### Why do this problem :

In

this
problem it seems something of a surprise that the square areas
which can be constructed should apparently be constrained in this
way (you cannot make an area of $4n + 3$ square units). It feels as
if there ought to be more freedom. This problem encourages analysis
and the forming of conjectures, and especially justifying or
accounting for pattern.

### Possible approach :

Ask the group to draw a variety of squares using paper with
dots, to calculate the area of each and to keep a record.

Ask the group to organise their record in a way they think
useful, and invite conjecture about the situation.

Ask the group to look at the way they decided to organise their
results and to decide what additional results might usefully be
acquired next.

Ask members of the group to share their thoughts.

Once the proposed additional work has been done again invite
conjecture about the situation: any comment, or things to try.

Suggest to the students that they look at the remainder when the
area values are divided by 4, leave some thinking time before again
inviting conjecture.

### Key questions :

- What are the freedoms the problem gives you and what are the
constraints?
- What does this problem invite you to explore?
- How can you organise your exploration so that analysis can
follow most easily?

### Possible extension :

Just
Opposite
### Possible support :

Tilted
Squares