This problem challenges students to use what they know about
gradients to make general statements about the gradients of
perpendicular lines.

This is a suitable preparation for students who are about to
move on to:

- An exploration of Pythagoras' Theorem (see for example Tilted Squares)
- Straight line graphs (see for example Parallel Lines and Perpendicular Lines)

This problem follows on from How
Steep is the Slope?

Introduce students to the existence of tilted squares, by
playing the game Square
it or using the activities Eight
Hidden Squares and Ten
Hidden Squares.

Hand out this
8 by 8 square dotty grid and ask students, perhaps working in
pairs, to find all the different tilted squares which can be drawn
with vertices at the dots. Clarify that two squares are the same if
one could be cut out and placed exactly on top of the other. There
are twelve different tilted squares. There is no need to share this
information with the students; ideally challenge the students to
justify that they have found all the possible tilted squares.

Once students have had a chance to find most of the tilted
squares, ask them to report back. Agree a way of describing the
squares they have found, perhaps by reference to the way you move
to get from one vertex to the next. The interactivity can be used
to check that the shapes identified are indeed squares. Keep a
record of the squares found, perhaps in a table that can be added
to later.

Once all twelve have been identified, and there is agreement
that there are no more, ask students to find the gradients of
adjacent sides in each square. Add this information to the table
with the square descriptors.

Bring the class together to discuss what they have noticed
about the gradients. Can they predict the gradient of a side of a
square if they are given the gradient of the adjacent side?

Ask students to describe a method for checking whether two
lines are perpendicular to each other.

Now set the task at the end of the problem where students must
decide whether two lines given by their coordinates are
perpendicular. Students could follow this up by making up their own
sets of lines (which must contain some which are perpendicular and
some which are not) and challenging their partners to identify the
perpendicular ones.

How do you think the computer decides whether a shape is a
square?

How do you decide whether two lines are perpendicular or
not?

This problem could be followed up by Square
Coordinates and then Tilted
Squares for a geometrical introduction to Pythagoras'
Theorem.

Alternatively, students can move into algebra by investigating
the relationship between the equations of parallel and
perpendicular lines on a co-ordinate grid in Parallel
Lines and Perpendicular
Lines

If students are struggling to draw tilted squares, they could
work at computers using the interactivity provided in the problem.
They could complete squares of different sizes and list how to get
from one vertex to the next. These results could be used in the
next stage of the lesson.

If students are struggling to describe gradients, they could
take a look at How
Steep Is the Slope?