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.

 

Descriptive Statement:  Data analysis, probability, and discrete mathematics are important interrelated areas of applied mathematics.  Each provides students with powerful 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 collecting and organizing data, and in presenting it using tables, charts, and graphs.  As they progress, they should gather data using sampling, and should increasingly be expected to analyze and make inferences from data, as well as to analyze data and inferences made by others.

 

Probability.  Students need to understand the fundamental concepts of probability so that they can interpret weather forecasts, avoid unfair games of chance, and make informed decisions about medical treatments whose success rate is provided in terms of percentages.  They should regularly be engaged in predicting and determining probabilities, often based on experiments (like flipping a coin 100 times), but eventually based on theoretical discussions of probability that make use of systematic counting strategies.  High school students should use probability models and solve problems involving compound events and sampling.

 

Discrete Mathematics—Systematic Listing and Counting.  Development of strategies for listing and counting can progress through all grade levels, with middle and high school students using the strategies to solve problems in probability.   Primary students, for example, might find all outfits that can be worn using two coats and three hats; middle school students might systematically list and count the number of routes from one site on a map to another; and high school students might determine the number of three-person delegations that can be selected from their class to visit the mayor.

 

Discrete Mathematics—Vertex-Edge Graphs and Algorithms.  Vertex-edge graphs, consisting of dots (vertices) and lines joining them (edges), can be used to represent and solve problems based on real-world situations.  Students should learn to follow and devise lists of instructions, called “algorithms,” and use algorithmic thinking to find the best solution to problems like those involving vertex-edge graphs, but also to solve other problems.

 

These topics provide students with insight into how mathematics is used by decision-makers in our society, and with important tools for modeling a variety of real-world situations.  Students will better understand and interpret the vast amounts of quantitative data that they are exposed to daily, and they will be able to judge the validity of data-supported arguments.

 

Cumulative Progress Indicators

 

By the end of Grade 2, students will:

 

A.     Data Analysis

 1.         Collect, generate, record, and organize data in response to questions, claims, or curiosity.

·        Data collected from students’ everyday experiences

·        Data generated from chance devices, such as spinners and dice

 2.         Read, interpret, construct, and analyze displays of data.

·        Pictures, tally chart, pictograph, bar graph, Venn diagram

·        Smallest to largest, most frequent (mode)

 

B.     Probability

 1.         Use chance devices like spinners and dice to explore concepts of probability.

·        Certain, impossible

·        More likely, less likely, equally likely

 2.         Provide probability of specific outcomes.

·        Probability of getting specific outcome when coin is tossed, when die is rolled, when spinner is spun (e.g., if spinner has five equal sectors, then probability of getting a particular sector is one out of five)

·        When picking a marble from a bag with three red marbles and four blue marbles, the probability of getting a red marble is three out of seven

 

C.     Discrete Mathematics—Systematic Listing and Counting

 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).

 

D.    Discrete Mathematics—Vertex-Edge Graphs and Algorithms

 1.         Follow simple sets of directions (e.g., from one location to another, or from a recipe).

 2.         Color simple maps with a small number of colors.

 3.         Play simple two-person games (e.g., tic-tac-toe) and informally explore the idea of what the outcome should be.

 4.         Explore concrete models of vertex-edge graphs (e.g. vertices as “islands” and edges as “bridges”).

·        Paths from one vertex to another

 

 

Link to Standard 4.4 Grade 3

 

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