GRID ALGEBRA

At the Joint LEA Mathematics Conference, Dave Hewitt led a session on Grid Algebra. I had met this idea in the early stages of my training and had used it with some of the groups I taught. In the first lesson I would introduce an imaginary number line running round the room. I would identify the starting number, tap the wall successfully and ask pupils for the number I had landed on. Dave Hewitt places emphasis on the steps taken to get to the new position (the journey)in an attempt to shift pupils’ attention on the operations. I do not feel I had focused on this enough. In the next stage of the lesson I would encourage pupils to record the journeys taken. I would draw a grid select a starting point and ask different pupils to tap journeys and other pupils to record the route. The structure was kept the same: starting point, operations and new position. Later numerical positions are dropped for algebraic one. “If I start here and tap, tap, tap. What is this cell called?” initially pupils were concerned with the fact that I had not stated what the starting cell was called. However, when asked what it could be some students suggested it could be anything and agreed using a letter. A few examples using algebraic notation, I would had a second row. This was the double of the one above it. The vertical downward movement is ‘multiply by two’ and upwards is ‘divide by 2’. I would then focus pupils’ attention on finding the two quickest routes to move from one cell to a specified one. My aim was for pupils to appreciate that for example. By the end of the lesson some students would identify a pattern; "oh you double both numbers”. I was then was able to ask students to consider what would happen for. Some students would correctly identify the pattern make new conjectures linked to changing the initial number and after a bit of practice we could move on. Some students would be able to work on similar questions at a later stage while others were not.

As otherteaching models became available to me (such as the grid method to expand brackets) however, I slowly abandoned Grid Algebra. I did not do so deliberately but I seemed to favour an area model to expand brackets and the ‘multiplication grid’ lent itself to this.However, I began to notice that my students appeared to be able to expand double brackets with ease but when presented with they did not appreciate that this is still a problem about calculating the area of a rectangle with sides respectively measuring 4 units and units. This is probably a consequence of the fact that I did not provide many of these examples to work on.

The opportunity to revisit Grid algebra presented it self once more. I did not want it to become another one of lesson, a gimmick for pupils to work on for the day and forget once an algorithm had been memorised. I wanted this experience to be more meaningful and lasting for most pupils (ideally for all pupils). I decide to work with my year 8 set two out of five. They are keen to show their ideas but not too keen to listen to others and hence develop their understanding further.

Lesson 1started in the same way: moving along an empty number line identifying the journey and the final destination. This time however, I was able to make use of the interactive white board and the ‘Grid algebra’ programme. I introduced a more complex grid and as a class we agreed on the rules for movement across the first lien and from row to row. Pupils began to suggest that “the second row would then move up in 2s and the third in 3s and so on”. Using numerical examples they convinced themselves that this was true. Using ICT provided students with immediate feedback to their conjectures.

1 / 3 / Goes up in 1
2 / Goes up in 2
3 / ∆ / Goes up in 3
4
5
6

Using the calculating function various pupils were invited to the board to mark answers/ results of journeys from one cell toanother. So for example pupils were able to identify ∆ as 18. As pupils appeared confident I moved the class on to looking at equivalent expressions. The software provides this function and a choice of grid with single or multiple rows. I chose to start with a single one. Using one row pupils were asked to identify different ways of moving from 11 to ∆.

1 / 11 / ∆

Pupils suggested ∆ and became more adventurous. The pupils were keen to complete the task and show off their efforts to the rest of the group. Introducing more than one row was a natural step to progression. Pupils were then asked to consider different routes to move from 11 to •

1 / 11
2
3 / •

Very quickly we considered different ways of getting from 21 to •

1 / 21
2
3 / •

It was interesting to see pupils create equivalent routes and not just the two quickest routes. Their emphasis was clearly on the journey and not on naming the cell. The opportunity to outdo each other was powerful in engaging pupils with the task. As pupils were able to engage with ease with the numerical situation I invited them to find possible routes that take us from a to •

1 / a / •

Pupils were not concerned with the meaning of a and continued to work on the task in the same way as they had done with numerical starting points. For homework they had to identify five different routs leading from b to •. In the second lesson we focused on examples from pupils to move from to using equivalent expressions. Once pupils appeared confident working with one row we considered multiple rows.

1 /
2
3 / •

I did not expect pupils to produce more than two routes but the extensive work on movement on one line led to pupils being more creative and adventurous. I allowed this at this point and focused their attention on the fact that all the statements were equivalent. In lesson three I began to focus pupils attention on the two quickest routes of getting from say cell to cell • and from cell to cell ∆.

•
∆

I then asked pupils to draw a grid for which and another for which . I wanted pupils to question this model and not to see it as a string of instructions to follow. I wanted them to consider how the grid was constructed and how its properties were reflected in their instructions/routes. Discussion focused on where to start, what did pupils notice and could they make any generalizations .Their justifications focused on knowing how big the grid needed to be and how wide it needed to be to accommodate the number of steps across. So for example students suggested that for the grid needed to have at least two rows as it required to double and at least four across for theand add three. I was also intrigued to see that it was not the usually confident students making their contributions but students who in other occasions had required reassurance. Their homework was set. Find the other route for

Pupils arrived to lesson four able to expand the brackets but unsure if their factorised questions were correct (noticed that this language has not yet been introduced to the students and I am still talking about routes). For this reason I decided to encourage pupils to continue to imagine what the grid would look like and draw it.One weeklater I returned to this through starter questions.I asked students to draw a grid and find the other route for. Pupils attempted these questions with confidence and in following lessons the starter included questions like: draw a grid and find the other route for,and. I was also surprised to see that overall pupils did not appear to have issues with and. In fact after an initial puzzlement one student suggested that that was the name of the initial cell and the others accepted this.

The algebraic expressions therefore represent the name of their position as well as instructions to get from one location to the specified one. Hewitt deliberately focuses pupils’ attention on what is to be done and why, while at the same time developing algebraic notation. What matters is not the answer but how we are getting there. Algebra to represent a cell is only introduced when the particular number is no longer of importance. In a few months pupils may still not be able to expand and factorise expressions given to them, however, they may be able to generate a grid to represent the situation and successfully find one route.This model may not be as versatile as models used in MiC (Maths in Context) but in common with MiC empowers pupils to make some sense of the situation presented. Pupils appreciate that it is possible to take different routes and end up at some place and that this generates different expressions which must be equivalent because they identify the same place.I will need to return to problems like and encourage pupils to redesign the grid related to this. Some students were already making conjectures and had a clear sense of the fact that once again the 4 represented the number of rows and that as we where on the fourth row then we were going up in 4s. I feel that pupils benefited from developing the use of” Grid Algebra” as a model to represent a situation and as there was no pressure to abandoned this (in fact I deliberately asked pupils to continue to draw a grid to work on the questions), pupils should be in the position to attempt to make sense of the mathematical situation at a later stage. The software provided immediate feedback to pupils conjectures. However, due to issues with the school’s system, I did not make extensive use of the package. I will explore this more in detail in future and encourage teachers in my department to do the same.
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Marisa Bartoli, CheadleHulmeHigh School, Stockport, 25/02/08