Reasoning · Grade 5-1 Equal-Area Transformation (2)
Problem
Give the road's three points names.
Putting a letter on each of the three points that matter turns a vague picture into something you can talk about precisely - and there really are only three points that matter.
4.G.A.1Draw A DiagramThe bend is worth exactly one triangle.
Areas add and subtract: whatever one farm gains the other loses, so the whole problem shrinks to keeping one triangle's area the same.
3.MD.C.7Identify SubproblemsPoints on a parallel line make triangles of equal area.
Between two parallel lines the gap is the same everywhere, so sliding the tip along the parallel line shears the triangle sideways without making it any taller.
6.G.A.1Draw A DiagramThe bend can be straightened because a triangle can be swapped for another of the same area on the same base.
Two triangles on one base with their tips on a line parallel to that base always cover the same amount of ground.
Each side of the road is its pieces put together, so swapping one piece for an equal one leaves both areas exactly as they were.
Through the bend draw the line parallel to the chord.
Drawing a parallel through a given point is a ruler-and-set-square move, so the whole construction works on a figure with no numbers on it at all.
4.G.A.2Draw A DiagramWhere it meets the bottom edge gives the new road.
One triangle was swapped for another of exactly the same size, so the total each farmer owns is untouched even though the boundary moved.
6.G.A.1Identify SubproblemsWith numbers, both areas come out unchanged.
Putting easy numbers on a figure that had none is the fastest way to be sure a construction really works before trusting the general argument.
6.G.A.1Solve An Easier Related ProblemGoing the other way does the same job.
The parallel line through the bend crosses both the top and the bottom edge, and each crossing gives one legal straight road - so listing them keeps you from thinking there is only one right answer.
4.G.A.1Make A Systematic ListSlide the tip of the wedge triangle along a line parallel to its base until it reaches the edge of the field: same area, but now the border is straight.