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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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