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

 

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

 

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, generate questions about, and draw inferences from displays of data.

·        Pictograph, bar graph, line plot, line graph, table

·        Average (mean), most frequent (mode), middle term (median)

 

B.     Probability

 1.         Use everyday events and chance devices, such as dice, coins, and unevenly divided spinners, to explore concepts of probability.

·        Likely, unlikely, certain, impossible, improbable, fair, unfair

·        More likely, less likely, equally likely

·        Probability of tossing “heads” does not depend on outcomes of previous tosses

 2.         Determine probabilities of simple events based on equally likely outcomes and express them as fractions.

 3.         Predict probabilities in a variety of situations (e.g., given the number of items of each color in a bag, what is the probability that an item picked will have a particular color).

·        What students think will happen (intuitive)

·        Collect data and use that data to predict the probability (experimental)

·        Analyze all possible outcomes to find the probability (theoretical)

 

C.     Discrete Mathematics—Systematic Listing and Counting

 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)

 

D.    Discrete Mathematics—Vertex-Edge Graphs and Algorithms

 1.         Follow, devise, and describe practical sets of directions (e.g., to add two 2-digit numbers).

 2.         Play two-person games and devise strategies for winning the games (e.g., “make 5" where players alternately add 1 or 2 and the person who reaches 5, or another designated number, is the winner).

 3.         Explore vertex-edge graphs and tree diagrams.

·        Vertex, edge, neighboring/adjacent, number of neighbors

·        Path, circuit (i.e., path that ends at its starting point)

 4.         Find the smallest number of colors needed to color a map or a graph.

 

Link to Standard 4.4 Grade 3

 

Link to Standard 4.4 Grade 5

 

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