a one dimensional (length) growth results in a linear nth term
two dimensional (area) growth results in a quadratic nth term
you can ask some of the beneficial 'growing shapes' questions:
- what will the next one/two look like?
- how would you describe the growth pattern to someone?
- if you were asked to count the squares, how might you do so?
this can lead to a generalisation that sigma (8n) is 4n^2 + 4n and hence to a rule for the sum of the first 'n' natural numbers
the second task is one provided by Mimi Yang on her blog
it might be worth noting that both the above and this linear growth sequence diagrams can be reformed into hollow rectangles
students might be able to give reasons for the linear growth rule involving 6 as a multiplier rather than 8 in the previous task
they might be able to see a way to 'reform' the second generalisation diagrams to identify the sum of two of the same square numbers
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