ALG 04

Week 4: In this set of tasks, we look at patterns in number grids: rcctangular arrays of cells containing consecutive numbers. We start with tasks where the patterns can be easily linked to numbers shown in the grid, but we then look at patterns for numbers that are more and more 'remote'.
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Monday: The grid is formed of consecutive numbers, with 8 numbers in each row. So for any given number, the number in the cell directly below is 8 bigger.

The number in the red square is 34. We can think of it as, for example, 18 + 8 + 8 or 17 + 8 + 8 + 1. Some pupils might simply fill in the consecutive squares until they get to the red square. If so, it is worth analysing this approach, by, for example, showing that it amounts to adding 8 (to move down one row) and another 8 (to move down another row); or adding 6 (to complete the given row), adding 8 (to complete the next row), and finally adding 2 (to get to the red square). You might also want to challenge such pupils to 'predict' the contents of a more remote square.
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As we move from one yellow cell to the next, we move 1 down and 1 to the right, which increases the number by +8 +1 = +9.
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The next yellow square is in the 8th row, 1st column (not, as one might easily think, the 7th row, 1st column).
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Tuesday: These tasks are relatively straightforward. They highlight the fact that the relation between numbers in adjacent rows depends on the length of the rows!

 The numbers in the yellow squares go up in
• 7+1 = 8 for the first number grid
• 6+1 = 7 for the second number grid
• 9–1 = 8 for the third number grid.
In the third grid, none of the yellow squares have numbers written in them. This provides a good indication of how readily pupils can work with 'general' rather than known numbers. "Do we need to know any of the numbers in the yellow squares?"
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Wednesday: Here we make quite a large leap from the given numbers, and from the visible but empty cells, to the cell we are interested in. How do we make the leap, and from where?

As we move from one row to the next, the numbers increase by 8.
We can move from 34 in the 5th row, 2nd column to the 21st row, 3rd column by jumping 16 rows and 1 column, giving 34 + 16×8 + 1 = 163.
Or, for example, we could start at 3 in the first row and add 20 lots of 8, giving 3 + 160 = 163.
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Thursday: We can solve this task without having to know what the numbers are in the tinted cells. The task is thus a good test of whether pupils are content to work in this way, or whether they feel compelled to find the numbers first.

One of Dan's numbers is 2×7 less than one of Ella's numbers, while the other is 2×7 more than Ella's other number. So the sum of Dan's numbers is the same as the sum of Ella's numbers.
There is endless scope for finding other such pairs. For example, if Ella keeps the same numbers, then if Dan changes one of his cells he just needs to change the other cell by the same steps in the opposite direction. Or if Ella changes one of her cells, Dan can just change his corresponding cell by the same steps.
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Friday: We have chosen large numbers here to discourage pupils from using some kind of empirical, trial and improvement method. On the other hand, the numbers are quite 'friendly' and so shouldn't cause much hindrance to pupils who opt for an analytic approach.

The difference, 30, between the given numbers tells us the row length of Ella's grid. So the second row started at 31 and the third row started at 61. We would have got to 67 by staying in the third row and moving along to the 7th column. The number 97 would have been in the 4th row, 7th column.