Undervisningsbeskrivelse
Stamoplysninger til brug ved prøver til gymnasiale uddannelser
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Termin(er)
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2024/25
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Institution
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X - Ikast-Brande Gymnasium
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Fag og niveau
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Matematik -
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Lærer(e)
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Anders Dalegaard
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Hold
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2024 MathAnSL/1 (1i MathAnSL/1)
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Oversigt over gennemførte undervisningsforløb
Beskrivelse af de enkelte undervisningsforløb (1 skema for hvert forløb)
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Titel
1
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Algebra 1
Unit Overview: Analysis and Approaches Standard Level
Unit Title: Core Algebra and Sequences
Topics Covered:
- Straight Lines
- Simultaneous Equations
- Radicals and Surds
- Rules of Exponents
- Arithmetic and Geometric Sequences and Series
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Learning Objectives:
By the end of this unit, students will:
- Understand and apply the fundamental properties of straight lines, including gradient, intercepts, and forms of linear equations.
- Solve simultaneous equations algebraically and graphically.
- Simplify, rationalize, and operate with radicals and surds.
- Apply the laws of exponents in simplifying expressions and solving equations.
- Analyze and solve problems involving arithmetic and geometric sequences and series, including the use of sigma notation.
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1. Straight Lines
- Key Concepts:
- Gradient and its interpretation.
- Forms of linear equations, such as slope-intercept form, point-slope form, and general form.
- Parallel and perpendicular lines.
- Applications: interpreting and modeling real-world data using linear equations.
- Skills:
- Derive the equation of a line given two points or a point and the gradient.
- Find the point of intersection of two lines.
- TOK Links:
- How do different representations (algebraic, graphical) of a line provide insight into mathematical concepts?
- How does the concept of "slope" connect to real-world interpretations of change?
- ATL Skills:
- Thinking Skills: Apply critical thinking to derive equations and analyze intersections.
- Communication Skills: Represent and explain relationships graphically and algebraically.
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2. Simultaneous Equations
- Key Concepts:
- Systems of linear equations in two variables.
- Graphical and algebraic methods (substitution and elimination).
- Consistency and dependence of systems (unique, infinite solutions, or no solution).
- Skills:
- Solve real-life problems using systems of equations.
- Use technology (e.g., graphing calculators) to visualize and solve systems.
- TOK Links:
- To what extent does the choice of method (graphical or algebraic) influence the way we approach solutions?
- What assumptions underlie the formulation of linear systems?
- ATL Skills:
- Research Skills: Use technology to model and solve systems.
- Collaboration Skills: Work in pairs or groups to solve systems of equations and validate solutions.
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3. Radicals and Surds
- Key Concepts:
- Definitions: radicals and surds (irrational square roots).
- Simplifying surds.
- Rationalizing denominators.
- Addition, subtraction, multiplication, and division of surds.
- Skills:
- Express radicals in simplified forms.
- Operate with radicals in equations and expressions.
- TOK Links:
- Why are some numbers considered "irrational," and how does this classification reflect the nature of mathematics?
- How do we decide the "simplest" form of a mathematical expression?
- ATL Skills:
- Self-Management Skills: Practice organization in step-by-step simplification processes.
- Critical Thinking: Analyze and justify the simplification of surds.
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4. Rules of Exponents
- Key Concepts:
- Laws of exponents, including rules for multiplication, division, powers, zero, and negative exponents.
- Exponential growth and decay.
- Simplifying and solving equations with exponents.
- Skills:
- Apply exponent laws in simplifications and problem-solving.
- Evaluate expressions involving fractional and negative exponents.
- TOK Links:
- How does the concept of infinity relate to exponents and their rules?
- To what extent do the rules of exponents depend on axiomatic systems?
- ATL Skills:
- Transfer Skills: Apply rules of exponents across different contexts.
- Information Literacy: Interpret exponential models in real-world scenarios.
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5. Arithmetic and Geometric Sequences and Series
- Key Concepts:
- Definitions of sequences and series.
- Arithmetic sequence: common difference and nth term formula.
- Geometric sequence: common ratio and nth term formula.
- Sum of n terms in arithmetic and geometric series.
- Infinite geometric series and conditions for convergence.
- Skills:
- Derive and apply nth term and sum formulas.
- Solve real-world problems involving sequences and series.
- Use sigma notation to represent and compute series.
- TOK Links:
- How does the concept of infinity influence our understanding of series?
- How do arithmetic and geometric sequences model patterns in nature and human activity?
- ATL Skills:
- Numeracy Skills: Interpret and calculate terms and sums accurately.
- Creative Thinking: Explore patterns and relationships in sequences.
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Assessment Opportunities:
- Formative Assessments:
- In-class practice problems.
- Group work on real-life applications (e.g., modeling with straight lines).
- Mini-quizzes on individual topics.
- Summative Assessments:
- Unit test covering all topics.
- Problem-solving tasks requiring integration of multiple topics (e.g., solving a real-world problem involving a sequence).
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Resources:
- Textbook chapters on algebra and sequences.
- Graphing calculators.
- Online tools like GeoGebra for visualization.
- IB Question Bank for past exam practice.
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Interdisciplinary Connections:
- Science: Modeling linear relationships in physics (e.g., velocity vs. time graphs).
- Economics: Analyzing growth patterns using sequences and series.
- Computer Science: Understanding algorithms involving arithmetic or geometric patterns.
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Differentiation Strategies:
- For advanced learners: Include proofs of formulas for sequences and encourage exploration of non-linear systems.
- For struggling learners: Provide scaffolding for simultaneous equations and additional practice with radicals and exponents.
- For visual learners: Use graphing software to illustrate key concepts.
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Estimated Duration:
4-6 weeks depending on student progress and integration of assessment tasks.
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Indhold
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Kernestof:
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Omfang
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Estimeret:
23,00 moduler
Dækker over:
19 moduler
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Særlige fokuspunkter
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Væsentligste arbejdsformer
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Titel
2
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Geometry 1
Unit Overview: Analysis and Approaches Standard Level
Unit Title: Geometry and Trigonometry Foundations
Topics Covered:
- Distance and Midpoints Between Points in 2D and 3D
- Area and Volume
- Right-Angled Trigonometry
- Non right-angled trigonometry
- Sine and cosine rule
- The Unit Circle
- Length of arc and area of sector
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Learning Objectives:
By the end of this unit, students will:
- Calculate distances and midpoints in two and three dimensions.
- Understand and apply formulas for the area of various shapes and volumes of solids.
- Solve problems using right-angled trigonometry, including real-world applications.
- Explore the unit circle and its role in defining trigonometric functions.
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1. Distance and Midpoints Between Points in 2D and 3D
- Key Concepts:
- Definitions of distance and midpoint.
- Relationship between geometry and coordinate systems.
- Extension from 2D to 3D coordinate geometry.
- Skills:
- Calculate the distance and midpoint between two points.
- Apply these concepts to solve geometric problems in 3D space.
- TOK Links:
- How does the abstraction of a coordinate system help in understanding spatial relationships?
- To what extent are distance and midpoint formulas dependent on the definitions of dimensions?
- ATL Skills:
- Thinking Skills: Analyze and solve spatial problems using coordinate geometry.
- Communication Skills: Represent spatial relationships clearly and accurately.
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2. Area and Volume
- Key Concepts:
- Areas of basic shapes: triangles, rectangles, circles.
- Volumes of solids: cubes, rectangular prisms, spheres, cylinders, and cones.
- Composite shapes and solids.
- Skills:
- Use appropriate formulas to find areas and volumes.
- Solve real-world problems involving areas and volumes.
- TOK Links:
- How do different cultures approach the calculation and representation of space?
- How are units of measurement standardized, and how does this affect our calculations?
- ATL Skills:
- Numeracy Skills: Accurately compute areas and volumes.
- Research Skills: Investigate applications of area and volume in real-world contexts.
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3. Right-Angled Trigonometry
- Key Concepts:
- Definitions of sine, cosine, and tangent.
- Relationships between sides and angles in right-angled triangles.
- Applications in solving triangles and modeling real-world problems.
- Skills:
- Solve for unknown sides or angles in right-angled triangles.
- Apply trigonometric ratios to practical scenarios.
- TOK Links:
- How do trigonometric concepts connect to other areas of mathematics and science?
- To what extent does trigonometry rely on the human-defined concept of an angle?
- ATL Skills:
- Critical Thinking: Analyze problems to identify appropriate trigonometric methods.
- Self-Management: Practice systematic approaches to solving trigonometric problems.
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4. The Unit Circle
- Key Concepts:
- Definition and significance of the unit circle.
- Sine and cosine as coordinates of points on the unit circle.
- Extension to all four quadrants and understanding periodicity.
- Skills:
- Relate angles to points on the unit circle.
- Interpret trigonometric functions in terms of the unit circle.
- TOK Links:
- How does the unit circle provide a unifying framework for understanding trigonometry?
- How are periodic phenomena modeled mathematically using the unit circle?
- ATL Skills:
- Transfer Skills: Apply unit circle concepts to solve trigonometric problems.
- Creative Thinking: Explore connections between geometry, algebra, and trigonometry.
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Assessment Opportunities:
- Formative Assessments:
- Practice problems on distances, midpoints, and areas.
- Group work on real-life trigonometric applications.
- Visual explorations of the unit circle using graphing tools.
- Summative Assessments:
- Unit test covering all topics.
- Investigative tasks combining multiple concepts, such as analyzing a geometric shape using area, volume, and trigonometry.
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Resources:
- Textbook chapters on geometry and trigonometry.
- Graphing calculators and software for visualizing concepts.
- IB Question Bank for past exam practice.
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Interdisciplinary Connections:
- Physics: Applications of trigonometry in forces and motion.
- Art: Understanding symmetry and proportion through geometric calculations.
- Geography: Using coordinate systems to interpret maps and spatial data.
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Differentiation Strategies:
- For advanced learners: Explore derivations of area and volume formulas and their connections to calculus.
- For struggling learners: Provide visual aids and step-by-step instructions for trigonometric and geometric concepts.
- For kinesthetic learners: Use hands-on activities to model and calculate areas and volumes.
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Estimated Duration:
4-6 weeks depending on student progress and integration of assessment tasks.
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Indhold
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Omfang
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Estimeret:
12,00 moduler
Dækker over:
9 moduler
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Særlige fokuspunkter
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Væsentligste arbejdsformer
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Titel
3
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3. Functions 1
Unit Title: Functions and Equations
Topics Covered:
2.1: Concept and notation of functions
2.2: Domain, range, and graph of functions
2.3: Composite functions
2.4: Inverse functions
2.5: Polynomial functions
2.6: Rational functions
2.7: Exponential and logarithmic functions
2.9: Transformations of graphs
2.10: Solving equations graphically
Objectives:
Understand and use the concept and notation of functions.
Determine the domain and range of functions and sketch their graphs.
Work with composite and inverse functions.
Analyze polynomial, rational, exponential, and logarithmic functions.
Apply transformations to function graphs.
Solve equations graphically.
Assessment:
Formative assessments through quizzes and homework assignments.
Summative assessment through a unit test covering all topics.
Links to Theory of Knowledge (TOK):
Mathematical Models: Discuss the role of mathematical models in representing real-world phenomena. How do models help us understand the world, and what are their limitations?
Function Notation: Explore the historical development of function notation and its impact on mathematical communication and understanding.
Graphical Solutions: Consider the reliability of graphical solutions in mathematics. How do different representations (algebraic vs. graphical) influence our understanding of solutions?
Approaches to Learning (ATL):
Thinking Skills: Develop critical thinking by analyzing and solving complex problems involving various types of functions.
Communication Skills: Enhance mathematical communication through the use of precise function notation and graphical representations.
Self-Management Skills: Foster self-management by encouraging students to organize their study schedules and manage their learning process effectively.
Research Skills: Promote research skills by investigating real-world applications of functions and equations.
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Indhold
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Kernestof:
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Omfang
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Estimeret:
23,00 moduler
Dækker over:
29 moduler
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Særlige fokuspunkter
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Væsentligste arbejdsformer
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Titel
4
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Statistics
Topics Covered:
4.1 Concepts of population, sample, and random sampling
4.2 Presentation of data (frequency tables, histograms, cumulative frequency graphs)
4.3 Measures of central tendency (mean, median, mode) and spread (range, interquartile range, standard deviation)
4.4 Correlation and linear regression (scatterplots, Pearson’s correlation coefficient)
4.10 Interpretation of the parameters of a linear model
Learning Objectives
By the end of this unit, students will be able to:
Distinguish between populations and samples and understand basic sampling methods.
Represent data appropriately using tables, histograms, and cumulative frequency curves.
Calculate and interpret measures of central tendency and spread.
Analyze data using scatterplots, describe correlation, and use linear regression to model relationships.
Interpret the slope and intercept of linear models in context.
Content Overview
1. Populations and Samples (4.1)
Understanding populations, samples, and the purpose of sampling.
Different sampling techniques (random, stratified, etc.).
2. Data Presentation (4.2)
Frequency distributions, cumulative frequency tables.
Histograms and cumulative frequency graphs.
Boxplots and understanding quartiles and outliers.
3. Measures of Central Tendency and Spread (4.3)
Mean, median, mode, range, interquartile range (IQR), and standard deviation.
Interpreting and comparing data sets using these measures.
4. Correlation and Linear Regression (4.4)
Scatterplots and describing relationships between two variables.
Pearson’s correlation coefficient, interpretation of strength and direction.
Line of best fit by least squares method.
5. Interpretation of Linear Models (4.10)
Meaning of slope and intercept in real-world contexts.
Using models to predict values and understanding limitations.
Theory of Knowledge (TOK) Links
How do we decide whether data truly represents reality?
What is the role of context in interpreting statistics?
How do models influence our interpretation of data, and when might they mislead us?
To what extent can correlation imply causation, and what are the dangers of assuming it does?
Approaches to Learning (ATL) Skills
Research Skills: Gather and interpret data sets.
Thinking Skills: Critically assess the appropriateness of statistical models.
Communication Skills: Present data visually and explain findings in context.
Self-Management Skills: Organize and manage data-handling processes systematically.
Information Literacy: Evaluate and use secondary data sources appropriately.
Assessment Opportunities
Formative Assessment:
Quizzes on calculating and interpreting measures of center and spread.
Data presentation projects (e.g., creating histograms, boxplots).
Small group data investigations and presentations.
Summative Assessment:
Unit test including both theoretical questions and practical applications (interpretation of regression outputs).
Data investigation task using technology (e.g., spreadsheet software or a GDC).
Resources
IB Mathematics: Analysis and Approaches SL textbook
Graphical Display Calculator (GDC)
Online tools: Desmos, GeoGebra Statistics Tool
IB Question Bank for past paper practice
Interdisciplinary Connections
Economics: Analyzing economic data trends.
Biology: Modeling population growth or environmental data.
Social Sciences: Interpreting survey data and trends.
Differentiation Strategies
For students needing support:
Step-by-step scaffolded practice on calculating statistics and constructing graphs.
Use real-life datasets that are more relatable or less complex.
For advanced learners:
Explore non-linear models briefly for comparison.
Investigate real-world datasets with multiple variables (introducing the idea of multiple regression informally).
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Indhold
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Omfang
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Estimeret:
6,00 moduler
Dækker over:
7 moduler
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Særlige fokuspunkter
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Væsentligste arbejdsformer
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