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 Instructional Focus DocumentMathematical Models with Applications
 TITLE : Unit 09: Probability SUGGESTED DURATION : 13 days

Unit Overview

This unit bundles student expectations that address the application of algebraic expressions and equations associated with linear and exponential relationships to describe mathematical patterns in probability. Concepts are incorporated into both mathematical and real-world problem situations. According to the Texas Education Agency, mathematical process standards including application, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace.

Prior to this unit, in Grade 7, students made predictions and determined theoretical and experimental probabilities in problem situations, including the introduction of the Fundamental Counting Principle. In Geometry, students studied combinations and permutations in contextual problems.

During this unit, students determine the number of ways an event may occur using combinations, permutations, and the Fundamental Counting Principle. Students perform experiements, demonstrating empirical probability, to predict the likelihood of an event occurring from the outcomes of the experiment. Students use the same events, calculating theoretical probability, to predict the likelihood of an event occurring using formulas and mathematical calculations without conducting an experiment. Students then compare the experimental outcomes to the calculated theoretical probability. After conducting multiple experiments, students realize as the number of trials in the experiment increases, the experimental probability of an event approaches the theoretical probability of the same event, known as the Law of Large Numbers. Students define binomial distribution and use the formula, with and without technology, to generate probabilities of various events. They perform various experiments using binomial distribution and compare the theoretical model to the experimental results to determine the reasonableness of the theoretical model. Students define geometric distribution and use the formula, with and without technology, to generate probabilities of various events. Students use the results of experiments and theoretical models to make general comparisons of theoretical to empirical probability.

After this unit, students will create a data collection and research project in which they will formulate a meaningful question, determine the data needed to answer the question, gather the appropriate data, analyze the data, and draw reasonable conclusions. Students will present their methods used, analyses conducted, and conclusions drawn through the use of one or more of the following: a written report, a visual display, an oral report, or a multi-media presentation.

This unit is supporting the development of the Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning B1; II. Algebraic Reasoning B1; V. Probabilistic Reasoning A1, B1, B2; VIII. Problem Solving and Reasoning; IX. Communication and Representation; X. Connections.

According to the Connections Standard for Grades 9-12 from the National Council of Teachers of Mathematics (NCTM), “Instructional programs from pre-kindergarten through grade 12 should enable students to:

• recognize and use connections among mathematical ideas;
• understand how mathematical ideas interconnect and build on one another to produce a coherent whole;
• recognize and apply mathematics in contexts outside of mathematics.

When students can see the connections across different mathematical content areas, they develop a view of mathematics as an integrated whole. As they build on their previous mathematical understandings while learning new concepts, students become increasingly aware of the connections among various mathematical topics. As students' knowledge of mathematics, their ability to use a wide range of mathematical representations, and their access to sophisticated technology and software increase, the connections they make with other academic disciplines, especially the sciences and social sciences, give them greater mathematical power” (NCTM, 2000, p. 354).

Education Policy Improvement Center (2009), Texas College and Career Standards, Austin, TX, University of Texas Printing.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics: Connections standard for grades 9-12. Reston, VA: National Council of Teachers of Mathematics, Inc.

OVERARCHING UNDERSTANDINGS and QUESTIONS

The probability or chance of an event occurring can be determined by various methods, interpreted for reliability, and used to make predictions and inferences in problem situations.

• Why is it important to understand and use probability?
• How do theoretical and empirical probability compare?
• Why is it important to understand and determine total possible outcomes?
• What methods can be used to determine probability?
• How can probability be represented?
• How is the probability of an event(s) used to make predictions and inferences in problem situations?
Performance Assessment(s) Overarching Concepts
Unit Concepts
Unit Understandings
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

Probabilistic Reasoning

• Conclusions/Predictions
• Events
• Permutations/Combinations
• Sample Spaces

Associated Mathematical Processes

• Application
• Tools and Techniques
• Problem Solving Model
• Communication
• Representations
• Relationships
• Justification

The number of ways an event may occur can be determined using combinations, permutations, and the Fundamental Counting Principle.

• How can combinations be used to determine the number of ways an event may occur?
• How can permutations be used to determine the number of ways an event may occur?
• How can it be determined whether to use combinations or permutations?
• When can the Fundamental Counting Principle be applied?
• How can the Fundamental Counting Principle be used to determine the number of ways an event may occur?
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

Algebraic Reasoning

• Ratios

Probabilistic Reasoning

• Conclusions/Predictions
• Events
• Probability of an Event
• Simulations
• Theoretical/Empirical Probability

Associated Mathematical Processes

• Application
• Tools and Techniques
• Problem Solving Model
• Communication
• Representations
• Relationships
• Justification

Probability is the likelihood of an event occurring from the total possible outcomes, and probability can be applied to solve mathematical and real-world problem situations.

• How can total possible outcomes or sample space be determined?
• How is probability of an event determined?
• How is probability of an event represented?
• How can probability based on area be used to solve contextual problems?
• How do with replacement or without replacement factor into determining if two events are independent or dependent?
• How is the probability of independent and dependent events applied in contextual problems?

Theoretical probability of an event occurring is calculated using formulas and empirical probability of an event occurring is determined by analyzing data collected experimentally.

• What is the purpose of determining the probability of the occurrence of an event?
• How do theoretical probability and empirical probability compare?
• What conditions might necessitate the use of theoretical probability?
• What conditions might necessitate the use of empirical probability?
• Why might the theoretical probability and empirical probability of a particular event differ?
• How does the number of trials in an experiment affect the validity of the empirical probability?
• How can a theoretical probability be determined from an area model?
• How can an empirical probability be simulated using an area model?

Experiments and simulations can be used to test the reasonableness of theoretical probability models.

• What types of theoretical probability models can be tested using experiments and simulations?
• How are experiments used to test the reasonableness of theoretical probability models?
• How are simulations used to test the reasonableness of theoretical probability models?
• What is meant by a binomial probability theoretical model?
• How is an experiment designed to determine the reasonableness of a binomial theoretical model for a particular situation?
• What is meant by a geometric area probability model?
• How is an experiment designed to determine the reasonableness of a geometric theoretical model for a particular situation?

MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may think they are to use the same ratio of events rather than changing the ratio when one is drawn or taken out and not replaced.

Underdeveloped Concepts:

• Some students may misunderstand the multiplication of ratios and want to add instead.

Unit Vocabulary

• Binomial probability – probability that an experiment with n trials results in exactly r successes, when the probability of success of each trial is p.
• Binomial probability formula: nCr  pr  q(nr), where n is the number of trials in the experiment, r is the number of successes, p is the probability of success on a trial, and q is the probability of failure on a trial.
• Combinations – number of different ways a set of objects can be selected without regard to a specific order
• Empirical (experimental) probability – probability of an event occurring predicted by the results of an experiment
• Formula for combinations: nCr = , where n is the total number of objects in the set and r is the number to be chosen.
• Formula for permutations: nPr, where n is the total number of objects in the set and r  is the number to be chosen.
• Geometric probability – probability that involves the comparison of geometric measures such as length or area.
• Geometric probability formula, which is the same as for finding the simple probability of an event. However, the outcomes will be geometric measurements.
• Law of large numbers – as the number of trials in an experiment increases, the experimental probability of an event approaches the theoretical probability of the same event, meaning the difference between the experimental and theoretical probability will be closer to zero
• Permutations – number of different ways a set of objects can be selected with regard to a specific order
• Theoretical probability – the likelihood of an event occurring predicted by using formulas and mathematical calculations without conducting an experiment

Related Vocabulary:

 Binomial model Combination Empirical Event Geometric model Outcome Permutation Probability Success Theoretical Trial
Unit Assessment Items System Resources Other Resources

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Unit Assessment Items that have been published by your district may be accessed through Search All Components in the District Resources tab. Assessment items may also be found using the Assessment Creator if your district has granted access to that tool.

System Resources may be accessed through Search All Components in the District Resources Tab.

Texas Higher Education Coordinating Board – Texas College and Career Readiness Standards (select CCRS from Standard Set dropdown menu)

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – STAAR Mathematics Resources

Texas Education Agency Texas Gateway – Revised Mathematics TEKS: Vertical Alignment Charts

Texas Education Agency Texas Gateway – Mathematics TEKS: Supporting Information

Texas Education Agency Texas Gateway – Interactive Mathematics Glossary

Texas Education Agency Texas Gateway – Resources Aligned to Mathematical Models with Applications Mathematics TEKS

Texas Instruments – Graphing Calculator Tutorials

TEKS# SE# TEKS Unit Level Specificity

• Bold black text in italics: Knowledge and Skills Statement (TEKS)
• Bold black text: Student Expectation (TEKS)
• Strike-through: Indicates portions of the Student Expectation that are not included in this unit but are taught in previous or future unit(s)
• Blue text: Supporting information / Clarifications from TCMPC (Specificity)
• Blue text in italics: Unit-specific clarification
• Black text: Texas Education Agency (TEA); Texas College and Career Readiness Standards (TxCCRS)
M.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
M.1A Apply mathematics to problems arising in everyday life, society, and the workplace.

Apply

MATHEMATICS TO PROBLEMS ARISING IN EVERYDAY LIFE, SOCIETY, AND THE WORKPLACE
Including, but not limited to:

• Mathematical problem situations within and between disciplines
• Everyday life
• Society
• Workplace

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• X. Connections
M.1B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.

Use

A PROBLEM-SOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEM-SOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION
Including, but not limited to:

• Problem-solving model
• Analyze given information
• Formulate a plan or strategy
• Determine a solution
• Justify the solution
• Evaluate the problem-solving process and the reasonableness of the solution

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VIII. Problem Solving and Reasoning
M.1C Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.

Select

TOOLS, INCLUDING REAL OBJECTS, MANIPULATIVES, PAPER AND PENCIL, AND TECHNOLOGY AS APPROPRIATE, AND TECHNIQUES, INCLUDING MENTAL MATH, ESTIMATION, AND NUMBER SENSE AS APPROPRIATE, TO SOLVE PROBLEMS
Including, but not limited to:

• Appropriate selection of tool(s) and techniques to apply in order to solve problems
• Tools
• Real objects
• Manipulatives
• Paper and pencil
• Technology
• Techniques
• Mental math
• Estimation
• Number sense

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VIII. Problem Solving and Reasoning
M.1D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

Communicate

MATHEMATICAL IDEAS, REASONING, AND THEIR IMPLICATIONS USING MULTIPLE REPRESENTATIONS, INCLUDING SYMBOLS, DIAGRAMS, GRAPHS, AND LANGUAGE AS APPROPRIATE
Including, but not limited to:

• Mathematical ideas, reasoning, and their implications
• Multiple representations, as appropriate
• Symbols
• Diagrams
• Graphs
• Language

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• IX. Communication and Representation
M.1E Create and use representations to organize, record, and communicate mathematical ideas.

Create, Use

REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS
Including, but not limited to:

• Representations of mathematical ideas
• Organize
• Record
• Communicate
• Evaluation of the effectiveness of representations to ensure clarity of mathematical ideas being communicated
• Appropriate mathematical vocabulary and phrasing when communicating mathematical ideas

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• IX. Communication and Representation
M.1F Analyze mathematical relationships to connect and communicate mathematical ideas.

Analyze

MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS
Including, but not limited to:

• Mathematical relationships
• Connect and communicate mathematical ideas
• Conjectures and generalizations from sets of examples and non-examples, patterns, etc.
• Current knowledge to new learning

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• X. Connections
M.1G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

Display, Explain, Justify

MATHEMATICAL IDEAS AND ARGUMENTS USING PRECISE MATHEMATICAL LANGUAGE IN WRITTEN OR ORAL COMMUNICATION
Including, but not limited to:

• Mathematical ideas and arguments
• Validation of conclusions
• Displays to make work visible to others
• Diagrams, visual aids, written work, etc.
• Explanations and justifications
• Precise mathematical language in written or oral communication

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• IX. Communication and Representation
M.8 Mathematical modeling in social sciences. The student applies mathematical processes to determine the number of elements in a finite sample space and compute the probability of an event. The student is expected to:
M.8A Determine the number of ways an event may occur using combinations, permutations, and the Fundamental Counting Principle.

Determine

THE NUMBER OF WAYS AN EVENT MAY OCCUR USING COMBINATIONS, PERMUTATIONS, AND THE FUNDAMENTAL COUNTING PRINCIPLE

Including, but not limited to:

• Event – a probable situation or condition
• Combinations – number of different ways a set of objects can be selected without regard to a specific order
• Formula for combinations: nCr, where n is the total number of objects in the set and r is the number to be chosen.
• Graphing calculator technology to determine combinations
• Permutations – number of different ways a set of objects can be selected with regard to a specific order
• Formula for permutations: nPr = , where n is the total number of objects in the set and r  is the number to be chosen.
• Graphing calculator technology to determine permutations
• Fundamental Counting Principle – if one event has a possible outcomes and a second independent event has b possible outcomes, then there are a b total possible outcomes for the two events together

Note(s):

• Grade 7 introduced the Fundamental Counting Principle.
• Geometry studied combinations and permutations in contextual problems.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• I. Numeric Reasoning
• B1 – Perform computations with real and complex numbers.
• II. Algebraic Reasoning
• B1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions.
• V. Probabilistic Reasoning
• A1 – Determine the nature and the number of elements in a finite sample space
• B1 – Compute and interpret the probability of an event and its complement
• B2 – Compute and interpret the probability of conditional and compound events
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
M.8B Compare theoretical to empirical probability.

Compare

THEORETICAL TO EMPIRICAL PROBABILITY

Including, but not limited to:

• Empirical (experimental) probability – the likelihood of an event occurring from the outcomes of an experiment
• Theoretical probability – the likelihood of an event occurring predicted by using formulas and mathematical calculations without conducting an experiment

Note(s):

• Grade 7 made predictions and determined theoretical and experimental probabilities in problem situations.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• I. Numeric Reasoning
• B1 – Perform computations with real and complex numbers.
• II. Algebraic Reasoning
• B1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions.
• V. Probabilistic Reasoning
• A1 – Determine the nature and the number of elements in a finite sample space
• B1 – Compute and interpret the probability of an event and its complement
• B2 – Compute and interpret the probability of conditional and compound events
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
M.8C Use experiments to determine the reasonableness of a theoretical model such as binomial or geometric.

Use

EXPERIMENTS OF A THEORETICAL MODEL SUCHAS AS BINOMIAL OR GEOMETRIC

Including, but not limited to:

• Binomial probability – probability that an experiment with n trials results in exactly r successes, when the probability of success of each trial is p
• Binomial probability formula: nCr • pr q(n-r), where n is the number of trials in the experiment, r is the number of successes, p is the probability of success on a trial, and q is the probability of failure on a trial.
• Geometric probability – probability that involves the comparison of geometric measures such as length or area
• Geometric probability formula: , which is the same as for finding the simple probability of an event. However, the outcomes will be geometric measurements.

To Determine

THE REASONABLENESS OF A THEORETICAL MODEL SUCH AS BINOMIAL OR GEOMETRIC

Including, but not limited to:

• Application of formulas
• Law of large numbers – as the number of trials in an experiment increases, the experimental probability of an event approaches the theoretical probability of the same event, meaning the difference between the experimental and theoretical probability will be closer to zero

Note(s):

• Grade 7 made predictions and determined theoretical and experimental probabilities in problem situations.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• I. Numeric Reasoning
• B1 – Perform computations with real and complex numbers.
• II. Algebraic Reasoning
• B1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions.
• V. Probabilistic Reasoning
• A1 – Determine the nature and the number of elements in a finite sample space
• B1 – Compute and interpret the probability of an event and its complement
• B2 – Compute and interpret the probability of conditional and compound events
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections