The Hole In The Triangle
Task 219 ... Years 4 - 10
SummaryThis task is related to 64 = 65, Cross & Square and Rectangle Nightmare, all of which are about an apparent loss or gain of a unit of area. This one appears to make the same size triangle in two different ways with the same pieces. The twist is that one of the triangles clearly has an area one square bigger than the other. It just can't be, but only mathematics can help us understand how we are being tricked and why the apparent difference is exactly one square unit of area.
IcebergA task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.
The two ways of arranging the pieces are:
and clearly the largest triangle in Diagram 2 is the same area as the largest triangle in Diagram 1, and yet it is also clearly one square bigger. What!??!
The clue is in the smaller triangles. The slope (or angle) of the hypotenuse of the yellow triangle is not the same as the slope (or angle) of the hypotenuse of the green triangle. Close ... but not the same.
Green Triangle: ... Rise/Run = 2/5 = 0.4
So, for a start, the apparent larger triangle is not a triangle at all. Further, the slope of the green triangle is steeper than that of the largest triangle and that of the yellow triangle is less. However this is still not sufficient knowledge to explain the extra square.
When we look at areas, the area of the four pieces is:
The paradox is happening because the three hypotenuses involved aren't quite at the same angle, so let's look more closely at the hypotenuses of the green and yellow triangles.
Look at the numbers involved:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55...might be jumping for joy about now. Could we create another hole in the triangle problem using the next set of related numbers in the Fibonacci sequence?
Whole Class InvestigationTasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.
To use this task as a whole class investigation, you will need to supply four pieces and a grid to each student. Using one centimetre as the dimensions of the unit square, two copies of the four pieces and the 5 x 13 grid will fit on an A4 page. Students carefully cut out the pieces and grid and then you set the challenge. Really, it's two challenges - discovering the problem and explaining the solution. Explaining to someone else is a significant aspect of a mathematician's work (see Working Mathematically), so encourage students to prepare a written report, poster, slideshow or video of their understanding.
At this stage, The Hole In The Triangle does not have a matching lesson on Maths300, however Lesson 132, 64 = 65 and Lesson 188, Missing Square Puzzle: 8=9, are related.
Is it in Maths With Attitude?Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.
The The Hole In The Triangle task is an integral part of: