58

Draw the bottom three-quarters of an oval: Flesh that out to make the bottom half of the strip: Connect one of the open ends at the top to the bottom on the other side: Now draw a straight line across the top: Finally fill in the last edge at the back: An advantage to this approach is that it highlights the fact that the mobius strip starts out as an ...


14

The middle third is obtained by trimming the edges off the original möbius loop. It is therefore simply a thinner möbius loop (the short loop). The outside thirds of the möbius loop are obtained by cutting the loop in half and trimming 1/3 off the edge that was not originally the outside edge. It is the same as cutting the strip in half: as the outside ...


13

You could explore maps such as the London Underground where the actual distance and locations of stops are unimportant only their relationships. Perhaps this could be turned to a practical lesson somehow? Kids with playing the part of stations and holding string? Design a similar map of their home city? Ranking maps? What makes a good map? You could take ...


13

Homotopy equivalence v. Homeomorphism. I believe an accessible difference between homotopy equivalence and homeomorphism is that one preserves an intuitive (though hard-to-define) topological invariant, while the other almost never does: the invariant of dimension. For instance, any $\mathbb R^n$ is homotopy equivalent to a point. Similarly, $\mathbb R^n - \...


12

I searched google images and found many nice renditions. Here one that you may prefer from this link (source: umich.edu) . I especially like that it shows the width of the paper and doesn't draw the strip as a line or piece of string. Having the students make them is always a hit.


11

Two ideas. (1) Form a Möbius strip out of paper. And then cut it down the middle. Restart and cut it at a third of its width. See, e.g., "Explanation for cutting a Möbius strip at one-third its width."             (Image from Kidzone.ws.) (2) You already cited the previous question about triangulated surfaces, presumably ...


8

I am not entirely sure on the best way to convey the difference between homotopy equivalence and isomorphic homology groups (or even isomorphic homotopy groups, though on CW-complexes I guess this isn't as big of a concern), except by way of examples. I remember my algebraic topology exam had an explicit example of spaces with all the same homology and ...


8

What you want is Omega Colored Chalk:                  "This non-toxic, low-dust chalk delivers extra-smooth and easily erasable writing in eight vibrant colors." I won't include a link, but if you search, you will find many office-supply outlets that carry it in the US, for about ...


8

Since I use Seifert's algorithm to construct surfaces a lot, I tend to go for the flat approach:


7

Thanks everyone for your tips! I thought I would share what we decided to go for in the end. We had far more material than this (it naturally extends to the investigation into mobius strips, but in an hour this is a good amount of material. Even for top set Year 8 students, who we also did this lesson with in the end). What surface is the game "snake" ...


6

First of all, let me echo all the comments -- the key point here is that these surfaces are homeomorphic, but this homeomorphism cannot be realized by an isotopy. This is an important distinction! It is probably worth taking some time to explain the issue. In particular, students should understand that topological spaces don't naturally come embedded in $\...


5

(Moved from comment...) Examples in the plane: a twice-punctured annulus shows the difference between homotopy and homology, because the fundamental group (free on two generators) has a much smaller abelianization. That is, for $s,t$ two loops, one around one hole, the other around the other, $sts^{-1}t^{-1}$ is non-zero in homotopy, but is homologous to $0$....


5

Instead of drawing on the blackboard, use a 3D model and pass it around when you're finished using it for demonstration. If the class is small enough, you can bring in a strip of paper for each student to make his/her own. The demonstration will be much more powerful this way and the lesson more likely to stick.


4

While the geometry of surfaces is a relatively quick starting topic it does not seem to me that it is the ideal way to show an 11 year old how much geometric insight can be obtained without metrical information. My favorite example for introducing topological ideas is "Euler's traversability" theorem which says that a graph (diagram of dots and lines) has a "...


4

I recall a cute introduction to the work of Miryam Mirzakhani that is written at the level of a younger student. Something like this could be adapted and incorporated into a 11-year-old's math curriculum. Here is the link to the infographic on Matific. It talks about surfaces and the genus of surfaces and is reasonably well explained. Keep in mind your 11-...


4

For a long time, until I transferred to a school with white boards and dry erase markers, I actually used sidewalk chalk like these It is cheap, lasts a long time, incredibly durable, and has nice bright colors. Sidewalk chalk also draws really thick lines making your writing more visible from further away. Additionally, many sidewalk chalk packs come with ...


4

I have used polydrons with 5th-grade students through to college students:           A store in Massachusetts used to sell them, but recently I've had to purchase them from England.


4

The outer two-thirds are the same piece; the inner third is a different piece from the outer two-thirds. There have to be two different pieces, because at all points along the cut line, the cut is separating a piece that "contains the centerline" from a piece that "contains the original edge". Because both edges are the same edge, the outer two-thirds form ...


3

You might try a couple of puzzles near the middle for them to think about, and see if they can come up with the answers. One of my favorites involves a (topological deformation of) two-holed doughnut with a door handle going through both holes, and attached to a door, naturally. The idea is to deform the doughnut so to free one of the holes. (You may be ...


3

A nice way to think about the result is to take a long strip of paper, cut it into three pieces, and then twist and join them. This results in the same configuration as cutting it after twisting and joining, but it lets you track each strand individually.


3

Sphere: No problem. Torus: Swim Ring. Double Torus: Figure Eight (rock climbing), but could be to small for your purposes. $n$-Torus: Craft them yourself fronm Polystyrene and cover them with paper to draw upon. Non-orientable, bounded surfaces: Hmm, not possible, if you really want to draw triangles upon them.


3

One thing that your son might consider talking about is the winding number. One example that I sometimes give involves garden hoses. When the hose is nicely coiled on the ground and you try to drag it (staying along the ground) to the garden, you develop kinks in the hose. One way of interpreting this is that the winding number is being preserved. Of course,...


3

How about giving them the five room puzzle on the plane and then on the torus? This is a rather visceral way to appreciate why "holes matter", and may segue nicely into topics in topology.


3

Check out Kinsey's book Topology of Surfaces. Kinsey proves the classification theorem of surfaces by considering triangulations of surfaces and then going through the algorithm of cutting-and-pasting to classify them. I'm pretty sure he doesn't talk about surfaces having to be locally Euclidean before this, or at least I don't remember it featuring in his ...


3

As suggested, I will turn my comment into an answer. Perhaps you could provide each student with a rubberband (or something similar) and provide different challenges for them by showing different shapes and asking which ones are possible and which ones aren't. You could even color the sides of the rubberbands different colors in case the different ...


2

You could do something in knot theory, for example: 3-colorability of a knot. The wikipedia article http://en.wikipedia.org/wiki/Tricolorability shows a proof of isotopy invariance which should be understandable to an 11 year old.


2

To explain the linking, consider three long parallel pieces of string attached to a long rectangular piece of paper. Joining the edges in a cylindrical fashion results in three disjoint loops, while using a Moebius twist joins two of the strings together, and does so in a linked fashion to the loop created by the middle string. From this image it is not ...


2

Alternatively, a pictorial explanation can be used (perhaps more effectively)


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