Mathematics Task CentresAuthor(s): Andy MartinSource: Mathematics in School, Vol. 29, No. 2 (Mar., 2000), pp. 21-23Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30212097 .

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mathematics

task

cCentres

by Andy Martin

Introduction A Task Centre is a collection of mathematical activities complete with the materials needed by pupils to solve the problem. The tasks require hands-on problem-solving skills and involve pupils working as individuals, small groups and on a whole class basis. All tasks provide a range of learning opportunities in mathematics that are not always obvious when the problem is read for the first time.

The Number Tiles problem is one of many that I use and describe as 'the tip of a mathematical iceberg'. It is a task that puts many mathematical skills into a context such that pupils in Key Stage 3 can access and understand them. It is also an activity that makes mathematical problem solving a hands-on learning experience that pupils enjoy. It can be fun and very motivating! The Problem

NUMBER TILES MATERIALS

Nine tiles numbered 1 to 9

1. Place the tiles onto the squares so that they make a correct addition sum. It can be done!

2. There are many solutions. Try to find five [51 more. C Curricuhlum Corporation, 1993

Classroom Organization I have tried working with this task in several different ways. In my experience this works very well when each pupil has

nine number tiles of their own to move around on the desk in front of them. Scrap paper is often the easiest, playing cards or blank playing cards numbered by the pupils are even better. The staff at Thorne use 'show-me' cards for many numeracy activities at the start of lessons, so these cards are often employed for more than one purpose.

When I first used this task in a classroom I gave pupils an A4 sheet of plain paper and asked them to fold it into thirds. We then tear down the folds and repeat with each strip. This reduces the A4 paper into nine equal rectangles ideal for this task when numbered. Pupils find folding into thirds very difficult, even when shown that pushing opposite edges together can create the solution. I have also found that discussing this issue with ITT students is very beneficial, often they can assume skills that experienced teachers know need developing.

I also keep a pile of scrap paper cut to A5 size. This is for pupils to record their solutions on and it allows me to display their answers on the noticeboard at the back of the classroom. Active use of display space is something that I try to encourage. Since the noticeboard now becomes a working area it can be used constructively, and actively, with pupils on other mathematical skills.

Solving the Problem Once pupils have the nine number tiles in front of them I explain the task. As a whole class we search for one solution. This usually takes several minutes since pupils need to realize that 'carrying' needs to take place in order to make the addition problem work. As soon as a solution is found we stop working. The pupil making the discovery writes their result on the board. The rest of the class now checks the arithmetic. This procedure can be repeated several times until the class has generated five or six different solutions.

Into the Iceberg To improve pupils' skills in mathematical problem solving they need to be encouraged to 'work mathematically'. This is an ideal task with which to begin the process. By working mathematically the problem-solving process can move in directions generated by the pupils themselves.

Mathematics in School, March 2000 21

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Example 1: Year 9 (class approximately all level 5)

This particular year 9 class was given this task following their Key Stage 3 SATs. The problem was to be solved on a whole class basis with the intention of providing some guidance on writing up coursework assignments when they moved into year 10.

One pupil, Juliet, noticed that the three digits in the answer always added up to 18. I asked whether they thought this was relevant. Another pupil, Daniel, sitting across the classroom, started to write some numbers down on his piece of paper. A few moments later he pointed out that there were only 42 possible ways of making 3-digit numbers that obey this condition. He knew that some could not be potential solutions to the original problem straightaway. Daniel explained that 189 satisfied Juliet's rule, but we couldn't add two figures together in the hundreds column and get 1 as an answer. The next step was to share out all of Daniel's possibilities amongst the class, so that each group of pupils had a smaller list of possible totals for the addition problem. The problem-solving activity had now changed. Pupils were now arranging six digits to give a predetermined 3-figure answer. The class were able to find many solutions to the original problem much quicker than working in isolation. A valuable lesson was learned. Problems can be made easier by breaking them down into smaller, more manageable tasks.

Example 2: Year 7 (mixed ability)

The task was introduced to this class in my usual way. Once all pupils had seen five or six different solutions we paused for thought. Could we use the skills we had been developing on ordering numbers to help solve the problem? One pupil, Sarah, then noticed that we could swap the figures in the units column to get another solution. She wrote on the board these results:

167 168

328 327

495 495

As a whole class we began to investigate what happens when the figures being added together in other columns were swapped. Sarah's total of 495 could be made in a number of different ways. The board finally contained the following contributions:

167 168 127 128 367 368 327 328

328 327 368 367 128 127 168 167

495 495 495 495 495 495 495 495

The pupils soon realized that each solution that had been found could be reordered in a similar way. The class agreed upon a suggestion made by a pupil called Christopher. Some of these answers were the same. If the larger number was always written first Sarah's total of 495 could be made in four different ways:

368 367 328 327

127 128 167 168

495 495 495 495

Christopher's explanation was based on a method he was using. Find a solution. Reorder the units column, return to

the first answer and reorder the tens column and finally return to the first answer and re-order both the tens and units columns. Each possible total could then be made in four ways. Any other reordering simply created one of these solutions again.

Back to the class. We used Christopher's method (or algorithm) to try to find all of the different solutions to the task. Again we worked together to solve the problem. Answers were recorded on the A5 scrap paper. Working with their Learning Support Assistant (LSA) two statemented pupils became responsible for displaying all of the discoveries. Using drawing pins they displayed each solution on the noticeboard 'in-order'. Their work had solutions from 495 up to 981. The problem-solving direction in this case had changed to that of:

* How many solutions are there? (The class went on to find over 100 themselves.)

* Can any total be made in more than four different ways? (When one was discovered it set off a flurry of rechecking!)

Example 3: Year 8 (top set) This class had just finished a module of work on algebra (using letters to represent numbers whose value was unknown). These pupils were familiar with this task from year 7. A chance to revisit the problem and learn something else. Often when work is revisited it is met with the usual chorus of 'we've done this before'. In this case the learning outcome is different; the task gives a context to the use of algebra.

Can the Number Tiles problem be solved without carrying a figure from one column to another? If yes, give an example. If not, can you prove it?

Allowing letters to represent numbers the problem can be written as: abc d ef where a-i are the numbers 1-9 in some order. ghi If no carrying of figures takes place the problem reduces to three distinct equations.

c+f=i b+e=h a+d=g

At this point it becomes easy for the pupils to lose sight of what it is they are trying to do. Here the context of the problem helps. This particular year 8 class were all comfortable with these algebraic equations and where they had come from. The next step is the one that requires the real problem-solving insight. Since the letters a-i represent the numbers 1-9 they can be added. This class had no problem in summing the first nine natural numbers. Again the algebraic representation was used.

a +b+ c +d+e + f +g+h + i = 45

By reordering the class was able to see where the three distinct equations could be used.

a + d + b + e + c +f+g + h + i = 45

g+h+i+g+h + i = 45

2g + 2h + 2i = 45

22 Mathematics in School, March 2000

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Since factorization had not been explicitly covered with these pupils at this stage I suggested dividing everything by two. So on the board appeared the statement:

g + h + i = 45/2

Now back to some whole class and small group discussion. The questions to be answered were:

* What does this statement suggest? (Three whole numbers can add up to a fraction)

* Is the algebra correct? (Yes) * Is there a mistake? If so, where?

This last point is one that really 'stretches' the mathematics for these pupils. The algebra is correct but the end result is nonsense. We must have used something that is incorrect in order to get to this stage. So the only possible mistake must be the assumption that 'no carrying of figures from one column to another takes place'. The problem- solving direction has now moved into algebra, and proof by contradiction using the algebra skills from year 8.

Professional Development for Staff At Thorne I have set up a Task Centre in order to encourage hands-on problem-solving and active classroom teaching into the Key Stage 3 mathematics curriculum. Taking pupils 'into the iceberg' only happens when the staff are prepared to experiment with different approaches and leave their own 'teacher comfort zone'. This is a brave step that requires confidence and encouragement. Problems of this type can then become a talking point between colleagues and display

space becomes much more active. Pupils are challenged and appear to have fun while they learn.

The staff here are in the process of integrating 100 different tasks into the school curriculum. These were purchased from Curriculum Corporation, Australia where the Mathematics Task Centre Project is a professional development initiative. Successes and failures are shared with an in-house teacher handbook updated on the basis of classroom experience. I would be grateful for other suggestions or feedback from people using problem-solving tasks in their teaching. I am now experimenting with mixed media activities, combinations of hands-on materials, ICT related software and traditional text resources. Sharing collective experience within a departmental team is an excellent way of ensuring professional development for all the staff.

Reference Further information about Curriculum Corporation can be found on the web at: http://www.curriculum.edu.au Further information about the Mathematics Task Centre Project can be found on the web at: http://www.mav.vic.edu.au/PSTC/index.html

Keywords: Problems; Task; Tiles.

Author

Andy Martin, Thorne Grammar School, St. Nicholas Road, Thorne, Doncaster, South Yorkshire DN8 5BQ. Fax: 01405 740315 e-mail: maths@thornegs.schoolzone.co.uk

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