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Enclosing Squares

Can you find sets of sloping lines that enclose a square?

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Parallel Lines

How does the position of the line affect the equation of the line? What can you say about the equations of parallel lines?

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Weekly Problem 35 - 2007

What is the area of the triangle formed by these three lines?

Perpendicular Lines

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Why do this problem?

This resource allows students to explore the connections between straight lines on graphs and the equations that represent or define them - equations of the form $y = mx + c$.

Possible approach

Take time to discuss how an equation represents a line by defining the set of points that lie along it. If this needs some reinforcement the related problem Parallel Lines might be a good place to start.

Show the interactivity, moving all four points to allow the group to see the freedoms the lines have. Draw attention to the equations of the lines showing beneath the graph.

Ask a student to move the lines so that they are perpendicular. Does everyone agree? How can we be sure? Can anyone set up a harder pair of perpendicular lines? Are they correct? Preliminary work looking at perpendicular lines is suggested in At Right Angles.

Ask students to work in pairs at computers, finding pairs of perpendicular lines and noting their equations. What do they notice about the equations of perpendicular lines? Encourage them to make and test their conjectures. Bring the class together to share insights and conclusions.

When they understand what is going on, ask them to set challenges for each other - either for their partner, or on the board for the whole group:
e.g. find three pairs of perpendicular lines which go through $(2,3)$, find perpendicular lines where one goes through $(-1,3)$ and the other through $(6,2)$...

Key questions

What does perpendicular mean?
How do the equations of perpendicular lines relate to each other and why is that?

Possible extension

Ask students to suggest equations for sets of fourlines that define a rectangle/square/parallelogram, etc. This is described in the problem Enclosing Squares . It would be useful to have graph plotting software available.

Possible support

Concentrate on the work in Parallel Lines
When discussing perpendicularity, ask students to draw tilted squares on the interactivity in Square Coordinates, and/or to do some work drawing them on squared paper. This topic might also provide a good excuse to play the game Square It