|
|
|
|
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 12, students will:
A. Data Analysis 1. Use surveys and sampling techniques to generate data and draw conclusions about large groups. · Advantages/disadvantages of sample selection methods (e.g., convenience sampling, responses to survey, random sampling) 2. Evaluate the use of data in real-world contexts. · Accuracy and reasonableness of conclusions drawn · Bias in conclusions drawn (e.g., influence of how data is displayed) · Statistical claims based on sampling 3. Design a statistical experiment, conduct the experiment, and interpret and communicate the outcome. 4. Estimate or determine lines of best fit (or curves of best fit if appropriate) with technology, and use them to interpolate within the range of the data. 5. Analyze data using technology, and use statistical terminology to describe conclusions. · Measures of dispersion: variance, standard deviation, outliers
B. Probability 1. Calculate the expected value of a probability-based game, given the probabilities and payoffs of the various outcomes, and determine whether the game is fair. 2. Use concepts and formulas of area to calculate geometric probabilities. 4. Determine probabilities in complex situations. · Dependent and independent events 5. Estimate probabilities and make predictions based on experimental and theoretical probabilities. C. Discrete Mathematics—Systematic Listing and Counting 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.
D. Discrete Mathematics—Vertex-Edge Graphs and Algorithms 1. Use vertex-edge graphs and algorithmic thinking to 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) · Determining how many Student Council representatives each class (9th, 10th, 11th, and 12th grade) gets when the classes have unequal sizes (apportionment)
|
|
