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, record, and organize data in response to questions, claims, or curiosity.

·        Data collected from students’ everyday experiences

·        Data generated from chance devices, such as spinners and dice

2.         Read, interpret, construct, and analyze displays of data.

·        Pictures, tally chart, pictograph, bar graph, Venn diagram

·        Smallest to largest, most frequent (mode)

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

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)

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

·        Range, median, and mean

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.

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

Sample ECR Item: Janet recorded the number of math problems she did for homework each night for twelve days. Her data is shown below:

• Draw a box-and-whisker plot to represent Janet's data.
• E xplain what your box-and-whisker plot shows about the data.

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

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.

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

5.         Analyze data using technology, and use statistical terminology to describe conclusions.

·        Measures of dispersion:  variance, standard deviation, outliers

·        Correlation coefficient

·        Normal distribution (e.g., approximately 95% of the sample lies between two standard deviations on either side of the mean)

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 chance devices like spinners and dice to explore concepts of probability.

·        Certain, impossible

·        More likely, less likely, equally likely

2.         Provide probability of specific outcomes.

·        Probability of getting specific outcome when coin is tossed, when die is rolled, when spinner is spun (e.g., if spinner has five equal sectors, then probability of getting a particular sector is one out of five)

·        When picking a marble from a bag with three red marbles and four blue marbles, the probability of getting a red marble is three out of seven

 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

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

1.         Determine probabilities of events.

·        Event, probability of an event

·     Probability of certain event is 1 and of impossible event is 0

 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

Sample MC Item: 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. 2/5

* c. 3/5

d. 1

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  

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

2.     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 SCR Item: If a computer randomly chooses a letter in the word “mathematics,” what is the probability that it chooses the letter “a”? (Answer: 2/11)

Sample MC Item: Four friends are playing a game with 4 different spinners. Carol has a spinner with 3 equal sections numbered 1 to 3. Maria has a spinner with 5 equal sections numbered 1 to 5. Linda has a spinner with 6 equal sections numbered 1 to 6. Julie has a spinner with 8 equal sections numbered 1 to 8. Everyone spins at the same time. The scoring is 10 points for an odd number and 5 points for an even number. Who has the best chance of getting the highest score?

* a. Carol

b. Maria

c. Linda

d. Julie

• "Discuss" here means "explain."

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.