ALG 10

Week 10: This week we look at ways of structuring geometric patterns, and of representing the structure with drawings and numerical expressions. We start with individual, 'isolated' dot patterns but move on to patterns of dots or matchsticks that can be thought of as 'generic', ie as representative of a whole family of patterns. We thus come very close to writing general expressions, though we stop short of using letters to represent key numbers in the expressions.
Some of the later tasks are quite challenging and you will need to judge how far to take them with particular pupils and classes. You will also need to decide how much scaffolding to provide, even though the tasks are already quite highly structured. [Of course, it would also be nice to see what structurings pupils come up with spontaneously, and you might decide to adopt a more open approach, at least at first, with some pupils.]
Even if you don't pursue all the tasks in depth, they allow us to think about where the work might go with older classes.
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Monday: Here we have an arrangement of 11 dots, in which the dots have been grouped in a way that can be represented by the expression 2×4 + 3. We are then given other expressions for the dot pattern and asked to determine a grouping that each expression could represent.

These are perhaps the neatest ways of grouping the dots for the three expressions, although we can find other (slightly different) groupings that fit the expressions:

A nice way to extend the task is to ask pupils for their own ways of structuring the pattern, and to show this my means of drawings and accompanying expressions.
 [Note: We are treating the given pattern of 11 dots as an 'isolated' pattern. We could decide to treat it as a member of a family of dots, for example by continually adding a column of 2 dots to the bottom left and top right of the pattern. However, we are not doing so here, so the stucturings shown by the given expressions and by the drawings of the surrounded dots are in no sense general. However, in later tasks we do encounter patterns that can more readily be thought of as members of a family so that we could, if we wished, also think of the resulting structurings as general.]
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Tuesday: There are several nice ways of surrounding the dots to fit the given expressions....

This task is very similar to Monday's task, except that there is a somewhat greater number of (nice) ways of surrounding the dots for each expression. It is worth divising more such tasks, perhaps involving richer dot patterns so that they can be structured in a greater variety of ways.
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Wednesday: This is a challenging task and you might need to help pupils unpack what each number in the given expression stands for.

In the given expression, the 3 stands for the number of squares formed by the matches, the 4 stands for the number of matches needed to form a single square, and the 2 stands for the number of 'overlapped' matches, ie the number of matches being surrounded twice. To add to the complexity, note that the 2 is related to the 3: it is 1 less than the number of squares.
In the second version of the pattern, the matches form 10 squares, so instead of 3×4 – 2, we now have 10×4 – 9. Or 10×4 – (10–1) if we want to be more explicit. Here we essentially have a general formula, since for any other such chain of matchsticks we simply replace the 10 by the number of squares formed by the matches.
You might at some stage want to ask, "How many matches would we need for a chain of n squares?" Of course, this might well leave pupils confused, so you might want to go back to asking for the expression for other specific chains of matches.
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Thursday: Again, you might need to scaffold the task by helping students see what each number in the given expression refers to. You might also need to address the fact that the expression is equal to 4+6=10, not 6×3=18. It might help to write the expression like this: 4 + (2×3).

By bearing in mind what the 4, 2 and 3 stand for in the given expression, we can write this for the row of 10 matchstick squares: 4 + 9×3. Or 4 + (10–1)×3 if we want to be more explicit. Again we essentially have a general expression here, or at least we are now in the position of being able to write an expression for any other size row of matchstick squares.
Again, bear in mind that this is challenging work, so don't expect it to simply fall into place!
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Friday: Here is another pattern that can readily be seen as part of a family, so that there is scope for generalising a rule from the case of the 10 by 10 square to any other such square. Again, this can be challenging and we don't explicitly consider other size squares here (though you might want to pursue this with some pupils). Rather, we focus on finding alternative ways of structuring the 10 by 10 pattern.
At first sight, pupils might suppose that the number of dots is give by 4×10 for the 10 by 10 square. The schematic diagram and given expression indicate that this is not the case, but pupils might need some time to think this through.

We can write this for diagram A: 4×9. Or 4×(10–1) if we want to spell out the structure fully (which would allow to us generalise the expression in a simple, direct way).
We can write this sort of thing for B: 2×8 + 2×10, or 2×(8+10). Or something like 2×((10–2)+10) if we want to express the structure fully.
You might need to help pupils with diagram C: it is meant to represent the difference between two arrays of dots, namely a 10 by 10 and an 8 by 8 array. The expression can be written as 10×10 – 8×8, or 10² – 8². Or 10² – (10–2)².
For each of our expressions, it is worth checking that they reduce to 36! With some pupils, you might also want to consider the equivalence of the various expression. For example, we can expand the expression 4×(10–1) for diagram A, to give our original expression 4×10 – 4.