MIAMI-DADE COUNTY PUBLIC SCHOOLS District …mrcobb.net/wp-content/uploads/2015/01/Topic-10-Probability.pdfMIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide ... Prentice Hall

MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide

Algebra II Course Code: 12003301

Office of Academics and Transformation Page 1 of 9 Topic X_Fourth Nine Weeks

Conceptual Category: S: Statistics and Probability Topic X: Probability

COMMON CORE STATE STANDARD(S) & MATHEMATICAL PRACTICE (MP)

ESSENTIAL CONTENT OBJECTIVES

MACC.912.S-CP.1.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”). (MP.1,MP.2, MP.4, MP.6, MP.7) MACC.912.S-CP.1.2: Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. (MP.1,MP.2,MP.3, MP.4, MP.6, MP.7) MACC.912.S-CP.1.3: Understand the conditional probability of A given B as P (A and B)/P (B), and interpret independence of A and B as saying that the conditional probability A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. (MP.1,MP.2, MP.4, MP.6, MP.7) MACC.912.S-CP.1.4: Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect: data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. (MP.1,MP.2,MP.3, MP.4, MP.5, MP.6, MP.7, MP.8) MACC.912.S-CP.2.6: Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. (MP.1, MP.4, MP.5, MP.7) MACC.912.S-CP.2.7: Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model. (MP.1, MP.4, MP.5, MP.6, MP.7)

A. Independence and Conditional Probability.

B. Probability of Compound Events.

C. Evaluating Outcomes of Decisions.

To describe probability relationships by creating a

probability distribution

To interpret data on a probability distribution in

conducting a simulation

To determine conditional probabilities

To interpret data and relationships by using formulas and

tree diagrams

To calculate measures of central tendency

To draw, analyze, and interpret box-and-whisker plots

To calculate the standard deviation of a set of data

To use standard deviations to help interpret

representations of real-world situations

Pacing Date(s) Traditional 14 days 03/31/14 – 04/17/14

Block 7 days 03/31/14 – 04/17/14




INSTRUCTIONAL TOOLS

Core Text Book: Prentice Hall Algebra 2. Honors Gold Series Florida

Benchmark Suggested Lessons Teacher Notes

MACC.912.S-CP.1.1

MACC.912.S-CP.1.2

MACC.912.S-CP.1.3

MACC.912.S-CP.1.4

MACC.912.S-CP.2.6

MACC.912.S-CP.2.7

N/A

.

Online Sample Lessons:

Mathematics Assessment Project http://map.mathshell.org/materials/index.php Evaluating Statements About Probability

Modeling Conditional Probabilities 2

NCTM Illuminations http://illuminations.nctm.org/

Stick or Switch?

Vocabulary:

Frequency table, cumulative probability, probability distribution, conditional probability, measures of central tendency, mean, median, mode, bimodal, quartiles, box-and-whisker

plot, percentiles, outlier, measures of variation, range of a data set, interquartile range, standard deviation, z-score

Instructional Strategies:

The Standard for Mathematical Practice, precision is important for working with conditional probability. Attention to the definition of an event along with the writing and use of probability function notation are important requisites for communication of that precision. For example: Let A: Female and B: Survivor, then P(A|B) =. The use of a vertical line for the conditional “given” is not intuitive for students and they often confuse the events B|A and A|B. Moreover, they often find identifying a conditional difficult when the problem is expressed in words in which the word “given” is omitted. For example, find the probability that a female is a survivor. The standard Make sense of problems and persevere in solving them also should be employed so students can look for ways to construct conditional probability by formulating their own questions and working through them such as is suggested in standard 4 above. Students should learn to employ the use of Venn diagrams as a means of finding an entry into a solution to a conditional probability problem. It will take a lot of practice to master the vocabulary of “or,” “and,” “not” with the mathematical notation of union (∪), intersection ( ∩ ), and whatever notation is used for complement.

http://map.mathshell.org/materials/index.php

http://map.mathshell.org/materials/lessons.php?taskid=225

http://map.mathshell.org/materials/lessons.php?taskid=405&subpage=concept

http://illuminations.nctm.org/

http://illuminations.nctm.org/LessonDetail.aspx?ID=L377




INSTRUCTIONAL TOOLS

Instructional Strategies (Cont.):

The independence of two events is defined in Standard 2 using the intersection. It is far more intuitive to introduce the independence of two events in terms of conditional probability (stated in Standard 3), especially where calculations can be performed in two-way tables. Probabilities of conditional events are to be found using a two-way table wherever possible. Using a two-way table begins with calculation of marginal probabilities. Conditional probabilities and determination of independent events follow. However, tree diagrams may be a helpful tool for some students. The difficulty is realizing that the second set of branches is conditional probabilities. . Identifying that a probability is conditional when the word “given” is not stated can be very difficult for students. For example, if a balanced tetrahedron with faces 1, 2, 3, 4 is rolled twice, what is the probability that the sum is prime (A) of those that show a 3 on at least one roll (B)? Whether what is asked for is P(A and B), P(A or B), or P(A|B) can be problematic for students. Showing the outcomes in a Venn Diagram may be useful. The calculation to find the probability that the sum is prime (A) given at least one roll shows 3 (B) is to count the elements of B by listing them if possible, namely in this example, there are 7 paired outcomes (31, 32, 33, 34, 13, 23, 43). Of those 7 there are 4 whose sum is prime (32, 34, 23, 43). Hence in the long run, 4 out of 7 times of rolling a fair tetrahedron twice, the sum of the two rolls will be a prime number under the condition that at least one of its rolls shows the digit 3. Note that if listing outcomes is not possible, then counting the outcomes may require a computation technique involving permutations or combinations. In the above example, if the question asked were what is the probability that the sum of two rolls of a fair tetrahedron is prime (A) or at least one of the rolls is a 3 (B), then what is being asked for is P(A or B) which is denoted as P(A B) in set notation. Again, it is often useful to appeal to a Venn Diagram in which A consists of the pairs: 11, 12, 14, 21, 23, 32, 34, 41, 43; and B consists of 13, 23, 33, 43, 31, 32, 34. Adding P(A) and P(B) is a problem as there are duplicates in the two events, namely 23, 32, 34, and 43. So P(A or B) is 9/16 + 7/16 – 4/16 = 12/16 or 3/4, so 3/4th of the time, the result of rolling a fair tetrahedron twice will result in the sum being prime, or at least one of the rolls showing a 3, or perhaps both will occur.




COMMON CORE STATE STANDARDS

MATHEMATICAL PRACTICES

DESCRIPTION

MACC.K12.MP.1 (back to page 1)

Make sense of problems and persevere in solving them.

Mathematically proficient students will be able to:

Explain the meaning of a problem and looking for entry points to its solution.

Analyze givens, constraints, relationships, and goals.

Make conjectures about the form and meaning of the solution and plan a solution pathway.

Consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution.

Monitor and evaluate their progress and change course if necessary.

Explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends.

Check answers to problems using a different method, and continually ask, “Does this make sense?”

Identify correspondences between different approaches.


Reason abstractly and quantitatively.


Make sense of quantities and their relationships in problem situations.

Decontextualize—to abstract a given situation and represent it symbolically.

Contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols

Create a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them.

Know and be flexible using different properties of operations and objects.


Construct viable arguments and critique the reasoning of

others.


Understand and use stated assumptions, definitions, and previously established results in constructing arguments.

Make conjectures and build a logical progression of statements to explore the truth of their conjectures.

Analyze situations by breaking them into cases, and can recognize and use counterexamples.

Justify their conclusions, communicate them to others, and respond to the arguments of others.

Reason inductively about data, making plausible arguments that take into account the context from which the data arose.

Compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is.

Determine domains to which an argument applies.


Model with mathematics.


Apply the mathematics they know to solve problems arising in everyday life, society, and the workplace.

Use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another.

Apply what they know and feel comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later.

Identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas.

Analyze relationships mathematically to draw conclusions.

Interpret mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.




COMMON CORE STATE STANDARDS

MATHEMATICAL PRACTICES

DESCRIPTION


Use appropriate tools strategically.


Consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software.

Make sound decisions about when each of the tools appropriate for their grade or course might be helpful, recognizing both the insight to be gained and their limitations. Example: High school students analyze graphs of functions and solutions using a graphing calculator.

Detect possible errors by strategically using estimation and other mathematical knowledge.

Know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data.

Identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems.

Use technological tools to explore and deepen their understanding of concepts


Attend to precision.


Communicate precisely to others.

Use clear definitions in discussion with others and in their own reasoning.

State the meaning of the symbols they choose, including using the equal sign consistently and appropriately.

Be careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem.

Calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context.


Look for and make use of structure.


Discern a pattern or structure. Example: In the expression x2 + 9x + 14, students can see the 14 as 2 × 7 and the 9 as 2 + 7.

Recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. Step back for an overview and shift perspective.

See complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. Example: They can see 5 – 3(x – y)2 as

5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.


Look for and express regularity in repeated

reasoning.


Notice if calculations are repeated, and look both for general methods and for shortcuts. Example: Noticing the regularity in the way terms cancel when expanding (x-1)(x+1),(x-1)(x2+x+1),and(x-1)(x3 +x2+x+1)might lead them to the general formula for the sum of a geometric series.

Maintain oversight of the process, while attending to the details as they work to solve a problem.

Continually evaluate the reasonableness of their intermediate results.




Common Core State Standards for Mathematics

STANDARD CODE STANDARD DESCRIPTION

MACC.912.S-CP.1.1

(back to page 1)

Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections,

or complements of other events (“or,” “and,” “not”).

Cognitive Complexity: Level 1: Recall

Belongs to: Understand independence and conditional probability and use them to interpret data

Remarks/Examples:

(none)

MACC.912.S-CP.1.2

(back to page 1)

Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this

characterization to determine if they are independent.

Cognitive Complexity: Level 1: Recall


Remarks/Examples:

(none)

MACC.912.S-CP.1.3

(back to page 1)

Understand the conditional probability of A given B as P (A and B)/P (B), and interpret independence of A and B as saying that the conditional

probability A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.

Cognitive Complexity: Level 2: Basic Application of Skills & Concepts


Remarks/Examples:

(none)

http://www.cpalms.org/Standards/PublicPreviewIdea1557.aspx






Common Core State Standards for Mathematics

STANDARD CODE STANDARD DESCRIPTION

MACC.912.S-CP.1.4

(back to page 1)

Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table

as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect: data from a random sample of

students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your

school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.



Remarks/Examples:

(none)

MACC.912.S-CP.2.6

(back to page 1)

Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model.


Belongs to: Use the rules of probability to compute probabilities of compound events in a uniform probability model

Remarks/Examples:

(none)

MACC.912.S-CP.2.7

(back to page 1)

Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.


Belongs to: Use the rules of probability to compute probabilities of compound events in a uniform probability model

Remarks/Examples:

(none)







TECHNOLOGY TOOLS

GIZMO CORRELATION

GIZMO TITLE

TOPIC X DISCOVERY EDUCATION CORRELATION

VIDEO TITLE

Probabilities of Compound Events

Methods of Determining Probability

Joint Probability: Understanding the Odds

Conditional Probability: The Monty Hall Problem

MATH OVERVIEW

Algebra II: Dependent and Independent Probabilities

Algebra II: Basic Probability

MATH EXPLANATION TITLE

Algebra II: Compound Events: Probability of a Compound Event

Algebra II: Compound Events: Applications of Combined Probabilities

Algebra II: Dependent and Independent Probabilities: Probability of Independent Events

http://app.discoveryeducation.com/player/view/assetGuid/2C9FE8B3-D188-48DD-A61B-30AB0229EC1F

http://app.discoveryeducation.com/player/view/assetGuid/15293E55-5F8C-4922-8648-F70211A10FB3

http://app.discoveryeducation.com/player/view/assetGuid/8C319C8C-E5DC-4EF2-B218-5F4EB3C40596

http://app.discoveryeducation.com/player/view/assetGuid/A27ECAE0-C67B-455F-B141-15AF4912A49D

http://app.discoveryeducation.com/player/view/assetGuid/804F4B3A-71CA-438A-B157-41254B154F76

http://app.discoveryeducation.com/player/view/assetGuid/5E68590A-FD15-4AB7-831E-19CCF2D8FA20

http://app.discoveryeducation.com/player/view/assetGuid/D1D9E3EE-D07D-43ED-BF0E-2650E4C96EE8

http://app.discoveryeducation.com/player/view/assetGuid/4B6821D8-E0FE-44A0-8C43-FB406C035C2B

http://app.discoveryeducation.com/player/view/assetGuid/6B7291FF-998A-4999-8797-AC99AAB3BC00




Date Pacing Guide

Standards Data Driven Standard(s)

Activities Assessment(s) Strategies

03/31/14 – 04/17/14

T: 14 B: 7

MACC.912. S-CP.1.1

MACC.912. S-CP.1.2

MACC.912. S-CP.1.3

MACC.912. S-CP.1.4

MACC.912. S-CP.2.6

MACC.912. S-CP.2.7

Documents

MIAMI-DADE COUNTY PUBLIC SCHOOLS District …mrcobb.net/wp-content/uploads/2015/01/Topic-10-Probability.pdfMIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide ... Prentice Hall