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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

9 0 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 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

16 0 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 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

www.ade.state.az.us/standards/math/Articulated08/Gradeleveldocs/MathGrade...

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Microsoft Word - Front Cover Page

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Glossary 43 Coordinates: A pair of numbers used to describe the location of a point on a coordinate plane. Counterexample: An example that proves a statement false. Cube: A three-dimensional figure that has six square faces. Customary Systems: A measuring system used mainly in the United States using units such as inches and feet. Data: Information, facts, or numbers used to describe something. Data Analysis: A process of collecting and organizing data in order to identify

50 0 http://www.doe.nv.gov/Standards/Mathematics/New_Nevada_Math_Standards_Complete_Document_6.29.06_PDF.pdf#page=50 www.doe.nv.gov/Standards/Mathematics/New_Nevada_Math_Standards_Complete_D...

Glossary 43 Coordinates: A pair of numbers used to describe the location of a point on a coordinate plane. Counterexample: An example that proves a statement false. Cube: A three-dimensional figure that has six square faces. Customary Systems: A measuring system used mainly in the United States using units such as inches and feet. Data: Information, facts, or numbers used to describe something. Data Analysis: A process of collecting and organizing data in order to identify

www.doe.nv.gov/Standards/Mathematics/New_Nevada_Math_Standards_Complete_D...

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Speaking Standard 3

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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

18 0 http://www.nj.gov/education/cccs/2004/s4_math.pdf#page=18 www.nj.gov/education/cccs/2004/s4_math.pdf#page=18

dimensional shapes. • Vertex, edge, face, side • 3D figures – cube, rectangular prism, sphere, cone, cylinder, and pyramid • 2D figures – square, rectangle, circle, triangle • Relationships between three- and two-dimensional shapes (i.e., the face of a 3D shape is a 2D shape) 3. Describe, identify and create instances of line symmetry. 4. Recognize, describe, extend and create designs and patterns with geometric objects of different shapes and colors. 4.2.2 B. Transforming Shapes 1. Use simple shapes to

36 0 http://www.nj.gov/education/cccs/2004/s4_math.pdf#page=36 www.nj.gov/education/cccs/2004/s4_math.pdf#page=36

mathematical perspectives on everyday phenomena and with important examples of how mathematics is used in the modern world. Two important areas of discrete mathematics are addressed in this standard; a third area, iteration and recursion, is addressed in Standard 4.3 (Patterns and Algebra). Data Analysis (or Statistics). In today’s information-based world, students need to be able to read, understand, and interpret data in order to make informed decisions. In the early grades, students should be involved in

38 0 http://www.nj.gov/education/cccs/2004/s4_math.pdf#page=38 www.nj.gov/education/cccs/2004/s4_math.pdf#page=38

describe practical sets of directions (e.g., to add two 2-digit numbers). 2. Explore vertex-edge graphs • Vertex, edge • Path 3. Find the smallest number of colors needed to color a map. Building upon knowledge and skills gained in preceding grades, by the end of Grade 4, students will: 4.4.4 A. Data Analysis 1. Collect, generate, organize, and display data in response to questions, claims, or curiosity. • Data collected from the school environment 2. Read, interpret, construct, analyze

39 0 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 for 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 of rectangles in a grid, find the number of rectangles of each size and add the results) 4.4.4 D. Discrete Mathematics—Vertex-Edge Graphs and Algorithms 1. Follow, devise, and

40 0 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-Edge Graphs and Algorithms 1. Devise strategies for winning simple games (e.g., start with two piles of objects, each of two players in turn removes any number of objects from a single pile, and the person to take the last group of objects wins) and express those strategies as sets of directions. Building upon knowledge and skills gained in preceding grades, by the end of Grade 6, students will: 4.4.6 A. Data Analysis 1. Collect

41 0 http://www.nj.gov/education/cccs/2004/s4_math.pdf#page=41 www.nj.gov/education/cccs/2004/s4_math.pdf#page=41

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

42 0 http://www.nj.gov/education/cccs/2004/s4_math.pdf#page=42 www.nj.gov/education/cccs/2004/s4_math.pdf#page=42

shortest network connecting specified sites • Finding the shortest route on a map from one site to another • Finding the shortest circuit on a map that makes a tour of specified sites Building upon knowledge and skills gained in preceding grades, by the end of Grade 8, students will: 4.4.8 A. Data Analysis 1. Select and use appropriate representations for sets of data, and measures of central tendency (mean, median, and mode). • Type of display most appropriate for given data • Box-and-whisker plot

44 0 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 edge 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 for making fair decisions. • Combining individual preferences into a group decision (e.g., determining winner of an election or selection process

www.nj.gov/education/cccs/2004/s4_math.pdf#page=41

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GRADE CWR

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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

6 0 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 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, 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-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

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-edge graphs to model and solve problems related to paths, circuits, networks, and relationships among a finite number of 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 information, determine how to represent the problem, and identify

www.ade.state.az.us/standards/math/Articulated08/Gradeleveldocs/MathGrade...

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GRADE HS

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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

6 0 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 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

18 0 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 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

www.ade.state.az.us/standards/math/Articulated08/Gradeleveldocs/MathGrade...

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GRADE 7

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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

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 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

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 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

www.ade.state.az.us/standards/math/Articulated08/Gradeleveldocs/MathGrade...

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Microsoft Word - K-8 Mathematics Curriculum Framework.doc

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experiment Explicit A formula whose dependent variable is defined in terms of the independent variable Ex. y = 2x – 3 Exponential form A quantity expressed as a number raised to a power (In exponential form, 32 can be written as 2 5 .) Face A two-dimensional side of a three-dimensional figure Ex. The faces of a cube are squares. Factor One of two or more numbers that are multiplied together to get a product (13 and 4 are both factors of 52 because 13 • 4 = 52.) Flip (Reflection) (See

70 0 http://arkansased.org/teachers/pdf/k8_math_may05.pdf#page=70 arkansased.org/teachers/pdf/k8_math_may05.pdf#page=70

experiment Explicit A formula whose dependent variable is defined in terms of the independent variable Ex. y = 2x – 3 Exponential form A quantity expressed as a number raised to a power (In exponential form, 32 can be written as 2 5 .) Face A two-dimensional side of a three-dimensional figure Ex. The faces of a cube are squares. Factor One of two or more numbers that are multiplied together to get a product (13 and 4 are both factors of 52 because 13 • 4 = 52.) Flip (Reflection) (See

arkansased.org/teachers/pdf/k8_math_may05.pdf#page=70

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GRADE 3

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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

15 0 http://www.ade.state.az.us/standards/math/Articulated08/Gradeleveldocs/MathGrade3.pdf#page=15 www.ade.state.az.us/standards/math/Articulated08/Gradeleveldocs/MathGrade...

19 0 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 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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Illinois Mathematics Assessment Framework PSAE Grade 11

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10C 1% 2% 0% Note: The mathematics portion of the PSAE is a combination of the ACT Assessment Mathematics component and the WorkKeys Applied Mathematics Assessment component. Note Added on 3/18/08: A new scoring methodology was announced that affects only reading and mathematics. For more information, please see Probability http://www.isbe.net/assessment/listserv/2008/mar14.htm. *These percents are typical, based on an analysis of forms from multiple years. These percents in this chart were derived

10 0 http://www.isbe.state.il.us/assessment/pdfs/iaf_math_PSAEFINAL.pdf#page=10 www.isbe.state.il.us/assessment/pdfs/iaf_math_PSAEFINAL.pdf#page=10

10C 1% 2% 0% Note: The mathematics portion of the PSAE is a combination of the ACT Assessment Mathematics component and the WorkKeys Applied Mathematics Assessment component. Note Added on 3/18/08: A new scoring methodology was announced that affects only reading and mathematics. For more information, please see Probability http://www.isbe.net/assessment/listserv/2008/mar14.htm. *These percents are typical, based on an analysis of forms from multiple years. These percents in this chart were derived

www.isbe.state.il.us/assessment/pdfs/iaf_math_PSAEFINAL.pdf#page=10

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apply the given formula to find area and circumference of circles, area and perimeter of polygons, and volume of regular solids; use appropriate measurements in collecting data for a real world situation At Level 3, the student is able to solve real world problems given logarithmic and exponential formulas (e.g. Ph scale, Richter scale.). Sample Task: Construct a regular geometric solid and determine the surface area, volume, and edge length. Linkages: Science, art, construction

5 0 http://www.state.tn.us/education/ci/math/doc/MA_3134.pdf#page=5 www.state.tn.us/education/ci/math/doc/MA_3134.pdf#page=5

apply the given formula to find area and circumference of circles, area and perimeter of polygons, and volume of regular solids; use appropriate measurements in collecting data for a real world situation At Level 3, the student is able to solve real world problems given logarithmic and exponential formulas (e.g. Ph scale, Richter scale.). Sample Task: Construct a regular geometric solid and determine the surface area, volume, and edge length. Linkages: Science, art, construction

www.state.tn.us/education/ci/math/doc/MA_3134.pdf#page=5

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