linear & quadratic growths

it can be helpful to provide students with two tasks that involve (i) a linear and (ii) a quadratic generalisation - so that they might better appreciate some of the reasons for these rules                     

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?

what seems neat about this task (above, though not the one below) is that both generalisations are not too demanding (on a good day) and, as students do observe, the first sequence 'loops' fit together, Russian doll like to create the second sequence

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