Frame Geometry: An Example in Posing and Solving Problems

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Frame Geometry: An Example in Posing and Solving ProblemsAuthor(s): Marion WalterSource: The Arithmetic Teacher, Vol. 28, No. 2 (October 1980), pp. 16-18Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41189691 .

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Frame Geometry: An Example in Posing and Solving Problems By Marion Walter

"Close your eyes and try to see your favorite picture. Then open them and draw it." Children who do this will usually enclose their pictures in a rectangle or frame and often take great pride in the details of their drawings (fig- 1).

After talking about the drawings, you could discuss how picture frames look and how they are made. We will ignore, however, the width of the frames and problems like mitering the corners - at least at the beginning. Imagine that the frames are made from precut, thin sticks, which the frame maker obtains from a stick factory. Usually the sticks can be ordered precut in any length but, alas, at the mo- ment the factory can supply sticks of only three different lengths: 3 cm, 5

Marion Walter is an associate professor in the De- partment of Mathematics at the University of Ore- gon in Eugene, where she teaches mathematics to prospective elementary school teachers. Her inter- est and work in informal geometry dates back to the early 1960s. She is the author of Boxes, Squares and Other Things, an NCTM pub- lication.

cm, and 8 cm. (Obviously the frames are for miniature pictures, perhaps model houses, stages, or doll houses.)

One Starting Problem Assume that the frame maker has as many sticks of lengths 3 cm, 5 cm, and 8 cm as we want. How many different shapes of frames can be made?

I purposely do not explain what is meant by different at the beginning, but the students soon ask. Is a frame that is 3 cm by 5 cm the same as one that is 5 cm by 3 cm? Students may, of course, pose this question just by pointing to two pictures (fig. 2, a and b) and ask- ing, Are they the same? What will you and they decide? Is the hook already at the back of the picture frame? Some- one may suggest that two hooks be put on the back of the picture so that it can be hung both ways. Such ideas can lead to fruitful discussions and further problems. For now we will decide that the two frames in figure 2 will be considered the same.

Try to solve the problem yourself, before reading on.

How do children solve the problem?

You may wish to supply the children with precut sticks. Older children could measure and cut the sticks them- selves. You can use wooden sticks and plastic joints, or straws and pipecleaner joints. I have also used very thin strips of card and glue, and even chains of large paper clips (fig. 3). In any case, one way of solving the problem is to make all the possible frames of some material. Some children do this in a haphazard way and perhaps do not obtain all the possible frames; other children may use a system to obtain them all. One bonus that the children who use the sticks or straws method reap is that they are surprised to find that the model is not rigid!

On another level, children may try to draw all possible frames and using grid paper helps them. Again, some children draw the different frames in random order, while others try some system.

More advanced children may just write out the dimensions of each

16 Arithmetic Teacher

frame. They, also, may not find all the frames at first trial. Figure 4a shows a list of a student who used no system, and 4b and 4c show lists of two students who used a system. Can you see what systems were used in lists b and c? Note that in c the student listed all the squares first; then, all the nonsquare rectangles.

Some students try and solve the problem "just by thinking." They mul- tiply three by three to get nine. Why does this not give the correct answer to "our" problem? To what problem is it the correct answer?

Children who have had some experi- ence with problem solving and pattern examination sometimes will solve the problem by first considering the case of only one available length, and then two and three. They first may count all the squares and then all nonsquare frames to obtain a table. (See the first three lines in table 1.) Some children go on to find how many frames can be made if four lengths are available. They see that there are four squares, and that the fourth new length can be combined with each of the three old ones to give three new nonsquare frames. For five lengths we have five squares and four new nonsquare frames, giving a total of ten nonsquare frames and fifteen in all. Some young- sters go on to solve the problem for larger numbers of sticks and for the general case of n sticks. It is a nice problem that leads to experiences with triangular numbers. Many students no- tice how the numbers increase, as well as other patterns in the table. No doubt your students and you will find other ways to solve the problem.

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Some children will realize that you can find out how may frames are possible without knowing the actual lengths of the three sticks. That is, if you repeat the problem with any three different lengths - say 6 cm, 1 1 cm, and 19 cm - the answer will be the same. For such children, it may be a good time to introduce the notation a, b, c for the given lengths.

Variations

How many frames are possible if a 3- by-5 frame is considered different from a 5-by-3 frame? That is, we consider the two frames shown in figure 2 as different. The results are shown in table 2. Here we have the square numbers ap- pearing!

Someone may suggest that each side of a frame can be made of two or more sticks. (You must decide whether you investigate or worry about whether two sticks would make a rigid side.) If you limit the conditions to two sticks per side, what lengths can be made now from 3-cm, 5-cm, and 8-cm sticks? No- tice, for example, that although 3 + 3 = 6 and 3 + 8=11 give new lengths, 3 + 5 = 8 does not. What numbers could you choose for starting lengths so that all sets of two sticks give a length different from 3,5, and 8? If we choose 3, 5, and 7 for starting lengths, we get all lengths different from 3, 5, and 7, but 3 + 7 and 5 + 5 give the same length. What starting lengths could be combined in pairs to give all new and different lengths? Here we have a rich arithmetic problem - espe- cially if you allow the combining of three lengths per side, or consider four starting stick lengths!

Table 1 Table 2 Numbers of- Numbers of- different different non- lengths squares squares Total lengths frames

¡ Ï Ö 1 i ¡ 2 2 1 3 2 4 3 3 3 6 3 9 4 4 3 + 3 = 6 10 4 16 5 5 6 + 4=10 15 i i i i i ; n n2

n n (k - l)(/i) n(n + 1) 2 2

October 1980 17

Fig. 5 Fig. 6

a. b. c.

Another problem may arise. Note that a 3-by-8 frame has the same shape as a (3 + 3)-by-(8 + 8) frame, and we can discuss frames that have different sizes but the same shape.

Still another problem that sometimes is suggested is to start off with a long stick of given length (say 20 cm) and ask how many ways can such sticks be cut into four pieces to make frames of different shapes and sizes (each side a whole number of centime- ters). You or the children will have to decide whether the whole stick must be used.

If you consider the amount of paper inside the frame, the matting, or the size of the picture, you are led naturally into considering area as well as perimeter problems. Of course, you can repeat the problems with wide frames and consider inside and outside dimensions! (At a recent workshop, a teacher started exploring how such frames fit snugly inside each other - see fig. 5. Such nesting frames would store very economically.)

Just when you think you have done enough with the subject of frames, a student will suggest exploring triangular, trapezoidal or other polygonal frames. Before long, children are inves- tigating which sets of three sticks can make a triangle - they are surprised that a 3-cm, a 5-cm, and an 8-cm stick do not. We had no such problems with rectangles! Who said that smaller numbers always involve easier problems! How many different triangles will sticks of length a, b, and c make if all combinations of the three chosen

lengths make triangles? Children will have to decide whether the two triangles in figs. 6a and 6b are "the same." So here we will have to consider not only turns, as in figure 2a and 2b and in 6a and 6c, but also flips. And what if you considered only equilateral or isosceles triangles?

Conclusion This article is only a brief outline of the paths that might be followed in working with one unit in informal geometry with children from a wide range of age and ability. The unit leaves plenty of scope for children to not only come up with new ideas, but to work through some of them. (For a way to extend the work into three dimensions, see Kuper and Walter 1976.)

The final "right" answer is not the only thing that matters. Posing and solving problems, clarifying problems, and finding different ways of solving the problems are also important activi- ties. When children get different an- swers because they have interpreted a problem in different ways, rich and useful discussions can follow. In that way mathematics is seen as more than an endless series of worksheets, often done in isolation and checked rou- tinely by the teacher or by students with an answer-book. Children see geometry as more than learning a bunch of definitions, identifying and classifying shapes, and applying for- mulas. To give just one example, in this unit a discussion about the differ- ence between a square and a rectangle comes up naturally because that dis-

tinction is significant in the discussion; it is more than one, narrow objective - "to learn what a square is" - on a page. Note also that the arithmetic problems emerge freely from some geometric no- tion and need not be isolated into different chapters.

You may find it worthwhile to jot down some of the many mathematical concepts involved in working with this unit. Your list may make you mistrust the narrow behavioral objectives given on each page of so many textbooks and may make you question whether the structures of such texts are not restrict- ing the content and activity in informal geometry in elementary and junior high school. Reference

Kuper, M., and M. Walter. "From Edges to Sol- ids." Mathematics Teaching 74 (March 1976): 20-23. m

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18 Arithmetic Teacher

Frame Geometry: An Example in Posing and Solving Problems

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