From point to point¶
About this lesson plan¶
This is a lesson plan for indented for realization during 2h lesson activities.
It has been developed during work in iCSE4school project based on lesson carried out in 2015-2017 at The Stefan Batory High School in Chorzów.
It was prepared by Krzysztof Oleś based on his lesson.
Attention!
In each of the “code” cells you can change any number, text or instruction. In order to return to the original version refresh the webpage. Sometimes the next code depends on variables defined from the previous one, so one has to execute cells in order of apperance.
From point to point¶
Let’s think about a graph connected with a function, i.e. \(f(x)=\log_x\left|4\sin\left(\frac{\pi}{2}-3x\right)-6\right|\). That’s of cource a continuous line. Really?
Using SAGE for the preparation of drawings, we can rely on:
the description of the function
equation, which we can call “entangled”
polar formula
Sometimes we can see some similarities among the effects of using these different approaches: in each of these cases a set of points appears on computer screen - it is fitted in more or less complex formula.
Maybe should we use the simplest method: from point to point? Using recursion?
Let’s think about the point placed in the coordinate system - it gives the opportunity to trace the connection between geometry and numerical operations. Important for us is the possibility of student’s experimentation and computer fun - examples are presented in their programming layer easy.
So we start by placing the point on the screen.
Apparently nothing - so let’s put on the screen five points…
After this small planned error we try to add points.
Note that even with ctr+c+ctrl+v it takes time and it’s scary to think about boredom of the placement in this way a hundred points - in a situation where we can see a certain REGULARITY in the second coordinate points considered. Therefore, let’s use it.
Change the point size, playing tinge.
Do not forget about the possibility of placing the loop in the loop.
Looking at the obtained effect we see a problem with the “left” apex of a triangle - try to remove it properly manipulating numbers.
If we did, then we can go to the draw.
In the example above, you can see a kind of chaos… Can you control over the points?
Imagine a situation in which the specified starting point \((a,b)\) is transformed in one of eight randomly selected transformations. Each of them consists of two parts: a linear operation on the first coordinate (three numbers \(a_i,b_i,c_i\)), and linear operations on the second coordinate (three numbers \(d_i,e_i,f_i\)). After the transformation we obtain a new point \((a,b)\), which is thrown into the described formulas again - and of course, not satisfied with two points, we (computer?) repeat it hundreds of times.
Let look carefully at the source code (specially on lines containing #).
With a hundred repetitions figure seems chaotic - therefore we’ve repeated our experiment more times (you should change d again).
Do our points put us in something familiar?
With the seeming chaos should emerge Sierpinski’s carpet.
But let’s try to experiment and answer the following not easy questions:
Does built figure depend on the choice of the starting point?
What happens to the built figure, if we change \(a_i,b_i,c_i,d_i,e_i,f_i\)?
What happens to the built figure, if one of the eight maps we exclude - how can we quickly in the source code do it?
Why is carpet colored in such a way and not another?
We believe that the answers to these questions - based on tested assumptions - will be surprising…
Perhaps more than a carpet fascinates us known (almost all) leaf.
Perhaps the carpet and the leaf ATTRACT us to further experiment, in which we will try to arrange the coefficients in the tables (various methods for introducing the coefficients in the above two examples encourage such arrangement).
Let’s go back to the Sierpinski’s carpet. Or is it rather a kind of line or something like connected squares? Is repeating iterations indefinitely (in our head…) move closer to a more normal squares? What does “more” mean?
Look at the blue line below - we want to measure it with a green ruler.
Let’s estimate the length of the blue line. Let \(M(\epsilon)\) means the length of the measured curve by a ruler legth of \(\epsilon\), and \(L(\epsilon)\) number of touchdowns rulers into the curve. Note that the smaller \(\epsilon\), the estimation more accurate. Note that \(M(\epsilon)\approx\epsilon\cdot L(\epsilon)\) and
(if the ruler is shorter, the more times we have to apply it). If we repeat this reasoning, considering the area instead of the length, a “ruler” would be a square with a side length of \(\epsilon\) and
What about the volume? Perhaps a “ruler” woud be a cube and
So
and \(d=1\) (when we try to estimate the length), \(d=2\) (when we try to estimate the area), \(d=3\) (when we try to estimate the volume).
Let’s try to get to \(d\).
and
maybe can we write a formula like this
(are there any mistake in replacing signs: \(\sim, \approx,=\) above?).
It looks quite dramatically. Let’s see how this works in the case of the Sierpinski’s carpet. This figure we can (SURELY?!) cover by 1 square with a side length of 1, 8 squares with a side length of \(\frac{1}{3}\), 64 squares with a side length of \(\frac{1}{9}\),…, \(8^n\) squares with a side length \(\left(\frac{1}{3}\right)^n\) and
Sierpinski’s carpet is something between a line and a square - perhaps we came a little closer to the concept of dimension…
Summary¶
In the text above three words are bold. Finally, we would like to return to them.
REGULARITY Actions based on SAGE can help students explore recursion (a kind of regularity repeated over and over again, thanks to computers very many times).
ATTRACT It is worth mentioning in the context of using by students SAGE attention to the concept of attractor (not only in the mathematical sense: fractals are often attractors) - a computer may able to attract them to small discoveries.
SURELY It should be emphasized that the above considerations concerning dimension are only a signal of a problem - but they can build a student intuition (which should not be immediately deal with the problem of the existence \(\lim_{\epsilon\to0}\dots\)).
And by the way the last word: where is the limit between a student’s intuitive fantasizing and a strict mathematical waffle?
