Den dansk-franske Skoles Blog

The top 3 math-didactical fails. Ever.

It was back towards the end of last century, and most of us didn’t really grasp what we were supposed to make of it. It didn’t seem to be of any concern whether the teaching was motivating or not, nor whether the time the children invested in the learning eventually was worth it. In the math lessons we were told that…

1. the cosine is used to calculate the length of the edges in a triangle…

… which no-one really has done for ages.

What you actually do use the cosine and sine for is to describe circular movements: cos(t) and sin(t) are respectively the x and the y coordinate of a point having moved t length units along the unit circle. From a practical perspective that can be used for instance to write a program that displays a clock.

Playing around with the code in line 7 and 8 gives a good hands-on understanding of the parameters in the expression

C + A·cos(ωt + φ),

i.e. the meaning of:

the amplitude A: the length of the hands of the watch
the frequency ω: the speed at which they rotate (one of them is 12 times faster than the other) ¹
the phase φ: at what time they start (in this case 12 o’clock, at -90°)
the offset C: where the center of the clock is.

Pushing it a little further, the concept can be used to convey an understanding of signals, frequencies and spectra, be it sound signals, light waves, simple alternating currents, rotating wheels and much more.

Interestingly, the kids are quite receptive to the notion of a (Fourier) spectrum. For instance, the spectrum of an F major chord on our piano looks like this:

You can also explain the children what a low-pass filter is by playing a high-pitched tone (~13kHz) on a computer: the kids will hold their ears in pain while most adults won’t be able to hear it at all. Very funny!

2. a function is something involving a graph…

…something with “ax+b”, which is not completely wrong per se, but nonetheless a serious under-statement of the importance of the concept of a function.

Indeed, considering functions as the fundamental building blocks of modeling – the Swiss army knife of abstract thinking so to speak – completely changes the perspective. A function associates an output with an input, no more, no less. When you have formulated your function – your model – you have identified what is relevant, and discarded what was irrelevant. That can be unfathomably important:

From then on you will begin to see relations of causality, predict events and progress, compare the predicted with the observed, chain functions together, understand what is happening, act intelligently, adapt and improve, share your understanding with others.

Within the natural sciences the concept is omnipresent. Physicist tend to refer to it as operators.

Large organizations are barely manageable without it nowadays. In business language functions are often referred to as processes.

A great tool for introducing the concept of a function to children (4+) is the function builder, a tool allowing to visually build and play with functions that transforms an input (on the left) to an output (on the right):

At the school we sometimes organize a function-Pictionary:

“If the input is… [drawing] a bread, and the output is… [drawing] slices, then the function is probably… a knife!”

3. differential equations are solved using obscure rules learned by heart…

… and aren’t of much use after all, as most people who learned about them in high-school will recall (hint: “coefficients in a complex-valued function”). What should have been the great epiphany concluding 13 painful years on the school bench ended up as a complete anti-climax for the vast majority.

But in reality it’s quite simple: once you have formulated your model using functions, you generally want to know how it behaves under different circumstances (inputs). The interesting models are often so complex that simply entering the input into a formula won’t work – one has to work out the final result in small pieces at the time. These small different pieces (differences) are added up (integrated) into a solution, nowadays in practice always using a computer.

From a pedagogical perspective the classical Euler method is quite suitable and can be used from age ~6 and up, see the first order example (x’ = ±1) below²:

In the example, the velocity v=1 of the cloud is constant until the cloud reaches sufficiently far out on the right (x>250), after which the velocity changes sign until it reaches sufficiently far left (x<150), after which it changes sign and so forth. In each iteration of the draw loop, v is added to the position x, giving the new position ³.

The slightly older children can use differential equations to answer truly relevant questions, such as calculating the time it takes a space ship to travel to Mars:

In most of the examples above, one skill has turned out to significantly leverage the fun and relevance of math: programming. And so it turns out that math can be easy after all!

So stop whining.

Start coding :)

Footnotes:

¹ Strictly speaking, ω is the angular frequency, i.e. ω=2πf where f is the frequency expressed in Hz.

² The example isn’t actually a true first order differential equation (DE), since there doesn’t exist a function f such that x’=f(x), since for a given value of x ∈ [150, 250], x’ can take on two values, 1 and -1. As it would often be the case when solving actual real-world problems, we are cheating a little: here we are using an extra variable (v) to store more information than what a first order system can hold. In order to solve the problem within a strict DE framework, one would need a second order equation involving Dirac delta functions, as in the flying pig example (code here).

³ Formally, rather than adding v we would need to add Δx, the distance traveled during a short time Δt, i.e. Δx=v·Δt. Here it works because we assume Δt=1.

Lærer i de humanistiske fag søges

Lilleskole søger engageret og frisindet underviser til dansk, engelsk, historie, samfundskundskab o.l.

Til besættelse af deltidsstilling og på sigt heltidsstilling søges snarest en underviser med mod på at prøve sine grænser af og undervise under usædvanlige vilkår.

Om skolen

Den dansk-franske Skole og Børnehave kendetegnes ved at:

anvende en Freinet-inspireret pædagogik i et aldersintegreret setup
inddrage det omgivende samfund, også fagligt, i form af en aktiv støttegruppe
undervise på fransk allerede fra børnehaven

Visionen med skolen er at skabe et fagligt og personligt udviklingsmiljø, hvor den enkelte har mulighed for og forventes at realisere sit fulde og alsidige potentiale.

Kandidaten

Den dansk-franske Skole søger en lærer der:

kan personificere en dygtighedskultur og ønsker at indgå i et intenst, krævende og udviklende livslangt læringsforløb
kan arbejde selvstændigt og kreativt, en ildsjæl
anerkender barnets vilje til egenudvikling og formår at udnytte barnets og gruppens egen motivation til at drive læringen
er bekendt med og ønsker at praktisere en Freinet-lignende pædagogik
er i stand til at indse og udnytte synergien mellem faglig og social udvikling
er fagligt og tværfagligt meget velfunderet, og er dygtig til at arbejde i et tætknyttet team

Uddannelse

besidder en kandidatgrad og/eller er pædagog/læreruddannet (evt. under uddannelse)
kan flydende dansk, helst på modersmålsniveau
er interesseret i at lære fransk eller kan det i forvejen, gerne på modersmålsniveau

Miljøet

Skolens personale består af en håndfuld højtuddannede og engagerede undervisere og gør ikke brug af vikarer. Den tæller ~40 elever fra 0. til 9. og en børnehave med ~20 elever.

Alle undervisere varetager børnehaven mindst ca. 1 dag om ugen.

Den ugentlige arbejdstid er 37 timer og 6 ugers årlig ferie.

Praktisk information

Ansøgningsfrist: 24. september 2018

Ansættelse iht. lærernes overenskomst.

Skolen ligger på Tagensvej 188, København NV, med gode transport- og udflugtsmuligheder.

For yderligere information se skolens hjemmeside http://www.ecolefrancodanoise.dk.

Ansøgning sendes til ecolefrancodanoise@gmail.com.

Den stærke gymnasieelev

Det er oftest elevernes faglige udvikling der vægtes højst når forældre skal vælge den helt rigtige skole til deres børn, men vi burde måske vægte personlig og social udvikling lige så højt?

Hvert år i august har jeg fornøjelsen af at lære en ny gruppe søde og unge 1. g’ere at kende. Det er typisk en blanding af elever fra forskellige baggrunde, som alle på hver deres måde bidrager til klasserummet, men som alligevel ikke befinder sig på samme niveau hverken fagligt, personligt og socialt. Mit fornemmeste job som lærer er derfor at finde et fagligt udgangspunkt hvor alle kan være med, og hvor alle ligeledes bliver udfordret både fagligt og personligt.

Det burde ikke være noget større problem, da alle elever i gymnasiet jo forinden er blevet erklæret “uddannelsesparate”, og alle har derfor et fagligt minimumsniveau og ligeledes nogle personlige og sociale forudsætninger for at kunne gennemføre en ungdomsuddannelse. Drømmesceneriet er selvfølgelig at eleverne er stærke på alle tre punkter, men da det sjældent er tilfældet, er der visse kvaliteter jeg som lærer sætter større pris på end andre.

Det er klart at et meget højt fagligt niveau faciliterer en rigtig god opstart, men hvis eleven hverken er selvstændig i sit arbejde, motiveret for at lære endnu mere eller tager ansvar for at opsøge ny viden og ikke kan samarbejde med andre, bliver overgangen mellem 9. klasse og 1.g en del sværere.

Forventningen til eleverne er jo at de kan arbejde selv, selvstændigt opsøge ny viden, arbejde i grupper og dermed også lære af hinanden, og hvis alle disse kompetencer allerede er der fra start, er det ikke en lige så stor udfordring at opnå de faglige mål i gymnasiet.

Jeg oplever ofte at de elever der kommer fra skoler med undervisningsmetoder som bunder i en tro på menneskets vilje til at lære, og en tro på at eleverne bør være aktive i læringsprocessen, er mere selvstændige, ansvarlige og motiverede. Disse elever besidder, udover de intellektuelle minimumskrav, en stor samarbejdsevne, kreative kompetencer og er gode til at tænke projekt- og procesorienteret, hvilket gør at de er selvstændige i deres stillingstagen. De er både lydhøre, men formår også at tænke kritisk og at være initiativrige. Mange elever der kommer direkte fra folkeskolen har et fint fagligt niveau, men har også svært ved at træffe valg, at arbejde selvstændigt og er meget resultatorienterede. Det lærer de selvfølgelig i løbet af gymnasietiden, men det er måske et unødvendigt benspænd.

Det er på baggrund af min erfaring med mine gymnasieelevers styrker og svagheder, at jeg har valgt en alternativ skolegang for mine børn. Her lærer de at fordybe sig i emner der interesserer dem, at reflektere, at tage ansvar og at være vedholdne. De bliver dagligt støttet i deres lyst til læring gennem trygge relationer og tillid, hvilket ikke kun styrker deres faglighed men i høj grad også de sociale og personlige kompetencer som jeg værdsætter hos mine gymnasieelever. Jeg er overbevist om at frihedsbaseret læring, som vi netop ser på Montessori-, Steiner- og Freinetskolerne spiller en stor rolle i elevernes udvikling. Friheden til arbejde, udvikling og vækst indenfor nogle klare grænser i et sundt miljø, hjælper eleverne til at kende deres styrker og svagheder, at samarbejde og at udvikle deres selvdisciplin og koncentration. Disse selvstændige elever med stor selvindsigt er, efter min mening, langt bedre rustet til ungdomsuddannelserne og i endnu højere grad også til de videregåendeuddannelser.

Mette Blicher, gymnasielærer

Coming to America

(When we landed in Boston, I was both excited and tired. Later on the evening after we had setteled in, we ate at a Mexican place. The next day we set out to find a supermarket so we could buy food and supplies for the week. Including a basketball ;)

(The next day, we started by playing a game called “Timeline*”. Then we went to the Museum of Science where there was a fantastic show about electricity)

There after we went for a driving in a duck but we had to wait for 30 minutes before the next duck drove. We went into the museum again and went up and looked at what we could buy. Basil bought a key ring with LOL written on it. Was after we were in the duock and drove then was a funny lady who spoke hell time because she should say where we were and who had built what and all that. After maybe 30-40 minutes we would sail in the water. Basil, Sacha, Fouad and Lydia went on shifts and tried to steer the boat.

When we were done we went home.

Sunday we should go for fifteen kilometers. It was hard for us and when we came home we bought an ice cream on the way .

Monday was shopping day. We were out and shope for the day. Sacha bought some shoes for $ 100 Alix bought a shoulder bag that cost 45 $ Lydia also bought a bag for $ 60. Then we went to a store we had been before to shop a little more and because then there was more cheap clothes. It was also here that Basil bought a pair of cool red and black pants. Then we went in to the store next door. I bought two jerseys for my little sister and a big coke .Sacha bought a celtic sweater and after we went home and ate pasta.

Around nine o’clock all the boys left and played baseball and went home after approx. 20-30 minutes. When we were home we went to sleep.

Now it was Tuesday. We were going to HARVARD. When we went over there we had to go to the office know when the tour begain. They said that the roundabout started in fifteen minutes. We went for a walk and came back. When we went to HAVARD, the guide told us , among other things, that Bill Gates had gone to HAVARD. We then went to a library that she also said she also worked in. after that should we eat in a burger resturan where I ordered a vegetarian burger that tasted really good. It was now raining me and Sacha, Basil ran home but the others took a taxi home.

the next day we were going to MIT der was a gay name krish he toll os abard MIT he was also a student in MIT he toll also some tricks to get ind MIT He said that you do not have to be smart to go in MIT They place emphasis on whether you can change the world he showed de small clasromm and the big clasroom

*Timeline is a game of cards with a certain discovery/invention/theory or idea on the front side and a date on the backside of the card. The point of the game is to locate the card where you think it belongs on the timeline.

Om at være til stede

Lige siden vi startede i 2010, har tilstedeværelseskravet til lærerne været helt centralt hos os.

Det er der en række grunde til.

Den første er, at mennesker lærer bedst i fællesskab. De fleste har nok prøvet at kæmpe med et problem eller et spørgsmål i timevis, ja måske endda dagevis eller årevis for til sidst tilfældigvis at støde på nogen, der med en enkel forklaring kunne løse det på få minutter. De gode råd gør en kæmpe forskel og kommer ofte, når man mindst forventer det. Den vindende strategi må derfor være, at man forbedrer sine chancer for sådanne Aha-oplevelser ved at omgive sig med dygtige, engagerede mennesker.

For voksne som for børn sker den vigtigste læring på en sådan osmostisk måde – man optager løbende viden fra det miljø man færdes i. Derfor er det afgørende at man gør hvad man kan for at færdes i et miljø, der er så rigt på viden, idéer, inspiration mm. som muligt. Her spiller de voksne en vigtig rolle, for de har som regel en del af den slags guldkorn at dele ud af, må man antage – og de har en meget strukturerende rolle for miljøet.

Den anden er, at tilstedeværelse er en forudsætning for nærvær, sjovt nok. Mennesker er som regel meget socialt anlagte, det skaber god stemning, når der er andre omkring een. Det, ens medmennesker implicit signalerer med deres tilstedeværelse, er, at de finder lige præcis dette sted så relevant og spændende lige på dette tidspunkt, at de simpelthen har valgt at være her. Det er et meget motiverende signal at sende til flokken – og meget betryggende.

Den tredje er, at det rent organisatorisk og administrativt er meget lettere at lede en gruppe, der er til stede. Fællesbeskeder kan gives og diskuteres løbende, kommunikationen lettes gevaldigt. Man kan vikariere for hinanden, og man undgår således at skulle hyre folk på ad hoc basis, folk som ikke kender børnegruppen og ikke har været med i læringsflowet.

Den fjerde er, at det at være på sin arbejdsplads i hele sin arbejdstid forstærker fokus. På arbejdspladsen kan psyken forholdsvis let forholde sig til, at det er her og nu det sker. Her er det meget lettere at være på, end hvis man er i gang med at hente børnene fra børnehave, forberede aftensmad og hvad der ellers måtte være af familiemæssige forpligtelser, som naturligt opstår, hvis man er den, der har tiden til dem.

Den femte er, at man med en fokuseret indsats kan løfte svære opgaver. Man kan fx differentiere sin undervisning, inspirere de særligt begavede og inkludere de mindre medgørlige, bruge gruppens dynamikker til at skabe en god kultur – en helt reel og anerkendelsesværdig ledelsesopgave. Når man således begynder at lykkes med en svær og meget vigtig opgave, begynder man at høste anerkendelse ude i samfundet. På sigt kunne man endda forestille sig, at lærerfaget kunne blive et tilløbsstykke for dygtige studerende, ligesom i Finland.

Derfor forstår jeg ikke, at det ikke er lærerne selv, der stiller tilstedeværelseskravet til sig selv og til hinanden. Det ville signalere engagement, fokus og styr på sagerne. Politikerne og resten af befolkningen ville klappe i deres små hænder. Lærerne ville have ryggen fri til at bygge en vision, videreuddanne sig, forske, udvikle professionen – og give børnene de bedste muligheder for at begå sig i og bidrage til morgendagens samfund.

Journée francophone du 25 février 2018 – Programmation et équations différentielles

Programmation et équations différentielles

Journée francophone, dimanche 25 février 2018

1) Pour commencer:

Dans le navigateur internet, aller sur

https://www.khanacademy.org/computer-programming/new/pjs

2) Dans la documentation, trouver comment dessiner un rectangle

En dessiner trois. Ou autre chose, comme on veut :)

3) Animer un des objets dessinés

Créer une variable, par exemple

var x = 0;

Créer une fonction

draw = function() {

background(255, 255, 255);

rect(x, 100, 100, 100);

x = x + 1;

}

4) Faire rebondir l’objet sur les rebords:

En début de programme, faire une variable v (pour vitesse):

var v = 1;

Dans la fonction draw, ajouter:

if (x < 0) {

v = 1;

}

if (x > 400) {

v = -1;

}

changer x = x + 1 en x = x + v

5) Au choix

ajouter une variable y (position verticale) et une vitesse verticale
ajouter une accélération

Exemple de résultat final.

On how we construct trust

Baby taking a nap outside while mom is shopping.

83%.

“Can most people be trusted?”. If you ask the Danes, they will answer yes 8.3 times out of 10, on average. This makes them the most trusting population in the world according to the OECD 2017 “How’s life” annual survey.

The probably complex historical and social context that has led to this state of affairs, the question whether the population’s homogeneity is a factor, or whether the Danes are simply too lazy to mistrust other people… will not be the subject of this article.

Nor will we address the question of whether the record high tax pressure is a prerequisite for the welfare society and a high trust score. Nor, conversely, whether the high trust level simply makes society so extraordinarily efficient that it can accommodate a large and inefficient public sector as well as a largely sub-optimal allocation of resources to tasks on a societal level, in part compensated for by undeclared work (which 40% of Danes make use of), not to mention widespread crab mentality.

This article is specifically about how – and why – we construct trust at The Danish-French School of Copenhagen.

First we will consider the importance of the main tool that we are using – relationships and the approx. 17 practical principles we use to build them. They are summarized later in the next section, but might be worthwhile a read to better understand the article.

Then we will apply an information-theoretic perspective to each of the principles in order to illustrate how the process of reducing the amount of information addresses a fundamental need, in turn strengthening relations and trust.

Finally we present a simple model that illustrates why trust deserves a very particular focus as a value in society.

Relationships and information condensation

Trust is about relations between people, hence the most important of our 17 principles, “Establish relationships”.

An important aspect that has turned out to be recurring in many of the 17 principles is the overarching idea of information reduction, or rather information condensation. The brain continuously processes enormous amounts of information and the task it accomplishes when distinguishing relevant from irrelevant – condensing the information – is a truly formidable effort. We see it as a sort of extra layer at the bottom of Maslow’s hierarchy of needs: the ability to give oneself a direction in the super-high-dimensional space of potentialities conveyed by our senses, but in essence akin to what the simplest organisms must be experiencing when following say a nutrient gradient.

From a didactical and a relationship-building perspective this also means that whenever one can help the child “find the gradient”, it is a fundamental need that is being addressed, resulting in a significant strengthening of the relationship and the trust.

This information-theoretic perspective sheds a new light on each of the 17 principles:

	Principle	How the principle acts as information condensing
1	Establish relationships	When you have a relation, you know what to expect from each other, allowing to disregard an infinity of other options that one then doesn’t need to spend attention on. Relations create psychological safety that allows you to relax and focus on moving forward. Self-esteem results from the fact that the relation conveys recognition and reflection.
2	Agreements	A clear and condensed formulation of what to expect.
3	Rules, recognizing justice and co-ownership	Like in principle no. 2 but even more condensed. Also, fairness and justice imply a high degree of predictability.
4	Noise and quiet	Clearly distinguish the information-carrying signals from the noise.
5	Simple explanations	Information-dense nuggets.
6	High expectations	A clear gradient to follow- which probably explains why they are so efficient.
7	Help others – social capital	Supports in building relations, see principle no. 1.
8	Keep track of what the individual is doing / finishing things	Staying focused on a gradient.
9	Children need to test	Clarifying the limits of the playing field.
10	Repetition makes master	Repetition is probably the most common information-condensing mechanism.
11	Set limits to the tasks, not the time spent	Staying focused on a gradient.
12	Give options	Reducing the high-dimensional space of possibilities to a few options.
13	Correct mistakes as early as possible – give and demand feedback	Feedback condenses the vast “what-I-might-have-been-perceived-like” to the simple “what-actually-was”.
14	Do not steal the children’s play	Favors relations, see principle 1.
15	Avoid “Shh!” and instead give specific messages	Specific messages are obviously much more information-dense and actionable.
16	Only one adult at a time	One responsible adult creates a simpler field of expectations than multiple persons.
17	Clear communication and honesty	Should be self-explanatory.

And so it appears that information condensation is indeed a governing trait of our pedagogy.

A few aspects that are not covered by the 17 principles, but are relevant in a trust-building context:

Long-lasting relations

The multi-aged structure of the school favors long-lasting relationships. Our teachers typically follow the children for many years – potentially from when they are 2 to 15. This means that teachers and children can get to know each other very well, simply due to the sheer time they spend together. But it also means that all the group members are in it for the long run, further strengthening the incentives to construct strong relations.

Impediments to trust

A direct impediment to trust typically occurs when the members of the group need to compete for a scarce resource, be it two children wanting the same toy, or two employees wanting the same job, etc. The guiding principles we have successfully applied in those cases were:

share the resource, for instance taking turns. Very often it will from a global perspective be a better solution that the two members each get ~50% of the resource, rather than splitting 0% – 100%.
in case the resource is not shareable, allocate the resource according to what is most aligned with the group’s mission. Discuss what best serves the greater purpose and apply that. Sometimes you just need to take one for the team.
in either case, be conscious of and open about the conflict. When the solution is fair, it is much easier to accept it even if it is not at one’s own advantage.

Another impediment to trust stems from the average human being’s relative perception of success. When you sit in a train and the train next to you starts moving, you can get the impression that you are moving backwards. That is similar to when your friend has success and you feel it as your failure. Both impressions are factually wrong, resulting from a flaw in how our perception works. Suffice to imagine that you are seeing the situations from an absolute observer position and it becomes clear that your friend’s success is in part your success, and that the winning strategy is to help your friends, to energize your network. We teach that from an early age.

The why

We have covered the question of how we construct trust at the school. The reason why should become very clear when considering the following model:

Given some relatively obvious assumptions of a somewhat mathematical nature, we will conclude that it is worthwhile to act honestly and trustingly in the sense that the growth you can expect to experience in return depends exponentially on it.

The simulation works in the following way: a population consisting of a number of individuals undergo a number of transactions (iterations). Each individual is described by its capital, its honesty and its trust in each of the other individuals. Each transaction occurs between two randomly picked individuals A and B in the following way:

A invests a value in B, where value = capital of A x trust between A and B.
B generates some added value from the investment.
a die is rolled and is compared to the honesty of B: B either returns the investment (successful transaction) or keeps the full investment (unsuccessful transaction).
in case of a successful transaction, the trust between A and B is increased. Conversely it is decreased in case of an unsuccessful transaction.
pick two new individuals A and B and go to 1 (unless the intended number of iterations has been reached)

The resulting simulated relations of trust in a sample of four different populations look like this after 200 iterations (The thickness of the line expresses the trust score, between 0 and 1):

The corresponding simulated total value of the population looks like this:

That… is a semilog plot.

It turned out that the y-axis had to be logarithmic to properly express how much of a winning strategy it is for a group of individuals to adopt an honest and trustful culture.

Conclusion

We have described that constructing trust amounts to building relations and how we do it. We emphasized the information-theoretic perspective, as we are uncovering that helping the children to condense the information, or understand, is a fantastic catalyst for trust.

We then saw that, perhaps surprisingly, growth depends exponentially on trust. Few other factors, if any, have that type of positive impact on society, suggesting that trust be the dearest treasure of any nation.

The Loop

In my previous article I described how one can use a differential equation to calculate the trajectory of the Earth around the Sun. But at the end it was quite tedious so I decided to use a computer instead. Because they are good at repeating things. One way to make a computer do the same things again and again is to use a loop. In the program below we will show how a for loop works:

//position
var x = 100;
var y = 0;
var xp = 0;
var yp = 30;
var G = 1;
var m = 100000;

for (var i = 0; i < 18; i+=1) {
    var d = Math.sqrt(xx + yy);
    var a = Gm/(dd);
    var xpp = -x/da;
    var ypp = -y/da;
    xp = xp + xpp;
    yp = yp + ypp;
    y = y + yp;
    x = x + xp;

fill(72, 54, 214);
ellipse(x + 200, - y + 200, 10, 10);


}
fill(255, 234, 0);
ellipse(200,200,20,20);

What the program does is to draw a planet 18 times using a for loop.

In each iteration of the loop first of all you calculate the distance from the sun to the planet, after you need to calculate the acceleration then you should find the speed of the planet at the end you can find the position of the planet.

This video explains about the code above: