Mathematics for science units

Grade 12 advanced

Contents

12AS.1 Algebra 1 379 12AS.9 Integration 1 455

12AS.2 Trigonometry 397 12AS.10 Measures 2 465

12AS.3 Algebra 2 403 12AS.11 Integration 2 471

12AS.4 Vectors 1 413 12AS.12 Vectors 3 481

12AS.5 Differentiation 1 421 12AS.13 Differential equations 487

12AS.6 Measures 1 433 Optional Complex numbers 495

12AS.7 Differentiation 2 439 Optional Numerical methods 505

12AS.8 Vectors 2 449

Mathematics units: Grade 12 advanced: mathematics for science 135 teaching hours

25%

1st semester70 hours

2nd semester65 hours

Reasoning and problem

solving should be integrated into each unit

UNIT 12AS.1: Algebra 1Techniques of algebraAlgebraic manipulation: multiplication, factorisation,simplification, combining and simplifying algebraicfractions, partial fractionsAlgebraic division: division of polynomial by first or secondorder polynomial, remainder and factor theoremsIndices and logarithms: exponents, roots, rules oflogarithms, change of base, use of calculatorBinomial theorem revision; nPr and nCr; binomial series20 hours

UNIT 12AS.3: Algebra 2Functions, graphs and equationsExponential, natural logarithm and modulus functions;composite functions; inverse functionsContinuity; parabolic functions and parametric formsQuadratic, exponential, logarithmic and trigonometricequations14 hours

UNIT 12AS.5: Differentiation 1Higher order derivatives; stationary points, maxima andminima, inflexions; second derivative testIncreasing and decreasing functions; standard functions16 hours

75%

UNIT 12AS.0: Grade 11A revision3 hours

UNIT 12AS.9: Integration 1Integration as inverse of differentiationIndefinite integrals, definite integrals, areas under curves12 hours

UNIT 12AS.11: Integration 2Integration techniques and applications12 hours

UNIT 12AS.7: Differentiation 2Derivative of sum, difference, product and quotient of twofunctions; derivative of composite function; derivative ofimplicitly defined function; applications; numericalmethods15 hours

UNIT 12AS.13: Differential equationsSolution of simple differential equations; experimentalmodels; simple harmonic motion10 hours

Optional unit: Complex numbersComplex numbers and functions of a complex variable[10 hours]

Optional unit: Numerical methodsTaylor series; iteration; Newton-Raphson method[8 hours]

UNIT 12AS.8: Vectors 2Vector equation of a lineApplications using vectors torepresent physical situations5 hours

UNIT 12AS.2: TrigonometryRevision of radian measureTrigonometric identitiesTrigonometric equations6 hours

UNIT 12AS.4: Vectors 1Components, unit vectors;position and displacementRules of vector algebraMagnitude, scalar product,applications of scalar product8 hours

UNIT 12AS.6: Measures 1Compound measuresRate of change3 hours

UNIT 12AS.12: Vectors 3Vector kinematicsSolving dynamics problems bydifferentiating or integratingvectors5 hours

UNIT 12AS.10: Measures 2Using integration to calculateareas and volumes6 hours

379 | Qatar mathematics scheme of work | Grade 12 advanced: mathematics for science | Unit 12AS.1 | Algebra 1 Education Institute 2005

GRADE 12AS: Algebra 1

Logarithms, binomial series and partial fractions

About this unit This is the first of two units on algebra for Grade 12 advanced: mathematics for science. It builds on the algebra units for Grade 11 advanced and comprises a collection of techniques covering algebraic operations, logarithms, combinatorics, the binomial series and partial fractions.

The unit is designed to guide your planning and teaching of mathematics lessons. It provides a link between the standards for mathematics and your lesson plans.

The teaching and learning activities should help you to plan the content and pace of lessons. Adapt the ideas to meet your students needs. Supplement the activities where necessary with appropriate tasks and exercises from textbooks and other resources, including ICT.

For consolidation activities, look at the units for Grade 11 advanced; for extension or enrichment, consider activities in the Grade 12 advanced units for quantitative mathematics, or on those websites referred to in the text.

Introduce the unit to students by summarising what they will learn and how this builds on earlier work. Review the unit at the end, drawing out the main learning points, links to other work and real-world applications.

Previous learning To meet the expectations of this unit, students should already be able to add, subtract and multiply two functions. They should understand exponents and nth roots, and be able to apply the laws of indices. They should be able to solve for x the equation y = ax and use the log function (logarithm in base 10) on a calculator. They should understand and use factorial notation and know the binomial theorem expansion of (1 + x)n for positive integer n. They should be able to find permutations and combinations.

Expectations By the end of the unit, students will break problems into smaller tasks, and set up and perform relevant manipulations. They will identify and use connections between mathematical topics. They will develop and explain chains of logical reasoning, using correct mathematical notation and terms, including logic symbols. They will generalise when possible and remark on special cases. They will approach problems systematically. They will work to expected degrees of accuracy. They will continue to develop skills of algebraic manipulation through further work on factorisation, exponents and logarithms, partial fractions and combinatorics. They will understand and use the remainder theorem and the factor theorem. They will expand and use the binomial series (1 + x)n for any rational value of n. Students who progress further will become more fluent in the logical analysis of combinatorics. They will be aware of the limitations of the methods they use, and more imaginative in finding ways round them.

Resources The main resources needed for this unit are: overhead projector (OHP) Internet access, computer and data projector spreadsheet software such as Microsoft Excel graph plotting software such as:

Autograph (see www.autograph-math.com) Graphmatica (free from www8.pair.com/ksoft)

computers with Internet access, spreadsheet and graph plotting software for students

graphics calculators for students

Key vocabulary and technical terms Students should understand, use and spell correctly: logarithm, exponential, dividend, divisor, quotient, remainder, factor domain of convergence, series expansion in ascending and descending

powers partial fraction, improper fraction, identity, linear and repeated factor

UNIT 12AS.1 20 hours

380 | Qatar mathematics scheme of work | Grade 12 advanced: mathematics for science | Unit 12AS.1 | Algebra 1 Education Institute 2005

Standards for the unit

20 hours SUPPORTING STANDARDS Grade 10A and 11A standards CORE STANDARDS Grade 12AS standards

EXTENSION STANDARDS

12AS.1.3 Identify and use interconnections between mathematical topics.

12AS.1.4 Break down complex problems into smaller tasks.

12AS.1.5 Use a range of strategies to solve problems, including working the problem backwards and redirecting the logic forwards; set up and solve relevant equations and perform appropriate calculations and manipulations; change the viewpoint or mathematical representation; and introduce numerical, algebraic, graphical, geometrical or statistical reasoning as necessary.

12AS.1.6 Develop chains of logical reasoning, using correct terminology and mathematical notation, including symbols for logical implication.

12AS.1.7 Explain their reasoning, both orally and in writing.

12AS.1.8 Understand and generate mathematical proofs, and discuss exceptional cases, knowing the importance of a counter-example.

12AS.1.9 Generalise whenever possible.

12AS.1.10 Approach complex problems systematically, recognising when it is important to enumerate all outcomes.

12AS.1.13 Work to expected degrees of accuracy, and know when an exact solution is appropriate.

11A.5.14 Add, subtract and multiply two functions.

12AS.2.1 Multiply, factorise and simplify expressions and divide a polynomial by a linear or quadratic expression.

12AS.2.2 Combine and simplify rational algebraic fractions.

12AS.2.3 Decompose a rational algebraic fraction into partial fractions (with denominators not more complicated than repeated linear terms).

12AS.2.4 Understand and use the remainder theorem.

12AS.2.5 Understand and use the factor theorem.

10A.3.1 Understand exponents and nth roots, and apply the laws of indices.

12AS.3.1 Understand exponents and nth roots, and apply the laws of indices to simplify expressions involving exponents; use the xy key and its inverse on a calculator.

3 hours

Algebraic fractions

3 hours

The remainder theorem and its uses

2 hours

Logic and implication

3 hours

Permutations and combinations

3 hours

Logarithms in theory and practice

3 hours

From the binomial theorem to the binomial series

3 hours

Partial fractions 11A.5.18 12AS.3.2 Know the definition of a logarithm in number base a (a > 0), and the rules

of combination of logarithms, including change of base.

12AS.5.1 Use a graphics calculator to plot exponential functions of the form y = ekx; describe these functions, distinguishing between cases when k is positive or negative, and the special case when k is zero.

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