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.

Big Idea Data Analysis: Reading, understanding, interpreting, and communicating data are critical in modeling a variety of real-world situations, drawing appropriate inferences, making informed decisions, and justifying those decisions.
Big Idea Probability: Probability quantifies the likelihood that something will happen and enables us to make predictions and informed decisions.
Big Idea Discrete Mathematics: Discrete mathematics consists of tools and strategies for representing, organizing, and interpreting non-continuous data.

4.4.6 A. Data Analysis

Descriptive Statement: 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.

Essential Questions

Enduring Understandings

- How can the collection, organization, interpretation, and display of data be used to answer questions? (4.5A4; 4.5A6; 4.5E1; 4.5E2; 4.5F1; 4.5F6)

- The message conveyed by the data depends on how the data is collected, represented, and summarized. (4.5A6; 4.5D6; 4.5E1; 4.5E2; 4.5E3)

- The results of a statistical investigation can be used to support or refute an argument. (4.5D1; 4.5D3; 4.5D5; 4.5E2; 4.5E3; 4.5F6)

Areas of Focus/Cumulative Progress Indicators

Comments and Examples

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.

4.4.6 B. Probability

Descriptive Statement: 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.

Essential Questions

Enduring Understandings

- How can experimental and theoretical probabilities be used to make predictions or draw conclusions? (4.5D5; 4.5D6)

- Experimental results tend to approach theoretical probabilities after a large number of trials.

Areas of Focus/Cumulative Progress Indicators

Comments and Examples

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  

4.4.6 C. Discrete Mathematics - Systematic Listing And Counting

Descriptive Statement: 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.

Essential Questions

Enduring Understandings

- How can attributes be used to classify data/objects?

- What is the best way to solve this? What counting strategy works best here?

- Grouping by attributes (classification) can be used to answer mathematical questions. (4.5E1; 4.5E3)

- Algorithms can effectively and efficiently be used to quantify and interpret discrete information.

Areas of Focus/Cumulative Progress Indicators

Comments and Examples

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)

Sample MC Item: At the Morse family reunion, everyone made ice cream sundaes for dessert. The following sundae "makings" were available:

How many different sundaes could be made using one choice from each column, if you start with ice cream, then add sauce, and then add a topping?

a. 6

b. 8

c. 12

* d. 18

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).

4.4.6 D. Discrete Mathematics - Vertex-Edge Graphs And Algorithms

Descriptive Statement: 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.

Essential Questions

Enduring Understandings

- How can visual tools such as networks (vertex-edge graphs) be used to answer questions? (4.5E1; 4.5E3)

- How can algorithmic thinking be used to solve problems?

- Optimization is finding the best solution within given constraints.

- Algorithms can effectively and efficiently be used to quantify and interpret discrete information.

Areas of Focus/Cumulative Progress Indicators

Comments and Examples

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