I should've asked this question two years ago when my son (at that time, 9 years old) came to me and said: "Dad, today in school our teacher drew a line on a paper and said this is a straight line, it goes from both directions and doesn't meet itself". I answered, "Okay, what is wrong with that". He said, "but I think it meets itself". I told, "how come?" Then, he drew a line on a piece of paper, made a cylinder with the paper and showed me how. At that time, I knew there is an element of truth in his idea, but being afraid of destroying the originality of his thinking I didn't add anything and instead promise him to think of his idea. I didn't keep my promise! But last night, he (now eleven) came back to me with the same idea thinking of what happens on a sphere. Again, I promised him that we discussed it later on (that is supposed to be today). The question is how can I help him to develop his ideas. The question for you is: Are there any sources available?
[Math] Non-Euclidean Geometry for Children
educationnoneuclidean-geometryreference-request
Related Solutions
Triangle similarity
He could have solved the question by drawing a figure such as the one below. If he knew about similar triangles at the age of $12$, he could easily set up the equation $ \dfrac{x}{4} = \dfrac{9}{x} $. From here, if he knew how to solve equations like this, he could solve $ x^2=36 $. If he did not, he could simply try a few different values of $x$ until he found one that works.
Colored triangles are similar. The problems with this explanation are:
- This is essentially drawing a time-position plot. Since this is a $12$ year old who does not know multivariate algebra, it's a bit of a stretch to assume he would know how to use kinematic plots to solve problems. But then again, who knows, maybe he was imagining the women walk and the plot seemed intuitive.
- It requires him to be able to set up and somehow find the solution of the quadratic equation.
- What does this have to do with "scaling arguments, dimensional analysis or toric variety theory"?
Trial and error
He could have simply tried a bunch of possible values until he found a solution.
Let's say he decided to try $9$ am: That means woman $A$ walked the first distance in $3$. So the ratio of distances is $\dfrac{3}{4}$. But if woman $B$ walked the first part in $3$ hours, then the ratio comes out $$\dfrac{3}{9+3}=\dfrac{3}{12}!$$ So $9$ am isn't right.
Luckily the solution is an integer, and somewhere between $4$ and $9$ hours before noon, since the quick woman would have walked more than half the distance in the morning and the slow one would have walked less than half (drawing a $1$D diagram makes this obvious). Even if the solution was not an integer, after exhausting integers he could have tried half hours, then quarter hours, and so on. I'm sure a binary search type of strategy would become obvious if he kept track of how much each sunrise hour was off by. Since he spent "all day" on it, there's plenty of time for numeric solutions.
The problem is that solving it by brute force teaches you absolutely nothing (well, it's arithmetic practice, and you do end up discovering binary search). There's also the question of what this has to do with scaling arguments, dimensional analysis or toric variety theory. It's also the sort of solution that you would expect from perhaps a future clerk, not mathematician.
Incidentally, after trying a bunch of numbers like this, the $\dfrac{x}{4}=\dfrac{9}{x}$ equation does suggest itself.
Sort of trial and error
It's not too great a leap to realize that the distance itself doesn't matter, so let's say he decided to let it be 50 km. Then the speed of woman $A$ is $\dfrac{50}{x+4}$. The speed of woman $B$ is $\dfrac{50}{x+9}$. We know that the ratio of their speed must be $\dfrac{x}{9}$ based on how long it took both women to walk the distance between $A$ and the meeting point. So $\dfrac{\dfrac{50}{x+4}}{\dfrac{50}{x+9}} = \dfrac{9}{x}$ which after some basic manipulation simplifies also to $x^2=36$. When the $50$s cancel, he would see right away that his hunch about distance not mattering was correct.
The problem with this is that requires a $12$ year old to reason about kinematics (ratio of speeds from ratio of times, deriving speed from time and distance) without having the mathematical vocabulary for doing so. It also requires him to not be daunted by the ugly looking equations that come out. Last, it requires him to have a hunch about the distance, otherwise he has to use a variable to represent distance and at that point we are back at multivariate algebra.
This method also does not appear to have anything to do with scaling arguments, dimensional analysis, or toric variety theory, except perhaps in the slightest sense.
Conclusion
None of these options really satisfy me. For instance, I can't really imagine myself using any of these solutions when I was $12$. The only one I would have comprehended would be the straight trial and error, which I would have been too lazy to actually carry out. Though then again, I was never very good at math.
The problem seems to be a simple linear equation system, so I don't see what it has to do with the concepts he refers to. Perhaps he was making a metaphorical point about how this sort of problem is the "tip of the iceberg" of linear algebra that children first get exposed to? Or perhaps he actually figured out linear equations over the course of that day, and that was the revelation?
That the sum of the angles of a triangle in the hyperbolic plane is less than 180° is a fact that depends only on the axioms chosen and is completely independent from the model we use to visualize the said plane in the euclidean plane.
For instance, the Poincaré disk model does not distort the angles: the euclidean perspective (in this model) is not different from the hyprlerbolic perspective. However, in this model the straights look like parts of euclidean circles. Or in the Klein model the hyperbolic straights look like euklidean segments and the triangles are like as if the sum of their angles were 180°.
What we see in an euclidean model does not have much to do with what we would see if we -- boom! -- turned to be hyperbolic. Would we see straights? It depends on the physical thing that we would cosider straight. The path of a light ray there? Then yes, if the light rays would be considered straight, and if they behaved like straights (in the hyperbolic sense) then we would consider everithing straight that would behave like light rays.
Best Answer
A classic introduction to some aspects of non-Euclidean geometry is Edwin Abbott's Flatland. It would be good for your son now because it doesn't require any technical apparatus.
(Flatland was used as inspiration for this scene in Carl Sagan's Cosmos. Sagan mentions Abbott by name.)
Another book is Geometry, Relativity and the Fourth Dimension. If you are relatively competent in mathematics yourself this book will give you lots of fodder: the explanation of the five Euclidean axioms, what happens when we throw out the parallel axiom, ... The diagrams in the first couple of chapters alone should give you things to talk to your son about. I have used it with teenage children of friends.