Curriculum (Germany)

Mathematics — Grade 10

Grade 10 Mathematics: Completion of lower secondary education — powers and roots, Pythagorean theorem, circles and calculation of solids as a review before the intermediate school leaving certificate. Ready-made printable worksheets are available for every topic.

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What do you learn in 10th grade mathematics?

Grade 10 completes lower secondary education. Students consolidate working with powers and roots, apply the Pythagorean theorem in the plane and in space, and calculate circles as well as the volume and surface area of solids — this is the material covered in the exams for the intermediate school-leaving certificate.

Important note on educational standards: The KMK educational standards describe what is achieved by the end of lower secondary education or upon obtaining the general higher education entrance qualification (Abitur) — not year by year. The assignment to specific grade levels is our decision based on the core curricula of the federal states (starting with North Rhine-Westphalia).

Status of this page: The exercises for this grade are complete as pure calculation problems (problem statement and formula); exercises with diagrams and real-world contexts will follow.

Curriculum scope

KMKS1.ZO (our numbering) · Teaching content · checked against the act

Number and Operations

our translation · original wording (DE): Zahl und Operation

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Zahl und Operation
KMKS1.ZO.1 (our numbering) · Teaching content · checked against the act

use meaningful concepts of rational numbers, especially natural numbers, integers, and fractions, according to the need for application

our translation · original wording (DE): nutzen sinntragende Vorstellungen von rationalen Zahlen, insbesondere von natürlichen, ganzen und gebrochenen Zahlen entsprechend der Verwendungsnotwendigkeit

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Zahl und Operation, Spiegelstrich 1 (obie kolumny)
KMKS1.ZO.2 (our numbering) · Teaching content · checked against the act

use meaningful mental models of real numbers (e.g. completeness of the number line)

our translation · original wording (DE): nutzen sinntragende Vorstellungen von reellen Zahlen (z. B. Vollständigkeit der Zahlengerade)

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Zahl und Operation, Spiegelstrich 2 (kolumna MSA)

Estimating square roots mentally (we teach in grade 9-10)
KMKS1.ZO.3 (our numbering) · Teaching content · checked against the act

use meaning-bearing mental representations of operations with rational numbers (e.g. step-by-step, semi-written procedures)

our translation · original wording (DE): nutzen sinntragende Vorstellungen von Operationen rationaler Zahlen (z. B. schrittweiser, halbschriftlicher Verfahren)

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Zahl und Operation, Spiegelstrich 3 (obie kolumny)
KMKS1.ZO.4 (our numbering) · Teaching content · checked against the act

investigate numbers for their factors, in simple cases without digital mathematics tools

our translation · original wording (DE): untersuchen Zahlen nach ihren Faktoren, in einfachen Fällen ohne digitale Mathematikwerkzeuge

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Zahl und Operation, Spiegelstrich 4 (obie kolumny)
KMKS1.ZO.5 (our numbering) · Teaching content · checked against the act

represent numbers appropriately for the situation, e.g. including in scientific notation (powers of ten)

our translation · original wording (DE): stellen Zahlen der Situation angemessen dar, z.B. unter anderem in Zehnerpotenzschreibweise

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Zahl und Operation, Spiegelstrich 5 (obie kolumny)

Representing numbers in scientific notation (we teach in grade 9-10)
KMKS1.ZO.6 (our numbering) · Teaching content · checked against the act

calculate with natural numbers, integers, and rational numbers that occur in daily life, both for verification and mentally, and explain the meaning of the arithmetic operations

our translation · original wording (DE): rechnen mit natürlichen, ganzen und rationalen Zahlen, die im täglichen Leben vorkommen, sowohl zur Kontrolle als auch im Kopf und erklären die Bedeutung der Rechenoperationen

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Zahl und Operation, Spiegelstrich 6 (obie kolumny)
KMKS1.ZO.8 (our numbering) · Teaching content · checked against the act

describe the necessity of number system extensions from ℕ to ℤ and ℚ as well as from ℚ to ℝ using examples

our translation · original wording (DE): beschreiben die Notwendigkeit von Zahlbereichserweiterungen von ℕ nach ℤ und ℚ sowie von ℚ nach ℝ an Beispielen

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Zahl und Operation, Spiegelstrich 8 (kolumna MSA)
KMKS1.ZO.9 (our numbering) · Teaching content · checked against the act

explain using examples the different concepts of fractions (in particular parts of one or more wholes, relative parts)

our translation · original wording (DE): erläutern an Beispielen die verschiedenen Vorstellungen zum Bruchbegriff (insbesondere Teile eines oder mehrerer Ganzer, relative Anteile)

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Zahl und Operation, Spiegelstrich 9 (obie kolumny)
KMKS1.ZO.10 (our numbering) · Teaching content · checked against the act

use arithmetic laws (e.g. commutative, associative, distributive laws), also for mental calculation strategies

our translation · original wording (DE): nutzen Rechengesetze (z. B. Kommutativ-, Assoziativ -, Distributivgesetz), auch zum vorteilhaften Rechnen

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Zahl und Operation, Spiegelstrich 10 (obie kolumny)
KMKS1.ZO.11 (our numbering) · Teaching content · checked against the act

use rough calculations for orientation and verification

our translation · original wording (DE): nutzen Überschlagsrechnungen zur Orientierung und zur Kontrolle

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Zahl und Operation, Spiegelstrich 11 (obie kolumny)
KMKS1.ZO.12 (our numbering) · Teaching content · checked against the act

round numbers sensibly according to the context

our translation · original wording (DE): runden Zahlen dem Sachverhalt entsprechend sinnvoll

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Zahl und Operation, Spiegelstrich 12 (obie kolumny)
KMKS1.ZO.13 (our numbering) · Teaching content · checked against the act

check and interpret results, also in real-world situations

our translation · original wording (DE): prüfen und interpretieren Ergebnisse, auch in Sachsituationen

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Zahl und Operation, Spiegelstrich 13 (obie kolumny)
KMKS1.ZO.14 (our numbering) · Teaching content · checked against the act

explain using examples the relationship between arithmetic operations and their inverses and use these relationships

our translation · original wording (DE): erläutern an Beispielen den Zusammenhang zwischen Rechenoperationen und deren Umkehrungen und nutzen diese Zusammenhänge

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Zahl und Operation, Spiegelstrich 14 (obie kolumny)
KMKS1.ZO.15 (our numbering) · Teaching content · checked against the act

use percentage and interest calculations based on mental models (e.g., percentage strips) and appropriately

our translation · original wording (DE): verwenden Prozent - und Zinsrechnung vorstellungsbasiert (z. B. Prozentstreifen) und sachgerecht

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Zahl und Operation, Spiegelstrich 15 (obie kolumny)
KMKS1.ZO.17 (our numbering) · Teaching content · checked against the act

explain powers and roots and calculate simple powers and roots

our translation · original wording (DE): erläutern Potenzen und Wurzeln und berechnen einfache Potenzen und Wurzeln

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Zahl und Operation, Spiegelstrich 17 (kolumna MSA)

Calculating powers with fractions and integers (we teach in grade 9-10) · Calculating square and cube roots with confidence (we teach in grade 9-10) · Determining missing exponents using exponent rules (we teach in grade 9-10)
KMKS1.ZO.19 (our numbering) · Teaching content · checked against the act

choose, describe and evaluate approaches and procedures that are based on algorithms or calculi, respectively, and carry them out (e.g. written arithmetic operations as well as for roots and powers)

our translation · original wording (DE): wählen, beschreiben und bewerten Vorgehensweisen und Verfahren, denen Algorithmen bzw. Kalküle zu Grunde liegen und führen diese aus (z. B. schriftliche Rechenoperationen sowie bei Wurzeln und Potenzen)

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Zahl und Operation, Spiegelstrich 19 (kolumna MSA)
KMKS1.ZO.20 (our numbering) · Teaching content · checked against the act

implement an algorithmic procedure (e.g. Heron's method for determining square roots, nested intervals) with digital mathematics tools

our translation · original wording (DE): implementieren ein algorithmisches Verfahren (z. B. Heron-Verfahren zur Bestimmung von Quadratwurzeln, Intervallschachtelung) mit digitalen Mathematikwerkzeugen

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Zahl und Operation, Spiegelstrich 20 (kolumna MSA)
KMKS1.ZO.21 (our numbering) · Teaching content · checked against the act

carry out systematic counting principles in concrete situations (e.g. number of handshakes when shaking hands with each person)

our translation · original wording (DE): führen in konkreten Situationen systematische Zählprinzipien aus (z. B. Anzahl Händeschütteln, wenn man jeder Person die Hand gibt)

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Zahl und Operation, Spiegelstrich 21 (kolumna MSA)
KMKS1.ZO.22 (our numbering) · Teaching content · checked against the act

continue number sequences, also using variables as a generalized number

our translation · original wording (DE): führen Zahlenfolgen fort, auch unter Verwendung von Variablen als allgemeine Zahl

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Zahl und Operation, Spiegelstrich 22 (kolumna MSA)
KMKS1.GM (our numbering) · Teaching content · checked against the act

Quantities and measurement

our translation · original wording (DE): Größen und Messen

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Größen und Messen
KMKS1.GM.1 (our numbering) · Teaching content · checked against the act

use the basic principle of measurement as comparison with (standard) units, e.g. when determining lengths, areas and volumes, also in real-world situations

our translation · original wording (DE): nutzen das Grundprinzip des Messens als Vergleichen mit (Standard-) Einheiten, z. B. bei der Bestimmung von Längen, Flächeninhalten und Volumina, auch in Sachsituationen

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Größen und Messen, Spiegelstrich 1 (obie kolumny)
KMKS1.GM.2 (our numbering) · Teaching content · checked against the act

select units of measurement appropriate to the situation (in particular for time, mass, money, length, area, volume and angle) and convert them if necessary

our translation · original wording (DE): wählen Einheiten von Größen situationsgerecht aus (insbesondere für Zeit, Masse, Geld, Länge, Fläche, Volumen und Winkel) und wandeln sie ggf. um

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Größen und Messen, Spiegelstrich 2 (obie kolumny)
KMKS1.GM.3 (our numbering) · Teaching content · checked against the act

estimate measurements using mental representations of suitable reference objects (e.g. typical object for a standard measurement) and also use this to check for plausibility

our translation · original wording (DE): schätzen Größen mit Hilfe von Vorstellungen über geeignete Repräsentanten (z. B. typisches Objekt für eine Standardgröße) und nutzen dies auch zur Plausibilitätsprüfung

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Größen und Messen, Spiegelstrich 3 (obie kolumny)
KMKS1.GM.5 (our numbering) · Teaching content · checked against the act

calculate area and perimeter of rectangles, triangles, and circles as well as figures composed of them, also with the help of digital mathematics tools

our translation · original wording (DE): berechnen Flächeninhalt und Umfang von Rechteck, Dreieck und Kreis sowie daraus zusammengesetzten Figuren, auch mit Hilfe digitaler Mathematikwerkzeuge

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Größen und Messen, Spiegelstrich 5 (kolumna MSA)

Calculating the circumference and area of circles (we teach in grade 9-10)
KMKS1.GM.7 (our numbering) · Teaching content · checked against the act

calculate volume and surface area of prisms, pyramids, cylinders, cones, and spheres as well as solids composed thereof, also with the help of digital mathematical tools

our translation · original wording (DE): berechnen Volumen und Oberflächeninhalt von Prisma, Pyramide, Zylinder, Kegel und Kugel sowie daraus zusammengesetzten Körpern, auch mit Hilfe digitaler Mathematikwerkzeuge

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Größen und Messen, Spiegelstrich 7 (kolumna MSA)

Volume and surface area of triangular prisms (we teach in grade 9-10) · Volume and Surface Area of Square Pyramids (we teach in grade 9-10) · Calculating the volume of a cylinder, cone, and sphere (we teach in grade 9-10)
KMKS1.GM.8 (our numbering) · Teaching content · checked against the act

take targeted measurements in their environment, also using digital media (as a source of information or measuring instrument), extract measurements from source material, perform calculations with them, and evaluate the results as well as the chosen approach in relation to the real-world situation

our translation · original wording (DE): nehmen in ihrer Umwelt gezielt Messungen vor, auch mit Hilfe digitaler Medien (als Informationsquelle oder Messinstrument), entnehmen Maßangaben aus Quellenmaterial, führen damit Berechnungen durch und bewerten die Ergebnisse sowie den gewählten Weg in Bezug auf die Sachsituation

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Größen und Messen, Spiegelstrich 8 (obie kolumny)
KMKS1.GM.10 (our numbering) · Teaching content · checked against the act

calculate segment lengths and angle sizes, also using the Pythagorean theorem, trigonometric relationships, and similarity relationships, also with the help of digital mathematics tools

our translation · original wording (DE): berechnen Streckenlängen und Winkelgrößen, auch unter Nutzung des Satzes von Pythagoras, von trigonometrischen Beziehungen und Ähnlichkeitsbeziehungen, auch mit Hilfe digitaler Mathematikwerkzeuge

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Größen und Messen, Spiegelstrich 10 (kolumna MSA)

Applying the Pythagorean theorem (we teach in grade 9-10)
KMKS1.SF (our numbering) · Teaching content · checked against the act

Structures and functional relationships

our translation · original wording (DE): Strukturen und funktionaler Zusammenhang

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Strukturen und funktionaler Zusammenhang
KMKS1.SF.1 (our numbering) · Teaching content · checked against the act

use variables depending on the context as a fixed number, as an arbitrary number from a number range, and as a varying quantity in a specific range, and can give examples of the different uses of variables

our translation · original wording (DE): verwenden Variablen je nach Kontext als eine feste Zahl, als eine beliebige Zahl aus einem Zahlbereich und als Veränderliche in einem bestimmten Bereich und können Beispiele für die unterschiedliche Verwendung von Variablen nennen

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Strukturen und funktionaler Zusammenhang, Spiegelstrich 1 (obie kolumny)
KMKS1.SF.3 (our numbering) · Teaching content · checked against the act

set up expressions to describe general relationships in context, transform them, and interpret them

our translation · original wording (DE): stellen Terme auf, um allgemeine Zusammenhänge im Sachkontext zu beschreiben, formen sie um und interpretieren sie

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Strukturen und funktionaler Zusammenhang, Spiegelstrich 3 (kolumna MSA)
KMKS1.SF.4 (our numbering) · Teaching content · checked against the act

use percentage calculation in growth processes (for example, in interest calculation), also using digital tools

our translation · original wording (DE): nutzen die Prozentrechnung bei Wachstumsprozessen (beispielsweise bei der Zinsrechnung), auch unter Verwendung digitaler Werkzeuge

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Strukturen und funktionaler Zusammenhang, Spiegelstrich 4 (obie kolumny)
KMKS1.SF.5 (our numbering) · Teaching content · checked against the act

use scales appropriately to the situation when reading and producing drawings

our translation · original wording (DE): nutzen Maßstäbe beim Lesen und Anfertigen von Zeichnungen situationsgerecht

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Strukturen und funktionaler Zusammenhang, Spiegelstrich 5 (obie kolumny)
KMKS1.SF.7 (our numbering) · Teaching content · checked against the act

recognize and use functional relationships and represent these in different representations (verbally, tabularly, graphically, algebraically) and can switch between these forms of representation, also with the help of digital mathematics tools

our translation · original wording (DE): erkennen und verwenden funktionale Zusammenhänge und stellen diese in verschiedenen Repräsentationen dar (sprachlich, tabellarisch, grafisch, algebraisch) und können zwischen diesen Darstellungsformen wechseln, auch mit Hilfe digitaler Mathematikwerkzeuge

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Strukturen und funktionaler Zusammenhang, Spiegelstrich 7 (kolumna MSA)
KMKS1.SF.9 (our numbering) · Teaching content · checked against the act

analyze, interpret and compare different functional relationships (linear, proportional and inversely proportional as well as quadratic functions), also with the help of digital mathematical tools

our translation · original wording (DE): analysieren, interpretieren und vergleichen unterschiedliche funktionale Zusammenhänge (lineare, proportionale und antiproportionale sowie quadratische Funktionen), auch mit Hilfe digitaler Mathematikwerkzeuge

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Strukturen und funktionaler Zusammenhang, Spiegelstrich 9 (kolumna MSA)
KMKS1.SF.10 (our numbering) · Teaching content · checked against the act

solve real-world problems in connection with linear, proportional, and inversely proportional relationships, if applicable also using the rule of three, also using digital mathematical tools

our translation · original wording (DE): lösen realitätsnahe Probleme im Zusammenhang mit linearen, proportionalen und antiproportionalen Zuordnungen, ggf. auch mit Hilfe des Dreisatzes, auch mit Hilfe digitaler Mathematikwerkzeuge

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Strukturen und funktionaler Zusammenhang, Spiegelstrich 10 (obie kolumny)
KMKS1.SF.12 (our numbering) · Teaching content · checked against the act

solve linear and quadratic equations as well as systems of linear equations numerically (systematic trial), algebraically (transforming), and graphically (using function graphs), also using digital mathematics tools, and compare the effectiveness of different solution methods with regard to the respective question or problem

our translation · original wording (DE): lösen lineare und quadratische Gleichungen sowie lineare Gleichungssysteme numerisch (systematisches Probieren), algebraisch (Umformen) und grafisch (mit Hilfe von Funktionsgraphen), auch unter Einsatz digitaler Mathematikwerkzeuge, und vergleichen die Eff ektivität verschiedener Lösungsverfahren im Hinblick auf die jeweilige Fragestellung oder das Problem

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Strukturen und funktionaler Zusammenhang, Spiegelstrich 12 (kolumna MSA)
KMKS1.SF.13 (our numbering) · Teaching content · checked against the act

investigate questions of solvability and the variety of solutions of linear and quadratic equations as well as systems of linear equations and formulate statements in this regard

our translation · original wording (DE): untersuchen Fragen der Lösbarkeit und der Lösungsvielfalt von linearen und quadratischen Gleichungen sowie linearen Gleichungssystemen und formulieren diesbezüglich Aussagen

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Strukturen und funktionaler Zusammenhang, Spiegelstrich 13 (kolumna MSA)
KMKS1.SF.15 (our numbering) · Teaching content · checked against the act

determine characteristic features of functions in the function rule, graph, and table of values and establish relationships between the representations

our translation · original wording (DE): bestimmen kennzeichnende Merkmale von Funktionen im Funktionsterm, Graph und der Wertetabelle und stellen Beziehungen zwischen den Darstellungen her

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Strukturen und funktionaler Zusammenhang, Spiegelstrich 15 (kolumna MSA)
KMKS1.SF.17 (our numbering) · Teaching content · checked against the act

apply in particular linear and quadratic functions as well as exponential functions of the form 𝑓(𝑥) = 𝑎 ∙ 𝑏𝑥 when describing and working on problems, also with the help of digital mathematical tools

our translation · original wording (DE): wenden insbesondere lineare und quadratische Funktionen sowie Exponentialfunktionen der Form 𝑓(𝑥) = 𝑎 ∙ 𝑏𝑥 bei der Beschreibung und Bearbeitung von Problemen an, auch mit Hilfe digitaler Mathematikwerkzeuge

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Strukturen und funktionaler Zusammenhang, Spiegelstrich 17 (kolumna MSA)
KMKS1.SF.18 (our numbering) · Teaching content · checked against the act

use the sine function in the form 𝑓(𝑥) = 𝑎 ∙ 𝑠𝑖𝑛(𝑏 ∙ 𝑥) to describe periodic processes using digital mathematics tools

our translation · original wording (DE): verwenden die Sinusfunktion in der Form 𝑓(𝑥) = 𝑎 ∙ 𝑠𝑖𝑛(𝑏 ∙ 𝑥) zur Beschreibung periodischer Vorgänge mit Hilfe digitaler Mathematikwerkzeuge

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Strukturen und funktionaler Zusammenhang, Spiegelstrich 18 (kolumna MSA)
KMKS1.SF.20 (our numbering) · Teaching content · checked against the act

describe changes in quantities using functions (also non-linear changes), also using digital mathematics tools

our translation · original wording (DE): beschreiben Veränderungen von Größen mittels Funktionen (auch nicht lineare Veränderungen), auch unter Verwendung digitaler Mathematikwerkzeuge

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Strukturen und funktionaler Zusammenhang, Spiegelstrich 20 (kolumna MSA)
KMKS1.SF.22 (our numbering) · Teaching content · checked against the act

give real-world situations for given functions that can be described using this function

our translation · original wording (DE): geben zu vorgegebenen Funktionen Sachsituationen an, die mit Hilfe dieser Funktion beschrieben werden können

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Strukturen und funktionaler Zusammenhang, Spiegelstrich 22 (kolumna MSA)
KMKS1.RF (our numbering) · Teaching content · checked against the act

Space and shape

our translation · original wording (DE): Raum und Form

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Raum und Form
KMKS1.RF.1 (our numbering) · Teaching content · checked against the act

name and describe geometric objects and relationships in the environment using mathematical models (points, angles, line segments, straight lines, surfaces, solids) and their connections

our translation · original wording (DE): benennen und beschreiben geometrische Objekte und Beziehungen in der Umwelt mit Hilfe mathematischer Modelle (Punkte, Winkel, Strecken, Geraden, Flächen, Körper) und ihre Zusammenhänge

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Raum und Form, Spiegelstrich 1 (obie kolumny)
KMKS1.RF.2 (our numbering) · Teaching content · checked against the act

develop mental representations in two- and three-dimensional space and operate mentally (e.g. translate, rotate, reflect) with the objects contained within it (points, line segments, surfaces, and solids)

our translation · original wording (DE): entwickeln Vorstellungen im zwei und dreidimensionalen Raum und operieren (z.B. verschieben, drehen, spiegeln) gedanklich mit den darin enthaltenen Objekten (Punkten, Strecken, Flächen und Körpern)

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Raum und Form, Spiegelstrich 2 (obie kolumny)
KMKS1.RF.3 (our numbering) · Teaching content · checked against the act

represent plane geometric figures (e.g. triangles, quadrilaterals) and elementary geometric transformations (e.g. translations, rotations, reflections, dilations) in a plane Cartesian coordinate system, also using digital mathematical tools

our translation · original wording (DE): stellen ebene geometrische Figuren (z. B. Dreiecke, Vierecke) und elementare geometrische Abbildungen (z. B. Verschiebungen, Drehungen, Spiegelungen, zentrische Streckungen) im ebenen kartesischen Koordinatensystem dar, auch mit Hilfe digitaler Mathematikwerkzeuge

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Raum und Form, Spiegelstrich 3 (obie kolumny)
KMKS1.RF.5 (our numbering) · Teaching content · checked against the act

produce nets, oblique drawings and models of selected solids (e.g. prism, pyramid) - also with the help of digital mathematics tools - and recognize solids from their corresponding representations

our translation · original wording (DE): fertigen Netze, Schrägbilder und Modelle von ausgewählten Körpern (z. B. Prisma, Pyramide) an - auch mit Hilfe digitaler Mathematikwerkzeuge - und erkennen Körper aus ihren entsprechenden Darstellungen

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Raum und Form, Spiegelstrich 5 (kolumna MSA)
KMKS1.RF.6 (our numbering) · Teaching content · checked against the act

analyze and classify geometric objects of the plane (in particular angles, triangles, quadrilaterals) and of space (in particular prisms, pyramids, cylinders, cones, sphere)

our translation · original wording (DE): analysieren und klassifizieren geometrische Objekte der Ebene (insbesondere Winkel, Dreiecke, Vierecke) und des Raumes (insbesondere Prismen, Pyramiden, Zylinder, Kegel, Kugel)

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Raum und Form, Spiegelstrich 6 (obie kolumny)
KMKS1.RF.8 (our numbering) · Teaching content · checked against the act

recognise, describe and justify properties and relationships of geometric objects (e.g. symmetry, idea of congruence, similarity, positional relationships) and use these in the context of problem-solving to analyse contextual situations

our translation · original wording (DE): erkennen, beschreiben und begründen Eigenschaften und Beziehungen geometrischer Objekte (z. B. Symmetrie, Idee der Kongruenz, Ähnlichkeit, Lagebeziehungen) und nutzen diese im Rahmen des Problemlösens zur Analyse von Sachsituationen

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Raum und Form, Spiegelstrich 8 (kolumna MSA)
KMKS1.RF.10 (our numbering) · Teaching content · checked against the act

apply theorems of plane geometry (in particular the Pythagorean theorem, Thales' theorem, similarity relationships and calculations, also with the help of digital mathematics tools, trigonometric relationships) in constructions, calculations, justifications and proofs, also with the help of digital mathematics tools

our translation · original wording (DE): wenden Sätze der ebenen Geometrie (insbesondere den Satz des Pythagoras, den Satz des Thales, Ähnlichkeitsbeziehungen und Berechnungen an, auch mit Hilfe digitaler Mathematikwerkzeuge, trigonometrische Beziehungen) bei Konstruktionen, Berechnungen, Begründungen und Beweisen an, auch mit Hilfe digitaler Mathematikwerkzeuge

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Raum und Form, Spiegelstrich 10 (kolumna MSA)

Applying the Pythagorean theorem (we teach in grade 9-10)
KMKS1.RF.11 (our numbering) · Teaching content · checked against the act

draw and construct geometric figures using appropriate media such as compasses, set square or digital mathematics tools

our translation · original wording (DE): zeichnen und konstruieren geometrische Figuren unter Verwendung angemessener Medien wie Zirkel, Geodreieck oder digitaler Mathematikwerkzeuge

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Raum und Form, Spiegelstrich 11 (obie kolumny)
KMKS1.RF.12 (our numbering) · Teaching content · checked against the act

investigate questions of solvability and variety of solutions of construction tasks and formulate statements in this regard

our translation · original wording (DE): untersuchen Fragen der Lösbarkeit und Lösungsvielfalt von Konstruktionsaufgaben und formulieren diesbezüglich Aussagen

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Raum und Form, Spiegelstrich 12 (kolumna MSA)
KMKS1.DZ (our numbering) · Teaching content · checked against the act

Data and chance

our translation · original wording (DE): Daten und Zufall

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Daten und Zufall
KMKS1.DZ.1 (our numbering) · Teaching content · checked against the act

evaluate graphical representations and tables of statistical surveys, also with the help of spreadsheets or stochastics tools

our translation · original wording (DE): werten grafische Darstellungen und Tabellen von statistischen Erhebungen aus, auch mit Hilfe von Tabellenkalkulation oder Stochastiktools

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Daten und Zufall, Spiegelstrich 1 (obie kolumny)
KMKS1.DZ.3 (our numbering) · Teaching content · checked against the act

use simulations to decide stochastic questions

our translation · original wording (DE): nutzen Simulationen, um stochastische Fragen zu entscheiden

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Daten und Zufall, Spiegelstrich 3 (kolumna MSA)
KMKS1.DZ.5 (our numbering) · Teaching content · checked against the act

plan statistical surveys, also under the aspects of sample selection and survey instrument

our translation · original wording (DE): planen statistische Erhebungen, auch unter den Aspekten Stichprobenauswahl und Erhebungsinstrument

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Daten und Zufall, Spiegelstrich 5 (kolumna MSA)
KMKS1.DZ.6 (our numbering) · Teaching content · checked against the act

systematically collect data (e.g. measurements, data from surveys or the internet), organize them in tables and represent them graphically, also using appropriate tools such as spreadsheets or stochastic tools

our translation · original wording (DE): sammeln systematisch Daten (z. B. Messwerte, Daten aus Befragungen oder Internet), organisieren sie in Tabellen und stellen sie grafisch dar, auch unter Verwendung geeigneter Hilfsmittel wie Tabellenkalkulation oder Stochastiktools

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Daten und Zufall, Spiegelstrich 6 (obie kolumny)
KMKS1.DZ.8 (our numbering) · Teaching content · checked against the act

determine and interpret statistical measures (e.g. minimum, maximum, arithmetic mean, median, range), arithmetic mean, median, range, quartiles)

our translation · original wording (DE): ermitteln und interpretieren Kenngrößen (z. B. Minimum, Maximum, arithmetisches Mittel, Median, Spannweite), arithmetisches Mittel, Median, Spannweite, Quartile)

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Daten und Zufall, Spiegelstrich 8 (kolumna MSA)
KMKS1.DZ.10 (our numbering) · Teaching content · checked against the act

create and interpret diagrams (e.g. column or bar chart, histograms, pie chart, line graph, box plot), also with the help of digital mathematical tools and justify the chosen form of representation

our translation · original wording (DE): erstellen und interpretieren Diagramme (z. B. Säulen - oder Balkendiagramm, Histogramme, Kreisdiagramm, Liniendiagramm, Boxplot), auch mit Hilfe digitaler Mathematikwerkzeuge und begründen die gewählte Darstellungsform

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Daten und Zufall, Spiegelstrich 10 (kolumna MSA)
KMKS1.DZ.11 (our numbering) · Teaching content · checked against the act

reflect, with the help of mathematical knowledge, on the handling and representation of data in media, such as with regard to the intention and possible effects of the representation

our translation · original wording (DE): reflektieren mit Hilfe der mathematischen Kenntnisse den Umgang mit und die Darstellung von Daten in Medien, etwa in Bezug auf die Absicht und mögliche Wirkungen der Darstellung

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Daten und Zufall, Spiegelstrich 11 (obie kolumny)
KMKS1.DZ.13 (our numbering) · Teaching content · checked against the act

describe random phenomena and interpret probability statements and their representations in the media

our translation · original wording (DE): beschreiben Zufallserscheinungen und interpretieren Wahrschein lichkeitsaussagen und ihre Darstellungen in Medien

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Daten und Zufall, Spiegelstrich 13 (kolumna MSA)
KMKS1.DZ.15 (our numbering) · Teaching content · checked against the act

Use and interpret, when conducting random experiments, the occurring relative frequencies as estimates of probabilities that improve with increasing sample size

our translation · original wording (DE): Nutzen und deuten bei der Durchführung von Zufallsexperimenten die auftretenden relativen Häufigkeiten als Schätzwerte von Wahrscheinlichkeiten, die bei wachsendem Stichprobenumfang besser werden

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Daten und Zufall, Spiegelstrich 15 (kolumna MSA)
KMKS1.DZ.17 (our numbering) · Teaching content · checked against the act

determine probabilities in single- or multi-stage random experiments, also with the help of appropriate visualizations (e.g. tree diagram, two-way table), without and with the help of digital mathematics tools

our translation · original wording (DE): bestimmen Wahrscheinlichkeiten bei ein- oder mehrstufigen Zufallsexperimenten, auch mit Hilfe entsprechender Visualisierungen (z. B. Baumdiagramm, Vierfeldertafel), ohne und mit Hilfe digitaler Mathematikwerkzeuge

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Daten und Zufall, Spiegelstrich 17 (kolumna MSA)
KMKS1.DZ.18 (our numbering) · Teaching content · checked against the act

use visualizations to recognize conditional probabilities in simple, everyday modelling, without and with the help of digital media

our translation · original wording (DE): nutzen Visualisierungen, um bei einfachen, alltagsnahen Modellierungen bedingte Wahrscheinlichkeiten zu erkennen, ohne und mit Hilfe digitaler Medien

KMK Bildungsstandards Mathematik ESA/MSA (23.06.2022), Leitidee Daten und Zufall, Spiegelstrich 18 (kolumna MSA)

Skills step by step

Applying the Pythagorean theorem

Young people use the Pythagorean theorem to calculate missing sides in a right-angled triangle and to check triangles for right angles.

Curriculum point KMKS1.GM.10 (our numbering) calculate segment lengths and angle sizes, also using the Pythagorean theorem, trigonometric relationships, and similarity relationships, also with the help of digital mathematics tools (our translation)
Curriculum point KMKS1.RF.10 (our numbering) apply theorems of plane geometry (in particular the Pythagorean theorem, Thales' theorem, similarity relationships and calculations, also with the help of digital mathematics tools, trigonometric relationships) in constructions, calculations, justifications and proofs, also with the help of digital mathematics tools (our translation)

In 10th grade, students specifically apply the relationship a² + b² = c². They square given side lengths, add or subtract them, and take the square root to determine either the hypotenuse or one of the two legs. In addition, they use given side measurements to verify mathematically whether a triangle actually has a right angle.

Common mistakes occur when rearranging the formula when solving for a leg. Out of habit, students often add the squares of the two known numbers instead of subtracting the square of the leg from the squared hypotenuse. Another typical stumbling block is forgetting the final step: the calculation ends prematurely with the squared value, without taking the final square root.

In practice exercises such as Calculate the hypotenuse or Calculate a leg, learners determine the missing measurement from two known side lengths. The exercise format Is the triangle right-angled? requires them to substitute three given lengths into the equation and test for equality, while formats like Hypotenuse — choose the answer train the quick identification and matching of the sought side.

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Task types: Find a leg of a right triangle · Find the hypotenuse · Hypotenuse — choose the answer · Is the triangle right-angled?

Calculating powers with fractions and integers

Students calculate the exact values of powers with integers and fractions as bases.

Curriculum point KMKS1.ZO.17 (our numbering) explain powers and roots and calculate simple powers and roots (our translation)

In 10th grade, students calculate powers whose base is either an integer or a fraction. As a targeted review before intermediate school-leaving exams, they reliably determine the numerical value of such expressions and consolidate their handling of rational numbers.

A typical mistake with fractions is accidentally applying the exponent only to the numerator while leaving the denominator unchanged. With integers, learners also frequently confuse exponentiation with multiplication and mistakenly calculate the base times the exponent (such as 2 times 4 instead of 2 to the power of 4).

In practice, students work through exercises such as Calculate the power of a fraction or Calculate the power of an integer, where the final result must be written as a number or a fully simplified fraction. For tasks following the pattern Value of a power — choose the answer, they identify the correct result in a multiple-choice format.

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Task types: Evaluate a power of a fraction · Evaluate a power of an integer · Value of a power — choose the answer

Calculating the volume of a cylinder, cone, and sphere

Students calculate the volume of cylinders, cones, and spheres using appropriate formulas in preparation for the intermediate school-leaving certificate.

Curriculum point KMKS1.GM.7 (our numbering) calculate volume and surface area of prisms, pyramids, cylinders, cones, and spheres as well as solids composed thereof, also with the help of digital mathematical tools (our translation)

In 10th grade, students confidently apply solid geometry formulas to determine the volume of curved solids. They use powers when calculating with the radius and multiply by the number pi to find exact or practically rounded volumes.

Typical errors mainly arise from confusing formula components: learners frequently substitute the given diameter directly instead of the radius. With cones, the fractional factor one-third is often forgotten, while in sphere calculations, the third power (r³) is repeatedly confused with the square (r²).

On the worksheets, students work with specific geometric measurements for individual solids:

Volume of a cylinder: Calculation of the volume using a given base area or radius and the height of the solid.
Volume of a cone: Determination of the volume incorporating the radius and height with the appropriate formula factor.
Volume of a sphere: Finding the volume solely on the basis of the radius or diameter.

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Task types: Volume of a cone · Volume of a cylinder · Volume of a sphere

Calculating square and cube roots with confidence

In grade 10, students learn to reliably determine square roots and cube roots and understand them as the inverse of square and cube numbers.

Curriculum point KMKS1.ZO.17 (our numbering) explain powers and roots and calculate simple powers and roots (our translation)

In 10th grade, students consolidate their skills in working with powers and roots in preparation for the intermediate secondary school certificate. They learn to determine square roots and cube roots precisely by specifically looking for the number whose second or third power yields the initial value.

A typical mistake is confusing taking roots with dividing by 2 or 3: for example, the result for the square root of 16 is often recorded as 8, or for the cube root of 27 as 9, instead of determining the required bases 4 and 3, respectively.

In practice, two types of exercises consolidate this skill:

Calculating square roots: Determining the value from known square numbers.
Calculating cube roots: Finding the number that yields the given radicand when multiplied by itself three times.

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Task types: Find the cube root · Find the square root

Calculating the circumference and area of circles

Students reliably calculate the circumference and area of circles based on the radius or diameter using the number pi.

Curriculum point KMKS1.GM.5 (our numbering) calculate area and perimeter of rectangles, triangles, and circles as well as figures composed of them, also with the help of digital mathematics tools (our translation)

In grade 10, students consolidate calculations on circular figures in preparation for the intermediate secondary school leaving certificate. They use the circle constant π (pi) to determine the circumference and the area from a given radius or diameter.

A typical error arises from confusing the radius and the diameter or mixing up the calculation methods: frequently, the radius is merely doubled instead of squared when calculating the area, or the squared formula is mistakenly used for the circumference.

In practice, students work on tasks from the templates Circumference of a circle and Area of a circle. In these exercises, one measurement is given from which the required value is calculated and then usually rounded to one or two decimal places.

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Task types: Area of a circle · Circumference of a circle

Volume and surface area of triangular prisms

The students confidently calculate the volume and surface area of right triangular prisms as a review before the intermediate school-leaving examination.

In 10th grade, students consolidate their skills in working with geometric solids and calculate the volume and surface area of right triangular prisms. To do this, they first determine the area of the base triangle and multiply it by the height of the prism to find the volume, or, for the surface area, calculate both the triangular base and top faces together with the three rectangular lateral faces.

A typical mistake lies in confusing the different heights: the height of the base triangle is often mixed up with the height of the entire prism. Furthermore, when calculating the surface area, students often forget to count the triangular face twice, or fail to correctly add all three rectangles of the lateral surface.

Typical task formats from the templates Volume of a Triangular Prism and Surface Area of a Triangular Prism provide measurements such as the side lengths of the base triangle and the height of the solid. Learners substitute these values into the appropriate geometric formulas, determining the desired measurement step by step.

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Task types: Volume of a triangular prism · Surface area of a triangular prism

Volume and Surface Area of Square Pyramids

Students learn to reliably calculate the volume and surface area of square pyramids using appropriate formulas.

In 10th grade, students calculate the volume and the net of square pyramids. They apply the formulas for base area, lateral area, and volume, and use the Pythagorean theorem when needed to determine missing lengths, such as the pyramid height or the slant height of the lateral faces, from given edges.

A typical stumbling block is confusing the two different heights: the vertical pyramid height inside the pyramid and the slant height on the outer triangles. If the pyramid height is accidentally used for the triangle area, or if the factor of one-third is forgotten when calculating the volume, this leads to calculation errors.

Typical practice problems focus on the formats Volume of a Square Pyramid and Surface Area of a Square Pyramid. In most cases, the base edge and one height are given, from which students determine the required surface areas or capacity step by step.

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Task types: Surface area of a square pyramid · Volume of a square pyramid

Understanding powers with negative exponents

Students learn to rewrite powers with negative exponents as fractions and reliably calculate their values.

In 10th grade, students consolidate their understanding of powers and roots before completing their intermediate secondary education. In doing so, they learn to understand powers with negative exponents as reciprocals and to convert them into fractions with positive exponents—for example, 2^-3 becomes the fraction 1/(2^3) and thus 1/8.

A typical mistake is confusing the minus sign in the exponent with the sign of the final result. Many learners mistakenly assume that a negative exponent automatically leads to a negative number (such as 2^-3 = -8), or they incorrectly multiply the base and the exponent to get -6. It takes practice to internalize that a negative sign in the exponent merely stands for the reciprocal.

Typical exercises from the template Power with a negative exponent require rewriting expressions with negative exponents such as 5^-2 or 10^-3 step by step into unit fractions and then calculating them exactly as a fraction or decimal number.

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Determining missing exponents using exponent rules

Students specifically apply the laws of exponents to correctly determine missing exponents in mathematical expressions in preparation for the intermediate school-leaving certificate.

Curriculum point KMKS1.ZO.17 (our numbering) explain powers and roots and calculate simple powers and roots (our translation)

In preparation for intermediate secondary school qualifications, 10th-grade students consolidate working with powers. They use fundamental laws of exponents when multiplying, dividing, and raising powers to powers to analyze relationships between bases and exponents and to calculate unknown exponents.

A typical stumbling block is confusing operations on exponents with operations on bases. Exponents are often multiplied instead of added when multiplying powers with the same base, or sign rules are applied incorrectly during division.

In task formats such as Missing Exponent — Laws of Exponents, learners work with incomplete equations. They determine the missing number in the exponent so that both sides yield the same value, for example when combining powers with the same base.

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Estimating square roots mentally

Students learn to place the value of non-perfect square roots between two consecutive integers and reasonably estimate it.

Curriculum point KMKS1.ZO.2 (our numbering) use meaningful mental models of real numbers (e.g. completeness of the number line) (our translation)

In 10th grade, students estimate the approximate value of square roots whose radicand is not a perfect square. They use familiar square numbers as reference points to narrow down the value: for example, for the square root of 50, they recognize that the result must lie between 7 and 8 because 50 lies between the square numbers 49 and 64.

A typical mistake is confusing taking a square root with halving a number. It is frequently assumed, incorrectly, that the square root of 50 is 25. In addition, some students find it difficult to assess which of the two adjacent numbers the result is closer to—in the case of 50, significantly closer to 7 than to 8.

On the worksheets for the template Estimating Square Roots, students usually enter such approximate values as a compound inequality or mark the estimated position on a number line. This reinforces a fundamental number sense when working with real numbers before the intermediate school-leaving certificate, entirely without a calculator.

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Representing numbers in scientific notation

Students learn to clearly represent and convert very large and very small numbers in scientific notation.

Curriculum point KMKS1.ZO.5 (our numbering) represent numbers appropriately for the situation, e.g. including in scientific notation (powers of ten) (our translation)

By the end of grade 10, students represent numbers appropriately for the situation by converting them into scientific notation using powers of ten. They convert standard numbers into a form consisting of a number between 1 and 10 multiplied by a power of ten, and also convert these forms back into standard decimal numbers.

Typical errors occur primarily when determining the exponent: often, students merely count the number of zeros instead of determining the actual number of places the decimal point moves. For very small numbers less than 1, the negative sign of the power is also frequently forgotten or confused with the sign of the number itself.

In practice, exercises for the template Scientific notation require numbers such as amounts in the millions or tiny decimal fractions to be rewritten into the appropriate form with powers of ten, or powers of ten to be written out again as complete numbers.

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Negative exponent