Mathematics Cumulative Progress Indicators (CPIs) for the end of the designated grade span

Place a "+" for an expectation that represents a strength & a "-" for a weakness

+ or -

STANDARD 4.4     (DATA ANALYSIS, PROBABILITY, AND DISCRETE MATHEMATICS)     ALL STUDENTS WILL DEVELOP AN UNDERSTANDING OF THE CONCEPTS AND TECHNIQUES OF DATA ANALYSIS, PROBABILITY, AND DISCRETE MATHEMATICS, AND WILL USE THEM TO MODEL SITUATIONS, SOLVE PROBLEMS, AND ANALYZE AND DRAW APPROPRIATE INFERENCES FROM DATA.

C.     Discrete Mathematics—Systematic Listing and Counting

By the end of Grade 2, students will:

 1.         Sort and classify objects according to attributes.

·        Venn diagrams

 2.         Generate all possibilities in simple counting situations (e.g., all outfits involving two shirts and three pants).

Building upon knowledge and skills gained in preceding grades, by the end of Grade 3, students will:

 1.         Represent and classify data according to attributes, such 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

Building upon knowledge and skills gained in preceding grades, by the end of Grade 4, students will:

 1.         Represent and classify data according to attributes, such 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)

Building upon knowledge and skills gained in preceding grades, by the end of Grade 5, students will:

 1.         Solve counting problems and justify that all possibilities have been enumerated without duplication.

·        Organized lists, charts, tree diagrams, tables

 2.         Explore the multiplication principle of counting in simple situations by representing all possibilities in an organized way (e.g., you can make 3 x 4 = 12 outfits using 3 shirts and 4 skirts).

Building upon knowledge and skills gained in preceding grades, by the end of Grade 6, students will:

 1.         Solve counting problems and justify that all possibilities have been enumerated without duplication.

·        Organized lists, charts, tree diagrams, tables

·        Venn diagrams

2.         Apply the multiplication principle of counting.

·        Simple situations (e.g., you can make 3 x 4 = 12 outfits using 3 shirts and 4 skirts).

·        Number of ways a specified number of items can be arranged in order (concept of permutation)

·        Number of ways of selecting a 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).

Building upon knowledge and skills gained in preceding grades, by the end of Grade 7, students will:

 1.         Apply the multiplication principle of counting.

·        Permutations:  ordered situations with replacement (e.g., number of possible license plates) vs. ordered situations without replacement (e.g., number of possible slates of 3 class officers from a 23 student class)

 2.         Explore counting problems involving Venn diagrams with three attributes (e.g., there are 15, 20, and 25 students respectively in the chess club, the debating team, and the engineering society; how many different students belong to the three clubs if there are 6 students in chess and debating, 7 students in chess and engineering, 8 students in debating and engineering, and 2 students in all three?).

3.         Apply techniques of systematic listing, counting, and reasoning in a variety of different contexts.

Building upon knowledge and skills gained in preceding grades, by the end of Grade 8, students will:

1.         Apply the multiplication principle of counting.

·        Permutations:  ordered situations with replacement (e.g., number of possible license plates) vs. ordered situations without replacement (e.g., number of possible slates of 3 class officers from a 23 student class)

·        Factorial notation

·        Concept of combinations (e.g., number of possible delegations of 3 out of 23 students)

 2.         Explore counting problems involving Venn diagrams with three attributes (e.g., there are 15, 20, and 25 students respectively in the chess club, the debating team, and the engineering society; how many different students belong to the three clubs if there are 6 students in chess and debating, 7 students in chess and engineering, 8 students in debating and engineering, and 2 students in all three?).

3.         Apply techniques of systematic listing, counting, and reasoning in a variety of different contexts.

Building upon knowledge and skills gained in preceding grades, by the end of Grade 12, students will:

 1.         Calculate combinations with replacement (e.g., the number of possible ways of tossing a coin 5 times and getting 3 heads) and without replacement (e.g., number of possible delegations of 3 out of 23 students).

 2.         Apply the multiplication rule of counting in complex situations, recognize the difference between situations with replacement and without replacement, and recognize the difference between ordered and unordered counting situations.

 3.         Justify solutions to counting problems.

 4.         Recognize and explain relationships involving combinations and Pascal’s Triangle, and apply those methods to situations involving probability.