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 A. Data Analysis

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 in response to questions, claims, or curiosity.

·        Data collected from the classroom environment

Instructional/Assessment Focus:
• The actual collection of data would be more a part of classroom instruction or performance assessment, rather than a part of statewide assessment.
• Assessment of this CPI is frequently within the context of CPI 4.4.3A2.

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

·        Pictograph, bar graph, table

4.4 B. Probability

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

·        Likely, unlikely, certain, impossible

·        More likely, less likely, equally likely

2.         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)

Sample Assessment Item:
• MC: Orlando has a bag of 10 marbles that contains 4 red marbles and 6 blue marbles. If Orlando reached into the bag without looking and picked one marble, what is the probability that he would pick a blue marble?
a. 1 out of 10
b. 4 out of 10
* c. 6 out of 10
d. 10 out of 10

4.4 C. Discrete Mathematics - Systematic Listing And Counting

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.      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

Sample Assessment Item:
MC: Roseanne has 3 sweatshirts: a grey one, a green one, and a red one. She also has 2 pairs of jeans: a blue pair and a black pair. If an outfit consists of one sweatshirt and one pair of jeans, how many different outfits can Roseanne make?
a. 8

* b. 6

c. 5

d. 3

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

2.         Explore vertex-edge graphs

·        Vertex, edge

·        Path

Instructional Focus:
• This content should be introduced at this grade level, but mastery of the content is not assessed in statewide assessment at this grade level.

3.         Find the smallest number of colors needed to color a map