chapter 1 the foundations: logic and proofs · chapter 1 the foundations: logic and proofs total...

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

http://www.mhhe.com/links/1256/1246/2083/2084/2086/index.html



http://alumni.cs.ucr.edu/~elenah/courses/CSCI217/Lecture1.pdf

https://www.youtube.com/watch?v=qV4htTfow-E



http://www.mhhe.com/math/advmath/rosen/student/webres/

http://highered.mheducation.com/sites/0073383090/student_view0/chapter1/extra_examples.html




http://highered.mheducation.com/sites/0073383090/student_view0/self_assessments.html




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


Lecture Notes

You Tube Boolean Basic Laws and Rules

Rosen Web Links

Rosen Extra Examples


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 test bank


Lecture Notes

You Tube DeMorgan's Law

Rosen Web Links





Note: APPENDIX

Learning Activites http://www.luc.edu/faculty/a


http://www.luc.edu/faculty/avande1/logic/worksheets-chapter1.htm



http://www.mhhe.com/links/1256/1246/2083/2084/2086/more2.html




https://www.youtube.com/watch?v=gj8QmRQtVao













http://cs.fit.edu/~wds/classes/adm/









https://www.youtube.com/watch?v=E_GzBO3psn8

https://www.youtube.com/watch?v=E_GzBO3psn8














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.




Text test bank


Lecture Notes

You Tube Predicates and Quantifiers 1

You Tube Predicates and Quantifiers 2

Web Links

Extra Examples




Note: APPENDIX


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.




Text test bank

Links Library Chapter 1-5

Lecture Notes

You Tube Nested Quantifiers 1

You Tube Nested Quantifiers 2

Web Links

Extra Examples




Note: APPENDIX

Learning Activites http://www.luc.edu/faculty/a







https://www.youtube.com/watch?v=DmCltf8ypks



https://www.youtube.com/watch?v=jNeISigUCo0




















https://www.youtube.com/watch?v=Vr0bEcxz7Io

https://www.youtube.com/watch?v=Vr0bEcxz7Io

https://www.youtube.com/watch?v=3gigcR1y3js

https://www.youtube.com/watch?v=3gigcR1y3js













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.




Text test bank


Lecture Notes

You Tube Rules of Inteference 1

You Tube Rules of Inteference 2

Web Links

Extra Examples




Note: APPENDIX


3d

1.7 Introduction to Proofs

Construct direct proofs. Construct proofs by

contraposition. Construct proofs by

contradiction. Recognize common

mistakes in proofs.




Text test bank


You Tube Direct and Indirect Proofs

Lecture Notes

Web Links

Extra Examples




Note: APPENDIX

Learning Activites http://www.luc.edu/faculty/avande1/logic/worksheets-






https://www.youtube.com/watch?v=B3L2Xe22j9o

https://www.youtube.com/watch?v=B3L2Xe22j9o

https://www.youtube.com/watch?v=Dsk6y4IQ9hs

https://www.youtube.com/watch?v=Dsk6y4IQ9hs

















https://www.youtube.com/watch?v=-nydLdXeYPA



http://cs.fit.edu/~wds/classes/adm/Handouts/Proofs.pdf














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.




Text test bank


You Tube Proof by Cases

Web Links

Extra Examples




Summative: Unit Test http://highered.mheducation.com/sites/0073383090/instructor_view0/printable_tests.htm

Note: APPENDIX


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




https://www.youtube.com/watch?v=fmAidFIgryU

https://www.youtube.com/watch?v=fmAidFIgryU












http://highered.mheducation.com/sites/0073383090/instructor_view0/printable_tests.htm







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

http://opencircuitdesign.com/

http://acs.ist.psu.edu/discrete-math/

http://www.maa.org/publications/periodicals/convergence/primary-historical-sources-in-the-classroom-discrete-mathematics-and-computer-science

http://www.maa.org/publications/periodicals/convergence/primary-historical-sources-in-the-classroom-discrete-mathematics-and-computer-science

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-

http://highered.mheducation.com/sites/0073383090/student_view0/index.html


http://www.district196.org/edsrv/Assessment%20Web%20Page/Math%20Applications/High%20School/Discrete%20Math/GHSPREPA.html

http://mathvids.com/topic/mathhelp/20-discrete-math

http://www.uow.edu.au/~bmaloney/wuct121/

http://www.cse.cuhk.edu.hk/~chi/csc2110/tutorial.html

http://www.inf.ed.ac.uk/teaching/courses/dmmr/schedule.html

http://cse.unl.edu/~choueiry/S06-235/

http://math.slu.edu/~freeman/325Kspring12.html

http://faculty.simpson.edu/lydia.sinapova/www/cmsc180/cmsc180-05/Sch180-05.htm


http://www.wou.edu/~kruczekk/Courses/Math_355_F09/MTH355_Coursepack

https://www.unf.edu/~wkloster/3100/problems.pdf

http://www.cs.gsu.edu/~ebullwinkel1/

http://www.sigcse.org/resources/reports/discrete/materials


https://www.math.lsu.edu/~verrill/teaching/discrete2020/Spring2005/

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

https://jasoninclass.wordpress.com/

http://www.rbjones.com/rbjpub/logic/

http://www.zweigmedia.com/RealWorld/logic/logicintro.html

http://webspace.ship.edu/deensley/DiscreteMath/flash/#chpt1

http://alumni.cs.ucr.edu/~elenah/courses/CSCI217/

http://mathvids.com/topic/20-discrete-math/major_subtopic/67/lesson/610-lecture-2-boolean-algebra-and-formal-logic/mathhelp

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

http://mathvids.com/topic/20-discrete-math/major_subtopic/67/lesson/610-lecture-2-boolean-algebra-and-formal-logic/mathhelp

http://facultypages.ecc.edu/perel/SurveyCourse/Chapter3_Logic/practice/translations.htm

Chapter 2 Basic Structures: Sets, Functions,

Sequences, Sums, and Matrices

Total Number of Days: 10-days Grade/Course: 12/Discrete mathematics


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.


RESOURCES


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,


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
















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

http://www.math.csusb.edu/notes/proofs/bpf/node5.html














http://www.mathsisfun.com/definitions/operation.html


















http://www.freemathhelp.com/functions.html

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















https://people.richland.edu/james/lecture/m116/sequences/sequences.html

















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


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.



















http://www.purplemath.com/modules/matrices.htm



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.)



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

http://www.mathgoodies.com/lessons/sets/

http://www.glencoe.com/sec/math/algebra/algebra2/pdf/alg2_10.pdf

http://www.learningfirst.org/helping-students-demonstrate-their-math-knowledge


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.


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):



http://highered.mheducation.com/sites/0073383090/instructor_view0/lecture_powerpoint_slides.htm?sessionId=1447444114112293863975a9eb81e027e4cd1813452dd777fe96b&existinguser=true

http://highered.mheducation.com/sites/0073383090/instructor_view0/lecture_powerpoint_slides.htm?sessionId=1447444114112293863975a9eb81e027e4cd1813452dd777fe96b&existinguser=true

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.


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/


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













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


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/













http://regentsprep.org/REgents/math/ALGEBRA/AE7/ExpDecayL.htm



http://www.eng.unt.edu/ian/books/free/lnoa.pdf
















http://bigocheatsheet.com/

http://bigocheatsheet.com/


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:


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)


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.


The twelve dots critical thinking making connections between multiple events B. Creativity and Innovation


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.


http://mathworld.wolfram.com/Logarithm.html

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


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


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.

http://www.mathwords.com/l/logarithm.htm


http://highered.mheducation.com/sites/0073383090/instructor_view0/powerpoint_slides_of_figures_and_tables.htm

Chapter 4 Number Theory and Cryptography

Total Number of Days: 15-days Grade/Course: 12/Discrete Math


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.


RESOURCES


NTS


(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.



PowerPoint Chapter4 Slides 4-17

Text test bank


Lecture Notes

You Tube Divisibility and Modular Arithmetic

Rosen Web Links



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




http://www.inf.ed.ac.uk/teaching/courses/dmmr/slides/13-14/Ch4.pdf

https://www.youtube.com/watch?v=2leHFuMKr9o











http://www.cse.chalmers.se/edu/year/2012/course/TDA351/impact/self1comm.html




https://docs.google.com/file/d/0B6LUxWheAPCnSS1KYUhpZGhuREE/edit?pli=1




http://www.maths.ed.ac.uk/~aboocher/courses/55/worksheets/w6.pdf



http://www.maths.ed.ac.uk/~aboocher/courses/55/worksheets/s6.pdf


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.




Text test bank


Lecture Notes

You Tube Conversions Between Number Systems

Rosen Web Links



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.




Text test bank


Lecture Notes

Lecture Notes Primes

Handout Euclidian Algorithm

You Tube Fundamental Theorem of Arithmetic


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





http://paws.kettering.edu/~jkwon/teaching/09-t4/ce-320l/Lecture02_handout-F09.pdf

https://www.youtube.com/watch?v=x5OpxRq_okQ












http://www.proprofs.com/quiz-school/story.php?title=binary-decimal-hexadecimal-number-systems






http://www.free-test-online.com/binary/binary_numbers.htm









http://platonicrealms.com/encyclopedia/Fundamental-Theorem-of-Arithmetic

https://primes.utm.edu/

https://primes.utm.edu/

https://www.rit.edu/~w-asc/documents/services/resources/handouts/DM%20-%206%20Euclidean%20Algorithm.pdf



https://www.youtube.com/watch?v=7sw6LdAfHgE








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.seethesolutions.net/practice-exams-topic/294/

Rosen Web Links


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


Text test bank


Lecture Notes

PDF Slides Chinese Remainder Theorem

You Tube Solving Linear Congruences

Rosen Web Links



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


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







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

http://web.stanford.edu/class/cs103x/pset2_sol.pdf




http://www.millersville.edu/~bikenaga/number-theory/linear-congruences/linear-congruences.html

http://www.math.harvard.edu/~knill/seminars/chinese/chinese.pdf




https://www.youtube.com/watch?v=U9Eo6Bsvm4M











http://quiz.thefullwiki.org/Linear_congruence_theorem



http://www.oxfordmathcenter.com/drupal7/node/200



http://www.math.ucsd.edu/~ibejenar/teaching/2013/109/HW9Solutions.pdf



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


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


Using Marzano’s Strategy 4: Students will use a Frayer Vocabulary Graphic Organizer. Strategy 5: Students will think-pair-share with vocabulary words.


9.4.D Business, Management & Administration Career Cluster 9.4.O(1) Engineering and Technology


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



assessments


10. Chunking assignments into smaller segments

11. Tape lectures or provide a peer note-taker



WEB LINKS 1. McGraw-Hill Education Teacher Resources (Textbook)



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

https://www.chatham.edu/pti/curriculum/units/2004/Amick.pdf

http://faculty.gvsu.edu/aboufade/web/enigma/School/Welcome.htm














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


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





8. Look for and express regularity in repeated reasoning.








http://www.math.rutgers.edu/~erowland/modulararithmetic.html

http://nrich.maths.org/4350

http://www.cimt.plymouth.ac.uk/projects/mepres/alevel/discrete_ch8.pdf


http://math.berkeley.edu/~ericp/teaching/Fall2013/55/chapter-4a.pdf

http://ocw.nctu.edu.tw/upload/classbfs1210032011184835.pdf

http://www.math.mtu.edu/mathlab/COURSES/holt/dnt/lincong.html

http://cobweb.cs.uga.edu/~cai/courses/discmath/spring2008/lecture-note3.pdf

http://math.berkeley.edu/~ericp/teaching/Fall2013/55/chapter-4a.pdf

Chapter 5 Induction and Recursion Total Number of Days: 5week Grade/Course: 12/Discreatemathmtics


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.


RESOURCES LEARNING


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/




. Learning Activities http://www.cs.cmu.edu/~rwh/introsml/techniques/indrec.htm

http://scienceblogs.com/goodmath/2007/01/23/basics-recursion-and-induction-1/















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



Formative: HW, Quiz , Exit Ticket


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
























http://www.sanfoundry.com/discrete-mathematics-mcqs-strong-induction-well-ordering/






















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


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


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)





http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-spring-2010/lecture-notes/MIT6_042JS10_lec19.pdf











Induction Activity use multiple strategies. B. Creativity and Innovation


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


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..


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..

http://scienceblogs.com/goodmath/2007/01/23/basics-recursion-and-induction-1




Chapter 6 COUNTING

Total Number of Days: 10-weeks Grade/Course: 12/Discreatemathmtics


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.


RESOURCES LEARNING


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


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




Summative: Unit Test .

Learning Activities http://www.math.ust.hk/~mabfchen/Math391I/Pigeonhole.pdf





http://www.aaamath.com/sta-basic-cntg.htm

















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



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



Formative: HW, Quiz , Exit Tickethttp://highered.mheducation.com/sites/0073383090/s



















https://www.khanacademy.org/math/precalculus/prob_comb/combinatorics_precalc/v/permutations-and-combinations-1


















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


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



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


























http://www.slideserve.com/cricket/generalized-permutations-and-combinations





3d

6.6 Generating Permutation and Combination

Model algorithms for generating permutations and combinations graph.


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


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,:


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 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.


Problems critical thinking B. Creativity and Innovation


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

http://www.cut-the-knot.org/probability.shtml




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


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 8 Advanced Counting Techniques

Total Number of Days: 12 days Grade/Course: 12/Discrete Math


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.


RESOURCES


NTS


(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 test bank


Lecture Notes Fibonacci

Lecture Notes Tower of Hanoi

Lecture Notes Recurrence

You Tube Recurrence Relations

Rosen Web Links

Rosen Extra


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




http://www.mathsisfun.com/numbers/fibonacci-sequence.html

http://www.mathsisfun.com/numbers/fibonacci-sequence.html

http://www.mathsisfun.com/games/towerofhanoi.html



http://www.math.uaa.alaska.edu/~afkjm/cs351/handouts/recurrence.pdf

http://www.math.uaa.alaska.edu/~afkjm/cs351/handouts/recurrence.pdf

https://www.youtube.com/watch?v=HzV6tY6OuBk









http://www.aber.ac.uk/~dcswww/Dept/Teaching/CourseNotes/2010-2011/CS20410/soln3Counting.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 test bank


You Tube Linear Homogeneous Recurrence Relations

You Tube Linear Non Homogeneous Recurrence Relations

Rosen Web Links



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 test bank


Lecture Notes

Lecture Notes

You Tube Divide and Conquer

Rosen Web Links

Rosen Extra


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





https://www.youtube.com/watch?v=TvFCDWEUjZs





https://www.youtube.com/watch?v=SM4GwdVMKIM












http://furthermathematicst.blogspot.com/2011/06/43-non-homogeneous-linear-recurrence.html




http://www.eecs.yorku.ca/course_archive/2008-09/S/1019/Website_files/21-linear-recurrences.pdf







http://www.math.ucsd.edu/~pcompeau/Ch07_DnC_LinearSpaceAlignment_Edited.pdf

http://courses.cs.vt.edu/cs5114/spring2009/lectures/lecture09-divide-and-conquer.pdf

https://www.youtube.com/watch?v=ATCYn9F3oUQ









http://www.atilim.edu.tr/~mcs401/MCS401%20Worksheet-2.pdf



http://www.cse.msu.edu/~torng/Classes/Archives/cse830.02spring/Homework/hw02.html




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 test bank


Lecture Notes

You Tube Power Series

Rosen Web Links



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


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


ACADEMIC VOCABULARY

Words: Recurrence relation: Fiobonacci Numbers: Tower of Hanoi: Dynamic Programming: Linear homogeneous recurrence relation with constant


http://goldman.cse.wustl.edu/crc2007/homework/hw1.pdf






http://www.math.cmu.edu/~af1p/Teaching/Combinatorics/Slides/Generating-Functions.pdf

https://www.khanacademy.org/math/integral-calculus/sequences_series_approx_calc/power-series-algebra/v/power-series-radius-interval-convergence

https://www.khanacademy.org/math/integral-calculus/sequences_series_approx_calc/power-series-algebra/v/power-series-radius-interval-convergence









http://www.webpages.uidaho.edu/~markn/395/pdf/rec-eq.pdf



http://math.berkeley.edu/~bernd/hw9sol.pdf

http://math.berkeley.edu/~bernd/hw9sol.pdf

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: 1. Less complex reading level

2. Shortened assignments 3. Different goals 4. IEP modifications for summative and formative

assessments


10. Chunking assignments into smaller segments

11. Tape lectures or provide a peer note-taker



9.4.O(1) Engineering and Technology


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


WEB LINKS 1. McGraw-Hill Education Teacher Resources (Textbook)



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


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

http://www.indiana.edu/~hmathmod/projects.html






















http://www.cs.utexas.edu/~eberlein/cs336/recurrence.pdf

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

=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)









http://www.wikihow.com/Solve-Recurrence-Relations

http://www.cs.columbia.edu/~cs4205/files/CM2.pdf

http://www.math.ucsd.edu/~pcompeau/Ch07_DnC_LinearSpaceAlignment_Edited.pdf

http://setur.fo/fileadmin/user_upload/documents/Undirvisingartilfar/NVD/QinX/Algorithms/A_4n.pdf

http://rutherglen.science.mq.edu.au/wchen/lndmfolder/dm14.pdf


CHAPTER 9 Relations



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?




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.




Text test bank


Lecture Notes

Lecture Notes

You Tube Binary Relations

Rosen Web Links



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




http://www.cs.uiuc.edu/class/fa07/cs173/notes/relations+functions.pdf

http://homepages.ius.edu/rwisman/C251/html/chapter8.htm

https://www.youtube.com/watch?v=h34hZ_hynzE











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





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


Lecture Notes Matrices

Lecture Notes

You Tube Representing Relations

Rosen Web Links



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.




Text test bank


Lecture Notes

You Tube Closure of Relations

Rosen Web Links



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




http://www.cs.rutgers.edu/~elgammal/classes/cs205/chapt73.pdf

http://www.cs.rutgers.edu/~elgammal/classes/cs205/chapt73.pdf


https://www.youtube.com/watch?v=r-EvSkZ6eMY










http://math.berkeley.edu/~mannisto/ws6_27.pdf



http://coitweb.uncc.edu/~tbarnes2/LogicAlgorithms/notes/pdf/HW8.pdf










http://people.cs.pitt.edu/~adamlee/courses/cs0441/lectures/lecture27-closures.pdf

https://www.youtube.com/watch?v=xuhZSt4Rw0k










http://www.cs.utsa.edu/~bylander/cs2233/graphshandout.pdf



https://math.berkeley.edu/~ralph42/hw11solns55.pdf



http://webpages.uncc.edu/tbarnes2/LogicAlgorithms/notes/pdf/Pak5.pdf




Equivalence of Relations

Equivalence classes. Equivalence classes

and partitions.

618



Text test bank

Library Chapter 9-5

Lecture Notes

Lecture Notes

You Tube Equivalence Relations

Rosen Web Links


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.




Text test bank


Lecture Notes

You Tube Partial Orderings

Rosen Web Links



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



http://www.math.uiuc.edu/~hildebr/347.summer14/relations.pdf

http://www.millersville.edu/~bikenaga/math-proof/equivalence-relations/equivalence-relations.html

https://www.youtube.com/watch?v=uOE7rSKTDbo







http://www.math.poly.edu/courses/ma2322/2322setB/solution4.pdf



http://www.math.vt.edu/people/mcquain/hwrelations_2534_07.pdf



http://folk.uib.no/mbr085/11MAT220/lecturenotes/worksheets.pdf






http://staff.scem.uws.edu.au/cgi-bin/cgiwrap/zhuhan/dmath/dm_readall.cgi?page=20

https://www.youtube.com/watch?v=EWCH79AKDcw







http://www.snow.edu/kenyonp/Links/Math1630/Worksheets/WS2-5.pdf




http://math.tutorvista.com/discrete-math/partially-ordered-set.html




https://www.math.vt.edu/people/mcquain/2534_sol_eqR_partR.pdf



https://www.ndsu.edu/pubweb/~bendunca/270worksheet13.pdf



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


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


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



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.




assessments





MULTIPLE CHAPTERS 1. McGraw-Hill Education Teacher Resources (Textbook)


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










http://freematheducation.com/graph/

http://www.maa.org/publications/ebooks/resources-for-teaching-discrete-mathematics

CHAPTER 10 Graphs



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?




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 test bank

Rosen Links Library Chapter 10

Lecture Notes

You Tube Graphs

Rosen Web Links



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.


Rosen Links Library





http://www.cs.princeton.edu/~rs/AlgsDS07/13DirectedGraphs.pdf

https://www.youtube.com/watch?v=gD4wwbPICUU

https://www.youtube.com/watch?v=gD4wwbPICUU












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


Text test bank

Chapter 10

Lecture Notes

You Tube Bipartite Graphs

Rosen Web Links


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.




Text test bank


Lecture Notes

You Tube Adjacency Matrix

Rosen Web Links



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



http://homepages.ius.edu/rwisman/C251/html/


https://www.cs.umd.edu/class/sum2005/cmsc451/bipartite.pdf

https://www.youtube.com/watch?v=68F3sTaejE0










http://www.willamette.edu/~cstarr/math130/worksheets/HoM/05-more.pdf




http://math.illinoisstate.edu/reu/2K5%20Module.pdf

http://math.illinoisstate.edu/reu/2K5%20Module.pdf




http://people.revoledu.com/kardi/tutorial/GraphTheory/Adjacency-Matrix.html

https://www.youtube.com/watch?v=fMuTwUYhxeY











http://www.math.unm.edu/~loring/links/graph_s05/hw2.pdf



http://www.utdallas.edu/~jwz120030/Teaching/PastCoursesUMBC/M221HS06/ProjectFiles/Adjacency.pdf









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 test bank

Lecture Notes

Rosen Web Links


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.




Text test bank


Lecture Notes

You Tube Hamilton

Rosen Web Links



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:


Lecture Notes


http://sites.saintmarys.edu/~cpeltier/Math251F13/Activities/M251Act

https://people.cs.umass.edu/~barring/cs250s13/lecture/16.pdf






http://www.math.ualberta.ca/~nastos/m322/chapter3Connectivity.pdf



http://users.encs.concordia.ca/~bui/pdf/slides6.pdf

http://users.encs.concordia.ca/~bui/pdf/slides6.pdf




http://www.cs.sfu.ca/~ggbaker/zju/math/euler-ham.html

https://www.youtube.com/watch?v=r5Xz2xUI2ok

https://www.youtube.com/watch?v=r5Xz2xUI2ok








http://faculty.mansfield.edu/hiseri/MA1115/1115Q21.pdf



http://langfordmath.com/126notes/VEF/EulerWS.pdf

http://langfordmath.com/126notes/VEF/EulerWS.pdf




http://www.cs.sfu.ca/~ggbaker/zju/math/euler-ham.html#tsp

http://sites.saintmarys.edu/~cpeltier/Math251F13/Activities/M251Act5F13.pdf



1-38(even) Advance 1-38(even) 42,43,44



Text test bank

You Tube Shortest Path

Rosen Web Links


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.


Text resources Text test

bank


Lecture Notes

You Tube Planar Graphs

Rosen Web Links



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


Lecture Notes

You Tube Graph Coloring



Formative: HW, Quiz , Exit Ticket http://www.math.uri.ed

https://www.youtube.com/watch?v=WN3Rb9wVYDY

https://www.youtube.com/watch?v=WN3Rb9wVYDY






http://castle.eiu.edu/~mathcs/mat3770/index/Webview/Homework/ShortPathsII.pdf










http://www.polymtl.ca/pub/sites/lagrapheur/docs/en/documents/NotesChap2.pdf

https://www.youtube.com/watch?v=u1a50tks2xQ

https://www.youtube.com/watch?v=u1a50tks2xQ





http://www.ms.uky.edu/~wrobinson/MA111_Fall2013/MA111%20--%20Worksheet%2015










http://www.geeksforgeeks.org/graph-coloring-applications/

https://www.youtube.com/watch?v=HoQY3GyDZ0w

https://www.youtube.com/watch?v=HoQY3GyDZ0w




http://www.math.uri.edu/~eaton/0131873814_MEb.pdf

1-38(even) 42,43,44

Text resources Text test

bank

Rosen Web Links


u/~eaton/0131873814_MEb.pdf

Note: APPENDIX

Learning Activites http://www.geom.uiuc.edu/~zarembe/grapht1.html


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


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:







http://www.geom.uiuc.edu/~zarembe/grapht1.html

http://www.geom.uiuc.edu/~zarembe/grapht1.html









assessments





MULTIPLE CHAPTERS 1. McGraw-Hill Education Teacher Resources (Textbook)


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




Identify structural similarities between integers and polynomials.













http://www.mathcove.net/petersen/lessons/get-lesson?les=1

http://www.inf.ed.ac.uk/teaching/courses/dmmr/slides/13-14/Ch10.pdf

http://www1.cs.columbia.edu/~zeph/3203s04/lectures.html

http://freematheducation.com/graph/

http://www.maa.org/publications/ebooks/resources-for-teaching-discrete-mathematics

Identify expressions as single entities, e.g. the difference of two squares. 8. Look for and express regularity in repeated reasoning.



Chapter 11 Trees Total Number of Days: 3days Grade/Course: 12/Discreatemathmtics


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.

.


RESOURCES LEARNING


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
















https://us.search.yahoo.com/yhs/search?hspart=Elex&hsimp=yhs-elex_v9&p=Introduction+to+Trees

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/



























http://algs4.cs.princeton.edu/43mst/

http://algs4.cs.princeton.edu/43mst/


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)




Graph theory and Trees critical thinking making connections between multiple events B. Creativity and Innovation


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.

http://academics.triton.edu/faculty/ebell/6%20-%20Graph%20Theory%20and%20Trees.pdf

http://math.mit.edu/~rothvoss/18.304.1PM/Presentations/1-Varun-MSTPresentation.pdf

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


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.


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)


4 Cs Three Part Objective Creativity: projects Behavior

http://graphics.stanford.edu/courses/cs448b-02-winter/lectures/treesgraphs/tree.graph.pdf

http://www.cs.cf.ac.uk/Dave/CM0167/Lectures/CM0167_Chap02_1_Trees.pdf


Critical Thinking: Math Journal Condition Collaboration: Teams/Groups/Stations Demonstration of Learning (DOL) Communication – Powerpoints/Presentations

chapter 1 the foundations: logic and proofs · chapter 1 the foundations: logic and proofs total...

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