Can you tile this big L shape with identical, but smaller copies of itself? If so, the shape is called a rep-tile (repeating tile).

However, that’s not what we are going to be looking at here. We are going to explore “irreptiles” which are shapes that can be tiled by different sized copies of themselves.

This L shape is an irreptile, but you will probably need some hints in order to make this a fun puzzle rather than a grueling ordeal.  See the next slide…

 

 

Tile the rest of the L with smaller, similarly proportioned copies of itself. The next slides will take you step by step through how I solved the problem…

 

First I tiled the top.

 

I kept working around the edge. This looks plausible.

 

I struggled a little bit before finding this one.

 

…but then this one was quite obvious. Now I looked and looked again. The final shape is an L, but it does not have the right dimensions – it is not a smaller version of the big L. It took me a little while before everything fell into place…

 

This was the secret…

 

Now the remaining L is a smaller version of the big L.

 

Finished.

Students should solve the irreptiles on the puzzle sheets to be found at the end of this slide show. Then they should make their own by googling complete solutions and then removing most of the tiles. It is easy to make extremely hard puzzles. It is more difficult to make puzzles that are a fun challenge for your classmates.

 

This rectangle is twice as long as it is tall. It is a special kind of irreptile because it can be covered with similarly proportioned rectangles – all different sizes!

I’ll give hints on the following slides. This is not how I solved it. I solved it by first finding how to fill the gap between the two given rectangles.

 

Pause whenever you think you know where to place another rectangle…

 

There are only four more to place.

 

There are only three more to place.

 

Well that gave it all away 😉

 

All rectangles have one dimension twice as long as the other and all are different sizes. Isn’t that beautiful!

Download puzzle sheets here.

Irreptiles

(MathPickle, 2012)

This puzzle is for students working on scaling shapes – maintaining the same proportions. Because it is an aesthetic puzzle you should be prepared to hand out additional puzzle-sheets for those students who want to beautify their answer. This is rarely a waste of time. Students should be encouraged to take pride in how their work looks.

Standards for Mathematical Practice

MathPickle puzzle and game designs engage a wide spectrum of student abilities while targeting the following Standards for Mathematical Practice:

 
MP1 Toughen up!

This is problem solving where our students develop grit and resiliency in the face of nasty, thorny problems. It is the most sought after skill for our students.

MP3 Work together!

This is collaborative problem solving in which students discuss their strategies to solve a problem and identify missteps in a failed solution. MathPickle recommends pairing up students for all its puzzles.

 
MP6 Be precise!

This is where our students learn to communicate using precise terminology. MathPickle encourages students not only to use the precise terms of others, but to invent and rigorously define their own terms.

MP7 Be observant!

One of the things that the human brain does very well is identify pattern. We sometimes do this too well and identify patterns that don't really exist.

(http://www.corestandards.org/Math/Practice/)

Please use MathPickle in your classrooms. If you have improvements to make, please contact us. We'll give you credit 😉

Gordon Hamilton

(MMath, PhD)

 

Lora Saarnio

(CEO)