分数が，数直線や長方形，円などでどのように表されるか見てみましょう．同じ分数が，図によってどのように見えるかくらべてみましょう．

﻿分数が，数直線や長方形，円などでどのように表されるか見てみましょう．同じ分数が，図によってどのように見えるかくらべてみましょう．

[i]次のしつもんに，自分で考えたり，友だちと話し合ったりして，勉強したことをもっとしっかり身につけましょう．[/i]

学んだことをまとめましょう 

分数を図で表そう

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Grafici e calcoli con oggetti e funzioni 3D

Esplora le funzioni, risolvi equazioni e costruisci oggetti geometrici

Esplora i concetti e le costruzioni geometriche in un ambiente dinamico

Visualizza equazioni e funzioni in grafici e diagrammi interattivi

Scopri la nostra app online per creare note e presentazioni contenenti grafici, immagini e molto altro ancora

Risolvi calcoli con frazioni, calcoli statistici e con funzioni esponenziali

Calcolatrici

Utilizzare i simboli per risolvere equazioni e definire modelli

Esercitazioni sulle operazioni elementari: addizione, sottrazione, moltiplicazione e divisione

Investigare successioni e serie, risolvere equazioni differenziali

Studiare le relazioni matematiche tra gli oggetti di due insiemi

Esplorare forme, dimensioni e relazioni tra oggetti matematici nello spazio

Quantificare e confrontare lunghezze, pesi e volumi

Comprendere i numeri, le relazioni tra di essi e applicare il ragionamento numerico

Applicare le operazioni elementari: addizione, sottrazione, moltiplicazione, divisione

Analizzare i concetti di probabilità e incertezza di eventi e risultati

Studiare angoli, triangoli, funzioni trigonometriche e rapporti

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Describe three-dimensional figures generated by rotations of plane figures in space.

3-Dimensional Shapes from 2-Dimensional Shapes

Ablesen von Daten aus Kreuztabellen und Vierfeldertafeln und Interpretieren dieser Daten.

Ablesen und Interpretieren von Daten aus Kreuztabellen und Vierfeldertafeln

Ablesen von Daten aus Tabellen und Diagrammen und Interpretieren dieser Daten.

Ablesen und Interpretieren von Daten aus Tabellen und Diagrammen

Ablesen von Koordinaten von Punkten

Ermitteln von einfachen Quadratwurzeln im Kopf; näherungsweises Berechnen und Abschätzen von Quadratwurzeln.

Abschätzen von Quadratwurzeln durch systematisches Probieren

Abschätzen von Rechenergebnissen und Ermitteln von Schranken.

Abschätzen von Rechenergebnissen

Evaluate absolute value expressions to solve number and word problems, including finding horizontal and vertical distances between points and lines.

Espressioni con valori assoluti

Determine the absolute value of a number with and without models in number and word problems.

Valore assoluto di numeri interi relativi

Ermitteln von Abständen zwischen zwei Punkten, zwischen Punkt und Gerade und zwischen zwei parallelen Geraden. 

Abstände zwischen Punkten und Geraden

Addieren von Bruchzahlen

Beschreiben und Anwenden des Additionsalgorithmus für natürliche Zahlen.

Addieren von natürlichen Zahlen

Addieren von nichtnegativen Dezimalzahlen.

Addieren von nichtnegativen Dezimalzahlen

Use addition and subtraction to find unknown measures of non-overlapping angles.

Sommare e sottrarre angoli

Add and subtract decimals using models and pictures to explain the process and record the results.

Sommare e sottrarre numeri decimali

Add and subtract fractions and mixed numbers with like denominators (without regrouping) in number and word problems.

Sommare e sottrarre frazioni o numeri misti con lo stesso denominatore

Add and subtract fractions and mixed numbers with unlike denominators in number and word problems.

Sommare e sottrarre frazioni e numeri misti con denominatori diversi

Add and subtract fractions and mixed numbers using models and pictures to explain the process and record the results in number and word problems.

Sommare e sottrarre frazioni e numeri misti

Model or compute with integers using addition or subtraction in number and word problems.

Sommare e sottrarre numeri interi

Somma algebrica e moltiplicazione di polinomi

Identify additive inverses (opposites) and multiplicative inverses (reciprocals) and use them to solve number and word problems.

Inverso rispetto alla somma e al prodotto

Rewrite or simplify algebraic expressions including the use of the commutative, associative, and distributive properties, and inverses and identities in number and word problems.

Algebraic Properties

Verbinden von Vorstellungen mit Maßeinheiten für Flächeninhalte; Schätzen von Flächeninhalten anhand von Stützpunktvorstellungen (Vergleich mit Größen aus dem Alltag).

Alltagsvorstellungen von Flächeninhalten

Verbinden von Vorstellungen mit Maßeinheiten für Längen; Schätzen von Längen anhand von Stützpunktvorstellungen (Vergleich mit Größen aus dem Alltag).

Alltagsvorstellungen von Längen

Verbinden von Vorstellungen mit Maßeinheiten für Massen; Schätzen von Massen anhand von Stützpunktvorstellungen (Vergleich mit Größen aus dem Alltag).

Alltagsvorstellungen von Massen

Verbinden von Vorstellungen mit Maßeinheiten für Rauminhalte; Schätzen von Rauminhalten anhand von Stützpunktvorstellungen (Vergleich mit Größen aus dem Alltag).

Alltagsvorstellungen von Rauminhalten

Use a variety of triangles, quadrilaterals, and other polygons to draw conclusions about the sum of the measures of the interior angles.

Somma degli angoli di triangoli, quadrilateri e poligoni

Identify the congruent and supplementary relationships of the angles formed by cutting parallel lines by a transversal.

Angles Formed by Parallel Lines

Identify and label the vertex, rays, and interior and exterior of an angle. Use appropriate naming conventions to identify angles.

Angoli

Anwenden des pythagoräischen Lehrsatzes für Berechnungen in ebenen Figuren; Aufstellen von Formeln.

Anwenden des Satzes des Pythagoras in ebenen Figuren

Berechnen einer Größe aus einer (einfachen) Formel, wenn alle anderen Größen gegeben sind.

Anwenden von Formeln

Lösen von Aufgaben mit direkten und indirekten Proportionalitäten; Beschreiben, Vergleichen und Begründen von Lösungswegen sowie kritisches Betrachten von Ergebnissen und deren Genauigkeit.  

Anwenden von Proportionalitäten in Sachsituationen

Berechnen aller Lösungen einer linearen Gleichung; Darstellen der Lösungen im Koordinatensystem. 

Anzahl der Lösungen von linearen Gleichungen

Use theorems about congruent chords and arcs and the relationships of a radius of a circle to a tangent to solve problems.

Proprietà di archi, corde e tangenti a una circonferenza

Use properties of circles to solve number and word problems involving arcs formed by central angles or inscribed angles.

Arc Properties of Circles

Estimate and calculate areas with scale drawings and maps.

Area nei grafici in scala e nelle mappe

Use various methods, including trigonometric relationships or Heron's Formula, to find the area of a triangle in number and word problems.

Area of a Triangle Using Trigonometry

Determine the area of rectangles, squares, and composite figures using nonstandard units, grids, and standard units in number and word problems.

Area di rettangoli, quadrati e figure composte

Use grids to develop the relationship between the total numbers of square units in a rectangle and the length and width of the rectangle (l x w); find area using the formula in number and word problems.

Area di rettangoli in una griglia

Recognize the effect of scale factors or ratios on areas and volumes of similar geometric figures; use formulas to solve number and word problems.

Area e volume di figure geometriche simili

Recognize and extend arithmetic sequences and geometric sequences. Identify the common difference or common ratio.

Successioni aritmetiche e geometriche

Kennen und Unterscheiden von verschiedenen Arten von Brüchen (Stammbruch, gemischte Zahl, uneigentlicher Bruch, unechter und echter Bruch).

Arten von Brüchen

Beschreiben und Anwenden der Eigenschaften von spitzwinkligen, rechtwinkligen, stumpfwinkligen, gleichschenkligen und gleichseitigen Dreiecken.

Arten von Dreiecken

Einteilen der Winkel ihrer Größe nach (spitzer, rechter, stumpfer, gestreckter, erhabener und voller Winkel); Erklären des Gradmaßes für Winkel.

Arten von Winkeln

Aufstellen von Gleichungen zu vorgegebenen Texten und Formulieren von Texten zu vorgegebenen Gleichungen; Anwenden von linearen Gleichungen in Sachsituationen. 

Aufstellen und interpretieren von Gleichungen

Aufstellen und Interpretieren von Termen in unterschiedlichen Kontexten.

Aufstellen und Interpretieren von Termen

Organize, display, and interpret information in bar graphs.

Diagrammi a barre

Berechnen von Quadrat- und Kubikwurzeln mittels Technologieeinsatz.

Berechnen von Quadrat- und Kubikwurzeln

Berechnen von relativen Häufigkeiten und deren Angabe in Bruch-, Dezimal- und Prozentdarstellung.

Berechnen von relativen Häufigkeiten

Erklären und Anwenden des Betrags einer Zahl; Deuten des Betrags auf der Zahlengeraden als Abstand vom Nullpunkt.

Betrag einer Zahl

Nachvollziehen eines Beweises des pythagoräischen Lehrsatzes.

Beweisen des Satzes des Pythagoras

Bilden und Benennen von Winkeln

Herleiten, grafisch Veranschaulichen und Anwenden der drei binomischen Formeln.

Binomische Formeln

Organize, display, and interpret information in box-and-whisker plots.

Diagrammi a scatola e baffi

Deuten von Brüchen als Anteile eines Ganzen bzw. als Anteile von mehreren Ganzen; Ergänzen von Brüche auf Ganze.

Brüche als Anteile von Ganzen

Estimate and calculate using numbers expressed in scientific notation.

Calcolare in notazione scientifica

Indicate the probability of a chance event with or without models as certain, impossible, more likely, less likely, or neither likely nor unlikely using benchmark probabilities of 0, 1/2, and 1.

Eventi certi, incerti, impossibili

Complete the square to identify characteristics of relations or functions and verify graphically.

Caratteristiche e grafici delle funzioni

Identify and use appropriate scales and intervals in graphs and data displays.

Scegliere scale e intervalli per i grafici

Organize, display, and interpret information in circle graphs using ratios and percents.

Grafici di circonferenze

Draw circles; identify and determine the relationships between the radius, diameter, chord, center, and circumference.

Circle and its Parts

Identify combinations of fractions that make one whole.

Operazioni con le frazioni

Compare data and distributions of data, numerical and contextual, to draw conclusions, considering the measures of center and measures of variability.

Confrontare distribuzioni di dati

Compare decimals (tenths and hundredths) with and without models.

Confrontare numeri decimali

Compare and order fractions using common numerators or denominators.

Confrontare frazioni

Compare and order integers with and without models.

Confrontare numeri interi

Compare and order numbers less than 10,000.

Confrontare numeri minori di 10.000

Write a ratio or rate to compare two quantities

Confrontare quantità con rapporti

Compare and order rational numbers with and without models.

Confrontare numeri razionali

Complementare di un evento

Define and identify complementary and supplementary angles.

Angoli complementari e supplementari

Solve quadratic equations by completing the square.

Risolvere equazioni di 2° grado con il completamento del quadrato

Frazioni non elementari

Find the inverse of a function or relation. Verify that two functions are inverses using their graphs or composition of functions.

Composizione e inversa di una funzione

Model the concept of percent and relate to the value in decimal or fractional form.

Convertire percentuali in frazioni o numeri decimali

Locate points in all quadrants of the coordinate plane using ordered pairs in number and word problems.

Piano cartesiano e coppie ordinate

Locate a point in Quadrant I of a coordinate grid given an ordered pair; name the ordered pair for a point in Quadrant I of a coordinate grid.

Piano cartesiano

Use models to make predictions and interpret the correlation coefficient of the model; distinguish between correlation and causation.

Coefficiente di correlazione, correlazione e causalità: differenze ed esempi

Use proportions to express relationships between corresponding parts of similar figures.

Parti corrispondenti di figure simili

Estimate and compute the cost of items greater than $1.00; make change.

Prezzi di oggetti e calcolo del resto

Determine the sample space for an event using counting strategies (include tree diagrams, permutations, combinations, and the fundamental counting principle).

Strategie di conteggio negli spazi campionari

Describe cross-sectional views of three-dimensional figures.

Cross Sections of Solids

Use geometric models and equations to investigate the meaning of the cube of a number and the relationship to its cube root.

Cubi e radici cubiche

Derive a quadratic or exponential function that models a set of data (curve of best fit) using calculators. Use the model to make predictions.

Curve di regressione

Darstellen von Daten in Kreuztabellen und Vierfeldertafeln. 

Darstellen von Daten in Kreuztabellen und Vierfeldertafeln

Darstellen von Daten in Tabellen und Diagrammen (Säulen- und Balkendiagramme, Piktogramme, Liniendiagramme).

Darstellen von Daten in Tabellen und Diagrammen

Grafisches Darstellen und Ablesen von natürlichen Zahlen am Zahlenstrahl.

Darstellen von natürlichen Zahlen

Wechseln zwischen der Funktionsgleichung und dem Funktionsgraph, insbesondere mithilfe von Steigungsdreiecken unter Verwendung der Parameter k und d; Interpretieren des Funktionsgraphen einer linearen Funktion als Gerade. 

Darstellungsformen linearer Funktionen

Nutzen von verschiedenen Darstellungsformen reeller Funktionen (Funktionsgleichung, Wertetabelle, Funktionsgraph); Wechseln zwischen den Darstellungsformen.

Darstellungsformen von Funktionen

Describe data using or selecting the appropriate measure of central tendency; choose a measure of central tendency based on the shape of the data distribution.

Dati, distribuzioni di dati e indici di tendenza centrale

Read, write, and compare numbers with decimal place values to the thousandths place or numbers greater than one million.

Numeri decimali maggiori di un milione

Determine the measure of an angle in degree mode or in radian mode.

Misura di angoli in gradi e radianti

Deuten von Brüchen als Quotienten

Deuten von Brüchen als relative Anteile und relative Häufigkeiten.

Deuten von Brüchen als relative Anteile und Häufigkeiten

Deuten des Potenzierens mit einem positiven ganzzahligen Exponenten als wiederholtes Multiplizieren derselben Zahl; Interpretieren und Anwenden der Potenzdarstellung mit positiven ganzzahligen Exponenten.

Deuten von Potenzen

Use dimensional analysis to rename quantities or rates

Analisi dimensionale

Write equations to represent direct variation and use direct variation to solve number and word problems.

Variazione diretta

Use the discriminant to determine the number and nature of the roots of a quadratic equation.

Discriminante e radici di un'equazione di 2° grado

Use the distance formula to find the distance between two points. Use the midpoint formula to find the coordinates of the midpoint of a segment.

Formula della distanza tra due punti e del punto medio di un segmento

Calculate distances from scale drawings and maps.

Distanza nei grafici in scala e nelle mappe

Use the distributive property to represent and simplify numerical expressions.

Proprietà distributiva

Dividieren von Brüchen; Deutung des Dividierens als Multiplizieren mit dem Kehrwert des Divisors.

Dividieren von Bruchzahlen

Beschreiben und Anwenden des Divisionsalgorithmus für natürliche Zahlen.

Dividieren von natürlichen Zahlen

Represent division of fractions and mixed numbers with and without models and pictures in number and word problems; describe the inverse relationship between multiplication and division.

Dividere frazioni e numeri misti

Divide two fractions or a fraction and a whole number in number or word problems.

Dividere le frazioni

Represent division of a unit fraction by a whole number or a whole number by a unit fraction using models to explain the process in number and word problems.

Dividere frazioni unitarie

Model division in a variety of ways including sharing equally, repeated subtraction, rectangular arrays, and the relationship with multiplication.

Modelli di divisione

Write and graph special functions (step, constant, and piecewise) and identify the domain and range.

Dominio e immagine di funzioni particolari

Draw and measure angles using a protractor. Understand that a circle measures 360 degrees.

Disegnare e misurare angoli con un goniometro

Durchführen der Primfaktorenzerlegung

Describe the effect of operations on size and order of numbers.

Effetto delle operazioni sui numeri

Identify outliers and determine their effect on the mean, median, and range of a set of data.

Effetto dei valori anomali (outlier) su media, mediana e campo di variazione di un insieme di dati

Determine the effects of changes in slope and/or intercepts on graphs and equations of lines.

Effetti della variazione di coefficiente angolare e ordinata all'origine delle rette

Beschreiben der Eigenschaften linearer Funktionen f mit f(x) = k ∙ x + d.

Eigenschaften linearer Funktionen

Beschreiben und Anwenden der Eigenschaften von besonderen Vierecken (Rechteck, Quadrat, Parallelogramm, Rhombus/ Raute, Drachenviereck/ Deltoid, Trapez). 

Eigenschaften von besonderen Vierecken

Beschreiben von grundlegenden Eigenschaften von Quadern mithilfe geeigneter geometrischer Begriffe (Ecken, Kanten, Flächen, parallel, normal). 

Eigenschaften von Quadern

Beschreiben von grundlegenden Eigenschaften von Punkten, Strecken, Strahlen und Geraden.

Eingeschaften von Strecke, Strahl, Gerade

Ermitteln von Laplace-Wahrscheinlichkeiten bei einstufigen Zufallsexperimenten; Interpretieren dieser Wahrscheinlichkeiten als Vorhersagewerte für relative Häufigkeiten.

Einstufige Zufallsexperimente

Einzeichnen von Punkten mit gegebenen Koordinaten auf den Achsen und in allen vier Quadranten des Koordinatensystems.

Einzeichnen von Punkten im Koordinatensystem

Misura del tempo

Write the equation of a line parallel or perpendicular to a given line through a given point.

Equation of a Line Parallel or Perpendicular to a Line

Use properties of circles to solve problems related to the equation of a circle, its center, and radius length.

Equation of Circles

Write an equation to describe the algebraic relationship between two defined variables in number and word problems, including recognizing which variable is dependent.

Equazioni in due variabili

Solve equations involving powers and roots by using inverse relationships.

Equazioni con potenze e radici

Use the definition of rational numbers to convert decimals and fractions to equivalent forms.

Forme equivalenti di numeri decimali e frazioni

Convert fractions and terminating decimals to the thousandths place to equivalent forms without models; explain the equivalence.

Frazioni e numeri decimali equivalenti

Use models to write equivalent fractions, including using composition or decomposition or showing relationships among halves, fourths, and eighths, and thirds and sixths.

Modelli di frazioni equivalenti

Write equivalent fractions with smaller or larger denominators.

Frazioni equivalenti

Use reasoning with equivalent ratios to solve number and word problems.

Rapporti equivalenti

Erkennen von direkten und indirekten Proportionalitäten in Sachsituationen; Begründen oder Widerlegen von Proportionalitäten.

Erkennen von Proportionalitäten in Sachsituationen

Ermitteln von absoluten Häufigkeiten

Determine precision unit, accuracy, and greatest possible error of a measuring tool. Apply significant digits in meaningful contexts.

Errore di uno strumento di misura

Erweitern und Kürzen von Brüche (auch mithilfe von Visualisierungen).

Erweitern und Kürzen von Brüchen

Erklären, dass durch die Erweiterung der rationalen Zahlen zu den reellen Zahlen die Zahlengerade lückenlos ausgefüllt wird.

Erweiterung der rationalen zu den reellen Zahlen

Estimate and compute sums and differences with decimals.

Stimare somme e differenze di numeri decimali

Estimate the measure of an object in one system given the measure of that object in another system.

Stimare le misure

Estimate, measure and compare capacity using appropriate tools and units in number and word problems

Stimare e misurare le capacità

Estimate, measure and compare length using appropriate tools and units in number and word problems

Stimare e misurare le lunghezze

Estimate, measure and compare weight using appropriate tools and units in number and word problems

Stimare e misurare i pesi

Estimate and compute products of whole numbers with multi-digit factors.

Stimare i prodotti di numeri interi

Estimate products and quotients of decimals or of mixed numbers.

Stimare prodotti e quozienti di numeri decimali e misti

Apply appropriate types of estimation for number and word problems that include estimating products and quotients.

Stimare prodotti e quozienti

Estimate and solve division problems with multi-digit divisors; explain solution.

Stimare i quozienti di numeri interi

Estimate the square root of a number between two consecutive integers with and without models. Use a calculator to estimate the square root of a number.

Approssimare radici quadrate al numero intero più vicino

Estimate sums and differences with fractions and mixed numbers.

Stimare somme e differenze di frazioni e numeri misti

Apply appropriate type of estimation for sums and differences.

Stimare somme e differenze

Evaluate algebraic expressions in number and word problems.

Risolvere espressioni algebriche

Solve exponential equations by rewriting expressions with like bases.

Equazioni esponenziali

Use exponential notation and repeated multiplication to describe and simplify exponential expressions.

Espressioni esponenziali

Write the equation of and graph exponential equations or functions, including f(x) = ab^x and f(x) = a(1+r)^x, in number and word problems; identify and interpret critical values.

Equazione, grafico e caratteristiche delle equazioni esponenziali

Evaluate expressions and use formulas to solve number and word problems involving exponential functions; classify exponential functions as exponential growth or decay.

Funzioni esponenziali

Simplify numerical expressions that may contain exponents.

Espressioni con esponenti

Factor quadratic polynomials, including special products.

Scomposizione di polinomi di 2° grado

Understand that many whole numbers factor in different ways.

Fattori dei numeri interi

Identify and make figures with line symmetry.

Figure con simmetrie

Calculate the areas of triangles, parallelograms, trapezoids, circles, and composite figures in number and word problems.

Calcolare l'area di triangoli, parallelogrammi, trapezi, cerchi e figure composte

Find the fractional part of a whole number or fraction with and without models and pictures.

Determinare la parte frazionaria

Identify or generate numerical and geometric patterns; correct errors in patterns or interpret pattern features.

Riconoscere la regola che definisce una sequenza numerica

Use models to find volume for prisms and cylinders as the product of the area of the base (B) and the height. Calculate the volume of prisms in number and word problems.

Calcolare il volume dati base e altezza

Näherungsweises Ermitteln des Flächeninhalts von Kreisen; Anwenden der Formel für die Berechnung des Flächeninhalts von Kreisen, insbesondere auf Halbkreise, Viertelkreise und Achtelkreise. 

Flächeninhalt eines Kreises

Ermitteln von Flächeninhalten von Rechtecken durch Zerlegen in passende Einheitsquadrate; Anwenden der Formel zur Berechnung des Flächeninhalts eines Rechtecks.

Flächeninhalt von Rechtecken

Anwenden und Begründen von Formeln für die Flächeninhalte von besonderen Vierecken (Parallelogramm, Trapez, Deltoid, Raute).

Flächeninhalte von besonderen Vierecken

Use models to develop the relationship between the total distance around a figure and the formula for perimeter; find perimeter using the formula in number and word problems.

Formula del perimetro

Use models to develop formulas for finding areas of triangles, parallelograms, trapezoids, and circles in number and word problems.

Formule delle aree di triangoli, parallelogrammi, trapezi e cerchi

Find the lateral area and surface area of a right prism, pyramid, right circular cylinder, and cone using formulas.

Formule dell'area della superficie laterale dei solidi

Formulieren des pythagoräischen Lehrsatzes für vorgegebene rechtwinklige Dreiecke unter Verwendung verschiedener Bezeichnungen.

Formulieren des Satzes des Pythagoras

Represent division of whole numbers as a fraction in number and word problems.

Frazioni come divisione

Use frequency tables, dot plots, and other graphs to determine the shape, center, and spread of a data distribution.

Tabelle di frequenza e diagrammi a punti

Interpret probability models for data from simulations or for experimental data presented in tables and graphs (frequency tables, line plots, bar graphs).

Tabelle di frequenza, grafici a linee e diagrammi a barre

Use and interpret function notation in number and word problems; determine a value of the function given an element of the domain.

Funzioni e notazione funzionale

Beschreiben und Interpretieren von Funktionen als eindeutige Zuordnungen; Erkennen und Begründen von funktionalen Zusammenhängen.

Funktionen als eindeutige Zuordnungen

Generate a set of ordered pairs using a rule which is stated in verbal, algebraic, or table form; generate a sequence given a rule in verbal or algebraic form.

Generare coppie ordinate e successioni

Bennenen und Untersuchen von geometrischen Körpern.

Geometrische Körper

Deuten von Prozent als "von Hundert"; grafische Darstellung von Prozenten.

Grafische Veranschaulichung von Prozenten

Lösen von linearen Gleichungssystemen durch grafische Darstellung.

Grafisches Lösen linearer Gleichungssysteme

Graph absolute value functions and their corresponding inequalities.

Grafici di funzioni e disequazioni con valori assoluti

Use ordered pairs derived from tables, algebraic rules, or verbal descriptions to graph linear functions.

Tracciare i grafici delle rette utilizzando coppie ordinate di valori

Graph quadratic functions. Identify and interpret the intercepts, maximum, minimum, and the axis of symmetry.

Tracciare i grafici delle funzioni di 2° grado

Graphically solve systems of linear equations.

Grafico di un sistema di equazioni lineari

Risolvere equazioni di 2° grado con il metodo grafico

Find and interpret the maximum, the minimum, and the intercepts of a quadratic function.

Grafici delle funzioni di 2° grado

Organize, display, and interpret multiple sets of information in the same graph including multiple bar, multiple stem-and-leaf, and multiple line graphs.

Grafici relativi a più insiemi di dati

Organize, display, and interpret information in graphs containing scales that represent multiple units.

Grafici con scale aventi unità di misura diverse

Find factors, common factors, and the greatest common factor of numbers; explain.

Massimo comun divisore

Organize and compare one set of information in histograms and bar graphs.

Istogramma o diagramma a barre?

Organize, display, and interpret information in histograms.

Istogrammi

Identify proportional parts of similar figures using rulers and protractors.

Identificare parti proporzionali tramite misurazione

Recognize that a statistical question is one that will require gathering data that has variability.

Identifying Statistical Questions

Describe the intervals for which a function is increasing or decreasing.

Funzioni crescenti e decrescenti

Graph or identify simple inequalities using symbol notation >, <, ≤, ≥, and ≠ in number and word problems.

Disequazioni e le notazioni simboliche >, <, ≤, ≥ e ≠

Make inferences about a population based on a sample and compare variation in multiple samples.

Inferenza statistica basata su dati campionari

Use concepts of positive numbers, negative numbers, and zero (e.g., on a number line, in counting, in temperature, in "owing") to describe quantities in number and word problems.

Numeri interi

Identify and interpret the intercepts of a linear relation in number and word problems.

Ordinata all'origine di una retta

Use properties, definitions, and theorems of polygons to solve problems related to the interior and exterior angles of a convex polygon.

Angoli interni ed esterni dei poligoni

Use remainders in problem-solving situations and interpret the remainder with respect to the original problem.

Interpretare i resti

Identify, draw, and name: points, rays, line segments, lines, and planes.

Introduzione alla Geometria

Use models to investigate the concept of the Pythagorean Theorem.

Alla scoperta del teorema di Pitagora

Erklären und Begründen, dass es Zahlen gibt, die nicht als Bruch darstellbar sind (irrationale Zahlen).

Irrationale Zahlen

Angeben von Klammer- und Vorrangregeln; Erklären und Anwenden von Klammer-Punkt-Strich-Regel.

Klammer- und Vorrangregeln

Konstruieren von besonderen Vierecken (mit Zirkel, Lineal und Geodreieck oder DGS).

Konstruieren von besonderen Vierecken

Konstruieren von Dreiecken (mit Zirkel, Lineal und Geodreieck oder DGS).

Konstruieren von Dreiecken

Skizzieren und Konstruieren von Rechecken und zusammengesetzten Figuren, die aus Rechtecken bestehen. 

Konstruieren von Rechtecken

Lösen von Konstruktionsaufgaben mit parallelen oder normalen Geraden, Strahlen und Strecken. 

Konstruktionsaufgaben

Beschreiben von grundlegenden Eigenschaften von Koordinatenachsen; Deuten von Koordinatenachsen als normal aufeinander stehende Zahlengeraden.

Koordinatenachsen

Herleiten und Anwenden von Formeln für den Umfang und Flächeninhalt von Kreissektoren.

Kreissektoren

Beschreiben, dass bei jedem Kreis das Verhältnis von Umfang zu Durchmesser die gleiche Zahl π ergibt; Experimentelles Ermitteln von Näherungswerten für π. 

Kreiszahl π 

Beschreiben der Lagebeziehung zwischen Kreisen, Punkten und Geraden; Konstruieren von Tangenten in Kreispunkten.

Lagebeziehung zwischen Kreis und Punkt, Gerade bzw. Kreis

Use rules of exponents to simplify numeric and algebraic expressions.

Proprietà delle potenze

Find multiples, common multiples, and the least common multiple of numbers; explain.

Minimo comune multiplo

Use measures of arcs or central angles to find arc length or sector area of a circle.

Length of Arc and Area of Sector of a Circle

Organize, display, and interpret information in line plots with a horizontal scale in fractional units.

Grafici a linee con scale frazionarie

Write a problem given a simple linear equation or inequality.

Problemi con equazioni e disequazioni

Model and solve linear inequalities using the properties of inequality in number and word problems.

Problemi con disequazioni lineari

Identify relations as directly proportional, linear, or nonlinear using rules, tables, and graphs.

Relazioni lineari, non lineari e di proporzionalità

Identify linear and nonlinear relationships in data sets.

Relazioni lineari e non lineari in un insieme di dati

Define and identify alternate interior, alternate exterior, corresponding, adjacent and vertical angles.

Rette e angoli

Derive a linear equation that models a set of data (line of best fit) using calculators. Use the model to make predictions.

Rette di regressione

Classify plane figures according to type of symmetry (line, rotational)

Assi di simmetria delle figure piane

Identify the number of lines of symmetry in a figure and draw lines of symmetry.

Assi di simmetria

Solve a literal equation for an indicated variable.

Literal Equation for an Indicated Variable

Divide using single-digit divisors with and without remainders.

Divisione in colonna

Lösen und Interpretieren von Abzählaufgaben

Lösen von Abzählaufgaben

Lösen von (einfachen) Gleichungen durch systematisches Probieren und Veranschaulichen.

Lösen von Gleichungen durch systematisches Probieren

Lösen von linearen Gleichungen durch Umkehren von Rechenoperationen.

Lösen von linearen Gleichungen durch Umkehroperationen

Konstruktives Lösen von Vermessungsaufgaben mithilfe maßstäblicher Zeichnungen.

Lösen von Vermessungsaufgaben

Maßumrechnungen von Raummaßen

Determine the mean absolute deviation (MAD) for one or more sets of data. Describe the meaning of MAD for given data sets.

Mean Absolute Deviation (MAD) of Data Sets

Mean of Data Sets

Select the appropriate measure of variability; choose a measure of variability based on the presence of outliers, clusters, and the shape of the data distribution.

Misure di variabilità

Mediana di un insieme di dati

Ermitteln von Laplace-Wahrscheinlichkeiten bei zweistufigen Zufallsexperimenten (mithilfe von Baumdiagrammen); Interpretieren dieser Wahrscheinlichkeiten; Erstellen und Interpretieren von Baumdiagrammen.

Mehrstufige Zufallsexperimente

Messen von Winkeln

Ordnen von Datenlisten; Ermitteln und Interpretieren von Minimum, Maximum sowie Spannweite.

Minimum, Maximum und Spannweite

Erheben und Interpretieren von Mittelwerten (arithmetischer Mittelwert, Modus, Median).

Mittelwerte

Moda di un insieme di dati

Approximate a linear model that best fits a set of data; use the linear model to make predictions.

Creare modelli di dati e fare previsioni

Construct or complete a table of values to solve problems associated with a given relationship.

Creare modelli con le tabelle

Determine the value of sets of coins and bills using cent sign and dollar sign appropriately. Create equivalent amounts with different coins and bills.

La matematica dei soldi

Add multi-digit numbers with regrouping in number and word problems.

Addizioni a più cifre con riporto

Subtract multi-digit numbers with regrouping in number and word problems.

Sottrazioni a più cifre con il prestito

Write a multiplication or a division sentence to represent a number or word problem; solve.

Espressioni con moltiplicazioni e divisioni

Multiplizieren von Bruchzahlen

Beschreiben und Anwenden des Multiplikationsalgorithmus für natürliche Zahlen.

Multiplizieren von natürlichen Zahlen

Multiplizieren von nichtnegativen Dezimalzahlen.

Multiplizieren von nichtnegativen Dezimalzahlen

Multiply a 1-digit number by a 2-digit multiple of 10.

Moltiplicare un numero a una cifra con uno a due cifre

Recognize and use patterns in powers of ten (with or without exponents) to multiply and divide whole numbers and decimals.

Moltiplicare e dividere per potenze di 10

Multiply and divide decimals using models and pictures to explain the process and record the results.

Modelli per moltiplicare e dividere numeri decimali

Multiply or divide two decimals or a decimal and a whole number in number and word problems.

Moltiplicare e dividere numeri decimali

Model or compute with integers using multiplication or division in number and word problems.

Moltiplicare e dividere numeri interi

Multiply or divide with mixed numbers in number and word problems.

Moltiplicare e dividere numeri misti

Multiply two fractions or a fraction and a whole number in number and word problems.

Moltiplicare frazioni

Multiply a multi-digit whole number by a 1-digit whole number or a 2-digit multiple of 10.

Moltiplicare numeri con più cifre

Solve multi-step number and word problems using the four operations.

Problemi non elementari

Identify the nets for prisms, pyramids, cylinders and cones in geometric and applied problems.

Sviluppi piani dei solidi

Erkennen von gegebenen Figuren als Netze eines Quaders; Skizzieren und Konstruieren von Netzen eines Quaders.

Netze von Quadern

Determine whether a linear equation has one solution, infinitely many solutions, or no solution.

Numero di soluzioni di un sistema di equazioni

Read and write word names for numbers from 1,000 to 9,999.

Numeri da 1.000 a 9.999

Read, write, and compare whole numbers from 10,000 to less than one million using standard and expanded notation.

Numeri da 10.000 a 1.000.000

Read and write word names for whole numbers from 101 to 999.

Numeri da 101 a 999

Read, write, or model numbers in expanded form using decimal fractions or exponents.

Numeri in forma polinomiale

Write whole numbers in scientific notation; convert scientific notation to standard form; investigate the uses of scientific notation.

La notazione scientifica dei numeri

Anwenden und Begründen von Formeln für die Berechnung von Oberflächeninhalten von Zylindern.

Oberflächeninahlt von Zylindern

Anwenden und Begründen von Formeln für die Berechnung von Oberflächeninhalten von Kegeln. 

Oberflächeninhalt von Kegeln

Anwenden und Begründen von Formeln für den Oberflächeninhalt von (geraden) Prismen; Untersuchen von Auswirkungen von Längenänderungen auf Oberflächeninhalte.

Oberflächeninhalt von Prismen

Anwenden von Formeln zur Berechnung der Oberflächeninhalte von Pyramiden; Untersuchen von Auswirkungen von Längenänderungen auf Oberflächeninhalte. 

Oberflächeninhalt von Pyramiden

Anwenden und Begründen von Formeln zur Berechnung des Oberflächinhalts von Quadern; Untersuchen von Auswirkungen von Längenänderungen auf Oberflächeninhalte.

Oberflächeninhalt von Quadern

Describe a data set by its number of observations, what is being measured, and the units of measurement.

Osservazione, misura e campionamento

Solve one-step linear equations and inequalities and graph solutions of the inequalities on a number line in number and word problems.

Equazioni e disequazioni lineari elementari

Perform multi-step operations with rational numbers (positive and negative) in number and word problems.

Operazioni con numeri razionali

Use order of operations including parentheses and other grouping symbols to simplify numerical expressions.

Ordine delle operazioni

Identify outliers and clusters in bivariate data in tables and scatter plots.

Valori anomali e aggregazione di dati nei grafici di dispersione

Use properties, definitions, and theorems of angles and lines to solve problems related to adjacent, vertical, complementary, supplementary, and linear pairs of angles.

Coppie di angoli

Determine and interpret the components of algebraic expressions including terms, factors, variables, coefficients, constants, and parts of powers in number and word problems.

Parti di una espressione algebrica

Identify parts of a numerical or algebraic expression.

Parti di un'espressione

Risolvere problemi e riconoscere modelli ripetitivi

Solve number and word problems involving percent increase and percent decrease.

Aumento e diminuzione percentuale

Calculate or estimate the percent of a number including discounts, taxes, commissions, and simple interest.

Percentuale di un numero

Solve number and word problems using percent proportion, percent equation, or ratios.

Problemi con le percentuali

Determine perimeter using concrete models, nonstandard units, and standard units in number and word problems.

Perimetri

Investigate and determine the relationship between the diameter and the circumference of a circle and the value of pi; calculate the circumference of a circle.

Pi greco come misura della relazione tra diametro e circonferenza

Valore posizionale delle migliaia

Indicate and compare the place value of each digit in a multi-digit whole number or decimal.

Valore posizionale

Use coordinate geometry to confirm properties of plane figures.

Plane Figures on the Coordinate Plane

Write the equation of and graph linear relationships given two points on the line.

Equazione della retta per due punti

Write the equation of and graph linear relationships given the slope and one point on the line.

Equazione del fascio di rette per un punto

Punti sulla retta dei numeri

Distinguish between a population and a sample and draw conclusions about the sample (random or biased).

Popolazione o campione?

Make predictions based on results from surveys and samples.

Previsioni basate su dati campionari

Identify prime and composite numbers less than 100.

Numeri primi e composti

Scomposizione in fattori primi

Erklären und Anwenden des Begriffs Primzahl.

Primzahlen

Determine a simple probability using geometric figures.

Probabilità in geometria

Describe the probability of a chance event using a fraction or ratio.

Probabilità di un evento come frazione o rapporto

Determine the probability of compound events (with and without replacement).

Probabilità di eventi composti

Use a coordinate grid to solve number and word problems. Describe the path between given points on the plane.

Problemi nel piano cartesiano

Recognize and apply algebra techniques to solve rate problems including distance, work, density, and mixture problems.

Problemi con tassi e rapporti

Use the commutative and associative properties to add or multiply numerical expressions.

Proprietà di addizione e moltiplicazione

Verify how properties and relationships of geometric figures are maintained or how they change through transformations.

Proprietà delle figure geometriche

Interpret and compare properties of linear functions, graphs, and equations.

Proprietà, grafici ed equazioni di funzioni lineari

Use manipulatives, pictorial representations, and appropriate vocabulary (e.g., polygon, side, angle, vertex, diameter) to identify and compare properties of plane figures.

Proprietà delle figure piane

Use the commutative, associative, and distributive properties, and inverses and identities to solve number and word problems with rational numbers.

Proprietà dei numeri razionali

Write a proportion to model a word problem; solve proportions.

Proporzioni

Anwenden von Prozenten in Sachsituationen und Bearbeitung von Fragestellungen im Zusammenhang mit der Prozentrechnung; Beschreiben und Vergleichen von Lösungswegen; Kritische Betrachtung von Ergebnissen und ihrer Genauigkeit.

Prozente in Sachsituationen

Bearbeiten von vertiefenden Aufgaben zur Prozentrechnung in ein- und mehrstufigen Situationen (unter Verwendung von Änderungsfaktoren).

Prozentuelle Vermehrung und Verminderung

Use the Pythagorean Theorem and its converse to solve number and word problems, including finding the distance between two points.

Teorema di Pitagora e suo inverso

Write a quadratic equation or quadratic function given its zeros.

Quadratic Equations and Functions from Zeros

Determine the quartiles or interquartile range for a set of data.

Quartiles and Interquartile Range of Data

Espressioni con radicali

Use rules of exponents to rewrite or simplify expressions with rational exponents or radicals and interpret their meaning.

Radicali e potenze a esponente razionale

Campo di variazione di un insieme di dati

Determine the ratio or rate of change of a relation given a table or graph.

Individuare un rapporto o un tasso a partire da una tabella o un grafico

Read and write word names for rational numbers in decimal form to the hundredths place or the thousandths place.

Forma decimale dei numeri razionali

Describe the relationship between corresponding terms in two or more numerical patterns or tables of ratios

Rapporti e proporzioni

Anwenden und Begründen von Formeln zur Berechnung des Rauminhalts von Pyramiden; Untersuchen von Auswirkungen von Längenänderungen auf Rauminhalte. 

Rauminhalt von Pyramiden

Ermitteln von Rauminhalten von Quadern durch Zerlegen in passende Einheitswürfel; Anwenden und Begründen der Formel zur Berechnung des Volumens eines Quaders; Untersuchen von Auswirkungen von Längenänderungen auf Oberflächeninhalte. 

Rauminhalt von Quadern

Anwenden und Begründen von Formeln zur Berechnung des Rauminhalts von Zylindern.

Rauminhalt von Zylindern

Describe, use, and compare real numbers. Use the definition of rational numbers to derive and distinguish irrational numbers.

Numeri reali, razionali e irrazionali

Describe graphically, algebraically and verbally real-world phenomena as functions; identify the independent and dependent variables and any constraints of the domain or range.

Modellare fenomeni di vita reale con le funzioni

Angeben, Erklären und Anwenden von Rechengesetze (Distributiv-, Assoziativ-, Kommutativgesetz).

Rechengesetze

Beschreiben, Anwenden und Begründen elementarer Rechenregeln für Potenzen.

Rechenregeln für Potenzen

Angeben und Anwenden der Regeln √(a ∙ b) = √a ∙ √b und √(a / b) = √a / √b angeben und anwenden; Begründen, dass im Allgemeinen √(a ± b) ≠ √a ± √b ist.

Rechenregeln

Kennenlernen von Grundbegriffen (Grundwert, Prozentanteil, Prozentsatz); Rechnen mit Prozenten in vielfältigen Situationen; Verwenden von additiven und multiplikativen Berechnungen und Begründen entsprechender Zusammenhänge.

Rechnen mit Prozenten

Lösen von linearen Gleichungssystemen durch Berechnung; insbesondere durch Lösungsverfahren (Einsetzungsverfahren, Gleichsetzungsverfahren, Eliminationsverfahren).

Rechnerisches Lösen linearer Gleichungssysteme

Find indicated terms, the common ratio, or the common difference using recursive sequence formulas; write recursive sequence formulas.

Formule delle successioni ricorsive

Locate, given the coordinates of, and graph plane figures which are the results of translations or reflections in all quadrants of the coordinate plane.

Simmetrie nel piano cartesiano

Define and distinguish between relations and functions, dependent and independent variables, and domain and range; identify whether relations are functions numerically and graphically.

Relazioni e funzioni

Relate a percent to its equivalent fraction or decimal.

Relazioni tra percentuali, frazioni e numeri decimali

Use models to investigate the relationship of the volume of a cone to a cylinder and a pyramid to a prism with the same base and height.

Relazione tra il volume di cono e cilindro o tra il volume di piramide e prisma

Identify and draw intersecting, parallel, skew, and perpendicular lines and line segments. Identify midpoints of line segments.

Relazioni tra due rette e punto medio di un segmento

Convert between different representations of relations and functions using tables, the coordinate plane, and algebraic or verbal statements.

Rappresentazioni di relazioni e funzioni

Represent fractions concretely and symbolically, including representing whole numbers as fractions.

Rappresentare le frazioni

Represent multiplication or division of mixed numbers with and without models and pictures.

Rappresentare il prodotto e la divisione di numeri misti

Determine or calculate residuals of a distribution.

Residui di una distribuzione

Rewrite and compare decimals to fractions (tenths and hundredths) with and without models and pictures.

Riscrivere frazioni e numeri decimali

Use properties, definitions, and theorems to solve problems about rigid transformations and dilations of plane figures.

Trasformazioni rigide e dilatazioni di figure piane

Locate, given the coordinates of, and graph plane figures which are the results of rotations (multiples of 90 degrees) with respect to a given point.

Rotazioni nel piano cartesiano

Round decimals to a given place value; round fractions and mixed numbers to a whole number or a given fractional place value.

Arrotondare numeri decimali, frazioni e numeri misti

Round whole numbers to a given place value.

Arrotondare i numeri interi

Given a list of ordered pairs in a table or graph, identify either verbally or algebraically the rule used to generate and record the results.

Regole per generare coppie ordinate

Runden von Zahlen

Lesen und Schreiben von Zahlen in römischer Zahldarstellung; Erklären der Vorteile (bei Berechnungen) mit dem dezimalen Stellenwertsystem im Vergleich zur römischen Zahldarstellung.

Römische Zahlen

Bearbeiten von Sachaufgaben zu Quadern, insbesondere Berechnen von Massen und Dichten.

Sachaufgaben zu Quadern

Darstellen von Sachverhalten (Neigungen, Drehungen) durch Winkel; Erkennen von Größenbeziehungen zwischen Winkeln in geometrischen Figuren.

Sachaufgaben zu Winkeln

Bearbeiten von Aufgaben in Sachsituationen zum Maßstab; Arbeiten mit Plänen und Landkarten.

Sachaufgaben zum Maßstab

Sammeln und Ordnen von Daten nach verschiedenen Gesichtspunkten.

Sammeln und Ordnen von Daten

Erklären, Anwenden und Begründen des Satzes von Thales. 

Satz von Thales

Determine and use scale factors to reduce and enlarge drawings on grids to produce dilations.

Fattori di scala

Organize, display, and interpret information in scatter plots. Approximate a trend line and identify the relationship as positive, negative, or no correlation.

Grafici di dispersione

Skizzieren von Schrägrissen von Quadern.

Schrägrisse von Quadern

Schätzen von Wahrscheinlichkeiten mithilfe empirisch gewonnener relativer Häufigkeiten.

Schätzen von Wahrscheinlichkeiten

Schätzen von Winkelgrößen

Describe, or graph plane figures which are the results of a sequence of transformations.

Composizione di trasformazioni nel piano cartesiano

Use set notation to describe domains, ranges, intersection, and union of sets. Identify cardinality of sets, equivalent sets, disjoint sets, complement, or subsets.

Teoria degli insiemi

Verwendung eines intuitiven Wahrscheinlichkeitsbegriff zur Quantifizierung von Sicherheit.

Sichere, wahrscheinliche, unwahrscheinliche und unmöglich Ereignisse

Use proportional reasoning to solve problems related to similar polygons.

Poligoni simili

Use properties of triangles to solve problems related to similar triangles and the relationships of their corresponding parts.

Triangoli simili ed elementi corrispondenti

Identify corresponding parts of similar and congruent figures.

Similitudine e congruenza

Determine a simulation, such as random numbers, spinners, and coin tosses, to model frequencies for compound events.

Simulazioni per stimare la probabilità di eventi composti

Describe the slope of a line given in the context of a problem situation; compare rates of change in linear relationships represented in different ways.

Coefficiente angolare come tasso di variazione

Write the equation of and graph linear relationships given the slope and y-intercept.

Equazione della retta in forma esplicita

Find the slope of a line given two points on a line, a table of values, the graph of the line, or an equation of the line in number and word problems.

Pendenza di una retta

Identify from a set of numbers which values satisfy a given equation or inequality.

Risolvere disequazioni

Solve linear equations using the associative, commutative, distributive, and equality properties and justify the steps used.

Risolvere le equazioni lineari

Determine algebraically or graphically the solutions of a linear inequality in two variables.

Risolvere disequazioni lineari

Solve systems of equations or inequalities algebraically and graphically that include nonlinear relationships.

Risolvere sistemi di equazioni e disequazioni non lineari

Determine whether a system of equations has one solution, multiple solutions, infinitely many solutions, or no solution using graphs, tables, and algebraic methods; compare solutions of systems of equations.

Risolvere sistemi di equazioni lineari

Use geometric models and equations to investigate the meaning of the square of a number and the relationship to its positive square root. Know perfect squares to 625.

Quadrati e radici quadrate

Determine measures of spread (standard deviation).

Deviazione standard

Relate standard and expanded notation to 3- and 4-digit numbers.

Notazione standard e polinomiale di numeri a 3 o 4 cifre

Contrast statements about absolute values of integers with statements about integer order.

Espressioni con valori assoluti di numeri interi relativi

Erklären und Anwenden des Stellenwertsystems für natürliche Zahlen.

Stellenwerte von natürlichen Zahlen

Erklären und Anwenden des dezimalen Stellenwertsystems; Erklären und Anwenden des Prinzips „Bündeln in Zehner-Schritten“.

Stellenwerte von rationalen Zahlen

Organize, display, and interpret information in stem-and-leaf plots.

Stem and Leaf Plots

Beschreiben von grundlegenden Eigenschaften von Streckensymmetralen (Mittelsenkrechten); Konstruieren von Streckensymmetralen (Mittelsenkrechten).

Streckensymmetrale (Mittelsenkrechte)

Subtract 2- and 3-digit numbers with and without models for number and word problems that require regrouping.

Sottrarre numeri a 2 o 3 cifre

Subtrahieren von Bruchzahlen

Subtrahieren von ganzen Zahlen

Beschreiben und Anwenden des Subtraktionsalgorithmus für natürliche Zahlen.

Subtrahieren von natürlichen Zahlen

Subtrahieren von nichtnegativen Dezimalzahlen

Use nets or formulas to find the surface area of prisms, pyramids, and cylinders in number and word problems.

Aree delle superfici dei solidi

Calculate the surface area and volume of a sphere in number and word problems.

Area della superficie a volume di una sfera

Write and solve systems of linear equations in two or more variables algebraically in number and word problems.

Sistemi di equazioni lineari in due o più variabili

Sistemi di disequazioni lineari

Organize, display, and interpret information in tables and graphs (frequency tables, pictographs, and line plots).

Tabelle e grafici

Angeben und Anwenden von Teilbarkeitsregeln; Begründen von Teilbarkeitsregeln; Begründen und Widerlegen von Aussagen zur Teilbarkeit.

Teilbarkeitsregeln

Beschreiben von grundlegenden Eigenschaften von Kreisen (Kreislinien, Kreisflächen) und Kreisteilen (Kreisbogen, Kreissektor, Kreissegment); Zeichnen von Kreisteilen.

Teile eines Kreises

Erkennen und Ermitteln von Teilern von natürlichen Zahlen; Ermitteln des größten gemeinsamen Teilers (mit Rechenverfahren).

Teiler von Zahlen

Determine the probability from experimental results or compare theoretical probabilities and experimental results.

Probabilità teorica o probabilità sperimentale?

Identify the place value of each digit in a multi-digit number to the thousandths place.

Numeri decimali con millesimi

Transform (translate, reflect, rotate, dilate) polygons in the coordinate plane; describe the transformation in simple algebraic terms.

Trasformazioni nel piano cartesiano

Identify transformations on nonlinear parent functions using function notation, algebraic equations, or graphs.

Trasformazioni di funzioni non lineari

Use models or points in the coordinate plane to illustrate, recognize, or describe rigid transformations (translations, reflections, and rotations) of plane figures.

Trasformare figure piane

Locate, given the coordinates of, and graph points which are the results of rigid transformations in all quadrants of the coordinate plane; describe the path of the motion using geometric models or appropriate terms.

Trasformare punti nel piano cartesiano

Translate between models or verbal phrases and algebraic expressions.

Tradurre espressioni algebriche

Write a linear equation or inequality to represent a given number or word problem; solve.

Traslazione di equazioni e disequazioni lineari

Translate between models or verbal phrases and numerical expressions.

Translating numerical expressions

Recognize conditions of side lengths that determine a unique triangle, more than one triangle, or no triangle.

Disuguaglianza triangolare

Use trigonometric ratios to represent relationships in right triangles to solve number and word problems.

Rapporti goniometrici

Solve two-step linear equations and inequalities and graph solutions of the inequalities on a number line.

Equazioni e disequazioni lineari non elementari

Write number sentences using any combination of the four operations that represent a two-step word problem; solve.

Problemi con le quattro operazioni

Construct and interpret a two-way table to display two categories of data from the same source.

Tabelle a doppia entrata

Identify and draw angles (acute, right, obtuse, and straight).

Tipi di angolo

Distinguish between types of events (conditional, mutually exclusive, independent, dependent, etc.). Use the appropriate formula to determine probabilities of random phenomena using addition rule, multiplication rule, or Venn diagrams.

Classificazione degli eventi (condizionati, incompatibili, dipendenti, indipendenti, ...)

Name polygons by the number of sides. Distinguish quadrilaterals based on properties of their sides or angles; relate quadrilaterals based upon their hierarchical attributes.

Tipi di quadrilatero

Identify and classify triangles according to the measures of the interior angles and the lengths of the sides; relate triangles based upon their hierarchical attributes.

Tipi di triangoli

Anwenden der Formel für die Berechnung des Umfangs von Kreisen, insbesondere auf Halbkreise, Viertelkreise und Achtelkreise.

Umfang eines Kreises

Anwenden und Begründen einer Formel für den Umfang von Rechtecken.

Umfang von Rechtecken

Umformen von Termen durch Ausmultiplizieren, Zusammenfassen, Herausheben oder Kürzen. 

Umformen von Termen

Lösen von Umkehraufgaben zu Kreisen, inbesondere durch Umformen von Formeln.

Umkehraufgaben zu Kreisen

Lösen von Umkehraufgaben zu Quadern, insbesondere durch Umformen von Formeln.

Umkehraufgaben zu Quadern

Lösen von Umkehraufgaben zu Rechtecken, insbesondere durch Umformen von Formeln.

Umkehraufgaben zu Rechtecken

Lösen von Umkehraufgaben zu Vierecken, insbsondere durch Umformen von Formeln. 

Umkehraufgaben zu Vierecken

Umschreiben eines Vierecks mit einem Kreis und Einschreiben eines Kreises in ein Viereck. Konstruieren von Umkreis und Innkreis von Rechteck und Quadrat.

Umkreis und Innkreis von Vierecken

Umrechnen von Flächenmaßen

Umrechnen von Längenmaßen

Umrechnen von Massenmaßen

Ausführen von Maßumrechnungen von Geldbeträgen.

Umrechnung von Geldbeträgen

Umrechnung von Zeitmaßen

Convert measures of length, area, capacity, weight, and time expressed in a given unit to other units in the same measurement system in number and word problems.

Conversione di unità di misura di lunghezza, area, capacità, peso e tempo

Use models to represent a fraction as a product of a whole number and a unit fraction in number and word problems.

Frazioni unitarie

Given a proportional relationship represented by tables, graphs, models, or algebraic or verbal descriptions, identify the unit rate (constant of proportionality)

Tasso unitario e costante di proporzionalità

Calculate unit rates in number and word problems, including comparison of unit rates.

Tasso unitario

Calculate unit rates of ratios that include fractions to make comparisons in number and word problems

Tasso unitario e frazioni

Use benchmark numbers (zero, one-half, one) and models to compare and order fractions.

Utilizzare i numeri di riferimento con le frazioni

Solve quadratic equations using properties of equality.

Risolvere equazioni di 2° grado utilizzando le proprietà

Use slopes to determine if two lines are parallel or perpendicular.

Utilizzare il coefficiente angolare per determinare rette parallele e perpendicolari

Solve quadratic equations using the quadratic formula.

Risolvere equazioni di 2° grado con la formula risolutiva

Use definitions and theorems of angles formed when a transversal intersects parallel lines.

Teoremi sugli angoli formati da rette parallele e trasversali

Make predictions based on theoretical probabilities or experimental results.

Utilizzare la probabilità teorica e sperimentale per fare previsioni

Find the value of a variable in a number sentence.

Valore di una variabile in un'espressione numerica

Analyze graphs, identify situations, or solve problems with varying rates of change.

Tasso di variazione variabile

Venn Diagrams

Veranschaulichen der Addition; Deutung des Addierens als Zusammenfügen von oder Weiterzählen bei natürlichen Zahlen; Deuten und Darstellen der Addition der Form z + n mit  z ϵ ℤ und n ϵ ℕ als Bewegungen auf der Zahlengeraden; diese Deutung verwenden, um entsprechende Rechenaufgaben zu lösen.

Veranschaulichung und Deutung der Addition

Veranschaulichen der Division; Deutung des Dividierens als Teilen, als Messen oder als Umkehren des Multiplizierens.

Veranschaulichung und Deutung der Division

Veranschaulichen der Multiplikation; Deutung des Multiplizierens mit natürlichen Zahlen als wiederholtes Addieren derselben Zahl oder als Vervielfachen; Deutung des Multiplizierens mit Bruchzahlen als Teilen und nachfolgendes Vervielfachen, als Vervielfachen und nachfolgendes Teilen oder als Ermitteln eines relativen Anteils.

Veranschaulichung und Deutung der Multiplikation

Veranschaulichen der Subtraktion; Deutung des Subtrahierens als Abziehen, als Ergänzen, als Ermitteln des Unterschieds oder als Umkehren des Addierens; Deuten und Darstellen der Subtraktion der Form z - n mit  z ϵ ℤ und n ϵ ℕ als Bewegungen auf der Zahlengeraden; diese Deutung verwenden, um entsprechende Rechenaufgaben zu lösen.

Veranschaulichung und Deutung der Subtraktion

Vergleichen und Messen von Flächeninhalten.

Vergleichen und Messen von Flächen

Vergleichen und Messen von Geldbeträgen.

Vergleichen und Messen von Geldbeträgen

Vergleichen und Messen von Längen

Vergleichen und Messen von Massen

Vergleichen und Messen von Rauminhalten

Vergleichen und Messen von Zeitspannen

Vergleichen und Ordnen von ganzen Zahlen (mithilfe der Zahlengerade); Interpretieren von ganzen Zahlen auch in außermathematischen Situationen.

Vergleichen und Ordnen von ganzen Zahlen

Vergleichen und Ordnen natürlicher Zahlen sowie nichtnegativer Dezimal- und Bruchzahlen.

Vergleichen und Ordnen von nichtnegativen Dezimal- und Bruchzahlen

Vergleichen und Ordnen von rationalen Zahlen; Darstellen von rationalen Zahlen auf der Zahlengeraden.

Vergleichen und Ordnen von rationalen Zahlen

Erkennen und Ermitteln von gemeinsamen Vielfache von natürlichen Zahlen; Ermitteln des kleinsten gemeinsame Vielfachen (mit Rechenverfahren).

Vielfache von Zahlen

Model the concept of the volume of a solid figure using cubic units.

Volume di un cubo e unità di volume

Determine the volume of composite figures in number and word problems.

Volume di figure composte

Calculate the volume of cylinders, pyramids, and cones in number and word problems.

Volume dei solidi

Wechseln zwischen Bruch- und (endlicher bzw. periodischer) Dezimaldarstellung.

Wechsel zwischen Bruch- und Dezimalzahldarstellung

Wechseln zwischen Bruch-, Dezimal- und Prozentdarstellung.

Wechsel zwischen Bruch-, Dezimalzahl- und Prozentdarstellung

Begründen und anwenden, dass die Winkelsumme in jedem Dreieck 180° beträgt.

Winkelsumme im Dreieck

Describe the meaning of an unknown in the context of a word problem.

Problemi con una incognita

Darstellen von Zahlen unter Verwendung von Zehnerpotenzen und Anwenden der Gleitkommadarstellung.

Zehnerpotenzen und Gleitkommadarstellung

Zeichnen von Winkeln

Identify and interpret zeros of a quadratic function using factoring in algebraic and word problems.

Zeros of Quadratic Functions

Beschreiben grundlegender Eigenschaften von ähnlichen Figuren; Interpretieren und Anwenden von Ähnlichkeitsfaktoren. 

Ähnlichkeit

Überprüfen von Achsensymmetrie einer ebenen Figur. 

Überprüfen von Achsensymmetrie

Überprüfen, ob ein Dreieck rechtwinklig ist durch Umkehrung des pythagoräischen Lehrsatzes. 

Überprüfen von Rechtwinkligkeit in Dreiecken

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Algebra illustration featuring a computer displaying graph of a mathematical function.

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Discover {1} free and ready-to-use GeoGebra resources for grades 4-8 to enhance your understanding of converting measures of length, area, capacity, weight, and time to different units. Learn to effectively measure and compare lengths, capacity, and weight through practical exercises in number and word problems.

Learn Units and Measurement - GeoGebra Math Resources

Learn and practice calculating the volume of solids with interactive resources from GeoGebra.

Discover more than {1} free and high-quality GeoGebra resources for grades 4-8 to practice finding and calculating the volume of 3D shapes such as cylinders, prisms, pyramids, cones, and composite figures.

分数を図で表そう

分数を図で表そう

学んだことをまとめましょう

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