Explore simplifying square roots by factoring out perfect squares and using the product property.

[i]Answer these open ended questions on your own or with others to form deeper math connections.[/i]

Putting It All Together

Simplifying Square Roots

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探索由全球 GeoGebra 社区精心打造的超过100万个数学与科学活动, 练习及课程资源库, 深入这片浩瀚的知识海洋, 畅享其中蕴藏的无限机遇.

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在 3D 中绘制函数图象并进行运算

探究函数性质, 求解方程根式, 构建几何图形

在动态环境中探究几何概念和尺规作图

借助交互式图像与绘图, 直观呈现方程和函数

探索在线笔记应用, 支持交互图像, 滑动条, 图片等更多功能.

执行分数运算, 统计分析及指数函数计算

计算器

练习基本运算, 如加法, 减法和除法

数学中自变量与因变量的对应关系

研究数学中的形状, 大小和空间关系

量化并比较长度、质量、体积等量属性

理解数字, 它们的关系和数字推理

执行数学运算, 比如加法, 减法, 除法

分析事件与结果的不确定性和可能性

研究角度, 三角形, 三角函数和比例

立即使用 GeoGebra, 开启你的数学探索之旅吧!

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This resource was reviewed for quality by the GeoGebra Team.

多项选择题 {1}

开放式问题 {1}

PDF 附件 {1}

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.

绝对值表达式

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

整数的绝对值

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.

小数的加减法

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

同分母分数或带分数的加减法

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

异分母分数与带分数的加减法

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

分数与带分数的加减

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

整数加法与减法

Adding, Subtracting, and Multiplying Polynomials

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

相反数与倒数

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.

三角形、四边形与多边形的内角和性质

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

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.

圆的弧、弦与切线的性质

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 in Scale Drawings and Maps

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.

长方形、正方形与组合图形的面积

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.

相似几何图形的面积与体积

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.

条形图

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.

箱线图

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.

科学记数法运算

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.

统计图的刻度与区间选取

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

Circle Graphs

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.

分数的混合运算

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.

比较小数大小

Compare and order fractions using common numerators or denominators.

比较分数大小

Compare and order integers with and without models.

整数大小比较

Compare and order numbers less than 10,000.

万以内数比较大小

Write a ratio or rate to compare two quantities

Comparing Quantities with Ratios

Compare and order rational numbers with and without models.

比较有理数大小

对立事件

Define and identify complementary and supplementary angles.

余角与补角

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.

坐标平面与有序数对

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.

平面直角坐标系

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).

样本空间计数方法

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.

立方与立方根

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 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.

数据, 数据分布, 集中趋势

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

大于一百万的小数

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

量纲分析

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

正比例

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.

距离公式与中点公式

Calculate distances from scale drawings and maps.

Distance in Scale Drawings and Maps

Use the distributive property to represent and simplify numerical expressions.

分配律

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.

分数与带分数的除法

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

分数除法

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.

单位分数除法

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.

特殊函数的定义域与值域

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

用量角器画角与量角

Durchführen der Primfaktorenzerlegung

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

运算对数的影响

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

异常值对数据集平均数、中位数、极差的影响

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

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

时长

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.

二元方程

Solve equations involving powers and roots by using inverse relationships.

含幂与根式的方程

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

小数与分数的等值形式

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.

等值分数

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.

指数式

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.

整数的因数

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.

三角形、平行四边形、梯形、圆与组合图形的面积

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.

寻找数字规律

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.

利用底和高求体积

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.

周长公式

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

三角形, 平行四边形, 梯形与圆的面积公式

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

立体图形的侧面积公式

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.

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).

频数表, 线图与条形图

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.

生成有序数对与数列

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.

用有序数对绘制一次函数图像

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

Graphing Quadratic Functions

Graphically solve systems of linear equations.

一次方程组的图像解法

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.

多组数据统计图

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.

最大公因数

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

直方图 vs. 条形图

Organize, display, and interpret information in histograms.

直方图

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.

用符号 >、<、≤、≥、≠ 表示不等式

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

根据样本作出推断

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.

整数

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

一次关系式的截距

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

多边形的内角与外角

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.

几何简介

Use models to investigate the concept of the Pythagorean Theorem.

探究勾股定理

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.

幂的运算法则

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

最小公倍数

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.

Line Plots with Fractional Scales

Write a problem given a simple linear equation or inequality.

一次方程与一次不等式应用题

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

应用题中的一次不等式

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

线性, 非线性, 正比例关系

Identify linear and nonlinear relationships in data sets.

数据集里的线性与非线性关系

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

直线和角

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.

对称轴

Solve a literal equation for an indicated variable.

Literal Equation for an Indicated Variable

Divide using single-digit divisors with and without remainders.

竖式除法

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.

Measures of Variability

数据集的中位数

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.

钱币计算

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

多位数进位加法

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.

乘法算式与除法算式

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.

一位数与两位数相乘

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

10 的幂的乘除运算

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.

小数的乘除法

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

整数的乘除

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

带分数的乘除运算

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

分数乘法

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

多位数乘法

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

多步计算题

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

立体图形展开图

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.

方程组解的数量

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

1000至9999的数

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

10,000 至 1,000,000 的数

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

101 至 999 的数

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

按位展开表示数

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.

观察, 测量, 抽样

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.

有理数的运算

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

运算顺序

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

散点图中的异常值与聚集点

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.

代数式的组成项

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.

百分数应用题

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

周长

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.

圆周率: 直径与周长的关系

Place Value with Thousands

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

Place Value

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.

直线的两点式

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

直线的点斜式

数轴上的点

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

总体 vs. 样本

Make predictions based on results from surveys and samples.

基于样本作出预测

Identify prime and composite numbers less than 100.

质数与合数

质因数分解

Erklären und Anwenden des Begriffs Primzahl.

Primzahlen

Determine a simple probability using geometric figures.

几何概率

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

用分数或比表示事件的概率

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

复合事件的概率

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

坐标平面问题

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.

加法与乘法的运算律

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

几何图形的性质

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

一次函数及其图像、方程的性质

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

平面图形的性质

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

有理数的性质

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

比例

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.

勾股定理及其逆定理

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

数据集的极差

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.

有理数的小数形式

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

比与比例

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.

实数, 有理数与无理数

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

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.

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.

分数的表示

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.

分数与小数互化

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

平面图形的全等变换与位似变换

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.

坐标平面内的旋转

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

小数、分数、带分数的四舍五入

Round whole numbers to a given place value.

整数取近似值

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.

生成有序数对的法则

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.

缩放因子

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

散点图

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.

平面直角坐标系中的图形变换序列

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

集合论

Verwendung eines intuitiven Wahrscheinlichkeitsbegriff zur Quantifizierung von Sicherheit.

Sichere, wahrscheinliche, unwahrscheinliche und unmöglich Ereignisse

Use proportional reasoning to solve problems related to similar polygons.

相似多边形

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

相似三角形与对应关系

Identify corresponding parts of similar and congruent figures.

相似与全等

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.

直线的斜截式

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.

直线的斜率

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

解不等式

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

解一次方程

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.

平方与平方根

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.

两位数与三位数减法

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.

立体图形的表面积

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

球的表面积与体积

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

二元及多元一次方程组

Systems of Linear Inequalities

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

表格和图表

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.

理论概率与实验概率

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

小数的千分位

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

平面直角坐标系中的变换

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.

平面图形的变换

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.

一次方程与一次不等式的转化

Translate between models or verbal phrases and numerical expressions.

列式转换

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

三角形不等式定理

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.

两步一次方程与不等式

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

运用四则运算解决两步问题

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

双向表

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

角的分类

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.

Types of Events (Conditional, Mutually Exclusive, Independent, Dependent, etc.)

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 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.

长度、面积、容积、质量与时间的单位换算

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

单位分数

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.

利用等式的性质解一元二次方程

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.

理论概率与实验概率的预测应用

Find the value of a variable in a number sentence.

算式中变量的值

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.

正方体的体积与体积单位

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

组合图形的体积

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

立体图形的体积

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.

含一个未知数的应用题

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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