Path Arithmetic
John Mason
2011
Aim
To focus attention on structural relationships rather than on calculation.
To rehearse addition and subtraction, with emphasis on doing (adding) and undoing
(subtracting), while at the same time introducing brackets.
Primary Version
On a grid of say 5 by 6 cells, choose one cell as the starting point and put a 7 in it. Choose a different cell
as the target. See example below.
A path consists of a sequence of horizontal and vertical moves leading from the starting cell to the
designated cell. Moving horizontally one cell to the right adds 3; moving horizontally one cell to the left
subtracts 3; moving vertically one cell up adds 2, and moving vertically down subtracts 2.
Paths & Values
What values can the target cell get?
Develop a notation for a sequence of actions so that someone can workout what path you took.
In how many ways can you get a route to the target using only additions? Using exactly two subtractions?
What effect does changing the action-values (the 2 and the 3) make to the final result? What about changing the
starting value (7)?
Try moving the target cell relative to the starting cell.
Comment
Perhaps the most important aspect is justifying the invariance of the target independent of the path taken, while at
the same time exploring the numbers of paths between cells.
Secondary Version
On the same grid, let horizontal movements be adding (or subtracting) 3, but let vertical movement be
multiplying (or dividing) by 2.
Paths & Values 2
What are the smallest and largest values obtainable by paths involving only addition and multiplication. What if you
permit a specified number of undoings?
Extend your notation so as to be able to work out the path associated with a sequence of calculations.
What effect does changing the action-values have on the maximum and the minimum achievable?
Try moving the target cell relative to the starting cell.
Comment
Here the underlying structure changes drammatically.
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