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Education
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 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: Vertex-Edge Graphs Understand and apply vertex-edge 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
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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—Vertex-Edge 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
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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
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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., who-eats-whom 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: Vertex-Edge Graphs Understand and apply vertex-edge 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: Vertex-Edge Graphs Understand and apply vertex-edge 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 vertex-edge 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
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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
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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 real-world and abstract settings. Connections: MCWR-S2C4-01, MCWR- S2C4-03, MCWR-S2C4-04, SSHS-S4C1- 03 MCWR-S5C2-01. 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...
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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: Vertex-Edge Graphs Understand and apply vertex-edge graphs. In Grade 7, students use vertex-edge
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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
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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>
GRADE 3
previously solved problems, and identify possible strategies for solving the problem. M03-S5C2-08. Make and test conjectures based on data (or information) collected from explorations and experiments. Examples: • County map of Arizona • Map of United States...
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previously solved problems, and identify possible strategies for solving the problem. M03-S5C2-08. Make and test conjectures based on data (or information) collected from explorations and experiments. Examples: • County map of Arizona • Map of United States PO 2. Investigate properties of vertex- edge graphs • circuits in a graph, • weights on edges, and • shortest path between two vertices. Connections: M03-S1C1-02, M03-S1C2- 01, M03-S1C2-02, SS3-S4C1-03 M03-S5C2-02. Identify relevant
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http://www.ade.state.az.us/standards/math/Articulated08/Gradeleveldocs/MathGrade3.pdf#page=15
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
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http://www.ade.state.az.us/standards/math/Articulated08/Gradeleveldocs/MathGrade3.pdf#page=19
www.ade.state.az.us/standards/math/Articulated08/Gradeleveldocs/MathGrade...
previously solved problems, and identify possible strategies <span class="highlight">for</span> solving the problem. M03-S5C2-08. Make and test conjectures based on data (or <span class="highlight">information</span>) collected from explorations and experiments. Examples: • County map <span class="highlight">of</span> Arizona • Map <span class="highlight">of</span> United States PO 2. Investigate properties <span class="highlight">of</span> vertex- <span class="highlight">edge</span> graphs • circuits in a graph, • weights on edges, and • shortest path between two vertices. Connections: M03-S1C1-02, M03-S1C2- 01, M03-S1C2-02, SS3-S4C1-03 M03-S5C2-02. Identify relevant
GRADE 2
Explanations and Examples Students are expected to: PO 2. Build vertex-edge graphs using concrete materials and explore simple properties of vertex-edge graphs • number of vertices and edges, • neighboring vertices, and • paths in a graph. Connections: M...
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Explanations and Examples Students are expected to: PO 2. Build vertex-edge graphs using concrete materials and explore simple properties of vertex-edge graphs • number of vertices and edges, • neighboring vertices, and • paths in a graph. Connections: M02-S2C4-03 A vertex-edge graph is a collection of vertices and edges. A vertex is a point/dot that represents an object or location. An edge connects two vertices and represents some relationship between them. The vertex-edge graph below has 4
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http://www.ade.state.az.us/standards/math/Articulated08/Gradeleveldocs/MathGrade2.pdf#page=17
www.ade.state.az.us/standards/math/Articulated08/Gradeleveldocs/MathGrade...
Explanations and Examples Students are expected to: PO 2. Build vertex-<span class="highlight">edge</span> graphs using concrete materials and explore simple properties <span class="highlight">of</span> vertex-<span class="highlight">edge</span> graphs • number <span class="highlight">of</span> vertices and edges, • neighboring vertices, and • paths in a graph. Connections: M02-S2C4-03 A vertex-<span class="highlight">edge</span> graph is a collection <span class="highlight">of</span> vertices and edges. A vertex is a point/dot that represents an object or location. An <span class="highlight">edge</span> connects two vertices and represents some relationship between them. The vertex-<span class="highlight">edge</span> graph below has 4
GRADE HS
Discrete Mathematics This strand requires students to use data collection, data analysis, statistics, probability, systematic listing and counting, and the study of graphs. This prepares students for the study of discrete functions as well as to make valid inferences, dec...
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Discrete Mathematics This strand requires students to use data collection, data analysis, statistics, probability, systematic listing and counting, and the study of graphs. This prepares students for the study of discrete functions as well as to make valid inferences, decisions, and arguments. Discrete mathematics is a branch of mathematics that is widely used in business and industry. Combinatorics is the mathematics of systematic counting. Vertex-edge graphs are used to model and solve problems involving
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http://www.ade.state.az.us/standards/math/Articulated08/Gradeleveldocs/MathGradeHS.pdf#page=6
www.ade.state.az.us/standards/math/Articulated08/Gradeleveldocs/MathGrade...
Discrete Mathematics 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
GRADE 1
Mathematics This strand requires students to use data collection, data analysis, statistics, probability, systematic listing and counting, and the study of graphs. This prepares students for the study of discrete functions as well as to make valid inferences, decisions, a...
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Mathematics This strand requires students to use data collection, data analysis, statistics, probability, systematic listing and counting, and the study of graphs. This prepares students for the study of discrete functions as well as to make valid inferences, decisions, and arguments. Discrete mathematics is a branch of mathematics that is widely used in business and industry. Combinatorics is the mathematics of systematic counting. Vertex-edge graphs are used to model and solve problems involving paths
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http://www.ade.state.az.us/standards/math/Articulated08/Gradeleveldocs/MathGrade1.pdf#page=8
www.ade.state.az.us/standards/math/Articulated08/Gradeleveldocs/MathGrade...
Mathematics 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
GRADE 8
of graphs. This prepares students for the study of discrete functions as well as to make valid inferences, decisions, and arguments. Discrete mathematics is a branch of mathematics that is widely used in business and industry. Combinatorics is the mathematics of s...
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of graphs. This prepares students for the study of discrete functions as well as to make valid inferences, decisions, and arguments. Discrete mathematics is a branch of mathematics that is widely used in business and industry. Combinatorics is the mathematics of systematic counting. Vertex-edge graphs are used to model and solve problems involving paths, networks, and relationships among a finite number of objects. Concept 1: Data Analysis (Statistics) Understand and apply data collection
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http://www.ade.state.az.us/standards/math/Articulated08/Gradeleveldocs/MathGrade8.pdf#page=7
www.ade.state.az.us/standards/math/Articulated08/Gradeleveldocs/MathGrade...
<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 collection
Microsoft Word - 2007 MS Math Framework Competencies and Objectives 9-18-07.doc
graphing. A variety of problem-solving techniques and technology will be used when applying these concepts, which will enable students to solve real-life application, routine word, and non- routine problems. Technology should be a component of the instruction. The instruction...
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graphing. A variety of problem-solving techniques and technology will be used when applying these concepts, which will enable students to solve real-life application, routine word, and non- routine problems. Technology should be a component of the instruction. The instructional approach should provide opportunities for students to work together collaboratively and cooperatively as they solve routine and non-routine problems. Communication strategies should include reading, writing, speaking, and critical
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http://www.mde.k12.ms.us/acad/id/curriculum/math/2007_framework/2007%20MS%20Math%20Framework%20Competencies%20and%20Objectives%209-18-07.pdf#page=43
www.mde.k12.ms.us/acad/id/curriculum/math/2007_framework/2007%20MS%20Math...
graphing. A variety <span class="highlight">of</span> problem-solving techniques and technology will be used when applying these concepts, which will enable students to solve real-life application, routine word, and non- routine problems. Technology should be a <span class="highlight">component</span> <span class="highlight">of</span> the instruction. The instructional approach should provide opportunities <span class="highlight">for</span> students to work together collaboratively and cooperatively as they solve routine and non-routine problems. Communication strategies should include reading, writing, speaking, and critical
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http://www.mde.k12.ms.us/acad/id/curriculum/math/2007_framework/2007%20MS%20Math%20Framework%20Competencies%20and%20Objectives%209-18-07.pdf#page=46
www.mde.k12.ms.us/acad/id/curriculum/math/2007_framework/2007%20MS%20Math...
about the characteristics <span class="highlight">of</span> graphs and their associated equations, expanding the techniques used to solve equations, and applying properties in real-world applications, routine word, and non-routine problems. Technology should be a <span class="highlight">component</span> <span class="highlight">of</span> the instruction. The instructional approach should provide opportunities <span class="highlight">for</span> students to work together collaboratively and cooperatively as they solve routine and non-routine problems. Communication strategies should include reading, writing, speaking, and
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http://www.mde.k12.ms.us/acad/id/curriculum/math/2007_framework/2007%20MS%20Math%20Framework%20Competencies%20and%20Objectives%209-18-07.pdf#page=49
www.mde.k12.ms.us/acad/id/curriculum/math/2007_framework/2007%20MS%20Math...
equations. Concepts and computations with matrices are introduced. The <span class="highlight">analysis</span> <span class="highlight">of</span> graphs includes scatter plots. Written and oral justifications to support solution methods and solutions are required. Technology should be a <span class="highlight">component</span> <span class="highlight">of</span> the instruction. The instructional approach should provide opportunities <span class="highlight">for</span> students to work together collaboratively and cooperatively as they solve routine and non-routine problems. Communication strategies should include reading, writing, speaking, and critical
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