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Investigation: Border battles

Lesson

 

Suppose an island is divided as follows:

However, we want to straighten the border without changing the area of either territory.

Discussion question

Is it possible to straighten the border without changing the area of either territory?

Activity

  • Individually for five minutes, try to straighten the border without changing the area of either territory.
    • If you can do it one way, try to do it another way too.
  • In pairs or small groups, discuss and figure out a way to straighten the border without changing the area of either territory.

Follow-up Questions

  1. Is there more than one way to straighten the border without changing the area of either territory? If so, find another way to draw the border.
  2. Use the applet below to show your new straight border. Slide the blue dots to make a new border and see how the areas change. Reveal the straight line to see if your points are indeed all on a straight line.

 

  1. Compare your new border with one of your peers.
  2. We have already seen that the area of the triangle is calculated by the following formula:
    $A=\frac{1}{2}bh$A=12bh
    Knowing this, is there a way to change the shape of the triangle without changing the area of the triangle?

     

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