The purpose of this document is to provide suggestions on
instructional techniques and philosophies. It is not meant to dictate policy and
would likely be more useful to instructors who are relatively new to teaching
the course (or ones similar to it) than those with more experience with such
courses.
LOGIC
SECTION 8.1
Do not do example six (or problems like it) because quantifiers are not part of
the course. Therefore, leave out the Sets of Real Numbers presentation on page
456. However, it would be nice to introduce it immediately before section 1.6.
SECTION 8.3
Although circuits are not part of the course, some students many benefit from
seeing the Equivalent Statements box on page 478. Going over them could aid the
students with their general understanding of the statements. Com prehension
should take precedence over memory.
SECTION 8.4
Knowledge should be required of the names of all four statements in the blue box
on page 483. Stress the equivalence or non‐equivalence of the converse, inverse,
and contra‐ positive statements to the condition statement.
SECTION 8.6
Have students understand that a logical argument with n premises is valid if and
only if the statement
“If [(premise 1) and (premise 2) and … (premise n)], then (conclusion)” is a
tautology.
SETS
SECTION 9.1
1) Students sometimes confuse the symbols Ø,0, and {0}.
Provide clarification.
2) Make connections between set intersection, set union, and set complement with
the respective conjunction, disjunction,
and negation statements from logic.
3) Other relationships of set theory to logic can be made. An example is the
connection between the set statement
and the logic statement A∪U= U and the logic statement p∨T=T.
4) Students sometimes believe that (A∪B)′is A′∪B′ and that (A∩B) ′is A′∩B′
. Show that neither is true using
counterexamples. It would help to again use logic to make connections. True
statements are
(A∪B)′=A′∩B′and (A∩B) ′= A′∪B′These are known as the DeMorgan’s Laws
for Sets.
SECTION 9.2
1) Be sure to place emphasis on the material of hw problems 19‐39.
2) Stress comprehension of the Union Rule for Counting on page 528 rather
than strict memory.
Make liberal use of Venn Diagram when teaching set theory. Venn Diagrams are
very effective visual aids.
PROBABILITY
This section should be presented using a conceptual rather than a formulaic one .
Yes, the formulas are important but it is
necessary for them to be understood as
well as memorized.
Section 9.3
1) Stress the terms experiment, outcomes, event, sample space, and
equally likely events.
2) Show how an experiment can have more than one sample space and point out that
sample spaces with equally likely events
are the most useful. That will become more apparent once the Basic Probability
Formula on page 537 is given. This can be
called the Classical Definition of the Probability of an Event.
3) Explain how by definition, all possible probability values range from 0 to 1.
Grading is up to the discretion of the instructor
and I make it my discretion to award no credit for negative probabilities or
probabilities larger than one.
SECTION 9.4
1) Make connections between the probability rules of this section and set
theory.
2) Avoid using the notation of the blue box on page 547. It can be confusing to
students.
3) Problems like example 10 on page 547 and hw problems #59‐62 can be solved
using Venn Diagrams. Students tend to like
that method of solution.
4) A common mistake of students is the forgetting of subtracting the probability
of A and B when using the Addition Rule (page
541). The error stems from not understanding the rule. Show how a probability
larger than one could arise if that error is
made. One can use the sample space S= {a,b,c,d,e,f} with events A={a,b,e,f}
and B= {b,d,e} demonstrate.
SECTION 9.5
1) Conditional Probability causes much difficulty for
students. It would help to stress the sample space restriction concept.
2) Deriving 
from
can be useful.
3) Trees are very helpful in this section. One can use trees without ever having
to directly memorize the Multiplication Rule.
4) The concepts of independent events and mutually exclusive events
can be confused by students.
5) Be sure to test on problems like #57‐62 on page 565. On
the test, do not state that the problem is on independent events.
Part of the testing process is determining if the students are able to figure
out that independent events are involved.
6) The derivation of the multiplication rule for independent events P(E∩F)=P(E)
(F) if and only E and F are
independent from the definition P(E｜F)=P(E) aids in student understanding.
SECTION 9.6
The material of this section should be treated as a part of 9.5. DO NOT state
the Bayes Theorem Formula! Rather, teach these
problems using trees. An alternative to the tree diagram method is the use
of charts
Counting, Probability Distributions, and Further Topics
in Probability
Section 10.1
Be sure that students are aware that the expected value of a random variable X
represents an average. Therefore, expected
cannot be less than the
minimum X or larger than the maximum. Students have been known to give such
responses on test questions. Doing so indicates a lack of diligence on the part
of the student.
Section 10.2
1) It can be easier for students to use the Multiplication Principle to evaluate
instead of the formula given on page 396. nrP
Also, 
can be evaluated as
, which students sometimes prefer over
.
2) It helps to make clear to students that (n -r)! is not n! - r! . Use
specific values of n and r to demonstrate that.
3) The ability to decide if permutations or combinations are to be used cannot
be overstressed. Permutations take order into
conside ration and combinations do not. Key words for permutations are
arrangement, rank, and sequence. Key
combinations words are selection, sample,
group, and subset. Have students look for forms of such words when
encountered with situations involving counting techniques.
Section 10.3
1) This material tends to be one of the more difficult ones for the students.
Frequent practice and asking for help when
needed are the keys. Stress finding the size of the sample space first when
forming the probability fraction. That will help
minimize the possibility of giving a probability of larger than one.
2) Be sure to ask an expected value problem or two when counting techniques are
useful in finding the probabilities
Section 10.4.
1) Mention that fact that the prefix bi of the word binomial indicates the
number two. That could help bring out the concept of
the two mutually exclusive outcomes success and failure.
2) Give the classic coin toss experiment example and use
outcomes such as heads/tails, yes/no, blue/not blue, five/not five as
illustrations of possible outcomes of binomial experiments.
3) Show several examples of problems using the phrases “at least”, “no more
than”, “no less than”, “less than”, and “more
than”.
4) The “plausibility argument” on page 617 of the expected value formula for a
binomial distribution should be supported
mathematically. To accomplish that, one can compute the expected value in
example two on page 616 using the definition
from section 10.1. A general proof requires the Binomial Theorem and is beyond
the scope of the course. However, it is
good for the students to at least see how the short formula E(x)=np arises
mathematically from the section 10.1 def.
Introduction to Statistics
Section 11.1
1) Ask finding the median of a data set with an even number of data
points.
2) Discuss the advantages and disadvantages of the mean, median, and mode as
data set representations. Place strong
emphasis on the median being more characteristic of a data set with extreme
values (outliers).
3) As stated in the departmental syllabus, you can leave out the mean and modal
class of grouped data.
Section 11.2
1) It is highly discouraged to cover the variance and standard deviation of
grouped data. That material is not worth all the
tedious arithmetic that it involves.
2) There are advantages of the standard deviation over the variance that should
be mentioned:
a) Unless the variance is less than or equal to one, the standard deviation is a
smaller number than the variance . That is
especially convenient when the
variance is large.
b) The variance can contain unfeasible units such as “ squared dollars ” or
“squared units”. Since the standard deviation is the
square root of the variance, there will be no presence of unusual “squared
units” in the standard deviation.
c) The standard deviation has a high level of importance in higher level
statistics but the variance does not.
Section 11.3
1) Do a good variety of examples and be sure to include those of determining a
data point from a given probability. The
equation 
allows for going from data points to
probabilities and the other way around. If desired, one can present
the equivalent formula 
as the conversion tool
from a probability to the corresponding data point x. That would
apply to hw problems 15‐18 and 37‐39.
2) Remind students that negative numbers or numbers
greater than one are impossible results for hw problems like #9‐14, and
22‐36.
3) Talking about the five‐number summary and boxplots is fine, but do not
require the construction of boxplots.
Section 11.4
Do not overlook the np≥5 and n(1-p)≥5 requirement given before example two on
page 685. This material is one of the tougher sections of the course but is an
important application of the normal distribution and should not be glossed over.
First‐Degree equations
1) hw problems like 15‐26 and 39‐48 need not be done.
2) Applications are important. Be sure to assign at least some problems on page
55 and do problems like example 11 in class.
3) Discourage “guessing and checking” for solving problems. The algebraic
manipulations are necessary and all students who
earn a college degree should have the ability to solve a basic linear equation
using algebra . Require that your students show
work for full credit.
Graphs, Lines and Inequalities
1) It could be true that many of your students have covered this material in
high school, but have done so using graphing
calculators and/or open‐note, open book, formula sheet‐type assessment. As a
result, they may not have a genuine
understanding of the material. Thus, this material should be taught from the
point of view of the students never having seen
it before!
2) Applications of linear equations cannot be overstressed. However, do
not cover material from this section involving
square roots, quadratic functions ( parabolas ), cubic functions , or ellipses .
3) As stated in the Departmental Syllabus, for section 2.5, only problems like
exercises 2‐26 and 57‐62 are required.
Systems of Linear Equations and Linear Inequalities
1) In section 6.1, do the 2 x 2 linear systems using substitution and
elimination. As in the chapter 2 material, teach section 6.1
as if the students have not seen the material before.
2) When covering systems of linear inequalities, be sure that students are clear
on the usage of dashed lines and solid lines.
3) Linear Programming applications (section 7.3) should not be treated lightly.
One of these problems should be on your final
exam and the semester test on the material.
Optional Topics
The requirement is that exactly one of Matrix Operations, the
Gauss‐Jordan Method, and Markov Chains be covered. Which one of the three is
your choice. Points to consider are:
1) Matrix Operations is the easiest of the three but the least useful in
applications.
2) Gauss‐Jordan is very tedious.
3) Markov Chains require some matrix multiplication.
NOTE: The course coordinator of MGF 1106 reserves the right to disallow the
assignment of 1106 sections to any adjunct who demonstrates deviation from the
requirements of material coverage and/or calculator and assessment policies
stated in the MGF 1106 Departmental Syllabus.
