The Hole In The Triangle

Task 219 ... Years 4 - 10

Summary

This 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.
 

Materials

  • 4 shapes for each person
  • 1 grid

Content

  • algebra, linear
  • concept of proof
  • decimals, calculations
  • measurement, angle
  • measurement, area
  • patterns, Fibonacci / Golden Ratio
  • ratio & proportion
  • recording mathematics
  • sequences & series
The Hole In The Triangle

Iceberg

A 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:


Diagram 1


Diagram 2

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
Yellow Triangle: ... Rise/Run = 3/8 = 0.375
Largest Triangle: ... Rise/Run = 5/13 = 0.3846... (if it were actually made)

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:

  • 7 + 8 + 5 + 12 = 32 square units.
When we look at the area of the correct triangle on the grid (the half rectangle), it is:
  • (13 x 5) = 32.5 square units
This half unit difference must surely have something to do with the extra unit square.

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.

Diagram 1

  • Use the bottom left corner of the grid as (0,0).
  • Assume the green and yellow triangles are correctly placed on the grid intersection where they meet at: (8, 3).
  • Is (8, 3) on the hypotenuse of the largest triangle?
  • Co-ordinate geometry tells us the equation of this hypotenuse is: y = 5/13 x, so if x = 8 in this equation the y value is just over 3.
  • So the green and yellow triangles run inside the line of the hypotenuse and form two sides of a long thin parallelogram. (There is a similar parallelogram in 64 = 65.)
Diagram 2
  • Use the bottom left corner of the grid as (0,0).
  • Assume the green and yellow triangles are correctly placed on the grid intersection where they meet at: (5, 2).
  • Is (5, 2) on the hypotenuse of the largest triangle?
  • Co-ordinate geometry tells us the equation of this hypotenuse is: y = 5/13 x, so if x = 5 in this equation the y value is just less than 2.
  • So the green and yellow triangles run outside the line of the hypotenuse and form the other two sides of the parallelogram.

So:

  • In Diagram 1 the two triangle pieces 'push back' into the half-rectangle-triangle by half a square unit along the true diagonal and 'cover up' the half unit difference in area.
  • In Diagram 2 the two triangle pieces 'pull out' beyond the half-rectangle-triangle by half a square unit along the true diagonal and 'stretch out' an extra half unit above the half unit difference in area.
    Leaving behind a unit square created from two half squares: + = 1.
What a wonderful task! Especially at middle secondary years, it gives meaning to:
  • conservation of area
  • understanding and application of gradient
  • co-ordinate geometry
  • area of a triangle
  • concept of proof
and it doesn't end there.

Look at the numbers involved:

  • Rectangle: 5 x 13
  • Green triangle: base = 5, height = 2
  • Yellow triangle: base = 8, height = 3
Those familiar with the Fibonacci Sequence:
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 Investigation

Tasks 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:

  • MWA Chance & Measurement Years 9 & 10

Green Line
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