Discrete Mathematics
A Discrete Math Unit Designed for Grade 8 Students
Jessica Stuewe
[email protected]
Lynnea Salscheider
[email protected]
MATH 6200
Bemidji State University
2010 Summer Institute
with Dr. Todd Frauenholtz and Craig Ripkema
Executive Summary!
3
Unit Overview!
4
State and National Standards !
5
Sample Standardized Test Questions !
11
Lesson 1: License Plate Combinations!
13
Lesson 2: Flag Trademarks !
16
Lesson 3: Best of Three, Best of Five...!
19
Lesson 4: Hamburger Toppings!
21
Lesson 5: Bridges of Konigsberg!
23
Lesson 6: Supreme Court Welcome !
26
Lesson 7: Traveling Salesman!
29
Lesson 8: Loops!
31
Discrete Mathematics Test!
34
References !
37
2
Executive Summary
About the unit design:
This unit is designed to meet state and national standards concerning discrete
mathematics in the 8th grade. All of the lessons in this unit make use of the discrete ideas
of non-continuous counting (countable solutions) to solve problems. Discrete topics
include: counting principle, combinations and permutations, using vertex-edge graphs to
solve problems, and using tables and tree diagrams to develop recursive equations. Many
of these lessons include algebraic extensions and options to develop both recursive and
explicit equations.
The Discrete Math Test that is included at the end of this unit is designed to be used as
both a pre and post test. The test will be administered at the beginning of the unit, and
then re-administered after the unit is taught. No changes will be made to the test to allow
for paired t-test scores for comparison of student growth. Test answers will be reviewed
after grading for the unit is complete.
About the teachers:
Jessica Stuewe teaches seventh and eighth grade mathematics in Detroit Lakes,
Minnesota. Detroit Lakes Middle School schedules 43-minute class periods. Jess teaches
two sections of seventh grade mathematics, two sections of eighth grade mathematics
(which will be Eighth Grade Algebra as of autumn, 2010) and one section of Algebra
(which will be Eighth Grade Advanced Algebra as of autumn, 2010).
Lynnea Salscheider teaches eighth grade mathematics in Piedmont, South Carolina. She
teaches on a 70-minute block schedule and has four sections of eighth grade
mathematics, one of which is a special education inclusion class.
Both teachers wrote this unit while completing coursework at the Bemidji State University
in Bemidji, Minnesota during the summer institute of 2010 in pursuit of their master
degrees in mathematics education.
3
Unit Overview
Lesson
Days
Big Ideas
“License Plates”
1-2
“Flag Trademarks”
2
“Best of Three”
1-2
•Enumerating sample space
•Tree diagrams
•Combination
•Combination formula
“Hamburger
Toppings”
1-2
•Enumerating sample space
•Tree diagrams
•Combination
•Combination formula
•Permutation formula
“Bridges of
Konigsberg”
1-2
•Vertex-edge graphs
•Completion of circuits (Euler Circuit)
•Practical applications and problem solving
“Supreme Court
Welcome”
2-4
•Recognizing patterns
•Exploring various representations to solve problems
-numeric
-geometric (vertex-edge graph)
-algebraic (recursive relationships)
•Use formulas to extend patterns in a spreadsheet and graph those
relationships
“Traveling
Salesman”
2-4
•Use weighted vertex-edge graphs to solve real life problems
•Use the “nearest neighbor” and the “greedy” algorithms to solve
problems
•Sort a list of data in ascending order
•Use factorial to determine the total number of possible solutions
“Loops”
1-2
•Recognizing patterns
•Using recursive formulas to predict the next outcome
•Counting principle
•Factorial notation
•Enumerating (counting) sample space (Number of Outcomes)
•Tree diagrams
•Counting principle
•Factorial notation
4
State and National Standards
Lesson
NCTM Standard(s)
http://
standards.nctm.org/
document/chapter1/
index.htm
MN Standard(s)
http://education.state.mn.us/MDE/
Academic_Excellence/
Academic_Standards/
Mathematics/index.html
“License
Plates”
•understand the concepts
of sample space and
probability distribution
and construct sample
spaces and distributions in
simple cases
•develop an understanding
of permutations and
combinations as counting
techniques
•build new mathematical
knowledge through
problem solving
•organize and consolidate
their mathematical
thinking through
communication;
•communicate their
mathematical thinking
coherently and clearly to
peers, teachers, and
others;
•use the language of
mathematics to express
mathematical ideas
precisely.
•create and use
representations to
organize, record, and
communicate
mathematical ideas
6.4.1.1 Determine the sample space
(set of possible outcomes) for a
given experiment and determine
which members of the sample space
are related to certain events. Sample
space may be determined by the use
of tree diagrams, tables or pictorial
representations.
7.4.3.2 Calculate probability as a
fraction of sample space or as a
fraction of area. Express probabilities
as percents, decimals and fractions.
9.4.3.1 Select and apply counting
procedures, such as the
multiplication and addition principles
and tree diagrams, to determine the
size of a sample space (the number
of possible outcomes) and to
calculate probabilities.
Counting
SC Standard(s)
http://ed.sc.gov/agency/
Standards-and-Learning/
Academic-Standards/old/
cso/standards/math/
7.6-8 Use the fundamental
counting principle to
determine the number of
possible outcomes for a
multistage event
5
Lesson
NCTM Standard(s)
http://
standards.nctm.org/
document/chapter1/
index.htm
MN Standard(s)
http://education.state.mn.us/MDE/
Academic_Excellence/
Academic_Standards/
Mathematics/index.html
“Flag
Trademarks”
•understand the concepts
of sample space and
probability distribution
and construct sample
spaces and distributions in
simple cases
•develop an understanding
of permutations and
combinations as counting
techniques
•build new mathematical
knowledge through
problem solving
•organize and consolidate
their mathematical
thinking through
communication;
•communicate their
mathematical thinking
coherently and clearly to
peers, teachers, and
others;
•use the language of
mathematics to express
mathematical ideas
precisely.
•create and use
representations to
organize, record, and
communicate
mathematical ideas
6.4.1.1 Determine the sample space
(set of possible outcomes) for a
given experiment and determine
which members of the sample space
are related to certain events. Sample
space may be determined by the use
of tree diagrams, tables or pictorial
representations.
7.4.3.2 Calculate probability as a
fraction of sample space or as a
fraction of area. Express probabilities
as percents, decimals and fractions.
9.4.3.1 Select and apply counting
procedures, such as the
multiplication and addition principles
and tree diagrams, to determine the
size of a sample space (the number
of possible outcomes) and to
calculate probabilities.
Counting
SC Standard(s)
http://ed.sc.gov/agency/
Standards-and-Learning/
Academic-Standards/old/
cso/standards/math/
7.6-8 Use the fundamental
counting principle to
determine the number of
possible outcomes for a
multistage event
6
Lesson
NCTM Standard(s)
http://
standards.nctm.org/
document/chapter1/
index.htm
MN Standard(s)
http://education.state.mn.us/MDE/
Academic_Excellence/
Academic_Standards/
Mathematics/index.html
SC Standard(s)
http://ed.sc.gov/agency/
Standards-and-Learning/
Academic-Standards/old/
cso/standards/math/
“Best of
Three”
•understand the concepts
of sample space and
probability distribution
and construct sample
spaces and distributions in
simple cases
•develop an understanding
of permutations and
combinations as counting
techniques
•build new mathematical
knowledge through
problem solving
•organize and consolidate
their mathematical
thinking through
communication;
•communicate their
mathematical thinking
coherently and clearly to
peers, teachers, and
others;
•use the language of
mathematics to express
mathematical ideas
precisely.
•create and use
representations to
organize, record, and
communicate
mathematical ideas
6.4.1.1 Determine the sample space
(set of possible outcomes) for a
given experiment and determine
which members of the sample space
are related to certain events. Sample
space may be determined by the use
of tree diagrams, tables or pictorial
representations.
7.4.3.2 Calculate probability as a
fraction of sample space or as a
fraction of area. Express probabilities
as percents, decimals and fractions.
9.4.3.1 Select and apply counting
procedures, such as the
multiplication and addition principles
and tree diagrams, to determine the
size of a sample space (the number
of possible outcomes) and to
calculate probabilities.
7.6-8 Use the fundamental
counting principle to
determine the number of
possible outcomes for a
multistage event
8.2-2 Understand the effect
of multiplying and dividing
a rational number by
another rational number.
Counting
7
Lesson
NCTM Standard(s)
http://
standards.nctm.org/
document/chapter1/
index.htm
MN Standard(s)
http://education.state.mn.us/MDE/
Academic_Excellence/
Academic_Standards/
Mathematics/index.html
SC Standard(s)
http://ed.sc.gov/agency/
Standards-and-Learning/
Academic-Standards/old/
cso/standards/math/
“Hamburger
Toppings”
•understand the concepts
of sample space and
probability distribution
and construct sample
spaces and distributions in
simple cases
•develop an understanding
of permutations and
combinations as counting
techniques
•build new mathematical
knowledge through
problem solving
•organize and consolidate
their mathematical
thinking through
communication;
•communicate their
mathematical thinking
coherently and clearly to
peers, teachers, and
others;
•use the language of
mathematics to express
mathematical ideas
precisely.
•create and use
representations to
organize, record, and
communicate
mathematical ideas
6.4.1.1 Determine the sample space
(set of possible outcomes) for a
given experiment and determine
which members of the sample space
are related to certain events. Sample
space may be determined by the use
of tree diagrams, tables or pictorial
representations.
7.4.3.2 Calculate probability as a
fraction of sample space or as a
fraction of area. Express probabilities
as percents, decimals and fractions.
9.4.3.1 Select and apply counting
procedures, such as the
multiplication and addition principles
and tree diagrams, to determine the
size of a sample space (the number
of possible outcomes) and to
calculate probabilities.
9.2.2.4 Express the terms in a
geometric sequence recursively and
by giving an explicit (closed form)
formula, and express the partial sums
of a geometric series recursively.
7.6-8 Use the fundamental
counting principle to
determine the number of
possible outcomes for a
multistage event
8.2-2 Understand the effect
of multiplying and dividing
a rational number by
another rational number.
Counting
8
Lesson
NCTM Standard(s)
http://
standards.nctm.org/
document/chapter1/
index.htm
MN Standard(s)
http://education.state.mn.us/MDE/
Academic_Excellence/
Academic_Standards/
Mathematics/index.html
SC Standard(s)
http://ed.sc.gov/agency/
Standards-and-Learning/
Academic-Standards/old/
cso/standards/math/
“Bridges of
Konigsberg”
•build new mathematical
knowledge through
problem solving
•organize and consolidate
their mathematical
thinking through
communication;
•communicate their
mathematical thinking
coherently and clearly to
peers, teachers, and
others;
•use the language of
mathematics to express
mathematical ideas
precisely.
•create and use
representations to
organize, record, and
communicate
mathematical ideas
8.2.2.4 Represent arithmetic
sequences using equations, tables,
graphs and verbal descriptions, and
use them to solve problems.
8.2.2.5 Represent geometric
sequences using equations, tables,
graphs and verbal descriptions, and
use them to solve problems.
9.3.1.5 Make reasonable estimates
and judgments about the accuracy of
values resulting from calculations
involving measurements.
7.3-2 Analyze tables and
graphs to describe the rate
of change between and
among quantities.
7.3-6 Represent
proportional relationships
with graphs, tables, and
equations.
8.2-1 Apply an algorithm to
add, subtract, multiply, and
divide integers.
•apply and adapt a variety
of appropriate strategies to
solve problems
•select and use various
types of reasoning and
methods of proof
8.2.2.4 Represent arithmetic
sequences using equations, tables,
graphs and verbal descriptions, and
use them to solve problems.
8.2.2.5 Represent geometric
sequences using equations, tables,
graphs and verbal descriptions, and
use them to solve problems.
9.3.1.5 Make reasonable estimates
and judgments about the accuracy of
values resulting from calculations
involving measurements.
7.3-6 Represent
proportional relationships
with graphs, tables, and
equations.
8.2-1 Apply an algorithm to
add, subtract, multiply, and
divide integers.
8.3-5 Classify relationships
between two variables in
graphs, tables, and/or
equations as either linear or
nonlinear.
•apply and adapt a variety
of appropriate strategies to
solve problems
•select and use various
types of reasoning and
methods of proof
8.2.2.4 Represent arithmetic
sequences using equations, tables,
graphs and verbal descriptions, and
use them to solve problems.
8.2.2.5 Represent geometric
sequences using equations, tables,
graphs and verbal descriptions, and
use them to solve problems.
9.3.1.5 Make reasonable estimates
and judgments about the accuracy of
values resulting from calculations
involving measurements.
7.3-6 Represent
proportional relationships
with graphs, tables, and
equations.
8.2-1 Apply an algorithm to
add, subtract, multiply, and
divide integers.
8.6-2 Organize data in
matrices or scatterplots as
appropriate.
Vertex-Edge
Graphs
“Supreme
Court
Welcome”
Vertex-Edge
Graphs
“Traveling
Salesman”
Vertex-Edge
Graphs
9
Lesson
NCTM Standard(s)
http://
standards.nctm.org/
document/chapter1/
index.htm
MN Standard(s)
http://education.state.mn.us/MDE/
Academic_Excellence/
Academic_Standards/
Mathematics/index.html
SC Standard(s)
http://ed.sc.gov/agency/
Standards-and-Learning/
Academic-Standards/old/
cso/standards/math/
“Loops”
•represent, analyze, and
generalize a variety of
patterns with tables,
graphs, words, and, when
possible, symbolic rules;
•relate and compare
different forms of
representation for a
relationship;
•identify functions as
linear or nonlinear and
contrast their properties
from tables, graphs, or
equations
8.2.1.3 Understand that a function is
linear if it can be expressed in the
form
or if its graph is a
8.3-1 Translate among
verbal, graphic, tabular, and
algebraic representations of
linear functions.
8.3-2 Represent algebraic
relationships with equations
and inequalities.
8.3-5 Classify relationships
between two variables in
graphs, tables, and/or
equations as either linear or
nonlinear.
Recursive and
Explicit
Equations
straight line.
8.2.2.1Represent linear functions
with tables, verbal descriptions,
symbols, equations and graphs;
translate from one representation to
another.
8.2.2.4 Represent arithmetic
sequences using equations, tables,
graphs and verbal descriptions, and
use them to solve problems.
10
Sample Standardized Test Questions
From Minnesota (MCA) and South Carolina (PASS) Standardized Test Specs.
Minnesota MCA Sampler Questions
http://education.state.mn.us/MDE/Accountability_Programs/
Assessment_and_Testing/Assessments/MCA/Samplers/index.html
11
South Carolina PASS Sampler Questions
http://ed.sc.gov/agency/Accountability/Assessment/documents/
Grade8_Apr2010.pdf
12
Lesson 1: License Plate Combinations
Adapted from class notes: Math 6200, Bemidji State University
(Summer Institute, 2010)
(Image of license from Minnesota Driver and Vehicle Services
website: http://www.dps.state.mn.us/dvs/PlBrochure/PlateFrame.htm).
Objectives:
Students will use the counting principle to determine how many license plate
combinations are possible in the state of Minnesota.
Materials:
Internet connection
Pencil and paper
Activity 1: Lock Combinations
Launch:
Ask students to raise their hand if they have have a bike lock. Call on a
couple of students to describe what their lock is like (combination, 4-# lock,
3-# lock, etc.). Draw a bike 2-# lock on the board for discussion. How many
combinations are possible for a lock with only three numbers, assuming
that we are not allowed to repeat digits (use 0-9)?
Explore/Share:
1) Allow students to work in small groups of 2-3. Ask students explore this problem and
propose an answer. (Make sure students show work and are prepared to share ideas
with the class.)
2) Circulate among student groups. Encourage students to show their thinking
methodically (perhaps with lists or tree diagrams). When all students have had enough
time to come up with a guess, ask students to share their thinking. How many
combinations are there if we can repeat digits? What method of problem solving did
they use? Ask students to display student work for the class to discuss.
3) Summarize student responses (there are 10·9 = 90 different combinations if we do not
repeat digits. If we can repeat digits, there are 10·10 = 100 different combinations.
4) Repeat this activity with a three-digit combination (there are 10·9·8 = 720 different
combinations if we donʼt repeat digits, there are 10·10·10 = 1000 different
combinations if we can repeat digits).
5) Ask, “How many combinations would be possible if we had a three digit lock that used
letters instead of numbers, and we can repeat the letters?” (26·26·26).
Summarize:
Discuss the tree diagrams for this problem. Tree diagrams are a useful tool for organizing
data and finding the amount of outcomes for a problem.
We also used the counting principle to solve this problem. For a two number lock, there
were two “bins.” If we could repeat digits, each bin had 10 possibilities. If we could not
13
repeat, the first bin had 10 possibilities, the second bin had 9. To find the total number of
possible combinations, multiply the two bins together.
So, for any bicycle lock, each stop is a “bin.” In order to find the total number of possible
combinations, we find the possible inputs for each bin and multiply the bins together. We
call this the “counting principle.”
Extend:
• Find the possible lock combinations for the above questions if we include letters.
• Use the counting principle to find the number of combinations possible on a typical
locker combination lock.
• “Address Number Combinations” Lesson from Illuminations: http://
illuminations.nctm.org/LessonDetail.aspx?ID=L718
Activity 2: License Plates
Launch:
Display the website: http://www.worldlicenseplates.com/hp.html (License Plates of the
World) and explore the look of license plates around the world with students. Look for
interesting license plates and unique qualities of plates.
Click to the Minnesota page and discuss the different looks of license plates that
Minnesota has had. Ask students, “Using our current license plate, how many different
plates could the state of Minnesota make?” Remind students to break the problem into
two parts: three number “bins” and three letter “bins.”
"
_____ _____ _____ "
_____ _____ _____
Explore/Share:
1) Partner students and ask students think about how they want to approach this problem
(tree diagram, list, counting principle...). Each group should have a problem solving
method in mind. Quickly survey the class to find out what direction students are going
to take.
2) Set students to the task to find the amount of possible license plates.
3) Discuss student responses. Where any of the problem solving methods more efficient
than others?
4) Multiply the possibilities together to calculate the number of possible outcomes (26
possibilities for each letter bin, 10 digits for each number bin = 17,576,000 license
plates could be produced).
5) Use the internet to look up Minnesotaʼs population: http://quickfacts.census.gov/qfd/
states/27000.html. Compare the number of license plates to the population.
6) Ask students to find the possible number of plates if letters or numbers had to be
different on the license plates (26⋅25⋅24⋅10⋅9⋅8).
Summarize:
14
Using “bins” and multiplication helped us solve this problem. This method of counting is
called the “counting principle” and is useful in many situations when the number of
outcomes is a large number or when the number of bins is larger than three.
Extend:
• Use the counting principle to find the number of combinations for license plates from
different states/countries/years.
15
Lesson 2: Flag Trademarks
Adapted from Navigating Through Discrete Mathematics in Grades
6-12 (Chapter 1), © 2008 NCTM.
Objectives:
In this investigation, students will solve counting problems using three
different models: enumerating, using a tree diagram, and using the counting
principle to determine the possible outcomes of an event.
Students will represent, analyze, and solve counting problems and apply the counting
principle, using algebra notation (including factorials) to find the possible outcomes of an
event.
Activity 1 is a review of tree diagrams and the counting principle. Activity 2 looks closer at
eliminating repeated outcomes and introduces the combination formula.
*Note: before starting this lesson, students should have some experience with tree
diagrams (activity 1 below is a warm-up for the Flag Trademark activity).
Materials:
• Copy of the “Flag Trademarks” activity sheet (on the CD included with Navigations)
• Scissors
• Glue or tape
• Poster board (something to display flag cut-outs)
• Markers
• A jar containing four colored balls: one red, one green, one blue, and one yellow
• Three copies of the “Flag Cutouts” template (one on red, one on white, one on blue
paper) for each pair of students (p. 118 in Navigations)
Activity 1: Possible Outcomes with Replacement and Without Replacement
Launch:
Hold up the jar with the four colored balls inside and ask students, “How many outcomes
are possible after drawing out two balls if after each draw, I put the ball back in the jar?”
Elicit student responses and brainstorm possible outcomes. Make a list of the outcomes
on the board.
Explore/Share:
1) Have students work in partners to draw a tree digram to prove that all possible
outcomes were listed.
2) Point out that the tree diagram shows the multiplication principle if we read it across:
4x4=16 possible outcomes. If read just the outcomes, we can use the addition
principle: 4+4+4+4=16 possible outcomes.
3) Ask students to use the tree diagram to find the answers to the multiple situations:
• how many outcomes produce exactly one red ball? (Three branches of red + one
branch from each of the others produces 3+1+1+1=6 of the 16 outcomes produce
one red ball.)
• how many outcomes produce first a red ball, then another color? (3)
• how many outcomes produce one green ball and one red ball? (2)
16
• how many outcomes produce first a green ball then a red ball? (1)
Launch:
Ask students, “how would the tree diagram be different if we did not allow replacement?”
Brainstorm ideas.
Explore/Share:
1) Have students work in partners to draw a tree digram to prove that all possible
outcomes were listed. Ask students to verbalize how this new tree diagram is different
from the previous tree diagram. (This tree diagram shows shows the multiplication
principle if we read it across: 4x3=12 possible outcomes. If read just the outcomes, we
can use the addition principle: 3+3+3+3=12 possible outcomes.)
2) Ask students to use the tree diagram to compare the outcomes from this tree diagram
to the previous one. Which questions have the same answers? Which ones have
different answers?
Extend:
Explore the tree diagram for the following situation: draw out three balls without
replacement. Discuss the counting principle related to the tree diagram and introduce
factorial notation for this problem: 4⋅3⋅2 or 4!
"
"
"
"
"
"
(4-3)!"
Summarize:
After students are comfortable with each tree diagram, discuss the counting principle for
each diagram. Ask students to verbalize why the counting is different for each tree
diagram, even though they both utilize the same jar of balls.
Activity 2: Flag Trademarks Investigation
Launch:
A luggage company is considering two rectangular flag patterns (fig. 1.4 on p. 21,
Navigations) with three possible colors. The winning design will be chosen to be used as
a trademark for the company. How many outcomes are possible? Does one of the
patterns have more possible outcomes than the other?
Explore/Share:
1) Before starting, discuss when the models are the same, and when they are different.
For example, if a half-turn produces the same model, then it is not considered another
outcome.
2) Pair students in partners or groups of three. Distribute the “Flag Cutout” sheets to each
group. Students will cut the models to create a poster that displays the possible
outcomes for the patterns A and B.
3) Ask students to prove they have a picture of all possible outcomes with a tree diagram
(some of the branches on the tree diagram will have to be eliminated to avoid
duplication of patterns).
17
4) Encourage students to explore what counting principle might help calculate the
possible outcomes for each pattern (pattern B has 6 outcomes, pattern A has 9).
Summarize:
Students should conclude that forming tree diagrams is helpful for visualizing the counting
principles, but tree diagrams become cumbersome when the number of outcomes
becomes too large. Students should be able to apply the counting principle directly to
problems like this.
Explain that using the counting formula is also a form of what we call “Permutations.” Use
the permutation formula (below) to illustrate for students.
n = number of choices
r = number of choices we are interested in (in this case, win or loose = 2 choices).
n! = sorts all n items into “bins”
(n-r)! = the number of “bins” that are not being used
Extensions:
• How many flag patterns are possible if we can use four colors from a choice of ten
colored cloths? (10⋅9⋅8⋅7 or 10!/(10-6)! = 5040 possible outcomes)
18
Lesson 3: Best of Three, Best of Five...
Adapted from class notes: Math 6200, Bemidji State University (Summer
Institute, 2010)
Objectives:
Students will create a sample space of possible win/loss situations for a “best of
three” situation and tree diagrams to check for sample space correctness.
Based on the sample space and tree diagram, students will calculate how many ways
there are to win the “best of three” situation. Students will then use the combination
formula to calculate the number of possible ways to win. “Best of five” and “best of seven”
situations are both extensions for this activity.
*Note: It is helpful if students have some experience with the counting principle and
factorials before starting this lesson.
Materials:
Paper and pencil
Whiteboard/chalkboard to display/discuss student work
Launch:
We play many games of chance that are “best of three” -type situations. Rock, Paper,
Scissors, for example, is usually played until one player has won two of the three games.
Do you know any other best of three games? How many ways can you win a best of three
game?
Explore/Share:
1) Have students play a number of rounds of “Rock, Paper, Scissors” in partners or
groups of three and keep track of how they win (for example: WW, LWW, WLW...etc.).
2) Discuss the class findings as a class and develop a list of the sample space. Ask
students “how can we be sure we have all of the possible combinations?”
3) Have students work in groups of 2-3 to come up with a method for being 100% sure all
possibilities are accounted for. (If students struggle here, hint toward a tree diagram).
4) Allow time for students to share their groupʼs thoughts. Display answers on the board
or on student white-boards (students should have a total sample space of eight
outcomes, but after eliminating duplicate samples, only three different ways to win).
19
Summarize:
We made a sample space of possible outcomes and found there were eight outcomes.
Why are there only three ways to win? (Discuss why WWW and WWL are really the same
thing).
Launch:
So, if we can calculate the number of ways to win in a best of three situation, can we
apply the same ideas to best of five (need to win three of five to win)? When might a best
of five situation happen in real life?
Explore/Share:
1) Ask students to develop a sample space for a best of five (W vs. L) situation and a
diagram to prove they have all of the samples of the population.
2) Have students work in small groups to find how many combinations yield a win. Ask
each group to share their results (32 possible outcomes, 10 combinations yield a win).
Summarize:
Review student results. A tree diagram works for small sample spaces, but there must be
an easier method to find the possible combinations for larger sample spaces. Discuss
student ideas. Compare the total number of outcomes to the combinations that yield a
win.
Motivate students toward discovery of the combination formula:
n = number of choices (in a best of three, n = 3. For best of 5, n = 5).
r = number of choices we are interested in (in this case, win or loose = 2 choices).
n! = sorts all n items into “bins”
r! = takes out duplicates (when order doesnʼt matter)
(n-r)! = the number of “bins” that are not being used
Discuss the formula and where these numbers come from. Why divide?
Extend:
• Find the possible number of wins for a best of 7 situation (like basketball playoffs and
baseball world series).
Information about Permutations and Combinations: http://www.mathsisfun.com/
combinatorics/combinations-permutations.html
Printable Tree Diagram Graphic Organizers
http://www.enchantedlearning.com/graphicorganizers/tree/
20
Lesson 4: Hamburger Toppings
Adapted from class notes: Math 6200, Bemidji State University
(Summer Institute, 2010)
Objectives:
Students will calculate the number of hamburger outcomes are
possible given the choice of five different toppings. Students will create a sample space,
use a tree diagram to check sample space correctness, and use permutation and
combination formulas to find the possible number of outcomes.
Materials:
Paper and pencil
Student whiteboards to discuss answers/display work
Launch:
Play the McDonaldʼs Big Mac commercial: http://www.youtube.com/watch?
v=en4muUSIRT4&feature=related then ask students what toppings they like to have on
their hamburgers. Give a choice of four toppings, how many different burgers could be
made?
Explore/Share:
1) After posing the question, allow students time to ponder and attempt the problem
(students will have had practice with sample space, tree diagrams and the counting
principle previous to this lesson). Partner students or group students of small groups of
three. Ask students to work together to solve the problem and put their thoughts on a
student whiteboard to share with the class.
2) Compare student answers. Sample space (lists) and tree diagrams should yield 16
different hamburger combinations. (With tree diagrams, start with the burger then add
one topping at a time. The first “bin” will have, for example, ketchup or no ketchup.) A
counting principle will also work: 2⋅2⋅2⋅2 = 16 combinations.
3) Use the tree diagram to discuss the following questions:
• How many ways can I start with four toppings, and choose only one?
• How many ways can I start with four toppings, and choose two?
• How many ways can I start with four toppings, and choose three?
• How many ways can I start with four toppings, and choose four?
4) Discuss the outcomes. Since the order of the toppings doesnʼt matter (ketchup, onion
is the same as onion, ketchup), we eliminate some of the outcomes.
5) Have students work in partners to work this observation into a formula (the combination
formula:
where n = number of choices (4) and r = how many
toppings we are choosing.
21
6) Discuss where the numbers in the numerator and denominator are coming from.
• The numerator is the factorial of the number of choices available...similar to the
counting principle.
• The denominator is a product of two things: how many bins we are filling (r!) and
what we need to eliminate because of duplicate answers (n-r)!.
7) Alter the problem: If order IS important (ketchup, onion is a different choice than onion,
ketchup), how does that change your answer?
• Choose 0 toppings = (4!)/(4-0)! = 1 way to order no toppings
• Choose 1 toppings = (4!)/(4-3)! = 4 ways to order 1 topping
• Choose 2 toppings = (4!)/(4-2)! = 12 ways to order 2 toppings
• Choose 3 toppings = (4!)/(4-3)! = 24 ways to order 3 toppings
• Choose 4 toppings = (4!)/(4-4)! = 4! = 24 ways to order 4 toppings (This
permutation is tricky, because the denominator seems to be zero. However, in
this case, the zero denominator simply means there is nothing to cancel out, so
all four bins are available, and no options are duplicated or cancelled).
• There is a total of 65 outcomes possible.
Summarize:
When order of the variables (in this case the toppings) does not matter (the order of
variables can be reversed, and the outcome is the same), we calculate the outcomes
using a combination method or formula. When the order is important (changing the order
of the variables yields a new outcome), we calculate the outcomes using a permutation
formula and do not cancel out duplicate outcomes.
Extend:
• Find combinations/permutations for ordering a group of five students.
• Find combinations/permutations for electing the class president/vice president.
Information on Permutations: http://theory.cs.uvic.ca/amof/e_permI.htm
22
Lesson 5: Bridges of Konigsberg
Adapted from Investigations into Mathematics, Unit 3B, © 2008 Glencoe and The
Beginnings of Topology... from http://mathforum.org/isaac/problems/bridges1.html
Objectives:
Students will explore practical applications of vertex-edge graphs by using the graphs to
solve a topological problem. Students will know that vertex-edge graphs are composed of
vertices (points) and edges (segments) and the connections form paths. Students will use
vertex-edge graphs to analyze a map of the bridges of Konigsberg to determine whether
citizens of the town could walk across each bridge without re-crossing any bridge.
Materials:
Internet connection (optional)
Ice cream pail lids (optional-for extension activity)
Launch:
In Germany, there is a city called Konigsberg that has a river that runs through it. The
river splits around an island, and soon after, it splits into two parts as it runs out of the
city. In order for the citizens of Konigsberg to cross the river efficiently, seven bridges
were built. The map of the bridges is below:
The citizens of the city loved to walk from end to end of their beloved Konigsberg. They
began to wonder if it was possible to walk across each of the bridges without re-crossing
any bridge. What do you think?
Explore/Share:
1) Partner students or have students work in small groups of three. Try the problem.
Sketch a map of the city on a small white board or on paper and try to sketch a path in
such a way that each bridge is traced over exactly one time with one continuous pencil
stroke.
2) After students have a chance to try the problem, share student results. Each group
should share their findings with the class. When all student opinions are aired, discuss
the problem-solving methods students used.
3) Discuss the solution:
23
4) Ask students what changes they would like to make to the map of the city in order to
make this problem work. Have groups brainstorm ideas to make it possible to cross
each bridge once (hint: students may need to add or remove bridges).
5) Discuss students findings. Then discuss the solution:
6) Go to the website: http://mathforum.org/isaac/problems/bridges2.html for Eulerʼs
solution using vertex-edge graphs.
7) Explore some of the other problems at the bottom of the webpage. Determine if each of
the networks are “traceable.” Ask students to theorize why some are and some are not
possible.
Summarize:
Vocabulary
• Vertex = a point on the vertex-edge graph
• Edge = a segment or curve connecting the vertices
• Odd Vertex = a vertex with an odd number of edges leading to it
• Even Vertex = a vertex with an even number of edges leading to it
24
• Euler path = a continuous path connecting all vertices that passes through every edge
exactly once
• Euler circuit = a Euler path that starts and ends at the same vertex
Eulerʼs Theorems
• If a graph has more than two odd vertices, then it does not have an Euler path (closed
circuit).
• If a graph has two or fewer odd vertices, then it has at least one Euler path.
• If a graph has any odd vertices, then it cannot have an Euler circuit.
• If every vertex in a graph is even, then it has at least one Euler circuit.
Extend:
• Group students into groups of 4-5. Have each group create a vertex-edge graph using
ice cream pail lids as vertices and their arms as edges. As a circle, the group
completes a Euler circuit, and all vertices have even degrees (that is, all vertices have
an even number of edges leading to it). Explore this idea of vertex degree and closed
circuits by manipulating the group:
1) Every vertex must have an odd amount of edges leading to it. Can the circuit be
closed?
2) Every other person must let go of one vertex and move to another vertex. Can a
closed circuit be created?
• Students can explore this topic further with puzzles at http://www.mathmaniacs.org/
lessons/12-euler/PencilPuzzles.html
• Complete the “Paths at Camp Graffinstuff” (Navigating Through Discrete Mathematics
in Grades 6-12, p. 49, ©2008, NCTM.)
Additional enrichment opportunities to explore include:
• Explore vertex-edge graphs at the following Web links:
"
http://illuminations.nctm.org/ActivityDetail.aspx?ID=20
"
http://mathforum.org/isaac/problems/bridges1.html
"
www.mathmaniacs.org/lessons/12-euler/PencilPuzzles.html
• Information on Euler in Historical Connections in Mathematics, Volume 1, pp. 65–72.
25
Lesson 6: Supreme Court Welcome
Adapted from Illuminations at http://illuminations.nctm.org/
LessonDetail.aspx?ID=U168
Objectives:
Students will use the context of handshakes to discover patterns and generate geometric
and algebraic representations that show the relationship is non-linear. Students will
generalize the number of handshakes for any group size and develop an expression for
triangular numbers.
Materials:
Internet connection/projection
Handshake activity sheet
http://illuminations.nctm.org/Lessons/Handshake/Supreme-AS-Handshakes.pdf
Graph paper or graphing calculator
Handshake spreadsheet
Algebra tiles
Triangular numbers activity sheet
Sum of integers spreadsheet
http://illuminations.nctm.org/LessonDetail.aspx?ID=L631
Activity 1: Supreme Court Handshake
Launch:
Project a picture of the supreme court justices on the front board (http://en.wikipedia.org/
wiki/Supreme_Court_of_the_United_States). Ask students what they know about the
Supreme Court. After students share what they know, discuss one tradition that will be
used in class today: every justice shakes hands with each of the other justices each time
they gather for a meeting. Chief Justice Melville W. Fuller (1888‑1910) started this
custom, saying that it shows ʻthat the harmony of aims, if not views, is the courtʼs guiding
principle.ʼ
Then present the problem (display on the board for students to read): Since there are
nine justices on the Supreme Court, how many handshakes occur if each shakes hands
with every other justice exactly one time?
Explore/Share:
1) Group students in groups of four and ask each group to brainstorm a plan for solving
the problem.
2) Each group will execute their plan and present their solution to the class (remind
students to keep record of their work).
3) After each group has had some time to work on the problem, ask each group to share
what they have so far (do not summarize findings yet---students will be generalizing
results in the next steps).
4) Distribute the handshake activity sheet to help with student investigation. Students may
use a variety of methods to work out this problem (including acting it out, using
pictures, tables, vertex-edge graphs, or lists).
26
5) After some time, ask students to discuss their findings with the class. Display lists,
tables, and vertex-edge graphs for students to consider. Students may need overhead
sheets or whiteboards to show their work to the class
6) Summarize by showing illustrations on the website: http://illuminations.nctm.org/
LessonDetail.aspx?ID=L630. There are 36 (8 + 7 + 6 + 5 + 4 + 3 + 2 + 1) handshakes
in all.
7) Ask, “How many handshakes occur when there are 30 people?” Review student
problem-solving methods used so far. Students will realize that list-making and vertexedge graphs are too cumbersome for the new problem and that there must be a more
efficient method to get to the answer.
8) To help generalize this problem, use the interactive applet at http://
illuminations.nctm.org/ActivityDetail.aspx?ID=126.
9) Work toward the equation:
"
"
n(n-1)
"
"
2
10) The formula for this problem is not linear (non-linear), which means that the
relationship is not a straight line. What will the graph look like? (Graph the problem on
graph paper or using graphing calculators).
Summarize:
This relationship between the number of people and the number of handshakes is a nonlinear relationship, because there is not a constant rate of change. The graph of this
relationship is a curve, not a straight line.
Review the representations that provided information to help solve this problem.
Extend:
See the “Extensions” link on the webpage http://illuminations.nctm.org/
LessonDetail.aspx?ID=L630
Activity 2: Beyond Handshake
Launch:
Yesterday, we explored the handshake problem and discovered that the relationship
between the number of people and the number of handshakes is non-linear. Today we are
going to plot our information in a spreadsheet and try to represent our data in another
way.
Explore/Share:
1) Display a spreadsheet program and check for background knowledge of students. Use
the handshake spreadsheet (http://illuminations.nctm.org/LessonDetail.aspx?ID=L631)
to review the handshake problem (see the website for directions).
2) Work with students to develop a formula for growing the table to look like the lists from
the day before (a 1 is listed in cell A3, but the formula we need to get a 2 in A4 and a 3
in A5 etc. should be “=A3 + 1”). Drag the formula to fill all of the necessary cells.
3) Insert a 0 into the cell B3 and then the formula “=(A4*A3)/2” into cell B4. Click and drag
the formula to fill the cells below.
27
4) Use the table to make a graph of the data. Discuss the graph and compare it to the
graphs from the first lesson.
5) Distribute algebra tiles an the Triangular Numbers activity sheet to each group.
Students can use the table or the manipulatives to build the fifth triangular number, and
so on.
6) Discuss the results as a class. The numbers 1, 3, 6, and 10 show up on the table and
should be familiar as they also showed up in the previous lesson.
7) Have students enter the new data into the spreadsheets and create a graph of the
information and discuss the new graphs.
8) Ask students to examine the Sum of Integers spreadsheet. Students should be able to
predict this graph will not be linear, and that the nth sum will be n(n+1)/2.
Summarize:
• The pattern is to multiply successive numbers and divide by 2. The handshake problem
multiplies by the previous number, but the triangular numbers multiply by the next
number.
• A curve will not have a constant rate of change from term to term, which is why it is not
linear.
• The generalized form of the sum of the first n positive numbers is [n(n+1)/2].
Extend:
• Examine the sum of the first n odd positive integers.
• Explore Pascalʼs Triangle: http://mathforum.org/workshops/usi/pascal/
pascal_lessons.html
• Other “Handshake Problem” resources:
Lesson: http://math.about.com/cs/weeklyproblem/a/q2.htm
Quick Answer Key: http://dwb4.unl.edu/calculators/activities/middle/shake.html
Another Resource: http://www.wcer.wisc.edu/NCISLA/teachers/
teacherResources.html
28
Lesson 7: Traveling Salesman
Adapted from Illuminations at http://illuminations.nctm.org/
LessonDetail.aspx?id=L749
Objective:
Students will plan a road trip starting in Cleveland, then traveling
through Cincinnati, Pittsburgh, Baltimore, and Boston. Students will
investigate three different methods to determine the shortest route: the
Nearest Neighbor method, the Cheapest Link method, and the Brute
Force method.
Materials:
Calculator
Overhead projector or Web connection and projection capability
Activity sheets from http://illuminations.nctm.org/LessonDetail.aspx?id=L749
Launch:
Project the Road Trip Overhead (or map from the website). Present the problem: they are
traveling salespeople on a road trip for their company. They will start in Cleveland, then
travel through Cincinnati, Pittsburgh, Baltimore, and Boston (not necessarily in that order).
Ask students, “How could we plan this trip so that our traveling time is the shortest
possible route?” (Some ideas might include: drive to the closest city you havenʼt visited,
pick the lowest values from the table and use those to form a route, or calculate all
possible routes and pick the shortest one.) Leave the Road Trip Overhead up for
students to view while working through the activity.
Explore/Share:
1) Group students in small groups of 2-3. Students will first work through the “Nearest
Neighbor” algorithm activity sheet. Students will calculate the shortest route by
choosing to drive to the closest city they havenʼt visited. Compare answers as a class
before moving on.
2) Distribute the Cheapest Link activity sheet (also called the “Greedy” algorithm).
Students will use weighted vertex-edge graphs to calculate the shortest distance by
finding all of the shortest distances and connecting as many cities as possible before
using longer distances. Discuss the vertex-edge graphs and student answers before
moving on. Ask students:
• “Why canʼt you have 3 routes going to and from the same city?”
• “Why canʼt you have a route that creates a “mini-tour” of cities (a closed circuit)?
3) Distribute the Brute Force activity sheet. Work through question 1 as a class, and stop
students to work through question 4 as a class as well. Ask students these questions:
• “In the 4 city example, how many choices did we have for the next city as we left
city A?”
• After that city, how many choices for the next?
• After that, how many choices remained?
• For n cities, we find the formula = (n-1)!/2 (see website for more explanation).
29
4) Distribute the Best Route activity sheet. Depending on class understanding and
pacing, this may be assigned as homework. As a class, decide on which method gave
the “best” answer for this problem.
Summarize:
Review the algorithms used in this problem: Nearest Neighbor, Cheapest Link, and Brute
Force (or “Greedy”). Ask students to briefly describe each.
Discuss with students: besides a traveling salesman career, when else is this type of
problem-solving (for finding the shortest route) useful?
Which algorithm did they think was most useful and why? What factors might change their
choice?
Extend:
• Given current gas prices and the mileage of an average car, have students estimate
the amount of money saved by taking the shortest route vs. taking the longest route.
•
Have students generate their own list of five cities and use web resources or maps to
calculate the shortest route.
•
"
Another look at Traveling Salesman:
!
Hart, Eric W. Navigating through Discrete Mathematics in Grades 6-12. Reston, VA: National
Council of Teachers of Mathematics, 2008. Print.
!
!
Touring Washington, D.C.
Schad, B., Georgeson, J., & Slater, S. (2008). Problem solvers: touring washington d.c.. Teaching
Children Mathematics, 26-27.
•
"
"
More work with Matrices:
Cryptography Inquiry Lesson
Dr. Antonio R. Quesada. A Quesada Director Project. Project AMP. Web. 14 July !2010.
<www.math.uakron.edu/.../Discrete/Cryptography.../Cryptography%20Inquiry%20LessonQ.doc>.
"
"
!
Matrix Systems:
Dr. Antonio R. Quesada. A Quesada Director Project. Project AMP. Web. 14 July !2010. Web. 14
July 2010. <www.math.uakron.edu/amc/Discrete/IBNetwork_Matrices_Other.doc>.
30
Lesson 8: Loops
Adapted from class notes: Math 6200, Bemidji State University
(Summer Institute, 2010)
Objective:
Students will cut loops of string to make a table of data comparing the number of loops to
the number of pieces of string after making one cut through the loops of string. Students
will develop recursive and explicit equations to describe the relationships.
Materials:
Scissors
String
Paper and pencil
Poster paper (optional)
Launch:
Say to students, “Today we are going to take a closer look at patterns. We all are familiar
with the game “rock, paper, scissors...”
Ask students to make small groups of three and assign a role to each person in the group
(one student will be “rock,” another “paper,” and the third will be “scissors”).
Assign the rock student to be the “go-getter” for the material needed today, the scissors
student to be the person who uses the scissors for the day, and the paper student to be
the record keeper for the group.
Explore/Share:
1) Ask the rock student to fetch a long piece of string and a scissors for their group, and
the paper student to have a piece of paper and pencil ready for record keeping.
2) Model for students how to loop the string and make one cut:
• Take a long piece of string and make a loop around the open end of a scissors.
• Make one cut and count the number of strings that are left after the cut.
3) The paper students should record what happens. Model cutting a string with two loops.
When students seem to understand the process, encourage them to make as many
loops and cuts as they need to see a pattern in their table (the must make at least four
loops/cuts).
Loops (L)
Pieces of String
After 1 Cut (P)
0
2
1
3
P=L+1
Next = L + 1
31
4) Circulate among the student groups. When all students seem to be done cutting string,
ask them to take a closer look at their table. What patterns do they notice? What is
happening to the number of pieces of string with each cut? Encourage students to
summarize their findings with a formula or equation. Groups should be prepared to
present their ideas to the class.
5) After groups have had enough time to either form an equation or get close to an
equation, ask each group to write their findings on the front board or on poster paper
around the room.
6) Share student results. Explore the patterns students found. (Students should find that
the number strings after each cut goes up by one. The recursive formula for this
relationship is Next = L +1. Students might also find the explicit relationship between
the number of loops and the pieces left after the cut. The explicit formula is y = L + 2).
7) After the recursive and explicit formulas are discovered and discussed as a class,
change the question and model the first one or two cuts:
• This time, make a circle with the string and tie a knot on one end (groups may need
more string at this point in the activity).
• Fold the circle in half (flatten it) and loop the new string around one end of the open
scissors in the same fashion as before, then make a cut.
8) Once students understand the new cutting method, remind them to make a table and
set groups to task on finding recursive and explicit equations for the new situation.
Loops (L)
Pieces of String
After 1 Cut (P)
0
2
1
4
...
...
P = 2L + 2
Next = L + 2
9) After groups have had enough time to either form an equation or get close to an
equation, ask each group to write their new findings on the front board or on poster
paper around the room.
10) Share student results. Explore the patterns students found. (Students should find that
the number strings after each cut goes up by two. The recursive formula for this
relationship is Next = L +2. Students might also find the explicit relationship between
the number of loops and the pieces left after the cut. The explicit formula is y = 2L + 2).
Summarize:
Review the main idea for todayʼs lesson: Students will develop recursive and explicit
equations to describe the relationships. Ask students to explain the difference between
recursive formulas and explicit formulas based on the lesson from today. Students should
be able to see that recursive formulas use only one of the columns (in this case, the
32
pieces of strings left after a cut). In order to find the next number in the sequence, we
need to know the previous terms. Explicit formulas relate the two columns together (loops
and pieces).
Make connections with algebra: when we work with x/y tables for graphing lines, we
typically see explicit formulas that relate y to x. Both of these relationships are linear,
which means that if we graph the data from the table, the graph will be a straight line.
Visit the website: http://www.algebralab.org/lessons/lesson.aspx?
file=Algebra_SeqSeriesIntro.xml and review the definition of recursive formulas. Scroll
down the page to the “Examples” and have students write recursive formulas for each
pattern.
Extend:
• Repeat the string activity with folds instead of loops and compare the new table to the
previous tables. Develop a recursive formula for the new situation:
•
•
•
•
•
•
•
Folds (F)
Pieces of String
After 1 Cut (P)
0
2
1
3
2
5
3
9
P = 2F + 1
Next = 2Now - 1
Graph the data from the tables using graphing calculators and examine the shapes of
the graphs. Ask students to describe the differences in the graphs and make
connections to the formulas discussed in class.
Try the “Rent-a-Car” activity for exploring linear relationships: http://math.rice.edu/
~lanius/Algebra/rentacar.html
Use patterns from “Building Bridges” for finding recursive formulas: http://
illuminations.nctm.org/LessonDetail.aspx?id=L247
Try some more discrete math problems: http://webspace.ship.edu/deensley/
DiscreteMath/flash/index.html
Josephus Problem: http://webspace.ship.edu/deensley/DiscreteMath/flash/ch1/
sec1_1/josephus.html
Opening lockers, online locker problem from MSU Math: http://www.mth.msu.edu/
~nathsinc/java/Lockers/
Opening lockers, online locker problem from Connected Math: http://
connectedmath.msu.edu/CD/Grade6/Locker/index.html
33
Discrete Mathematics Test
Name: ______________________ Date: __________
Answer the following questions to the best of your ability. Be sure to show all of your
work.
Your family is taking is taking a road trip to Canada. At the border patrol, you notice the
license plate of the car in front of you. It looks like this:
You get to wondering how many license plates can be made in Canada if all of the license
plates look like this one.
1) How many different license plates can be produced if the digits on the plate can not be
repeated on a plate?
_____________ different license plates are possible.
2) How many license plates can be produced if digits can be repeated on a plate?
_____________ different license plates are possible.
3) You and your brother are playing rock, paper, scissors to see who has to take out the
trash this week. The first to win three games gets out of the task. Use a tree diagram to
show how many ways you can win RPS.
_____________ ways to win rock paper scissors.
34
4) You get to go to Burger Time for lunch today, but in order to get back to school in time
for your next class, you have to be ready with your order ahead of time. Burger time
has five toppings today: ketchup, onions, pickles, lettuce, and cheese. How many
burger combinations are possible to order with three of the toppings?
_____________ different combinations with three toppings are possible.
5) Is it possible to trace the following figure without passing through any vertex twice and
without lifting your pencil? Explain your reason.
Circle one:" "
Possible"
"
Not possible
Explain your reasoning:
Your family wants to take a road trip to visit some of your relatives. Your relatives live in
St. Cloud, Duluth, and Shakopee. Use the matrix below to help answer the following
questions.
Detroit Lakes
Detroit Lakes
X
St. Cloud
136
Duluth
207
Shakopee
216
St. Cloud
136
X
144
78
Duluth
207
144
X
178
Shakopee
216
78
178
X
6) Draw a vertex-edge graph connecting all of the cities together. Make the vertices
represent the towns and the edges represent the distances between the towns.
7) Use the “nearest neighbor” algorithm to find the shortest route.
35
8) Use the table below to write a recursive formula for the number of pieces of string left.
Folds (F)
Pieces of String
After 1 Cut (P)
0
2
1
3
2
5
3
9
Next =
________________
9) Draw a graph using the data from the table above. Is the graph linear? ___________
10)In the combination formula :
What does the (n-r)! represent?
36
References
Lesson
“License Plates”
Reference(s)
Adapted from class notes: Math 6200, Bemidji State University (Summer
Institute, 2010).
LICENSE PLATES OF THE WORLD. Web. 14 July 2010. <http://
www.worldlicenseplates.com/>.
“Minnesota QuickFacts from the US Census Bureau.” State and County
QuickFacts. Web.14 July 2010. <http://quickfacts.census.gov/qfd/states/
27000.html>.
“Flag Trademarks”
Hart, Eric W. Navigating through Discrete Mathematics in Grades 6-12. Reston,
VA: National Council of Teachers of Mathematics, 2008. Print.
“Best of Three”
Adapted from class notes: Math 6200, Bemidji State University (Summer
Institute, 2010).
“Hamburger
Toppings”
Adapted from class notes: Math 6200, Bemidji State University (Summer
Institute, 2010).
“Bridges of
Konigsberg”
“Extension-Vertex-Edge Graphs.” Investigations into Mathematics. Glencoe,
2008. Montgomery County Public Schools, Rockville, MD. Web. 14 July 2010. <
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“Supreme Court
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