Sliding Tiles

Iceberg A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.

This task is totally consistent with learning to work like a mathematician because (a) it is a problem and (b) students find it interesting. Mathematicians tell us their work begins with an interesting problem so that is what makes this task the work of a mathematician, not whether it contains skill based content. The content of 'reasoning' is content enough and the focus in this case is on: playing with the problem developing and testing hypotheses attempting to explain 'how to do it'. The nice thing about the task for younger children is that there is a 50:50 chance of the dice providing a starting point that will work, so the problem can provide early success. The more common form of sliding tile puzzle is 15 tiles sliding within a 4x4 frame. The user is challenged to slide the tiles into certain configurations and is also given 'impossible' challenges. Our task has been created by applying the strategy of 'try a simpler case' and it can be simplified still further by covering up the two right hand positions and only using the tiles numbered 1 to 3. Suppose now that the challenge is to finish with the sequence 1, 2, 3, blank, that is 1    2 3    __ Starting with this simpler puzzle might be necessary to help students find keys to solution of the larger version. No matter how the dice falls to place the tiles, in the end there are only 24 possible starting points for this puzzle. These come about because if we place from the top left corner first and remember that blank also has a place, then there are 4 choices for the the first tile, leaving 3 for the second, 2 for the third and only 1 for the fourth; 4x3x2x1 = 24 and students can be encouraged to write out these 24 possibilities. Here is a start. 1    2 3    __ 1    2 __    3 1    __ 2    3 1    __ 3    2 1    3 __    2 1    3 2    __ Now begin the ordered setting out again with 2, then 3, then blank in the top left. (This reasoning is the same as in Task 118, Ice Cream Flavours and others which can be found by searching 'combination theory' in the Task Cameo Content Finder. Attempting to solve each one of those above shows that the first, second and fourth can be changed to the required solution (the first because it already is what is asked for). Trying all 24 possibilities in this way yields exactly 12 solutions. Further, if the solutions are organised into those that solve and those that don't solve students might see that each member of the 'solve set' has a partner in the 'don't solve set' which is its reflection. For example, this one does solve: 1    2 3    __ but if we reflect this pattern in a line running down the right hand edge of the puzzle we get... 2    1 __    3 ...and this arrangement doesn't solve. Equally, if we begin with an arrangement that doesn't solve: 1    __ 2    3 and reflect it in the same way we get... __    1 3    2 ...and this arrangement does solve. Further, looking for a way to organise the 12 solutions can lead to seeing that each is obtained from the previous one by rotation. This analysis gives clues to using the 3 x 2 frame with 5 tiles. There are 6! = 6x5x4x3x2x1 = 720 ways of placing the tiles with the blank. We can expect half of these (360) to yield solutions and half not to. We can also expect that each arrangement will have a reflected partner (and that each arrangement within the 360 is like a 'rotation' of others). So, if the tiles are placed at random then there is a 50:50 chance that they can be moved into any given arrangement. Students may or may not be capable of this level of reasoning, but may still be able to solve the puzzles on the card. As they become more confident in doing so, encourage them to first orally explain, and then try to write or draw, how to do it so that someone else can learn.

Whole Class Investigation Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.

With very little effort using a word processor or drawing package, you can prepare a printable version of this puzzle which the students cut to begin the investigation. The objective of doing so is to model how to work like a mathematician, in this case emphasising: playing with the problem to collect and organise data keeping a journal making and testing hypotheses problem solving strategies such as 'trying a simpler problem' and 'listing every possible case' explaining/publishing a solution as outlined above. In a whole class investigation it is an advantage to the teacher to have the insights of all the students to draw from to develop the investigation and an advantage to the students to be able to explore without feeling that all the responsibility for the solution is 'on their shoulders'. When the class is feeling confident with the problem, a web search on 'sliding tile puzzles' will reveal several software versions which provide additional challenge and tutorials for solving particular forms of them. Perhaps these will encourage your students to publish solutions and 'how to do it' articles (slideshows, videos?) for our task. If so, we would be happy to make them available through our site. (If you choose to publish elsewhere on the web, then please acknowledge our task as the source.) At this stage Sliding Tiles does not have a matching Maths300 lesson.

Is it in Maths With Attitude? Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.

The Sliding Tiles task is an integral part of: MWA Space & Logic Years 5 & 6 MWA Space & Logic Years 9 & 10

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