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Stony Brook: What is a System of Linear Equations?
Graphs Nicely Drawing Trees Planarity Detection and Embedding 1.5 Graph Problems  hard problems Clique Independent Set Vertex Cover Traveling Salesman Problem Hamiltonian Cycle Graph Partition Vertex Coloring Edge Coloring Graph Isomorphism Steiner Tree Feedback Edge/Vertex...
Student Portfolios as Visual Organizers for Journey North
cycle information, characteristics, and migration maps Drawings, diagrams, charts, graphs, and analysis of data Concept maps or KWL charts Initial predictions and revised predictions along with explanations for revisions Calculations and "worksheets" Compact i...
Development of Performance Level Descriptors for the California Standards Tests (CSTs) and...
Appendix A: Mathematics PLDs Specific Descriptors: Statistics, Data Analysis, and Probability Advanced students: • Identify best representation for a given set of data. • Find probability of independent and dependent events. Proficient students: • Fi...
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Appendix A: Mathematics PLDs Specific Descriptors: Statistics, Data Analysis, and Probability Advanced students: • Identify best representation for a given set of data. • Find probability of independent and dependent events. Proficient students: • Find all possible outcomes of a situation to determine the probability of a particular outcome. • Find the mean, median, and mode of a data set. Basic students: • Interpret a graph to evaluate information. • Identify the possible outcomes for
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http://www.cde.ca.gov/ta/tg/sr/documents/pldreport.pdf#page=109
www.cde.ca.gov/ta/tg/sr/documents/pldreport.pdf#page=109
Appendix A: Mathematics PLDs Specific Descriptors: Statistics, Data <span class="highlight">Analysis</span>, and Probability Advanced students: • Identify best representation <span class="highlight">for</span> a given set <span class="highlight">of</span> data. • Find probability <span class="highlight">of</span> <span class="highlight">independent</span> and dependent events. Proficient students: • Find all possible outcomes <span class="highlight">of</span> a situation to determine the probability <span class="highlight">of</span> a particular outcome. • Find the mean, median, and mode <span class="highlight">of</span> a data set. Basic students: • Interpret a graph to evaluate <span class="highlight">information</span>. • Identify the possible outcomes <span class="highlight">for</span>
Univ. of Tennessee: Graph Theory Glossary
have different colors. See the Four Color Theorem. circuit A circuit is a path which ends at the vertex it begins (so a loop is an circuit of length one). complete graph A complete graph with n vertices (denoted Kn) is a graph with n vertices in which each vertex is connected to each...
Univ. of Tenn. at Martin:Graph Theory Glossary  D
have different colors. See the Four Color Theorem. circuit A circuit is a path which ends at the vertex it begins (so a loop is an circuit of length one). complete graph A complete graph with n vertices (denoted Kn) is a graph with n vertices in which each vertex is connected to each...
A Rainbow of Colors!
Evaluation Checklist Prerequisites (skills or background needed): Basic knowledge of computer skills including how to use a mouse and keyboard and how to access and search teacherselected Web sites Basic knowledge of how to use spreadsheet software Procedure Teacher Compon...
GRADE 5
Arizona Mathematics Standard Articulated by Grade Level The bulleted items within a performance objective indicate the specific content to be taught. Explanations and Examples Updated 1.19.09 Grade 5 Arizona Department of Education: Standards and Assessment Division 16 Approved 6.24.08 St...
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Arizona Mathematics Standard Articulated by Grade Level The bulleted items within a performance objective indicate the specific content to be taught. Explanations and Examples Updated 1.19.09 Grade 5 Arizona Department of Education: Standards and Assessment Division 16 Approved 6.24.08 Strand 2: Data Analysis, Probability, and Discrete Mathematics Concept 4: VertexEdge Graphs Understand and apply vertexedge graphs. In Grade 5, students continue to develop their
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http://www.ade.state.az.us/standards/math/Articulated08/Gradeleveldocs/MathGrade5.pdf#page=9
www.ade.state.az.us/standards/math/Articulated08/Gradeleveldocs/MathGrade...
and the study <span class="highlight">of</span> graphs. This prepares students <span class="highlight">for</span> the study <span class="highlight">of</span> discrete functions as well as to make valid inferences, decisions, and arguments. Discrete mathematics is a branch <span class="highlight">of</span> mathematics that is widely used in business and industry. Combinatorics is the mathematics <span class="highlight">of</span> systematic counting. Vertex<span class="highlight">edge</span> graphs are used to model and solve problems involving paths, networks, and relationships among a finite number <span class="highlight">of</span> objects. Concept 1: Data <span class="highlight">Analysis</span> (Statistics) Understand and apply data
16
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http://www.ade.state.az.us/standards/math/Articulated08/Gradeleveldocs/MathGrade5.pdf#page=16
www.ade.state.az.us/standards/math/Articulated08/Gradeleveldocs/MathGrade...
Arizona Mathematics Standard Articulated by Grade Level The bulleted items within a performance objective indicate the specific content to be taught. Explanations and Examples Updated 1.19.09 Grade 5 Arizona Department <span class="highlight">of</span> Education: Standards and Assessment Division 16 Approved 6.24.08 Strand 2: Data <span class="highlight">Analysis</span>, Probability, and Discrete Mathematics Concept 4: Vertex<span class="highlight">Edge</span> Graphs Understand and apply vertex<span class="highlight">edge</span> graphs. In Grade 5, students continue to develop their
Speaking Standard 3
slate of officers from a class (e.g., if there are 23 students and 3 officers, the number is 23 x 22 x 21) 3. List the possible combinations of two elements chosen from a given set (e.g., forming a committee of two from a group of 12 students, finding how many handshakes...
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slate of officers from a class (e.g., if there are 23 students and 3 officers, the number is 23 x 22 x 21) 3. List the possible combinations of two elements chosen from a given set (e.g., forming a committee of two from a group of 12 students, finding how many handshakes there will be among ten people if everyone shakes each other person’s hand once). 4.4.6 D. Discrete Mathematics—VertexEdge Graphs and Algorithms 1. Devise strategies for winning simple games (e.g., start with two piles of objects
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http://www.nj.gov/education/cccs/2004/s4_math.pdf#page=39
www.nj.gov/education/cccs/2004/s4_math.pdf#page=39
as shape or color, and relationships. • Venn diagrams • Numerical and alphabetical order 2. Represent all possibilities <span class="highlight">for</span> a simple counting situation in an organized way and draw conclusions from this representation. • Organized lists, charts, tree diagrams • Dividing into categories (e.g., to find the total number <span class="highlight">of</span> rectangles in a grid, find the number <span class="highlight">of</span> rectangles <span class="highlight">of</span> each size and add the results) 4.4.4 D. Discrete Mathematics—Vertex<span class="highlight">Edge</span> Graphs and Algorithms 1. Follow, devise, and
40
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http://www.nj.gov/education/cccs/2004/s4_math.pdf#page=40
www.nj.gov/education/cccs/2004/s4_math.pdf#page=40
shirts and 4 skirts). 4.4.5 D. Discrete Mathematics—Vertex<span class="highlight">Edge</span> Graphs and Algorithms 1. Devise strategies <span class="highlight">for</span> winning simple games (e.g., start with two piles <span class="highlight">of</span> objects, each <span class="highlight">of</span> two players in turn removes any number <span class="highlight">of</span> objects from a single pile, and the person to take the last group <span class="highlight">of</span> objects wins) and express those strategies as sets <span class="highlight">of</span> directions. Building upon knowledge and skills gained in preceding grades, by the end <span class="highlight">of</span> Grade 6, students will: 4.4.6 A. Data <span class="highlight">Analysis</span> 1. Collect
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http://www.nj.gov/education/cccs/2004/s4_math.pdf#page=41
www.nj.gov/education/cccs/2004/s4_math.pdf#page=41
slate <span class="highlight">of</span> officers from a class (e.g., if there are 23 students and 3 officers, the number is 23 x 22 x 21) 3. List the possible combinations <span class="highlight">of</span> two elements chosen from a given set (e.g., forming a committee <span class="highlight">of</span> two from a group <span class="highlight">of</span> 12 students, finding how many handshakes there will be among ten people if everyone shakes each other person’s hand once). 4.4.6 D. Discrete Mathematics—Vertex<span class="highlight">Edge</span> Graphs and Algorithms 1. Devise strategies <span class="highlight">for</span> winning simple games (e.g., start with two piles <span class="highlight">of</span> objects
44
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http://www.nj.gov/education/cccs/2004/s4_math.pdf#page=44
www.nj.gov/education/cccs/2004/s4_math.pdf#page=44
represent and solve practical problems. • Circuits that include every <span class="highlight">edge</span> in a graph • Circuits that include every vertex in a graph • Scheduling problems (e.g., when project meetings should be scheduled to avoid conflicts) using graph coloring • Applications to science (e.g., whoeatswhom graphs, genetic trees, molecular structures) 2. Explore strategies <span class="highlight">for</span> making fair decisions. • Combining individual preferences into a group decision (e.g., determining winner <span class="highlight">of</span> an election or selection process
GRADE CWR
Concept 4: VertexEdge Graphs Understand and apply vertexedge graphs. In Grades 11 and 12, students extend their understanding of networks to devise, analyze, and apply algorithms for solving problems related to circuits, shortest paths, minimum weight spanning trees,...
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Concept 4: VertexEdge Graphs Understand and apply vertexedge graphs. In Grades 11 and 12, students extend their understanding of networks to devise, analyze, and apply algorithms for solving problems related to circuits, shortest paths, minimum weight spanning trees, and adjacency matrices. Performance Objectives Process Integration Explanations and Examples Students are expected to: PO 1. Study the following topics related to vertexedge graphs: Euler circuits, Hamilton circuits, the
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http://www.ade.state.az.us/standards/math/Articulated08/Gradeleveldocs/MathGradeCWR.pdf#page=6
www.ade.state.az.us/standards/math/Articulated08/Gradeleveldocs/MathGrade...
This strand requires students to use data collection, data <span class="highlight">analysis</span>, statistics, probability, systematic listing and counting, and the study <span class="highlight">of</span> graphs. This prepares students <span class="highlight">for</span> the study <span class="highlight">of</span> discrete functions as well as to make valid inferences, decisions, and arguments. Discrete mathematics is a branch <span class="highlight">of</span> mathematics that is widely used in business and industry. Combinatorics is the mathematics <span class="highlight">of</span> systematic counting. Vertex<span class="highlight">edge</span> graphs are used to model and solve problems involving paths, networks, and
14
0
http://www.ade.state.az.us/standards/math/Articulated08/Gradeleveldocs/MathGradeCWR.pdf#page=14
www.ade.state.az.us/standards/math/Articulated08/Gradeleveldocs/MathGrade...
Concept 4: Vertex<span class="highlight">Edge</span> Graphs Understand and apply vertex<span class="highlight">edge</span> graphs. In Grades 11 and 12, students extend their understanding <span class="highlight">of</span> networks to devise, analyze, and apply algorithms <span class="highlight">for</span> solving problems related to circuits, shortest paths, minimum weight spanning trees, and adjacency matrices. Performance Objectives Process Integration Explanations and Examples Students are expected to: PO 1. Study the following topics related to vertex<span class="highlight">edge</span> graphs: Euler circuits, Hamilton circuits, the
15
0
http://www.ade.state.az.us/standards/math/Articulated08/Gradeleveldocs/MathGradeCWR.pdf#page=15
www.ade.state.az.us/standards/math/Articulated08/Gradeleveldocs/MathGrade...
and Examples Students are expected to: PO 2. Understand, analyze, and apply vertex<span class="highlight">edge</span> graphs to model and solve problems related to paths, circuits, networks, and relationships among a finite number <span class="highlight">of</span> elements, in realworld and abstract settings. Connections: MCWRS2C401, MCWR S2C403, MCWRS2C404, SSHSS4C1 03 MCWRS5C201. Analyze a problem situation, determine the question(s) to be answered, organize given <span class="highlight">information</span>, determine how to represent the problem, and identify
GRADE 7
Arizona Mathematics Standard Articulated by Grade Level The bulleted items within a performance objective indicate the specific content to be taught. Explanations and Examples Updated 1.19.09 Grade 7 Arizona Department of Education: Standards and Assessment Division 18 Approved 6.24.08...
1
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Arizona Mathematics Standard Articulated by Grade Level The bulleted items within a performance objective indicate the specific content to be taught. Explanations and Examples Updated 1.19.09 Grade 7 Arizona Department of Education: Standards and Assessment Division 18 Approved 6.24.08 Strand 2: Data Analysis, Probability, and Discrete Mathematics Concept 4: VertexEdge Graphs Understand and apply vertexedge graphs. In Grade 7, students use vertexedge
10
0
http://www.ade.state.az.us/standards/math/Articulated08/Gradeleveldocs/MathGrade7.pdf#page=10
www.ade.state.az.us/standards/math/Articulated08/Gradeleveldocs/MathGrade...
counting, and the study <span class="highlight">of</span> graphs. This prepares students <span class="highlight">for</span> the study <span class="highlight">of</span> discrete functions as well as to make valid inferences, decisions, and arguments. Discrete mathematics is a branch <span class="highlight">of</span> mathematics that is widely used in business and industry. Combinatorics is the mathematics <span class="highlight">of</span> systematic counting. Vertex<span class="highlight">edge</span> graphs are used to model and solve problems involving paths, networks, and relationships among a finite number <span class="highlight">of</span> objects. Concept 1: Data <span class="highlight">Analysis</span> (Statistics) Understand and apply data
18
0
http://www.ade.state.az.us/standards/math/Articulated08/Gradeleveldocs/MathGrade7.pdf#page=18
www.ade.state.az.us/standards/math/Articulated08/Gradeleveldocs/MathGrade...
Arizona Mathematics Standard Articulated by Grade Level The bulleted items within a performance objective indicate the specific content to be taught. Explanations and Examples Updated 1.19.09 Grade 7 Arizona Department <span class="highlight">of</span> Education: Standards and Assessment Division 18 Approved 6.24.08 Strand 2: Data <span class="highlight">Analysis</span>, Probability, and Discrete Mathematics Concept 4: Vertex<span class="highlight">Edge</span> Graphs Understand and apply vertex<span class="highlight">edge</span> graphs. In Grade 7, students use vertex<span class="highlight">edge</span>

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