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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
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 HS
find Hamilton paths, and discuss implications for ranking the players in the tournament. In grades 9 and 10, students can expand their initial methods of analysis to now include a matrix method to rank the players in the tournament. Grade 8 Example: • Four players (Dom, N...
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find Hamilton paths, and discuss implications for ranking the players in the tournament. In grades 9 and 10, students can expand their initial methods of analysis to now include a matrix method to rank the players in the tournament. Grade 8 Example: • Four players (Dom, Nate, Ryan, & Zach) are playing in a round-robin tennis tournament, where every player plays every other player. Dom beats Nate and Ryan, Nate beats Zach, Ryan beats Nate and Zach, and Zach beats Dom. o Represent this
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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
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http://www.ade.state.az.us/standards/math/Articulated08/Gradeleveldocs/MathGradeHS.pdf#page=18
www.ade.state.az.us/standards/math/Articulated08/Gradeleveldocs/MathGrade...
find Hamilton paths, and discuss implications <span class="highlight">for</span> ranking the players in the tournament. In grades 9 and 10, students can expand their initial methods <span class="highlight">of</span> <span class="highlight">analysis</span> to now include a matrix method to rank the players in the tournament. Grade 8 Example: • Four players (Dom, Nate, Ryan, & Zach) are playing in a round-robin tennis tournament, where every player plays every other player. Dom beats Nate and Ryan, Nate beats Zach, Ryan beats Nate and Zach, and Zach beats Dom. o Represent this
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
colored using the same color. Many students believe a point is considered an edge. Realizing that a point is not an edge does not happen without exploration and discussion. Students should have multiple opportunities to color a variety of simple pictures and maps. The goal is ...
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colored using the same color. Many students believe a point is considered an edge. Realizing that a point is not an edge does not happen without exploration and discussion. Students should have multiple opportunities to color a variety of simple pictures and maps. The goal is for students to use the fewest number of colors; some possible strategies that young children may use include a guess-and-check strategy, use many colors and try to eliminate one color at a time; or use few colors and insert
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http://www.ade.state.az.us/standards/math/Articulated08/Gradeleveldocs/MathGrade2.pdf#page=15
www.ade.state.az.us/standards/math/Articulated08/Gradeleveldocs/MathGrade...
colored using the same color. Many students believe a point is considered an <span class="highlight">edge</span>. Realizing that a point is not an <span class="highlight">edge</span> does not happen without exploration and discussion. Students should have multiple opportunities to color a variety <span class="highlight">of</span> simple pictures and maps. The goal is <span class="highlight">for</span> students to use the fewest number <span class="highlight">of</span> colors; some possible strategies that young children may use include a guess-and-check strategy, use many colors and try to eliminate one color at a time; or use few colors and insert
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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 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 K
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 K Arizona Department of Education: Standards and Assessment Division 7 Approved 6.24.08 S...
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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 K Arizona Department of Education: Standards and Assessment Division 7 Approved 6.24.08 Strand 2: Data Analysis, Probability, and Discrete
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http://www.ade.state.az.us/standards/math/Articulated08/Gradeleveldocs/MathGradeK.pdf#page=7
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 K Arizona Department <span class="highlight">of</span> Education: Standards and Assessment Division 7 Approved 6.24.08 Strand 2: Data <span class="highlight">Analysis</span>, Probability, and Discrete
GRADE 4
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 4 Arizona Department of Education: Standards and Assessment Division 18 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 4 Arizona Department of Education: Standards and Assessment Division 18 Approved 6.24.08 Strand 2: Data Analysis
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http://www.ade.state.az.us/standards/math/Articulated08/Gradeleveldocs/MathGrade4.pdf#page=11
www.ade.state.az.us/standards/math/Articulated08/Gradeleveldocs/MathGrade...
Probability, and 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
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http://www.ade.state.az.us/standards/math/Articulated08/Gradeleveldocs/MathGrade4.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 4 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>
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
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