Algorithmique du cycle 4 - 3e

Niveau 2 : variables

Exercice 1 : Déplacement véhicule - Décrémentation différente de 1 de variables (3/5)

Votre véhicule est équipé d'un réservoir contenant 10 litres d'essence, plein au démarrage.

Après une mesure précise, vous remarquez qu'en réalité votre véhicule consomme 2 litres à chaque mouvement.

Votre jauge étant toujours cassée, modifier l'algorithme suivant pour que la variable représente le niveau d'essence présent et diminue de 2 litres à chaque mouvement.

{"options": {"toolbox": [{"value_1": {"var_name": "essence", "type": "base_varvalue"}, "value_2": {"type": "base_numberinput", "value": 1.0}, "type": "base_simplecalculus", "op": "MINUS"}, {"type": "base_numberinput", "value": 1.0}, {"var_name": "essence", "type": "base_setvar"}], "blocks": {"base_start": {"inputs": [{"fields": [{"text": "Au d\u00e9marrage", "type": "text"}], "align": "left", "type": "dummy_input"}], "on_event": "start", "creatable": false, "tooltip": "D\u00e9marrage de l'algorithme", "colour": 65, "next_statement": true}, "base_numberinput": {"inputs": [{"fields": [{"default": 0.0, "type": "number_input", "name": "value"}], "align": "left", "type": "dummy_input"}], "colour": 260, "tooltip": "Fourni un nombre", "output": "Number"}, "road_forward": {"inputs": [{"fields": [{"text": "Avancer de 1 case", "type": "text"}], "align": "left", "type": "dummy_input"}], "colour": 160, "next_statement": true, "tooltip": "Avancer d'une case tout droit", "previous_statement": true}, "base_varvalue": {"inputs": [{"fields": [{"default": "x", "type": "variable", "name": "var_name"}], "align": "left", "type": "dummy_input"}], "colour": 330, "tooltip": "Fourni la valeur d'une variable", "output": true}, "base_simplerepeattimes": {"colour": 120, "tooltip": "Execute une suite d'instructions un certain nombre de fois", "previous_statement": true, "compound": {"blocks": {"base_repeattimes": {"inputs": [{"fields": [{"text": "R\u00e9p\u00e9ter", "type": "text"}], "align": "left", "type": "value_input", "name": "ntimes", "check": "Number"}, {"fields": [{"text": "fois", "type": "text"}], "align": "left", "type": "dummy_input"}, {"fields": [], "align": "left", "type": "statement_input", "name": "code_repeat", "check": null}], "colour": 120, "tooltip": "Execute une suite d'instructions un certain nombre de fois", "previous_statement": true, "inline": true, "next_statement": true}, "base_numberinput": {"inputs": [{"fields": [{"default": 0.0, "type": "number_input", "name": "value"}], "align": "left", "type": "dummy_input"}], "colour": 260, "tooltip": "Fourni un nombre", "output": "Number"}}, "ast": {"type": "base_repeattimes", "ntimes": {"alias": {"value": "ntimes"}, "type": "base_numberinput"}}}, "inline": true, "next_statement": true}, "base_setvar": {"inputs": [{"fields": [{"text": "mettre", "type": "text"}, {"default": "x", "type": "variable", "name": "var_name"}, {"text": "\u00e0", "type": "text"}], "align": "left", "type": "value_input", "name": "value", "check": null}], "colour": 330, "next_statement": true, "tooltip": "Affecter une valeur \u00e0 une variable", "previous_statement": true}, "base_simplecalculus": {"inputs": [{"fields": [], "align": "left", "type": "value_input", "name": "value_1", "check": "Number"}, {"fields": [{"type": "dropdown", "values": [["+", "PLUS"], ["-", "MINUS"], ["\u00d7", "TIMES"], ["\u00f7", "DIV"]], "name": "op"}], "align": "left", "type": "dummy_input"}, {"fields": [], "align": "left", "type": "value_input", "name": "value_2", "check": "Number"}], "colour": 260, "inline": true, "tooltip": "Effectue un calcul simple", "output": "Number"}}}, "ast": [[{"editable": false, "type": "base_start", "deletable": false}, {"code_repeat": [{"type": "road_forward", "deletable": false}], "type": "base_simplerepeattimes", "deletable": false, "ntimes": 4.0}]]}

Exercice 2 : Algorithme d'une longueur dans un triangle rectangle

Compléter le programme suivant permettant de trouver la longueur \(BC\) connaissant \(AC\) et \(AB\), dans le triangle \(ABC\) rectangle en \(A\).

Par exemple si l'utilisateur rentre \(AB=3\), \(AC=4\) votre programme doit afficher en sortie la valeur de \(BC\), soit \(5\)

{"options": {"runner": {"environment": "input/output", "test_cases": [[3, 4, 5], [6, 8, 10], [5, 12, 13], [9, 12, 15], [8, 15, 17], [12, 16, 20], [7, 24, 25], [15, 20, 25], [10, 24, 26], [20, 21, 29], [18, 24, 30], [16, 30, 34], [21, 28, 35], [12, 35, 37], [15, 36, 39], [24, 32, 40], [9, 40, 41], [27, 36, 45], [14, 48, 50], [30, 40, 50], [24, 45, 51], [20, 48, 52], [28, 45, 53], [33, 44, 55], [40, 42, 58], [36, 48, 60], [11, 60, 61], [16, 63, 65], [25, 60, 65], [33, 56, 65], [39, 52, 65], [32, 60, 68], [42, 56, 70], [48, 55, 73], [24, 70, 74], [21, 72, 75], [45, 60, 75], [30, 72, 78], [48, 64, 80], [18, 80, 82], [13, 84, 85], [36, 77, 85], [40, 75, 85], [51, 68, 85], [60, 63, 87], [39, 80, 89], [54, 72, 90], [35, 84, 91], [57, 76, 95], [65, 72, 97], [28, 96, 100], [60, 80, 100]]}, "toolbox": [{"type": "base_numberinput"}, {"type": "base_simplecalculus"}, {"type": "base_sqrt"}, {"var_name": "AB", "type": "base_varvalue"}, {"var_name": "AC", "type": "base_varvalue"}, {"var_name": "BC", "type": "base_varvalue"}], "blocks": {"base_start": {"inputs": [{"fields": [{"text": "Au d\u00e9marrage", "type": "text"}], "align": "left", "type": "dummy_input"}], "on_event": "start", "creatable": false, "tooltip": "D\u00e9marrage de l'algorithme", "colour": 65, "next_statement": true}, "base_numberinput": {"inputs": [{"fields": [{"default": 0.0, "type": "number_input", "name": "value"}], "align": "left", "type": "dummy_input"}], "colour": 260, "tooltip": "Fourni un nombre", "output": "Number"}, "base_varvalue": {"inputs": [{"fields": [{"default": "x", "type": "variable", "name": "var_name"}], "align": "left", "type": "dummy_input"}], "colour": 330, "tooltip": "Fourni la valeur d'une variable", "output": true}, "base_setvar": {"inputs": [{"fields": [{"text": "mettre", "type": "text"}, {"default": "x", "type": "variable", "name": "var_name"}, {"text": "\u00e0", "type": "text"}], "align": "left", "type": "value_input", "name": "value", "check": null}], "colour": 330, "next_statement": true, "tooltip": "Affecter une valeur \u00e0 une variable", "previous_statement": true}, "simple_input_output_print": {"inputs": [{"fields": [{"text": "Afficher", "type": "text"}], "align": "left", "type": "value_input", "name": "value", "check": null}], "colour": 160, "next_statement": true, "tooltip": "Afficher une valeur", "previous_statement": true}, "base_simplecalculus": {"inputs": [{"fields": [], "align": "left", "type": "value_input", "name": "value_1", "check": "Number"}, {"fields": [{"type": "dropdown", "values": [["+", "PLUS"], ["-", "MINUS"], ["\u00d7", "TIMES"], ["\u00f7", "DIV"]], "name": "op"}], "align": "left", "type": "dummy_input"}, {"fields": [], "align": "left", "type": "value_input", "name": "value_2", "check": "Number"}], "colour": 260, "inline": true, "tooltip": "Effectue un calcul simple", "output": "Number"}, "simple_input_output_readnumber": {"inputs": [{"fields": [{"text": "demander un nombre", "type": "text"}], "align": "left", "type": "dummy_input"}], "colour": 160, "tooltip": "Demander un nombre \u00e0 l'utilisateur", "output": true}, "base_sqrt": {"inputs": [{"fields": [{"text": "Racine carr\u00e9e de", "type": "text"}], "align": "left", "type": "value_input", "name": "value", "check": null}], "colour": 260, "output": true}}}, "ast": [[{"editable": false, "type": "base_start", "deletable": false}, {"var_name": "AB", "type": "base_setvar", "deletable": false, "value": {"type": "simple_input_output_readnumber", "deletable": false}}, {"var_name": "AC", "type": "base_setvar", "deletable": false, "value": {"type": "simple_input_output_readnumber", "deletable": false}}, {"var_name": "BC", "type": "base_setvar", "deletable": false}, {"type": "simple_input_output_print", "deletable": false, "value": {"var_name": "BC", "type": "base_varvalue"}}]]}

Exercice 3 : Simplifier un algorithme grâce à un développement d'expression littérale

On s'intéresse à un algorithme qui permet à un fabricant de bonbons de savoir combien lui coûte un bonbon.

Le fabricant sait que chaque bonbon lui coûte 7 en sucre.
Il sait que chaque bonbon est emballé dans un papier qui lui coûte 8.
Il met les bonbons dans des paquets de 40 bonbons.
Les paquets sont des sachets plastiques qui eux-même coûtent 5.

Enfin, l'usine lui coûte 9000 tous les mois peu importe le nombre de bonbons qui sont produits.

On veut produire \(x\) bonbons pendant 1 mois.

Quel est le coût associé au sucre en fonction de \(x\) ?
On omettra le signe €.

Quel est le prix des emballages pour \(x\) bonbons ?
On omettra le signe €.

Quel est le prix des sachets plastiques pour \(x\) bonbons ?
On supposera que le prix des paquets est linéaire en fonction du nombre de bonbons.
On omettra le signe €.

Quel est le prix de l'usine pour \(x\) bonbons pendant 1 mois ?
On omettra le signe €.

Quelle est la formule qui permet de calculer le coût total en appelant \(x\) le nombre de bonbons que l'on souhaite ?
On omettra le signe €.

Développer puis réduire l'expression trouvée pour le coût total.
On écrira le résultat avec des fractions ou des entiers, pas de nombre à virgule.

Voici un algorithme qui calcule ce prix selon le nombre de bonbons à produire par mois. {"options": {"blockly": {"readOnly": true}, "blocks": {"base_numberinput": {"output": "Number", "colour": 260, "tooltip": "Fourni un nombre", "inputs": [{"type": "dummy_input", "fields": [{"name": "value", "default": 0.0, "type": "number_input"}], "align": "left"}]}, "base_start": {"next_statement": true, "colour": 65, "tooltip": "D\u00e9marrage de l'algorithme", "inputs": [{"type": "dummy_input", "fields": [{"text": "Au d\u00e9marrage", "type": "text"}], "align": "left"}], "creatable": false, "on_event": "start"}, "base_setvar": {"previous_statement": true, "next_statement": true, "colour": 330, "tooltip": "Affecter une valeur \u00e0 une variable", "inputs": [{"type": "value_input", "fields": [{"text": "mettre", "type": "text"}, {"name": "var_name", "default": "x", "type": "variable"}, {"text": "\u00e0", "type": "text"}], "align": "left", "name": "value", "check": null}]}, "base_varvalue": {"output": true, "colour": 330, "tooltip": "Fourni la valeur d'une variable", "inputs": [{"type": "dummy_input", "fields": [{"name": "var_name", "default": "x", "type": "variable"}], "align": "left"}]}, "base_simplecalculus": {"output": "Number", "colour": 260, "tooltip": "Effectue un calcul simple", "inputs": [{"type": "value_input", "fields": [], "align": "left", "name": "value_1", "check": "Number"}, {"type": "dummy_input", "fields": [{"name": "op", "values": [["+", "PLUS"], ["-", "MINUS"], ["\u00d7", "TIMES"], ["\u00f7", "DIV"]], "type": "dropdown"}], "align": "left"}, {"type": "value_input", "fields": [], "align": "left", "name": "value_2", "check": "Number"}], "inline": true}, "simple_input_output_readnumber": {"output": true, "colour": 160, "tooltip": "Demander un nombre \u00e0 l'utilisateur", "inputs": [{"type": "dummy_input", "fields": [{"text": "demander un nombre", "type": "text"}], "align": "left"}]}, "simple_input_output_print": {"previous_statement": true, "next_statement": true, "colour": 160, "tooltip": "Afficher une valeur", "inputs": [{"type": "value_input", "fields": [{"text": "Afficher", "type": "text"}], "align": "left", "name": "value", "check": null}]}}}, "ast": [[{"type": "base_start", "deletable": false, "editable": false}, {"type": "base_setvar", "value": {"type": "base_numberinput", "value": 7.0}, "var_name": "prix du sucre"}, {"type": "base_setvar", "value": {"type": "base_numberinput", "value": 8.0}, "var_name": "prix du papier"}, {"type": "base_setvar", "value": {"type": "base_numberinput", "value": 5.0}, "var_name": "prix du paquet"}, {"type": "base_setvar", "value": {"type": "base_numberinput", "value": 9000.0}, "var_name": "prix de l'usine"}, {"type": "base_setvar", "value": {"type": "base_numberinput", "value": 40.0}, "var_name": "bonbons par paquets"}, {"type": "base_setvar", "value": {"type": "simple_input_output_readnumber"}, "var_name": "nombre de bonbons"}, {"type": "base_setvar", "value": {"type": "base_simplecalculus", "value_1": {"type": "base_varvalue", "var_name": "prix du sucre"}, "value_2": {"type": "base_varvalue", "var_name": "nombre de bonbons"}, "op": "TIMES"}, "var_name": "total sucre"}, {"type": "base_setvar", "value": {"type": "base_simplecalculus", "value_1": {"type": "base_varvalue", "var_name": "prix du papier"}, "value_2": {"type": "base_varvalue", "var_name": "nombre de bonbons"}, "op": "TIMES"}, "var_name": "total papier"}, {"type": "base_setvar", "value": {"type": "base_simplecalculus", "value_1": {"type": "base_varvalue", "var_name": "prix du paquet"}, "value_2": {"type": "base_simplecalculus", "value_1": {"type": "base_varvalue", "var_name": "nombre de bonbons"}, "value_2": {"type": "base_varvalue", "var_name": "bonbons par paquets"}, "op": "DIV"}, "op": "TIMES"}, "var_name": "total paquet"}, {"type": "base_setvar", "value": {"type": "base_simplecalculus", "value_1": {"type": "base_varvalue", "var_name": "total sucre"}, "value_2": {"type": "base_simplecalculus", "value_1": {"type": "base_varvalue", "var_name": "total papier"}, "value_2": {"type": "base_simplecalculus", "value_1": {"type": "base_varvalue", "var_name": "total paquet"}, "value_2": {"type": "base_varvalue", "var_name": "prix de l'usine"}, "op": "PLUS"}, "op": "PLUS"}, "op": "PLUS"}, "var_name": "total"}, {"type": "simple_input_output_print", "value": {"type": "base_varvalue", "var_name": "total"}}]]}

Grâce à l'expression du coût total développée, écrire un nouveau programme qui calcule le prix plus simplement.

En pratique, dans la vie courante il peut être pratique de garder la forme plus longue de l'algorithme, notamment pour changer plus facilement le prix du sucre lorsqu'on change de fournisseur par exemple. En revanche, certains algorithmes sont beaucoup trop longs à calculer sous leur forme "naïve". Il faut donc simplifier les algorithmes pour les rendre plus rapides à calculer.

{"options": {"runner": {"environment": "input/output", "test_cases": [[0, 9000.0], [1, 9015.125], [2, 9030.25], [3, 9045.375], [4, 9060.5], [5, 9075.625], [6, 9090.75], [7, 9105.875], [8, 9121.0], [9, 9136.125], [10, 9151.25], [11, 9166.375], [12, 9181.5], [13, 9196.625], [14, 9211.75], [15, 9226.875], [16, 9242.0], [17, 9257.125], [18, 9272.25], [19, 9287.375]]}, "blocks": {"base_setvar": {"previous_statement": true, "next_statement": true, "colour": 330, "tooltip": "Affecter une valeur \u00e0 une variable", "inputs": [{"type": "value_input", "fields": [{"text": "mettre", "type": "text"}, {"name": "var_name", "default": "x", "type": "variable"}, {"text": "\u00e0", "type": "text"}], "align": "left", "name": "value", "check": null}]}, "base_simplecalculus": {"output": "Number", "colour": 260, "tooltip": "Effectue un calcul simple", "inputs": [{"type": "value_input", "fields": [], "align": "left", "name": "value_1", "check": "Number"}, {"type": "dummy_input", "fields": [{"name": "op", "values": [["+", "PLUS"], ["-", "MINUS"], ["\u00d7", "TIMES"], ["\u00f7", "DIV"]], "type": "dropdown"}], "align": "left"}, {"type": "value_input", "fields": [], "align": "left", "name": "value_2", "check": "Number"}], "inline": true}, "base_numberinput": {"output": "Number", "colour": 260, "tooltip": "Fourni un nombre", "inputs": [{"type": "dummy_input", "fields": [{"name": "value", "default": 0.0, "type": "number_input"}], "align": "left"}]}, "simple_input_output_readnumber": {"output": true, "colour": 160, "tooltip": "Demander un nombre \u00e0 l'utilisateur", "inputs": [{"type": "dummy_input", "fields": [{"text": "demander un nombre", "type": "text"}], "align": "left"}]}, "simple_input_output_print": {"previous_statement": true, "next_statement": true, "colour": 160, "tooltip": "Afficher une valeur", "inputs": [{"type": "value_input", "fields": [{"text": "Afficher", "type": "text"}], "align": "left", "name": "value", "check": null}]}, "base_start": {"next_statement": true, "colour": 65, "tooltip": "D\u00e9marrage de l'algorithme", "inputs": [{"type": "dummy_input", "fields": [{"text": "Au d\u00e9marrage", "type": "text"}], "align": "left"}], "creatable": false, "on_event": "start"}, "base_varvalue": {"output": true, "colour": 330, "tooltip": "Fourni la valeur d'une variable", "inputs": [{"type": "dummy_input", "fields": [{"name": "var_name", "default": "x", "type": "variable"}], "align": "left"}]}}, "toolbox": [{"type": "base_numberinput"}, {"type": "base_setvar", "var_name": "total"}, {"type": "base_simplecalculus"}, {"type": "base_varvalue", "var_name": "prix par bonbons"}, {"type": "base_varvalue", "var_name": "prix de l'usine"}, {"type": "base_varvalue", "var_name": "nombre de bonbons"}, {"type": "base_varvalue", "var_name": "total"}, {"type": "simple_input_output_print"}]}, "ast": [[{"type": "base_start", "deletable": false, "editable": false}, {"type": "base_setvar", "var_name": "prix par bonbons", "deletable": false}, {"type": "base_setvar", "var_name": "prix de l'usine", "deletable": false}, {"type": "base_setvar", "value": {"type": "simple_input_output_readnumber"}, "var_name": "nombre de bonbons", "deletable": false}]]}

Exercice 4 : Calcul d'un côté dans une figure de Thalès

Compléter le programme suivant permettant de trouver la longueur \( AC \) connaissant \( AE \), \( BC \) et \( DE \) dans la figure de Thales suivante :

Par exemple si l'utilisateur rentre \( AE = 6 \), \( BC = 2 \) et \( DE = 6 \), votre programme doit afficher en sortie la valeur de \( AC \), soit \( 2 \).

{"options": {"runner": {"environment": "input/output", "test_cases": [[11, 15, 33], [12, 11, 12], [15, 18, 45], [15, 13, 30], [12, 11, 48], [8, 8, 16], [11, 9, 11], [15, 8, 15], [19, 19, 38], [8, 13, 16], [18, 15, 90], [13, 13, 13], [13, 15, 65], [19, 8, 76], [11, 19, 22]]}, "blocks": {"base_varvalue": {"output": true, "colour": 330, "tooltip": "Fourni la valeur d'une variable", "inputs": [{"type": "dummy_input", "fields": [{"name": "var_name", "default": "x", "type": "variable"}], "align": "left"}]}, "base_numberinput": {"output": "Number", "colour": 260, "tooltip": "Fourni un nombre", "inputs": [{"type": "dummy_input", "fields": [{"name": "value", "default": 0.0, "type": "number_input"}], "align": "left"}]}, "simple_input_output_print": {"previous_statement": true, "next_statement": true, "colour": 160, "tooltip": "Afficher une valeur", "inputs": [{"type": "value_input", "fields": [{"text": "Afficher", "type": "text"}], "align": "left", "name": "value", "check": null}]}, "simple_input_output_readnumber": {"output": true, "colour": 160, "tooltip": "Demander un nombre \u00e0 l'utilisateur", "inputs": [{"type": "dummy_input", "fields": [{"text": "demander un nombre", "type": "text"}], "align": "left"}]}, "base_simplecalculus": {"output": "Number", "colour": 260, "tooltip": "Effectue un calcul simple", "inputs": [{"type": "value_input", "fields": [], "align": "left", "name": "value_1", "check": "Number"}, {"type": "dummy_input", "fields": [{"name": "op", "values": [["+", "PLUS"], ["-", "MINUS"], ["\u00d7", "TIMES"], ["\u00f7", "DIV"]], "type": "dropdown"}], "align": "left"}, {"type": "value_input", "fields": [], "align": "left", "name": "value_2", "check": "Number"}], "inline": true}, "base_setvar": {"previous_statement": true, "next_statement": true, "colour": 330, "tooltip": "Affecter une valeur \u00e0 une variable", "inputs": [{"type": "value_input", "fields": [{"text": "mettre", "type": "text"}, {"name": "var_name", "default": "x", "type": "variable"}, {"text": "\u00e0", "type": "text"}], "align": "left", "name": "value", "check": null}]}, "base_start": {"next_statement": true, "colour": 65, "tooltip": "D\u00e9marrage de l'algorithme", "inputs": [{"type": "dummy_input", "fields": [{"text": "Au d\u00e9marrage", "type": "text"}], "align": "left"}], "creatable": false, "on_event": "start"}}, "toolbox": [{"type": "base_numberinput"}, {"type": "base_simplecalculus"}, {"type": "base_varvalue", "var_name": "AB"}, {"type": "base_varvalue", "var_name": "AC"}, {"type": "base_varvalue", "var_name": "AD"}, {"type": "base_varvalue", "var_name": "AE"}, {"type": "base_varvalue", "var_name": "BC"}, {"type": "base_varvalue", "var_name": "DE"}]}, "ast": [[{"type": "base_start", "deletable": false, "editable": false}, {"type": "base_setvar", "value": {"type": "simple_input_output_readnumber"}, "var_name": "AE"}, {"type": "base_setvar", "value": {"type": "simple_input_output_readnumber"}, "var_name": "BC"}, {"type": "base_setvar", "value": {"type": "simple_input_output_readnumber"}, "var_name": "DE"}, {"type": "base_setvar", "var_name": "AC", "deletable": false}, {"type": "simple_input_output_print", "value": {"type": "base_varvalue", "var_name": "AC"}, "deletable": false}]]}

Exercice 5 : Afficher une table de multiplication (boucle et variables)

Écrire un algorithme qui permet d'afficher les tables de multiplications.
Le programme doit commencer par demander le nombre de la table à afficher puis les 10 valeurs de la tables (la multiplication de ce nombre par les nombres de 1 à 10 compris).
Par exemple si l'utilisateur rentre \(10\), le programme doit afficher :

\(10\)
\(20\)
\(30\)
\(40\)
\(50\)
\(60\)
\(70\)
\(80\)
\(90\)
\(100\)

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