64 = 65
Task 155 ... Years 4 - 10
Summary64 does not equal 65! Or does it? This jigsaw puzzle suggests that it does The paradox is one of a genre of 'missing square puzzles'. If you begin with an 8 x 8 square and cut the four pieces shown, then their total area must be 64 square units. But when placed in a 5 x 13 frame, they appear to fit and therefore show an area of 65 square units. Where did the extra square come from?
The explanation of this extra square paradox can be attempted at many year levels. However, noticing that the key numbers involved (5, 8 and 13) are successive terms of the Fibonacci sequence is the starting point for an extended investigation.
IcebergA task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.
There are two levels of explanation for the missing square - 'by eye' and by calculation. By eye, if the 'diagonal' of the rectangle is closely examined, there is a long thin parallelogram in the middle ... and you can guess the area of this parallelogram.
If necessary, to help students line up the rectangle a 5 x 13 frame matching the puzzle pieces is supplied. If you have the eTask Package, the frame will match the pieces if, when printing, you are careful that the Page Scaling option is set to none. If making your own puzzle pieces, design your 8 x 8 square first then the correct size 5 x 13. The calculation alternative for the extra square calls on mathematician's tools such as Pythagoras' Theorem and Trigonometry. The dissection of the 8 x 8 square of unit squares is:
Close! Do we conclude therefore that the rectangle can be made because if only two decimal places are used, the lengths would appear to be the same?
We have to turn to trigonometry, or ratio, for more information. There are four pieces in the above dissection, but they come in pairs - 2 trapeziums and 2 triangles. One from each pair makes half of the supposed 13 x 5 rectangle like this:
Rise/Run = 3/8 = 0.375Triangle section of trapezium
Rise/Run = 2/5 = 0.4Almost the same, but just different enough to prove that the 13 x 5 rectangle can only exist if there is a long thin parallelogram 'up the diagonal'.
ExtensionsA major extension of the problem, which can turn it into an extended investigation, comes from noticing a link in the key numbers of the problem. The square is 8 x 8 and the rectangle is 13 x 5. The key numbers are 5, 8 and 13. A student might notice that the first two sum to the third. It is not much data, but it might lead to investigating whether 3 other numbers might exist that could generate this 'extra square phenomenon'.
It is a property of the Fibonacci sequence that for any three consecutive terms of the pattern, the product of the first and third differs from the square of the middle term by 1. For example:
An interesting pattern within this is that the square of the middle term alternates between being bigger by 1 and smaller by 1; a connection perhaps with the 'disappearing' unit of area. Could it be therefore that a dissection like that of the 8 x 8 square in the puzzle can be reproduced for any three numbers in the sequence? A hypothesis worthy of testing.
At least, to visually highlight the parallelogram aspect of the puzzle, try the dissection on a 5 x 5 square and 'prove' that 25 = 24. The parallelogram (overlapping in this case) is clearly visible.
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.
This is easy to state and easy to start as a whole class lesson. Each pair needs graph paper and scissors and you need an image of the dissection for all to see.
All I want you to do to start the lesson today is carefully cut an 8 x 8 square from the graph paper.Check that the students know where the pieces came from:
...And what was the area of the original square?and record this fact.
Next I want you to rearrange the four pieces to make a rectangle.Record this fact too ... and wait for the discussion to begin!
For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 132, 64 = 65 which includes using Trigonometry and the Sine Rule to calculate the area of the parallelogram.
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 64 = 65 task is an integral part of: