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around the parable of the plank

two crafts that I used at the beginning of the school year, to ask the question: "what is the plotted curves are parabolas with ways to understand quite different.

Act 1: in module:

The first craft uses the fact that the parabola is the locus of points equidistant from a straight line and a point outside this line.
On a plate, screw a curtain rail "rail" at the top.
Then, cut the hook with a plastic knife to have a flat surface.

On the plate, tape a sheet of paper that will be replaced after each use or Velleda white adhesive film. With four bars
meccano, make a diamond. Two opposite vertices are made with screws and nuts, the other two are manufactured using bugs that are fixed on the hook amended, the other on the board.
Hang a plumb line on the pin of the hook and a rubber band between the two screws.

It will probably add a level to ensure that the wire is perpendicular to the rod.
felt drawn to the point of intersection between the elastic and the plumb.
The pin on the rod is movable. While moving it, find some other points of intersection between the elastic and the plumb.

Once it has few or a dozen, we can ask of the curve formed if we placed all these points of intersection.
is the first time this year that it begs the question of the locus and questions to be answered first: what is fixed in this item? what is mobile? What point does one consider the position?
The shape evokes the parable, but it must justify that indeed obtained a parabola.
Two approaches, one digital, the other geometric we will work on Geogebra. Geometric approach

But before working on geogebra, he should probably just roughing the figure to bring out important points.
A major obstacle to understanding the figure is the fact that bars Meccano almost hide the rest of the figure, as he must first reflect on the nature of the line carried by the elastic. Reflecting together, we arrive at that point M is on the bisector of the segment [HF] between the two pins, then the point M is equidistant from the right and to the point F. We arrive at the definition of the curve and head home.
It then tries to imagine this building without the physical constraints of the length of the rod and the length of the bar meccano. The curve is limited?
Once it laid flat, the students will model the curve using GeoGebra. It is relatively straightforward.
Easier than inserting applets geogebra anyway.

Numerical approach.
How do we define a parabola? For now, students have a digital design is a curve with equation y = a x ² + bx + c. We must find a, b and c. But it only makes sense when viewed in a frame. We must choose.
Fixed objects are fixed in this frame. One can choose according to this benchmark thereof. Posing as the rod axis and the pin as the point coordinates F (0 - 2), for example. Mobile
points are defined from the second bug, since it enables you to find the points of the parabola. Since it belongs to the rail, the ordinate is 0 and its coordinates are H (x, 0), with x real and purpose of the calculation is to find the point of abscissa x M such that HM = FM.

Intermission:
As homework, I asked the FIG to build the simplest possible geogebra and save under their name in the directory of the LCS class, and determine the algebraic equation this curve by placing the x-axis on the curtain rod and the pin at coordinates (0 - 2).
In another exercise of this duty, I asked students to determine the equation of the parabola passing through three given points not aligned.
After rendering duties, we can conclude that by three points not aligned, you can find one that matches trinomen and therefore there is only one dish.

Act 2:
In seeking a halyard between two points at the same height, one obtains a curve. This curve is it a parable? Justify.
A debate is established.
Student responses:
If it is a parable, we can determine the equation had to choose a landmark.
can use a software image processing to be coordinated;
Or geometry software.
Once we can find the coordinates of 3 points, we can get the trinomial. If the other points satisfy the equation found, then it is a parable, or else ...

I took a photo of this coube, and I put it in their directory.
By opening a file and inserting geogebra image, students can place this image in a frame and put some points on this coube in order to determine the coordinates of a rather specific. Students
comfortable chose their mark so that the coordinates are as simple as possible, others have chosen a random marker and had to be faced with a very complicated system of equations.
By moving the benchmark and changing the lines with the wheel, we can get this:

When they found the values of a, b, c for the curve passes through three points, they can noted that a fourth point does not pass by this parable.

In fact, this is called a catenary curve, and the equation used in the functions they will end.

On two occasions, students were asked "is that the curve is a parabola?" and the response method is not the same: in the first case, a parabola, the formula we will find is a trinomial for all points (proved in general) in the second This is not a parable, and there is a cons-example.