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

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

By the end of Grade 4:

1.         Collect, generate, organize, and display data in response to questions, claims, or curiosity.

·        Data collected from the school environment

Instructional/Assessment Focus:
• Assessment will focus on organization and display of data, more than the collecting or generating of data. The actual gathering of data may appropriately receive additional attention during instruction.
• Assessment of this CPI is frequently within the context of CPI 4.4.4A2.

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

·        Pictograph, bar graph, line plot, line graph, table

·        Average (mean), most frequent (mode), middle term (median)

4.4 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

By the end of Grade 4:

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

·        Likely, unlikely, certain, impossible, improbable, fair, unfair

·        More likely, less likely, equally likely

·        Probability of tossing “heads” does not depend on outcomes of previous tosses 

Instructional/Assessment Focus:
• The exploration using dice, coins, and unevenly divided spinners is largely instructional, and generally assessed indirectly on statewide assessments.
• Familiarity with the concepts and vocabulary in the bullets is frequently assessed within the context of CPI 4.4.4B3.

2.         Determine probabilities of simple events based on equally likely outcomes and express them as fractions.

Instructional/Assessment Focus:
• Assessment of this CPI is generally within the context of CPI 4.4.4B3.

3.        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)

·        Analyze all possible outcomes to find the probability (theoretical)

Instructional/Assessment Focus:
• In fourth grade, students are expected to write probabilities as fractions, although they are not necessarily expected to reduce those fractions until fifth grade.

Sample Assessment Items:
• SCR: If there are seven marbles in a bag, three red and four green, what is the probability that a marble picked from the bag will be red? (Answer: 3/7 or 3 out of 7)
•MC: Cynthia has a bag of 10 marbles that contains 4 red marbles and 6 blue marbles. If Cynthia reached into the bag without looking and picked one marble, what is the probability that she would pick a blue marble?
a. 1/10

b. 4/10

* c. 6/10

d. 10/10

•MC: Joanne has a bag of marbles that contains 5 blue marbles, 4 red marbles, 2 white marbles, and 1 yellow marble. If Joanne wants to pick a marble out of the bag without looking, what is the probability that she will pick a red or yellow marble?

a. 1/12

b. 4/12

*c. 5/12

d. 7/12

4.4 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

By the end of Grade 4:

1.        Represent and classify data according to attributes, such as shape or color, and relationships.

·        Venn diagrams

·        Numerical and alphabetical order

Instructional/Assessment Focus:
This is an area of focus in grade 3 and may be assessed at a higher level of understanding in grade 4.

2.        Represent all possibilities for a simple counting situation in an organized way and draw conclusions from this representation.

·        Organized lists, charts, tree diagrams

·        Dividing into categories (e.g., to find the total number of rectangles in a grid, find the number of rectangles of each size and add the results)

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

By the end of Grade 4:

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

2.         Play two-person games and devise strategies for winning the games (e.g., “make 5" where players alternately add 1 or 2 and the person who reaches 5, or another designated number, is the winner).

Instructional/Assessment 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.

3.         Explore vertex-edge graphs and tree diagrams.

·        Vertex, edge, neighboring/adjacent, number of neighbors

·        Path, circuit (i.e., path that ends at its starting point)

Instructional/Assessment 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.

4.         Find the smallest number of colors needed to color a map or a graph.