**Year 12 IB SL Application and Interpretation**

**Year 12 IB SL Application and Interpretation**

**12 SL AI: Summer revision list**

You should revise the following topics (chapter references in brackets):

· Sequences and series (5.3, 10.1, 10.2)

· Functions (5.1)

· Linear functions (5.2)

· Quadratic functions (9.1, 9.2)

· Exponential functions (10.3, 10.4)

· Differentiation (12.1 – 12.3)

· Coordinate geometry (4.1 – 4.4)

· Volume and surface area of 3D shapes (2.3)

· Pythagoras and trigonometry (1.5, 1.6, 2.1, 2.2)

· Sampling (3.2)

· Measures of central tendency and dispersion (3.1)

· Presentation of data (3.3)

· Compound interest, annuities and amortization (10.2)

· Cubic models (9.3)

**Prior Knowledge **

NUMBER AND ALGEBRA: Approximation and estimation, Standard form

1.5 Laws of exponents with integer exponents. SI Units, Percentage Error,

GEOMETRY: Coordinates: points, lines, midpoints, Equations of lines, parallel and perpendicular, length & gradient of lines joining points

3.2, 3.3 Right-angled trigonometry and sine and cosine rule, applications, angles of elevation and depression

3.1 3-D shapes, volume and surface area: cuboid, prism, right-pyramid, cylinder, sphere, hemisphere, right-cone

3.1 angle between 2 lines, angle between line and plane

STATISTICS: Scatter diagrams, Probability

CALCULUS: Differentiation of xn, Identification of stationary points

FUNCTION NOTATION

1.5 Laws of exponents with integer exponents. SI Units, Percentage Error,

GEOMETRY: Coordinates: points, lines, midpoints, Equations of lines, parallel and perpendicular, length & gradient of lines joining points

3.2, 3.3 Right-angled trigonometry and sine and cosine rule, applications, angles of elevation and depression

3.1 3-D shapes, volume and surface area: cuboid, prism, right-pyramid, cylinder, sphere, hemisphere, right-cone

3.1 angle between 2 lines, angle between line and plane

STATISTICS: Scatter diagrams, Probability

CALCULUS: Differentiation of xn, Identification of stationary points

FUNCTION NOTATION

**Year 12 Autumn Term - First Half**

**1.5 Law of exponents with integer exponents [2]**

Introduction to logarithms with base 10 and e. Awareness that a^

*x*=

*b*is equivalent to log_a b = x, that b > 0, and log_e x = ln x. Numerical evaluation of logarithms using technology.

**1.2, 1.3, 1.4 Sequences and series [8]**

Arithmetic sequences and series. Use of the formulae for the nth term and the sum of the first n terms of the sequence. Use of sigma notation for sums of arithmetic sequences. Geometric sequences and series. Use of the formulae for the nth term and the sum of the first n terms of the sequence. Use of sigma notation for the sums of geometric sequences. Application of the above, including compound interest, depreciation and population growth.

**2.2, 2.3, 2.4 Functions [8]**

Concept of a function, domain, range and graph.

Function notation, for example f(x) , v(t) , C(n)

The concept of a function as a mathematical model.

Informal concept that an inverse function reverses or undoes the effect of a function.

Inverse function as a reflection in the line y = x, and the notation f.

The graph of a function; its equation y = f(x)

Creating a sketch from information given or a context, including transferring a graph from screen to paper.

Using technology to graph functions including their sums and differences. Determine key features of graphs. Finding the point of intersection of two curves or lines using technology.

**2.1 Linear functions [2] Review from IGCSE**

Different forms of the equation of a straight line.

Gradient; intercepts.

Lines with gradients m1 and m2.

Parallel lines m1 = m2.

Perpendicular lines m1 × m2 = … 1.

**2.5 Quadratic Functions [6]**

The quadratic function f(x) = ax2 + bx + c ; a ≠ 0.

Axis of symmetry, vertex, zeros and roots.

The solution of ax2 + bx + c = 0, a ¹ 0.

The quadratic formula.

The form of a(x – h)2 + k : vertex (h, k).

The form of a( x – p)(x – q): x-intercepts ( p, 0) and ( q, 0).

The discriminant

**Test on Sequences, Series and Functions [2**

**]**

**Year 12 Autumn Term - Second Half**

Exponential growth and decay models.

f(x) = ka^x + c

f(x) = ka^-x + c (for a > 0)

f(x) = ke^rx + c

Equation of a horizontal asymptote.

Modelling with Quadratic and Exponential functions

Introduction to the concept of a limit.

Derivative of sums of integer powers of x

Derivative interpreted as gradient function and as rate of change.

Increasing and decreasing functions.

Graphical interpretation of f ′(x) > 0, f ′(x) = 0, f ′(x) < 0.

Derivative of f(x) = ax^n is f ′(x) = anx^n-1 , n ∈ ℤ

The derivative of functions of the form f x = ax^n + bx^n-1 where all exponents are integers.

Values of

Maximum and minimum points (include second derivative) and optimization problems in context

Tangents and normals at a given point, and their equations.

TEST on everything so far with modelling and algebra heavy questions.

Review of test and splitting of groups as required

**2.5 Exponential functions [4]**Exponential growth and decay models.

f(x) = ka^x + c

f(x) = ka^-x + c (for a > 0)

f(x) = ke^rx + c

Equation of a horizontal asymptote.

**2.6 Modelling [4]**Modelling with Quadratic and Exponential functions

**5.1,5.2,5.3,5.4,5.6,5,7 Calculus [10]**Introduction to the concept of a limit.

Derivative of sums of integer powers of x

Derivative interpreted as gradient function and as rate of change.

Increasing and decreasing functions.

Graphical interpretation of f ′(x) > 0, f ′(x) = 0, f ′(x) < 0.

Derivative of f(x) = ax^n is f ′(x) = anx^n-1 , n ∈ ℤ

The derivative of functions of the form f x = ax^n + bx^n-1 where all exponents are integers.

Values of

*x*where the gradient of a curve is zero. Solution of f ′(x) = 0.Maximum and minimum points (include second derivative) and optimization problems in context

Tangents and normals at a given point, and their equations.

TEST on everything so far with modelling and algebra heavy questions.

Review of test and splitting of groups as required

## **Year 12 Spring Term - First Half**

The distance between two points in three-dimensional space, and their midpoint.

Volume and surface area of three-dimensional solids including right-pyramid, right cone, sphere, hemisphere and combinations of these solids.

The size of an angle between two intersecting lines or between a line and a plane.

Use of sine, cosine and tangent ratios to find the sides and angles of right-angled triangles.

The sine rule

The cosine rule

Area of a triangle as 1/2 absinC.

Applications of right and non-right angled trigonometry, including Pythagoras’ theorem.

Angles of elevation and depression.

Construction of labelled diagrams from written statements.

4.1 Introduction to sampling [2]

Concepts of population, sample, random sample, discrete and continuous data.

Reliability of data sources and bias in sampling.

Interpretation of outliers.

Sampling techniques and their effectiveness.

4.3 Measures of central tendency [6]

Measures of central tendency (mean, median and mode).

Estimation of mean from grouped data.

Modal class

Measures of dispersion (interquartile range, standard deviation and variance).

Effect of constant changes on the original data.

Quartiles of discrete data

4.2 Presentation of data [2] Review from IGCSE

Presentation of data (discrete and continuous): frequency distributions (tables).

Histograms.

Cumulative frequency; cumulative frequency graphs; use to find median, quartiles, percentiles, range and interquartile range (IQR).

Production and understanding of box and whisker diagrams.

**3.1 Geometry [6]**The distance between two points in three-dimensional space, and their midpoint.

Volume and surface area of three-dimensional solids including right-pyramid, right cone, sphere, hemisphere and combinations of these solids.

The size of an angle between two intersecting lines or between a line and a plane.

**3.2,3.3 Sine rule, Cosine rule, ½ ab sinc [8]**Use of sine, cosine and tangent ratios to find the sides and angles of right-angled triangles.

The sine rule

The cosine rule

Area of a triangle as 1/2 absinC.

Applications of right and non-right angled trigonometry, including Pythagoras’ theorem.

Angles of elevation and depression.

Construction of labelled diagrams from written statements.

4.1 Introduction to sampling [2]

Concepts of population, sample, random sample, discrete and continuous data.

Reliability of data sources and bias in sampling.

Interpretation of outliers.

Sampling techniques and their effectiveness.

4.3 Measures of central tendency [6]

Measures of central tendency (mean, median and mode).

Estimation of mean from grouped data.

Modal class

Measures of dispersion (interquartile range, standard deviation and variance).

Effect of constant changes on the original data.

Quartiles of discrete data

4.2 Presentation of data [2] Review from IGCSE

Presentation of data (discrete and continuous): frequency distributions (tables).

Histograms.

Cumulative frequency; cumulative frequency graphs; use to find median, quartiles, percentiles, range and interquartile range (IQR).

Production and understanding of box and whisker diagrams.

**Year 12 Spring Term - Second Half**

1.7 Finiancial [2]

Amortization and annuities using technology.

1.8 Solving systems of equations [2]

Use technology to solve:

Systems of linear equations in up to 3 variables

Polynomial equations

2.5, 2.6 Cubic models [4]

f(x) = ax^3 + bx^2 + cx + d.

Applications

2.5, 2.6 Sinusoidal models [4]

f x = asin(bx) + d, f x = acos(bx) + d.

Applications

4.4 Bivariate Data [6]

Linear correlation of bivariate data.

Pearson’s product-moment correlation coefficient, r.

Scatter diagrams; lines of best fit, by eye, passing through the mean point.

Equation of the regression line of y on x.

Use of the equation of the regression line for prediction purposes. Interpret the meaning of the parameters, a and b, in a linear regression y = ax + b.

Amortization and annuities using technology.

1.8 Solving systems of equations [2]

Use technology to solve:

Systems of linear equations in up to 3 variables

Polynomial equations

2.5, 2.6 Cubic models [4]

f(x) = ax^3 + bx^2 + cx + d.

Applications

2.5, 2.6 Sinusoidal models [4]

f x = asin(bx) + d, f x = acos(bx) + d.

Applications

4.4 Bivariate Data [6]

Linear correlation of bivariate data.

Pearson’s product-moment correlation coefficient, r.

Scatter diagrams; lines of best fit, by eye, passing through the mean point.

Equation of the regression line of y on x.

Use of the equation of the regression line for prediction purposes. Interpret the meaning of the parameters, a and b, in a linear regression y = ax + b.

**Year 12 Summer Term - First Half**

Internal exams [4]

Go through papers [2]

5.5 Integration [10]

Introduction to integration as anti-differentiation of functions of the form f(x) = axn + bxn …1 + .., where n ∈ ℤ, n ≠ … 1.

Anti-differentiation with a boundary condition to determine the constant term.

Definite integrals using technology.

Area of a region enclosed by a curve y = f(x) and the x-axis, where f(x) > 0.

Approximating areas using the trapezoidal rule.

1.05j tanx=sinx/cosx and sin^2+cos^2=1. Solving trigonometric equations and simple trigonometric proofs.

1.05b Sine & cosine rules, including bearings and ambiguous case.

1.05c . A=1/2 ab Sin C

1.05o Solving trigonometric equations in an interval, including quadratics and equations involving multiples of x . e.g. sinx=0.5 for 0<=x <360, 6sin^x+cosx-4=0 for 0<=x <360, tan3x=-1 for -180<x<180 . Extending this knowledge to include radians and further trigonometric identities.

EE week [4]

Go through papers [2]

5.5 Integration [10]

Introduction to integration as anti-differentiation of functions of the form f(x) = axn + bxn …1 + .., where n ∈ ℤ, n ≠ … 1.

Anti-differentiation with a boundary condition to determine the constant term.

Definite integrals using technology.

Area of a region enclosed by a curve y = f(x) and the x-axis, where f(x) > 0.

Approximating areas using the trapezoidal rule.

1.05j tanx=sinx/cosx and sin^2+cos^2=1. Solving trigonometric equations and simple trigonometric proofs.

1.05b Sine & cosine rules, including bearings and ambiguous case.

1.05c . A=1/2 ab Sin C

1.05o Solving trigonometric equations in an interval, including quadratics and equations involving multiples of x . e.g. sinx=0.5 for 0<=x <360, 6sin^x+cosx-4=0 for 0<=x <360, tan3x=-1 for -180<x<180 . Extending this knowledge to include radians and further trigonometric identities.

EE week [4]

**Year 12 Summer Term - Second Half**

Exploration [2]

3.4 Circle, arc, sector [4]

The circle: length of an arc; area of a sector.

3.5, 3.6 Voronoi diagrams[6]

Equations of perpendicular bisectors.

Voronoi diagrams: sites, vertices, edges, cells

Addition of a site to an existing Voronoi diagram.

Nearest neighbour interpolation.

Applications of the “toxic waste dump” problem.

Concepts of trial, outcome, equally likely outcomes, relative frequency, sample space (U) and event.

The probability of an event A is P(A) = n(A) n(U) .

The complementary events A and A′ (not A).

Expected number of occurrences.

Use of Venn diagrams, tree diagrams, sample space diagrams and tables of outcomes to calculate probabilities.

Combined events: P(A ∪ B) = P(A) + P(B) … P(A ∩ B). Mutually exclusive events: P (A ∩ B) = 0

Conditional probability: P(A|B) = P(A ∩ B)/ P(B) .

Independent events: P(A ∩ B) = P(A)P(B).

3.4 Circle, arc, sector [4]

The circle: length of an arc; area of a sector.

3.5, 3.6 Voronoi diagrams[6]

Equations of perpendicular bisectors.

Voronoi diagrams: sites, vertices, edges, cells

Addition of a site to an existing Voronoi diagram.

Nearest neighbour interpolation.

Applications of the “toxic waste dump” problem.

**4.5 Probability [6]**Concepts of trial, outcome, equally likely outcomes, relative frequency, sample space (U) and event.

The probability of an event A is P(A) = n(A) n(U) .

The complementary events A and A′ (not A).

Expected number of occurrences.

**4.6 Conditional Probability****[6]**Use of Venn diagrams, tree diagrams, sample space diagrams and tables of outcomes to calculate probabilities.

Combined events: P(A ∪ B) = P(A) + P(B) … P(A ∩ B). Mutually exclusive events: P (A ∩ B) = 0

Conditional probability: P(A|B) = P(A ∩ B)/ P(B) .

Independent events: P(A ∩ B) = P(A)P(B).