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### Why do this problem?

Many students are accustomed to using number patterns in order to
generalise.

This problem
offers an alternative approach, challenging students to consider
multiple ways of looking at the structure of the problem. The
powerful insights from these multiple approaches can help us to
derive general formulae, and can lead to students' appreciation of
the equivalence of different algebraic expressions.

### Possible approach

Display

this
image of a full size $5$ box of chocolates, and ask students to
work out how many chocolates there are, without speaking or writing
anything down. Compare solutions and share approaches.

Mention that mathematicians like to find efficient methods
which can be used not only for simple cases but also when the
numbers involved are very large. Explain that the pictures of
Penny's, Tom's and Matthew's partially-eaten chocolates could be
used by a mathematician as a starting point for finding an
efficient method for counting the total number of chocolates.

Click

here for
the three pictures.

Hand out

these chocolate
box templates and ask students to show how the images of the
partially-eaten chocolates can be used to calculate the
total.

Ask students to report back, explaining the methods which have
emerged.

Then ask students to use all three methods, along with any
methods they devised for themselves, to work out the number of
chocolates in a size $10$ box, and verify that all methods
agree.

Challenge students to express each method for finding the number of
chocolates in any size of box, perhaps introducing some algebra and
the idea of a size $n$ box if appropriate.

Bring the class together to share findings. Compare the different
"formulae" which have emerged, and ask students to explain why they
are equivalent.

### Key questions

How does each image help you to count the total number of
chocolates quickly?

Can you demonstrate the equivalence of different algebraic
expressions?

### Possible extension

The problems

Summing Squares
and

Picture
Story lead to formulae for some intriguing sequences through
analysis of the structure of the contexts.

### Possible support

The problem

Seven
Squares gives lots of simple contexts where formulae emerge by
looking at structure rather than number sequences.