19

A possibility is to show your students Google's implementation of the game Snake. If you enter the term snake into Google's search engine, the there is a box at the top showing If your particular Google bubble doesn't show this result, I think that there is a direct link. If you hit "Play", you can play a game of Snake, using the standard rules. ...


8

I would use a number line. This is the most straight forward way to explain being "in between" integers while giving some intuition with a visual. It is possible that a school-age child would be familiar with it, too. The number line is particularly useful because you can demonstrate that the distance between a point near the end of 2020 and a ...


8

Introduction I wasn't taught the partial fractions decomposition (PFD) in calculus. We didn't cover it in high school, and when I went to college, they assumed we all knew it. Somehow it was when I read the proof in van der Waerden's Modern Algebra that I understood why it was called partial fractions. I looked at it again a couple days ago, and ...


7

One thing that I like about the Pacman analogy is that, if you draw Pacman's eye (or Ms. Pacman's bow, which is my usual choice), then Pacman is not mirror symmetric. This means you can talk about nonorientable surfaces and draw how Pacman goes around a Mobius strip/Klein bottle/projective plane and comes back reversed.


6

I don't think it's necessary or a good idea to invoke a specific game, because any given game will be one that only a small fraction of your students have played. Just say, "You know how in some video games, if you go off one side of the screen, you reappear on the other side? Say you have a game where this happens both left/right and top/bottom."


5

This isn't exactly what you are looking for, but it's pretty close: the book "Surreal Numbers" by Donald Knuth builds up the theory of surreal numbers by telling a fictional story of two people discovering and building the correct axioms. You should check it out. It's a neat read, and if I remember correctly it actually has exercises too. From the ...


5

Let me build on the idea of Steven Gubkin in his comments. One way to visualize this scenario is to use Ford circles. The standard picture is to plot a circle tangent to the $x$-axis at $\frac{p}{q}$ with radius $\frac{1}{2q^2}$ where we always assume $p$ and $q$ are relatively prime. This gives an intriguing family of circles with special tangency ...


4

I'm wondering how to introduce the idea in a way that is intuitive or geometric. How about an introduction along the lines of adding two shapes together. Start with adding fractions to have a picture in mind: For example, one could add the fractions $\frac{1}{5}$ and $\frac{1}{7}$ by first drawing a rectangle with width $\frac{1}{5}$ and height $1 = \frac{...


3

I think associativity is the most natural property to encounter. This is because transformations are associative. For transformations $f,g,h$ with compatible domains and ranges, $(f\circ g)\circ h=f\circ(g\circ h)$. Transformations are not necessarily commutative. In your question, I sense you want to focus on motivating the foundational properties of the ...


3

Part of the answer is that mathematicians stumbled upon structures that had an "addition" and a "multiplication" that behaved (more or less) like integers do (rationals, reals, complex numbers, the residue rings, matrices, polynomials, power series, rational functions, ...). So it made sense to just assume a set and operations with those ...


3

My approach might be to try to use smaller units of time to draw comparisons. I would look at a clock make sure the child knew about seconds or minutes. We could talk about how long one second is, and then how long one minute is, and how on a standard digital clock, the time is usually displayed up to the minute. We could even watch the time change as it ...


3

When you're picking out a movie on Netflix, there's a grid of movies & TV available. If you keep scrolling to the right in any one of the rows, you eventually come back to the first movie in the row. And if you keep scrolling down through the categories, eventually you come back to the first category you started out with. (At least, that's how the ...


2

I have (once) used a square of felt with two strips of velcro glued to the felt. Then just "identify" one opposing pair of edges in an orientation preserving way via velcro. Demonstrate that one obtains a finite cylinder. Then "identify" the two circle edges in an orientation preserving way. This also gave some room to talk about ...


1

East to west on a world map The world map is usually what I use to explain toroidal periodic boundary conditions. The analogy isn't perfect because the Earth isn't a torus, but it gets the important ideas across in a way everyone can understand. Unfortunately, it doesn't work north-to-south. I used to use Asteroids, but few students are familiar with the ...


1

I'm not sure you need an analogy, I think you could explain it nicely just stating how to construct it from a square, but if you do, you could buy a couple of bicycle tubes (preferable for a children’s bike, which are smaller and wider). Then cut it open so you have a rectangle. You can then show the intact and the cut version side by side. (It might not be ...


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