The Farmers' Field Boundary

Why do this problem?

This problem presents a geometrical situation which can be solved using constructions. The interactive environment gives students the opportunity to explore and make conjectures which they can go on to justify, using what they know about areas of triangles. 

Possible approach

Explain the context of two farmers who wish to redraw their boundary, keeping the field areas the same, but making the boundary a single straight line. Add the extra constraint that one end of the line must be by the tap. Then give students some time to explore the interactive environment in pairs. They can set up various configurations of the two fields and then try to position the boundary correctly. Once they think they have positioned the boundary in the right place, there is the option to reveal the actual areas to see how close they are to the desired boundary.

When everyone has had a chance to explore, encourage them to start thinking about how to find the solution using geometrical reasoning rather than just guesswork. One possible approach is to encourage them to collect screenshots of correct solutions so they can look at several examples at once. The interactivity offers the option to draw the line AB, which might prompt some students to imagine lines parallel to AB that provide useful information when looking for a way to conserve the original areas.

The problem Shear Magic might offer students useful insights about families of triangles with the same base and a third point moving along a line parallel to the base. These insights can then be used to justify their method for constructing the new boundary. 

Once students have come up with a method for solving this type of problem, they could have a go at constructing solutions away from the computer.
Here is a worksheet with three examples of fields that students could construct the new boundary on: Farmers' Fields
There is also a version without grids: Farmers' Fields (without grid)
Here is sheet showing the answers to the three examples: Farmers' Fields Answers

Key questions

Approximately where does the line need to go?
Are there any parallel lines we can draw that might help?
What do we know about the area of triangles with the same base and height?
Can we transform the triangle with base AB into an equivalent triangle with the same area?

Possible support

It might be worth working on Shear Magic before trying this problem.
Encourage students to start with rectangular field boundaries like the first example on the Farmers' Fields worksheet. 

Possible extension

Triangle in a Triangle and the problems that follow it require similar geometrical reasoning but provide a more challenging exploration.