Variation Grid is a two
dimensional grid of cells.
Each cell has two parts, a form or calculation (yellow background), and
a content or result (blue background).
Treating the two parts of a cell as a statement of equality, each cell
is a particular case of some general formula or relationship.
The visible grid can be thought of as a window onto an effectively
infinite grid extending in all directions.
Grids can be on paper or in electronic format. On paper,
attention can be directed to various lines of cells, so that they form
a sequence to generalise. On an e-screen, grid cells can be
activated by clicking to reveal sequences to generalise.
The idea is to
provoke learners into using their natural powers to detect patterns, to
imagine and express predictions, to generalise, at first for particular
eventually for any-all cells in general.
By following patterns of natural numbers or sequences related to
natural numbers, learners can be encouraged to ‘go with the
grain’, to ‘follow the flow’ and so to
anticipate and predict.
By connecting and relating the two parts of cells, learners can be
encouraged to ‘go across the grain’, to experience
structure, in order to make mathematical sense, both in
particular (in particular cells) and in general.
Put another way, relationships between yellow and blue parts of cells
can be seen and expressed as properties which hold between the cell
parts of any cell in the grid. Learners can then use the
structure of the yellow cell entries in order to justify their
predictions for the blue cell entries. In this way they are
experiencing mathematical structure (as displayed in the grid),
learning to reason on the basis of stated properties, and making
mathematical sense. They are also appreciating that
mathematics does make sense.
After exposure to several grids, learners may begin to think about
mathematical structure in terms of grids: the route between two cells
does not matter, for the transformations involved in moving to adjacent
cells are structurally related.
You can download
the grids (see
below) by opening one, saving it, then running it with flashplayer or
from any browser. Downloaded grids can be adjusted to fill the screen.
Starting at the bottom left and proceeding
along the bottom
row, click on the blue parts of cells.
Do the same in the
next row up, then the next, and so on. Expose enough so that
you can predict the entry in the upper right corner.
Now click on the right-pointing arrow a
times, and the up-pointing arrow a few times. With the entry
in the blue part of the bottom left cell showing, predict and check the
entries in the upper right hand 2 by 2 corner.
clear (round blue button) and restore (rewind arrow), and repeat with
the yellow parts of cells.When
you are confident in predicting cell entries to the right and above,
restore the gird and then click on the left and down pointing arrows
in the upper right hand corner, establish patterns in the blue cells
going to the left, then going down, in both cases going far enough to
encounter negative numbers. Do the same for the yellows.Finally,
follow patterns down from the top left, and leftward from the bottom
right to see that the patterns in the sequences always lead to the same
predictions concerning the product of negative numbers.
Working With SVGrids
are designed to be used with one or more learners under the direction
of a teacher. The teacher is needed to expose certain parts
certain cells and then to prompt learners to conjecture and justify
entries in other cells.
basic idea is to expose a sequence of cells in a row or column, to
predict and generalise (going with the grain), and to make structural
sense by connecting the patterns between the entries in yellow cells,
and patterns in the entries between corresponding blue cells..
it is sensible to start by exposing the blue cells in some line, in
sequence, so that learners can use their natural powers of pattern
spotting to anticipate and predict the next entry, and then to express
The direction arrows
shift the window so that further cells in the sequence can be predicted
and exposed. Parallel
lines of cells can
be used to expose related sequences, and then the two dimensionality
can be used to predict the contents of any such cell.
all the blue cells, you can then do the same pattern-detecting and cell
prediction with sequences of yellow cells.
both blue and yellow cells are readily predicted, attention can be
drawn to whole cells (blue and yellow parts) and the relationship they
express when treated as being equal.
is sometimes sensible to start with exposing a yellow cell, and then,
after a pause, the corresponding blue cell. Then another
to it, and then another adjacent to that, so that a line of cells is
Learners are invited to anticipate and then predict the contents of
other cells, and eventually, of any cell in the grid.
is drawn both to relationships between cells, and relationships between
blue and yellow parts of cells. Some of these relationships
be taken as properties, and then used to justify the equality of the
blue and yellow parts.
on a cell
reveals either a calculation (bottom) or a result or alternative format
Clicking on a large arrow moves the window one position in that
so that it is as if there is a small window looking onto an infinite
Clicking on a solid yellow or blue button reveals the cells in the
or where there is only one button, all the lower (upper) cells.
Clicking on a 'sun behind cloud' reveals parameters that can be
clicking again hides those parameters.
the numbers obtainable from a bilinear function
I am embarking on new versions using Cinderella, which makes it easier to programme changes.
To use the Download version, open the zipped file, then open the html file within the folder.
Revised versions make it easier to modify the grid yourself.
the numbers obtainable from a bilinear function (each row and column forms an arithmetic progression).
There are notes about the grid and why it is called a sieve, in the download version.
The grid offers general formulae as well as specific numbers, against which to check conjectures.
Steps in the genesis of
– An early
experience of a silent mathematics lesson;
– Experience with offering learners sequential
– Desire to exploit more than one dimension of
variation. For more details
Download PME-NA 2005
paper from which these notes are drawn but extended.