Resource description: Tarsia is a software program that enables relatively simple ‘jigsaws’ to be produced. There is one question for each internal edge in the puzzle (the other side of the edge is an answer). Outer edges are blank. Tarsia puzzles come in many shapes including squares and rectangles (made up of square pieces) and rhombuses, parallelograms and regular hexagons (made up of triangular pieces).

I put together a sheet outlining the task and asked students to investigate the number of questions that appeared in grid shapes of different sizes. Most students began by examining the square grid shape but a few students struggled to begin with so I suggested to them examining rectangular shaped grids with a fixed width. Obviously, these will lead to linear number sequences being produced and this approach enabled the students who lacked confidence to find a way into the problem. However, this situation was far from ideal as some of the key parts of the process have been undertaken by myself rather than the students themselves; breaking the problem down, creating logical steps and organising work on paper are all fundamental aspects of working on a investigative piece and teacher intervention immediately reduces the students’ opportunities to develop this aspect of their mathematics.

Teacher comment:
My class had recently used Tarsia to explore number sequences and, as a result of this work, were able to find the nth term of a linear sequence (see Using Tarsia). As part of the discussion that stemmed from that piece of work, I asked a student in the class how many linear sequences were included in the puzzle. As he ran through the questions, he lost count, so we had a chat about possible other approaches that could be used to find the total number of questions contained in the puzzle. It dawned on me that the number of questions in a Tarsia puzzle could form the basis of an investigation.

What I did:
I have always had a problem with the idea of formal mathematical investigations. I have also frequently felt dissatisfied with the way my classes have approached them. My experience tends to be:

Introduce a task to the class, seemingly unconnected to anything they have recently covered, usually couched in a strange context or totally abstract.

Set them off, only to be faced by a barrage of ‘I don’t get it’ comments.

Go back to the board and ‘point’ the students in the right direction (ie do their thinking for them), usually by suggesting how to break the task down and then suggest (or get them to suggest) logical next steps.

Students then carry out what is, in effect, a closed task consisting of generating number sequences and finding nth terms.

Although there have been exceptions to the outlined mode of work (one of the reasons why I find Spirals such a fantastic investigation is that it tends to avoid this pattern of behaviour), historically students in my class have not taken control for the investigative aspect of the task – they wait for me to do the thinking for them. Usually this ends up with a class full of very similar looking pieces which, I believe, is the antithesis of what an investigation should produce.

The Tarsia investigation I set the class generated a very different response and led to an extremely varied set of reports, with a wide range of thinking and strategic decision making in evidence. I put this down to a couple of things:

The students were already familiar with the context of the work and confident of working within it. The question posed was not artificial or contrived in their eyes; they could see the worth in exploring it due to the work they had recently carried out.

I suggested the students examined a square grid as the first shape to investigate as opposed to the more accessible rectangular grid (with a fixed width). Because this led to a quadratic number sequence being generated, the students tended to consider the structure of the grid when constructing their formula rather than examining the number sequence. It was surprising (and gratifying) how many of the class were comfortable doing this and also how many alternative approaches were utilised amongst the class (see outcomes section below).

Reflections/Outcomes:
The task revealed itself to be particularly rich in the number of different approaches the students used to generate formulae.

Some students ‘built’ their formulae by considering the geometry of the grids. One approach used by a number of students was:

One student used a different approach to construct a formula for the total number of questions in a square grid:

Another student approached the same problem in a different way:

A common approach that used the structure of the square grid to generate the formula was to consider the number of vertical and horizontal questions as two separate entities and find a formula for each separately.

Some students looked for number sequences within number sequences. For example with square grids:

This led students into constructing the formula:

y((yx2)-2) = Number of questions in a square grid

It is rare for a class of mine to use so many alternative approaches in evidence from a piece of investigative work and the ‘organic’ manner in which the task evolved and the nature of the starting point, I feel, were instrumental in leading to this.