A Problem Solving Activity

A Problem Solving ActivityAuthor(s): Ed VaughanSource: Mathematics in School, Vol. 15, No. 3 (May, 1986), pp. 2-6Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30214073 .

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A Problem

Solvino Act0

by Ed Vaughan North Riding College

The teachers following our Mathematical Association Diploma course now take it for granted that problem- solving is important - if for no other reason than the tutors keep saying so! We have certainly devoted a significant part of the course to this aspect of Cockcroft 243. We have solved a number of problems and we have thought about some of the processes that characterise problem-solving activity. Furthermore we felt we should learn a great deal by considering how it might be possible to create our own problems. We were particularly interested in problem- solving activities suitable for use in school.

As a way in, it was agreed we should take a broad view of the many ideas that already exist and that are designed to encourage children in problem-solving activity and investigational work. Maybe we would be able to classify the vast number of different problems we found. If we could, it would clearly help in our aim to construct problems of our own. Following on from this introductory stage, we envisaged two directions we could pursue: We could start with specified materials and try to design problems related to these. Or, we could try to develop our own problems from specific mathematical ideas.

Looking for Problems There were a number of sources. Some problems appealed to us more than others, for various reasons. We immediately became selective. An important criterion for us, in selecting a problem, was whether it could be easily presented to children; those problems arising from a simple starting point were favoured.' Also, we are all involved in teaching mathematics at various levels (infant to lower secondary). So we were anxious to categorise the problems according to age-range suitability.

We were struck by the many different kinds of problems available. Of course, this is how it should be, for we were

aware that problems are not likely to give rise to true problem-solving activity unless they present unfamiliar situations.

"Problem solving is defined as working in an unfamiliar situation; it necessarily involves both content and process aspects". " .. .the general activity of problem-solving - the essence of which is dealing with unfamiliar, non-routine situations." 1

Ought we really to classify them? There could well be dangers if we were to use such a classification to aid our teaching of problem solving skills. Yet in order for us to gain a greater feel for this wide variety of problems, we decided at this stage it would be helpful.

Our attempts at sorting the selected problems resulted in our adopting the following classification:

C1. Combinatorial problems C2. Geometrical situations giving rise to number

patterns C3. Challenges in Number C4. Number chains/sequences produced by

algorithms CS. Shapes constructed under stated conditions C6. Shape puzzles; closed problems in work on

Shape C7. Logic games and puzzles

Problems falling into these classes were regarded as "contrived" as opposed to "real-life, everyday" problems. Mathematical problems arising out of "everyday situations" were not considered.

It would be foolish to claim that every problem in our collection could be thought of as a member of one of the classes C1 to C7 above; it was often difficult to decide to which category a particular problem should belong; sometimes a problem seemed to fit more than one category. Yet the process of our trying to match problem to category did help us to understand more clearly the relationships between different problems. The following set of well-known - even classical problems - serve to exemplify each of the classes C1 to C7.

C1. How many different pentominoes are there?2 C2. Into how many parts is space divided by five planes?3 C3. Which of the even numbers can be expressed as the sum

of two primes?4 (Goldbach's conjecture) C4. Input a whole number. If it is even then halve it;

otherwise treble it and add 1. If this gives 1 then stop; otherwise return to previous step. Investigate the string of values produced corresponding to particular input values.s

C5. What class of ~Number of pairs of equal sides quadrilaterals

corresponds to

Number 1 2 each cell in the table?6

of 1

right 2

angles 3

4

C6. Use the seven tangram pieces to construct two squares of the same size.7

C7. A farmer is on one side of a stream with a boat, a dog, a chicken and a bag of grain. When he is present, the dog will not eat the chicken, and the chicken will not eat the grain; but as soon as he leaves to cross the stream their behaviour changes. On any crossing there is only room for himself and one of his possessions. Can he transport everything across the stream?8

2 Mathematics in School, May 1986



Creating Problems We were now ready to create our own problems. We were to work within the constraints of a "mathematical environment" which we had to define. This term was preferred to the term "set of materials" for it would allow us to include, for example, numbers or symbols or some abstract geometrical configuration; we did not wish to be restricted solely to concrete materials. Of course we had to be prepared to modify the environment as we developed a problem solving situation. This modification meant either introducing further constraints into the environment or relaxing some constraint.

We chose as our first mathematical environment simply a small set of centicubes. (Later we allowed the use of larger wooden cubes.) We attempted to think of a problem related to the cubes that also fell into the classes considered above. We dealt with C1, C2, ..., C7 in turn. In creating the following set of problems, we were influenced either by the problems already encountered in our stage 1, or by other similar cube problems, or we were inspired by ideas that just seemed to occur to us.

Cubes Cul. (a) How many different solid shapes can you make by

fixing together four white centicubes? (b) Given four red and four blue (wooden) cubes, how

many ways can you build a 2 x 2 x 2 cube? (c) In how many different ways couldyou build a cuboid

using no more than 12 unit cubes?

A problem in this class essentially entails counting a finite number of cases through an exhaustive procedure. We must try to ensure that the problem contains a manageable number of cases to be counted. It will involve the solver in searching for a systematic way of dealing with these cases. This will necessarily entail clarification of the problem, some classifying or ordering, and a means of recording the cases to be counted. Children trying this kind of problem will be encouraged to move away from the trial and error approach to a way of working which requires a degree of analytical thought.

Cu2. The fifth case of a sequence of constructions is shown. Apply the question given to this case, and also to the general case.

(a) 125 unit cubes make this solid cube.

How many unit cubes have 3 faces exposed? 2 faces exposed? 1 face exposed?

(b) External dimensions of box are the same as the solid cube above. Box is open at top, closed at base. Walls are 1 unit thick. How many unit cubes altogether?

(c) The bottom layer of the pyramid contains 5 x 5 = 25 unit cubes. The next layer 4 x 4, the next 3 x 3, and so on. How many cubes are hidden from the outside? How many cubes have only one face, two faces,..., five faces visible?

(d) A "skeletal" cube. How many unit cubes are needed to build it?

Problems that lead directly into dealing with number patterns provide an opportunity for children to make mathematical generalisations. This important process offers a difficulty for most children in their mathematics learning. In solving problems of this type the stage of generalisation is approached through processes of analysing pattern, pre- dicting, proposing and testing conjectures. This particular approach, leading to children making their own generalisations, is considered by many teachers an effective way of improving the children's ability to handle generalisation in

Mathematics in School, May 1986 3



mathematics. The nature of the cube problems presented here is such as to allow older pupils to apply methods of proving the generalisations discovered. These methods do not necessarily require an ability to work with algebraic symbols.

Cu3. (a) (Follow up to Cul(c)). How many different cuboids, each containing 100 unit cubes could you make? What number of cubes, less than 100, gives rise to the greatest number of different cuboids?

(b) If you had 1000 unit cubes, how many different sized cubes could you make from these?

The main objective behind this kind of problem is to develop a child's ability to apply his/her knowledge of Number. In an opposite manner, through pursuing these kinds of investigations it is believed that children will increase their awareness of number relations and generally develop their "feel for Number". The problems above could be re-stated without any mention of cubes. Eg. Cu3(a) corresponds to: "Express 100 as the product of three factors in as many different ways as you can. Etc." In this case the problem-solver would be exploring the set of whole numbers up to 100 - the "environment" for this newly framed question.

Cu4. Start with a sequence of, say, four cubes.

Eg. RED BLUE GREEN YELLOW

Interchange the

frst pair of cubes

Interchange the second pair of cubes

Interchange the last pair of cubes what happens?

These are normally very open investigations. Often it may be difficult to detect pattern, let alone deal with it. But children should be encouraged to regard as important the recording and the mathematical representation of their findings if a greater understanding of the situation is sought.

CuS. What solid shapes can you make with centicubes so that every cube has four, and only four, faces visible? (Next consider five or three faces instead of four). Consider cases which take into account whether the two "unseen" faces of each cube are adjacent or opposite.

The teaching approach suggested by this class of problem is favoured by those teachers who believe in the value of children themselves becoming involved in constructing and classifying - in this case, shapes. Problems on Shape employed in this teaching approach would serve not only to develop general problem solving skills but also help to improve the children's understanding of geometrical properties and generalisations concerning shapes.

Cu6. (a) Can you build a cube by fitting together a collection of identical shapes each comprising three unit cubes as illustrated?

(b) This solid comprising 10 unit cubes may be bisected to form two congruent shapes thus:

Can you find another way of doing this?

(c) What is the greatest number of dzifferent sized cuboids that can be made to fit into a box measuring 4 cm by 4 cm by 4 cm. The dimensions of each cuboid must each be a whole number of centimetres.

A typical problem here can arise out of a geometrical dissection. In Cu6(a) the dissected pieces are given and are to be assembled - like a jigsaw; in Cu6(b) a particular dissection has to be found. The skills required to solve problem Cu6(c) clearly depend on those required to solve Cul(c). As children progress in their learning of problem solving skills they may proceed from a pure trial and error approach towards one that involves a great deal of analysis and reasoning. The difference between the skills necessary to solve Cul(c) and Cu6(c) highlights a significant step within this progression.

Cu7. You are given 27 cubes of the same size, 9 red, 9 blue, and 9 yellow. With these, can you build a 3 x 3 x 3 cube so that (i) each face of this large cube contains 3 red, 3 blue, and 3 yellow squares, (ii) the six faces of the large cube each conatin a different number of red squares?

Often, we tended to place problems in this category if they seemed not to fit any of the others. A positive test for inclusion, however, is to check whether the problem solving activity contains a need for some form of logical reasoning: the problem-solver should gain from thinking, "If..., then..."

Overall, the task was aimed at improving our problem- creating skills. Of course, the problems we were trying to create had to be appropriate to children in school; they had to be not too open, not too closed; they must not involve mathematical concepts with which children are unfamiliar; they should be capable of engaging young problem-solvers in those processes essential to problem solving and of developing their ability to employ general strategies that characterise investigational work in mathematics.

Having created a problem - or what appeared to be a problem - we needed to solve it. Inevitably, it would require modification and then more solving. Sometimes we asked ourselves "What would happen if...?" Or even, " Can we really make a problem out of this?" Further restrictions on the environment would be imposed or some restrictions lifted, another problem posed, and a solution tried. It occurred to us that we were well and truly into mathematical investigation. The need to create suitable problems, that would lead children into worthwhile problem-solving and provide starting points for them to create their own problems, predominated in our approach. But without this, the nature of our work would resemble closely that of an investigation which we might imagine our pupils undertaking.

We went on to look at other possible environments. The geoboard, for example, was considered in a similar way to that for the cubes. But after this, we were quite happy to invent one or two problems for a particular environment without regard to the category to which these might belong, although we were ready to refer consciously to the problem- creating approaches we had developed. Here are a few suggestions for starting points:

Matchsticks, squared paper, square tiles, domino shapes, isometric paper, a set of numerals, a set of coins, dominoes, dice. Further examples of environments will be met in the next section.

Problems from Specific Topics Our starting points for problems now became specific areas of mathematics. The first topic we decided on was 2D shape: in particular, work concerned with the recognition, naming and properties of shapes. The work was divided




into three sections - triangles, quadrilaterals, polygons. Under each of these headings, appropriate environments were contrived in which were embodied relevant mathematical ideas, and out of which, it was felt, suitable problems could be devised. It became clear that the type of problem most usefully employed in this area was C5. Through the use of this type of problem we envisaged children learning by exploration and discovery and also by their being allowed the freedom to create their own shapes.

Triangles T1. Triangles on a Geoboard

(a) What kinds of triangle can you make on a geoboard?

(b) Restricted to a square array of nine pins, how many ways can you draw the same triangle?

[Geoboards, dotty paper, special 9-dot sheets]

T2. Triangles with Matchsticks

How many different kinds of triangle can you make using no more than 10 sticks for each triangle?

[10 matchsticks]

T3. Building Shapes from Triangles. Fitfour triangles together so that wherever two come into contact their edges coincide. How many different shapes can you make? [Plastic equilateral triangles, isometric paper, scissors]

T4. Reflecting Triangles What symmetrical shapes can you make by fitting together two triangles of the same shape and size? (Extend to paper folding - 1, 2, 4 folds - and cutting out triangles).

[Triangle templates, coloured paper, scissors]

T5. Triangles from strips

How many different triangles can you construct using one strip for each side of a triangle?

[At least three strips of each colour: Red (length 3cm), blue (4cm), green (5cm), yellow (6cm)]

Quadrilaterals Q1. Quadrilaterals from Two Triangles

Choose two triangles and place them together to make a quadrilateral. What kinds of quadrilateral are possible made this way?

[Triangles in card/paper; dimensions: Red (1,1,1); blue (1,1, 3 ); green(,/,//,2); yellow (1,2,./5)]

Q2. Quadrilaterals on Isometric Paper

(a) Draw quadrilaterals by following the lines on isometric paper. What sort of quadrilaterals can you draw?

(b) On triangular dotty paper, join four points to make quadrilaterals. What different kinds can you make?

[Isometric paper, scissors; triangular dotty paper]

Q3. Quadrilaterals from Straws

Choose four of the straws to make the sides of a quadrilateral. What kinds of quadrilateral can you make with these? Try a different set of four straws.

[Straws of two different lengths - say 3cm and 6cm pieces]

Q4. Diagonals of a quadrilateral Take two sticks - either two red, or two blue, or one of each colour and lie one across the other on a piece of paper. Mark the four ends of the sticks with a pencil. Take away the sticks and join up the four points to make a quadrilateral. What other kinds of quadrilateral can you make this way?

I I [Red and blue sticks (pipe cleaners); reds 8cm long, blues 5cm, say]

QS. Rectangles in a Square Grid Draw a 3 x 3 square grid. How many different rectangles can you find inside this figure?

[Squared paper]

Polygons P1. Cutting Rectangles

(a) You are allowed to divide a square or oblong into two pieces by means of one straight line cut. What shapes can you make this way? Consider varying the number of cuts and the number of pieces.

(b) As (a), but just one cut after a single fold. [Small gummed paper rectangles, scissors]

P2. Assemble Four Pieces

From a square of paper make four triangular pieces by cutting along the diagonals. Can you put thefour pieces together to make a triangle, then an oblong, then a parallelogram? What other shapes can you make?

[Gummed paper squares, scissors]

P3. Shapes "inside" a Hexagon Draw lines joining the vertices of a regular hexagon. How many dzifferent shapes can you find whose sides are described by a set of these lines?

[Hexagon stencil]

Mathematics in School, May 1986 6



P4. The Intersecting Polygon Draw two identical polygons on separate pieces of tracing paper. Place one sheet over the top of the other. What kinds of shape can be made where the two polygons overlap? [Tracing paper. Stencils of various shapes]

PS. Explode a Polygon Cut up a regular polygon into a number of congruent shapes. In what ways is this possible? [Templates - regular hexagon, pentagon, octagon, etc. Paper, paste]

Now the second topic:

Fractions Fl. How many ways are there of cutting a square in half?

[Materials: 4 x 4 squared paper; scissors]

F2. What fractions of a circle can you make by folding and cutting?

[Filter papers or bun cases]

F3. Whatfractions of an equilateral triangle can you make?

[Several equilateral triangles in paper]

F4. Given a set of 12 counters, divide them into three sets, each representing a dzifferent fraction of the 12 counters. In what different ways can you do this?

F5. Given eight wholes and eight quarters, make a set of five dzifferent mixed numbers.

[10 squares of paper, two of them cut into quarters]

25

F6. You have a whole-strip of card, a half-, a third-, and a quarter-strip. By placing them together or on top of one another, What fractions can you display?

[Four lengths of card; whole-, 1-, -, strips]

F7. Find three different unit fractions that together give a whole. Alternatively expressed: Find whole numbers x,

1 I 1 y, z such that x >y > z and - +- + - = 1

x y z

F8. Both the numerator and denominator of a fraction are numbers from 1 to 6. How may such fractions can you form so that together they represent different parts of a whole?

F9. An L-shape is made by removing a unit square from a 2 2 square. This L-shape is -th of a whole rectangle. What is the size of the rectangle?

F10. Consider in a game of chess the queen positioned at a particular square on the board. What fraction of the total number of squares could it move to? This fraction describes the "power" of the piece. Compare the powers of different pieces.

These two topics - Properties of 2D Shapes and Frac- tions - correspond to major objectives in primary mathematics. If we were to select other topics for consideration we could do worse than consult the lists of objectives set out in "Mathematics from 5 to 16"9. Above all, the list given under the heading "Conceptual Structures" applies here (Section 2C, Appendix 1C(10, 11) of Mathematics from 5 to 16). Presenting to children problems of the kind designed in this section aims to enable them not only to employ general strategies (see Section 2D of Mathematics from 5 to 16) but also to encounter and examine the conceptual structures embodied in the environments defined by the problems. An approach that relies on a programme of structured learning is traditionally regarded as the means of teaching this latter kind of objective. Yet teachers admit to the difficulties that often exist in finding a variety of suitable pieces of structural apparatus, in setting up and guiding the practical work and recording (particularly in mixed ability groups), and furthermore in conveying to the children the relevance of the activity. A problem-solving approach can make a contri- bution to this area of learning. Children are more likely to find relevance in a problem or challenge; they are allowed to work at a level appropriate to their stage of mathematical development as they solve problems in their own particular way; as they become more skilled and trained in problem- solving they should begin to demand less support from the teacher.

How about more sets of problems to do with Place Value, Decimals, Money, Probability and Statistics, 3D-shape, etc? For us to extend our investigation we would require a lot more time - and this we do not find easily! We would like to think we shall continue to invent our own problems and encourage our pupils to do the same, but hopefully this will take place as a result of "inspiration in the classroom". However, we would like to see the appearance in teachers' and pupils' books of more problem-solving ideas, collected together under specific mathematical topics, rather than according to classes of problem-solving skills. To repeat our worries expressed above, authors who take this latter point of view may be in danger of turning problem-solving into a subject concerned with the application of routine skills. But also, they may lose an opportunity to make a statement about using problem-solving in a teaching approach aimed at improving children's understanding of key concepts in mathematics.

References 1. Watson, F. R. (1983) "Investigation" Educational Analysis, 5:3. 2. Golomb, S. W. (1966) Polyominoes, Allen & Unwin. 3. Polya, G. (1971) Induction and Analogy in Mathamatics, Vol 1, Princeton

UP. 4. Mason, J. (1982) Thinking mathematically, Addison-Wesley. 5. Fielker, D. (1984) Removing the Shackles of Euclid, ATM. 6. Read, R. C. (1965) Tangrams,.Dover Publications.

S7. Lingard, D. (1980) Mathematical Invstigations in the Classroom. ATM. 8. MEP (1983) Micro Primer Pack 1, Tecmedia. 9. DES (1985) Mathematics from 5 to 16, HMSO.




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A Problem Solving Activity