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Agile Algebra

Observe symmetries and engage the power of substitution to solve complicated equations.

Graphs of Changing Areas

Age 16 to 18 Challenge Level:

Why do this problem?

This problem offers an ideal opportunity to begin thinking about graphs of simple rational functions. Students can begin to make sense of concepts such as symmetry and asymptotes with the security of a concrete example on which to hang their understanding.

Possible approach

Display the graph or hand out this worksheet.

"What could the graph represent?"
"If I told you that $x$ and $y$ represented the length and width of a family of rectangles, what could you say about all the rectangles?"
All the rectangles have the same area.

Now display the questions from the problem, or hand out this sheet.

Give students some time to work in pairs to answer the questions. Encourage them to switch between algebraic thinking and reasoning based on their geometrical understanding of the properties of rectangles.

Finally allow some time for students to share their solutions.

Key questions

How does the rectangle with length $x$ and width $y$ relate to the rectangle with width $x$ and length $y$?

What does it mean when the line $y=\frac{1}{2}P-x$ intersects with the curve $y=\frac{10}{x}$?

Possible extension

Students could be invited to consider representations in three dimensions of cuboids with equal volume.

Possible support

Set students the stage 3 problem Can They Be Equal? as a warm-up before beginning this task.