Rectangle NightmareTask 84 ... Years 4  10SummaryRectangle Nightmare was designed by Geoff Giles, a well known Scottish maths educator. In essence the problem is:We have a frame into which we seem to exactly fit a set of shapes. We remove one shape (a significant rectangle), rearrange and the remaining pieces still fit into the frame! Now you see it  now you don't. How can this paradox be explained? This cameo has a From The Classroom section which provides Geoff's own paper on the mathematics behind the design of this intriguing puzzle and additional comments to assist with challenging students who are trying it. 
Materials
Content

IcebergA task is the tip of a learning iceberg. There is always more to a task than is recorded on the card. 
We have a conservation of area problem here which is part of a wellloved set of 'missing square' puzzles. In this case, the square has become a rectangle and Geoff supplies us with all the background in an article included in this cameo. The pieces can be fitted into the frame as shown.
However, the way the material is displayed in the photograph above suggests there is another way to fit the six shapes into the frame.
At the first level of analysis there must be a very narrow space between the shapes in the first solution and the frame in which they fit. That space must be the same as the space taken up by the additional rectangle. Yes, but:

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. 
There are at least two ways to convert this task to a whole class investigation so that all students have the opportunity to struggle with the spatial challenge.
At this stage, Rectangle Nightmare does not have a matching lesson on Maths300, however, Lesson 132, 64 = 65, is a companion missing square puzzle that is also related to Fibonacci Numbers. For more ideas and discussion about 64 = 65, open a new browser tab (or page) and visit Maths300. 
Is it in Maths With Attitude?Maths With Attitude is a set of handson learning kits available from Years 310 which structure the use of tasks and whole class investigations into a week by week planner. 
The Rectangle Nightmare task is an integral part of:

Constructing

Rectangle Nightmare was designed by Geoff Giles, a well known Scottish mathematics educator who is perhaps most recognised for the challenging materials of the DIME Project. Geoff contributed several tasks to the Mathematics Task Centre Project and was also responsible for developing the educational use of Plastazote, the colourful, almost indestructible foam from which this task, and several others, are made.
Now deceased, Geoff, prepared the article below in the latter years of his retirement in response to our request for enlightenment about Rectangle Nightmare. It is reproduced here with his permission, albeit slightly modified. 
Figure 1 
Figure 2 
But this means that, while the area of the rectangle is clearly 15 × 22 = 330 sq. cm., the area of the triangle is in fact ˝ × 24 × 27 = 324 sq. cm. So where has the extra area disappeared to?
The answer turns out to be very simple. If we look closely at Figure 2 we see that the sides of the triangle don't look quite straight.
Look at point A. We find that the sloping side rises vertically 5 as it moves horizontally 2, making a gradient of 2·5. But when it passes the horizontal line marked 10  10, the gradient then becomes 22/10, or 2·2.Now if these shapes are cut out of a soft, spongy material like Plastazote, the small difference will not be noticed at all, and the deception will then be complete.So clearly these two lines do not form exactly the straight line they seem to. The triangle is slightly fatter than it should be.
Our second example takes a rectangle of size 3 x 8 (with an area equal to 24) and makes it appear to have the same area as a square of side 5 (with an area of 25).
Instead of presenting it as a finished puzzle, we now show how it was all worked out from the beginning so that you can see how the 'sleight of hand' was achieved.
Figure 3a 
Figure 3b 

Figure 4 

Figure 5 

Figure 6 
Your students won't have the information above when they try the task. Just completing the two 'jigsaws' involved may provide headache enough for them, let alone explaining the paradox. Some questions you might use if students become frustrated are:
Explaining the paradox can be as straightforward as drawing attention to the thin space around the solution to Qu.3 which is isn't there in the solution to Qu.4. Or it can be as complex as Geoff's explanation above.
One final thing  because a mathematician is never finished with a problem. Could it be that there is something special about the 5 : 8 ratio that was chosen to distort the original 1cm squares above? 5 and 8 appear in the Fibonacci series:
1, 1, 2, 3, 5, 8, 13, 21, ...