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Chapter 1 The Foundations: Logic and Proofs
Total Number of Days: 18-days Grade/Course: 12/Discrete Math
ESSENTIAL QUESTIONS ENDURING UNDERSTANDINGS
What are the symbols used in logic? How can the truth value of a proposition or set of
propositions be determined? What makes a proof or an argument logically valid? Is there a best way to write a logically valid proof?
There is more than one correct way to write logical proof. Logical proofs carry more validity than baseless arguments. Common language is often riddled with fallacies (sometimes
when we communicate, our assumed meaning is not necessarily the logical meaning)
PACING CONTENT SKILLS STANDARDS (CCCS/MP)
RESOURCES
LEARNING ACTIVITIES/ASSESSME
NTS
Basic Topic Description of what students will be able to do Mc Graw Hill OTHER
(e.g., tech)
1d Course Pre-Assessment
Assess math skills. Algebra 1 Final Exam
1d Review Review of axioms for the real numbers and the positive integers
N.Q.2
Text – Appen 1 Pgs A-1 to A-6
1d 1.1 Propositional Logic
Understand the basic terminology of propositional logic, including logical connectives.
Construct truth tables. Illustrate the
importance of logic with applications.
The study of logic through logic puzzles
N.Q.2
Text pgs 12 -16 Basic 1-28(even) Average: 1-38(even) Advance 1-38(even) 42,43,44
Text resources
Rosen Links Library Chapter 1-1
Lecture Notes
You Tube Propositional Logic
Rosen Web Links
Rosen Extra
Assessments Diagnostic: Do now,
http://www.eecs.yorku.ca/course_archive/2006-07/W/3341/C1q.html
Formative: HW, Quiz , Exit Ticket http://highered.mheducation.com/sites/0073383090/student_view0/self_assessments.html
Note: APPENDIX
Learning Activites
and system specifications.
Lesson PowerPoint Chapter1p1 Slides 4-24
Examples http://www.luc.edu/faculty/avande1/logic/worksheets-chapter1.htm
2d
1.2 Applications of Propositional Logic
Translate English sentences into logical statements.
Use Boolean searches. Apply propositional
logic to situations. Work with logic
puzzles .
N.Q.2 Text pgs 22 -24
Text resources Lesson
PowerPoint Chapter1p1 Slides 25-39
Figures PowerPoint Ch 1 Slides 6-10, 13
Text test bank
Rosen Links Library Chapter 1-2
Lecture Notes
You Tube Boolean Basic Laws and Rules
Rosen Web Links
Rosen Extra Examples
Assessments Diagnostic: Do now,
http://people.cs.pitt.edu/~milos/courses/cs441/lectures/Class2.pdf
Formative: HW, Quiz , Exit Ticket http://highered.mheducation.com/sites/0073383090/student_view0/self_assessments.html http://cs.fit.edu/~wds/classes/adm/
Note: APPENDIX
Learning Activites http://www.luc.edu/faculty/avande1/logic/worksheets-chapter1.htm
2d
1.3 Propositional Equivalences
Show that compound propositions are logically equivalent.
Use truth tables to verify mathematical laws (use DeMorgan’s Laws).
Show a conditional statement is a tautology using truth tables.
Show two compound propositions are equivalent.
Show a compound
N.Q.2 Text pgs 25-36
Text resources Lesson
PowerPoint Chapter1p1 Slides 40-63
Text test bank
Rosen Links Library Chapter 1-3
Lecture Notes
You Tube DeMorgan's Law
Rosen Web Links
Rosen Extra Examples
Assessments Diagnostic: Do now,
http://people.cs.pitt.edu/~milos/courses/cs441/lectures/Class2.pdf
Formative: HW, Quiz , Exit Ticket http://highered.mheducation.com/sites/0073383090/student_view0/self_assessments.html http://cs.fit.edu/~wds/classes/adm/
Note: APPENDIX
Learning Activites http://www.luc.edu/faculty/a
proposition is satisfiable or not.
vande1/logic/worksheets-chapter1.htm
2d
1.4 Predicates and Quantifiers
Given a statement determine the truth value.
Translate quantifications into English.
Express statements in terms of functions, quantifiers and logical connectives.
Given the domain, determine the truth value of statements.
Translate statements into logical expressions using predicates, quantifiers, and logical connectives.
N.Q.2 Text pgs 36-57
Text resources Lesson
PowerPoint Chapter1p2 Slides 3-40
Text test bank
Rosen Links Library Chapter 1-4
Lecture Notes
You Tube Predicates and Quantifiers 1
You Tube Predicates and Quantifiers 2
Web Links
Extra Examples
Assessments Diagnostic: Do now,
http://people.cs.pitt.edu/~milos/courses/cs441/lectures/Class2.pdf
Formative: HW, Quiz , Exit Ticket http://highered.mheducation.com/sites/0073383090/student_view0/self_assessments.html http://cs.fit.edu/~wds/classes/adm/
Note: APPENDIX
Learning Activites http://www.luc.edu/faculty/avande1/logic/worksheets-chapter1.htm
2d
1.5 Nested Quantifiers
Understand statements involving nested quantifiers.
Translate mathematical statements into statements involving nested quantifiers.
Translate from nested quantifiers into English and English sentences into logical expressions.
Negate nested quantifiers.
N.Q.2 Text pgs 57-68
Text resources Lesson
PowerPoint Chapter1p2 Slides 41-57
Text test bank
Links Library Chapter 1-5
Lecture Notes
You Tube Nested Quantifiers 1
You Tube Nested Quantifiers 2
Web Links
Extra Examples
Assessments Diagnostic: Do now,
http://people.cs.pitt.edu/~milos/courses/cs441/lectures/Class2.pdf
Formative: HW, Quiz , Exit Ticket http://highered.mheducation.com/sites/0073383090/student_view0/self_assessments.html http://cs.fit.edu/~wds/classes/adm/
Note: APPENDIX
Learning Activites http://www.luc.edu/faculty/a
vande1/logic/worksheets-chapter1.htm
2d 1.6 Rules of Inference
Understand valid arguments in propositional logic.
Use rules of inference for propositional logic to build arguments.
Use the resolution rule of inference.
Recognize fallacies in incorrect arguments.
Use rules of inference for qualified statements.
Combine rules of inference for propositions and qualified statements.
N.Q.2 Text pgs 69-80
Text resources Lesson
PowerPoint Chapter1p3 Slides 3-31
Text test bank
Links Library Chapter 1-6
Lecture Notes
You Tube Rules of Inteference 1
You Tube Rules of Inteference 2
Web Links
Extra Examples
Assessments Diagnostic: Do now,
http://people.cs.pitt.edu/~milos/courses/cs441/lectures/Class2.pdf
Formative: HW, Quiz , Exit Ticket http://highered.mheducation.com/sites/0073383090/student_view0/self_assessments.html http://cs.fit.edu/~wds/classes/adm/
Note: APPENDIX
Learning Activites http://www.luc.edu/faculty/avande1/logic/worksheets-chapter1.htm
3d
1.7 Introduction to Proofs
Construct direct proofs. Construct proofs by
contraposition. Construct proofs by
contradiction. Recognize common
mistakes in proofs.
N.Q.2 Text pgs 80-92
Text resources Lesson
PowerPoint Chapter1p3 Slides 32-49
Text test bank
Links Library Chapter 1-7
You Tube Direct and Indirect Proofs
Lecture Notes
Web Links
Extra Examples
Assessments Diagnostic: Do now,
http://people.cs.pitt.edu/~milos/courses/cs441/lectures/Class2.pdf
Formative: HW, Quiz , Exit Ticket http://highered.mheducation.com/sites/0073383090/student_view0/self_assessments.html http://cs.fit.edu/~wds/classes/adm/
Note: APPENDIX
Learning Activites http://www.luc.edu/faculty/avande1/logic/worksheets-
chapter1.htm
2d 1.8 Proof Methods and Strategy
Using different methods for constructing proofs.
Exhaustive proof and proof by cases.
Existence proofs. Uniqueness proofs. Proof strategies. Look for
counterexamples.
N.Q.2 Text pgs 92-109
Text resources Lesson
PowerPoint Chapter1p3 Slides 50-71
Text test bank
Links Library Chapter 1-7
You Tube Proof by Cases
Web Links
Extra Examples
Assessments Diagnostic: Do now,
http://people.cs.pitt.edu/~milos/courses/cs441/lectures/Class2.pdf
Formative: HW, Quiz , Exit Ticket http://highered.mheducation.com/sites/0073383090/student_view0/self_assessments.html http://cs.fit.edu/~wds/classes/adm/
Summative: Unit Test http://highered.mheducation.com/sites/0073383090/instructor_view0/printable_tests.htm
Note: APPENDIX
Learning Activites http://www.luc.edu/faculty/avande1/logic/worksheets-chapter1.htm
INSTRUCTIONAL FOCUS OF UNIT
Introduction to propositional logic. To introduce the basic terminology of propositional logic, including logical connectives, to
show how to construct truth tables, to illustrate the importance of logic with applications, and to motivate the study of logic
through logic puzzles and system specifications.
Applications of propositional logic. To introduce some important applications of propositional logic, including many important
applications in computer science. Also, to work with logic puzzles, which provide an entertaining way to learn and enjoy
propositional logic.
Propositional equivalences. To show how propositional equivalences are established and to introduce the most important such
equivalences.
Predicates and quantifiers. To introduce predicate logic, especially existential and universal quantification. Moreover, to explain
how to translate between English sentences (or mathematical statements) and logical expressions.
Nested quantifiers. This section explains how to work with nested quantifiers and makes clear that the order of quantification
matters. This section helps students gain maturity working with complicated logical expressions involving multiple quantifiers.
Rules of Inference. To introduce the notion of a valid argument and rules of inference for propositional logic. To explain how to
use rules of inference to build correct arguments in propositional calculus. Moreover, to introduce rules of inference for predicate
logic and how to use these rules of inference to build correct arguments in predicate logic. To show how rules of inference for
propositional calculus and predicate calculus can be combined. Finally, to learn how to distinguish between correct and incorrect
arguments.
Introduction to proofs. To introduce the notion of proof and basic methods of proof, including direct proof, proof by
contraposition, and proof by contradiction. Furthermore, to learn how to distinguish between correct and incorrect arguments, and
to understand and construct basic types of proofs.
Proof methods and strategy. To learn important methods of proofs including proof by cases and existence proofs, supplementing
the basic methods introduced in Section 1.7. To introduce key strategies for proving theorems, to understand the roles of
conjectures and counterexamples, and to learn about some important open problems.
TEXTBOOK
Discrete Mathematics and Its Applications 7th Edition Kevin Rosen
ACADEMIC VOCABULARY Words: Proposition: Truth value: Converse: Inverse: Biconditional: Boolean variable: Tautology: Contrapositive: Contradiction: Contingency: Compound
proposition: Predicate
Marzano’s Six Strategies for Teaching Vocabulary: 1. YOU provide a description, explanation or example. (Story, sketch, power point) 2. Ask students to restate or re-explain meaning in their own words. (Journal, community circle, turn to your neighbor) 3. Ask students to construct a picture, graphic or symbol for each word.
4. Engage students in activities to expand their word knowledge. (Add to their notes, use graphic organizer format) 5. Ask students to discuss vocabulary words with one another (Collaborate) 6. Have students play games with the words. (Bingo with definitions, Pictionary, Charades, etc.)
Using Marzano’s Strategy 5: Tell a friend- one student tells the other a story using 3 related terms (terms: meteor, meteorite, meteoroid) Strategy 4: Venn Diagram- use Venn Diagram to compare terms (protoplanet and protosun)
21ST CENTURY SKILLS (4Cs & CTE Standards)
21st Century Life and Careers 9.4.O(1) Engineering and Technology
9.4.12.O.(1).1 Apply the concepts, processes, guiding principles, and standards of school mathematics to solve science, technology, engineering, and mathematics problems.
9.4.12.O.(1).7 Use mathematics, science, and technology concepts and processes to solve problems in projects involving design and/or production (e.g., medical, agricultural, biotechnological, energy and power, information and communication, transportation, manufacturing, and construction).
Activity: Use logic with circuit design. Free circuit design software http://opencircuitdesign.com/ Link for activity http://acs.ist.psu.edu/discrete-math/
Activity: Research paper on history of logic Link for activity http://www.maa.org/publications/periodicals/convergence/primary-historical-sources-in-the-classroom-
discrete-mathematics-and-computer-science
MODIFICATIONS/ACCOMMODATIONS
Modifications: 1. Less complex reading level 2. Shortened assignments 3. Different goals 4. IEP modifications for summative and formative
assessments
Accommodations: 1. Preferential seating 2. Have students work in pairs 3. Assistive technologies 4. Three options on multiple choice exams 5. Larger print 6. Fewer problems on each page 7. More time 8. Test administered in a quieter setting 9. Tests read orally
10. Chunking assignments into smaller segments 11. Tape lectures or provide a peer note-taker
Extensions: 1. Alternative assignments 2. Independent studies 3. Mentoring of other students
APPENDIX (Teacher resource extensions)
http://www.cs.gsu.edu/~ebullwinkel1/courses/Chapter1p1g.pdf WEB LINKS 1. McGraw-Hill Education Teacher Resources (Textbook)
Login at http://highered.mheducation.com/sites/0073383090/student_view0/index.html Weblinks for textbook http://www.mhhe.com/math/advmath/rosen/student/webres/
2. Discrete Math Applications (Performance Tasks) http://www.district196.org/edsrv/Assessment%20Web%20Page/Math%20Applications/High%20School/Discrete%20Math/GHSPREPA.html
3. Math Vids for Discrete Math (Videos) http://mathvids.com/topic/mathhelp/20-discrete-math 4. WUCT121 Discrete Mathematics (Lectures and Assignments) http://www.uow.edu.au/~bmaloney/wuct121/ 5. CSC2110 Discrete Mathematics (Tutorials) http://www.cse.cuhk.edu.hk/~chi/csc2110/tutorial.html 6. University of Edinburgh (Lecture pdfs from Rosen textbook) http://www.inf.ed.ac.uk/teaching/courses/dmmr/schedule.html 7. University of Nebraska (Lectures and Assignments) http://cse.unl.edu/~choueiry/S06-235/ 8. St. Louis University (Lectures and Assignments) http://math.slu.edu/~freeman/325Kspring12.html
9. Discrete Mathematics (Lectures, Quizzes, and Assignments) http://faculty.simpson.edu/lydia.sinapova/www/cmsc180/cmsc180-05/Sch180-05.htm
10. Western Oregon University (Lectures) http://www.wou.edu/~kruczekk/Courses/Math_355_F09/MTH355_Coursepack 11. University of North Florida (Assignments) https://www.unf.edu/~wkloster/3100/problems.pdf 12. Georgia State University (Quiz and Test Keys) http://www.cs.gsu.edu/~ebullwinkel1/ 13. SIGCSE http://www.sigcse.org/resources/reports/discrete/materials 14. Florida Institute of Technology (Lectures, Quizzes, Handouts, and Assignments) http://cs.fit.edu/~wds/classes/adm/ 15. LSU (Lectures, Quizzes, and Assignments) https://www.math.lsu.edu/~verrill/teaching/discrete2020/Spring2005/ 16. Discrete Mathematics Handouts
http://www.instructables.com/files/orig/FPE/EZYW/HMMFA5GF/FPEEZYWHMMFA5GF.pdf http://www.cheatography.com/dois/cheat-sheets/discrete-math/ http://www.rit.edu/~w-asc/math-handouts.php
CHAPTER 1 17. Discrete Mathematics for Dummies https://jasoninclass.wordpress.com/
18. Logic http://www.rbjones.com/rbjpub/logic/ 19. Finite Mathematics http://www.zweigmedia.com/RealWorld/logic/logicintro.html 20. Discrete Math Resources Flash Applications http://webspace.ship.edu/deensley/DiscreteMath/flash/#chpt1 21. Problems http://alumni.cs.ucr.edu/~elenah/courses/CSCI217/
22. Lecture 2: Boolean Algebra And Formal Logic http://mathvids.com/topic/20-discrete-math/major_subtopic/67/lesson/610-lecture-2-
boolean-algebra-and-formal-logic/mathhelp
23. Translating into logic form http://facultypages.ecc.edu/perel/SurveyCourse/Chapter3_Logic/practice/translations.htm
APPENDIX (Mathematical Practices)
1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision.
Communicate the precise answer to a real-world problem. 7. Look for and make use of structure.
Identify structural similarities between integers and polynomials. Identify expressions as single entities, e.g. the difference of two squares.
8. Look for and express regularity in repeated reasoning. Notes to teacher (not to be included in your final draft):
4 Cs Three Part Objective Creativity: projects Behavior Critical Thinking: Math Journal Condition Collaboration: Teams/Groups/Stations Demonstration of Learning (DOL) Communication – Powerpoints/Presentations
Chapter 2 Basic Structures: Sets, Functions,
Sequences, Sums, and Matrices
Total Number of Days: 10-days Grade/Course: 12/Discrete mathematics
ESSENTIAL QUESTIONS ENDURING UNDERSTANDINGS
How are sets useful in organizing data? In what ways can sets be represented visually? If the symbol “≤" were used in the set builder
notation instead of “<” what other changes need to be made to the notation?
A collection of distinct objects is called set. Matrices can be used to organized data. Using matrices can be make it easier to perform calculation on data. Matrices can be add, or subtract or multiply by scale.
PACING CONTENT SKILLS STANDARDS (CCCS/MP)
RESOURCES
LEARNING ACTIVITIES/ASSESSME
NTS
Basic Topic Description of what
students will be able to do Mc Graw Hill OTHER (e.g., tech)
1d 2.1 Sets Assess the basic terminology of set theory.
A.REI.1 Understand solving equations as a process of reasoning and explain the reasoning
Text: P 125-126 Basic 1-15(even) Average: 1-30(even) Advance 1-30(even) 45,46,47
http://www.mathsisfun.com/sets/sets-introduction.html
Assessments Diagnostic:Do now,
http://highered.mheducation.com/sites/0073383090/student_view0/chapter2/extra_examples.html
Formative: HW, Quiz , Exit Ticket http://highered.mheducation.com/sites/0073383090/student_view0/chapter2/extra_examples.html Note: APPENDIX
Learning Activites http://www.onlinemathlearning.com/math-sets.html
1.5 d 2.2 Set Operations
Categorize how set identities are established and to introduce the most important such identities.
A.REI.3 Solve equations and inequalities in one
variable real.
Text: P:136-138 Basic 1-20(odd) Average: 32-43
Advance 32-43, 63-65,59 48, 50, 56, 57, 73-89
http://www.math.csusb.edu/notes/proofs/bpf/node5.html.
Assessments Diagnostic
Do now, http://highered.mheducation.com/sites/0073383090/student_view0/chapter2/extra_examples.html
Formative: HW Quiz , Exit Ticket, http://highered.mheducation.com/sites/0073383090/student_view0/chapter2/extra_examples.html
Learning Activites http://www.mathsisfun.com/definitions/operation.html
2d.
2.3 Functions
Differentiate the concept of a function, the notion of one-to-one functions, onto functions, and the floor and ceiling functions.
F.IF.1,2 Understand a
function…assigns to each element of the domain
exactly one element of the
range
Text: P 152-155 Basic:1-20 Average: 46-61 Advance 46-61,72,79,80
Function http://www.math.niu.edu/~rusin/known-math/index/03EXX.html
Assessments DiagnosticDo
now, http://highered.mheducation.com/sites/0073383090/student_view0/chapter2/extra_examples.htmlFormative: HW, Quiz , Exit Ticket http://highered.mheducation.com/sites/0073383090/student_view0/chapter2/extra_examples.html Learning activities http://www.freemathhelp.com/functions.html
1.5d
2.4 Sequences and Summations
Evaluate terminology used for sequences and summations and to introduce the concept of count ability.
F.IF.3 Recognize that sequence are function, Sometime defined recursively , whose domain is a subset of the integers.
Text: P167-169 Basic: 5-20(odd) Average: 5-25(odd) Advance 5-25(odd) 35,37,38,43
http://www.research.att.com/~njas/sequences/index.html
Assessments Diagnostic : Do now, http://highered.mheducation.com/sites/0073383090/student_view0/chapter2/extra_examples.html
Formative: HW, Quiz , Exit Ticket http://highered.mheducation.com/sites/0073383090/student_view0/chapter2/extra_examples.html Learning Activities https://people.richland.edu/james/lecture/m116/sequences/sequences.html
2d
2.5 Cardinality of Sets
Distinguish what it means for two infinite sets to have the same cardinality, providing us with a way to measure the relative sizes of infinite sets.
Text: P 176-177 Basic:5-10 Average: 5-16(odd) Advance 5-16(odd) 26,32,39
https://www.youtube.com/watch?v=K8whYBarK8M
Assessments Diagnostic: Do now, http://highered.mheducation.com/sites/0073383090/student_view0/chapter2/extra_examples.htmlFormative: HW, Quiz , Exit Ticket http://highered.mheducation.com/sites/0073383090/student_view0/chapter2/extra_examples.html Learning Activities http://planetmath.org/cardinality
2 days
2.6 Matrices
Analyze basic properties of matrices and matrix arithmetic, including Boolean operations on zero-one matrices.
N.VM.6,7,8,9,10,11,12 Perform operations on matrices and use matrices in applications
Text: P 183-185 Basic: 1-10 Average: 1-20(even) Advance 1-20(even)
17,19,21odd, 39-69, 78-90
http://www.mathsisfun.com/algebra/matrix-introduction.html
Assessments Diagnostic: Do now, http://highered.mheducation.com/sites/0073383090/student_view0/chapter2/extra_examples.html
Formative: HW, Quiz , Exit Ticket http://highered.mheducation.com/sites/0073383090/student_view0/chapter2/extra_examples.html
Summative: Unit Test http://highered.mheducation.com/sites/0073383090/instructor_view0/printable_tests.htm Learning Activities http://www.purplemath.com/modules/matrices.htm
INSTRUCTIONAL FOCUS OF UNIT
Student should distinguishing between the sets 0 and {0}, so explain that the empty set is the set with no elements and that it is a
subset of every set because when we talk about things in sets, we call the things contained in the sets elements. So we can be more
precise and say the empty set is a unique set with no elements.
Students should know how to express the definitions of one-to-one and onto in terms of quantifiers as such, the graph of a function
is a type of relation
Students should understand how matrix multiplication is defined and know that it is not commutative.
Students should work with the different forms of notation and with shifting indices in summations ,should also understand that
sequences and strings are just special types of functions.
ACADEMIC VOCABULARY Words: Set: Axiom: Element: Venn diagram: Finite set: Infinite set: Universal set: Member ship table: Function from A to B: Matrix: Sequence: Activity:
Marzano’s Six Strategies for Teaching Vocabulary: 7. YOU provide a description, explanation or example. (Story, sketch, power point) 8. Ask students to restate or re-explain meaning in their own words. (Journal, community circle, turn to your neighbor) 9. Ask students to construct a picture, graphic or symbol for each word. 10. Engage students in activities to expand their word knowledge. (Add to their notes, use graphic organizer format) 11. Ask students to discuss vocabulary words with one another (Collaborate) 12. Have students play games with the words. (Bingo with definitions, Pictionary, Charades, etc.)
Using Marzano’s Strategy 5: Tell a friend- one student tells the other a story using 3 related terms (terms: meteor, meteorite, meteoroid) Strategy 4: Venn Diagram- use Venn Diagram to compare terms (protoplanet and protosun)
21ST CENTURY SKILLS (4Cs & CTE Standards)
21st Century Life and Careers 9.1 21st Century Life & Career Skills: All students will demonstrate the creative, critical thinking, collaboration, and problem-solving skills needed to function successfully as both global citizens and workers in diverse ethnic and organizational cultures.
A. Critical Thinking and Problem Solving 9.1.12.A.1: Apply critical thinking and problem-solving strategies during structured learning experiences.
goodies role of density in problem solving. B. Creativity and Innovation
9.1.12.B.1: Present resources and data in a format that effectively communicates the meaning of the data and its implications for solving problems, using multiple perspectives.
The bus schedule 9.4 21st Century Career and Technical Education:
O. Science, Technology, Engineering & Mathematics Career Cluster 9.4.12.O.2: Demonstrate mathematics knowledge and skills required to pursue the full range of postsecondary education and career opportunities math knowledge
MODIFICATIONS/ACCOMMODATIONS
Small Group Activities- when students are given group guided practice like, “Diagrams of function machines can be used to represent both simple and complex functions, such as mod, div, exponential and logarithmic functions. Functions can be expressed in set notation or in mapping form, to allow students to see different arrangements of the data. Drawing arrows from domain to range can aid in determining whether or not a relation is a function. “
Sets can be represented with three-dimensional objects, as elements listed on index cards, or as abstract items in a list. The use of color (red and blue blocks, colored index cards, colored pencils or markers) can aid in comparing sets. Making the objects tangible can make it easier to bridge into representing the sets with proper notation and operations.
Modified assessments and assignments(class work , homework. Quizzes/tests) as needed.
APPENDIX (Teacher resource extensions)
24. E-Text, Interactive Digital Resources, Teacher Resources Login at http://www.mhhe.com/links/1256/1246/2083/2084/2091/index.html :http://highered.mheducation.com/sites/0073383090/instructor_view0/lecture_powerpoint_slides.htm?sessionId=1447444114112293863975a9eb81e027e4cd1813452dd777fe96b&existinguser=true CCSS. Mathematical Practices MP1: Make Sense of problems and persevere in solving them. MP2: Reason abstractly and quantitatively. MP3: Construct viable arguments and critique the reasoning of others. MP4: Model with mathematics. MP5: Use appropriate tools strategically. MP6: Attend to precision. MP7: Look for and make use of structure.
MP8 :Look for and express regularity in repeated reasoning.)
Notes to teacher (not to be included in your final draft):
4 Cs Three Part Objective Creativity: projects Behavior Critical Thinking: Math Journal Condition Collaboration: Teams/Groups/Stations Demonstration of Learning (DOL) Communication – Powerpoints/Presentations
Chapter 3 Algorithms Total Number of Days: 5week Grade/Course: 12/Discreatemathmtics
ESENTIAL QUESTIONS ENDURING UNDERSTANDINGS
How can algorithmic thinking be used to solve problems?
How does the logarithmic form of an exponential equation compare to the original equation?
How can you use the properties of exponents to evaluate a logarithm?
Measures the largest number of basic operations requires to execute and algorithm.
The big-O notation is measure of the growth of functions and often used to measure the complexity of algorithms.
In optimization problems, algorithms that use the best choice at each step are called greedy algorithms.
PACING CONTENT SKILLS STANDARDS (CCCS/MP)
RESOURCES LEARNING
ACTIVITIES/ASSESSMENTS
Basic Topic Description of what students will
be able to do Mc Graw
Hill OTHER
(e.g., tech)
1.5d 3.1 Algorithms Apply the concept and basic properties of an algorithm
F.IF4 Interpret functions that arise in applications in terms of a context (linear and exponential
Text: P 125-126 Basics 1-10 Average: 26,27,41,42,47-49 Advance 26,27,41,42,47-49,56,58,60,61
http://cg.scs.carleton.ca/~morin/misc/sortalg/
Assessments Diagnostic: Do now, http://highered.mheducation.com/sites/0073383090/student_view0/chapter3/extra_examples.html
Formative: HW, Quiz , Exit Ticket http://highered.mheducation.com/sites/0073383090/student_view0/chapter3/extra_examples.html Learning Activities http://www.matrixlab-examples.com/algorithm-examples.html
1.5d
3.2 The Growth of
Functions .
Create big-0 and related notation (Q, 0) and to show how to estimate the size of functions using this notation.
F.LE.4 For exponential model, express as a logarithm the solution to 𝑎𝑏𝑐𝑡 = 𝑑 where a, c and d are numbers and the base b is 2, 10 or evaluate the logarithm using technology.
Text: P 136-138 Basic: 1-10 Average: 12-22(even) Advance 12-22(even) 61-69,72
http://www.cs.sfu.ca/~ggbaker/zju/math/growth.html
Assessments Diagnostic: Do now, http://highered.mheducation.com/sites/0073383090/student_view0/chapter3/extra_examples.html
Formative: HW, Quiz , Exit Ticket http://highered.mheducation.com/sites/0073383090/student_view0/chapter3/extra_examples.html Learning Activities http://regentsprep.org/REgents/math/ALGEBRA/AE7/ExpDecayL.htm
2d
3.3 Complexity of Algorithms
Pinpoint computational complexity analysis and Euclid’s algorithm.
Text: P 152-155 Basic: 1-10 Average:1-4 Advance,1-4 12-14
http://www.eng.unt.edu/ian/books/free/lnoa.pdf.
Assessments Diagnostic: Do now, http://highered.mheducation.com/sites/0073383090/student_view0/chapter3/extra_examples.html Formative: Quiz , E.T http://highered.mheducation.com/sites/0073383090/student_view0/chapter3/extra_examples.html Summative: Unit Test . http://highered.mheducation.com/sites/0073383090/instructor_view0/prin
table_tests.htm Learning Activities http://bigocheatsheet.c
om/
INSTRUCTIONAL FOCUS OF UNIT
The algorithm for finding the largest element in a finite sequence of integers provides a good example of an algorithm since it is simple and it solves a useful problem.
Students have trouble with big-0 notation. Often they cannot decide how to choose the witnesses C and k in the definition. Show them how different pairs of constants can be used as witnesses. Give several different examples to illustrate the concept. Show how the definition of this notation involves the use of existential and universal quantifiers.
Complexity of algorithms is an important mathematical part of computer science. We define different types of complexity, but concentrate on time
complexity. Marzano's Academic vocabulary
Words:
Algorithm: Time complexity: Space complexity: Greedy algorithm: Tractable problem: Intractable problem: Solvable problem: Unsolvable problem:
Marzano’s Six Strategies for Teaching Vocabulary: 1. YOU provide a description, explanation or example. (Story, sketch, power point) 2. Ask students to restate or re-explain meaning in their own words. (Journal, community circle, turn to your neighbor) 3. Ask students to construct a picture, graphic or symbol for each word. 4. Engage students in activities to expand their word knowledge. (Add to their notes, use graphic organizer format) 5. Ask students to discuss vocabulary words with one another (Collaborate) 6. Have students play games with the words. (Bingo with definitions, Pictionary, Charades, etc.)
Using Marzano’s Strategy 2: Teach a friend- pair of students gets 3 terms each, individually read definintion, read example use in sentence, and small paragraph. Students will alternate teaching their partner their words (terms: set 1-corona, photosphere, chromosphere set 2 sunspot, prominence, solar flare)
Strategy 3: Graph-Ic- Using two similar terms, create a graphic for each term focusing on the differences (terms: fission and fusion)
21ST CENTURY SKILLS (4Cs & CTE Standards)
1 21st Century Life & Career Skills: All students will demonstrate the creative, critical thinking, collaboration, and problem-solving skills needed to function successfully as both global citizens and workers in diverse ethnic and organizational cultures.
A. Critical Thinking and Problem Solving 9.1.12.A.1: Apply critical thinking and problem-solving strategies during structured learning experiences.
The twelve dots critical thinking making connections between multiple events B. Creativity and Innovation
9.1.12.B.1: Present resources and data in a format that effectively communicates the meaning of the data and its implications for solving problems, using multiple perspectives.
wolfram organize and graph data F. Accountability, Productivity, and Ethics
9.1.12.F.2: Demonstrate a positive work ethic in various settings, including the classroom and during structured learning experiences.
mathwords students must work together to measure 9.1.12.F.6: Relate scientific advances (e.g., advances in medicine) to the creation of new ethical dilemmas.
Example: STEAM project regarding global warming and the competing views regarding how to address it. 9.4 21st Century Career and Technical Education:
O. Science, Technology, Engineering & Mathematics Career Cluster 9.4.12.O.2: Demonstrate mathematics knowledge and skills required to pursue the full range of postsecondary education and
MODIFICATIONS/ACCOMMODATIONS
Teacher directly instruction by providing students with more necessary steps in order to solve the problems. Small Group Activities- when students are given group guided practice like
“Diagrams of function machines can be used to represent both simple and complex functions, such as mod, div, exponential and logarithmic functions. Functions can be expressed in set notation or in mapping form, to allow students to see different arrangements of the data. Drawing arrows from domain to range can aid in determining whether or not a relation is a function.”
IEP/504 Modification: Modified assessments and assignments(class work , homework. Quizzes/tests) as needed
APPENDIX (Teacher resource extensions)
E-Text, Interactive Digital Resources, Teacher Resources Login at http://www.mhhe.com/links/1256/1246/2083/2084/2091/index.html
http://highered.mheducation.com/sites/0073383090/instructor_view0/powerpoint_slides_of_figures_and_tables.htm 1. E-Text, Interactive Digital Resources, Teacher Resources Login at http://www.mhhe.com/links/1256/1246/2083/2084/2091/index.html CCSS. Mathematical Practices: MP1: Make Sense of problems and persevere in solving them. MP2: Reason abstractly and quantitatively. MP3: Construct viable arguments and critique the reasoning of others. MP4: Model with mathematics. MP5: Use appropriate tools strategically. MP6: Attend to precision. MP7: Look for and make use of structure. MP8 :Look for and express regularity in repeated reasoning.
Chapter 4 Number Theory and Cryptography
Total Number of Days: 15-days Grade/Course: 12/Discrete Math
ESSENTIAL QUESTIONS ENDURING UNDERSTANDINGS
Why and how is coding used? What are some of the benefits of coding? How do patterns work with coding?
In addition to the base 10 number system, there are various number systems present in everyday life.
Mathematics is used in coding for efficiency. Patterns exist throughout coding.
PACING CONTENT SKILLS STANDARDS (CCCS/MP)
RESOURCES
LEARNING ACTIVITIES/ASSESSME
NTS
Basic Topic Description of what students will be able to do Mc Graw Hill OTHER
(e.g., tech)
3d 4.1 Divisibility and Modular Arithmetic
Use the rules of divisibility.
Apply the division algorithm.
Introduction to modular arithmetic.
Properties and uses of modulo m.
N.Q.2 Text pgs 237-245
Text resources Lesson
PowerPoint Chapter4 Slides 4-17
Text test bank
Rosen Links Library Chapter 4-1
Lecture Notes
You Tube Divisibility and Modular Arithmetic
Rosen Web Links
Rosen Extra Examples
Assessments Diagnostic: Do now,
http://betterlesson.com/community/lesson/12368/apply-divisibility-rules
Formative: HW, Quiz , Exit Ticket http://www.cse.chalmers.se/edu/year/2012/course/TDA351/impact/self1comm.html https://docs.google.com/file/d/0B6LUxWheAPCnSS1KYUhpZGhuREE/edit?pli=1
Note: APPENDIX
Learning Activites http://www.maths.ed.ac.uk/~aboocher/courses/55/worksheets/w6.pdf http://www.maths.ed.ac.uk/~aboocher/courses/55/wor
ksheets/s6.pdf
3d 4.2 Integer Representations and Algorithms
Representation of integers in other bases.
Conversions between binary, octal, and hexadecimal.
Algorithms for integer operations.
Modular exponentiation.
N.Q.2 Text pgs 245-257
Text resources Lesson
PowerPoint Chapter4 Slides 18-33
Text test bank
Rosen Links Library Chapter 4-2
Lecture Notes
You Tube Conversions Between Number Systems
Rosen Web Links
Rosen Extra Examples
Assessments Diagnostic: Do now,
http://community.iisme.org/lessons/display.cfm?lessonid=2153
Formative: HW, Quiz , Exit Ticket http://www.proprofs.com/quiz-school/story.php?title=binary-decimal-hexadecimal-number-systems http://www.free-test-online.com/binary/binary_numbers.htm
Note: APPENDIX
Learning Activites http://community.iisme.org/lessons/display.cfm?lessonid=2153
3d 4.3 Primes and Greatest Common Divisors
The fundamental theorem of arithmetic.
Trial division. The Seive of
Eratosthenes. Conjectures and open
problems about primes.
Greatest common divisors and least common multiples.
The Euclidean Algorithm.
N.Q.2 Text pgs 257-274
Text resources Lesson
PowerPoint Chapter4 Slides 34-59
Text test bank
Rosen Links Library Chapter 4-3
Lecture Notes
Lecture Notes Primes
Handout Euclidian Algorithm
You Tube Fundamental Theorem of Arithmetic
Assessments Diagnostic: Do now,
https://www.engageny.org/resource/grade-6-mathematics-module-2-topic-d-overview
Formative: HW, Quiz , Exit Ticket http://quiz.thefullwiki.org/Fundamental_theorem_of_arithmetic http://contacts.ucalgary.ca/info/math/files/info/unitis/courses/MATH271/F2009/LEC1/MATH271-F09-LEC1-Quiz-2-Solutions.pdf http://www.seethesoluti
Rosen Web Links
Rosen Extra Examples
ons.net/practice-exams-topic/294/
Note: APPENDIX
Learning Activites http://www.mathgoodies.com/webquests/number_theory/PDF/unit3_wks2.pdf http://www.mathgoodies.com/webquests/number_theory/PDF/unit3_wks2_key.pdf http://web.stanford.edu/class/cs103x/pset2_sol.pdf
4d 4.4 Solving Congruences
Linear congruences. The Chinese
Remainder Theorem. Computer arithmetic
with large integers. Fermat’s Little
Theorem. Pseudoprimes. Primative roots and
discrete logarithms.
N.Q.2 Text pgs 274-287 Text resources Lesson
PowerPoint Chapter4 Slides 60-77
Text test bank
Rosen Links Library Chapter 4-4
Lecture Notes
PDF Slides Chinese Remainder Theorem
You Tube Solving Linear Congruences
Rosen Web Links
Rosen Extra Examples
Assessments Diagnostic: Do now,
https://www.artofproblemsolving.com/Store/products/intro-numtheory/exc2.pdf
Formative: HW, Quiz , Exit Ticket http://quiz.thefullwiki.org/Linear_congruence_theorem http://www.oxfordmathcenter.com/drupal7/node/200
Note: APPENDIX
Learning Activites http://www.math.ucsd.edu/~ibejenar/teaching/2013/109/HW9Solutions.pdf
INSTRUCTIONAL FOCUS OF UNIT
Divisibility and modular arithmetic. To introduce some fundamental concepts from number theory, including the division
algorithm, congruences, and the rules of modular arithmetic.
Integer representations and algorithms. To study representations of integers in different bases, including binary, octal, and
hexadecimal representations, and to introduce algorithms involving integers based on these representations.
Primes and greatest common factors. To introduce some fundamental concepts from number theory, including primality,
prime factorization, and greatest common divisors. To introduce some important conjectures about primes.
Solving congruences. To learn how to solve linear congruences and simultaneous systems of linear congruences. To introduce
Fermat’s little theorem, pseudoprimes, primitive roots, and discrete logarithms.
TEXTBOOK
Discrete Mathematics and Its Applications 7th Edition Kevin Rosen
ACADEMIC VOCABULARY Words: Modular arithmetic: Prime: Composite: Divisibility: Modulo : Binary: Octal: Hexidecimal: Fundamental Theorem of Arithmetic:
Seive of Eratosthenes: The Euclidean Algorithm: Linear congruences: The Chinese Remainder Theorem: Fermat’s Little Theorem: Pseudoprimes
Marzano’s Six Strategies for Teaching Vocabulary: 1. YOU provide a description, explanation or example. (Story, sketch, power point) 2. Ask students to restate or re-explain meaning in their own words. (Journal, community circle, turn to your neighbor) 3. Ask students to construct a picture, graphic or symbol for each word. 4. Engage students in activities to expand their word knowledge. (Add to their notes, use graphic organizer format) 5. Ask students to discuss vocabulary words with one another (Collaborate) 6. Have students play games with the words. (Bingo with definitions, Pictionary, Charades, etc.)
Using Marzano’s Strategy 4: Students will use a Frayer Vocabulary Graphic Organizer. Strategy 5: Students will think-pair-share with vocabulary words.
21ST CENTURY SKILLS (4Cs & CTE Standards)
9.4.D Business, Management & Administration Career Cluster 9.4.O(1) Engineering and Technology
9.4.12.O.(1).1 Apply the concepts, processes, guiding principles, and standards of school mathematics to solve science, technology, engineering, and mathematics problems.
9.4.12.O.(1).7 9.4.O(2) Science and Mathematics
9.4.12.O.(2).2 Apply science and mathematics when developing plans, processes, and projects to find solutions to real world problems.
Activity: Cryptology.
Links for activities https://www.chatham.edu/pti/curriculum/units/2004/Amick.pdf http://faculty.gvsu.edu/aboufade/web/enigma/School/Welcome.htm
MODIFICATIONS/ACCOMMODATIONS
Modifications: 1. Less complex reading level 2. Shortened assignments 3. Different goals 4. IEP modifications for summative and formative
assessments
Accommodations: 1. Preferential seating 2. Have students work in pairs 3. Assistive technologies 4. Three options on multiple choice exams 5. Larger print 6. Fewer problems on each page 7. More time 8. Test administered in a quieter setting 9. Tests read orally
10. Chunking assignments into smaller segments
11. Tape lectures or provide a peer note-taker
Extensions: 1. Alternative assignments 2. Independent studies 3. Mentoring of other students
APPENDIX (Teacher resource extensions)
WEB LINKS 1. McGraw-Hill Education Teacher Resources (Textbook)
Login at http://highered.mheducation.com/sites/0073383090/student_view0/index.html Weblinks for textbook http://www.mhhe.com/math/advmath/rosen/student/webres/
2. Discrete Math Applications (Performance Tasks) http://www.district196.org/edsrv/Assessment%20Web%20Page/Math%20Applications/High%20School/Discrete%20Math/GHSPREPA.html
3. Math Vids for Discrete Math (Videos) http://mathvids.com/topic/mathhelp/20-discrete-math 4. WUCT121 Discrete Mathematics (Lectures and Assignments) http://www.uow.edu.au/~bmaloney/wuct121/ 5. CSC2110 Discrete Mathematics (Tutorials) http://www.cse.cuhk.edu.hk/~chi/csc2110/tutorial.html 6. University of Edinburgh (Lecture pdfs from Rosen textbook) http://www.inf.ed.ac.uk/teaching/courses/dmmr/schedule.html 7. University of Nebraska (Lectures and Assignments) http://cse.unl.edu/~choueiry/S06-235/ 8. St. Louis University (Lectures and Assignments) http://math.slu.edu/~freeman/325Kspring12.html 9. Discrete Mathematics (Lectures, Quizzes, and Assignments)
http://faculty.simpson.edu/lydia.sinapova/www/cmsc180/cmsc180-05/Sch180-05.htm 10. Western Oregon University (Lectures) http://www.wou.edu/~kruczekk/Courses/Math_355_F09/MTH355_Coursepack 11. University of North Florida (Assignments) https://www.unf.edu/~wkloster/3100/problems.pdf
12. Georgia State University (Quiz and Test Keys) http://www.cs.gsu.edu/~ebullwinkel1/ 13. SIGCSE http://www.sigcse.org/resources/reports/discrete/materials 14. Florida Institute of Technology (Lectures, Quizzes, Handouts, and Assignments) http://cs.fit.edu/~wds/classes/adm/ 15. LSU (Lectures, Quizzes, and Assignments) https://www.math.lsu.edu/~verrill/teaching/discrete2020/Spring2005/ 16. Discrete Mathematics Handouts
http://www.instructables.com/files/orig/FPE/EZYW/HMMFA5GF/FPEEZYWHMMFA5GF.pdf http://www.cheatography.com/dois/cheat-sheets/discrete-math/ http://www.rit.edu/~w-asc/math-handouts.php
CHAPTER 4
17. Modular Arithmetic http://www.math.rutgers.edu/~erowland/modulararithmetic.html 18. An Introduction To Modular Arithmetic http://nrich.maths.org/4350 19. Codes in Everyday Use http://www.cimt.plymouth.ac.uk/projects/mepres/alevel/discrete_ch8.pdf
20. Theory of Codes http://www.cimt.plymouth.ac.uk/projects/mepres/alevel/discrete_ch9.pdf
21. Number Theory and Cryptology http://math.berkeley.edu/~ericp/teaching/Fall2013/55/chapter-4a.pdf
22. Basic Number Theory http://ocw.nctu.edu.tw/upload/classbfs1210032011184835.pdf
23. Solving Congruences http://www.math.mtu.edu/mathlab/COURSES/holt/dnt/lincong.html
24. The Fundamentals http://cobweb.cs.uga.edu/~cai/courses/discmath/spring2008/lecture-note3.pdf
25. Berkley http://math.berkeley.edu/~ericp/teaching/Fall2013/55/chapter-4a.pdf
APPENDIX (Mathematical Practices)
1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision.
Communicate the precise answer to a real-world problem. 7. Look for and make use of structure.
Identify structural similarities between integers and polynomials. Identify expressions as single entities, e.g. the difference of two squares.
8. Look for and express regularity in repeated reasoning.
Notes to teacher (not to be included in your final draft):
4 Cs Three Part Objective Creativity: projects Behavior Critical Thinking: Math Journal Condition Collaboration: Teams/Groups/Stations Demonstration of Learning (DOL) Communication – Powerpoints/Presentations
Chapter 5 Induction and Recursion Total Number of Days: 5week Grade/Course: 12/Discreatemathmtics
ESENTIAL QUESTIONS ENDURING UNDERSTANDINGS
How are explicit formulas used to represent sequences?
What are some applications of sequences? What are some instances or recursive functions in
computer science?
Sequences can be used to represent patterns that exist in mathematics and the world around us.
Induction has two parts, a basis step, where we show that P(1) is true, and an inductive step, where we show that for all positive integers k, if P(k) is true, then P(k + 1) is true.
PACING CONTENT SKILLS STANDARDS (CCCS/MP)
RESOURCES LEARNING
ACTIVITIES/ASSESSMENTS
Basic topics Description of what
students will be able to do
Mc Graw Hill OTHER
(e.g., tech)
1.5d 5.1 Induction and Recursion
Compare and construct proofs of a variety of theorems using various forms of mathematical induction.
A.CED.1.2.3.4
Create equations that describe numbers or relationships
Text: P 329-333 Basic: 1-10 Average: 3-46(odd) Advance 3-46(odd) 52-55,60-65,72,73,83
http://scienceblogs.com/goodmath/2007/01/23/basics-recursion-
and-induction-1/
Assessments Diagnostic
Do now, http://highered.mheducation.com/sites/0073383090/student_view0/chapter5/extra_examples.html
Formative: HW, Quiz , Exit Ticket http://highered.mheducation.com/sites/0073383090/student_view0/chapter5/extra_examples.html
. Learning Activities http://www.cs.cmu.edu/~rwh/introsml/techniques/indrec.htm
1.5d
5.2 Strong Induction and Well-Ordering
Classify how to construct proofs of a variety of theorems using strong induction and the
well-ordering property..
Text: P 341-344 Basic:1-10 Average: 1-10,14-20(even) Advance 1-10,14-20(even) 22,23,29,30,32
http://math.stackexchange.com/questions/536404/proof-by-induction-for-a-recursive-sequence-and-a-formula
Assessments Diagnostic
Do now, http://highered.mheducation.com/sites/0073383090/student_view0/chapter5/extra_examples.html
Formative: HW, Quiz , Exit Ticket
http://highered.mheducation.com/sites/0073383090/student_view0/chapter5/extra_examples.html
Learning Activities
http://www.sanfoundry.com/discrete-
mathematics-mcqs-strong-induction-well-
ordering/ 2d 5.3 Recursive
Definitions and Structural Induction
Describe how functions, sequences, and sets can be defined recursively and to show how to use various forms of induction, including structural induction, to prove properties of such entities.
F-IF-3 Recognize that sequences are functions, sometimes defines recursively, whose domain is sub set of integers. F-LE2 Construct linear and exponential function, including arithmetic and geometric sequences, given a graph, a description of relationship, or two input-out put pairs.
Text: P 357-360 Basic: 1-10 Average: 12-19, 32,33,36,43 Advance 12-19, 32,33,36,43 47-58
http://www.eecs.yorku.ca/course_archive/2008-09/S/1019/Website_files/17-recursive-definitions-and-structural-induction.pdf
Assessments Diagnostic
Do now, http://highered.mheducation.com/sites/0073383090/student_view0/chapter5/extra_examples.html
Formative: HW, Quiz , Exit Ticket http://highered.mheducation.com/sites/0073383090/student_view0/chapter5/extra_examples.html
Summative: Unit Test .
http://highered.mheducation.com/sites/0073383090/instructor_view0/printable_t
ests.htm Learning Activities http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-spring-2010/lecture-notes/MIT6_042JS10_lec19.pdf
INSTRUCTIONAL FOCUS OF UNIT
Carefully explain the steps that make up a proof by mathematical induction. Explain why the basis step can begin at any integer. It helps students when you structure proofs by mathematical induction.
Introduce some notions of computational geometry and show how strong induction is used in the proof of Theorem 1, which shows that every simple polygon can be triangulated.
Introduce the technique of structural induction and provide some examples of how it is used to prove results about recursively defined sets.
Marzano's Academic vocabulary Words;
Sequence: Structural induction: Recursive algorithm: Merge sort: Iteration : Sequence: loop invariant: final assertion: merge sort: arithmetic progression
Marzano’s Six Strategies for Teaching Vocabulary: 1. YOU provide a description, explanation or example. (Story, sketch, power point) 2. Ask students to restate or re-explain meaning in their own words. (Journal, community circle, turn to your neighbor) 3. Ask students to construct a picture, graphic or symbol for each word. 4. Engage students in activities to expand their word knowledge. (Add to their notes, use graphic organizer format) 5. Ask students to discuss vocabulary words with one another (Collaborate) 6. Have students play games with the words. (Bingo with definitions, Pictionary, Charades, etc.)
Using Marzano’s Strategy 1: Create a story board of the steps in the scientific method (terms: scientific method) Strategy 6: Students play $10,000 pyramid game where they have to give their partners clues to guess words (terms: aphelion, astronomical unit, force, perihelion, ellipse, eccentricity, foci, inertia, orbital period, light-year)
21ST CENTURY SKILLS (4Cs & CTE Standards)
21st Century Life and Careers 9.1 21st Century Life & Career Skills: All students will demonstrate the creative, critical thinking, collaboration, and problem-solving skills needed to function successfully as both global citizens and workers in diverse ethnic and organizational cultures.
A. Critical Thinking and Problem Solving 9.1.12.A.1: Apply critical thinking and problem-solving strategies during structured learning experiences.
Induction Activity use multiple strategies. B. Creativity and Innovation
9.1.12.B.1: Present resources and data in a format that effectively communicates the meaning of the data and its implications for solving problems, using multiple perspectives.
strog induction F. Accountability, Productivity, and Ethics
9.1.12.F.2: Demonstrate a positive work ethic in various settings, including the classroom and during structured learning experiences. 9.1.12.F.6: Relate scientific advances (e.g., advances in medicine) to the creation of new ethical dilemmas.
9.4 21st Century Career and Technical Education: O. Science, Technology, Engineering & Mathematics Career Cluster
activity
MODIFICATIONS/ACCOMMODATIONS
Small Group Activities- when students are given group guided practice like, “Students can use a wooden Tower of Hanoi puzzle to experience the steps necessary to move the disks according to the rules of the game. This hands-on experience can bring to life the concept of solving smaller problems (start with 1 disk, then 2, then 3, etc.) to develop a strategy for solving a larger problem. This concept can be repeated with a coin problem in which the penny is replaced with a 3 cent piece. Students can use chips or blocks to represent a 3 cent piece and a 5 cent piece and see what values of change are and are not possible to make. Using the manipulative will make it easier to solve several simpler problems before attempting a large one. In this way the concepts of pattern, induction, and recursion are made more tangible.”
IEP/504 Modification: Modified assessments and assignments (class work , homework. Quizzes/tests) as needed..
APPENDIX (Teacher resource extensions)
1. E-Text, Interactive Digital Resources, Teacher Resources Login at http://www.mhhe.com/links/1256/1246/2083/2084/2091/index.html CCSS. Mathematical Practices: MP1: Make Sense of problems and persevere in solving them. MP2: Reason abstractly and quantitatively. MP3: Construct viable arguments and critique the reasoning of others. MP4: Model with mathematics. MP5: Use appropriate tools strategically..
Chapter 6 COUNTING
Total Number of Days: 10-weeks Grade/Course: 12/Discreatemathmtics
ESENTIAL QUESTIONS ENDURING UNDERSTANDINGS
How can the product rule be used to find the number of functions from a set with m elements to a set with n elements?
Explain how the pigeonhole principle can be used to
show that among any 11 integers, at least two must have the same last digit.
For a polynomial function, how are factors and roots related?
How can a row of Pascal’s triangle be produced from the one above it?
How do you determine which counting principle is used to solve a problem?
How are combinations related to subsets of a set?
Algorithms can effectively and efficiently be used to quantify and interpret discrete information. Counting methods can be used to find the number of possible ways to choose objects with and without regard to
order.
Permutation and combination notation can be used to represent real world situation.
Pascal’s identity shows that when two adjacent binomial coefficients in this triangle are added, the binomial coefficient in the next row between these two coefficients is produced.
PACING CONTENT SKILLS STANDARDS (CCCS/MP)
RESOURCES LEARNING
ACTIVITIES/ASSESSMENTS
Description of what students
will be able to do Mc Graw Hill Pearson OTHER
(e.g., tech
1 6.1 The Basic of Counting
Describe the basic counting rules and to show how they are used to solve a variety of counting
S.IC.1,2 Understand and evaluate random process underlying statistical experiments.
Text Pg. 396-398 Basic: 1-10 Average 41-45,48-61,64-69 Advance 41-45,48-61,64-69 ,73,50,51,70
http://www.mathsisfun.com/data/basic-counting-principle.html
Assessments Diagnostic: Do now, http://highered.mheducation.com/sites/0073383090/student_view0/chapter6/extra_examples.html
Formative: HW, Quiz , Exit Ticket http://highered.mhe
ducation.com/sites/0073383090/student_view0/chapter6/extra_examples.html
Learning Activities http://www.aaamath.com/sta-basic-cntg.htm
1.5d
6.2 The Pigeonhole Principle
Identify the pigeonhole principle and show how to use it in enumeration and in proofs.
Text: p 405-406 Basic:1-10 Average: 8,24-30 Advance 8,24-30 11,23,24,25,40,42
https://www.math.hmc.edu/funfacts/ffiles/10001.4.shtml
Assessments Diagnostic
Do now, http://highered.mheducation.com/sites/0073383090/student_view0/chapter6/extra_examples.html
Formative: HW, Quiz , Exit Ticket http://highered.mheducation.com/sites/0073383090/student_view0/chapter6/extra_examples.html
Summative: Unit Test .
Learning Activities http://www.math.ust.hk/~mabfchen/Math391I/Pigeonhole.pdf
1d 6.3 Permutation and Combination
Model permutations and combinations, to solve counting problems using them, and to show how theorems are proved by combinatorial arguments.
S.CP.9 Use permutations and combination to compute probabilities of compound event and solve problems
Text: P 413-415 Basic: 1-10 Average: 8-45(odd) Advance: 8-45(0dd) 44,46
http://www.mathsisfun.com/combinatorics/combinations-permutations.html
Assessments Diagnostic
Do now, http://highered.mheducation.com/sites/0073383090/student_view0/chapter6/extra_examples.html
Formative: HW, Quiz , Exit Tickethttp://highered.mheducation.com/sites/0073383090/student_view0/chapter6/extra_examples.html Learning Activities https://www.khanacademy.org/math/precalculus/prob_comb/combinatorics_precalc/v/permutations-and-combinations-1
1d
6.4 Binomial Coefficients And Identities
Describe the binomial theorem and to show how combinatorial identities can be proved by combinatorial arguments
S.CP.9 Use permutations and combination to compute probabilities of compound
event and solve problems.
Text: P 421-423 Basic:1-10 Average: 14-17,21,22,27-30, Advance 14-17,21,22,27-30,32,38,39
http://mathworld.wolfram.com/BinomialCoefficient.html
Assessments Diagnostic
Do now, http://highered.mheducation.com/sites/0073383090/student_view0/chapter6/extra_examples.html
Formative: HW, Quiz , Exit Tickethttp://highered.mheducation.com/sites/0073383090/s
tudent_view0/chapter6/extra_examples.html • Learning
Activities http://mathworld.wolfram.com/BinomialCoefficient.html
1.5d . 6.5 Generalized Permutation and Combination
Solve counting problems involving permutations and combinations with repetition allowed and permutations where objects may be indistinguishable
S.CP.9 Use permutations and combination to compute probabilities of compound event and solve problems
Text: P 432-434 Basic:1-10 Average: 1-13, 15-16,30-37,50-59 Advance 1-13, 15-16,30-37,50-59 47,48,49,61,63
http://www.mhhe.com/math/advmath/rosen/r5/instructor/shared/transparencies5/data/rdm4_5.pdf
Assessments Diagnostic
Do now, http://highered.mheducation.com/sites/0073383090/student_view0/chapter6/extra_examples.html
Formative: HW, Quiz , Exit Ticket http://highered.mheducation.com/sites/0073383090/student_view0/chapter6/extra_examples.html Learning Activities http://www.slideserve.com/cricket/generalized-permutations-and-combinations
3d
6.6 Generating Permutation and Combination
Model algorithms for generating permutations and combinations graph.
S.CP.9 Use permutations and combination to compute probabilities of compound event and solve problems
Text: P 432-434 Basic:1-10 Average: 14-17 Advance 1-13(even)14-17
http://www.cis.uoguelph.ca/~sawada/2910/notes/generating-1x2.pdf
Assessments Diagnostic Do now, http://highered.mheducation.com/sites/0073383090/student_view0/chapter6/extra_examples.html Formative: HW, Quiz , Exit Ticket http://highered.mheducation.com/sites/0073383090/student_view0/chapter6/extra_examples.html Summative:
Unit Test . http://highered.mheducation.com/sites/0073383090/instructor_view0/printable_
tests.htm Learning Activities
http://en.wikipedia.org/wiki/Permutation
INSTRUCTIONAL FOCUS OF UNIT
Show how counting problems (such as enumerating valid passwords on a computer system, discussed in Example 16, or counting Internet addresses, discussed in Example 17) can be solved using a combination of the two rules.
Students need to understand clearly that a combination involves an unordered selection of objects from a set with no repetition allowed, while a permutation involves an ordered selection of objects from a set with no repetition allowed.
Students have trouble drawing valid conclusions from the pigeonhole principle. You can clarify this using Figure 1, which illustrates what you can and cannot conclude from the pigeonhole principle, namely that if there are more pigeons than pigeonholes, some pigeonhole contains more than one pigeon, but some pigeonholes may contain no pigeons, and others may contain many pigeons. Example 4 provides an interesting application of the pigeonhole principle.
Marzano's Academic vocabulary Words:
Combinatory: Enumeration: Permutation: Pascal’s triangle: Binomial Co-efficient (𝑛𝑟
): Stirling number, r-permutation, r-combination : pigeonhole,:
Marzano’s Six Strategies for Teaching Vocabulary: 1. YOU provide a description, explanation or example. (Story, sketch, power point) 2. Ask students to restate or re-explain meaning in their own words. (Journal, community circle, turn to your neighbor) 3. Ask students to construct a picture, graphic or symbol for each word. 4. Engage students in activities to expand their word knowledge. (Add to their notes, use graphic organizer format) 5. Ask students to discuss vocabulary words with one another (Collaborate) 6. Have students play games with the words. (Bingo with definitions, Pictionary, Charades, etc.)
Using Marzano’s Strategy 2: Think-Restate-Pair-Share – Using textbook definition and explanation, come up with a definition that a 2nd grader could understand (terms: galactic cluster, galactic disk, galactic bulge)
Strategy 6: Name that structure- students will play game using dice and a board that had diagrams of galaxy structure. Where the dice lands, the student has to identify the structure. The student advances the number of spaces on the dice if they get it correct. (terms: barred-spiral galaxy, irregular galaxy, elliptical galaxy, galactic bulge, galactic center, galactic disk, galactic halo, galactic nucleus)
21ST CENTURY SKILLS (4Cs & CTE Standards)
21st Century Life and Careers
9.1 21st Century Life & Career Skills: All students will demonstrate the creative, critical thinking, collaboration, and problem-solving skills needed to function successfully as both global citizens and workers in diverse ethnic and organizational cultures.
A. Critical Thinking and Problem Solving 9.1.12.A.1: Apply critical thinking and problem-solving strategies during structured learning experiences.
Problems critical thinking B. Creativity and Innovation
9.1.12.B.1: Present resources and data in a format that effectively communicates the meaning of the data and its implications for solving problems, using multiple perspectives.
The game of five 9.4 21st Century Career and Technical Education:
O. Science, Technology, Engineering & Mathematics Career Cluster 9.4.12.O.2: Demonstrate mathematics knowledge and skills required to pursue the full range of postsecondary education and career opportunities
The board game
MODIFICATIONS/ACCOMMODATIONS
Small group Activities-when students are given group guided practice like, “Manipulative such as cards, coins, dice and blocks can be used for students to physically create arrangements of items as well as subsets. Results from problems with small amounts of elements such as 2, 3, 4, or 5 items, can be recorded in a table and discussed as a class. Diagrams of arrangements in a row or in a circle can be pre-made for students to fill in with blocks or by coloring them in. “
Modified assessments and assignments (class work , homework. Quizzes/tests) as needed
APPENDIX (Teacher resource extensions)
1. E-Text, Interactive Digital Resources, Teacher Resources Login at http://www.mhhe.com/links/1256/1246/2083/2084/2091/index.html CCSS. Mathematical Practices: MP1: Make Sense of problems and persevere in solving them. MP2: Reason abstractly and quantitatively. MP3: Construct viable arguments and critique the reasoning of others. MP4: Model with mathematics. MP5: Use appropriate tools strategically. MP6: Attend to precision. MP7: Look for and make use of structure. MP8 :Look for and express regularity in repeated reasoning.
Notes to teacher (not to be included in your final draft):
4 Cs Three Part Objective Creativity: projects Behavior Critical Thinking: Math Journal Condition Collaboration: Teams/Groups/Stations Demonstration of Learning (DOL) Communication – Powerpoints/Presentations
CHAPTER 8 Advanced Counting Techniques
Total Number of Days: 12 days Grade/Course: 12/Discrete Math
ESSENTIAL QUESTIONS ENDURING UNDERSTANDINGS
How can counting problems be modeled using recurrence relations?
How can generating functions be used to solve be used to solve counting problems?
Recurrence relations can be used to represent patterns that exist in mathematics and the world around us.
PACING CONTENT SKILLS STANDARDS (CCCS/MP)
RESOURCES
LEARNING ACTIVITIES/ASSESSME
NTS
Basic Topic Description of what students will be able to do Mc Graw Hill OTHER
(e.g., tech)
3d
8.1 Applications of Recurrence Relations
Fiobonacci Numbers
Tower of Hanoi
Model counting through recurrence relations.
Use dynamic programming to solve optimization problems.
F.1F.A1 F.1F.A3
Text pgs 501-514
Text resources Lesson
PowerPoint Chapter8 Slides 3-15
Text test bank
Rosen Links Library Chapter 8-1
Lecture Notes Fibonacci
Lecture Notes Tower of Hanoi
Lecture Notes Recurrence
You Tube Recurrence Relations
Rosen Web Links
Rosen Extra
Assessments Diagnostic: Do now,
http://disi.unitn.it/~ldkr/ml2013fall/MLexercises.pdf
Formative: HW, Quiz , Exit Ticket http://www.aber.ac.uk/~dcswww/Dept/Teaching/CourseNotes/2010-2011/CS20410/soln3Counting.pdf
Note: APPENDIX
Learning Activites http://www.knightswoodsecondary.org.uk/personal/Resources/Hillhead/Higher_Worksheets/Unit1_RecurrenceRelations.pdf
Examples 3d
8.2 Solving Linear Recurrence Relations
Solving linear homogeneous recurrence relations with constant coefficients.
Finding solutions to
linear nonhomogeneous recurrence relations with constant coefficients.
F.1F.A1 F.1F.A3
Text pgs 514-527
Text resources Lesson
PowerPoint Chapter8 Slides 16-32
Text test bank
Rosen Links Library Chapter 8-2
You Tube Linear Homogeneous Recurrence Relations
You Tube Linear Non Homogeneous Recurrence Relations
Rosen Web Links
Rosen Extra Examples
Assessments Diagnostic: Do now,
http://www.csee.umbc.edu/~stephens/203/PDF/8-3.pdf
Formative: HW, Quiz , Exit Ticket http://furthermathematicst.blogspot.com/2011/06/43-non-homogeneous-linear-recurrence.html
Note: APPENDIX
Learning Activites http://www.eecs.yorku.ca/course_archive/2008-09/S/1019/Website_files/21-linear-recurrences.pdf
3d 8.3 Divide and Conquer Algorithms and Recurrence Relations
Divide-and-conquer recurrence relations.
F.1F.A1 F.1F.A3
Text pgs 527-536
Text resources Lesson
PowerPoint Chapter8 Slides 33-44
Text test bank
Rosen Links Library Chapter 8-3
Lecture Notes
Lecture Notes
You Tube Divide and Conquer
Rosen Web Links
Rosen Extra
Assessments Diagnostic: Do now,
http://courses.csail.mit.edu/6.046/spring04/handouts/ps1-sol.pdf
Formative: HW, Quiz , Exit Ticket http://www.atilim.edu.tr/~mcs401/MCS401%20Worksheet-2.pdf http://www.cse.msu.edu/~torng/Classes/Archives/cse830.02spring/Homework/hw02.html
Note: APPENDIX
Learning Activites
Examples http://goldman.cse.wustl.edu/crc2007/homework/hw1.pdf
3d
8.4 Generating Functions
Useful facts about power series.
Solve counting problems by generating functions.
Use generating functions to solve recurrence relations.
Proving identities by generating functions.
F.1F.A1 F.1F.A3
Text pgs 537-552
Text resources Lesson
PowerPoint Chapter8 Slides 45-52
Text test bank
Rosen Links Library Chapter 8-4
Lecture Notes
You Tube Power Series
Rosen Web Links
Rosen Extra Examples
Assessments Diagnostic: Do now,
https://www.math.ust.hk/~mabfchen/Math3343/Recurrence-Relation-Generating-Function.pdf
Formative: HW, Quiz , Exit Ticket http://www.webpages.uidaho.edu/~markn/395/pdf/rec-eq.pdf
Note: APPENDIX
Learning Activites http://math.berkeley.edu/~bernd/hw9sol.pdf
INSTRUCTIONAL FOCUS OF UNIT
Applications of Recurrence Relations. To show how counting problems can be modeled using recurrence relations. To illustrate
how recurrence relations can be used in dynamic programming algorithms.
Solving Linear Recurrence Relations. To solve linear recurrence relations with constant coefficients.
Divide and Conquer Algorithms and Recurrence Relations. To study the complexity of divide-and-conquer algorithms with
functions that satisfy a special kind of recurrence relation.
Generating Functions. To introduce the notion of a generating function, to show how generating functions can be used to model
and solve counting problems, and to show how generating functions can be used to solve recurrence relations.
TEXTBOOK
Discrete Mathematics and Its Applications 7th Edition Kevin Rosen
ACADEMIC VOCABULARY
Words: Recurrence relation: Fiobonacci Numbers: Tower of Hanoi: Dynamic Programming: Linear homogeneous recurrence relation with constant
coefficients: Linear nonhomogeneous recurrence relation with constant coefficients: Divide-and-conquer algorithm: Generating function of a sequence: Power Series
Marzano’s Six Strategies for Teaching Vocabulary: 1. You provide a description, explanation or example. (Story, sketch, power point) 2. Ask students to restate or re-explain meaning in their own words. (Journal, community circle, turn to your neighbor) 3. Ask students to construct a picture, graphic or symbol for each word. 4. Engage students in activities to expand their word knowledge. (Add to their notes, use graphic organizer format) 5. Ask students to discuss vocabulary words with one another (Collaborate) 6. Have students play games with the words. (Bingo with definitions, Pictionary, Charades, etc.)
Using Marzano’s Strategy 3: Ask students to construct a picture, symbol, or graphic representing each word. Strategy 4: Students should make flash card sets with solutions on the back and use them with other students to practice the
vocabulary.
MODIFICATIONS/ACCOMMODATIONS
Modifications: 1. Less complex reading level
2. Shortened assignments 3. Different goals 4. IEP modifications for summative and formative
assessments
Accommodations: 1. Preferential seating 2. Have students work in pairs 3. Assistive technologies 4. Three options on multiple choice exams 5. Larger print 6. Fewer problems on each page 7. More time 8. Test administered in a quieter setting 9. Tests read orally
10. Chunking assignments into smaller segments
11. Tape lectures or provide a peer note-taker
Extensions: 1. Alternative assignments 2. Independent studies 3. Mentoring of other students
21ST CENTURY SKILLS (4Cs & CTE Standards)
9.4.O(1) Engineering and Technology
9.4.12.O.(1).1 Apply the concepts, processes, guiding principles, and standards of school mathematics to solve science, technology, engineering, and mathematics problems.
9.4.12.O.(1).7 Use mathematics, science, and technology concepts and processes to solve problems in projects involving design and/or production (e.g., medical,
agricultural, biotechnological, energy and power, information and communication, transportation, manufacturing, and construction).
Activity: Mathematical Modeling (link has 40 different projects) Links for activities http://www.indiana.edu/~hmathmod/projects.html
APPENDIX (Teacher resource extensions)
WEB LINKS 1. McGraw-Hill Education Teacher Resources (Textbook)
Login at http://highered.mheducation.com/sites/0073383090/student_view0/index.html Weblinks for textbook http://www.mhhe.com/math/advmath/rosen/student/webres/
2. Discrete Math Applications (Performance Tasks) http://www.district196.org/edsrv/Assessment%20Web%20Page/Math%20Applications/High%20School/Discrete%20Math/GHSPREPA.html
3. Math Vids for Discrete Math (Videos) http://mathvids.com/topic/mathhelp/20-discrete-math 4. WUCT121 Discrete Mathematics (Lectures and Assignments) http://www.uow.edu.au/~bmaloney/wuct121/ 5. CSC2110 Discrete Mathematics (Tutorials) http://www.cse.cuhk.edu.hk/~chi/csc2110/tutorial.html 6. University of Edinburgh (Lecture pdfs from Rosen textbook) http://www.inf.ed.ac.uk/teaching/courses/dmmr/schedule.html 7. University of Nebraska (Lectures and Assignments) http://cse.unl.edu/~choueiry/S06-235/ 8. St. Louis University (Lectures and Assignments) http://math.slu.edu/~freeman/325Kspring12.html
9. Discrete Mathematics (Lectures, Quizzes, and Assignments) http://faculty.simpson.edu/lydia.sinapova/www/cmsc180/cmsc180-05/Sch180-05.htm
10. Western Oregon University (Lectures) http://www.wou.edu/~kruczekk/Courses/Math_355_F09/MTH355_Coursepack 11. University of North Florida (Assignments) https://www.unf.edu/~wkloster/3100/problems.pdf 12. Georgia State University (Quiz and Test Keys) http://www.cs.gsu.edu/~ebullwinkel1/ 13. SIGCSE http://www.sigcse.org/resources/reports/discrete/materials 14. Florida Institute of Technology (Lectures, Quizzes, Handouts, and Assignments) http://cs.fit.edu/~wds/classes/adm/ 15. LSU (Lectures, Quizzes, and Assignments) https://www.math.lsu.edu/~verrill/teaching/discrete2020/Spring2005/ 16. Discrete Mathematics Handouts
http://www.instructables.com/files/orig/FPE/EZYW/HMMFA5GF/FPEEZYWHMMFA5GF.pdf http://www.cheatography.com/dois/cheat-sheets/discrete-math/ http://www.rit.edu/~w-asc/math-handouts.php
CHAPTER 8 17. Recurrence Relations http://www.cs.utexas.edu/~eberlein/cs336/recurrence.pdf 18. Using Trains to Model Recurrence Relations
http://books.google.com/books?id=05DEJ8Kh67AC&pg=PA55&lpg=PA55&dq=using+trains+to+model+recurrence+relations&source=bl&ots
=pRGHbGrwlM&sig=C7tZBv4286gOfZ7ZuSugOTCFAik&hl=en&sa=X&ei=DqNNVKS6FbSTsQS3toL4DQ&ved=0CB4Q6AEwAA#v=onepage&q=using%20trains%20to%20model%20recurrence%20relations&f=false
19. Solving Linear Recurrence Relations http://www.eecs.yorku.ca/course_archive/2008-09/S/1019/Website_files/21-linear-recurrences.pdf
20. Wiki How http://www.wikihow.com/Solve-Recurrence-Relations
21. Solving Recurrences http://www.cs.columbia.edu/~cs4205/files/CM2.pdf
22. Divide and Conquer Algorithms http://www.math.ucsd.edu/~pcompeau/Ch07_DnC_LinearSpaceAlignment_Edited.pdf
23. Algorithms and Data Structures http://setur.fo/fileadmin/user_upload/documents/Undirvisingartilfar/NVD/QinX/Algorithms/A_4n.pdf
24. Generating Functions http://rutherglen.science.mq.edu.au/wchen/lndmfolder/dm14.pdf
25. Generating Functions http://www.mhhe.com/math/advmath/rosen/r5/instructor/shared/transparencies5/data/rdm6_4.pdf
APPENDIX
(Mathematical Practices)
1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision.
Communicate the precise answer to a real-world problem. 7. Look for and make use of structure.
Identify structural similarities between integers and polynomials. Identify expressions as single entities, e.g. the difference of two squares.
8. Look for and express regularity in repeated reasoning.
CHAPTER 9 Relations
Total Number of Days: 17 days Grade/Course: 12/Discrete Math
ESSENTIAL QUESTIONS ENDURING UNDERSTANDINGS
What are the symbols used in logic? How can the truth value of a proposition or set of propositions
be determined? What makes a proof or an argument logically valid? Is there a best way to write a logically valid proof?
There is more than one correct way to write logical proof. Logical proofs carry more validity than baseless arguments. Common language is often riddled with fallacies (sometimes
when we communicate, our assumed meaning is not necessarily the logical meaning)
PACING CONTENT SKILLS STANDARDS (CCCS/MP)
RESOURCES
LEARNING ACTIVITIES/ASSESSM
ENTS
Basic Topic Description of what students will be able to do
Mc Graw Hill OTHER (e.g., tech)
3d
9.1 Relations and Their Properties
Functions as binary relations.
Functions as relations
on a set. Properties of relations. Combining relations.
N.Q.2 Text pgs 573-583
Text resources Lesson
PowerPoint Chapter9 Slides 3-16
Text test bank
Rosen Links Library Chapter 9-1
Lecture Notes
Lecture Notes
You Tube Binary Relations
Rosen Web Links
Rosen Extra Examples
Assessments Diagnostic: Do now,
http://www.millersville.edu/~bikenaga/math-proof/relations/relations.html
Formative: HW, Quiz , Exit Ticket https://www.cs.umd.edu/class/fall2003/cmsc250/hw/hw14.pdf http://faculty.simpson.edu/lydia.sinapova/www/cmsc180/LN180_Johnsonbaugh-07/L16-Properties.htm
Note: APPENDIX
Learning Activites http://cseweb.ucsd.edu/classes/fa13/cse20-a/hw_8_fa13_sol.pdf
3d
9.3 Representing Relations
Representing relations using matrices.
Representing relations using digraphs.
N.Q.2 Text pgs 591-597 Text resources Text test
bank
Rosen Links Library Chapter 9-3
Lecture Notes Matrices
Lecture Notes
You Tube Representing Relations
Rosen Web Links
Rosen Extra Examples
Assessments Diagnostic: Do now,
http://www.math.drexel.edu/~jsteuber/spr08/Ch2_Relations_HW.htm
Formative: HW, Quiz , Exit Ticket http://math.berkeley.edu/~mannisto/ws6_27.pdf http://coitweb.uncc.edu/~tbarnes2/LogicAlgorithms/notes/pdf/HW8.pdf
Note: APPENDIX
Learning Activites http://www.math.drexel.edu/~jsteuber/spr08/Ch2_Relations_HW.htm
3d
9.4 Closures of Relations
Reflexive and diagonal closures.
Paths in directed
graphs. Transitive closures. Warshall’s algorithm.
N.Q.2 Text pgs 597-607
Text resources Lesson
PowerPoint Chapter9 Slides 17-31
Text test bank
Rosen Links Library Chapter 9-4
Lecture Notes
You Tube Closure of Relations
Rosen Web Links
Rosen Extra Examples
Assessments Diagnostic: Do now,
http://homepage.cs.uiowa.edu/~hzhang/c19/ch10b.pdf
Formative: HW, Quiz , Exit Ticket http://www.cs.utsa.edu/~bylander/cs2233/graphshandout.pdf https://math.berkeley.edu/~ralph42/hw11solns55.pdf
Note: APPENDIX
Learning Activites http://webpages.uncc.edu/tbarnes2/LogicAlgorithms/notes/pdf/Pak5.pdf
3d 9.5 Equivalence relations. N.Q.2 Text pgs 607- Rosen Links Assessments
Equivalence of Relations
Equivalence classes. Equivalence classes
and partitions.
618
Text resources Lesson
PowerPoint Chapter9 Slides 32-42
Text test bank
Library Chapter 9-5
Lecture Notes
Lecture Notes
You Tube Equivalence Relations
Rosen Web Links
Rosen Extra Examples
Diagnostic: Do now, http://www.math.poly.edu/courses/ma2322/2322setB/solution4.pdf
Formative: HW, Quiz , Exit Ticket http://www.math.vt.edu/people/mcquain/hwrelations_2534_07.pdf
Note: APPENDIX
Learning Activites http://folk.uib.no/mbr085/11MAT220/lecturenotes/worksheets.pdf
3d 9.6 Partial Orderings
Lexicographic order. Hasse diagrams. Maximal and minimal
elements. Lattices. Topological sorting.
N.Q.2 Text pgs 618-633
Text resources Lesson
PowerPoint Chapter9 Slides 43-52
Text test bank
Rosen Links Library Chapter 9-6
Lecture Notes
You Tube Partial Orderings
Rosen Web Links
Rosen Extra Examples
Assessments Diagnostic: Do now,
http://www.snow.edu/kenyonp/Links/Math1630/Worksheets/WS2-5.pdf http://math.tutorvista.com/discrete-math/partially-ordered-set.html
Formative: HW, Quiz , Exit Ticket https://www.math.vt.edu/people/mcquain/2534_sol_eqR_partR.pdf https://www.ndsu.edu/pubweb/~bendunca/270worksheet13.pdf
Note: APPENDIX
Learning Activites http://www.math.utah.edu/~schwede/math311/WS5.p
df INSTRUCTIONAL FOCUS OF UNIT
Relations and Their Properties. To introduce the concept of a relation and basic properties of relations, including the reflexive,
symmetric, antisymmetric, and transitive properties.
Representing Relations. To show how relations can be represented using zero–one matrices and directed graphs.
Closures of Relations. To introduce the concept of the closure of a relation with respect to a property, and to develop algorithms
for constructing transitive closures.
Equivalence of Relations. To study equivalence relations and their equivalence classes.
Partial Orderings. To study partial orderings and their properties and applications. You may want to cover topological sorting and
scheduling. You may also want to discuss lattices and their application to information flow.
TEXTBOOK
Discrete Mathematics and Its Applications 7th Edition Kevin Rosen
ACADEMIC VOCABULARY Words: binary relation from A to B: reflexive: symmetric: antisymmetric: transitive: directed graph or digraph: loop: closure of a relation R with respect to
a property P: path in a digraph: circuit: equivalence relation: equivalent: partition of a set S: partial ordering: lexicographic order: Hasse diagram: topological sort: poset
Marzano’s Six Strategies for Teaching Vocabulary: 1. YOU provide a description, explanation or example. (Story, sketch, power point) 2. Ask students to restate or re-explain meaning in their own words. (Journal, community circle, turn to your neighbor) 3. Ask students to construct a picture, graphic or symbol for each word. 4. Engage students in activities to expand their word knowledge. (Add to their notes, use graphic organizer format) 5. Ask students to discuss vocabulary words with one another (Collaborate) 6. Have students play games with the words. (Bingo with definitions, Pictionary, Charades, etc.)
Using Marzano’s Strategy 5: Tell a friend- one student tells the other a story using 3 related terms Strategy 4: Venn Diagram- use Venn Diagram to compare terms
21ST CENTURY SKILLS (4Cs & CTE Standards)
21st Century Life and Careers 9.4.O(1) Engineering and Technology
9.4.12.O.(1).1
Apply the concepts, processes, guiding principles, and standards of school mathematics to solve science, technology, engineering, and mathematics problems.
9.4.12.O.(1).7 Use mathematics, science, and technology concepts and processes to solve problems in projects involving design and/or production (e.g., medical, agricultural, biotechnological, energy and power, information and communication, transportation, manufacturing, and construction).
MODIFICATIONS/ACCOMMODATIONS
Modifications: 1. Less complex reading level 2. Shortened assignments 3. Different goals 4. IEP modifications for summative and formative
assessments
Accommodations: 1. Preferential seating 2. Have students work in pairs 3. Assistive technologies 4. Three options on multiple choice exams 5. Larger print 6. Fewer problems on each page 7. More time 8. Test administered in a quieter setting 9. Tests read orally
10. Chunking assignments into smaller segments 11. Tape lectures or provide a peer note-taker
Extensions: 1. Alternative assignments 2. Independent studies 3. Mentoring of other students
APPENDIX (Teacher resource extensions)
MULTIPLE CHAPTERS 1. McGraw-Hill Education Teacher Resources (Textbook)
Login at http://highered.mheducation.com/sites/0073383090/student_view0/index.html Weblinks for textbook http://www.mhhe.com/math/advmath/rosen/student/webres/
2. Discrete Math Applications http://www.district196.org/edsrv/Assessment%20Web%20Page/Math%20Applications/High%20School/Discrete%20Math/GHSPREPA.html
3. Math Vids for Discrete Math http://mathvids.com/topic/mathhelp/20-discrete-math 4. WUCT121 Discrete Mathematics http://www.uow.edu.au/~bmaloney/wuct121/ 5. CSC2110 Discrete Mathematics http://www.cse.cuhk.edu.hk/~chi/csc2110/tutorial.html 6. University of Edinburgh – Lecture pdfs from Rosen textbook http://www.inf.ed.ac.uk/teaching/courses/dmmr/schedule.html 7. University of Nebraska – Lecture pdfs and worksheets http://cse.unl.edu/~choueiry/S06-235/ 8. St. Louis University – Lecture pdfs and worksheets http://math.slu.edu/~freeman/325Kspring12.html
9. Discrete Mathematics http://faculty.simpson.edu/lydia.sinapova/www/cmsc180/cmsc180-05/Sch180-05.htm
10. Problems http://www.wou.edu/~kruczekk/Courses/Math_355_F09/MTH355_Coursepack 11. Problems https://www.unf.edu/~wkloster/3100/problems.pdf
CHAPTER 9
12. Discrete Math: Intro to Relations https://www.youtube.com/watch?v=h34hZ_hynzE 13. Discrete Mathematics http://www.win.tue.nl/~hansc/dw/notes.pdf 14. Hasse Diagrams http://staff.scem.uws.edu.au/cgi-bin/cgiwrap/zhuhan/dmath/dm_readall.cgi?page=20&part=2
15. Order Theory http://math.stackexchange.com/questions/353105/discrete-math-hasse-diagrams
16. Definition Poset https://proofwiki.org/wiki/Definition:Poset 17. Rutgers http://www.cs.rutgers.edu/~elgammal/classes/cs205/chapt76.pdf
Graph Theory (Directory) http://freematheducation.com/graph/ 18. http://www.maa.org/publications/ebooks/resources-for-teaching-discrete-mathematics
APPENDIX (Mathematical Practices)
1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision.
Communicate the precise answer to a real-world problem. 7. Look for and make use of structure.
Identify structural similarities between integers and polynomials. Identify expressions as single entities, e.g. the difference of two squares.
8. Look for and express regularity in repeated reasoning.
Notes to teacher (not to be included in your final draft):
4 Cs Three Part Objective Creativity: projects Behavior Critical Thinking: Math Journal Condition Collaboration: Teams/Groups/Stations Demonstration of Learning (DOL) Communication – Powerpoints/Presentations
CHAPTER 10 Graphs
Total Number of Days: 17 days Grade/Course: 12/Discrete Math
ESSENTIAL QUESTIONS ENDURING UNDERSTANDINGS
What are the symbols used in logic? How can the truth value of a proposition or set of propositions
be determined? What makes a proof or an argument logically valid? Is there a best way to write a logically valid proof?
There is more than one correct way to write logical proof. Logical proofs carry more validity than baseless arguments. Common language is often riddled with fallacies (sometimes
when we communicate, our assumed meaning is not necessarily the logical meaning)
PACING CONTENT SKILLS STANDARDS (CCCS/MP)
RESOURCES
LEARNING ACTIVITIES/ASSESSM
ENTS
Basic Topic Description of what students will be able to do
Mc Graw Hill OTHER (e.g., tech)
1d
10.1 Graphs and Graph Models
Definition of a graph. Directed graphs Graph terminology. Graph models.
N.Q.2 Text pgs 573-583 Basic 1-28(even) Average: 1-38(even) Advance 1-38(even) 42,43,44
Text resources Lesson
PowerPoint Chapter10 Slides 3-21
Text test bank
Rosen Links Library Chapter 10
Lecture Notes
You Tube Graphs
Rosen Web Links
Rosen Extra Examples
Assessments Diagnostic: Do now,
http://infolab.stanford.edu/~ullman/focs/ch09.pdf
Formative: HW, Quiz , Exit Ticket http://infolab.stanford.edu/~ullman/focs/ch09.pdf
Note: APPENDIX
Learning Activites http://infolab.stanford.edu/~ullman/focs/ch09.pdf
2d
10.2 Graph Terminology
Basic graph terminology.
N.Q.2 Text pgs 591-597
Rosen Links Library
Assessments Diagnostic: Do now,
and Special Types of Graphs
Special simple graphs.
Bipartite graphs
Bipartite graphs and
matchings
Applications of graphs
Basic 1-28(even) Average: 1-38(even) Advance 1-38(even) 42,43,44 Text resources Lesson
PowerPoint Chapter10 Slides 3-21
Text test bank
Chapter 10
Lecture Notes
You Tube Bipartite Graphs
Rosen Web Links
Rosen Extra Examples
http://www.whitman.edu/mathematics/cgt_online/section05.04.html
Formative: HW, Quiz , Exit Ticket http://www.willamette.edu/~cstarr/math130/worksheets/HoM/05-more.pdf
Note: APPENDIX
Learning Activites http://math.illinoisstate.edu/reu/2K5%20Module.pdf
3d
10.3 Representing Graphs and Graph Isomorphism
Representing graphs. Adjacency matrices. Incidence matrices. Isomorphism of
graphs.
N.Q.2 Text pgs 597-607 Basic 1-28(even) Average: 1-38(even) Advance 1-38(even) 42,43,44
Text resources Lesson
PowerPoint Chapter10 Slides 2-41
Text test bank
Rosen Links Library Chapter 10
Lecture Notes
You Tube Adjacency Matrix
Rosen Web Links
Rosen Extra Examples
Assessments Diagnostic: Do now,
http://www.uow.edu.au/~bmaloney/wuct121/GraphsWeek10Lecture2.pdf
Formative: HW, Quiz , Exit Ticket http://www.math.unm.edu/~loring/links/graph_s05/hw2.pdf
Note: APPENDIX
Learning Activites http://www.utdallas.edu/~jwz120030/Teaching/PastCoursesUMBC/M221HS06/ProjectFiles/Adjacency.pdf
2d
10.4 Connectivity
Paths. Connectedness in
N.Q.2 Text pgs 607-618 Basic
Rosen Links Library Chapter 10
Assessments Diagnostic: Do now,
http://homepages.ius.edu/rwisman/C251/html/
undirected graphs. How connected is a
graph?
Connectedness in directed graphs.
Paths and
Isomorphism
Counting paths between vertices
1-28(even) Average: 1-38(even) Advance 1-38(even) 42,43,44
Text resources Lesson
PowerPoint Chapter10 Slides 42-57
Text test bank
Lecture Notes
Rosen Web Links
Rosen Extra Examples
chapter9.htm Formative:
HW, Quiz , Exit Ticket http://www.math.ualberta.ca/~nastos/m322/chapter3Connectivity.pdf
Note: APPENDIX
Learning Activites http://users.encs.concordia.ca/~bui/pdf/slides6.pdf
2d 10.5 Euler and Hamilton Paths
Euler paths and circuits.
Hamilton paths and
circuits. Applications of
Hamilton circuits.
N.Q.2 Text pgs 618-633 Basic 1-28(even) Average: 1-38(even) Advance 1-38(even) 42,43,44
Text resources Lesson
PowerPoint Chapter10 Slides 58-72
Text test bank
Rosen Links Library Chapter 10
Lecture Notes
You Tube Hamilton
Rosen Web Links
Rosen Extra Examples
Assessments Diagnostic: Do now,
http://faculty.cord.edu/andersod/Worksheet_day2.pdf
Formative: HW, Quiz , Exit Ticket http://faculty.mansfield.edu/hiseri/MA1115/1115Q21.pdf
Note: APPENDIX
Learning Activites http://langfordmath.com/126notes/VEF/EulerWS.pdf
2d 10.6 Shortest Path Problems
Shortest path algorithm.
The traveling
salesperson problem.
N.Q.2 Text pgs 618-633 Basic 1-28(even) Average:
Rosen Links Library Chapter 10
Lecture Notes
Assessments Diagnostic: Do now,
http://sites.saintmarys.edu/~cpeltier/Math251F13/Activities/M251Act
1-38(even) Advance 1-38(even) 42,43,44
Text resources Lesson
PowerPoint Chapter10 Slides 73-88
Text test bank
You Tube Shortest Path
Rosen Web Links
Rosen Extra Examples
5F13.pdf Formative:
HW, Quiz , Exit Ticket http://castle.eiu.edu/~mathcs/mat3770/index/Webview/Homework/ShortPathsII.pdf
Note: APPENDIX
Learning Activites http://www.cimt.plymouth.ac.uk/projects/mepres/alevel/discrete_ch2.pdf
2d 10.7 Planar Graphs
Euler formula. Kuratowski’s theorem.
N.Q.2 Text pgs 618-633 Basic 1-28(even) Average: 1-38(even) Advance 1-38(even) 42,43,44
Text resources Text test
bank
Rosen Links Library Chapter 10
Lecture Notes
You Tube Planar Graphs
Rosen Web Links
Rosen Extra Examples
Assessments Diagnostic: Do now,
http://www.math.uiuc.edu/~dipasqu1/Math181/Worksheets/September22/WS0922.pdf
Formative: HW, Quiz , Exit Ticket http://www.ms.uky.edu/~wrobinson/MA111_Fall2013/MA111%20--%20Worksheet%2015
Note: APPENDIX
Learning Activites http://www.cimt.plymouth.ac.uk/projects/mepres/alevel/discrete_ch6.pdf
2d 10.8 Graph
Coloring The four color
theorem. Applications of graph
coloring.
N.Q.2 Text pgs 618-633 Basic 1-28(even) Average: 1-38(even) Advance
Rosen Links Library Chapter 10
Lecture Notes
You Tube Graph Coloring
Assessments Diagnostic: Do now,
http://faculty.cord.edu/andersod/Worksheet_day4.pdf
Formative: HW, Quiz , Exit Ticket http://www.math.uri.ed
1-38(even) 42,43,44
Text resources Text test
bank
Rosen Web Links
Rosen Extra Examples
u/~eaton/0131873814_MEb.pdf
Note: APPENDIX
Learning Activites http://www.geom.uiuc.edu/~zarembe/grapht1.html
INSTRUCTIONAL FOCUS OF UNIT
Graphs and Graph Models. To introduce the notion of a graph and to show how to build graph models and to demonstrate the
wide applicability of graph models.
Graph Terminology and Special Types of Graphs. To introduce some of the basic terminology of graph theory and some basic
results about graphs. To describe some important families of graphs and to introduce the notion of a bipartite graph.
Representing Graphs and Graph Isomorphism. To show how to represent graphs and to study isomorphism of graphs.
Connectivity. To introduce the notions of paths and circuits in graphs and to define connectivity of graphs.
Euler and Hamilton Paths. To develop necessary and sufficient conditions for the existence of Euler circuits and paths, to give
algorithms for constructing them, and to study Hamilton paths and circuits.
Shortest Path Problems. To present an algorithm for finding a shortest path in a weighted graph, and to discuss the traveling
salesman problem.
Planar Graphs. To introduce the concept of planarity of graphs and to develop tools to decide whether a graph is planar.
Graph Coloring. To introduce the concept of the coloring of a graph and give applications of graph colorings.
TEXTBOOK
Discrete Mathematics and Its Applications 7th Edition Kevin Rosen
ACADEMIC VOCABULARY Words: undirected edge: directed edge: multiple edges: multiple directed edges: loop: undirected graph: simple graph: multigraph: pseudograph: directed
graph: directed multigraph: simple directed graph: adjacent: bipartite graph: regular graph: adjacency matrix: incidence matrix: isomorphic simple graphs: simple path: circuit: connected graph: Euler path: Euler circuit: Hamilton path: Hamilton circuit: weighted graph: shortest-path problem: traveling salesperson problem: planar graph: graph coloring: chromatic number:
Marzano’s Six Strategies for Teaching Vocabulary: 7. YOU provide a description, explanation or example. (Story, sketch, power point) 8. Ask students to restate or re-explain meaning in their own words. (Journal, community circle, turn to your neighbor) 9. Ask students to construct a picture, graphic or symbol for each word. 10. Engage students in activities to expand their word knowledge. (Add to their notes, use graphic organizer format) 11. Ask students to discuss vocabulary words with one another (Collaborate) 12. Have students play games with the words. (Bingo with definitions, Pictionary, Charades, etc.)
Using Marzano’s Strategy 5: Tell a friend- one student tells the other a story using 3 related terms (terms: meteor, meteorite, meteoroid) Strategy 4: Venn Diagram- use Venn Diagram to compare terms (protoplanet and protosun)
21ST CENTURY SKILLS (4Cs & CTE Standards)
21st Century Life and Careers 9.4.O(1) Engineering and Technology
9.4.12.O.(1).1 Apply the concepts, processes, guiding principles, and standards of school mathematics to solve science, technology, engineering, and mathematics problems.
9.4.12.O.(1).7 Use mathematics, science, and technology concepts and processes to solve problems in projects involving design and/or production (e.g., medical, agricultural, biotechnological, energy and power, information and communication, transportation, manufacturing, and construction).
MODIFICATIONS/ACCOMMODATIONS
Modifications: 5. Less complex reading level 6. Shortened assignments 7. Different goals 8. IEP modifications for summative and formative
assessments
Accommodations: 12. Preferential seating 13. Have students work in pairs 14. Assistive technologies 15. Three options on multiple choice exams 16. Larger print 17. Fewer problems on each page 18. More time 19. Test administered in a quieter setting 20. Tests read orally
21. Chunking assignments into smaller segments 22. Tape lectures or provide a peer note-taker
Extensions: 4. Alternative assignments 5. Independent studies 6. Mentoring of other students
APPENDIX (Teacher resource extensions)
MULTIPLE CHAPTERS 1. McGraw-Hill Education Teacher Resources (Textbook)
Login at http://highered.mheducation.com/sites/0073383090/student_view0/index.html Weblinks for textbook http://www.mhhe.com/math/advmath/rosen/student/webres/
2. Discrete Math Applications http://www.district196.org/edsrv/Assessment%20Web%20Page/Math%20Applications/High%20School/Discrete%20Math/GHSPREPA.html
3. Math Vids for Discrete Math http://mathvids.com/topic/mathhelp/20-discrete-math 4. WUCT121 Discrete Mathematics http://www.uow.edu.au/~bmaloney/wuct121/ 5. CSC2110 Discrete Mathematics http://www.cse.cuhk.edu.hk/~chi/csc2110/tutorial.html 6. University of Edinburgh – Lecture pdfs from Rosen textbook http://www.inf.ed.ac.uk/teaching/courses/dmmr/schedule.html 7. University of Nebraska – Lecture pdfs and worksheets http://cse.unl.edu/~choueiry/S06-235/ 8. St. Louis University – Lecture pdfs and worksheets http://math.slu.edu/~freeman/325Kspring12.html 9. Discrete Mathematics http://faculty.simpson.edu/lydia.sinapova/www/cmsc180/cmsc180-05/Sch180-05.htm 10. Problems http://www.wou.edu/~kruczekk/Courses/Math_355_F09/MTH355_Coursepack 11. Problems https://www.unf.edu/~wkloster/3100/problems.pdf
CHAPTER 10
12. Graph Theory Lessons: http://www.mathcove.net/petersen/lessons/get-lesson?les=1 13. Graphs: http://www.inf.ed.ac.uk/teaching/courses/dmmr/slides/13-14/Ch10.pdf 14. Discrete Math Graphs: http://www1.cs.columbia.edu/~zeph/3203s04/lectures.html Graph Theory (Directory) http://freematheducation.com/graph/ 15. http://www.maa.org/publications/ebooks/resources-for-teaching-discrete-mathematics
APPENDIX (Mathematical Practices)
1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision.
Communicate the precise answer to a real-world problem. 7. Look for and make use of structure.
Identify structural similarities between integers and polynomials.
Identify expressions as single entities, e.g. the difference of two squares. 8. Look for and express regularity in repeated reasoning.
Notes to teacher (not to be included in your final draft):
4 Cs Three Part Objective Creativity: projects Behavior Critical Thinking: Math Journal Condition Collaboration: Teams/Groups/Stations Demonstration of Learning (DOL) Communication – Powerpoints/Presentations
Chapter 11 Trees Total Number of Days: 3days Grade/Course: 12/Discreatemathmtics
ESENTIAL QUESTIONS ENDURING UNDERSTANDINGS
How many vertices does a full m-ary tree have if it has i internal vertices? How many leaves does the tree have?
What is a minimum spanning tree of a connected weighted graph?
A tree is a connected undirected graph with no simple circuits.
A minimum spanning tree in a connected weighted graph is a spanning tree that has the smallest possible sum of weights of its edges.
.
PACING CONTENT SKILLS STANDARDS (CCCS/MP)
RESOURCES LEARNING
ACTIVITIES/ASSESSMENTS
Basic Topic
Description of what students will be able to do Mc Graw Hill
Pearson OTHER
(e.g., tech)
1.5d 11.1 Introduction
to Trees
Differentiate between the concept of a tree, to present basic terminology for trees, and to develop relationships among the number of vertices of different kinds and the number of edges in trees.
S.IC.1,2
Understand and evaluate random processes underlying statistical experiments
Text:755-757 Basic: 1-10 Average: 1-10,(0dd)11-13, Advance: 1-10,(0dd)11-15,24,30,48
http://www.nhest.org/trees.html
Assessments Diagnostic Do now, http://highered.mheducation.com/sites/0073383090/student_view0/chapter11/extra_examples.html Formative: HW, Quiz , Exit Ticket http://highered.mheducation.com/sites/0073383090/student_view0/chapter11/extra_examples.html Learning Activities https://us.search.ya
hoo.com/yhs/search?hspart=Elex&hsimp=yhs-elex_v9&p=Introduction+to+Trees
1.5d
11.5 Minimum Spanning Trees
Study minimum spanning trees and produce algorithms for generating them
F.LE.4 Construct and compare
linear, quadratic and
exponential models with
logarithms as solutions
Text:802-803 Basic: 1-10 Average: 11-15,24-31, 34,35 Advance: 11-15,24-31, 34,35,21,30,43
https://www.cse.ust.hk/~dekai/271/notes/L07/L07.pdf
Assessments Diagnostic Do now http://highered.mheducation.com/sites/0073383090/student_view0/chapter11/extra_examples.html Formative: HW, Quiz , Exit Tickethttp://highered.mheducation.com/sites/0073383090/student_view0/chapter11/extra_examples.html Summative:
Unit Test
http://highered.mheducation.com/sites/0073383090/instructor_view0/printa
ble_tests.htm Learning Activities http://algs4.cs.princeton.edu/43mst/
INSTRUCTIONAL FOCUS OF UNIT
We begin by defining un rooted trees; then we define rooted trees and show how an un rooted tree can be rooted by choosing any vertex as the root. Go over one or two uses of trees in modeling, such as those discussed in Examples 5, 6, 7, and 8(Section11.1). These involve chemistry, business, and
computer science. The construction of minimum spanning trees is required for many applications, so algorithms for their construction have been studied extensively. We
present the two best known such algorithms: Prim’s algorithm and Kruskal’s algorithm. It is useful to go over both of them to illustrate that the same problem can be solved in different ways. Proving that Prim’s algorithm produces a spanning tree is rather subtle, so explain the proof slowly.
Marzano's Academic vocabulary Words
Tree: Forest: Rooted tree: Subtree: Minmax strategy: weighted graph: prefix code: , inorder traversal:, m-ary tree:, binary tree Marzano’s Six Strategies for Teaching Vocabulary:
1. YOU provide a description, explanation or example. (Story, sketch, power point) 2. Ask students to restate or re-explain meaning in their own words. (Journal, community circle, turn to your neighbor) 3. Ask students to construct a picture, graphic or symbol for each word. 4. Engage students in activities to expand their word knowledge. (Add to their notes, use graphic organizer format) 5. Ask students to discuss vocabulary words with one another (Collaborate) 6. Have students play games with the words. (Bingo with definitions, Pictionary, Charades, etc.)
Using Marzano’s Strategy 2: Teach a friend- pair of students gets 3 terms each, individually read definintion, read example use in sentence, and small paragraph. Students will alternate teaching their partner their words (terms: set 1-corona, photosphere, chromosphere set 2 sunspot, prominence, solar flare)
Strategy 3: Graph-Ic- Using two similar terms, create a graphic for each term focusing on the differences (terms: fission and fusion)
21ST CENTURY SKILLS (4Cs & CTE Standards)
21st Century Life and Careers 9.1 21st Century Life & Career Skills: All students will demonstrate the creative, critical thinking, collaboration, and problem-solving skills needed to function successfully as both global citizens and workers in diverse ethnic and organizational cultures.
A. Critical Thinking and Problem Solving 9.1.12.A.1: Apply critical thinking and problem-solving strategies during structured learning experiences.
Graph theory and Trees critical thinking making connections between multiple events B. Creativity and Innovation
9.1.12.B.1: Present resources and data in a format that effectively communicates the meaning of the data and its implications for solving problems, using multiple perspectives.
spanning tree organize and graph data F. Accountability, Productivity, and Ethics
9.1.12.F.2: Demonstrate a positive work ethic in various settings, including the classroom and during structured learning experiences.
Trees and graph students must work together to measure 9.4 21st Century Career and Technical Education:
O. Science, Technology, Engineering & Mathematics Career Cluster Trees and application 9.4.12.O.2: Demonstrate mathematics knowledge and skills required to pursue the full range of postsecondary education and
MODIFICATIONS/ACCOMMODATIONS
Teacher directly instruction by providing students with more necessary steps in order to solve the problems. Small Group Activities- when students are given group guided practice like
“Drawings of trees, networks, circuits, and bins can be pre-made and provided to students. Students can focus on the concepts rather than making the diagrams. Colored pencils can be used for vertex coloring to aid in determining the least number of colors to be used.”
IEP/504 Modification: Modified assessments and assignments (class work , homework. Quizzes/tests) as needed.
APPENDIX (Teacher resource extensions)
1. E-Text, Interactive Digital Resources, Teacher Resources Login at http://www.mhhe.com/links/1256/1246/2083/2084/2091/index.html CCSS. Mathematical Practices: MP1: Make Sense of problems and persevere in solving them. MP2: Reason abstractly and quantitatively. MP3: Construct viable arguments and critique the reasoning of others. MP4: Model with mathematics. MP5: Use appropriate tools strategically. MP6: Attend to precision. MP7: Look for and make use of structure.
MP8 :Look for and express regularity in repeated reasoning.. 1. Look for and make use of structure. 2. Look for and express regularity in repeated reasoning.
All of the content presented in this course has connections to the standards for mathematical practices. *This course includes the exponential and logarithmic functions as modeling tools. (PARCC Model Content Frameworks)
Notes to teacher (not to be included in your final draft):
4 Cs Three Part Objective Creativity: projects Behavior