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Why do this problem?
develops the links between visualisation, verbal
description and algebraic representation.
Kick off by talking about odd numbers:
What do the first $10$ odd numbers add up to?
What do the first $5$ odd numbers add up to?
What do the first $12$ odd numbers add up to?
What do you expect the first $20$ odd numbers will add up to?
The first $50$?
What is the $50$th odd number anyway? The $100$th?
$125$ is an odd number. Which is it?
Show students this image or the interactivity
Ask for comments on the arrangement of dots. How can this help us
explain the relationship between square numbers and the sum of odd
How many more dots will I need to add to make the next square? And
the next? And the next?
How many more dots will I need to go from the $100$th square to the
Set students off to work in pairs on the questions set in the main
body of the problem.They can be printed off from here
What is the $5$th, $10$th, $455$th odd number?
What is the sum of the first $10$, $20$, $50$, ... $n$ odd
A suitable extension task is provided in this worksheet
This task could be used as a context for working hard on odd
numbers and their structure, practising doubling numbers and mental
addition. Tasks could include adding sets of odd numbers, imagining
the last layer on the $30$th square, the $57$th square, working out
which square would have $43$ as its last layer.
To prepare students for looking closely at other sequence
pattern diagrams, the interactivity could support discussion
between students - how they imagine the next diagram will look,
whether different students see it differently.
For another problem that uses a similar idea go to
Picturing Triangle Numbers