Introduction

At last! thisfinal sequence will crown the efforts and perseverance that were yours all along this project.

Congratulations. It is time to use the data you have carefully collected, either through your own survey, or through the other classrooms by Internet. We are now going to measure together the length of your meridian and finally find the size of our planet.

Note : This last step is the most tricky in the project, that's why we opted for a progressive action, in order to lead your pupils to the final calculation.

- Experimental and technological sciences :

• - The subject : horizontal and vertical plane : interest for some technical devices.

• - The sky and the Earth :

• - light and shadows

• - the apparent movement of the Sun.

- Mathematics :

• - Space and geometry :

• - use of maps and plans.

• - use of tools (rule, set square, compasses) and techniques (foldings, tracing paper, cross-ruled paper).

• - geometric relations and properties : alignment, perpendicularity, parallelism.

• - Magnitude and measures :

• - keeping track of time and duration (year, month, week, day, hour, minute, second) and their relations.

• - angles : comparison, reproduction.

• - Exploitation of numerical data :

• - questions of ratio.

• - use of data in lists and tables.

• - View of the world : compare the global representations of the Earth (globes, planispheres) and the world (maps).

Specific skills :

• ? To know that locations are relative : a place is east (or north) of another, but west (or south) of a third.

• ? To be able to imagine the apparent course of the Sun in the sky and its changes during the year. To know that it is at its shortest on the winter solstice (the Sun is then low on the horizon) and at its longest on the summer solstice (the Sun is then high in the sky)..

• ? To be able to use a calendar to determine the characteristics of each season and the dates that mark the beginning of each.

• ? use a plane or a map to locate an object, anticipate or make a move, evaluate a distance

Comments :

The measures taken by groups of pupils will give an opportunity to compare the results and tackle with the question of accuracy of a measure. It is not necessary to introduce the idea of uncertainty and you should certainly not use the according formalism. We only need, in link with the mathematical field about decimals, to suggest a thought about the number of figures it should be reasonable to use to express an experimental result.

First : Eratosthenes' measure

Our Greek scientist was at the end of his considerations, and he tells us the results of his experiment. Give the following text to your pupils :

"Having measured the angle between the sunrays and the vertical -the obelisk in his city of Alexandria-, Eratosthenes drew on the ground a section of the Earth following a meridian. He put on it the cities of Syrene and Alexandria and drew the sunrays getting to these two cities. He compared the angles of these rays with the vertical in each of these towns and extended the sunray from Syene to the centre of the Earth, and had the sudden idea to measure the circumference of our planet. Soon, he understood that something was necessary to complete his project : the distance between Alexandria and Syene. He knew that caravans going through the desert were used to measure the distances between cities. Men who were called "bematists" walked beside the camels and counted their steps. Knowing the average length of a step, they deduced the distance they had walked by multiplying this length by the number of steps to make their trip! There was approximately one million steps between Alexandria and Syene... That meant approximately 5000 Egyptian stadiums (a unit they used at that time).

Eratosthenes rapidly discovered after a few simple calculations that the circumference of the Earth was exactly 250 000 stadiums. He told that to his colleagues scientists and geographers and the news spread all around the Greek world that a scientist named Eratosthenes had for the first time measured the size of our planet."

Now, it's your turn ! Try to reproduce the figure taht made Eratosthenes famous in the whole world and discover how he could measure the circumference of the Earth. The, use your own measures and those of a partner school to find the size of our planet by yourself.

1- How did Eratothenes measure the circumference of the Earth

a - The vertical at the scale of the Earth

The pupils, having read the text in the classroom, are going to ask various questions. The first thing to make them understand is the notion of vertical in the two cities of Syene and Alexandria. If they have understood the activities about the notions of verticality and horizontality, they will certainly have a good idea of their local vertical. But what happens when it comes to the scale of our planet?

Ask them the following question :

"When the gnomons are adjusted (see the part about this activity if necessary and the notes they had taken in their notebooks at that time), how are they with regard to the horizontal support?" They will readily answer that they are vertical and as such perpendicular to the horizontal ground.

"What would happen then for gnomons all around the Earth?"

They will discuss about that and note in their books their hypothesis, to be checked by an experiment. For that, they can use a simple strip of stiff paper on which they will stick small shafts or pins, perpendicular to the sheet. The gnomons as they see them around them when they are adjusted. Since they are convinced that the Earth is not flat, they curve the strip and see that the gnomons are not parallel anymore, but that the directions to which they point are changing. If they close the strip upon itself, they will see the gnomons radiate from the surface of the Earth. They can draw on their notebook this strange figure of Earth covered with shafts just like a hedgehog.

Seeing that all these gnomons are the vertical for each point of the Earth, ask them what happens when they extend by imagination all those shafts inside the Earth? They converge all to the centre of the Earth! It can be checked if you use the experiment, and replace the shafts by long needles or skewers.

They conclude that the vertical to each point of the Earth also points to the centre of our planet, and as such, the gnomons in two far away cities are not parallels but that their directions make an angle. Then, draw on the blackboard a circle for the Earth and ask them how they could place on that figure the cities of Syene and Alexandria, using what they know about Eratosthenes measures.

b - To discover Eratosthenes' secret

The question is difficult andf you will help them find the answer that will lead them to the famous figure made by Eratosthenes!

The small historical texts told us that in Syene, on the 21st of June at solar noony, the sunrays got down to the bottom of the wells and that vertical objects had absolutely no shadow. As such, they were perfectly vertical! On a great sheet (A4 or A3), they will draw a circle for the Earth just as the first sketch of figure 2. They draw several sunrays (parallel, of course), and one following that vertical.

(You can begin to draw on a great sheet of paper the parallel sunrays and then cut in a coloured sheet a circle for the Earth, put it on the sheet and pierce it with something to fasten it on its center, and make it turn until the sunrays fall vertically on Syene).

How do you place Alexandria now? Ask them what Eratosthenes measure this same day, at the same time? "The angle between the sunrays and the obelisk... so the angles these rays make with the vertical !" They will need to recover the value of this angle (7.2 degrees), and draw a schema, just as explained in fig. 2. The pupils will use tracing paper, on which they will draw the angle for Alexandria and they will make it slide on their figures until the ray on the obelisk becomes parallel to the others.

Eratosthenes secret :

When they have written down with a black pen the position of Alexandria, they draw the vertical going through this city that goes to the centre of the Earth. Ask them what is the angle between this vertical and the one of Syene? "It looks strangely equal to the angular sector, the one measured by Eratosthenes".

To check that, they return the tracing paper and superimpose the angle to the one at the centre of the Earth. It works! This would be Eratosthenes' secret ! Make them check for another angle (twice that value, for example), and they get the same result. They create a new Alexandria at an angle of 14 degrees, draw the vertical and measure the new angle at the centre of the Earth. They can also use the protractor to check these hypothesis.

Then, they reproduce on their notebooks the figure without its irrelevant lines, proud to have discovered that by themselves. They will also note the conclusion : the famous secret discovered by Eratosthenes : the angle measured between the sunrays and the vertical in Alexandria is exactly the angle between Alexandria and Syene at the centre of the Earth. They will then see the "Z of Zorro" that will surely help them remember this incredible result!

One more question : what would happen if the Earth was turned so as to make sunrays no more vertical in Syene? (turn the circle of Earth slightly counter-clockwise).

They try the experiment, draw the angles, compare them, and see that the angles are not equal anymore !!! A new angle has appeared in Syene, changing everything.

Tracing the new angles appeared in Syene and Alexandria between the rays and the vertical, maybe they will discover after a few tries that the angle between the two cities at the centre of the Earth (see the first model) is equal to the difference of the angles measured in the two cities between the sunrays and the vertical (they can also measure them with a protractor and look for the relationships between the three angles : in Syene, in Alexandria and in the centre of the Earth between the two cities). It can also be easily seen with the tracing papers.

They just have extended the conclusion to the cases (the most current!) when sunrays do not fall vertically. They will write down this discovery on their notebook, because it will be useful to reproduce Aratosthenes' figure with their own measures. (Note that this conclusion also applies ont the 21st of June between Syene and Alexandria, but that one of the angles is null... The difference is then equal to the angle measured in Alexandria.)

Now, your pupils are ready to measure the circumference of the Earth in any case!

c- To measure the length of the meridian passing through Syene and Alexandria

To make them understand the rule of three (or rules of proportions), a rule they will need absolutely to measure the meridian, tell them to think about the following scenario :

Imagine that Eratosthenes measured a different angle in Alexandria. Imagine that Syene and Alexandria are in fact on an Earth similar to a pie cut for example in 8 equal pieces, the two cities being just as on figure 4. If you know the length of the rim of a piece of the pie, how could you find the length of the circumference of that pie?

"It's easy, you only need to multiply the length by 8!" Are you sure? You can tell them to check :

make a great circle, divide it in 8 equal sections and measure with two threads the length of the rim of a piece and the length of the entire circumferen. They will find a difference of 8 to 1 between the lengths of the two threads.

That's exactly what Eratosthenes told himself, but how many pieces are there in the "pie"?

They can suggest several experiments to try to discover it. You can divide the pupils in groups, in order to try all the proposals :

- they can use the angle of 7.2 degrees and make it turn around the centre of the Earth to see how many "pieces" they need to fill the Earth (or half the Earth, and then multiply the result by two).

- They can use a thread and compare the length of the rim for the "piece" Syene-Alexandria (on the real figure made before) and compare it to the length of the circumference of the Earth.

- Those who prefer could divide 360 degrees (a whole circle) by 7.2 degrees (the angle at the centre).

They will find a factor of 50 precisely (through calculation, at least !).

They only need to use the rules of proportionality since Eratosthenes tells us that the distance between Syene and Alexandria is of 5000 Egyptian stadiums. Multiply by 50 and find : 250 000 exactly !!! Just as the great Greek scientist. The puzzle is now solved.

But what was the Egyptian stadium distance value in the metric system? Final quest that will take them to encyclopedias or internet search engines. They will find the following answer :

1 Egyptian stadium = 157.5 meters, which gives a circumfere of 39 375 km for the Earth : Compare it to the values found in your dictionaries and you will be astonished by the preciseness of this measure.

2- Work process

The European schools must be registered on the eTwinning plateform and will use the Eratosthenes Twinspace as working space.

a- Check your school data

The teachers check the shared google sheet with : city, country, name of school, coordinates and Google map of the school. They post a message in teacher bulletin if there is an error or new school to register.

b- Share your measure in the data google sheet

From a copy and paste of the "check your school" sheet: identify the school, enter date, gnomon and shadow. The angle of the sun is calculated automatically ATAN (shadow / gnomon)

c- Post of the measurement in project journal

The teacher or admin student post : photo + data (school, date, time, gnomon, shadow,

d- Choose the best partnership

The second tab of the shared google sheet (calculation) calculates the distance between the latitudes of the cities, and the circumference

e- Draw the geometric figure

You can easily draw the figure using one of the 3 Geogebra models