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Math Common Core Standard, http://www.corestandards.org
CCSS.Math.Content_K:Ecole maternelle et élémentaire (K)
CCSS.Math.Content_K_CC:Compter et Dénombrer (CC)
CCSS.Math.Content_K_CC_A1:Compter jusqu'à 100 de 1 en 1 et de 10 en 10.
CCSS.Math.Content_K_CC_A2:Compter en augmentant à partir d'un nombre donné (au lieu de commencer par 1).
CCSS.Math.Content_K_CC_A3:Ecrire les nombres de 0 à 20. Représenter un certain nombre d'objets avec un nombre écrit entre 0 et 20 (0 représentant le nombre d'objet lorsqu'il n'y a pas d'objet).
CCSS.Math.Content_K_CC_B4:Comprendre la relation entre un nombre et une quantité; relier le fait de compter au dénombrement.
CCSS.Math.Content_K_CC_B4a:Lorsqu'on compte des objets, dire le nom des nombres dans l'ordre, en associant chaque objet à un et un seul nom de nombre et chaque nom de nombre à un et un seul objet.
CCSS.Math.Content_K_CC_B4b:Comprendre que le dernier nombre prononcé donne le nombre d'objets comptés. Le nombre d'objets est le même indépendemment de leur disposition ou de l'ordre dans lequel ils ont été comptés.
CCSS.Math.Content_K_CC_B4c:Comprendre que le nombre successeur fait référence à une quantité augmentée de un.
CCSS.Math.Content_K_CC_B5:Compter pour répondre aux questions "combien" pour 20 objets disposés en ligne, en une surface rectangulaire ou en cercle, ou pour 10 objets dispersés; Etant donné un nombre entre 1 et 20, déterminer un ensemble avec autant d'objets.
CCSS.Math.Content_K_CC_C6:Déterminer si le nombre d'objets dans un ensemble est plus grand, plus petit ou égal au nombre d'objets dans un autre ensemble, par exemple en utilisant des stratégies de comptage ou de correspondance.
CCSS.Math.Content_K_CC_C7:Comparer deux nombres entre 1 et 10 écrits en lettres.
CCSS.Math.Content_K_G:Géométrie
CCSS.Math.Content_K_G_A1:Décrire des objets à proximité en utilisant des noms de formes, et décrire les positions relatives de ces objets en utilisant des termes comme <i>au dessus de</i>, <i>au dessous de</i>, <i>à côté de</i>, <i>devant</i>, <i>derrière</i>, and <i>proche de</i>.
CCSS.Math.Content_K_G_A2:Nommer correctement des formes indépendamment de leur orientation ou de leur taille.
CCSS.Math.Content_K_G_A3:Identifier des formes ayant deux dimensions (reposant par morceaux sur un plan, "pouvant être découpé en morceaux plat"), ou 3 dimensions ("volumes").
CCSS.Math.Content_K_G_B4:Analyser et comparer des formes de deux et trois dimensions, de différentes tailles et orientations, en utilisant un language informel pour décrire leurs similarités, leurs différences, leurs parties, (par exemple le nombre de cotés, de sommets), et d'autres caractéristiques (par exemple avoir des côtés de même longueur).
CCSS.Math.Content_K_G_B5:Faire des maquettes des formes du monde en construisant des assemblages (par exemples des batonnets et des boules de pate à modeler) et dessiner des formes.
CCSS.Math.Content_K_G_B6:Composer des formes simples pour constituer des formes plus grandes. <i>Par exemple, "pouvez-vous assembler ces deux triangles avec deux côtés se touchant entièrement pour faire un rectangle</i>?"
CCSS.Math.Content_K_MD:Grandeurs et gestion de données
CCSS.Math.Content_K_MD_A1:Décrire des caractéristiques mesurables d'objets, comme la longueur ou le poids. Décrire plusieurs caractéristiques mesurables d'un même objet.
CCSS.Math.Content_K_MD_A2:Comparer directement deux objets avec une caractérique mesurable en commun, pour voir quel objet a "plus que", "moins que", et décrire la différence.<i>Par exemple, comparer les tailles de deux enfants, et préciser lequel est plus grand ou plus petit</i>.
CCSS.Math.Content_K_MD_B3:Classer des objets dans des catégories données; compter le nombre d'objets dans chaque catégorie et ordonner les catégories par les nombres obtenus.
CCSS.Math.Content_K_NBT:Nombres et Opérations en base dix.
CCSS.Math.Content_K_NBT_A1:Composer et décomposer des nombres de 11 à 19 en dix uns et quelques uns, par exemple en utilisant des objets ou des dessins, et garder chaque composition et décomposition en dessinant ou avec une équation (comme 18=10+8); Comprendre que ces nombres sont composés de dix uns, et de un, deux, trois, quatre, cinq, six, sept, huit ou neuf 1.
CCSS.Math.Content_K_OA:Opérations et raisonnement algébrique.
CCSS.Math.Content_K_OA_A1:Donner du sens à une addition ou une soustraction à l'aide de schémas, d'objets, d'images mentales, de sons (par exemple en tapant dans ses mains), de ses doigts, de situations représentatives, d'explications orales, d'expressions ou d'équations.
CCSS.Math.Content_K_OA_A2:Résoudre un problème additif (addition ou soustraction) donné sous forme de texte et ajouter ou soustraire à partir de 10, par exemple en utilisant des objets ou des schémas représentant le problème.
CCSS.Math.Content_K_OA_A3:Décomposer les nombres inférieurs ou égaux à 10 en somme de deux termes de plusieurs façons, par exemple en utilisant des objets ou des schémas ; et expliciter chaque décomposition par un schéma ou une équation (par exemple, 5 = 2 + 3 et 5 = 4 + 1).
CCSS.Math.Content_K_OA_A4:Pour chaque nombre de 1 à 9 trouver le complément à 10, par exemple en utilisant des objets ou des schémas ; et expliciter la réponse par un schéma ou une équation.
CCSS.Math.Content_K_OA_A5:Ajouter et soustraire 5 rapidement
CCSS.Math.Content_1:Grade 1
CCSS.Math.Content_1_G:Géométrie
CCSS.Math.Content_1_G_A1:Distinguer propriétés propres (un triangle est un polygone a trois côtés) et aspect (couleur, position, taille) ; construire et dessiner des figures possédant des propriétés propres.
CCSS.Math.Content_1_G_A2:Utiliser des figures planes (rectangles, carrés, trapèzes, triangles, demi-cercles et quart de cercles) ou des solides (cubes, prismes droits,cônes et cylindres de révolution) pour créer un nouvel objet. Créer de nouveaux objets à l'aide de ceux obtenus.
CCSS.Math.Content_1_G_A3:Partager un cercle ou un rectangle en deux ou quatre éléments superposables. Utiliser les mots « moitié », « quart » et les phrases « est la moitié de », « est le quart de ». Savoir décrire l'élément de départ comme étant « le double de » ou le « le quadruple de » chaque part. Comprendre à partir de ces exemples que dans ce type de décomposition &#58; plus le nombre de parts est grand, plus les parts sont petites.
CCSS.Math.Content_1_MD:Grandeurs et gestion de données
CCSS.Math.Content_1_MD_A1:Ordonner trois objets en fonction de leurs longueurs. Comparer indirectement la longueur de deux objets en utilisant celle d'un troisième.
CCSS.Math.Content_1_MD_A2:Exprimer la longueur d'un objet comme un multiple entier d'une longueur unitaire en utilisant une juxtaposition d'un élément de longueur unitaire. Comprendre que la longueur d'un objet et le nombre d'objets de longueur unitaire nécessaire pour le recouvrir sans chevauchement.
CCSS.Math.Content_1_MD_B3:À partir d'horloges digitale ou analogique, lire et écrire l'heure en terme d'heures et demi-heures.
CCSS.Math.Content_1_MD_C4:Organiser, représenter et interpréter des données relatives à au plus trois catégories. Poser et répondre à des questions concernant le nombre total de données, la répartition de ces données selon les catégories, la comparaison des effectifs de chaque catégorie.
CCSS.Math.Content_1_NBT:Numération et opérations en base 10
CCSS.Math.Content_1_NBT_A1:Compter jusqu'à 120 à partir de n'importe quel entier inférieur à 120. Lire et écrire tout entier naturel dans cet intervalle et associer à un ensemble d'objets le nombre d'objets qui le composent.
CCSS.Math.Content_1_NBT_B2:Comprendre que les deux chiffres d'un entier à deux chiffres représentent respectivement le nombre de dizaines et d'unités.
CCSS.Math.Content_1_NBT_B2a:10 peut être interprété comme 10 unités.
CCSS.Math.Content_1_NBT_B2b:Les nombres compris entre 11 et 19 sont composés d'une dizaine et d'une, deux, trois, quatre, cinq, six, sept, huit, neuf unités.
CCSS.Math.Content_1_NBT_B2c:Les nombres 10, 20, 30, 40, 50, 60, 70, 80, 90 s'interprètent comme une, deux, trois, quatre, cinq, six, sept, huit, neuf dizaines (et zéro unité).
CCSS.Math.Content_1_NBT_B3:Ordonner deux entiers à deux chiffres en comparant leur chiffre des dizaines et celui des unités. Traduire le résultat en utilisant les symboles > ; = ; <.
CCSS.Math.Content_1_NBT_C4:Ajouter deux entiers inférieurs à 100 ; en particulier un entier à deux chiffres et un entier à un chiffre, un entier à deux chiffres et un multiple de 10, en utilisant des modèles pratiques, des dessins ou des stratégies fondées sur la position des chiffres, les propriétés des opérations et/ou la relation entre addition et soustraction ; formuler la démarche par écrit et expliciter le raisonnement utilisé. Comprendre que dans l'addition de deux tels entiers, on ajoute les dizaines, les unités et qu'il est parfois nécessaire de composer une dizaine.
CCSS.Math.Content_1_NBT_C5:Retrancher ou ajouter mentalement 10 à un entier à deux chiffres donné, sans avoir à compter. Expliciter sa démarche.
CCSS.Math.Content_1_NBT_C6:Soustraire un multiple de 10 compris entre 10 et 90 à un multiple de 10 compris entre 10 et 90 (différence positive ou nulle), en utilisant des modèles pratiques, des dessins ou des stratégies fondées sur la position des chiffres, les propriétés des opérations et/ou la relation entre addition et soustraction ; formuler la démarche par écrit et expliciter le raisonnement utilisé.
CCSS.Math.Content_1_OA:Opérations et raisonnement algébrique
CCSS.Math.Content_1_OA_A1:Utiliser l'addition et la soustraction de nombres entiers inférieurs à 20 pour la résolution d'un problème littéral faisant appel à la notion « d'ajouter à », « enlever à », « rassembler », « séparer » et « comparer » ; « l'inconnue » pouvant se situer à n'importe quel niveau, en utilisant par exemple des dessins, des objets et des équations où un symbole représente le nombre cherché.
CCSS.Math.Content_1_OA_A2:Résoudre un problème littéral impliquant l'addition de trois entiers dont la somme est inférieure à 20 en utilisant par exemple des dessins, des objets et des équations où un symbole représente le nombre cherché.
CCSS.Math.Content_1_OA_B3:Utiliser les propriétés des opérations comme stratégie d'addition et de soustraction. Exemples &#58; sachant que 8 + 3 = 11, on peut affirmer que 3 + 8 = 11 (commutativité de l'addition). 2 + 6 + 4 = 2 + 10 = 12 (associativité de l'addition).
CCSS.Math.Content_1_OA_B4:Comprendre la soustraction comme une addition à trous. Par exemple calculer 10 &ndash; 8  revient à déterminer le nombre qu'il faut ajouter à 8 pour obtenir 10.
CCSS.Math.Content_1_OA_C5:Relier compter à ajouter et soustraire.
CCSS.Math.Content_1_OA_C6:Ajouter et soustraire des nombres entiers inférieurs à 20, avec une aisance particulière pour les entiers inférieurs à 10. Développer des stratégies basées sur 10 (par exemple, 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14), sur la relation entre addition et soustraction (par exemple sachant que 8+4=12,  on peut en déduire que 12 - 8 = 4 ) ou encore décomposer une somme en sommes plus élémentaires (par exemple  6 + 6 + 1 = 12 + 1 = 13).
CCSS.Math.Content_1_OA_D7:Comprendre la signification du symbole = et déterminer si une équation impliquant une addition ou une soustraction est vraie ou fausse. Par exemple, parmi les équations suivantes déterminer celles qui sont vraies et celles qui sont fausses &#58; 6 = 6, 7 = 8 &ndash; 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.
CCSS.Math.Content_1_OA_D8:Déterminer l'entier inconnu dans une addition ou une soustraction à trou. Par exemple 8 + ? = 11, 5 = ? - 3;   ou encore  6 + 6 = ?.
CCSS.Math.Content_2:Cours élémentaire 1
CCSS.Math.Content_2_G:Géométrie
CCSS.Math.Content_2_G_A1:Reconnaître et dessiner des figures ayant des propriétés spécifiques, par exemple un nombre donné d'angles ou de côtés superposables. Identifier triangles, quadrilatères, pentagones, hexagones et cubes
CCSS.Math.Content_2_G_A2:Décomposer un rectangle en lignes et colonnes de même dimension et savoir dénombrer une telle décomposition.
CCSS.Math.Content_2_G_A3:Décomposer disques et rectangles en deux, trois ou quatre parties superposables, décrire chacune de ces parties en utilisant le vocabulaire "moitié", "tiers", "une moitié de", "un tiers de" etc.. et se référer au tout  comme "deux moitiés", "trois tiers", "quatre quarts". Observer que l'on peut obtenir des décompositions similaires n'ont pas nécessairement les mêmes formes.

CCSS.Math.Content_2_MD:Grandeurs et gestion de données.
CCSS.Math.Content_2_MD_A1:Mesurer une longueur en utilisant un outil approprié comme une règle graduée, un mètre, un mètre ruban.
CCSS.Math.Content_2_MD_A2:Mesurer une longueur de deux manières différentes en utilisant des outils avec des unités différentes. Établir une relation entre mesure et unité utilisée.
CCSS.Math.Content_2_MD_A3:Estimer une longueur en utilisant comme unité les "pouces", les "pieds", les "centimètres" et les "mètres".
CCSS.Math.Content_2_MD_A4:Utiliser des mesures pour comparer les longueurs de deux objets, et exprimer leur différence en utilisant une unité donnée.
CCSS.Math.Content_2_MD_B5:Utiliser l'addition et la soustraction des entiers inférieurs à 100 pour la résolution de problèmes littéraux impliquant des longueurs données dans la même unité, en utilisant par exemple une représentations de règle graduée et une équation où un symbole représente le nombre cherché.
CCSS.Math.Content_2_MD_B6:Représenter des nombres entiers, leur somme et leur différence (lorsqu'elles sont inférieures à 100) en utilisant une droite graduée.
CCSS.Math.Content_2_MD_C7:À partir d'une horloge digitale ou analogique lire et écrire l'heure à 5 minutes près.
CCSS.Math.Content_2_MD_C8:Résoudre des problèmes concrets impliquant billets, pièces de monnaie, rendre la monnaie etc ..
CCSS.Math.Content_2_MD_D9:À partir de mesures, arrondies à l'entier le plus proche, d'un même ou de plusieurs objets créer un tableau de données et le représenter graphiquement, l'axe horizontal étant gradué en nombres entiers.
CCSS.Math.Content_2_MD_D10:Représenter une série de données comprenant au plus quatre modalités par un diagramme en barres ou un diagramme utilisant des pictogrammes. À partir de ce type de représentations, résoudre des problèmes à l'aide d'additions, de soustractions ou de comparaisons.
CCSS.Math.Content_2_NBT:Nombres et opérations en base 10.
CCSS.Math.Content_2_NBT_A1:Comprendre que les chiffres utilisés pour écrire un nombre à trois chiffres représentent respectivement le nombre de centaines, dizaines et d'unités. Par exemple, l'écriture 706 signifie 7 centaines 0 dizaines et 6 unités.
CCSS.Math.Content_2_NBT_A1a:100 peut se comprendre comme 10 dizaines que l'on apelle une centaine.
CCSS.Math.Content_2_NBT_A1b:Les nombres 100, 200, 300, 400, 500, 600, 700, 800 et 900 peuvent se comprendre comme une, deux, trois, quatre, cinq, six, sept, huit et neuf centaines (et 0 dizaine et 0 unité).
CCSS.Math.Content_2_NBT_A2:Compter de 5 en 5, 10 en 10, 100 en 100 jusqu'à 1000.
CCSS.Math.Content_2_NBT_A3:Pour les entiers inférieurs à 1000, connaître l'écriture en base 10, savoir les écrire en toutes lettres et savoir les décomposer selon centaines, dizaines et unités.
CCSS.Math.Content_2_NBT_A4:Ordonner les entiers inférieurs à 1000 en comparant leur chiffre des centaines, dizaines et unités. Exprimer le résultat en utilisant les symboles <, = et >.
CCSS.Math.Content_2_NBT_B5:Ajouter et soustraire avec aisance des entiers inférieurs à 100 en utilisant des stratégies fondées sur la position des chiffres, les propriétés des opérations et/ou la relation entre addition et soustraction.
CCSS.Math.Content_2_NBT_B6:Ajouter jusqu'à quatre entiers inférieurs à 100 en utilisant des stratégies basées sur la position des chiffres et les propriétés des opérations.
CCSS.Math.Content_2_NBT_B7:Ajouter et soustraire des entiers inférieurs à 1000 en utilisant des modèles concrets, des dessins ou des stratégies fondées sur la position des chiffres, les propriétés des opérations et/ou la relation entre addition et soustraction ; formuler la démarche par écrit. Comprendre que lors d'une telle opération on ajoute/soustrait les centaines - dizaines - unités et qu'il est parfois nécessaire de fabriquer ou décomposer une centaine ou une dizaine.
CCSS.Math.Content_2_NBT_B8:Ajouter - soustraire mentalement 10 ou 100 à un entier compris entre 100 et 900.
CCSS.Math.Content_2_NBT_B9:Expliquer la pertinence des stratégies fondées sur l'addition et la soustraction en utilisant la position des chiffre et les propriétés de ces opérations.
CCSS.Math.Content_2_OA:Opérations et raisonnement algébrique.
CCSS.Math.Content_2_OA_A1:Utiliser l'addition et la soustraction des entiers inférieurs à 100 pour la résolution de problèmes concrets à deux étapes impliquant les notions de "ajouter à", "retrancher à", "rassembler", "séparer" et "comparer", l'inconnue pouvant se situer à n'importe quel niveau. Par exemple en utilisant des représentations et des équations où un symbole représente le nombre cherché.
CCSS.Math.Content_2_OA_B2:Ajouter et soustraire couramment deux entiers inférieurs à 20. À la fin du cours élémentaire 1 avoir mémoriser les sommes de deux entiers inférieurs à 10.
CCSS.Math.Content_2_OA_C3:Déterminer si le nombre d'éléments d'un ensemble (inférieur ou égale à 20) est pair ou impair par exemple en associant les objets par paires ou en comptant de 2 en 2 ; écrire une équation traduisant le fait qu'un nombre pair est la somme de deux entiers égaux.
CCSS.Math.Content_2_OA_C4:Utiliser une addition pour déterminer le nombre d'éléments d'un tableau dont chacune des dimensions est inférieure à 5. Écrire une équation exprimant le résultat comme la somme de termes égaux.
CCSS.Math.Content_3:Grade 3
CCSS.Math.Content_3_G:Géométrie
CCSS.Math.Content_3_G_A1:Comprendre que des figures de différentes formes (par exemple : des losanges, des rectangles, ou autres) peuvent avoir certaines propriétés communes (par exemple celle d'avoir quatre côtés), et que ces propriétés communes peuvent définir un ensemble plus général (par exemple celui des quadrilatères). Reconnaître des losanges, des rectangles et des carrés comme étant des exemples de quadrilatères et dessiner d'autres exemples de quadrilatères qui n'appartiennent à aucune de ces sous-ensembles.
CCSS.Math.Content_3_G_A2:Partager des figures en parties d'aires égales. Exprimer l'aire de chaque partie comme fraction unitaire de l'aire totale.<i>Par exemple, partager une figure en 4 parties de même aire et exprimer l'aire de chacune de ces parties comme étant égale à 1/4 de l'aire de la figure</i>.
CCSS.Math.Content_3_MD:Grandeurs et gestion de données
CCSS.Math.Content_3_MD_A1:Dire et écrire l'heure à la minute près et mesurer des intervalles de temps en minutes. Résoudre des problèmes donnés sous forme de phrases, comportant des additions ou soustractions d'intervalles de temps en minutes, par exemple grâce à une représentation du problème sur un diagramme linéaire.
CCSS.Math.Content_3_MD_A2:Mesurer et estimer des volumes de liquides et des masses d'objets en utilisant des grammes (g), des kilogrammes (kg) ou des litres (l). Additionner, soustraire, multiplier ou diviser pour résoudre des problèmes donnés sous forme de phrases, se résolvant en une étape et comportant des masses et des volumes donnés dans la même unité, par exemple en utilisant des schémas (comme un verre gradué) pour représenter le problème.
CCSS.Math.Content_3_MD_B3:Dessiner un diagramme en barre (ou utilisant des images à la place des barres) pour représenter une série comportant plusieurs modalités. Résoudre des problèmes en une ou deux étapes, posant des questions du type « combien de … de plus que ... » ou « combien de … de moins que ... » et faisant référence à des informations données sous forme d'un diagramme en barres.<i>Par exemple, dessiner un diagramme en barres dans lequel chaque carreau représente 5 animaux</i>.
CCSS.Math.Content_3_MD_B4:Collecter des données en mesurant des longueurs grâce à une règle graduée en unités, demis et quarts. Présenter ces données en dessinant une courbe pour laquelle l'axe des abscisses est gradué de façon appropriée&mdash; en unités, demis ou quarts.
CCSS.Math.Content_3_MD_C5:Considérer l'aire comme une propriété des figures planes et comprendre les concepts de mesure d'aire.
CCSS.Math.Content_3_MD_C5a:Un carré de côté 1 unité, appelé  « carré unité», est dit avoir une aire de 1 « unité d'aire» et peut être utilisé pour mesurer des aires.
CCSS.Math.Content_3_MD_C5b:Une figure plane qui peut être couverte sans trous ni recouvrements par <i>n</i> carrés unités est dit avoir une aire de <i>n</i> unités d'aire.
CCSS.Math.Content_3_MD_C6:Mesures d'aires par comptage de carrés-unités (cm carré, m carré, ou autres unités diverses).
CCSS.Math.Content_3_MD_C7:Mettre en relation le calcul d'aire avec les opérations de multiplication et d'addition.
CCSS.Math.Content_3_MD_C7a:Donner l'aire d'un rectangle dont les côtés ont des mesures entières, grâce à un pavage, et montrer que l'aire est égale au résultat trouvé en multipliant les longueur et largeur.
CCSS.Math.Content_3_MD_C7b:Multiplier les longueurs des côtés (longueurs entières) pour calculer l'aire de rectangles, dans le contexte de résolution de problèmes mathématiques ou de la « vie courante » ; et représenter des produits d'entiers comme des aires de rectangles dans des raisonnements mathématiques.
CCSS.Math.Content_3_MD_C7c:Utiliser des pavages pour montrer dans un cas concret que l'aire d'un rectangle aux longueurs de côtés entières <i>a</i> et <i>b</i> + <i>c</i> est la somme de <i>a</i> &times; <i>b</i> et <i>a</i> &times; <i>c</i>. Utiliser le modèle des aires pour représenter la propriété de distributivité dans des raisonnements mathématiques.
CCSS.Math.Content_3_MD_C7d:Considérer l'aire comme une notion additive. Donner des aires de figures recti-linéaires, en les décomposant en rectangles sans recouvrement et en additionnant les aires de ces rectangles ; utiliser cette technique pour résoudre des problèmes de la « vie courante ».
CCSS.Math.Content_3_MD_D8:Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.
CCSS.Math.Content_3_NBT:Numération et opérations en base 10
CCSS.Math.Content_3_NBT_A1:Use place value understanding to round whole numbers to the nearest 10 or 100.
CCSS.Math.Content_3_NBT_A2:Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and.or the relationship between addition and subtraction.
CCSS.Math.Content_3_NBT_A3:Multiply one-digit whole numbers by multiples of 10 in the range 10&ndash;90 (e.g., 9 &times; 80, 5 &times; 60) using strategies based on place value and properties of operations.
CCSS.Math.Content_3_NF:Nombres et opérations - Fractions
CCSS.Math.Content_3_NF_A1:Understand a fraction 1.<i>b</i> as the quantity formed by 1 part when <i>a</i> whole is partitioned into <i>b</i> equal parts; understand a fraction <i>a</i>.<i>b</i> as the quantity formed by a parts of size 1.<i>b</i>.
CCSS.Math.Content_3_NF_A2:Understand a fraction as a number on the number line; represent fractions on a number line diagram.
CCSS.Math.Content_3_NF_A2a:Represent a fraction 1.<i>b</i> on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into <i>b</i> equal parts. Recognize that each part has size 1.<i>b</i> and that the endpoint of the part based at 0 locates the number 1.<i>b</i> on the number line.
CCSS.Math.Content_3_NF_A2b:Represent a fraction <i>a</i>.<i>b</i> on a number line diagram by marking off a lengths 1.<i>b</i> from 0. Recognize that the resulting interval has size <i>a</i>.<i>b</i> and that its endpoint locates the number <i>a</i>.<i>b</i> on the number line.
CCSS.Math.Content_3_NF_A3:Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
CCSS.Math.Content_3_NF_A3a:Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.
CCSS.Math.Content_3_NF_A3b:Recognize and generate simple equivalent fractions, e.g., 1.2 = 2.4, 4.6 = 2.3. Explain why the fractions are equivalent, e.g., by using a visual fraction model.
CCSS.Math.Content_3_NF_A3c:Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. <i>Examples&#58; Express 3 in the form 3 = 3.1; recognize that 6.1 = 6; locate 4.4 and 1 at the same point of a number line diagram</i>.
CCSS.Math.Content_3_NF_A3d:Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols &gt;, =, or &lt;, and justify the conclusions, e.g., by using a visual fraction model.
CCSS.Math.Content_3_OA:Operations and Algebraic Thinking
CCSS.Math.Content_3_OA_A1:Interpret products of whole numbers, e.g., interpret 5 &times; 7 as the total number of objects in 5 groups of 7 objects each. <i>For example, describe a context in which a total number of objects can be expressed as 5 &times; 7</i>.
CCSS.Math.Content_3_OA_A2:Interpret whole-number quotients of whole numbers, e.g., interpret 56 &divide; 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. <i>For example, describe a context in which a number of shares or a number of groups can be expressed as 56 &divide; 8</i>.
CCSS.Math.Content_3_OA_A3:Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.
CCSS.Math.Content_3_OA_A4:Determine the unknown whole number in a multiplication or division equation relating three whole numbers. <i>For example, determine the unknown number that makes the equation true in each of the equations 8 &times; ? = 48, 5 = _ &divide; 3, 6 &times; 6 = ?</i>
CCSS.Math.Content_3_OA_B5:Apply properties of operations as strategies to multiply and divide. <i>Examples&#58; If 6 &times; 4 = 24 is known, then 4 &times; 6 = 24 is also known. (Commutative property of multiplication.) 3 &times; 5 &times; 2 can be found by 3 &times; 5 = 15, then 15 &times; 2 = 30, or by 5 &times; 2 = 10, then 3 &times; 10 = 30. (Associative property of multiplication.) Knowing that 8 &times; 5 = 40 and 8 &times; 2 = 16, one can find 8 &times; 7 as 8 &times; (5 + 2) = (8 &times; 5) + (8 &times; 2) = 40 + 16 = 56. (Distributive property.)</i>
CCSS.Math.Content_3_OA_B6:Understand division as an unknown-factor problem. <i>For example, find 32 &divide; 8 by finding the number that makes 32 when multiplied by 8</i>.
CCSS.Math.Content_3_OA_C7:Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 &times; 5 = 40, one knows 40 &divide; 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.
CCSS.Math.Content_3_OA_D8:Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
CCSS.Math.Content_3_OA_D9:Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. <i>For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends</i>.
CCSS.Math.Content_4:Grade 4
CCSS.Math.Content_4_G:Géométrie
CCSS.Math.Content_4_G_A1:Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.
CCSS.Math.Content_4_G_A2:Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.
CCSS.Math.Content_4_G_A3:Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.
CCSS.Math.Content_4_MD:Grandeurs et gestion de données
CCSS.Math.Content_4_MD_A1:Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table. <i>For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36), ...</i>
CCSS.Math.Content_4_MD_A2:Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.
CCSS.Math.Content_4_MD_A3:Apply the area and perimeter formulas for rectangles in real world and mathematical problems. <i>For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor</i>.
CCSS.Math.Content_4_MD_B4:Make a line plot to display a data set of measurements in fractions of a unit (1.2, 1.4, 1.8). Solve problems involving addition and subtraction of fractions by using information presented in line plots. <i>For example, from a line plot find and interpret the difference in length between the longest and shortest specimens in an insect collection</i>.
CCSS.Math.Content_4_MD_C5:Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement
CCSS.Math.Content_4_MD_C5a:An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1.360 of a circle is called a "one-degree angle," and can be used to measure angles.
CCSS.Math.Content_4_MD_C5b:An angle that turns through <i>n</i> one-degree angles is said to have an angle measure of <i>n</i> degrees.
CCSS.Math.Content_4_MD_C6:Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.
CCSS.Math.Content_4_MD_C7:Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.
CCSS.Math.Content_4_NBT:Numération et opérations en base 10
CCSS.Math.Content_4_NBT_A1:Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right.<i> For example, recognize that 700 &divide; 70 = 10 by applying concepts of place value and division</i>.
CCSS.Math.Content_4_NBT_A2:Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using &gt;, =, and &lt; symbols to record the results of comparisons.
CCSS.Math.Content_4_NBT_A3:Use place value understanding to round multi-digit whole numbers to any place.
CCSS.Math.Content_4_NBT_B4:Fluently add and subtract multi-digit whole numbers using the standard algorithm.
CCSS.Math.Content_4_NBT_B5:Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and.or area models.
CCSS.Math.Content_4_NBT_B6:Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and.or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and.or area models.
CCSS.Math.Content_4_NF:Nombres et opérations - Fractions
CCSS.Math.Content_4_NF_A1:Explain why a fraction <i>a</i>.<i>b</i> is equivalent to a fraction (<i>n</i> &times; <i>a</i>).(<i>n</i> &times; <i>b</i>) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.
CCSS.Math.Content_4_NF_A2:Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1.2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols &gt;, =, or &lt;, and justify the conclusions, e.g., by using a visual fraction model.
CCSS.Math.Content_4_NF_B3:Understand a fraction <i>a</i>.<i>b</i> with <i>a</i> &gt; 1 as a sum of fractions 1.<i>b</i>.
CCSS.Math.Content_4_NF_B3a:Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.
CCSS.Math.Content_4_NF_B3b:Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. <i>Examples&#58; 3.8 = 1.8 + 1.8 + 1.8 ; 3.8 = 1.8 + 2.8 ; 2 1.8 = 1 + 1 + 1.8 = 8.8 + 8.8 + 1.8</i>.
CCSS.Math.Content_4_NF_B3c:Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and.or by using properties of operations and the relationship between addition and subtraction.
CCSS.Math.Content_4_NF_B3d:Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.
CCSS.Math.Content_4_NF_B4:Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
CCSS.Math.Content_4_NF_B4a:Understand a fraction <i>a</i>.<i>b</i> as a multiple of 1.<i>b</i>. <i>For example, use a visual fraction model to represent 5.4 as the product 5 &times; (1.4), recording the conclusion by the equation 5.4 = 5 &times; (1.4)</i>.
CCSS.Math.Content_4_NF_B4b:Understand a multiple of a.b as a multiple of 1.b, and use this understanding to multiply a fraction by a whole number. <i>For example, use a visual fraction model to express 3 &times; (2.5) as 6 &times; (1.5), recognizing this product as 6.5. (In general, n &times; (a.b) = (n &times; a).b.)</i>
CCSS.Math.Content_4_NF_B4c:Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. <i>For example, if each person at a party will eat 3.8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?</i>
CCSS.Math.Content_4_NF_C5:Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100. <i>For example, express 3.10 as 30.100, and add 3.10 + 4.100 = 34.100</i>.
CCSS.Math.Content_4_NF_C6:Use decimal notation for fractions with denominators 10 or 100. <i>For example, rewrite 0.62 as 62.100; describe a length as 0.62 meters; locate 0.62 on a number line diagram</i>.
CCSS.Math.Content_4_NF_C7:Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols &gt;, =, or &lt;, and justify the conclusions, e.g., by using a visual model.
CCSS.Math.Content_4_OA:Operations and Algebraic Thinking
CCSS.Math.Content_4_OA_A1:Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 &times; 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.
CCSS.Math.Content_4_OA_A2:Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.
CCSS.Math.Content_4_OA_A3:Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
CCSS.Math.Content_4_OA_B4:Find all factor pairs for a whole number in the range 1&ndash;100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1&ndash;100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1&ndash;100 is prime or composite.
CCSS.Math.Content_4_OA_C5:Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. <i>For example, given the rule "Add 3" and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way</i>.
CCSS.Math.Content_5:Grade 5
CCSS.Math.Content_5_G:Géométrie
CCSS.Math.Content_5_G_A1:Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., <i>x</i>-axis and <i>x</i>-coordinate, <i>y</i>-axis and<i> y</i>-coordinate).
CCSS.Math.Content_5_G_A2:Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.
CCSS.Math.Content_5_G_B3:Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.
CCSS.Math.Content_5_G_B4:Classify two-dimensional figures in a hierarchy based on properties.
CCSS.Math.Content_5_MD:Grandeurs et gestion de données
CCSS.Math.Content_5_MD_A1:Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.
CCSS.Math.Content_5_MD_B2:Make a line plot to display a data set of measurements in fractions of a unit (1.2, 1.4, 1.8). Use operations on fractions for this grade to solve problems involving information presented in line plots. <i>For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally</i>.
CCSS.Math.Content_5_MD_C3:Recognize volume as an attribute of solid figures and understand concepts of volume measurement.
CCSS.Math.Content_5_MD_C3a:A cube with side length 1 unit, called a "unit cube," is said to have "one cubic unit" of volume, and can be used to measure volume.
CCSS.Math.Content_5_MD_C3b:A solid figure which can be packed without gaps or overlaps using <i>n</i> unit cubes is said to have a volume of <i>n</i> cubic units.
CCSS.Math.Content_5_MD_C4:Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.
CCSS.Math.Content_5_MD_C5:Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.
CCSS.Math.Content_5_MD_C5a:Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication.
CCSS.Math.Content_5_MD_C5b:Apply the formulas <i>V</i> = <i>l</i> &times; <i>w</i> &times; <i>h</i> and <i>V</i> = <i>b</i> &times; <i>h</i> for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems.
CCSS.Math.Content_5_MD_C5c:Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.
CCSS.Math.Content_5_NBT:Nombres et opérations en base 10
CCSS.Math.Content_5_NBT_A1:Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1.10 of what it represents in the place to its left.
CCSS.Math.Content_5_NBT_A2:Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
CCSS.Math.Content_5_NBT_A3:Read, write, and compare decimals to thousandths.
CCSS.Math.Content_5_NBT_A3a:Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 &times; 100 + 4 &times; 10 + 7 &times; 1 + 3 &times; (1.10) + 9 &times; (1.100) + 2 &times; (1.1000).
CCSS.Math.Content_5_NBT_A3b:Compare two decimals to thousandths based on meanings of the digits in each place, using &gt;, =, and &lt; symbols to record the results of comparisons.
CCSS.Math.Content_5_NBT_A4:Use place value understanding to round decimals to any place.
CCSS.Math.Content_5_NBT_B5:Fluently multiply multi-digit whole numbers using the standard algorithm.
CCSS.Math.Content_5_NBT_B6:Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and.or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and.or area models.
CCSS.Math.Content_5_NBT_B7:Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and.or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
CCSS.Math.Content_5_NF:Nombres et opérations - Fractions
CCSS.Math.Content_5_NF_A1:Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. <i>For example, 2.3 + 5.4 = 8.12 + 15.12 = 23.12. (In general, a.b + c.d = (ad + bc).bd.)</i>
CCSS.Math.Content_5_NF_A2:Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. <i>For example, recognize an incorrect result 2.5 + 1.2 = 3.7, by observing that 3.7 &lt; 1.2</i>.
CCSS.Math.Content_5_NF_B3:Interpret a fraction as division of the numerator by the denominator (<i>a</i>.<i>b</i> = <i>a</i> &divide; <i>b</i>). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. <i>For example, interpret 3.4 as the result of dividing 3 by 4, noting that 3.4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3.4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?</i>
CCSS.Math.Content_5_NF_B4:Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
CCSS.Math.Content_5_NF_B4a:Interpret the product (<i>a</i>.<i>b</i>) &times; <i>q</i> as a parts of a partition of <i>q</i> into <i>b</i> equal parts; equivalently, as the result of a sequence of operations<i> a</i> &times; <i>q</i> &divide; <i>b</i>. <i>For example, use a visual fraction model to show (2.3) &times; 4 = 8.3, and create a story context for this equation. Do the same with (2.3) &times; (4.5) = 8.15. (In general, (a.b) &times; (c.d) = ac.bd.)</i>
CCSS.Math.Content_5_NF_B4b:Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.
CCSS.Math.Content_5_NF_B5:Interpret multiplication as scaling (resizing), by
CCSS.Math.Content_5_NF_B5a:Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.
CCSS.Math.Content_5_NF_B5b:Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence <i>a</i>.<i>b</i> = (<i>n</i> &times; <i>a</i>).(<i>n</i> &times; <i>b</i>) to the effect of multiplying <i>a</i>.<i>b</i> by 1.
CCSS.Math.Content_5_NF_B6:Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
CCSS.Math.Content_5_NF_B7:Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.
CCSS.Math.Content_5_NF_B7a:Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. <i>For example, create a story context for (1.3) &divide; 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1.3) &divide; 4 = 1.12 because (1.12) &times; 4 = 1.3</i>.
CCSS.Math.Content_5_NF_B7b:Interpret division of a whole number by a unit fraction, and compute such quotients. <i>For example, create a story context for 4 &divide; (1.5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 &divide; (1.5) = 20 because 20 &times; (1.5) = 4</i>.
CCSS.Math.Content_5_NF_B7c:Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. <i>For example, how much chocolate will each person get if 3 people share 1.2 lb of chocolate equally? How many 1.3-cup servings are in 2 cups of raisins?</i>
CCSS.Math.Content_5_OA:Operations and Algebraic Thinking
CCSS.Math.Content_5_OA_A1:Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
CCSS.Math.Content_5_OA_A2:Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. <i>For example, express the calculation "add 8 and 7, then multiply by 2" as 2 &times; (8 + 7). Recognize that 3 &times; (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product</i>.
CCSS.Math.Content_5_OA_B3:Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. <i>For example, given the rule "Add 3" and the starting number 0, and given the rule "Add 6" and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so</i>.
CCSS.Math.Content_6:Grade 6
CCSS.Math.Content_6_EE:Expressions et équations
CCSS.Math.Content_6_EE_A1:Write and evaluate numerical expressions involving whole-number exponents.
CCSS.Math.Content_6_EE_A2:Write, read, and evaluate expressions in which letters stand for numbers.
CCSS.Math.Content_6_EE_A2a:Write expressions that record operations with numbers and with letters standing for numbers. <i>For example, express the calculation "Subtract y from 5" as 5 &ndash; y</i>.
CCSS.Math.Content_6_EE_A2b:Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. <i>For example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms</i>.
CCSS.Math.Content_6_EE_A2c:Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). <i>For example, use the formulas V = s and A = 6 s to find the volume and surface area of a cube with sides of length s = 1.2</i>.
CCSS.Math.Content_6_EE_A3:Apply the properties of operations to generate equivalent expressions. <i>For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y</i>.
CCSS.Math.Content_6_EE_A4:Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). <i>For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.</i>.
CCSS.Math.Content_6_EE_B5:Understand solving an equation or inequality as a process of answering a question&#58; which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
CCSS.Math.Content_6_EE_B6:Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
CCSS.Math.Content_6_EE_B7:Solve real-world and mathematical problems by writing and solving equations of the form <i>x</i> + <i>p</i> = <i>q</i> and<i> px</i> = <i>q</i> for cases in which <i>p</i>, <i>q</i> and <i>x</i> are all nonnegative rational numbers.
CCSS.Math.Content_6_EE_B8:Write an inequality of the form <i>x</i> &gt; <i>c</i> or <i>x</i> &lt; <i>c</i> to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form <i>x</i> &gt; <i>c</i> or<i> x</i> &lt; c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.
CCSS.Math.Content_6_EE_C9:Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.
CCSS.Math.Content_6_G:Géométrie
CCSS.Math.Content_6_G_A1:Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
CCSS.Math.Content_6_G_A2:Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas <i>V = l w h</i> and <i>V = b h</i> to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.
CCSS.Math.Content_6_G_A3:Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.
CCSS.Math.Content_6_G_A4:Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.
CCSS.Math.Content_6_NS:The Number System
CCSS.Math.Content_6_NS_A1:Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. <i>For example, create a story context for (2.3) &divide; (3.4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2.3) &divide; (3.4) = 8.9 because 3.4 of 8.9 is 2.3. (In general, (a.b) &divide; (c.d) = ad.bc.) How much chocolate will each person get if 3 people share 1.2 lb of chocolate equally? How many 3.4-cup servings are in 2.3 of a cup of yogurt? How wide is a rectangular strip of land with length 3.4 mi and area 1.2 square mi?</i>.
CCSS.Math.Content_6_NS_B2:Fluently divide multi-digit numbers using the standard algorithm.
CCSS.Math.Content_6_NS_B3:Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
CCSS.Math.Content_6_NS_B4:Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1&ndash;100 with a common factor as a multiple of a sum of two whole numbers with no common factor. <i>For example, express 36 + 8 as 4 (9 + 2).</i>.
CCSS.Math.Content_6_NS_C5:Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above.below zero, elevation above.below sea level, credits.debits, positive.negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.
CCSS.Math.Content_6_NS_C6:Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.
CCSS.Math.Content_6_NS_C6a:Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., &ndash;(&ndash;3) = 3, and that 0 is its own opposite.
CCSS.Math.Content_6_NS_C6b:Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
CCSS.Math.Content_6_NS_C6c:Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.
CCSS.Math.Content_6_NS_C7:Understand ordering and absolute value of rational numbers.
CCSS.Math.Content_6_NS_C7a:Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. <i>For example, interpret &ndash;3 &gt; &ndash;7 as a statement that &ndash;3 is located to the right of &ndash;7 on a number line oriented from left to right</i>.
CCSS.Math.Content_6_NS_C7b:Write, interpret, and explain statements of order for rational numbers in real-world contexts. <i>For example, write &ndash;3 <sup>o</sup>C &gt; &ndash;7 <sup>o</sup>C to express the fact that &ndash;3 <sup>o</sup>C is warmer than &ndash;7 <sup>o</sup>C</i>.
CCSS.Math.Content_6_NS_C7c:Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. <i>For example, for an account balance of &ndash;30 dollars, write |&ndash;30| = 30 to describe the size of the debt in dollars</i>.
CCSS.Math.Content_6_NS_C7d:Distinguish comparisons of absolute value from statements about order. <i>For example, recognize that an account balance less than &ndash;30 dollars represents a debt greater than 30 dollars</i>.
CCSS.Math.Content_6_NS_C8:Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.
CCSS.Math.Content_6_RP:Ratios and Proportional Relationships
CCSS.Math.Content_6_RP_A1:Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. <i>For example, "The ratio of wings to beaks in the bird house at the zoo was 2&#58;1, because for every 2 wings there was 1 beak." "For every vote candidate A received, candidate C received nearly three votes."</i>
CCSS.Math.Content_6_RP_A2:Understand the concept of a unit rate a.b associated with a ratio a:b with b &ne; 0, and use rate language in the context of a ratio relationship. <i>For example, "This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3.4 cup of flour for each cup of sugar." "We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger."</i>
CCSS.Math.Content_6_RP_A3:Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
CCSS.Math.Content_6_RP_A3a:Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.
CCSS.Math.Content_6_RP_A3b:Solve unit rate problems including those involving unit pricing and constant speed. <i>For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?</i>
CCSS.Math.Content_6_RP_A3c:Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30.100 times the quantity); solve problems involving finding the whole, given a part and the percent.
CCSS.Math.Content_6_RP_A3d:Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.
CCSS.Math.Content_6_SP:Statistiques et probabilités
CCSS.Math.Content_6_SP_A1:Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. <i>For example, "How old am I?" is not a statistical question, but "How old are the students in my school?" is a statistical question because one anticipates variability in students&rsquo; ages</i>.
CCSS.Math.Content_6_SP_A2:Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.
CCSS.Math.Content_6_SP_A3:Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.
CCSS.Math.Content_6_SP_B4:Display numerical data in plots on a number line, including dot plots, histograms, and box plots.
CCSS.Math.Content_6_SP_B5:Summarize numerical data sets in relation to their context, such as by
CCSS.Math.Content_6_SP_B5a:Reporting the number of observations.
CCSS.Math.Content_6_SP_B5b:Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.
CCSS.Math.Content_6_SP_B5c:Giving quantitative measures of center (median and.or mean) and variability (interquartile range and.or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.
CCSS.Math.Content_6_SP_B5d:Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.
CCSS.Math.Content_7:Grade 7
CCSS.Math.Content_7_EE:Expressions et équations
CCSS.Math.Content_7_EE_A1:Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
CCSS.Math.Content_7_EE_A2:Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. <i>For example, a + 0.05a = 1.05a means that "increase by 5%" is the same as "multiply by 1.05."</i>
CCSS.Math.Content_7_EE_B3:Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. <i>For example&#58; If a woman making $25 an hour gets a 10% raise, she will make an additional 1.10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3.4 inches long in the center of a door that is 27 1.2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation</i>.
CCSS.Math.Content_7_EE_B4:Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
CCSS.Math.Content_7_EE_B4a:Solve word problems leading to equations of the form <i>px</i> + <i>q</i> = <i>r</i> and <i>p</i>(<i>x</i> + <i>q</i>) = <i>r</i>, where <i>p</i>, <i>q</i>, and <i>r</i> are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. <i>For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?</i>
CCSS.Math.Content_7_EE_B4b:Solve word problems leading to inequalities of the form <i>px</i> + <i>q</i> &gt; <i>r</i> or <i>px</i> + <i>q</i> &lt; <i>r</i>, where <i>p</i>, <i>q</i>, and <i>r</i> are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. <i>For example&#58; As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions</i>.
CCSS.Math.Content_7_G:Géométrie
CCSS.Math.Content_7_G_A1:Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
CCSS.Math.Content_7_G_A2:Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
CCSS.Math.Content_7_G_A3:Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.
CCSS.Math.Content_7_G_B4:Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
CCSS.Math.Content_7_G_B5:Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.</li>
CCSS.Math.Content_7_G_B6:Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
CCSS.Math.Content_7_NS:The Number System
CCSS.Math.Content_7_NS_A1:Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.
CCSS.Math.Content_7_NS_A1a:Describe situations in which opposite quantities combine to make 0. <i>For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged</i>.
CCSS.Math.Content_7_NS_A1b:Understand <i>p</i> + <i>q</i> as the number located a distance |<i>q</i>| from <i>p</i>, in the positive or negative direction depending on whether <i>q</i> is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.
CCSS.Math.Content_7_NS_A1c:Understand subtraction of rational numbers as adding the additive inverse, <i>p</i> &ndash; <i>q</i> = <i>p</i> + (&ndash;<i>q</i>). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.
CCSS.Math.Content_7_NS_A1d:Apply properties of operations as strategies to add and subtract rational numbers.
CCSS.Math.Content_7_NS_A2:Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
CCSS.Math.Content_7_NS_A2a:Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (&ndash;1)(&ndash;1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.
CCSS.Math.Content_7_NS_A2b:Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If <i>p</i> and <i>q</i> are integers, then &ndash;(<i>p</i>.<i>q</i>) = (&ndash;<i>p</i>).<i>q</i> = <i>p</i>.(&ndash;<i>q</i>). Interpret quotients of rational numbers by describing real-world contexts.
CCSS.Math.Content_7_NS_A2c:Apply properties of operations as strategies to multiply and divide rational numbers.
CCSS.Math.Content_7_NS_A2d:Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.
CCSS.Math.Content_7_NS_A3:Solve real-world and mathematical problems involving the four operations with rational numbers.
CCSS.Math.Content_7_RP:Ratios and Proportional Relationships
CCSS.Math.Content_7_RP_A1:Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. <i>For example, if a person walks 1.2 mile in each 1.4 hour, compute the unit rate as the complex fraction <sup>1.2</sup>.<sub>1.4</sub> miles per hour, equivalently 2 miles per hour</i>.
CCSS.Math.Content_7_RP_A2:Recognize and represent proportional relationships between quantities.
CCSS.Math.Content_7_RP_A2a:Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
CCSS.Math.Content_7_RP_A2b:Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
CCSS.Math.Content_7_RP_A2c:Represent proportional relationships by equations. <i>For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn</i>.
CCSS.Math.Content_7_RP_A2d:Explain what a point (<i>x</i>, <i>y</i>) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, <i>r</i>) where r is the unit rate.
CCSS.Math.Content_7_RP_A3:Use proportional relationships to solve multistep ratio and percent problems. Examples&#58; simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.
CCSS.Math.Content_7_SP:Statistiques et probabilités
CCSS.Math.Content_7_SP_A1:Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.
CCSS.Math.Content_7_SP_A2:Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. <i>For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be</i>.
CCSS.Math.Content_7_SP_B3:Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. <i>For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable</i>.
CCSS.Math.Content_7_SP_B4:Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. <i>For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book</i>.
CCSS.Math.Content_7_SP_C5:Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1.2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.</li>
CCSS.Math.Content_7_SP_C6:Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. <i>For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times</i>.
CCSS.Math.Content_7_SP_C7:Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.
CCSS.Math.Content_7_SP_C7a:Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. <i>For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected</i>.
CCSS.Math.Content_7_SP_C7b:Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. <i>For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?</i>
CCSS.Math.Content_7_SP_C8:Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
CCSS.Math.Content_7_SP_C8a:Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.
CCSS.Math.Content_7_SP_C8b:Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., "rolling double sixes"), identify the outcomes in the sample space which compose the event.
CCSS.Math.Content_7_SP_C8c:Design and use a simulation to generate frequencies for compound events. <i>For example, use random digits as a simulation tool to approximate the answer to the question&#58; If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?</i>
CCSS.Math.Content_8:Grade 8
CCSS.Math.Content_8_EE:Expressions et équations
CCSS.Math.Content_8_EE_A1:Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3&nbsp;&times;&nbsp;3<sup>&ndash;5</sup> = 3<sup>&ndash;3</sup> = 1.3<sup>3</sup> = 1.27.
CCSS.Math.Content_8_EE_A2:Use square root and cube root symbols to represent solutions to equations of the form <i>x</i><sup>2</sup> = <i>p</i> and <i>x</i><sup>3</sup> = p, where <i>p</i> is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that &radic;2 is irrational.
CCSS.Math.Content_8_EE_A3:Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. <em>For example, estimate the population of the United States as 3 times 10<sup>8</sup> and the population of the world as 7 times 10<sup>9</sup>, and determine that the world population is more than 20 times larger</em>.
CCSS.Math.Content_8_EE_A4:Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology
CCSS.Math.Content_8_EE_B5:Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.
CCSS.Math.Content_8_EE_B6:Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation <i>y</i> = <i>mx</i> + <i>b</i> for a line intercepting the vertical axis at <i>b</i>.
CCSS.Math.Content_8_EE_C7:Solve linear equations in one variable.
CCSS.Math.Content_8_EE_C7a:Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form <i>x</i> = <i>a</i>, <i>a</i> = <i>a</i>, or <i>a</i> = <i>b</i> results (where <i>a</i> and <i>b</i> are different numbers).
CCSS.Math.Content_8_EE_C7b:Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.
CCSS.Math.Content_8_EE_C8:Analyze and solve pairs of simultaneous linear equations.
CCSS.Math.Content_8_EE_C8a:Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.
CCSS.Math.Content_8_EE_C8b:Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. <i>For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6</i>.
CCSS.Math.Content_8_EE_C8c:Solve real-world and mathematical problems leading to two linear equations in two variables. <i>For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair</i>.
CCSS.Math.Content_8_F:Fonctions
CCSS.Math.Content_8_F_A1:Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.
CCSS.Math.Content_8_F_A2:Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). <i>For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change</i>.
CCSS.Math.Content_8_F_A3:Interpret the equation <i>y = mx + b</i> as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. <i>For example, the function A = s<sup>2</sup> giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line</i>.
CCSS.Math.Content_8_F_B4:Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (<i>x, y</i>) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
CCSS.Math.Content_8_F_B5:Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
CCSS.Math.Content_8_G:Géométrie
CCSS.Math.Content_8_G_A1:Verify experimentally the properties of rotations, reflections, and translations
CCSS.Math.Content_8_G_A1a:Lines are taken to lines, and line segments to line segments of the same length.
CCSS.Math.Content_8_G_A1b:Angles are taken to angles of the same measure.
CCSS.Math.Content_8_G_A1c:Parallel lines are taken to parallel lines.
CCSS.Math.Content_8_G_A2:Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
CCSS.Math.Content_8_G_A3:Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
CCSS.Math.Content_8_G_A4:Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.
CCSS.Math.Content_8_G_A5:Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. <i>For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so</i>.
CCSS.Math.Content_8_G_B6:Explain a proof of the Pythagorean Theorem and its converse.
CCSS.Math.Content_8_G_B7:Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
CCSS.Math.Content_8_G_B8:Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
CCSS.Math.Content_8_G_C9:Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
CCSS.Math.Content_8_NS:The Number System
CCSS.Math.Content_8_NS_A1:Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
CCSS.Math.Content_8_NS_A2:Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., &pi;<sup>2</sup>). <i>For example, by truncating the decimal expansion of &radic;2, show that &radic;2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations</i>.
CCSS.Math.Content_8_SP:Statistiques et probabilités
CCSS.Math.Content_8_SP_A1:Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.
CCSS.Math.Content_8_SP_A2:Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.
CCSS.Math.Content_8_SP_A3:Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. <i>For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm.hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height</i>.
CCSS.Math.Content_8_SP_A4:Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. <i>For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?</i>
CCSS.Math.Content_HS:High School
CCSS.Math.Content_HS_A:Algebra
CCSS.Math.Content_HS_A_APR:Arithmetic with Polynomials and Rational Expressions
CCSS.Math.Content_HS_A_APR_A1:Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
CCSS.Math.Content_HS_A_APR_B2:Know and apply the Remainder Theorem&#58; For a polynomial <i>p</i>(<i>x</i>) and a number <i>a</i>, the remainder on division by <i>x &ndash; a</i> is <i>p</i>(<i>a</i>), so <i>p</i>(<i>a</i>) = 0 if and only if (<i>x &ndash; a</i>) is a factor of <i>p</i>(<i>x</i>).
CCSS.Math.Content_HS_A_APR_B3:Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.
CCSS.Math.Content_HS_A_APR_C4:Prove polynomial identities and use them to describe numerical relationships. <i>For example, the polynomial identity (x<sup>2</sup> + y<sup>2</sup>)<sup>2</sup> = (x<sup>2</sup> &ndash; y<sup>2</sup>)<sup>2</sup> + (2xy)<sup>2</sup> can be used to generate Pythagorean triples.</i></li>
CCSS.Math.Content_HS_A_APR_C5:(+) Know and apply the Binomial Theorem for the expansion of (<i>x</i> + <i>y</i>)<sup><i>n</i></sup> in powers of <i>x</i> and <i>y</i> for a positive integer <i>n</i>, where <i>x</i> and <i>y</i> are any numbers, with coefficients determined for example by Pascal&rsquo;s Triangle.
CCSS.Math.Content_HS_A_APR_D6:Rewrite simple rational expressions in different forms; write <sup><i>a</i>(<i>x</i>)</sup>.<sub><i>b</i>(<i>x</i>)</sub> in the form <i>q</i>(<i>x</i>) + <sup><i>r</i>(<i>x</i>)</sup>.<sub><i>b</i>(<i>x</i>)</sub>, where <i>a</i>(<i>x</i>), <i>b</i>(<i>x</i>), <i>q</i>(<i>x</i>), and <i>r</i>(<i>x</i>) are polynomials with the degree of <i>r</i>(<i>x</i>) less than the degree of <i>b</i>(<i>x</i>), using inspection, long division, or, for the more complicated examples, a computer algebra system.
CCSS.Math.Content_HS_A_APR_D7:(+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.
CCSS.Math.Content_HS_A_CED:Creating Equations*
CCSS.Math.Content_HS_A_CED_A1:Create equations and inequalities in one variable and use them to solve problems. <i>Include equations arising from linear and quadratic functions, and simple rational and exponential functions</i>.
CCSS.Math.Content_HS_A_CED_A2:Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
CCSS.Math.Content_HS_A_CED_A3:Represent constraints by equations or inequalities, and by systems of equations and.or inequalities, and interpret solutions as viable or nonviable options in a modeling context. <i>For example, represent inequalities describing nutritional and cost constraints on combinations of different foods</i>.
CCSS.Math.Content_HS_A_CED_A4:Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. <i>For example, rearrange Ohm&rsquo;s law V = IR to highlight resistance R</i>.
CCSS.Math.Content_HS_A_REI:Reasoning with Equations and Inequalities
CCSS.Math.Content_HS_A_REI_A1:Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
CCSS.Math.Content_HS_A_REI_A2:Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
CCSS.Math.Content_HS_A_REI_B3:Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
CCSS.Math.Content_HS_A_REI_B4:Solve quadratic equations in one variable.
CCSS.Math.Content_HS_A_REI_B4a:Use the method of completing the square to transform any quadratic equation in <i>x</i> into an equation of the form (<i>x</i> &ndash; <i>p</i>)<sup>2</sup> = <i>q</i> that has the same solutions. Derive the quadratic formula from this form.
CCSS.Math.Content_HS_A_REI_B4b:Solve quadratic equations by inspection (e.g., for <i>x</i><sup>2</sup> = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as <i>a</i> &plusmn; <i>bi</i> for real numbers <i>a</i> and <i>b</i>.
CCSS.Math.Content_HS_A_REI_C5:Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
CCSS.Math.Content_HS_A_REI_C6:Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
CCSS.Math.Content_HS_A_REI_C7:Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line <i>y</i> = &ndash;3<i>x</i> and the circle <i>x</i><sup>2</sup> + <i>y</i><sup>2</sup> = 3.
CCSS.Math.Content_HS_A_REI_C8:(+) Represent a system of linear equations as a single matrix equation in a vector variable.
CCSS.Math.Content_HS_A_REI_C9:(+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 &times; 3 or greater).
CCSS.Math.Content_HS_A_REI_D10:Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
CCSS.Math.Content_HS_A_REI_D11:Explain why the <i>x</i>-coordinates of the points where the graphs of the equations <i>y</i> = <i>f</i>(<i>x</i>) and <i>y</i> = <i>g</i>(<i>x</i>) intersect are the solutions of the equation <i>f</i>(<i>x</i>) = <i>g</i>(<i>x</i>); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where <i>f</i>(<i>x</i>) and.or <i>g</i>(<i>x</i>) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.<sup>&#9733;</sup>
CCSS.Math.Content_HS_A_REI_D12:Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
CCSS.Math.Content_HS_A_SSE:Seeing Structure in Expressions
CCSS.Math.Content_HS_A_SSE_A1:Interpret expressions that represent a quantity in terms of its context.<sup>&#9733;</sup>
CCSS.Math.Content_HS_A_SSE_A1a:Interpret parts of an expression, such as terms, factors, and coefficients.
CCSS.Math.Content_HS_A_SSE_A1b:Interpret complicated expressions by viewing one or more of their parts as a single entity. <i>For example, interpret P(1+r)<sup>n</sup> as the product of P and a factor not depending on P</i>.
CCSS.Math.Content_HS_A_SSE_A2:Use the structure of an expression to identify ways to rewrite it. <i>For example, see x<sup>4</sup> &ndash; y<sup>4</sup> as (x<sup>2</sup>)<sup>2</sup> &ndash; (y<sup>2</sup>)<sup>2</sup>, thus recognizing it as a difference of squares that can be factored as (x<sup>2</sup> &ndash; y<sup>2</sup>)(x<sup>2</sup> + y<sup>2</sup>)</i>.
CCSS.Math.Content_HS_A_SSE_B3:Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.<sup>&#9733;</sup>
CCSS.Math.Content_HS_A_SSE_B3a:Factor a quadratic expression to reveal the zeros of the function it defines.
CCSS.Math.Content_HS_A_SSE_B3b:Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
CCSS.Math.Content_HS_A_SSE_B3c:Use the properties of exponents to transform expressions for exponential functions. <i>For example the expression 1.15<sup>t</sup> can be rewritten as (1.15<sup>1.12</sup>)<sup>12t</sup> &asymp; 1.012<sup>12t</sup> to reveal the approximate equivalent monthly interest rate if the annual rate is 15%</i>.
CCSS.Math.Content_HS_A_SSE_B4:Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. <i>For example, calculate mortgage payments.</i><sup>&#9733;</sup>
CCSS.Math.Content_HS_F:Fonctions
CCSS.Math.Content_HS_F_BF:Building Functions
CCSS.Math.Content_HS_F_BF_A1:Write a function that describes a relationship between two quantities.<sup>&#9733;</sup>
CCSS.Math.Content_HS_F_BF_A1a:Determine an explicit expression, a recursive process, or steps for calculation from a context.
CCSS.Math.Content_HS_F_BF_A1b:Combine standard function types using arithmetic operations. <i>For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model</i>.
CCSS.Math.Content_HS_F_BF_A1c:(+) Compose functions. <i>For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time</i>.
CCSS.Math.Content_HS_F_BF_A2:Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.<sup>&#9733;</sup>
CCSS.Math.Content_HS_F_BF_B3:Identify the effect on the graph of replacing <i>f</i>(<i>x</i>) by <i>f</i>(<i>x</i>) + <i>k</i>,<i> k</i> <i>f</i>(<i>x</i>), <i>f</i>(<i>kx</i>), and <i>f</i>(<i>x</i> + <i>k</i>) for specific values of <i>k</i> (both positive and negative); find the value of <i>k</i> given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
CCSS.Math.Content_HS_F_BF_B4:Find inverse functions.
CCSS.Math.Content_HS_F_BF_B4a:Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. <i>For example, f(x) =2 x<sup>3</sup> or f(x) = (x+1).(x&ndash;1) for x &ne; 1</i>.
CCSS.Math.Content_HS_F_BF_B4b:(+) Verify by composition that one function is the inverse of another.
CCSS.Math.Content_HS_F_BF_B4c:(+) Read values of an inverse function from a graph or a table, given that the function has an inverse.
CCSS.Math.Content_HS_F_BF_B4d:(+) Produce an invertible function from a non-invertible function by restricting the domain.
CCSS.Math.Content_HS_F_BF_B5:(+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
CCSS.Math.Content_HS_F_IF:Interpreting Functions
CCSS.Math.Content_HS_F_IF_A1:Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If <i>f</i> is a function and <i>x</i> is an element of its domain, then <i>f</i>(<i>x</i>) denotes the output of <i>f</i> corresponding to the input <i>x</i>. The graph of <i>f</i> is the graph of the equation <i>y</i> = <i>f</i>(<i>x</i>).
CCSS.Math.Content_HS_F_IF_A2:Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
CCSS.Math.Content_HS_F_IF_A3:Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. <i>For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n &ge; 1</i>.
CCSS.Math.Content_HS_F_IF_B4:For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. <i>Key features include&#58; intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity</i>.<sup>&#9733;</sup>
CCSS.Math.Content_HS_F_IF_B5:Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. <i>For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.</i><sup>&#9733;</sup>
CCSS.Math.Content_HS_F_IF_B6:Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.<sup>&#9733;</sup>
CCSS.Math.Content_HS_F_IF_C7:Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.<sup>&#9733;</sup>
CCSS.Math.Content_HS_F_IF_C7a:Graph linear and quadratic functions and show intercepts, maxima, and minima.
CCSS.Math.Content_HS_F_IF_C7b:Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
CCSS.Math.Content_HS_F_IF_C7c:Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
CCSS.Math.Content_HS_F_IF_C7d:(+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
CCSS.Math.Content_HS_F_IF_C7e:Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
CCSS.Math.Content_HS_F_IF_C8:Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
CCSS.Math.Content_HS_F_IF_C8a:Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
CCSS.Math.Content_HS_F_IF_C8b:Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t.10, and classify them as representing exponential growth or decay.
CCSS.Math.Content_HS_F_IF_C9:Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). <i>For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum</i>.
CCSS.Math.Content_HS_F_LE:Linear, Quadratic, and Exponential Models*
CCSS.Math.Content_HS_F_LE_A1:Distinguish between situations that can be modeled with linear functions and with exponential functions.
CCSS.Math.Content_HS_F_LE_A1a:Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
CCSS.Math.Content_HS_F_LE_A1b:Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
CCSS.Math.Content_HS_F_LE_A1c:Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
CCSS.Math.Content_HS_F_LE_A2:Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
CCSS.Math.Content_HS_F_LE_A3:Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
CCSS.Math.Content_HS_F_LE_A4:For exponential models, express as a logarithm the solution to <i>ab</i><i><sup>ct</sup></i> = <i>d</i> where <i>a</i>, <i>c</i>, and <i>d</i> are numbers and the base <i>b</i> is 2, 10, or <i>e</i>; evaluate the logarithm using technology.
CCSS.Math.Content_HS_F_LE_B5:Interpret the parameters in a linear or exponential function in terms of a context.
CCSS.Math.Content_HS_F_TF:Fonctions trigonométriques
CCSS.Math.Content_HS_F_TF_A1:Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
CCSS.Math.Content_HS_F_TF_A2:Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
CCSS.Math.Content_HS_F_TF_A3:(+) Use special triangles to determine geometrically the values of sine, cosine, tangent for &pi;.3, &pi;.4 and &pi;.6, and use the unit circle to express the values of sine, cosine, and tangent for <i>x</i>, &pi; + <i>x</i>, and 2&pi; &ndash; <i>x</i> in terms of their values for <i>x</i>, where <i>x</i> is any real number.
CCSS.Math.Content_HS_F_TF_A4:(+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
CCSS.Math.Content_HS_F_TF_B5:Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.<sup>&#9733;</sup>
CCSS.Math.Content_HS_F_TF_B6:(+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.
CCSS.Math.Content_HS_F_TF_B7:(+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.<sup>&#9733;</sup>
CCSS.Math.Content_HS_F_TF_C8:Prove the Pythagorean identity sin<sup>2</sup>(&theta;) + cos<sup>2</sup>(&theta;) = 1 and use it to find  sin(&theta;), cos(&theta;), or tan(&theta;) given sin(&theta;), cos(&theta;), or tan(&theta;) and the  quadrant of the angle.
CCSS.Math.Content_HS_F_TF_C9:(+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.
CCSS.Math.Content_HS_G:Géométrie
CCSS.Math.Content_HS_G_C:Cercles
CCSS.Math.Content_HS_G_C_A1:Prove that all circles are similar.
CCSS.Math.Content_HS_G_C_A2:Identify and describe relationships among inscribed angles, radii, and chords. <i>Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
CCSS.Math.Content_HS_G_C_A3:</i>Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.</i>
CCSS.Math.Content_HS_G_C_A4:(+) Construct a tangent line from a point outside a given circle to the circle.
CCSS.Math.Content_HS_G_C_B5:Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
CCSS.Math.Content_HS_G_CO:Congruence
CCSS.Math.Content_HS_G_CO_A1:Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
CCSS.Math.Content_HS_G_CO_A2:Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
CCSS.Math.Content_HS_G_CO_A3:Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
CCSS.Math.Content_HS_G_CO_A4:Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
CCSS.Math.Content_HS_G_CO_A5:Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
CCSS.Math.Content_HS_G_CO_B6:Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
CCSS.Math.Content_HS_G_CO_B7:Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
CCSS.Math.Content_HS_G_CO_B8:Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
CCSS.Math.Content_HS_G_CO_C9:Prove theorems about lines and angles. <i>Theorems include&#58; vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment&rsquo;s endpoints</i>.
CCSS.Math.Content_HS_G_CO_C10:Prove theorems about triangles. <i>Theorems include&#58; measures of interior angles of a triangle sum to 180&deg;; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point</i>.
CCSS.Math.Content_HS_G_CO_C11:Prove theorems about parallelograms. <i>Theorems include&#58; opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals</i>.
CCSS.Math.Content_HS_G_CO_D12:Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). <i>Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line</i>.
CCSS.Math.Content_HS_G_CO_D13:Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
CCSS.Math.Content_HS_G_GMD:Geometric Measurement and Dimension
CCSS.Math.Content_HS_G_GMD_A1:Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. <i>Use dissection arguments, Cavalieri&rsquo;s principle, and informal limit arguments</i>.
CCSS.Math.Content_HS_G_GMD_A2:</i>(+) Give an informal argument using Cavalieri&rsquo;s principle for the formulas for the volume of a sphere and other solid figures.</i>
CCSS.Math.Content_HS_G_GMD_A3:Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.<sup>&#9733;</sup>
CCSS.Math.Content_HS_G_GMD_B4:Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
CCSS.Math.Content_HS_G_GPE:Expressing Geometric Properties with Equations
CCSS.Math.Content_HS_G_GPE_A1:Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.
CCSS.Math.Content_HS_G_GPE_A2:Derive the equation of a parabola given a focus and directrix.
CCSS.Math.Content_HS_G_GPE_A3:(+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.
CCSS.Math.Content_HS_G_GPE_B4:Use coordinates to prove simple geometric theorems algebraically. <i>For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, &radic;3) lies on the circle centered at the origin and containing the point (0, 2).
CCSS.Math.Content_HS_G_GPE_B5:</i>Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).</i>
CCSS.Math.Content_HS_G_GPE_B6:Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
CCSS.Math.Content_HS_G_GPE_B7:Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.<sup>&#9733;
CCSS.Math.Content_HS_G_MG:Modeling with Geometry
CCSS.Math.Content_HS_G_MG_A1:Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).<sup>&#9733;</sup>
CCSS.Math.Content_HS_G_MG_A2:Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).<sup>&#9733;</sup>
CCSS.Math.Content_HS_G_MG_A3:Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).<sup>&#9733;</sup>
CCSS.Math.Content_HS_G_SRT:Similarity, Right Triangles, and Trigonometry
CCSS.Math.Content_HS_G_SRT_A1:Verify experimentally the properties of dilations given by a center and a scale factor
CCSS.Math.Content_HS_G_SRT_A1a:A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
CCSS.Math.Content_HS_G_SRT_A1b:The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
CCSS.Math.Content_HS_G_SRT_A2:Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
CCSS.Math.Content_HS_G_SRT_A3:Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
CCSS.Math.Content_HS_G_SRT_B4:Prove theorems about triangles. <i>Theorems include&#58; a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
CCSS.Math.Content_HS_G_SRT_B5:</i>Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.</i>
CCSS.Math.Content_HS_G_SRT_C6:Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
CCSS.Math.Content_HS_G_SRT_C7:Explain and use the relationship between the sine and cosine of complementary angles.
CCSS.Math.Content_HS_G_SRT_C8:Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.<sup>&#9733;</sup>
CCSS.Math.Content_HS_G_SRT_D9:(+) Derive the formula <i>A</i> = 1.2 <i>ab</i> sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
CCSS.Math.Content_HS_G_SRT_D10:(+) Prove the Laws of Sines and Cosines and use them to solve problems.
CCSS.Math.Content_HS_G_SRT_D11:(+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).
CCSS.Math.Content_HS_M:Modeling
CCSS.Math.Content_HS_N:Nombres et quantités
CCSS.Math.Content_HS_N_CN:The Complex Number System
CCSS.Math.Content_HS_N_CN_A1:Know there is a complex number <i>i</i> such that <i>i</i><sup>2</sup> = &ndash;1, and every complex number has the form <i>a + bi</i> with <i>a</i> and <i>b</i> real.
CCSS.Math.Content_HS_N_CN_A2:Use the relation <i>i</i><sup>2</sup> = &ndash;1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
CCSS.Math.Content_HS_N_CN_A3:(+)Trouver le conjugué d'un nombre complexe ; utiliser les conjugués pour calculer le module et des quotiens de nombres complexes.
CCSS.Math.Content_HS_N_CN_B4:(+) Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.
CCSS.Math.Content_HS_N_CN_B5:(+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. <i>For example, (-1 + &radic;3 i)<sup>3</sup> = 8 because (-1 + &radic;3 i) has modulus 2 and argument 120&deg;.</i>
CCSS.Math.Content_HS_N_CN_B6:(+) Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.
CCSS.Math.Content_HS_N_CN_C7:Solve quadratic equations with real coefficients that have complex solutions.
CCSS.Math.Content_HS_N_CN_C8:(+) Extend polynomial identities to the complex numbers. <i>For example, rewrite x<sup>2</sup> + 4 as (x + 2i)(x &ndash; 2i)</i>.
CCSS.Math.Content_HS_N_CN_C9:(+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
CCSS.Math.Content_HS_N_Q:Quantities*
CCSS.Math.Content_HS_N_Q_A1:Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
CCSS.Math.Content_HS_N_Q_A2:Define appropriate quantities for the purpose of descriptive modeling.
CCSS.Math.Content_HS_N_Q_A3:Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
CCSS.Math.Content_HS_N_RN:The Real Number System
CCSS.Math.Content_HS_N_RN_A1:Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. <i>For example, we define 5<sup>1.3</sup> to be the cube root of 5 because we want (5<sup>1.3</sup>)<sup>3</sup> = 5<sup>(1.3)3</sup> to hold, so (5<sup>1.3</sup>)<sup>3</sup> must equal 5</i>.
CCSS.Math.Content_HS_N_RN_A2:Rewrite expressions involving radicals and rational exponents using the properties of exponents.
CCSS.Math.Content_HS_N_RN_B3:Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
CCSS.Math.Content_HS_N_VM:Vector and Matrix Quantities
CCSS.Math.Content_HS_N_VM_A1:(+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., <b><i>v</i></b>, |<b><i>v</i></b>|, ||<b><i>v</i></b>||, <i>v</i>).
CCSS.Math.Content_HS_N_VM_A2:(+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
CCSS.Math.Content_HS_N_VM_A3:(+) Solve problems involving velocity and other quantities that can be represented by vectors.
CCSS.Math.Content_HS_N_VM_B4:(+) Add and subtract vectors.
CCSS.Math.Content_HS_N_VM_B4a:Add vectors end-to.end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.
CCSS.Math.Content_HS_N_VM_B4b:Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
CCSS.Math.Content_HS_N_VM_B4c:Understand vector subtraction <b><i>v</i></b> &ndash; <b><i>w</i></b> as <b><i>v</i></b> + (&ndash;<b><i>w</i></b>), where &ndash;<b><i>w</i></b> is the additive inverse of <b><i>w</i></b>, with the same magnitude as <b><i>w</i></b> and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.
CCSS.Math.Content_HS_N_VM_B5:(+) Multiply a vector by a scalar.
CCSS.Math.Content_HS_N_VM_B5a:Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as <i>c</i>(<i>v</i><sub><i>x</i></sub>, <i>v</i><sub><i>y</i></sub>) = (<i>cv</i><sub><i>x</i></sub>, <i>cv</i><sub><i>y</i></sub>).
CCSS.Math.Content_HS_N_VM_B5b:Compute the magnitude of a scalar multiple <i>c</i><b><i>v</i></b> using ||<i>c</i><b><i>v</i></b>|| = |<i>c</i>|<b><i>v</i></b>. Compute the direction of <i>c</i><b><i>v</i></b> knowing that when |<i>c</i>|<b><i>v</i></b> &ne; 0, the direction of <i>c</i><b><i>v</i></b> is either along <b><i>v</i></b> (for <i>c</i> &gt; 0) or against <b><i>v</i></b> (for <i>c</i> &lt; 0).
CCSS.Math.Content_HS_N_VM_C6:(+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
CCSS.Math.Content_HS_N_VM_C7:(+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.
CCSS.Math.Content_HS_N_VM_C8:(+) Add, subtract, and multiply matrices of appropriate dimensions.
CCSS.Math.Content_HS_N_VM_C9:(+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.
CCSS.Math.Content_HS_N_VM_C10:(+) Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.
CCSS.Math.Content_HS_N_VM_C11:(+) Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.
CCSS.Math.Content_HS_N_VM_C12:(+) Work with 2 &times; 2 matrices as a transformations of the plane, and interpret the absolute value of the determinant in terms of area.
CCSS.Math.Content_HS_S:Statistiques et probabilités
CCSS.Math.Content_HS_S_CP:Conditional Probability and the Rules of Probability
CCSS.Math.Content_HS_S_CP_A1:Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ("or," "and," "not").
CCSS.Math.Content_HS_S_CP_A2:Understand that two events <i>A</i> and <i>B</i> are independent if the probability of <i>A</i> and <i>B</i> occurring together is the product of their probabilities, and use this characterization to determine if they are independent.
CCSS.Math.Content_HS_S_CP_A3:Understand the conditional probability of <i>A</i> given <i>B</i> as <i>P</i>(<i>A</i> and <i>B</i>).<i>P</i>(<i>B</i>), and interpret independence of <i>A</i> and <i>B</i> as saying that the conditional probability of <i>A</i> given <i>B</i> is the same as the probability of <i>A</i>, and the conditional probability of <i>B</i> given <i>A</i> is the same as the probability of <i>B</i>.
CCSS.Math.Content_HS_S_CP_A4:Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. <i>For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.</i>
CCSS.Math.Content_HS_S_CP_A5:Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. <i>For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.</i>
CCSS.Math.Content_HS_S_CP_B6:Find the conditional probability of <i>A</i> given <i>B</i> as the fraction of <i>B</i>&rsquo;s outcomes that also belong to <i>A</i>, and interpret the answer in terms of the model.
CCSS.Math.Content_HS_S_CP_B7:Apply the Addition Rule, P(A or B) = P(A) + P(B) &ndash; P(A and B), and interpret the answer in terms of the model.
CCSS.Math.Content_HS_S_CP_B8:(+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.
CCSS.Math.Content_HS_S_CP_B9:(+) Use permutations and combinations to compute probabilities of compound events and solve problems.
CCSS.Math.Content_HS_S_IC:Making Inferences and Justifying Conclusions
CCSS.Math.Content_HS_S_IC_A1:Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
CCSS.Math.Content_HS_S_IC_A2:Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. <i>For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model</i>?
CCSS.Math.Content_HS_S_IC_B3:</i>Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.</i>
CCSS.Math.Content_HS_S_IC_B4:Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.
CCSS.Math.Content_HS_S_IC_B5:Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.
CCSS.Math.Content_HS_S_IC_B6:Evaluate reports based on data.
CCSS.Math.Content_HS_S_ID:Interpreting Categorical and Quantitative Data
CCSS.Math.Content_HS_S_ID_A1:Represent data with plots on the real number line (dot plots, histograms, and box plots).
CCSS.Math.Content_HS_S_ID_A2:Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
CCSS.Math.Content_HS_S_ID_A3:Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).
CCSS.Math.Content_HS_S_ID_A4:Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.
CCSS.Math.Content_HS_S_ID_B5:Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
CCSS.Math.Content_HS_S_ID_B6:Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
CCSS.Math.Content_HS_S_ID_B6a:Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.
CCSS.Math.Content_HS_S_ID_B6b:Informally assess the fit of a function by plotting and analyzing residuals.
CCSS.Math.Content_HS_S_ID_B6c:Fit a linear function for a scatter plot that suggests a linear association.
CCSS.Math.Content_HS_S_ID_C7:Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
CCSS.Math.Content_HS_S_ID_C8:Compute (using technology) and interpret the correlation coefficient of a linear fit.
CCSS.Math.Content_HS_S_ID_C9:Distinguish between correlation and causation.
CCSS.Math.Content_HS_S_MD:Using Probability to Make Decisions
CCSS.Math.Content_HS_S_MD_A1:(+) Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.
CCSS.Math.Content_HS_S_MD_A2:(+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.
CCSS.Math.Content_HS_S_MD_A3:(+) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. <i>For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes.</i>
CCSS.Math.Content_HS_S_MD_A4:(+) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. <i>For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households?</i>
CCSS.Math.Content_HS_S_MD_B5:(+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.
CCSS.Math.Content_HS_S_MD_B5a:Find the expected payoff for a game of chance. <i>For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant.</i>
CCSS.Math.Content_HS_S_MD_B5b:Evaluate and compare strategies on the basis of expected values. <i>For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.</i>
CCSS.Math.Content_HS_S_MD_B6:(+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
CCSS.Math.Content_HS_S_MD_B7:(+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).