.1 had never heard of one, but the class agreed that it would help out a lot in the four "4's" game if we could find and use one. It was suggested that such a symbol might be called a "plustorial" and discussion turned to the origins of symbols. Some symbols look like what they signify. For example, "1" looks like a single object. Other symbols can be traced back to their obviously meaningful predecessors. Thus the symbol "2" may be derived from two horizontal lines, = , and the connect- ing stroke made when the pen moves from the top line to the bottom one. The Roman numeral V was derived from the shape of

308 The Arithmetic Teacher



1 3 6 10 15 21

Figure 1

an outstretched hand. Many symbols have been altered so much by usage that they cannot be traced back to an obvious source, a symbolic equivalent to an ono- matopoetic word, and, of course, many symbols are just invented helter-skelter without any reference to their meaning. Four factorial is a good example of this. The exclamation point used is often greeted with laughter when first introduced because, instead of pointing to the meaning of 4!, it points in another direction. Ohlooki A 4!

The class wanted a new symbol, a four "plustorial." "All right," I said, "you can have it if you can invent one that makes sense, one that has an obvious reference to the meaning of 4+3+2 + 1. " Less than a minute after the challenge had been given, the symbol seeker in the back of the room had his hand up. "I have a symbol," he said, "It's a triangle with a 4 inside it." I asked what he meant, and Johnny, coming to the board, drew the following figure:

If you take four dots and put three dots above them, and two above that, and one on top, then you have a picture of four

"plustorial" or /' . "Say, it's like bowl-

ing pins," said another child, and the class was off on a discussion that eventually encompassed taxicab geometry, the population growth in a rabbit pen, and the patterns of Pascal's triangle.

We had soon made a list of triangular numbers, first finding them by dot pic- tures (see Fig. 1) and also a list of square numbers (see Fig. 2). We made a chart of triangular and square numbers (see Fig. 3) so that we could look for patterns in them.

Here are some of the observations, made by the children, taken from the tape of the session:

"Start with 1 you add 2, get 3. Then you add 3, and you keep on adding one more and you end up with the next triangular number."

"Square numbers go odd-even odd- even • • • "

"The differences between square numbers go 3, 5, 7, 9, 11, • • • "

"The first two triangular numbers are odd, the next two are even, the next two odd, the next two even. • • • "

"1, 4, 9, 16 all have whole number square roots: 1, 2, 3, 4, • • • "

Figure 2

14 9 10 25

October 1962 309



Figure 3 1 3 6 10 15 21 28 36 45 55 • • • 1 4 9 16 25 36 49 64 81 100 •••

"I seem to notice a relationship between the square numbers and the triangular numbers. The square "1" and the triangular "3", there's a difference of 2. The difference between 4 and 6 is 2. The difference between 9 and 10 • • • Oh, it doesn't work."

"Well 36 could be a square or a triangular number."

"Sois 1."

"There is a relationship! 1 and 1, there's a zero relationship. 3 and 4 there's a 1 relationship. 6 and 9 there's a 3 relationship • • • " (We found out that he meant the successive differences between triangular and square numbers were the triangular numbers in order. The next remark below extended his idea.)

"Say, if you add the two triangular numbers you get the square numbers below. 1 and 3 are 4; 3 and 6 are 9; 6 and 10 are 16."

"If you take the differences of the square numbers and the differences of the triangular numbers and add them and then take the differences again it's always 3. I think you could check them this way." It turned out that what the last child meant was the scheme shown in Figure 4. A very important idea had been introduced into our discussion by this girl who observed that regularities of pattern could be used to check computations. She had chosen an extremely complicated way to check the accuracy of our work, and we

Figure 4

did go on to look at some of the simpler ways to check up on triangular and square numbers, but her observation was used again and again in the work that followed. What is the next square number in the list above? The class no longer needed to draw dot diagrams. "It's 81." Why? "Because the next difference should be 17, and 64 plus 17 is 81." "Hey, that's 9 times 9." And so our first session ended.

But before I could leave the classroom Johnny had to show me another new symbol, 4 "pyramidical." As he described it, "... the square in a triangle, well that would be a number within that square within a triangle . . . the square of four on down. ..." It turned out that he was thinking of the pyramid formed when, for example, you pile cannon balls on a square base. From above, they would look like a set of squares, and from the side, like triangles:

<> • • » • • •

| y*l " • • • • II • • (

The first layer would consist of a 4X4 square of 16 cannon balls; the next, a 3X3 square, then a 2X2 square, and finally the 1X1 square of one cannon ball on top. Johnny's symbol for this

sum, 16+9+4 + 1, or 30, was ^^. He

had been working out the patterns of the pyramidical series while the rest of us

Triangular numbers: 1, 3, 6, 10, 15, 21, 28, 36, •• Differences: 2 3 4 5 6 7 8

Square numbers: 1, 4, 9, 16, 25, 36, 49, 64, • • • Differences: 3 5 7 9 11 13 15

Sum of Differences: 5, 8, 11, 14, 17, 20, 23, •• Differences: 3 3 3 3 3 3




had been exploring simpler topics. I had wondered why he never joined in our discussion after he had started us off with his original discovery. Now I knew. Johnny is a poor student by conventional stand- ards. He is inattentive in class and doesn't have any friends. I wonder what else he thinks about while he appears to be day- dreaming there in the back of the room.

During the next week I wondered how best to develop this discovery of triangular numbers and their patterns. I de- cided that some work with Pascal's triangle might be worthwhile so I undertook some extra reading. I found that I was making discoveries myself. Pascal's triangle is an array of numbers :

1 1 1 1 2 1 13 3 1 14 6 4 1 1 5 10 10 5 1

Add any two adjacent numbers in any row and the sum yields the number below the second addend. For example, the 6 in the fifth row is the sum of the 3 above it and the 3 above and to the left. I had first met Pascal's triangle in high school al- gebra when we studied the binomial theorem. Each row of the triangle gives the coefficients of the terms in the expansion of (a+b)n where n is the second number in any row of the triangle. Thus, (a+6)2 equals a2+2ab+b2, the coefficients 1, 2, and 1 forming the third row of our triangle.

It did not seem feasible to discuss the binomial theorem with my sixth grade, at least not until they had had some experience with simpler algebraic theorems. However, I did try the following ap- proaches with this sixth-grade class and other fifth- and sixth-grade classes in the weeks that followed. While each of the paths outlined below was not above the intellectual level of average fifth and sixth graders, the last method worked best.

Pascal's triangle may be obtained by forming the combinations of a things taken b at a time, symbolized as aC&:

iCo iCi

2C0 2C1 2C2

3C0 3C1 3C2 3C3

For example, the number of combinations of three things, say A, B, and C, taken two at a time when repetitions are not considered, is 3: AB, AC, and BC.

One approach to n-dimensional geometry used by the instructor of a course I had taken years ago1 also yielded Pascal's triangle. He had started with the simplest geometric object, a point, or vertex, and by drawing the point out into one dimension, had formed a line segment with two vertices or endpoints. Hence, when a point is extended into another dimension a line segment results. If we take a third vertex in a second dimension, and draw out the line segment to it we get a triangle with three vertices, three edges or line seg- ments, and one new "thing," a two- dimensional object we will call a plane surface, or "face." Thus, the extension of an edge gives a face. If we now take a point or vertex in a third dimension and draw out our triangle, each vertex forms a new edge , each edge, a new face. Our new three-dimensional object has a volume. The triangular pyramid or tetra- hedron formed has four vertices, six edges (the three old ones from the triangle and three new ones from the drawn- out vertices), four faces and one volume. By simple extension we can go into four-, five-, and even n-dimensions this way. Although we may never see a four- dimensional triangle, we know the number of vertices, edges, faces, and volumes it must have. Figure 5 shows how Pascal's triangle is formed by this data. Only the first column is missing.

The n-dimensional construction of our triangle had a close relation to our original

1 E. Kasner at Columbia University, 1948.

October 1962 «*H



3-D 4-D Vertices Edges Faces things things

1 2 1 3 3 1 4 6 4 1 5 10 10 5 1

Figure 5

and simplest method because it showed geometrically how each number was the sum of the two numbers above and to the left of it.

In further reading2 I found taxicab geometry and made a discovery of my own. Taxicab space is like the lines on ordinary graph paper, or the pattern of the streets in some cities. In such a space you may not travel as the crow flies but must follow the roads which are equidistant from each other and always meet at right angles. Because of this there is usually more than one way to travel the shortest distance between two points. For example, to get from A to B in the figure below (Fig. 6) I might go two blocks north and one block east. But I could go one east and two north. Or, if I liked to turn corners, I could go one north, one east, and one north again. In each case, the distance traveled is just three blocks, indeed, each was a route that carried me over the shortest distance between A and£.

s Karl Menger, You Will Like Geometry (Chicago: Museum of Science and Industry, 1961), p. 5.

Figure 6

B__

Let us make a chart of our travels in taxicab space as follows: We will let D equal the shortest distance between two given points. V will equal the distance that has to be traveled in a due north direction no matter which route is taken. If our points are on the same east-west street then V is zero. In the example above, V is 2 and there are three shortest routes between the two points. We will let P stand for the number of shortest routes in each case (see Fig. 7 below). Notice that the values for P give successive rows of Pascal's triangle. Why is this so? Is there a similarity between this der- ivation of Pascal's triangle and the others? Look at the column enclosed in a box. Here our points are four blocks apart whichever route you take. The vertical distance to travel north is always two blocks whichever way you go. The figure below may be helpful in tracing the six possible shortest distances.

1 IB

M I Route 1 : 2 north and 2 east Route 2: 1 north, 1 east, 1 north, 1 east Route 3 : 1 north, 2 east, 1 north Route 4: 1 east, 2 north, 1 east Route 5: 1 east, 1 north, 1 east, 1 north Route 6 : 2 east, 2 north

Observe that when you have gone one block north you are in the same position as a person with 3 blocks to go, one of them in a due north direction. From our chart we know that a person in this position has 3 possible routes. Similarly, if you go one block east you have 3 blocks

Figure 7

D011222333344444 70010120123012 3 4 P111121133114641




left to go, but a vertical distance of 2 blocks, and from our chart we know that here, too, there are just three possibilities. So there are 3 plus 3 or six ways to go altogether. Finding the sum of two simpler routes gives us our answer.

My experience in working out these methods of constructing Pascal's triangle had been to return again and again to a basic idea, that of adding two numbers to obtain a third. I now realized that there were probably infinitely many ways to get the triangle, and only my lack of imagination prevented me from discover- ing more of them. The pattern, its uni- versality, became an object of wonder for me, and, I hope, for some of my pupils. Later, I read of yet another way to build up Pascal's triangle, literally, with Cuisenaire rods,3 and here the pattern of building on what is already built is most concretely clear.

The sixth graders noticed as many patterns within Pascal's triangle as they had in our first session. I thought that we had finished with our topic when further reading4 uncovered a dramatic extension of this basic mathematical idea. A thir- teenth-century Italian mathematician, Leonardo of Pisa, nick-named Fibonacci, had posed a problem about rabbits in 1202. If a pair of rabbits is placed in a pen, and if rabbits take one month to mature, and after maturing produce another pair of rabbits each month, and if none of the rabbits die or escape, how many pairs of rabbits will be found in the

1 P. Puig Adam, "Progressions arithmétiques d'ordre su- périeur," in Le Matériel pour V enseignement des mathématiques (Neuchâtel, Switzerland: Delachaux & Niestlé. 1958).

* N. N. Vorob'ev, Fibonacci Numbers (New York: Blaisdell Publishing Co., 1961), p. 13.

pen after one year? A census count at the end of each month yields the Fibonacci series: 1, 1, 2, 3, 5, 8, 13, 21, 34, • • . where each new number is the sum of its two predecessors. Many objects in nature take the shape they have because their growth is similar to that of the rabbit population. The number of branches on a tree, and the spirals of some shells and seeds can be shown to fit the pattern of the Fibonacci numbers. A recent film by Walt Disney, Donald in Mathemagic Land, makes excellent educational use of this relationship.

Since the Fibonacci numbers are obtained in a way similar to that of the numbers in Pascal's triangle, it seems reasonable to look for a relationship here, and indeed there is one. If you take the sum of the numbers along each diagonal of Pascal's triangle shown below you will obtain the Fibonacci series.

/¥Jf 6 4 1 /T 5 10 10 5 1

It is not so remarkable, once you con- sider it, that elementary-school children are as good as adults in detecting such relationships and patterns. Perhaps they are even better at it. The search for pattern plays a large role in modern mathematics instruction. School children are quick to detect it and capable of vivid insights and significant discoveries. Why should we not allow them to pursue some of these rich veins of mathematical gold? Perhaps we may follow their discoveries with some of our own.

"You can't stop people from thinking - but you can start them." - Origin unknown.

October 1962 3 13



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Taxicab geometry, rabbits, and Pascal's triangle—discoveries in a sixth-grade classroom