Determina la forma simplificada de una expresión racional con variables, expresada sin potencias negativas.

Simplificar expresiones racionales con variables

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Sekundarstufe 1

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Gå in i en lektion

Rita grafer gör beräkningar med 3D-funktioner

Utforska funktioner, lös ekvationer, konstruera geometriska former

Undersök geometriska begrepp och konstruktioner i en dynamisk miljö

Visualisera ekvationer och funktioner med interaktiva grafer

Undersök en anteckningsapp med verktyg för interaktiva grafer, bildspel, bilder och mycket mer

Utför beräkningar med bråk, statistik och exponentialfunktioner

Räknare

Använd symboler för att lösa ekvationer och se mönster

Träna på grundläggande operationer som addition, subtraktion and division

Undersök talföljder och serier, derivator och integraler, lös differentialekvationer.

Hitta relationer mellan indata och utdata i matematik

Undersöka former, storlekar och rumsliga relationer i matematik

Förstå tal, deras relationer och numeriskt resonemang

Utför matematiska beräkningar som addition, subtraktion, division

Analysera osäkerhet och sannolikheten för händelser

Studera vinklar, trianglar och trigonometriska förhållanden

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Flervalsfråga {1}

Öppen fråga {1}

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.

Absolute Value Expressions

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

Absolute Value of Integers

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.

Adding and Subtracting Angles

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

Adding and Subtracting Decimals

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

Adding and Subtracting Fractions or Mixed Numbers with Like Denominators

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

Adding and Subtracting Fractions and Mixed Numbers with Unlike Denominators

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

Adding and Subtracting Fractions and Mixed Numbers

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

Adding and Subtracting Integers

Adding, Subtracting, and Multiplying Polynomials

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

Additive and Multiplicative Inverse

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.

Angle Sum Properties of Triangles, Quadrilaterals, and Polygons

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.

Angles

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.

Arc, Chord, and Tangent Properties of Circles

Estimate and calculate areas with scale drawings and maps.

Area in Scale Drawings and Maps

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

Area of Rectangles, Squares, and Composite Figures

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 of Rectangles with Grids

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 and Volume of Similar Geometric Figures

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

Arithmetic and Geometric Sequences

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.

Bar Graphs

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

Nachvollziehen eines Beweises des pythagoräischen Lehrsatzes.

Beweisen des Satzes des Pythagoras

Bilden und Benennen von Winkeln

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

Box and Whisker Plots

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.

Calculating Using Scientific Notation

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.

Certain, Likely, Unlikely, Impossible Events

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

Characteristics of Functions and Their Graphs

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

Choosing Scales and Intervals for Graphs

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

Circle Graphs

Identify combinations of fractions that make one whole.

Combinations of Fractions

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

Comparing Data Distributions

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

Comparing Decimals

Compare and order fractions using common numerators or denominators.

Comparing Fractions

Compare and order integers with and without models.

Comparing Integers

Compare and order numbers less than 10,000.

Comparing Numbers Less Than 10,000

Write a ratio or rate to compare two quantities

Comparing Quantities with Ratios

Compare and order rational numbers with and without models.

Comparing Rational Numbers

Complement of an Event

Define and identify complementary and supplementary angles.

Complementary and Supplementary Angles

Solve quadratic equations by completing the square.

Completing the Square to Solve Quadratic Equations

Complex Fractions

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

Composition and Inverse of Functions

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

Converting Between Percents and Fractions or Decimals

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

Coordinate Plane and Ordered Pairs

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.

Coordinate Plane

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

Correlation Coefficient, Correlation and Causation- Differences, Design and Examples

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

Corresponding Parts of Similar Figures

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

Cost of Items and Making Change

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

Counting Strategies for Sample Spaces

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.

Cube and Cube Roots

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

Curves of Best Fit

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.

Data, Data Distribution, and Central Tendency

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

Decimals Greater Than One Million

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

Degree and Radian Measure of Angles

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

Dimensional Analysis

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

Direct Variation

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

Discriminant and Roots of a Quadratic Equation

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.

Distance Formula and Midpoint Formula

Calculate distances from scale drawings and maps.

Distance in Scale Drawings and Maps

Use the distributive property to represent and simplify numerical expressions.

Distributive Property

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.

Dividing Fractions and Mixed Numbers

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

Dividing Fractions

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.

Dividing Unit Fractions

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

Division Models

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

Domain and Range of Special Functions

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

Drawing and Measuring Angles with a Protractor

Durchführen der Primfaktorenzerlegung

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

Effect of Operations on Numbers

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

Effect of Outliers on Mean, Median, and Range of Data Sets

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

Effects of Changes in Slope and Intercepts of Lines

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

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

Einzeichnen von Punkten im Koordinatensystem

Elapsed Time

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

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

Equations in Two Variables

Solve equations involving powers and roots by using inverse relationships.

Equations Involving Powers and Roots

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

Equivalent Forms of Decimals and Fractions

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

Equivalent Fractions and Decimals

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

Equivalent Fractions Models

Write equivalent fractions with smaller or larger denominators.

Equivalent Fractions

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

Equivalent Ratios

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.

Error of a Measuring Tool

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.

Estimating Decimal Sums and Differences

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

Estimating Measurements

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

Estimating and Measuring Capacity

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

Estimating and Measuring Lengths

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

Estimating and Measuring Weights

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

Estimating Products of Whole Numbers

Estimate products and quotients of decimals or of mixed numbers.

Estimating Products and Quotients of Decimals and Mixed Numbers

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

Estimating Products and Quotients

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

Estimating Quotients of Whole Numbers

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.

Estimating Square Roots to the Nearest Integer

Estimate sums and differences with fractions and mixed numbers.

Estimating Sums and Differences of Fractions and Mixed Numbers

Apply appropriate type of estimation for sums and differences.

Estimating Sums and Differences

Evaluate algebraic expressions in number and word problems.

Evaluating Algebraic Expressions

Solve exponential equations by rewriting expressions with like bases.

Exponential Equations

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

Exponential Expressions

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.

Exponential Functions- Equation, Graph and Critical Values

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

Exponential Functions

Simplify numerical expressions that may contain exponents.

Expressions with Exponents

Factor quadratic polynomials, including special products.

Factoring Quadratic Polynomials

Understand that many whole numbers factor in different ways.

Factors of Whole Numbers

Identify and make figures with line symmetry.

Figures with Symmetry

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

Finding the Area of a Triangles, Parallelograms, Trapezoids, Circles, Composite Figures

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

Finding the Fractional Part

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

Finding the Rule of a Number Pattern

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.

Finding Volume with Base and Height

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 for Perimeter

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

Formulas for Area of Triangles, Parallelograms, Trapezoids, and Circles

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

Formulas for Lateral Surface Area of Solids

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

Fraction as Division

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

Frequency Tables and Dot Plots

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

Frequency Tables, Line Plots and Bar Graphs

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

Functions and Function Notation

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.

Generating Ordered Pairs and Sequences

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.

Graphing Absolute Value Functions and Inequalities

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

Graphing Linear Functions Using Ordered Pairs

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

Graphing Quadratic Functions

Graphically solve systems of linear equations.

Graphing Systems of Linear Equations

Graphing to Solve Quadratic Equations

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

Graphs of Quadratic Functions

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

Graphs with Multiple Sets of Information

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

Graphs with Scales That Represent Multiple Units

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

Greatest Common Factor

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

Histograms vs. Bar Graphs

Organize, display, and interpret information in histograms.

Histograms

Identify proportional parts of similar figures using rulers and protractors.

Identifying Proportional Parts Using Measurement

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.

Increasing and Decreasing Functions

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

Inequalities Using Symbolic Notation >, <, ≤, ≥, and ≠

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

Inferences From Samples

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.

Integers

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

Intercepts of Linear Relations

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

Interior and Exterior Angles of Polygon

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

Interpreting Remainders

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

Introduction to Geometry

Use models to investigate the concept of the Pythagorean Theorem.

Investigating the Pythagorean Theorem

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

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.

Laws of Exponents

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

Least Common Multiple

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

Line Plots with Fractional Scales

Write a problem given a simple linear equation or inequality.

Linear Equations and Inequalities in Word Problems

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

Linear Inequalities in Problems

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

Linear, Nonlinear, and Directly Proportional Relations

Identify linear and nonlinear relationships in data sets.

Linear and Nonlinear Relationships in Data Sets

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

Lines and Angles

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

Lines of Best Fit

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

Lines of Symmetry in Plane Figures

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

Lines of Symmetry

Solve a literal equation for an indicated variable.

Literal Equation for an Indicated Variable

Divide using single-digit divisors with and without remainders.

Long Division

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.

Measures of Variability

Median of Data Sets

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

Mode of Data Sets

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

Modeling Data and Making Predictions

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

Modeling with Tables

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

Money Math

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

Multi-digit Addition with Regrouping

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

Multi-digit Subtraction with Regrouping

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

Multiplication and Division Sentences

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.

Multiplying a 1-digit by a 2-digit Number

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

Multiplying and Dividing by Powers of 10

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

Multiplying and Dividing Decimals with Models

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

Multiplying and Dividing Decimals

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

Multiplying and Dividing Integers

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

Multiplying and Dividing Mixed Numbers

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

Multiplying Fractions

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

Multiplying Multi-digit Numbers

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

Multi-Step Problems

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

Nets of Solids

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.

Number of Solutions to System of Equations

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

Numbers from 1,000 to 9,999

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

Numbers from 10,000 to One Million

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

Numbers from 101 to 999

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

Numbers in Expanded Form

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

Numbers in Scientific Notation

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.

Observation, Measurement, and Sampling

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

One-step Linear Equations and Inequalities

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

Operations on Rational Numbers

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

Order of Operations

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

Outliers and Clusters in Scatter Plots

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

Pairs of Angles

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

Parts of an Algebraic Expression

Identify parts of a numerical or algebraic expression.

Parts of an Expression

Patterns and Problem Solving

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

Percent Increase and Decrease

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

Percent of a Number

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

Percent Problems

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

Perimeter

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 as a Relationship between Diameter and Circumference

Place Value with Thousands

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

Place Value

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

Point Point Form of a Line

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

Point Slope Form of Line

Points on a Number Line

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

Population vs. Sample

Make predictions based on results from surveys and samples.

Predictions From Samples

Identify prime and composite numbers less than 100.

Prime and Composite Numbers

Prime Factorization

Erklären und Anwenden des Begriffs Primzahl.

Primzahlen

Determine a simple probability using geometric figures.

Probabilities in Geometry

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

Probability of an Event as Fractions or Ratios

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

Probability of Compound Events

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

Problems in the Coordinate Plane

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

Problems Involving Rates

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

Properties of Addition and Multiplication

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

Properties of Geometric Figures

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

Properties of Linear Functions, Graphs, and Equations

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

Properties of Plane Figures

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

Properties of Rational Numbers

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

Proportions

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.

Pythagorean Theorem and its Converse

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

Radical Expressions

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

Radicals and Rational Exponents

Range of Data Sets

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

Ratio or Rate Given a Table or Graph

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

Rational Numbers in Decimal Form

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

Ratios and Proportions

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.

Real, Rational, and Irrational Numbers

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

Real-World Phenomena as Functions

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

Rechengesetze

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.

Recursive Sequence Formulas

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

Reflections in the Coordinate Plane

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

Relations and Functions

Relate a percent to its equivalent fraction or decimal.

Relationship Between Percent and Fraction or Decimal

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.

Relationship Between Volume of a Cone and a Cylinder or a Pyramid and a Prism

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

Relationships Between Two Lines and Midpoints of Line Segments

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

Representations of Relations and Functions

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

Representing Fractions

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

Representing Multiplying and Dividing Mixed Numbers

Determine or calculate residuals of a distribution.

Residuals of a Distribution

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

Rewriting Fractions and Decimals

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

Rigid Transformations and Dilations of Plane Figures

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.

Rotations in the Coordinate Plane

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

Rounding Decimals, Fractions, and Mixed Numbers

Round whole numbers to a given place value.

Rounding Whole Numbers

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.

Rules for Generating Ordered Pairs

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

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.

Scale Factors

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

Scatter Plots

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.

Sequence of Transformations on the Coordinate Plane

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

Set Theory

Use proportional reasoning to solve problems related to similar polygons.

Similar Polygons

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

Similar Triangles and Correspondence

Identify corresponding parts of similar and congruent figures.

Similarity and Congruence

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

Simulations to Estimate Probabilities of Compound Events

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.

Slope as Rate of Change

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

Slope Intercept Form of a Line

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.

Slope of a Line

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

Solving Inequalities

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

Solving Linear Equations

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

Solving Linear Inequalities

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

Solving Nonlinear Systems of Equations and Inequalities

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.

Solving Systems of Linear Equations

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.

Squares and Square Roots

Determine measures of spread (standard deviation).

Standard Deviation

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

Standard and Expanded Notation of 3 and 4-digit Numbers

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

Statements with Absolute Values of Integers

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.

Subtracting 2- and 3-digit Numbers

Subtrahieren von Bruchzahlen

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.

Surface Area of Solids

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

Surface Area and Volume of a Sphere

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

Systems of Linear Equations in Two or More Variables

Systems of Linear Inequalities

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

Tables and Graphs

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.

Theoretical vs. Experimental Probability

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

Thousandths Place in Decimals

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

Transformations in the Coordinate Plane

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

Transformations on Nonlinear Parent Functions

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

Transforming Plane Figures

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.

Transforming Points in the Coordinate Plane

Translate between models or verbal phrases and algebraic expressions.

Translating Algebraic Expressions

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

Translating Linear Equations and Inequalities

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.

Triangle Inequality Theorem

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

Trigonometric Ratios

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

Two-step Linear Equations and Inequalities

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

Two-Step Problems Using the Four Operations

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

Two-way tables

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

Types of Angles

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

Types of Quadrilaterals

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.

Types of Triangles

Anwenden und Begründen einer Formel für den Umfang von Rechtecken.

Umfang von Rechtecken

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

Umkehraufgaben zu Kreisen

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

Umkehraufgaben zu Rechtecken

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.

Unit Conversion for Length, Area, Capacity, Weight, and Time

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

Unit Fractions

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

Unit Rate and the Constant of Proportionality

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

Unit Rate

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

Unit Rates with Fractions

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

Using Benchmark Numbers with Fractions

Solve quadratic equations using properties of equality.

Using Properties of Equality to Solve Quadratic Equations

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

Using Slope to Determine Parallel and Perpendicular Lines

Solve quadratic equations using the quadratic formula.

Using the Quadratic Formula to Solve Quadratic Equations

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

Using Theorems for Angles Formed by Parallel Lines

Make predictions based on theoretical probabilities or experimental results.

Using Theoretical and Experimental Probability to Make Predictions

Find the value of a variable in a number sentence.

Value of a Variable in a Number Sentence

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

Variable Rate of Change

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 of a Cube and Cubic Units

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

Volume of Composite Figures

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

Volume of Solids

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.

Word Problems with One Unknown

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

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Learn Absolute Value - GeoGebra Math Resources

Learn and practice adding numbers and perfect your addition skills with interactive resources from GeoGebra.

Explore over {1} free and ready-to-use GeoGebra resources for adding numbers for grades 4-8. Improve your skills with commutative, associative, & identity properties.

Learn Addition - GeoGebra Math Resources

Learn and practice identifying, drawing and, measuring different types of angles and become an expert in angles with interactive resources from GeoGebra.

Discover over {1} free and ready-to-use GeoGebra resources for grades 4-8 to learn and practice different types of angles. Improve your skills in drawing and measuring angles with a protractor.

Learn Angles - GeoGebra Math Resources

Learn and practice finding the area of plane figures with interactive resources from GeoGebra.

Access over {1} free and ready-to-use GeoGebra resources for grades 4-8 to learn how to develop formulas for finding area of rectangles, squares, triangles, parallelograms, trapezoids, circles & composite shapes.

Learn Area - GeoGebra Math Resources

Learn and practice solving problems involving circles with interactive resources from GeoGebra.

Discover more than {1} free and high-quality GeoGebra resources for grades 4-12 to practice solving problems about circles and the relationships between their parts, including the center, radius, circumference, arcs, angles, and more.

Learn Circles - GeoGebra Math Resources

Lerne und übe die Konzepte zum Thema Kreis mit interaktiven Unterrichtsmaterialien von GeoGebra.

Entdecke über {1} kostenlose und sofort verwendbare GeoGebra-Unterrichtsmaterialien zum Thema Kreis für alle Schulstufen, perfekt zum Entdecken und Üben.

Kreis verstehen & anwenden - Interaktive Mathe-Materialien | GeoGebra

Entdecke Kegel und festige dein Wissen mit interaktiven Unterrichtsmaterialien von GeoGebra.

Entdecke über {1} kostenlose und sofort verwendbare GeoGebra-Unterrichtsmaterialien zum Thema Kegel für alle Schulstufen, perfekt zum Entdecken und Üben.

Kegel verstehen & anwenden - Interaktive Mathe-Materialien | GeoGebra

Learn and practice identifying similar and congruent figures with interactive resources from GeoGebra.

Discover over {1} free and ready-to-use GeoGebra resources to practice identifying corresponding parts of similar & congruent figures for grades 4-8. Learn to use proportions to express relationships.

Learn Congruence and Similarity  - GeoGebra Math Resources

Learn and practice locating points on a number line and in the coordinate plane with interactive resources from GeoGebra.

Access over {1} free and ready-to-use GeoGebra resources for grades 4-8 to practice locating and naming points in the coordinate grid quadrants and solving number & word problems.

Learn Coordinates - GeoGebra Math Resources

Entdecke Quader und festige dein Wissen mit interaktiven Unterrichtsmaterialien von GeoGebra.

Entdecke über {1} kostenlose und sofort verwendbare GeoGebra-Unterrichtsmaterialien zum Thema Quader für alle Schulstufen, perfekt zum Entdecken und Üben.

Quader verstehen & anwenden - Interaktive Mathe-Materialien | GeoGebra

Entdecke Zylinder und festige dein Wissen mit interaktiven Unterrichtsmaterialien von GeoGebra.

Entdecke über {1} kostenlose und sofort verwendbare GeoGebra-Unterrichtsmaterialien zum Thema Zylinder für alle Schulstufen, perfekt zum Entdecken und Üben.

Zylinder verstehen & anwenden - Interaktive Mathe-Materialien | GeoGebra

Learn and practice working with decimal numbers with interactive resources from GeoGebra.

Discover over {1} free and ready-to-use GeoGebra resources for grades 4-8 to practice comparing, adding, subtracting, multiplying & dividing decimal numbers.

Learn Decimals - GeoGebra Math Resources

Learn and practice using various diagrams, such as histograms, box & whisker plots, scatter plots, line plots, dot plots, and frequency tables with interactive resources from GeoGebra.

Discover over {1} free and ready-to-use GeoGebra resources for grades 4-8 to practice using different diagrams like histograms, box & whisker plots, scatter plots, line plots, dot plots & frequency tables to organize, display & interpret information.

Learn Diagrams - GeoGebra Math Resources

Learn and practice division skills and improve your understanding of dividing numbers and solving division problems with interactive resources from GeoGebra.

Explore over {1} free and ready-to-use GeoGebra resources for grades 4-8 to improve your skills in estimating and solving division problems and using rules for divisibility.

Learn Division - GeoGebra Math Resources

Learn and practice solving equations and systems of linear equations with interactive resources from GeoGebra.

Discover over {1} free and ready-to-use GeoGebra resources for grades 4-8 to practice solving linear equations, equations with powers/roots, and word problems with one unknown. Learn how to solve systems of linear equations.

Learn Equations - GeoGebra Math Resources

Entdecke und übe das Runden und Abschätzen mit interaktiven Unterrichtsmaterialien von GeoGebra.

Entdecke über {1} kostenlose und sofort verwendbare GeoGebra-Unterrichtsmaterialien zum Thema Runden und Abschätzen, perfekt zum Entdecken und Üben.

Runden und Abschätzen leicht gemacht - Interaktive Mathe-Materialien | GeoGebra

Learn and practice interpreting and graphing exponential functions with interactive resources from GeoGebra.

Explore over {1} free and ready-to-use GeoGebra resources for grades 4-12 to enhance your understanding of exponential functions including its graphs, equations, and behavior.

Learn Exponential Functions - GeoGebra Math Resources

Learn and practice evaluating exponents and exponent expressions with interactive resources from GeoGebra.

Access over {1} free and ready-to-use GeoGebra resources for grades 4-8 to learn and practice describing & simplifying exponential expressions using the rules of exponents.

Learn Exponents - GeoGebra Math Resources

Learn and practice simplifying expressions with properties with interactive resources from GeoGebra.

Discover over {1} free and ready-to-use GeoGebra resources for grades 4-8 to practice translating & evaluating math expressions and learning the commutative, associative, and distributive properties.

Learn Expressions - GeoGebra Math Resources

Learn and practice finding multiples, common multiples, factors, and common factors of numbers with interactive resources from GeoGebra.

Get access to over {1} free and ready-to-use GeoGebra resources for grades 4-8 to improve your skills in finding multiples, common multiples, factors, common factors, least common multiples, and greatest common factor of numbers.

Learn Factors and Multiples- GeoGebra Math Resources

Lerne und übe den Umgang mit Formeln mit interaktiven Unterrichtsmaterialien von GeoGebra.

Entdecke über {1} kostenlose und sofort verwendbare GeoGebra-Unterrichtsmaterialien zum Thema Formeln für alle Schulstufen, perfekt zum Entdecken und Üben.

Formeln verstehen & anwenden - Interaktive Mathe-Materialien | GeoGebra

Learn and practice operating with fractions with interactive resources from GeoGebra.

Explore over {1} free and ready-to-use GeoGebra resources for grades 4-8 to learn and practice adding, subtracting, multiplying, and dividing fractions.

Learn Operations with Fractions - GeoGebra Math Resources

Learn and practice fraction comparison and ordering skills with interactive resources from GeoGebra.

Access over {1} free and ready-to-use GeoGebra resources for grades 4-8 to learn and practice comparing, ordering fractions and identifying equivalent ones.

Learn Fraction Sense - GeoGebra Math Resources

Learn and practice identifying and graphing different types of functions with interactive resources from GeoGebra.

Discover over {1} free and ready-to-use GeoGebra resources for grades 4-8 to learn and practice the distinction between relations and functions. Learn to graph functions and solve number and word problems.

Learn Functions - GeoGebra Math Resources

Learn and practice solving and graphing inequalities with interactive resources from GeoGebra.

Discover {1} free and ready-to-use GeoGebra resources to practice solving one-step & two-step inequalities for grades 4-8. Graph solutions on number line in word & number problems.

Learn Inequalities - GeoGebra Math Resources

Learn and practice working with integers with interactive resources from GeoGebra.

Discover over {1} free and ready-to-use GeoGebra resources for grades 4-8 to practice comparing & ordering integers using positive & negative numbers in word & number problems.

Learn Integers - GeoGebra Math Resources

Learn and practice interpreting and graphing linear functions with interactive resources from GeoGebra.

Find over {1} free and ready-to-use GeoGebra resources for grades 4-12 to enhance your understanding of linear functions and the effects of changes in slope and intercepts.

Learn Linear Functions - GeoGebra Math Resources

Learn and practice identifying linear, non-linear, and directly proportional relationships with interactive resources from GeoGebra.

Explore over {1} free and ready-to-use GeoGebra resources for grades 4-8 to practice approximating linear models and identifying linear, non-linear, and directly proportional relations in data sets.

Learn Linear Regression - GeoGebra Math Resources

Entdecke und übe unterschiedliche Massenmaße mit interaktiven Unterrichtsmaterialien von GeoGebra.

Entdecke über {1} kostenlose und sofort verwendbare GeoGebra-Unterrichtsmaterialien zum Thema Massenmaße für alle Schulstufen, perfekt zum Entdecken und Üben.

Massenmaße verstehen & anwenden - Interaktive Mathe-Materialien | GeoGebra

Learn and practice working with mixed numbers with interactive resources from GeoGebra.

Discover over {1} free and ready-to-use GeoGebra resources to explore and practice working with mixed numbers for grades 4-8. Learn to model and identify mixed numbers, and develop skills in adding, subtracting, multiplying, and dividing them.

Learn Mixed Numbers - GeoGebra Math Resources

Entdecke und übe die Zusammenhänge zwischen Geld und Mathematik mit interaktiven Unterrichtsmaterialien von GeoGebra.

Entdecke über {1} kostenlose und sofort verwendbare GeoGebra-Unterrichtsmaterialien zum Thema Geld für alle Schulstufen, perfekt zum Entdecken und Üben.

Geld verstehen & anwenden - Interaktive Mathe-Materialien | GeoGebra

Learn and practice multiplying numbers and solving multiplication problems with interactive resources from GeoGebra.

Discover over {1} free and ready-to-use GeoGebra resources for grades 4-8 to practice multiplying multi-digit numbers & using commutative & associative properties of multiplication.

Learn Multiplication - GeoGebra Math Resources

Learn and practice working with natural numbers with interactive resources from GeoGebra.

Discover over {1} free and ready-to-use GeoGebra resources for grades 4-8 to practice reading, writing, and comparing natural numbers, identifying prime & composite numbers, and representing numbers in various forms

Learn Natural Numbers - GeoGebra Math Resources

Entdecke und übe das Zahlenverständnis mit interaktiven Unterrichtsmaterialien von GeoGebra.

Entdecke über {1} kostenlose und sofort verwendbare GeoGebra-Unterrichtsmaterialien zum Thema Zahlenverständnis für alle Schulstufen, perfekt zum Entdecken und Üben.

Zahlenverständnis vertiefen - Interaktive Mathe-Materialien | GeoGebra

Learn and practice order of operations with interactive resources from GeoGebra.

Discover {1} free and ready-to-use GeoGebra resources for grades 4-8 to practice order of operations and simplify expressions with parentheses. Enhance your mathematical skills now!

Learn Order of Operations - GeoGebra Math Resources

Learn and practice identifying and generating numerical and geometric patterns and enhance your pattern recognition skills with interactive resources from GeoGebra.

Enhance your understanding of patterns and relationships with our collection of over {1} free and ready-to-use GeoGebra resources for grades 4-8. Explore and practice identifying or generating numerical & geometric patterns. Use table of values to solve problems associated with a given relationship with our interactive and engaging resources.

Learn Algebraic Patterns - GeoGebra Math Resources

Learn and practice calculating percentages and solving number and word problems with interactive resources from GeoGebra.

Discover {1} free and ready-to-use GeoGebra resources for grades 4-8 to learn and practice calculating percentages, solving percent increase/decrease, percent proportion, percent equation, and ratios problems.

Learn Percentages - GeoGebra Math Resources

Learn and practice finding perimeter of figures & circumference of circles with interactive resources from GeoGebra.

Find over {1} free and ready-to-use GeoGebra resources for grades 4-8 to learn and practice perimeter & circumference of figures and circles, using formulas and the value of pi. Explore the relationship between diameter, circumference and pi.

Learn Perimeter - GeoGebra Math Resources

Learn and practice identifying place value of numbers with interactive resources from GeoGebra.

Discover {1} free and ready-to-use GeoGebra resources for grades 4-8 to learn & practice place value of multi-digit numbers and rounding whole numbers to a specified place value.

Learn Place Value - GeoGebra Math Resources

Learn and practice identifying, drawing and naming geometric shapes like points, rays, lines, line segments, and planes with interactive resources from GeoGebra.

Discover over {1} free and ready-to-use GeoGebra resources for grades 4-8 to learn and practice identifying, drawing and naming points, rays, lines, line segments and planes in geometry.

Learn Points, Lines, Segments, Rays, and Planes - GeoGebra Math Resources

Learn and practice identifying, naming, and distinguishing polygons with interactive resources from GeoGebra.

Discover {1} free and ready-to-use GeoGebra resources for grades 4-8 to learn and practice naming polygons by the number of sides and differentiate between quadrilaterals based on their properties.

Learn Polygons  - GeoGebra Math Resources

Learn and practice calculating polynomials with interactive resources from GeoGebra.

Find over {1} free and ready-to-use GeoGebra resources for grades 4-12 to learn & practice factoring and evaluating operations with polynomials such as addition, subtraction, multiplication, and division.

Learn Polynomials - GeoGebra Math Resources

Entdecke Prismen und festige dein Wissen mit interaktiven Unterrichtsmaterialien von GeoGebra.

Entdecke über {1} kostenlose und sofort verwendbare GeoGebra-Unterrichtsmaterialien zum Thema Prisma für alle Schulstufen, perfekt zum Entdecken und Üben.

Prisma verstehen & anwenden - Interaktive Mathe-Materialien | GeoGebra

Learn and practice finding event probabilities with interactive resources from GeoGebra.

Discover {1} free and ready-to-use GeoGebra resources to explore and practice probability for grades 4-8. Learn how to find probabilities from experiments, sample spaces, and compound events.

Learn Probability - GeoGebra Math Resources

Entdecke Pyramiden und festige dein Wissen mit interaktiven Unterrichtsmaterialien von GeoGebra.

Entdecke über {1} kostenlose und sofort verwendbare GeoGebra-Unterrichtsmaterialien zum Thema Pyramide, perfekt zum Entdecken und Üben.

Pyramide verstehen & anwenden - Interaktive Mathe-Materialien | GeoGebra

Learn and practice using Pythagorean theorem to solve math problems with interactive resources from GeoGebra.

Discover over {1}  free and ready-to-use GeoGebra resources for grades 4-8 to practice and explore the Pythagorean Theorem and its converse. Solve real-world problems using the theorem.

Learn Pythagorean Theorem - GeoGebra Math Resources

Learn and practice solving quadratic equations with interactive resources from GeoGebra.

Access over {1} free and ready-to-use GeoGebra resources for grades 4-12 to learn and practice solving quadratic equations using various techniques such as graphing, using properties of equality, applying the quadratic formula, and completing the square.

Learn Quadratic Equations - GeoGebra Math Resources

Learn and practice interpreting and graphing quadratic functions with interactive resources from GeoGebra.

Discover over {1} free and ready-to-use GeoGebra resources to improve your skills in rewriting quadratic functions into different forms and determining key characteristics such as the vertex, intercepts, minimum or maximum values, orientation, and axis of symmetry.

Learn Quadratic Functions - GeoGebra Math Resources

Learn and practice working with rational numbers with interactive resources from GeoGebra.

Discover over {1} free and ready-to-use GeoGebra resources for grades 4-8 for exploring and practicing comparing decimals to fractions, ordering rational numbers, and solving problems with properties of arithmetic operations.

Learn Rational Numbers - GeoGebra Math Resources

Learn and practice finding unit rates and solving proportions with interactive resources from GeoGebra.

Discover over {1} free and ready-to-use GeoGebra resources ready for use to learn and practice unit rate calculations and proportion solving in number and word problems for grades 4-8.

Learn Ratios and Rates - GeoGebra Math Resources

Learn and practice working with real numbers with interactive resources from GeoGebra.

Find over {1} free and ready-to-use GeoGebra resources for grades 4-8 to explore and practice real number concepts like classifying real numbers, comparing and ordering.

Learn Real Numbers - GeoGebra Math Resources

Learn and practice working with roots with interactive resources from GeoGebra.

Discover over {1} free and ready-to-use GeoGebra resources for grades 4-8 to practice finding square and cube roots and understanding the relationship between a number and its square or cube root. Explore geometric models and equations.

Learn Roots - GeoGebra Math Resources

Learn and practice writing and converting whole numbers in scientific notation with interactive resources from GeoGebra.

Discover over {1} free and ready-to-use GeoGebra resources for grades 4-8 to learn & practice writing & converting whole numbers in scientific notation. Enhance your math skills with interactive lessons.

Learn Scientific Notation of Numbers - GeoGebra Math Resources

Learn and practice generating ordered pairs from verbal, algebraic, and table rules with interactive resources from GeoGebra.

Discover {1} free and ready-to-use GeoGebra resources for grades 4-8 to learn how to create ordered pairs from verbal, algebraic or table rules.

Learn Sequences and Series - GeoGebra Math Resources

Learn and practice identifying solid figures with interactive resources from GeoGebra.

Find over {1} free and ready-to-use GeoGebra resources for grades 4-8 for exploring and practicing identifying the nets for prisms, pyramids, cylinders, and cones. Learn to calculate surface area and volume of spheres and describe cross-sectional views of 3D figures.

Learn Solids - GeoGebra Math Resources

Learn and practice standard deviation concepts with interactive resources from GeoGebra.

Discover over {1} free, ready-to-use GeoGebra resources to learn and practice the concept of standard deviation, including determining measures of spread.

Learn Standard Deviations - GeoGebra Math Resources

Learn and practice describing data with mean, median, and mode with interactive resources from GeoGebra.

Discover {1} free and ready-to-use GeoGebra resources to learn and practice data description with mean, median, and mode for grades 4-8. Explore and understand outliers and their impact on statistical characteristics.

Learn Statistical Characteristics - GeoGebra Math Resources

Learn and practice identifying statistical questions, measuring variability and making predictions from sample data with interactive resources from GeoGebra.

Explore and practice statistical concepts with {1} free and ready-to-use GeoGebra resources for grades 4-8. Identify statistical questions, measure variability, and make predictions from samples. Enhance your data analysis skills with GeoGebra.

Learn Statistics - GeoGebra Math Resources

Learn and practice subtracting numbers with interactive resources from GeoGebra.

Discover over {1} free and ready-to-use GeoGebra resources for grades 4-8 to practice and explore multi-digit subtraction, including estimating differences.

Learn Subtraction - GeoGebra Math Resources

Learn and practice finding the surface area of different types of solids with interactive resources from GeoGebra.

Discover {1} free and ready-to-use GeoGebra resources for grades 4-8 to practice finding the surface area of solids including prisms, pyramids, and cylinders in real-world math problems

Learn Surface Area - GeoGebra Math Resources

Learn and practice identifying lines of symmetry in shapes with interactive resources from GeoGebra.

Discover over {1} free and ready-to-use GeoGebra resources for grades 4-8 for exploring & practicing symmetry in figures: identify lines & create symmetrical shapes

Learn Symmetry - GeoGebra Math Resources

Learn and practice graphing and solving systems of equations and inequalities with interactive resources from GeoGebra.

Explore over {1} free and ready-to-use GeoGebra resources for grades 4-12 to enhance your understanding of how to solve and graph systems involving a combination of equations and/or inequalities.

Learn Systems of Equations and Inequalities - GeoGebra Math Resources

Entdecke und übe unterschiedliche Zeitmaße mit interaktiven Unterrichtsmaterialien von GeoGebra.

Entdecke über {1} kostenlose und sofort verwendbare GeoGebra-Unterrichtsmaterialien zum Thema Zeitmaße für alle Schulstufen, perfekt zum Entdecken und Üben.

Zeitmaße verstehen & anwenden - Interaktive Mathe-Materialien | GeoGebra

Learn and practice transforming polygons in the coordinate plane with interactive resources from GeoGebra.

Find over {1} free and ready-to-use GeoGebra resources for grades 4-8 to learn & practice transforming polygons in the coordinate plane.

Learn Transformations - GeoGebra Math Resources

Learn and practice identifying and classifying triangles with interactive resources from GeoGebra.

Find over {1} free and ready-to-use GeoGebra resources for grades 4-8 to explore and practice how to identify and classify triangles, learn about angle sum properties, and improve triangle knowledge and skills.

Learn Triangles - GeoGebra Math Resources

Learn and practice converting measures and using appropriate tools with interactive resources from GeoGebra.

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.