### Why do this problem?

Relationships can be discovered by making accurate paper folds or
diagrams, but there is also scope for some sophisticated
geometrical reasoning, manipulation of fractions and finding and
justifying general rules for the different fractions that can be
made.

### Possible approach

Learners could start by marking off quarters along one side of
a square, and making folds from the corner to these marks, as shown
in the diagrams. Pose the question "What fraction of the diagonal
do you think is formed by the line joining to a quarter of the way
along the side? Half way? Three quarters?"

Learners can measure the lengths of the lines on their diagram
to see if their conjectures appear to be right - they may be
surprised by the results.

Once they have built up a picture of what is happening with
quarters, they can investigate what happens when the side is
divided into eighths. As it is not always easy to measure
accurately enough and the fractions are not always obvious,
learners might think about how to work more systematically and/or
decide to use a more analytical approach. Moving them
towards more formal and less experimental methods could be
encouraged through discussion and sharing of ideas.

Groups may wish to present their findings through
posters.

### Key questions

How can you organise your work so that you are able
to identify any patterns that emerge?

How can you be sure of the fractions
that appear to be emerging when you measure?

What mathematics have you met before that might be useful
here?

What do you think would happen if we divided the side into
fifths? Sixths?

### Possible extension

If I divided the side of my square into $n$ equal portions,
what fraction of the diagonal would I get by folding to
$\frac{m}{n}$ of the way along the side?

What other quadrilaterials will this idea apply to and does
the rule need modification in any particular cases?

### Possible support

Start with

Take
a Square and build up ideas about halves and quarters before
trying to generalise.