I'm fairly new to teaching and am currently teaching high school students about vector geometry. I am currently making a test for them for next week and it is mostly about the scalar product and the vector product. For this, I went through my old exams and exercises (which are a lot) and chose the ones I thought were important to solve.

Now there is one exercise that I think would be a good exercise for them, however it has really really ugly numbers as a result whereas all other exercises in the exam have very nice numbers. It reads as follows:

The point $M = (6,1,6)$ is midpoint of the regular octahedron $ABCDEF$. Further, one of the edges is part of the line $g$ consisting of the points $P = (0,-6,6)$ and $Q = (9,0,3)$. Find the volume of the octahedron without calculating any futher points. Then, find the coordinates of the six vertices $A,B, C, D, E, F$.

What software would be suited to create such exercises? Basically, what I need is an octahedron with a "nice" volume, its vertices at nice points in $\mathbb{R}^3$ and one of its edges running through nice points. By nice, I mean points with coordinates in $\mathbb{Z}$ or maybe very simple fractions.

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    $\begingroup$ If all the other tasks have "very nice numbers", you should include this task with "not nice numbers". Thinking of a solution as wrong because it's ugly numbers is simply a heuristic that can go wrong. The students should be aware of this fact. $\endgroup$ – Toscho Mar 29 '14 at 13:02
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    $\begingroup$ Just a minor comment: the word "midpoint" is usually reserved for the center point of a line segment. The word "midpoint" in this question should be replaced by "center point" or something similar, e.g. "center" or "centroid". $\endgroup$ – Jim Belk Mar 29 '14 at 14:48

I'm not sure how software would be helpful here. To make a problem of this form, you need to:

  1. Choose any center point $M$.

  2. Choose two orthogonal vectors $\textbf{v}$ and $\textbf{w}$ of the same length.

  3. Compute $M + \textbf{v}$ and $M+\textbf{w}$.

  4. Find two other points on the line through $M+\textbf{v}$ and $M+\textbf{w}$.

The volume will end up being $\dfrac{4}{3}\|\textbf{v}\|^3$. If you want to avoid radicals during the intermediate calculations, the only important consideration is that $\textbf{v}$ and $\textbf{w}$ have integer lengths.

Here are some orthogonal triples of integer-length vectors of equal lengths that you can use to pick $\textbf{v}$ and $\textbf{w}$.

  • $(3,4,0)$, $(4,-3,0)$, and $(0,0,5)$. More generally, $(a,b,0)$, $(-b,a,0)$, and $(0,0,c)$ for any Pythagorean triple $(a,b,c)$.

  • $(1,2,2)$, $(2,1,-2)$, and $(-2,2,-1)$

  • $(2,3,6)$, $(6,2,-3)$, and $(3,-6,2)$

  • $(4,4,7)$, $(1,-8,4)$, and $(8,-1,-4)$

If you look for them, these triples can be found in many vector geometry problems.


Trial-and-Error solution

Model this situation with any Dynamic Geometry Software, e.g. GeoGebra, using parameters. Change the parameters, e.g. using slow animation, watching for "nice" solutions.

  • $\begingroup$ What do you mean by "watching"? Since it must be exact I am not so convinced this works; moreover "simple fractions" might not even be so easy to see. But perhaps I am wrong or misunderstand what you mean by "watching." Do you use this method? Did you try it? $\endgroup$ – quid Mar 29 '14 at 13:34
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    $\begingroup$ @quid I use this method, but I haven't tried it for your problem. "Watching" means: show the values, you want to be nice. Look at them while changing the other parameters e.g. via animation. When the values are nice, stop the animation. I can see fractions with denominators 1 to 10 except 7 and some multiples of 2 and 5. If you need more complicated "simple" fractions, multiply the values in question by 1001. That way, you can also catch denominators 7,11 and 13 (and multiples with the other denominators). $\endgroup$ – Toscho Mar 29 '14 at 15:04

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