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 Instructional Focus DocumentGeometry
 TITLE : Unit 10: Probability SUGGESTED DURATION : 12 days

#### Unit Overview

This unit bundles student expectations that address permutations, combinations, and various types of probability. Concepts are incorporated into both non-contextual 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 determined theoretical and experimental probabilities for simple and compound events. Students also described and found the complement of a probability.

During this unit, students define and develop formulas for permutations and combinations and apply permutations and combinations to solve contextual problems. Students determine number of possible outcomes of an event, including combinations, permutations, and the Fundamental Counting Principle. Students investigate and define probability in terms of possible outcomes and desired event. Students determine the theoretical and experimental probability based on area models in problem situations. Students identify events as being independent or dependent, and connect independence and dependence to with and without replacement. Students apply the concept of probabilities of independent and dependent events to solve contextual problems. Students are introduced to conditional probability, including notation, and apply conditional probability in contextual problems.

After this unit, the concepts of permutations, combinations, and probability will be applied in subsequent mathematics courses and real world situations.

This unit is supporting the development of Texas College Career Readiness Standards (TxCCRS): III. Geometric Reasoning C2; VIII. Problem Solving and Reasoning; IX. Communication and Representation; X. Connections.

According to Navigating through Probability in Grades 9 – 12 (2004), from the National Council of Teachers of Mathematics (NCTM), “Probability is an area of the school mathematics curriculum that high school courses often glossed over or sometimes even skip. However, the importance of probabilistic reasoning – particularly, reasoning about the relationship between data and probability – has received increased attention from national groups over the past several decades” (p. 1). Additionally, “The Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989) accorded to probability and statistics the same status in the K – 12 mathematics curriculum as algebra, geometry, measurement, and number and operations. Eleven years later, Principals and Standards for School Mathematics (NCTM, 2000), reaffirmed this assessment of the importance of data analysis and probability in the Pre-K – 12 mathematics curriculum” (as cited in NCTM, 2004, p. 1).

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics. (2004). Navigating through probability in 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 are the different types of probability distinguished?
• What methods can be used to determine probability?
• How is the probability of an event(s) used to make predictions and inferences in problem situations?

Combinatorics is a branch of discrete mathematics involving enumeration, combination, and permutation of sets of numbers and their application in problem situations.

• How can combinatorics be used to count the number of structures possible for different types and sizes of sets?
• How are combinatorics applied in probability?
• How can combinatorics be used to solve 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
• Independent/Dependent Events
• Outcomes
• Permutations/Combinations
• Probability of an Event
• Sample Space
• Simulations
• Conditional/Theoretical/Empirical Probability

Associated Mathematical Processes

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

Combinations and permutations are used to determine the number of total possible outcomes for a set and are applied to solve problem situations, including probability problems.

• How are combinations used to determine total possible outcomes for a set?
• What formula is used to model combinations?
• How are permutations used to determine total possible outcomes for a set?
• What formula is used to model permutations?
• How are combinations and permutations different?
• Why are combinations and permutations used in determining probability?
• How can combinations and permutations be used to solve contextual problems?

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 can two events be identified as independent?
• How is the probability of independent events applied in contextual problems?
• How do with replacement or without replacement factor into determining if two events are independent or dependent?
• How can conditional probabilities be identified?
• How is the probability of conditional events applied in contextual problems?

Theoretical probability is an expected probability calculated using formulas, whereas experimental probability is found by experimentation and/or data collection.

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

#### MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may confuse when to use permutations and when to use combinations to determine outcomes.
• Some students may not realize that an event is a subset of the total possible outcomes.
• Some students may confuse independent events and dependent events.
• Some students may think that if objects are not replaced in the set it does not impact the probability instead of realizing that the total outcomes for the next probability will be decreased.

#### Unit Vocabulary

• Combinations – number of different ways a set of objects can be selected without regard to a specific order
• Complement of an event – the probability of the non-occurrence of a desired outcome; the sum of the probability of the event and the probability of the non-occurrence of the event are equal to one
• Compound events – a set of outcomes from a combination of actions or activities where the outcomes can be subdivided (e.g., flipping a coin and rolling a number cube, drawing tiles out of a bag and spinning a spinner, etc.)
• Conditional probability – for two events, A and B, the probability of B given that A has already occurred; written as P(B|A); read as “the probability of B, given A”
• Dependent events – the outcome from one action or activity may affect the probability of the outcome(s) of any subsequent action(s) or activity(s); usually involves compound events
• Experimental probability – the likelihood of an event occurring from the outcomes of an experiment
• Independent events – the outcome from one action or activity does not affect the probability of the outcome(s) of any subsequent action(s) or activity(s); usually involves compound events
• Independent probability – if P(B|A) = P(B), then A and B are said to be independent
• Permutations – number of different ways a set of objects can be selected with regard to a specific order
• Probability – a ratio between the number of desired outcomes to the total possible outcomes, 0 ≤ p ≤ 1
• Sample space – a set of all possible outcomes of one or more events
• Theoretical probability – the likelihood of an event occurring predicted by using formulas and mathematical calculations without conducting an experiment

Related Vocabulary:

 Equally likely Event Experiment Factorial Favorable outcomes Outcome Tree diagram
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 Geometry 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)
G.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
G.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
G.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
G.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
G.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
G.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
G.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
G.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
G.13 Probability. The student uses the process skills to understand probability in real-world situations and how to apply independence and dependence of events. The student is expected to:
G.13A Develop strategies to use permutations and combinations to solve contextual problems.

Develop

STRATEGIES TO USE PERMUTATIONS AND COMBINATIONS

Including, but not limited to:

• Permutations – number of different ways a set of objects can be selected with regard to a specific order
• Formula for permutations: , where n is the total number of objects in the set and r is the number to be chosen.
• Combinations – number of different ways a set of objects can be selected without regard to a specific order
• Formula for combinations: , where n is the total number of objects in the set and r is the number to be chosen.
• Strategies to determine permutations and combinations
• Diagram
• Lists
• Formulas
• Graphing calculator technology

To Solve

CONTEXTUAL PROBLEMS

Including, but not limited to:

• Application of permutations to determine the number of ways an event can occur in contextual problem situations
• Application of combinations to determine the number of ways an event can occur in contextual problem situations

Note(s):

• Grade 7 introduced simple and compound events and applications of probability.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• III. Geometric Reasoning
• C2 – Make connections between geometry, statistics, and probability.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
G.13B Determine probabilities based on area to solve contextual problems.

Determine

PROBABILITIES BASED ON AREA TO SOLVE CONTEXTUAL PROBLEMS

Including, but not limited to:

• Probability – a ratio between the number of desired outcomes to the total possible outcomes, 0 ≤ p ≤ 1
• Sample space – a set of all possible outcomes of one or more events
• 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
• Complement of an event – the probability of the non-occurrence of a desired outcome; the sum of the probability of the event and the probability of the non-occurrence of the event are equal to one
• Representation of probability with an area model
• Representation of real-world problem situations with area models
• Representation of data sets with an area model (circle graph)
• Determination of probability from an area model
• Hands-on and technology to model and simulate events

Note(s):

• Grade 7 introduced simple and compound events and applications of probability.
• Geometry extends the concept of probabilities based on area.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• III. Geometric Reasoning
• C2 – Make connections between geometry, statistics, and probability.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
G.13C Identify whether two events are independent and compute the probability of the two events occurring together with or without replacement.

Identify

WHETHER TWO EVENTS ARE INDEPENDENT

Including, but not limited to:

• Compound events – a set of outcomes from a combination of actions or activities where the outcomes can be subdivided (e.g., flipping a coin and rolling a number cube, drawing tiles out of a bag and spinning a spinner, etc.)
• Independent events – the outcome from one action or activity does not affect the probability of the outcome(s) of any subsequent action(s) or activity(s); usually involves compound events(s)
• Dependent events – the outcome from one action or activity may affect the probability of the outcome(s) of any subsequent action(s) or activity(s); usually involves compound events

Compute

THE PROBABILITY OF THE TWO EVENTS OCCURRING TOGETHER WITH OR WITHOUT REPLACEMENT

Including, but not limited to:

• Probability – a ratio between the number of desired outcomes to the total possible outcomes, 0 ≤ p ≤ 1
• Compound independent events with replacement
• Compound dependent events without replacement

Note(s):

• Grade 7 introduced simple and compound events and applications of probability.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• III. Geometric Reasoning
• C2 – Make connections between geometry, statistics, and probability.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
G.13D Apply conditional probability in contextual problems.

Apply

CONDITIONAL PROBABILITY IN CONTEXTUAL PROBLEMS

Including, but not limited to:

• Conditional probability – for two events, A and B, the probability of B given that A has already occurred; written as P(B|A); read as “the probability of B, given A”
• Application of conditional probability in contextual problems

Note(s):

• Grade 7 introduced simple and compound events and applications of probability.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• III. Geometric Reasoning
• C2 – Make connections between geometry, statistics, and probability.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
G.13E Apply independence in contextual problems.

Apply

INDEPENDENCE IN CONTEXTUAL PROBLEMS

Including, but not limited to:

• Application of independence in contextual problems.
• Independent events in terms of conditional probability
• Conditional probability – for two events, A and B, the probability of B given that A has already occurred; written as P(B|A); read as “the probability of B, given A”
• Independent probability – if P(B|A) = P(B), then A and B are said to be independent
• Distinguishing between events that are independent and dependent using contextual problems involving conditional probability
• Computing compound probabilities based on conditional events, using formulas such as P(A and B) = P(A) • P(B|A).

Note(s):