Module 1: Counting

Text Chapter 11

1.  Use systematic listing, product tables and tree diagrams to represent the set of possible results of a single-step or multi-step task.

1. For a multi-step task

• Determine if the uniformity criterion applies.
• If the uniformity criterion applies use the fundamental counting theorem to determine the number of ways in which the task can be accomplished.

2. Determine the number of orders in which n objects can be arranged.

1. Recognize whether a given task in which a given number of objects are chosen from a given set involves permutations or combinations.

2. Explain how to reason out the number of permutations of r objects chosen from a set of n objects.

3. Explain how to reason out the number of combinations of r objects chosen from a set of n objects.

4. Explain how to reason out the number of r-element subsets of an n-element set.

1. Use Pascal's Triangle to find C(n, r) for given n and r.

2. Use the binomial theorem to expand a given binomial to a given power.

3. Recognize and verify patterns in Pascal's Triangle.

Module 2: Probability

Text Chapter 12

1. Explain the concept and give the definition of probability.

2. State and apply the formula for theoretical probability.

3. State and apply the formula for empirical probability.

4. State and give an example to illustrate the Law of Large Numbers.

5. Distinguish between odds and probability, and given one of these quantities find the other.

1. Know, determine the applicability of, explain in terms of examples and if applicable use to solve problems:

• the complements rule of probability
• the general addition rule of probability
• the special addition rule of probability

2. Use the probability distribution for a given random variable to determine probabilities of various events.

1.  Know and apply the definition of conditional probability.

2. Explain in terms of Venn diagrams the meaning of and formula for conditional probability.

3. Know and apply the definition of independent events.

4. Know and apply the rule for the probability that two independent events will both occur.

1.  For the binomial probability formula explain, generally or in terms of a nontrivial example, the meaning of each:

• C(n, x)
• p
• q
• p^x
• q^(n-x)
• p^x * q^(n - x)
• why the probability of exactly x successes in n attempts is not p^x * q^(n - x)

2. Apply the concept of and/or formula for binomial probability to solve mathematical or real-world problems.

1.  Explain, generally or in terms of a nontrivial example, the meaning of mathematical expectation.

2. Apply the concept of and/or formula for mathematical expectation to solve mathematical or real-world problems.

1.  Simulate a given process using coin flips or a table of random numbers.

Module 3: Statistics

Text Chapter 12

1.  Read and interpret frequency distributions using the following as requested, and express data given in one format in the other formats:

• listing
• grouped frequency distribution
• stem-and-leaf display
• bar graphs
• pie charts

1.  Calculate and interpret the mean, median and mode of a set of data items.

2. Calculate and interpret the weighted mean, median and mode of a frequency distribution.

3. Recognize when a frequency distribution is symmetric, skewed to the left, skewed to the right and/or bimodal.

1.  For a given set of data calculate and interpret the range and standard deviation.

2. For a given set of data verify Chebyshev's Theorem.

1.  For a given value of a random variable, given the mean and standard deviation of its distribution, find its z value.

2. Calculate percentiles, deciles and quartiles for a set of data items.

3. Construct a box-and-whisker plot of a given set of data.

1. Know the probabilities that a normally distributed random variable will take a value within 1, 2 and 3 standard deviations of the mean.

2. Sketch the standard normal curve and relabel for a given mean and standard deviation.

3. Using a sketch of the normal curve estimate the probabilities of various events.

4. Using a z-table determine the probabilities of various events.

1. Given a set of data, hand-sketch a scatter plot and estimate the slope and y-intercept of the best-fit straight line, as estimate the correlation coefficient.

2. Given a set of data and the necessary formulas, calculate the slope and y-intercept of the best-fit straight line as well as the correlation coefficient, and write the equation of the best-fit straight line.

Module 4: Geometry

Text Chapter 9

1. Know the meanings of points, lines and planes, and the associated notations.

2. Know and apply the definitions of

• line segments
• parallel lines
• intersecting lines
• parallel planes
• intersecting planes
• skew lines
• angle
• vertex
• acute, right and obtuse angles
• perpendicular lines
• vertical angles
• complementary and supplementary angles
• transversals and vertical angles

1. Define, sketch, recognize and apply:

• curves which are and are not closed
• curves which are and are not simple
• regions which are and are not convex
• closed figures which are polygons
• closed figures which are regular polygons
• acute, right, obtuse, equilateral, isosceles and scalene triangles
• various quadrilaterals:  trapezoids, parallelograms, rectangles, squares, rhombi
• circles

2. Know, apply and be able to demonstrate that sum of the angles of a triangle is 360 degrees.

3. Given the angles of a triangle find the measure of any specified exterior angle.

4. Know the theorem about an angle inscribed in a semicircle.

Know and apply the definitions of perimeter (or if more appropriate circumference) and area to solve problems involving triangles, the various quadrilaterals and circles.

1. Know and apply

• the congruence properties of triangles
• the properties of isosceles triangles
• properties of similarity of triangles
• the Pythagorean Theorem and its converse

1. Know and apply the definitions and properties of

• the regular polyhedra
• pyramids
• prisms
• cylinders
• cones
• tori (plural of 'torus')

2. Use the formulas for the area and circumference of a circle and the definition of volume to find the area and circumference of a right circular cylinder, and to verify the standard formulas for the area and volume.

3. Explain how to use the area of the base and the altitude of a prism or cylinder to find its volume.

4. Explain how to use the '1/3 principle' to find the volume of a pyramid or a cone, given its altitude and the area of its base.

5. Know and apply the formulas for the surface area and volume of a sphere.

1.  Explain in terms of an example why any pair of straight lines on a sphere must intersect, while it is possible to construct two non-intersecting straight lines on a plane.

2. Explain in terms of an example why the sum of the angles of a triangle constructed on a spherical surface is greater that 180 degrees.

3. Sketch a surface on which the sum of the angles of a triangle would be less than 180 degrees.

4. Identify the genus of a given object.

5. Explain why it is impossible to traverse a network having more than two odd vertices.

6. Given a network determine whether it is traversible, and if it is sketch a path that traverses it.

1.  Show how succesive iterations of the logistic equation can lead to convergence or chaos depending on the value of the parameter m.

2.  For a given construction of self-similar objects of increasing length, identify the scale factor for size that results from a convenient ratio of lengths.