This time around, I have changed the order of presentation. The course will begin with counting, as before. The second unit will now be the graphing unit. I'll be expanding this unit out a bit to include some work with Hamiltonian circuits, graph homeomorphisms, and graph coloring.
Below is the general philosophy of this problem-based course and the outline for the counting unit. All lesson slides are being posted under the course notes on my web site, so you can follow the progression through the course this semester.
I welcome any questions or suggestions as to course content and sequencing.
Overview
The focus of this course is to promote
mathematical thinking by exploring concepts and problems in discrete
mathematics. Students will be expected to reason through problems, to make
sense of the mathematics, and to justify their solutions. Part of the instruction
will help build foundational knowledge of algorithms and demonstrations of
proofs.
This course uses a problem-based approach to
presenting and learning the material.
Problem-based learning
typically follows prescribed steps:
1.
Presentation of an
"ill-structured" (open-ended, "messy") problem
2.
Problem definition or
formulation (the problem statement)
3.
Generation of a
"knowledge inventory" (a list of "what we know about the
problem" and "what we need to know")
4.
Generation of possible
solutions
5.
Formulation of learning
issues for self-directed and coached learning
6.
Sharing of findings and
solutions
Unit
1 – Counting
The first unit will explore concepts in
counting. These ideas will repeatedly come up as students explore areas of
discrete mathematics. Having a foundation for counting will enable students to
have more confidence tackling problems knowing they have these techniques and
abilities at their disposal.
1.
U1-01 What is discrete math
a.
Circle name activity
i.
Push for various solutions and results
b.
Overview of problem-based learning approach
c.
Exploring Pascal’s triangle
i.
Use problem-based approach
ii.
Connections
d.
What is discrete math
i.
Syllabus highlights
ii.
Web site
e.
Counting pre-assessment
i.
Use page 61 of Data Analysis and
Probability Workbook
1.
Project on smartboard
ii.
Go through results on page 20 of Data
Analysis and Probability Workbook Teacher’s Guide
f.
Exit ticket
i.
What connections are there between the circle name activity and Pascal’s
triangle?
ii.
What patterns in Pascal’s triangle surprised you?
iii.
What topics mentioned are you most interested in learning and why?
2.
U1-02a Figurative numbers
a.
Square numbers (1 day)
b.
Triangular numbers (1 day)
c. Finite Differences lesson U2-02o before proceeding (2 day plus)
d.
Figurative numbers (1 day)
i.
Define pentagonal numbers
ii.
Write general expressions pentagonal numbers
1.
Use table and finite difference algorithm
e.
Exit ticket
3.
U1-02b Finite Differences
a.
Explore linear relationships
i.
Example: y = 3x + 7
ii.
Have students explore y = -2x + 3 and y = 2x - 4
b.
Explore quadratic relationships
i.
Have students explore y = x2 and y = 3x2
ii.
Conjecture about relationship
iii.
Try conjecture
iv.
Discuss relationship
c.
Explore cubic relationships
i.
Have students explore y = x3 and y = 2x3
ii.
Have students conjecture about relationship
iii.
Have students check conjecture
iv.
Discuss relationship
d.
Generalize relation
i.
First conjecture and discuss general relations
ii.
test on square or triangular values
e.
Discuss algorithm for extracting equation from table values
i.
Determine degree and first coefficient
ii.
Subtract nth term value
iii.
Calculate new differences
iv.
Repeat process until entire equation complete
v.
Emphasize how this process is an algorithm
f.
Exit
i.
How can finite differences help you determine the relationship in your
data?
g.
Homework
i.
Use finite difference to find equation for triangular or square numbers
h.
Homework discussion
i.
Practice extracting equation
i.
Demonstrate with a quadratic, such as f(x) = 2x2 -3x + 5
ii.
Give students a cubic equation to work through
1.
X3 – 2x2 + x – 3 or something similar
4.
U1-02c polygonal numbers
a.
Figurative numbers (1 day)
i.
Define pentagonal numbers
ii.
Write general expressions pentagonal numbers
1.
Use table and finite difference algorithm
b.
Mastery quiz – write-up of pentagonal numbers
c.
Exit ticket
5.
U1-02d mathematical induction
a.
How do you know something will work for all cases?
b.
Ideas behind mathematical proof
i.
Based on accepted truths
ii.
Logical steps
iii.
Demonstrates true for all situations
iv.
Why are proofs important
1.
Once proven can now use as accepted fact
2.
Allows knowledge to build
3.
Do not have to question whether something will always work
c.
Mathematical induction
i.
One method of mathematical proof
ii.
Useful for formulas based on discrete values
d.
Parts of induction
i.
Demonstrate something is true for a starting point
ii.
Demonstrate that from any given point that you can advance one step
1.
Think of climbing a ladder
iii.
Why does this show something will always be true?
e.
Demonstrate with triangular numbers
i.
Demonstrate formula is true for a starting point
1.
N = 1, show 1(1+1)/2 = 1, yes it works
ii.
Show that from any given point you can advance one step
1.
Assume n(n+1)/2 works. Is it true for value n+1?
a.
Is (n+1)(n +1 + 1)/2 the right
form?
b.
Yes it is
iii.
Therefore, by mathematical induction, true for every k
f.
Have students use pentagonal numbers
6.
U1-A1 Assessment on figurate numbers and finite differences
a.
Give students a table of the first three hexagonal numbers and their
figures
i.
Draw the next hexagonal number
ii.
What is the value of H5 and H10?
iii.
Using finite differences, write an expression for Hn.
iv.
If eight people attend a party and greet all others by shaking hands,
how many handshakes occurred?
b.
Mastery quiz problem
i.
Write up investigation of pentagonal numbers
1.
Drawing
2.
Tables
3.
Finite differences
4.
Formula
7.
U1-03 Combinations and permutations (will cover 2 periods?)
a.
Class picture problem
i.
Many students know the answer is 30!
ii.
Ask them why this is the answer
b.
Faberge eggs
c.
Basketball teams
d.
Pizza combos
e.
Exit ticket
8.
U1-04 Pigeon Hole Principle
a.
Socks in a drawer
i.
Discuss solutions
b.
Gumball problem
i.
Discuss solutions
c.
Definition of pigeon hole principle
i.
3 pigeons and 2 pigeon coups imply on coup contains more than one pigeon
ii.
Does this imply that all coups are occupied?
d.
Three examples – students are to explain why the statements are true
i.
Five cards drawn from a deck implies at least one suit match
ii.
Five integers from 1-8 two must sum to nine
iii.
If train every day for 30 days and you have 45 total training sessions,
then you will have a set of consecutive days with exactly 14 training sessions
e.
What connections do you see across the problems we looked at today and
how does the pigeon hole principle apply?
9.
U1-05 Advanced counting
a.
2 flag problems
i.
8 stripes, 3 colors
1.
How many with at least 6 blue stripes
2.
How many with at least 2 blue stripes
b.
2 playing card problems
i.
4-card hands with two pair
ii.
5-card hands that are full houses
c.
2 people group problems
i.
12 people
1.
Break into one group of 2, one of 3, and one of 7
2.
Break into 4 groups of 3
10. U1-A2 Assessment
on Combinations/Permutations and Pigeon Hole
a.
Permutation
b.
Combination
c.
Pigeon hole
11. U1-06 Basic
probability rules (2 days?)
a.
What is the probability of being dealt a full-house?
i.
What do we need to know/do to calculate this probability
b.
Define P(A) = outcomes of A / total outcomes
i.
Calculate P(dealt full-house)
ii.
Calculate P(dealt 4 of a kind)
c.
Probability Rules
i.
Have students create list of rules they know
1.
Rules to have
a.
P(A or B) = P(A) + P(B) for mutually exclusive events
b.
P(A and B) = P(A) x P(B) for independent events
c.
0 <= P(A) <= 1
d.
P(not A) = 1 – P(A)
e.
sum of all probabilities = 1
f.
may be others
2.
Share out list and discuss
3.
Students create examples of each rule using playing cards
a.
Share and discuss
4.
Cover additional rules not previously identified
d.
Practice problems (use card decks as model)
i.
Probability dealt a black-jack
ii.
Probability dealt a flush
iii.
Probability dealt the queen of spades when four people playing hearts
12. U1-07 Conditional
probability and reversing probability
a.
Example probability problems involving conditional probability
b.
Common characteristics of problems
c.
Define conditional probability
d.
Calculate some conditional probabilities
e.
Define independence of two events
i.
Confirm probability relationship between independent events
f.
Use general multiplication rule to reverse probability
i.
Reference Bayes theorem
ii.
Give example
g.
Provide practice worksheet
i.
Downloaded from education.com
ii.
bayes reference – trouble downloading
h.
Exit
i.
Connections between traditional and conditional probability
ii.
Things to remember about reversing probability
13. U1-08 Counting
Review
a.
Concept map
b.
Practice problems
14. U1-09 Counting
Review
a.
Problem card sort
b.
Practice problems
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