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.

4.4.8 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.         Select and use appropriate representations for sets of data, and measures of central tendency (mean, median, and mode).

·        Type of display most appropriate for given data

·        Box-and-whisker plot, upper quartile, lower quartile

·        Scatter plot

·        Calculators and computer used to record and process information

·        Finding the median and mean (weighted average) using frequency data.

·        Effect of additional data on measures of central tendency

Sample MC Item:

The table above shows the ages of the students in Elaine’s class. To the nearest tenth of a year, what is the mean of the 30 students’ ages?

a. 13.0

b. 13.4

* c. 13.8

d. 14.0

 2.         Make inferences and formulate and evaluate arguments based on displays and analysis of data sets.

4.4.8 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

Assessment of this CPI is generally within the context of one or more of the other content CPIs.

Sample MC Item: Jeremy has a fair coin and a number cube with the sides labeled one through six. What is the probability of getting both a head on a toss of the coin and a four on a roll of the number cube?

a. 2/3

b. 1/2

c. 1/3

* d. 1/12

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

·        Frequency, relative frequency

Instructional Focus:
• This CPI is largely an instructional CPI. Assessment of this CPI is generally within the context of one or more of the other content CPIs.

Sample Classroom Performance Task: Design a spinner that has the following probabilities: P(red) = 3/8 P(blue) = 25 % P(yellow) = 12 ½ % P(white) = remaining section Design means to draw your spinner and label each section with its appropriate color and probability. • Is this a fair spinner? Why or why not? Explain your reasoning. • Devise a fair game using this spinner. Describe your game.

Instructional/Assessment Focus:
• Assessment will focus on analysis of the probabilities, more than the playing of the games. The actual playing of games may appropriately receive additional attention during instruction.

"Discuss" here means "explain."

4.4.8 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.         Apply the multiplication principle of counting.

·        Permutations:  ordered situations with replacement (e.g., number of possible license plates) vs. ordered situations without replacement (e.g., number of possible slates of 3 class officers from a 23 student class)

·        Factorial notation

·        Concept of combinations (e.g., number of possible delegations of 3 out of 23 students)

3.         Apply techniques of systematic listing, counting, and reasoning in a variety of different contexts.

4.4.8 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

1.         Use vertex-edge graphs and algorithmic thinking to represent and find solutions to practical problems.

·        Finding the shortest network connecting specified sites

·        Finding a minimal route that includes every street (e.g., for trash pick-up)

·        Finding the shortest route on a map from one site to another

·        Finding the shortest circuit on a map that makes a tour of specified sites

·        Limitations of computers (e.g., the number of routes for a delivery truck visiting n sites is n!, so finding the shortest circuit by examining all circuits would overwhelm the capacity of any computer, now or in the future, even if n is less than 100)