Holdet 2dMa/en (2025/26) - Undervisningsbeskrivelse - Lectio

Holdet 2dMa/en (2025/26) - Undervisningsbeskrivelse

Undervisningsbeskrivelse

Stamoplysninger til brug ved prøver til gymnasiale uddannelser

Termin(er)	2024/25 - 2025/26
Institution	Rødkilde Gymnasium
Fag og niveau	Matematik B
Lærer(e)	Mathias Fugmann
Hold	2024 Ma/d/en (1dMa/en, 2dMa/en)

Oversigt over gennemførte undervisningsforløb

Titel 1	Mathematical thinking
Titel 2	Exponential functions
Titel 3	Private economics
Titel 4	Descriptive statistics
Titel 5	Trigonometry
Titel 6	Power functions
Titel 7	Polynomials
Titel 8	Probabilities and combinatorics
Titel 9	Logarithms
Titel 10	Election math
Titel 11	Differential calculus
Titel 12	Plane geometry
Titel 13	Binomial distribution
Titel 14	Theory of differential calculus
Titel 15	Repetition and exam

Beskrivelse af de enkelte undervisningsforløb (1 skema for hvert forløb)

Titel 1	Mathematical thinking Content Short course consisting of creative problem solving working in groups at whiteboards. The problems worked on aimed to practice speaking about math in English as well as practicing basic algebra and abstract pattern recognition. No litterature.
Indhold
Omfang	Estimeret: Ikke angivet Dækker over: 4 moduler
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Titel 2	Exponential functions Contents Literature – Notes in OneNote. – D. Roegel (2010): "Briggs' 1624 table of logarithms (reconstruction)". Core substance – Formula, graph and table for exponential functions. – Exponential growth property. – Two-point formula for exponential functions. – Logarithm as inverse of exponential function (base 10 and e). Focus on application of logarithms to solve exponential equations. – Natural exponential function of base e and conversion to/from general form. – Domain and range of exponential and logarithmic functions. – Doubling time and half-life. Supplementary substance – Historical perspective on logarithms: multiplication and division using Briggs' logarithm table from 1624. Proofs – Two-point formula. – Doubling time and half-life.
Indhold
Omfang	Estimeret: Ikke angivet Dækker over: 10 moduler
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Titel 3	Private economics Contents Literature – Notes in OneNote – Teacher note: "Project private economics". Core substance – Compund interest – Annuity savings – Annuity loans Supplementary substance – Budget – Estimate calculations – Tax In the project the students made meal plans, grocery lists and went to Føtex to estimate their food/beverage expenses. They have researched housing prices, tax rates, SU-loan and deductables. The final product was a presentation of their project.
Indhold
Omfang	Estimeret: Ikke angivet Dækker over: 7 moduler
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Titel 4	Descriptive statistics Literature – Notes in OneNote – Mike Vandal Auerbach (2020): "Statistics" Chapter 1, 2 and 3 (except 2.4) Core substance – Statistical descriptors for grouped and ungrouped data sets. -- Mean value, standard deviation, five-number summary, frequency, relative frequency, cumulated frequency, cumulated relative frequency, range, outliers -- Graphical representations: bar chart, dot chart, boxplot, cumulated relative frequency graph, histogram – How to choose samples - representative, systematic errors, hidden variables.
Indhold
Omfang	Estimeret: Ikke angivet Dækker over: 8 moduler
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Titel 5	Trigonometry Contents Literature – Notes in OneNote. – Mike Vandal Auerbach (2019): Plane Geometry. Chapter 4 (except 4.4). Core substance – Right-angled triangles. – Pythagorean Theorem – Definition of sin(v), cos(v), tan(v) in the unit circle – Similar triangles and the multiplication factor – The area of the triangle using sin(v) – The law of sine – The law of cosine Proofs – The Pythagorean Theorem. – Law of sine – Area formula – Law of cosine
Indhold
Omfang	Estimeret: Ikke angivet Dækker over: 8 moduler
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Titel 6	Power functions Contents Literature – Notes in OneNote – Mike Vandal Auerbach (2019): Functions. Chapter 5, 5.1 and 5.2. Core substance – Formula – Power growth – Power regression – Two point formula Proofs – Two point formula
Indhold
Omfang	Estimeret: 3,00 moduler Dækker over: 3 moduler
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Titel 7	Polynomials Contents Covering primarily quadratic polynomials, polynomial regression and a few properties of general n-th degree polynomials. Literature – Teacher notes in OneNote. Core substance – Characteristics of quadratic polynomials. – Vertex. – Roots. – Zero-product property. – Factoring. – Polynomial regression. Supplementary substance – Number of roots in polynomial of degree n. – Uniqueness of polynomial regression of degree n+1 for n data points. – Shamir's Secret Sharing. Proofs – Formula for roots.
Indhold
Omfang	Estimeret: Ikke angivet Dækker over: 8 moduler
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Væsentligste arbejdsformer

Titel 8	Probabilities and combinatorics Contents Introduction to basic concepts in combinatorics and probability calculations. Concludes with a written test. Litterature – Teacher note: Combinatorics. 10 pages. – Teacher note: Probability theory. 11 pages. Core substance – Set notation. – Addition and multiplication principles. – Permutations of n objects taken r at a time. Formula derived via example. – Combinations of n objects taken r at a time, i.e. the binomial formula. – Pascal's triangle. – Probability space: sample space and probability distribution. – A priori and empirical probability. – Law of large numbers. – Events and probability of event: P(A)=favorable outcomes/possible outcomes – Complement events. – Independent events. – Random variables. – Mean, variance and standard deviation of random variable. Supplementary substance – Simulation of the Monty Hall problem.
Indhold
Omfang	Estimeret: Ikke angivet Dækker over: 11 moduler
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Titel 9	Logarithms Contents We return to logarithms, known from the topic of exponential functions, to further examine the logarithm rules and other properties fo logarithms, including logarithmic transformations of exponential and power functions. Literature – Teacher note: Logarithmic functions. 8 pages. – Virtual slide rule. URL: museum.syssrc.com/static/sliderule.html. – Roegel, Denis: A reconstruction of Briggs' Arithmetica logarithmica (1624). URL: locomat.loria.fr/briggs1624/briggs1624doc.pdf. A few pages used to look up logarithms. – Teacher notes in OneNote. Core substance – Definition of log and ln as inverse functions of 10^x and e^x. – Domain and range of logarithmic functions. – Graphs of logarithmic functions. – Calculation rules for logarithms. – Growth property. – Semi-log and log-log plots. Supplementary substance – Historical perspective: the use of logarithm tables and the slide rule. Proofs – Calculation rules for logarithms. – Transformation of exponential and power functions into linear functions.
Indhold
Omfang	Estimeret: Ikke angivet Dækker over: 3 moduler
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Titel 10	Election math Contents We take a look at different mathematical approaches of allocating seats after an election and discuss what is fair. Litterature – Teacher note on seat distribution methods. Translated after Vestergaard: Mandatfordelinger og retfærdighed. URL: www.matematiksider.dk/mandatfordelinger.html. 2 pages. – Teacher note on seat distribution in local elections in Denmark. 2 pages. – "The Electoral System in Denmark". Ministry of Social Affairs and the Interior, 2020. Pages 1–6. Supplementary substance – Collaboration with social studies on how elections work. – Working with authentic data from Statistics Denmark.
Indhold
Omfang	Estimeret: Ikke angivet Dækker over: 4 moduler
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Titel 11	Differential calculus Contents A practical introduction to differential calculus and how it's used to analyze functions. Concluded with a written test and a group project on optimization where the group designed a container and minimized use of material. Literature – Formula booklet. P. 36–41. – Notes in OneNote. Core substance – Definition of derivative as slope of tangent at a point. – Interpretation as rate of change. – Calculating derivatives. – Rules of differentiation: k·f, f±g, f·g. – Tangent equation. – Monotony analysis. – Optimization. Proofs None (see course "Theory of differential calculus").
Indhold
Omfang	Estimeret: Ikke angivet Dækker over: 13 moduler
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Titel 12	Plane geometry Contents Literature – Euclid: The Game. URL: https://euclid.findell.org/Level1/ – Teacher note: "Euclid's Elements and mathematical logic[/url]. 9 pages. Covers definitions, postulates, common notions and propositions I.1–I.3+I.47 from Euclid's Elements, examples of mathematical conditional statements and biconditional statements, the converse Pythagorean theorem, and a brief comment on the foundational crisis of mathematics. – Teacher note: "Analytical plane geometry". 15 pages. – Teacher notes in OneNote. Core substance – Cartesian coordinate system. – Distance between points. – Linear equation y=ax+b. – Orthogonal lines. – Angle of inclination. – Distance between point and line. – Equation of circle. – Completing the square. – Circle tangents. – Intersections. Supplementary substance – Euclid's Elements as a foundation of axiomatic-deductic structure. Proofs – Distance dist(P, l) from point to line. – Proposition I.1–I.3 and I.47 (the Pythagorean theorem). – The converse Pythagorean theorem.
Indhold
Omfang	Estimeret: Ikke angivet Dækker over: 15 moduler
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Titel 13	Binomial distribution Contents Literature – Mike Vandal Auerbach (2020): "The Binomial Distribution". In: Probability Theory, p. 19–24. URL: https://www.mathematicus.dk/matematik/english/Probability_Theory.pdf. – Teacher notes in OneNote. Core substance – Binomial trial, parameters n (number of trials) and p (probability of success). – Probability in a binomial distribution, X~b(n,p). – Mean value and standard deviation of a random variable in a binomial distribution. – Binomial test, two-sided. Supplementary substance – One-sided binomial test. – Binomial theorem. Proofs – Formula P(X=r) for probability of r successes in n repetitions. Generalized from example. – Formula for mean value in the case n=3.
Indhold
Omfang	Estimeret: Ikke angivet Dækker over: 7 moduler
Særlige fokuspunkter
Væsentligste arbejdsformer

Titel 14	Theory of differential calculus Contents Literature – Khan Academy: "Introduction to piecewise functions". URL: https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:absolute-value-piecewise-functions/x2f8bb11595b61c86:piecewise-functions/v/piecewise-function-example?referrer=share_link. Duration: 3:48 min. – Khan Academy: "Worked example: evaluating piecewise functions". URL: https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:absolute-value-piecewise-functions/x2f8bb11595b61c86:piecewise-functions/v/evaluating-piecewise-functions-example?referrer=share_link. Duration: 4:23 min. – Teacher note: "Properties of parabolas". Proof of relation between coefficients and parabola, and coordinates of vertex, using differential calculus. – Teacher note: "Growth of a power function". Account of the different cases of the exponent in a power function using differential calculus. – Teacher note: "Derivatives proofs". Proofs using the three-step method of x^2, quadratic polynomial, reciprocal function, square root, and x^3. – Teacher note: "Mean value theorem". Illustration of the mean value theorem and proof that sign of the derivative is equivalent with the function's monotonicity. – Teacher notes in OneNote. Core substance – Composite functions and the chain rule. – Piecewise functions (in context of continuous and differentiable functions). – Derivatives of simple functions using the three-step method. Supplementary substance – Interpretation of double derivative as acceleration/deceleration. – Analysis of coefficients in quadratic polynomial. – Mean value theorem. Proofs – Meaning of a, b and c for the graphs of a quadratic polynomial. – Rate of change and acceleration of power functions in the cases a<0, 0<a<1, and a>1. – Derivative of x^2, quadratic polynomial, reciprocal function, square root, and x^3. – Sign of derivative for increasing and decreasing functions.
Indhold
Omfang	Estimeret: Ikke angivet Dækker over: 8 moduler
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Væsentligste arbejdsformer

Titel 15	Repetition and exam Contents Allowed tools for part 2 of the written exam – WordMat. – Macro-enabled Excel sheet for binomial distribution. – GeoGebra Classic 6 (or 5). Online version might be necessary due to technical issues with regression analysis. URL: https://www.geogebra.org/classic. – Microsoft Excel. Allowed ressources – OneNote class notebook "Mathematics 2024d e-class" (collaboration space is turned off for the exam). – Any previous assignments and solutions to these. – Any documents, images, videos, etc. that are locally accessible.
Indhold	Kernestof: Make sure WordMat is updated.
Omfang	Estimeret: Ikke angivet Dækker over: 13 moduler
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