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 6, students will:

 

A.     Data Analysis

 1.         Collect, generate, organize, and display data.

·        Data generated from surveys

 2.         Read, interpret, select, construct, analyze, generate questions about, and draw inferences from displays of data.

·        Bar graph, line graph, circle graph, table, histogram

·        Range, median, and mean

·        Calculators and computers used to record and process information

 3.         Respond to questions about data, generate their own questions and hypotheses, and formulate strategies for answering their questions and testing their hypotheses.

 

B.     Probability

 1.         Determine probabilities of events.

·        Event, complementary event, probability of an event

·        Multiplication rule for probabilities

·        Probability of certain event is 1 and of impossible event is 0

·        Probabilities of event and complementary event add up to 1

 2.         Determine probability using intuitive, experimental, and theoretical methods (e.g., using model of picking items of different colors from a bag).

·        Given numbers of various types of items in a bag, what is the probability that an item of one type will be picked

·        Given data obtained experimentally, what is the likely distribution of items in the bag  

 3.         Explore compound events.

 4.         Model situations involving probability using simulations (with spinners, dice) and theoretical models.

 5.         Recognize and understand the connections among the concepts of independent outcomes, picking at random, and fairness.

 

C.     Discrete Mathematics—Systematic Listing and Counting

 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).

 

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.

 2.         Analyze vertex-edge graphs and tree diagrams.

·        Can a picture or a vertex-edge graph be drawn with a single line?  (degree of vertex)

·        Can you get from any vertex to any other vertex?  (connectedness)

 3.         Use vertex-edge graphs to find solutions to practical problems.

·        Delivery route that stops at specified sites but involves least travel

·        Shortest route from one site on a map to another

 

 

 

Link to Standard 4.4 Grade 5

 

Link to Standard 4.4 Grade 7

 

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