Level 2: Strange loops

Infinity and strange math · level 2 of 8 · rough age 7-8 (a guide for parents; anyone can start anywhere)

In this level the never-ending things from level 1 start to play tricks. You make a loop of paper with only one side and cut it in surprising ways, learn to compare groups by matching instead of counting, and use matching to find that there are just as many even numbers as counting numbers. You meet the googol and the googolplex, enormous numbers that are still not infinite, draw a picture that contains itself, and find out why the impossible triangle and the never-ending staircase work on paper but never in real life.

Lesson	You will be able to
The Möbius strip	Make a Möbius strip, show it has one side and one edge, predict what cutting it does, and explain the surprises with lanes
Matching without counting	Compare two groups by matching them one to one, read the answer from the leftovers, and explain why counting is matching
Patterns that go on forever	Match the counting numbers with the even numbers, the tens and the odd numbers, and explain why a part of an endless list can be as big as the whole
Googol and googolplex	Explain why a zero on the end makes a number ten times bigger, write a googol, and say why a googolplex is huge but still finite
Picture in a picture	Find the Droste effect around you, work out how fast the copies shrink, and draw your own picture in a picture
Impossible pictures	Explain why the Penrose triangle and the never-ending staircase cannot be built, by following the loop and counting the steps up

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