Download Topic 7 Descriptive Statistics

International Baccalaureate
Mathematical Studies SL
2010-2011
MATHEMATICAL STUDIES–SL
INTERNATIONAL BACCALAUREATE
Prerequisites: Algebra Ia and Ib, Algebra 1 and/or Geometry.
Open to: 11, 12
Credits: 2
This course is a two-year program available at standard level (SL)
only. It is intended for students with varied backgrounds and
abilities. More specifically, it is designed to build confidence and
encourage an appreciation of mathematics in students who do not
anticipate a need for mathematics in their future studies. Students
taking this course need to be already equipped with fundamental
skills and a rudimentary knowledge of basic processes.
This course concentrates on mathematics that can be applied to
contexts related as far as possible to other subjects being studied, to
common real-world occurrences and to topics that relate to home,
work, and leisure situations. The course requires students to
produce a project, a piece of written work based on personal
research, guided and supervised by the teacher. The project
provides an opportunity for students to carry out a mathematical
investigation in the context of another course being studied, a hobby
or interest of their choice using skills learned before and during the
course.
It is recommended that students in this course have a TI-84 or TI84+ graphing calculator.
Topics First Year
Topic 1 Introduction to the graphic display calculator
Arithmetic calculations, use of the GDC to graph a variety of functions.
Appropriate choice of “window”; use of “zoom” and “trace” (or equivalent) to
locate points to a given accuracy.
Explanations of commonly used buttons.
Entering data in lists.
Topic 2 Number Sets, Properties and Measurement
The sets of natural number, N; integers, Z; rational numbers, Q and real
numbers, R.
Approximation: decimal places, significant figures; Percentage errors;
Estimation.
Expressing numbers in the form a×10Λk, where 1≤a≤10 and kєZ. Operations
with numbers expressed in the form a×10Λk, where 1≤a≤10 and kєZ.
SI (Système International) and other basic units of measurement; for example,
grapm (g), metre (m), second (s), liter (l), metre per second (m s -1), Celsius
and Fahrenheit scales.
Topic 3 Sets and Venn Diagrams
Basic concepts of set theory: subsets; intersection; union; complement.
Venn diagrams and simple applications.
Sample space: event, A; complementary event, A′
Venn diagrams; tree diagrams; tables of outcomes.
Topic 4 The Rule of Pythagoras
Use of Pythagorean Theorem to find coordinate distances including lengths of
altitudes, medians and angle bisectors and points of concurrency.
Topic 5 Coordinate Geometry
Coordinates in two dimensions: points; lines; midpoints. Distances between
points.
Equation of a line in two dimensions: the forms y=mx+c and ax+by+d=0.
Gradient; intercepts. Points of intersection of lines; parallel lines;
perpendicular lines.
Lengths of lines joining vertices with vertices, vertices with midpoints and
midpoints with midpoints; sizes of angles between two lines and between
lines and planes.
Topic 6 Linear and Exponential Algebra
Solutions of pairs of linear equations in two variables by use of a GDC.
Concept of a function as a mapping.
Domain and range. Mapping diagrams.
Graphs and properties of exponential functions. Growth and decay; basic
concepts of asymptotic behaviour.
Accurate graph drawing.
Topic 7 Descriptive Statistics
Classification of data as discrete or continuous.
Simple discrete data: frequency tables; frequency polygons.
Grouped discrete or continuous data: frequency tables; mid-interval values;
upper and lower boundaries. Frequency histograms. Stem and leaf diagrams
(stem plots).
Cumulative frequency tables for grouped discrete data and for grouped
continuous data; cumulative frequency curves. Box and whisker plots (box
plots). Percentiles; quartiles.
Measures of central tendency.
For simple discrete data: mean; median; mode. For grouped discrete and
continuous data: approximate mean; modal group; 50th percentile.
Measures of dispersion: range; interquartile range; standard deviation.
Scatter diagrams; line of best fit, by eye, passing through the mean point.
Topic 8 Two Variable Statistics
Bivariate data: the concept of correlation.
Pearson’s product–moment correlation coefficient. Interpretation of positive,
zero and negative correlations.
The regression line for y on x. Use of the regression line for prediction
purposes.
The χ2 test for independence: formulation of null and alternative hypotheses;
significance levels; contingency tables; expected frequencies; degrees of
freedom; use of tables for critical values; p-values.
Topic 9 Financial mathematics
Currency conversions, Simple Interest, I =Crn/100 where C=capital, r =% rate,
n = number of time periods, I – interest.
Compound interest. Depreciation. The value of r can be positive or negative.
Construction and use of tables: loans and repayment schemes; investment and
saving schemes, inflation.
Topic 10 Probability
Laws of probability. Combined Events; Mutually exclusive events; Independent
events; Conditional probability. Solution of problems using “with
replacement” and “without replacement”.
Topic 11 Logic
Basic concepts of symbolic logic: definition of a proposition; symbolic notation
of propositions. Compound statements: implication, equivalence, negation,
conjunction, disjunction,exclusive disjunction . Translation between verbal
statements, symbolic form and Venn diagrams. Knowledge and use of the
“exclusive disjunction” and the distinction between it and “disjunction”.
Truth tables: the use of truth tables to provide proofs for the properties of
connectives; concepts of logical contradiction and tautology.
Definition of implication: converse; inverse; contrapositive. Logical
equivalence.
Topics Second Year
Topic 1 Quadratic Algebra
Solutions of quadratic equations: by factorizing; by use of a GDC.
Topic 2 Function Notation and Quadratic Functions
The graph of the quadratic function:
f (x)=ax^2+bx+c. Properties of symmetry; vertex; intercepts.
Topic 3 Exponential and Trigonometric Functions
Graphs and properties of the sine and cosine functions.
Amplitudes and periods.
Topic 4 Numerical Trigonometry
Right angled trigonometry. Use of the ratios of sine; cosine and tangent.
The sine rule; the cosine rule; area of a triangle; construction of labeled
diagrams from verbal statements.
Topic 5 Perimeter, Area and Volume
Geometry of three-dimensional shapes: cuboid; prism; pyramid; cylinder;
sphere; hemisphere; cone.
Topic 6 Sequences and Series
Arithmetic sequences and series, and their applications. Use of the formulae
for the nth term and the sum of the first n terms.
Geometric sequences and series, and their applications. Use of the formulae
for the nth term and the sum of n terms.
Topic 7 More Functions
Use of a GDC to sketch and analyse some simple, unfamiliar functions.
Use of a GDC to solve equations involving simple combinations of some simple,
unfamiliar functions.
Topic 8 Introductory differential calculus
Gradient of the line through two points, P and Q, that lie on the graph of a
function. Behaviour of the gradient of the line through two points, P and Q, on
the graph of the function as Q approaches P. Tangent to a curve.
The derivative functions.
Gradients of curves for given values of x. Values of x where f’(x) is given.
Equation of the tangent at a given point.
Increasing and decreasing functions.
Values of x where the gradient of a curve is 0 (zero); solution for f’(x) =0.
Local maximum and minimum points.