I got a good thirty journals, and I could edit the last activity. I realized that many students have trouble seeing in space. The manufacture four tetrahedra in the yard and was able to superimpose the levy from the interior volume remained empty. Students have nodes on the edges of the octahedron and could see in particular the lines parallel to each other.
The most amusing is that I felt my first module group needed to move to this construction, while the second, warned that it could be required to make in the yard and in the cold, thought he could be so interesting to make a figure in perspective and are arrived quickly to the result.
fact remains that the calculation is a bit annoying that the volume of the tetrahedron.
Students learned the formula: Volume of a pyramid
= one third of the area of the base times the height
But in a regular tetrahedron, what height, what is the bottom? and how to calculate this? In a
session course (probably a few, prefer a group session in half), I asked three students to construct a regular tetrahedron with my journals two meters (screw eyes at the end of the journals, and to link colson ), after pointing out the number of edges at each vertex.
After two minutes, the tetrahedron was constructed.
(I did not take pictures during my course, I would have liked to have presence of mind to do so in the heat of the moment, so I just took in my backyard, it will not reflect During this lively )
Question: while the height, what is it?
A student stands up, points to the height, took the rule and attempts to measure it.
then I go out a plumb and resumed his idea by attaching it to the top so that the tip touches the ground: here is the height. We can actually measure the rule.
can also be estimated by comparing with my size, because the top comes up to me, at about 1.65mètres. As I let them move around or even in the tetrahedron, the students can get an idea based on their own size.
But is there a way to know the exact value?
And besides, where the falls plummet?
responses flowing:
- in the center of the base (which center of the triangle?)
- the center of the circle circumscribed
- the center of gravity
- the center of the inscribed circle
- the orthocenter
- at the intersection of bisectors (how to draw?)
- at the intersection bisectors (trace how exactly?)
- at the intersection of heights (trace how exactly?)
- at the intersection of the medians (how to draw?)
- at the intersection of the medians (how to draw?)
Who is right among all these proposals?
- the triangle is equilateral, so everyone is right.
OK, we'll draw these lines so remarkable. I marked the middle of each edge. What can we trace exactly?
Some students use strings to represent the medians.
Can you calculate the length of the median?
- yes, because there is a triangle, so we can use the Pythagorean theorem.
With what lengths?
- the median coincides with the height, so one side is 1 meter and the hypotenuse is 2 meters.
Everyone is able to calculate the median?
And the center? Where is he?
Here, I had to remember that the center of gravity is located at 2 / 3 of each center, starting from the top.
And the height?
There I got a surprise. Nobody said he could see to put on a triangle. I pulled out my bracket and I turned around the plumb line, showing that this line was perpendicular to each line of the soil, especially medians plotted. The silence has been attentive to this point showed that something was happening, and that most students did not realize this fact.
Students were then set themselves to work to calculate all values. A figure
was marked with the names of points, they have the same calculation.
Those difficulties may remain around the tetrahedron to take all necessary measures. Those who want to check their calculation can also get up to measure. This course was interesting and lively. I think the students moving around and in a volume to calculate took another consciousness and developed their vision in space, the tetrahedron is to build this and build these lines provided to students. Some very average students first took the measurements used to calculate formulas an approximate value of the volume, and then have it validated their response. I then asked them to correct all calculations, with explanations. They are then returned to a good start in the problem.
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