Pedagogical interest 

Interest of the activity in understanding mathematical concepts - graph theory and dominating sets: Many real life situations can be abstracted into the form of a 'graph' or more commonly a drawing of networks connecting objects by edges. Graphs are of great interest for the development of algorithms because they allow to model relationships between a finite set of entities by considering both their properties and their interactions. The entities are the vertices or nodes of our networks represented in a simplified way. The image below is a very simple example of a graph drawing, where each number represents an entity, an object. The links between the nodes are the edges of the graph.

In order to illustrate the concept of graphs and dominating sets, we propose a simple activity that will allow us to understand the interactions between entities but also their limits. To do this, we consider a city, with several streets (our edges) that cross each other, creating crossroads (which will be represented by nodes), and a challenge: to create a network of urban farms that will allow each citizen to find fruit and vegetables in a short circuit close to their home. We use the map presented below to represent the city. As we can see, the city is made up of edges (our streets, in grey) which connect vertices (our crossroads represented by white dots). Thanks to the mathematical concept of graphs, we can represent our city by a mathematical model which simplifies the initial concepts. If you superimpose our plan and our graph, you will see that the two representations coincide perfectly, one being complex because it includes a lot of information that is not necessary to solve the problem (i.e. finding the optimal location of urban farms to serve all the citizens of our city). The second one is simpler, proposing only the useful information (the areas where the farms can be located i.e. the vertices, and the adjacent streets that will be served by these farms i.e. the edges).

In order to solve the problem, we will use the city graph to propose an optimal location of the farms. To do this, we propose a simple activity: using the crossroads (the vertices), locate the first farm. Once located, we can consider that the inhabitants living in the adjacent streets (i.e. those located on one of the edges touching the node) will be served. To do this, you can use the proposed graph and colour the vertice (or crossroad) where your farm is located and the adjacent edges (the streets) to visualise your choice, as in the diagram opposite. After this first step, you can continue with the positioning of a second, third, fourth.

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At this stage, the activity is simple, it would be enough to place a farm on each vertice to cover the whole city. Let's add an intuitive constraint to avoid redundancy of farms: we want to have a minimum of farms that cover the whole city (all streets). Can we easily find a minimum set of locations that can cover the whole city? It comes down to asking, what is the minimum number of farms that need to be positioned to allow every citizen to have easy access to food? Although easy to formulate, this is actually a surprisingly difficult problem to solve mathematically. Today, no one yet knows whether there is an algorithm for calculating this minimum set of locations that would be more efficient than a manual method of trial and error and/or iterations of strategies (called heuristics). 

We suggest that you get the pupils to work individually to start with, by providing them with the graph of the city - printable 2 (on transparent paper if you want to overlay the results or on normal paper) and asking them to cover their city with as few farms as possible without insisting on the method to be applied. To motivate the students, you can make this a class challenge. When all the pupils have proposed the solution they think is best, you can compare the results. To make it clear that behind each solution, there was a process (which we will later call an algorithm), ask the pupils who had good solutions to explain how they found them. Emphasise the choices, the repetitions and the strategies used to make the students understand the key concepts of algorithms. Based on this oral systematisation, ask the class to repeat the challenge to come up with the "best algorithm" possible. Through this activity, students will therefore cover several topics, including mapping, relationships, puzzle solving and iterative goal finding, combinatorics and graph theory. 

Simply approaching game theory: To make the activity more complex, you can also introduce the roles proposed below to approach game theory. Indeed, by repeating the game, each player can implement a strategy that will not follow the same logic from one round to the next. By introducing new role constraints into the game, each player will have conflicting objectives. We suggest that you introduce up to 4 roles and secretly propose the following objectives to the children:

• Citizens: To have access to a maximum of public and economic services in the city - Hidden objective: to increase the number of urban farms offered in the city territory

• Urban farmers: To offer a profitable service i.e. a good cost/benefit ratio in the installation of their harvesting and selling points - Hidden objective: to have a minimum of farms for a maximum of customers

• Real estate developers / Builders: Increase the value of their building through additional services - Hidden objective: offer a maximum of urban farm services in order to attract residents to the buildings

• Urban planner (add random bus stops to the graph): Find the best match between public and private services provided to citizens - Hidden objective: Locate farms near a bus stop to enhance urban transport planning.

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With regard to the different roles proposed, potential alliances can already be identified, for example between the farmer and the urban planner, or between the citizen and the real-estate developer. By encouraging discussions between each of these representatives (group work for this stage), the children will be able to choose their strategy, which will no longer correspond to a single constraint but to a plurality of sometimes antagonistic needs. They can thus decide either to play collectively, i.e. to cooperate, or to play individually by trying to influence the other members of the group, and thus to enter into a competitive logic. As a teacher/facilitator, you will be responsible for encouraging the search for a win-win strategy. This second phase of the game will allow you to work on the approach to citizenship and living together. In this context, game theory should enable us to learn that the game defines the players, but that in the end, it is we, the players, who define the game.

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Open discussion on food consumption and smart city practices: Finally, as with all the activities offered in the Unplugged Quest, we want to be able to enable teachers to open up discussion on societal issues in the classroom. In the "Farm in the City" game, the aim is to enable children to learn more about urban agriculture, local food systems and the importance of learning cities in facilitating more sustainable practices by citizens who support urban policies. In addition to the implementation of the game, it is also possible to take a look at the other resources provided in this document to initiate concrete agricultural projects with students and illustrate the need for these practices even through a mathematical activity.