ALG 6

In this week's set of tasks we look at figurative patterns and express their structure in generic terms by using a quasi variable (in this case, 20). We look at equivalent ways of construing/expressing the structure.
Some of the week's patterns turn out to have a linear structure and some a quadratic structure - which raises the question,  
How can we recognise this in the pattern itself, as well as in its symbolic representation?
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MONDAY: Here is the week's root task:

Note: We have chosen 20 here (and in later tasks) on the basis that it is big enough to deter most students from simply counting each individual dot. Also, if students can find an efficient method for counting a 20-chain, it is likely that they can do so for any length of chain, perhaps even a chain of n Y-shapes. So, for many students, 20 is taking on the role of a variable.
Note 2: Another reason for going for such a 'far generalisation' early on, is to encourage students to look at the structure of the 20-chain as a whole. Of course, it is still possible to use an 'incremental' (or scalar or term-to-term) approach by noting that the 2-chain has 4 more dots than the 1-chain (and so the 20-chain has 19×4 more dots than the 1-chain). It would be interesting to see how the task would work without showing the 2-chain. We decided to put it in to avoid students thinking that there might be a hidden dot underneath the dot where the Ys join (in which case a 2-chain would simply have 2×5 dots and a 20-chain would simply have 20×5 dots).
Note 3: The presence of the 2-chain does open up other interesting possibilities: some students might argue that the 20-chain will have 10 times as many dots as the 2-chain; others might simply focus on the given numerical information, 1→5 and 2→9, and look for a mapping-rule that fits both, which can then be applied to 20.
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TUESDAY: Here we are given two expressions for finding the number of dots in the 20-chain. The expressions are deliberately left open so as to show how the chain of Y-shapes can be structured.

It may be helpful to make an annotated sketch, as in the examples below, when describing each structure (ie when explaining Alma's and Bojan's methods).

The given expressions are of course equivalent and you might want to challenge students to show how one expression can be transformed into the other. It is possible to structure the pattern in other (equivalent) ways, resulting in other (equivalent) expressions 'in 20'. How many can you find?
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WEDNESDAY: Here we consider two further chains of Y-shapes, but this time the size of the Ys increases as the number of Ys in the chain increases. What does this do to the relationship between the number of Ys and the number of dots in a chain? Is it still linear? Can you tell from the pattern? Can you tell from the generic expression for the total number of dots?

 Note: Here we deliberately consider a far generalisation again, rather than ask for the number of dots in the 5-chain, say. The hope is that this will focus students' attention on the overall structure of the pattern rather than on the change from one chain to the next. Put another way, we are trying to see whether students will consider the pattern generically. Of course, it is perfectly legitimate to look at differences, ie to consider the pattern 'incrementally', but it's not particularly efficient here.

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THURSDAY: Here we look at some more dot patterns, but this time the Y-chains have morphed into single trees.

Yesterday, the relation between chain-length and the total number of dots turned out to be quadratic, in both cases. What about today? Can you tell just by looking at the pattern? Do the generic expressions for the 20-brown-dots trees confirm this?

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FRIDAY: Here we consider equivalent generic expressions for the number of dots in Fynn's 20-brown-dots tree. We try to relate them to the dot pattern - the first one is relatively straightforward, the second less so. We then try to transform one expression into the other.

We can show the structure contained in the expressions with annotated sketches like these:

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