Belleville Board of Education - … Belleville Board of Education Discrete Mathematics and...

Mathematics

Belleville Board of Education

Discrete Mathematics and Statistics

Belleville Board of Education

102 Passaic Avenue Belleville, NJ 07109

Submitted by: Annamaria Contella Charles Eccles Date: April 15, 2013

marissa.maldonado

Text Box

Board Approved April 29, 2013

Department – Philosophy (Mathematics Dept. Philosophy) Mathematical abilities enable one to think and reason logically, sequentially, and algorithmically. Students must be able to solve, understand, and express mathematical, algebraic, and geometric concepts; in independent written assignments, as well as meaningfully participate in classroom and cooperative-learning tasks. Students are expected to listen, record effectively, inquire, offer solutions, organize, evaluate, and apply information. Mathematics is progressive and integrative as skills in numerical operations, algebra, geometry, and data analysis must be interwoven to develop critical thinking and application to open-ended situations, the perspective job-place and other real-life phenomena. The ability to acquire math knowledge via Internet research or effective use of technology is equally important to thinking, reasoning, expressing, and applying mathematics to everyday life. Mathematics is more than the exclusive acquisition of numerical, algebraic, or geometric problem-solving skills. It is also the ability to recognize that there are various acceptable ways of representing one’s solution using a combination of some or all of these acquired skills in open-ended situations, and be able to use technological resources to achieve those purposes. Math educators must be willing and able to engage students with different learning styles through the use of various pedagogical methods of presenting the material. By welcoming students with different intelligence strengths, teachers may enhance the learning process by varying the types of classroom tasks for students to solve problem-situations. Students should be given the opportunity to express what they’ve learned in a variety of formats: for example; knowing, comprehending, writing, speaking, applying through sketching, modeling, role-playing, constructing, designing, organizing, analyzing, synthesizing, creating, and evaluating. Common Core Curriculum Standards for Mathematics describe the mathematical experience needed for students to grow intellectually, socially, and emotionally in classrooms across the curriculum. The standards are intended to promote students’ capacities to construct meaning in any arena, with others as well as on their own. If students learn to simply, solve, sketch, explain, apply, evaluate and analyze critically, and if they learn to use these mathematical concepts dependently and independently, they will develop the mathematical skills they need to solve personal and shared problem-situations throughout their lives.

Title of Unit Matrices, Sequences and Series Grade Level 11-12 Curriculum Area Discrete Mathematics and Statistics Time Frame 4-5 weeks Developed By: A. Contella and C. Eccles

Desired Results (Stage 1) Established Goals

Established Goals (G): Common Core State Standards/Technology Standards F-BF 2.

N-Q 1.

Write an arithmetic and geometric sequence both recursively and with an explicit formula, use them to model situations, and translate between the two forms. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

N-Q 2. Define appropriate quantities for the purpose of descriptive modeling. N-Q 3.

N-VM 6. N-VM 7. N-VM 8. A.REI 9.

Choose a level of accuracy appropriate to limitation on measurement when reporting quantities. Use matrices to represent and manipulate data, Multiply matrices by scalars to produce new matrices, Add, subtract, and multiply matrices of appropriate dimensions. Find the inverse of a matrix if it exists and use it to solve systems of linear equations

Primary Interdisciplinary Connections 8.1.12.A.3 Participate in online courses, learning communities, social networks, or virtual worlds and recognize them as resources for lifelong learning. 8.1.12.F.2 Analyze the capabilities and limitations of current and emerging technology resources and assess their potential to address educational, career,

personal, and social needs. 9.1.12.A.1 Apply critical thinking and problem-solving strategies during structured learning experiences. 9.1.12.A.2 Participate in online strategy and planning sessions for course-based, school-based, or outside projects. 9.1.12.B.1 Present resources and data in a format that effectively communicates the meaning of the data and its implications for solving problems, using multiple

perspectives. 9.1.12.C.4 Demonstrate leadership and collaborative skills when participating in online learning communities and structured learning experiences. 9.1.12.C.5 Assume a leadership position by guiding the thinking of peers in a direction that leads to successful completion of a challenging task or project. 9.1.12.F.2 Demonstrate a positive work ethic in various settings, including the classroom and during structured learning experiences.

21st Century Interdisciplinary Themes: X Global Awareness X Financial, economic, business, and entrepreneurial literacy Civic Literacy Health Literacy

Transfer Students will be able to use their learning to… T1. Communicate about mathematical ideas in a standard and understandable manner. T2. Students will relate the topics learned about organizing and manipulating data mathematically to things that they do in their everyday lives. T3. Analyze real life situations in a mathematical way. T4. Become proficient in daily skills involving mathematics.

1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 6. Attend to precision. 7. Look for and make use of structure.

Transfer: Common Core State Standards for Mathematical Practice

8. Look for and express regularity in repeated reasoning.

Meaning (Discrete Mathematics and Statistics - continued) Understandings Essential Questions

Students will understand that… Students will keep considering… U1. Mathematical associations, real-world relationships, and data

organization can be represented using matrices as a tool. Q1. How can matrices be used to organize and manage real-life data?

U2. Mathematics involves the use of different types of tools, such as matrices, to manipulate numbers and solve problems.

Q2. How can matrices be used to analyze complex real-world situations?

U3. Matrices are used to organize and compute large amounts of data. Q3.

What tools are used to organize data that support easy manipulations and problem solving?

U4.

U5.

Mathematical properties can be used to transform mathematical relationships. Relations and functions can be represented numerically, graphically, algebraically, and/or verbally.

Q4.

Q5.

Q6.

Why is it useful to represent real-life situations mathematically? How are patterns of change related to the behavior of functions? In what ways does the recursive nature of sequences generate formulas and effect applications of sequences and series?

Acquisition Knowledge Students will know…

Skills Students will be able…

K1. Organize matrices and their elements S1. Identify elements and dimensions in a matrix. K2. Matrix addition, subtraction and multiplication. S2. Perform matrix addition, subtraction and multiplication. K3. Transpose matrices. S3. Perform transformations on a matrix using correct algorithms. K4.

K5.

K6.

K7.

Solving system of equations. Sequences and series. Sum of finite arithmetic and geometric series. Sum of infinite geometric series.

S4.

S5.

S6.

S7.

S8.

Solve system of equations using various methods involving matrices. Identify arithmetic and geometric sequences and series. Model sequences and series recursively. Apply sigma notation, and find the sum of finite arithmetic, and finite and infinite geometric series. Apply sequences and series to areas of design and finance and investment planning.

Evidence (Stage 2 – Discrete Mathematics and Statistics - continued) Evaluation Criteria Checks for Alignment Performance is judged in terms of…

Assessment Evidence

Transfer Task(s)

Formative: Book problems, Worksheets, Cooperative Activities, Problem-solving Activities, Do-Now, Closure Activities Summative: (used at the end of identified topics to assess learning)

T1-T4 U1-U5 Q1-Q6

Blooms Taxonomy:

• Knowledge • Comprehension • Application • Analysis • Synthesis • Evaluation Teacher Created Rubric Quizzes, Tests, Projects, Presentations

Other Evidence

Formative: Essays Journals Rubrics

K1-K7 S1-S8

Blooms Taxonomy: • Knowledge • Comprehension • Application • Analysis • Synthesis • Evaluation Reports

Other Assessments

Summative:

Teacher Created Rubric: Pre-assessments:

Journal entries Informal observations

Other

Learning Plan (Stage 3) Checks for alignment and best practice

Summary of Key Learning Events and Instruction The teaching and learning needed to achieve the unit goals.

Required Activities Required Resources Student groups will be formed to examine the Projects& Papers problems in the text. Teachers can differentiate instruction by selecting exercises

T1-T4,K1-K7,S1-S8

from the text and use of mini-excursions.

Text: Excursions in Modern Mathematics with Mini-Excursions. 6th Edition. Peter Tannenbaum. Pearson-Prentice Hall.2007.

Title of Unit Graph Theory Grade Level 11-12 Curriculum Area Discrete Mathematics and Statistics Time Frame 5-6 weeks Developed By: A. Contella and C. Eccles


Established Goals (G): Common Core State Standards/Technology Standards N-Q 1. Use units as a way to understand problems and to guide the solution of multi-step problems. G-MG 1. Use geometric shapes, their measures, and their properties to describe objects. G-MG 3.

N-VM 6.

Apply geometric methods to solve design problems. Use matrices to represent and manipulate data e.g.to represent relationships in a network.

Primary Interdisciplinary Connections 8.1.12.A.3 Participate in online courses, learning communities, social networks, or virtual worlds and recognize them as resources for lifelong learning. 8.1.12.F.2 Analyze the capabilities and limitations of current and emerging technology resources and assess their potential to address educational, career,

personal, and social needs. 9.1.12.A.1 Apply critical thinking and problem-solving strategies during structured learning experiences. 9.1.12.A.2 Participate in online strategy and planning sessions for course-based, school-based, or outside projects. 9.1.12.B.1 Present resources and data in a format that effectively communicates the meaning of the data and its implications for solving problems, using

multiple perspectives. 9.1.12.C.4 Demonstrate leadership and collaborative skills when participating in online learning communities and structured learning experiences. 9.1.12.C.5 Assume a leadership position by guiding the thinking of peers in a direction that leads to successful completion of a challenging task or project. 9.1.12.F.2 Demonstrate a positive work ethic in various settings, including the classroom and during structured learning experiences.

21st Century Interdisciplinary Themes: X Global Awareness

Civic Literacy X Financial, economic, business, and entrepreneurial literacy

Health Literacy

Transfer

Students will be able to use their learning to… T1. Communicate about mathematical ideas in a standard and understandable manner. T2. Students will relate the topics learned about organizing and manipulating data mathematically to things that they do in their everyday lives.. T3. Analyze real life situations in a mathematical way. T4. Become proficient in daily skills involving mathematics.

1. Make sense of problems and persevere in solving them.





Students will understand that… Students will keep considering… U1.

Graphs can help organize data . Q1. How can networks be used to answer real-world questions?

U2. Graphs can be used to model scheduling problems and resolve conflicts.

Q2. What real-world events can be modeled using Euler and Hamilton paths?

U3. Networks can be used to promote efficiency. Q3.

Why can any planar map be colored in at most four colors?

U4. Networks can be used to construct optimal solutions. Q4.

Why is it useful to represent real-life situations mathematically?

Q5.

How can a situation be resolved in the most effective manner?



K1. Organize data using networks. S1. Construct graphs and recognize they are useful tools in modeling real-life situations.

K2. Terms describing networks including graph notation, valences, chromatic number, planarity, directed graphs.

S2. Find an Euler circuit or path if one exists in a network.

K3. Difference between Circuits and Paths, both Euler and Hamilton.

S3. Apply and create algorithms to solve problems involving Euler circuits, networks and paths.

K4.

K5.

Traveling salesperson problems. Conflict scheduling.

S4. Find a Hamilton circuit or path.

S5. S6.

S7. S8. S9. S10.

Differentiate between an Euler and Hamilton circuit or path using a correct algorithm. Model real-world problems with weighted graphs. Identify and solve Traveling Salesperson and Shortest Route Problems. Create a traveling Salesperson problem and find an optimal solution. Use the real life example of scheduling examination to resolve time conflicts via graph theory and vertex coloring. Use Graph Coloring to solve scheduling problems or problems involving conflicts

Evidence (Stage 2 – Discrete Mathematics and Statistics - continued) Evaluation Criteria Checks for Alignment Performance is judged in terms of…

Assessment Evidence

Transfer Task(s)


T1-T4 U1-U4 Q1-Q5

Blooms Taxonomy:

• Knowledge • Comprehension • Application • Analysis • Synthesis • Evaluation

Teacher Created Rubric

Quizzes, Tests, Projects, Presentations

Other Evidence

Formative:

K1-K5 S1-S10

Blooms Taxonomy: • Knowledge

Essays

Other Evidence

Formative: Essays Journals Rubrics Reports Other Assessments

Summative:

• Comprehension • Application • Analysis • Synthesis • Evaluation Teacher Created Rubric

Pre-assessments Journal entries Informal observations Other




T1-T4,K1-K5,S1-S10



Title of Unit Voting Methods, Fair Division and Apportionment

Grade Level 11-12

Curriculum Area Discrete Mathematics and Statistics Time Frame 5-6 weeks Developed By: A. Contella and C. Eccles


Established Goals (G): Common Core State Standards/Technology Standards A-CED 1. Create equations and inequalities in one variable and use them to solve problems. F-BF 1.

N-Q 2. Write a function that describes a relationship between two quantities. Define appropriate quantities for the purpose of descriptive modeling.

Primary Interdisciplinary Connections 8.1.12.A.3 Participate in online courses, learning communities, social networks, or virtual worlds and recognize them as resources for lifelong learning. 8.1.12.F.2 Analyze the capabilities and limitations of current and emerging technology resources and assess their potential to address educational,

career, personal, and social needs. 9.1.12.A.1 Apply critical thinking and problem-solving strategies during structured learning experiences. 9.1.12.A.2 Participate in online strategy and planning sessions for course-based, school-based, or outside projects. 9.1.12.B.1 Present resources and data in a format that effectively communicates the meaning of the data and its implications for solving problems, using



Transfer Students will be able to use their learning to… T1. Communicate about mathematical ideas in a standard and understandable manner. T2. Students will relate the topics learned about organizing and manipulating data mathematically to things that they do in their everyday lives.. T3. Analyze real life situations in a mathematical way. T4. Become proficient in daily skills involving mathematics.





Students will understand that… Students will keep considering… U1. A winner can vary depending on the voting method being used. Q1. How are the desires of many individuals combined to produce a single result?

U2. There is not an optimal voting method for all situations and

each method has its flaws. Q2. What are the advantages and disadvantages of using different

voting methods?

U3. Fair division can be applied to many real world events. Q3. How do we define fairness?

U4. The success of the estate division algorithm requires that each heir be capable of placing a value on each object in the estate.

Q4. When dividing food, land, and/or personal items are the methods we used considered fair?

U5.

U6.

A fair division problem may be discrete or continuous. The success of the estate division algorithm requires that each heir places a value on each object in the estate.

Q5.

Q6.

How can we divide an estate when individuals have different views of the values of an item? How can a population be represented fairly?

U7. Congressional representation determined by an apportionment method based on the last census.



K1. Election methods(Plurality, majority, Borda, Runoff, etc)

S1. Determine rankings by using several procedures including plurality, Borda, runoff, sequential runoff and Condorcet methods.

K2. Fairness criteria in both continuous and discrete events.

S2. Distinguish between continuous and discrete fair division problems.

K3. Cake division algorithms. S3. Apply appropriate cake division algorithm.

K4. Method of sealed bids. S4. Divide an estate fairly using the method of sealed bids. K5. Hamilton, Jefferson, Webster, & Hill Methods of apportionment. S5. Apportion votes based on population using Hamilton, Jefferson,

Webster-Wilcox, and Hill-Huntington methods.

S6. Compare and contrast different voting procedures.

Evidence (Stage 2 – Discrete Mathematics and Statistics - continued) Evaluation Criteria Checks for Alignment Performance is judged in terms

Assessment Evidence

of … Transfer Task(s)


T1-T4 U1-U7 Q1-Q6

BLOOMS TAXONOMY RUBRIC

Quizzes, Tests, Projects, Presentations Other Evidence


Summative:

K1-K5 S1-S6



Title of Unit Codes and Cryptography Grade Level 11-12 Curriculum Area Discrete Mathematics and Statistics Time Frame 4-5 weeks Developed By: A. Contella and C. Eccles


Established Goals (G): Common Core State Standards/Technology Standards Algebra: N-VM 6. Use matrices to represent and manipulate data, N-VM 7. Multiply matrices by scalars to produce new matrices, N-VM 8.

A-CED 1.

Add, subtract, and multiply matrices of appropriate dimensions.

Create equations and inequalities in one variable and use them to solve problems.

Primary Interdisciplinary Connections 8.1.12.A.3 Participate in online courses, learning communities, social networks, or virtual worlds and recognize them as resources for lifelong learning. 8.1.12.F.2 Analyze the capabilities and limitations of current and emerging technology resources and assess their potential to address educational,

career, personal, and social needs. 9.1.12.A.1 Apply critical thinking and problem-solving strategies during structured learning experiences. 9.1.12.A.2 Participate in online strategy and planning sessions for course-based, school-based, or outside projects. 9.1.12.B.1 Present resources and data in a format that effectively communicates the meaning of the data and its implications for solving problems, using



Transfer Students will be able to use their learning to… T1. Communicate about ideas in Algebra in a standard and understandable manner. T2. Relate the topics learned in Algebra to things that they do in their everyday lives. T3. Analyze real life situations in a mathematical way.

T4. Become proficient in daily skills involving mathematics. 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 6. Attend to precision. 7. Look for and make use of structure.




Students will understand that… Students will keep considering… U1. Each part of a code is part an identity of what is being coded. Q1. Why is encryption necessary?

U2. Codes play an important role in our technological society. Q2. What methods are used to encrypt data?

U3. Most codes have a check digit that can be calculated using a specific algorithm.

Q3. What do certain parts of codes mean?

U4. Material can be encoded and decoded using mathematics. Q4. How can we use mathematics to encode messages? Q5. How can mathematics be used to decode messages or break

codes?



K1. Identify zip codes , bar codes, routing numbers and check digits.

S1. Apply the zip code algorithm .

K2. Calculate check digits. S2. Interpret a bar code algorithm given a bar code.

K3. Describe and apply a coding algorithm S3. Demonstrate how error-detecting schemes work using binary codes as signals.

K4. Caesar ciphers and matrix multiplication. S4. Define parity-check-sums and apply them to correct any single

error. S5. Define and apply cryptology. S6. Use matrices as arrays of data to solve logic problem

Evidence (Stage 2 –- continued) Evaluation Criteria Checks for Alignment Performance is judged in terms

Assessment Evidence



T1-T4 U1-U4 Q1-Q5




Summative:

K1-K4 S1-S6





Required Activities Required Resources




Title of Unit Collecting Statistical Data Grade Level 11,12 Curriculum Area Discrete Math / Statistics Time Frame 5 weeks Developed By: A. Contella and C. Eccles


Established Goals (G): Common Core State Standards/Technology Standards Statistics HSS-IC.A.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that

population. HSS-IC.B.3 Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how

randomization relates to each. HSS-IC.B.4 Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of

simulation models for random sampling. HSS-IC.B.5 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between

parameters are significant. HSS-IC.B.6 Evaluate reports based on data.

Primary Interdisciplinary Connections 8.1.12.F.2 Analyze the capabilities and limitations of current and emerging technology resources and assess their potential to address educational,

career, personal, and social needs. 9.1.12.A.1 Apply critical thinking and problem-solving strategies during structured learning experiences. 9.1.12.B.1 Present resources and data in a format that effectively communicates the meaning of the data and its implications for solving problems, using

multiple perspectives. 9.1.12.C.4 Demonstrate leadership and collaborative skills when participating in structured learning experiences. 9.1.12.C.5 Assume a leadership position by guiding the thinking of peers in a direction that leads to successful completion of a challenging task or project. 9.1.12.F.2 Demonstrate a positive work ethic in various settings, including the classroom and during structured learning experiences.


Transfer

Students will be able to use their learning to… T1. Communicate about ideas in Statistics in a standard and understandable manner. T2. Relate the topics learned in Statistics to things that they do in their everyday lives. T3. Analyze real life situations in a mathematical way. T4. Become proficient in daily skills involving mathematics.




Meaning Understandings Essential Questions

Students will understand that… Students will keep considering… U1. The data collection process is very important, since the

analysis of the data later depends on the sample being representative of the entire population it is taken from.

Q1. What are the methods to insure proper data collection?

U2. Which methods of data collection are best suited t situations. Q2. Which data collection methods are best suited to a given situation?

U3. Measures need to be taken to minimize error during the data collection process.

Q3. What measures can be taken to minimize error in data collection?



K1. The importance of how statistical data is collected. S1. To identify the population to which the data apply. K2. What type of data collection method is best suited for a given

situation. S2. To understand and apply concepts of population, N-value, and

census. K3. How to select and apply a data collection method. S3. To understand and apply basic concepts of sampling: sampling

frame, selection bias, nonresponse bias, sampling error, sampling variability, and chance error.

K4. How to estimate large populations. S4. To understand and apply various sampling methods: quota sampling, simple random sampling, and stratified sampling.

K5. How to collect data to later determine cause and effect. S5. To estimate large populations by applying the capture-recapture method for sampling.

S6. To apply concepts of a clinical study to determine cause and effect.

S7. To understand key concepts of a clinical study: control group, treatment group, randomization, placebo effect, blind and double-blind studies.

Evidence (Stage 2) Evaluation Criteria Checks for Alignment Performance is judged in terms

Assessment Evidence


Formative: Book problems, worksheets, cooperative activities, problem-solving activities, Do-Now, closure activities Summative: (used at the end of identified topics to assess learning)

T1-T4 U1-U5 Q1-Q5



Formative: Essays Projects Rubrics Reports Other Assessments

Summative:

K1-K5 S1-S8


Pre-assessments Informal observations Other

Learning Plan (Stage 3)

Checks for alignment and best practice


Required Activities Required Resources Student groups will be formed to examine the Projects and Papers problems in the text. Teachers can differentiate instruction by selecting exercises

from the text and use of the mini-excursions.

Text: Excursions in Modern Mathematics with Mini-Excursions. 6th Edition. Peter Tannenbaum. Pearson-Prentice Hall. 2007.

Title of Unit Descriptive Statistics Grade Level 11,12 Curriculum Area Discrete Math / Statistics Time Frame 5 weeks Developed By: A. Contella and C. Eccles


Established Goals (G): Common Core State Standards/Technology Standards Statistics HSN-Q.A.1 Use units as a way to understand problems; choose and interpret the scale and the origin in graphs and data displays. HSN-Q.A.2 Define appropriate quantities for the purpose of descriptive modeling. HSN-Q.A.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. HSS-ID.A.1 Represent data with graphs. HSS-ID.A.2 Use statistics to compare center (mean, median) and spread (interquartile range, standard deviation) of two or more different

sets of data.





Transfer Students will be able to use their learning to… T1. Communicate about ideas in Statistics in a standard and understandable manner. T2. Relate the topics learned in Statistics to things that they do in their everyday lives. T3. Analyze real life situations in a mathematical way. T4. Become proficient in daily skills involving mathematics.





Students will understand that… Students will keep considering… U1. There is a need to be able to summarize and communicate

statistical data. Q1. What are effective and efficient ways to summarize data?

U2. There are a wide variety of applications of statistics. Q2. How can descriptive statistics be used to assess situations and help make decisions?

U3. Large sets of data can be made manageable and understandable by the use of descriptive statistics.

Q3. How can misuse of descriptive statistics be recognized and/or avoided?



K1. That there are two main ways to summarize data: graphically and numerically.

S1. To construct and interpret various graphical descriptions of data: bar graph, pictograph, histogram, and pie chart.

K2. There are a variety of graphing methods and which are best suited to particular situations.

S2. To understand how to properly represent data that is categorical, numerical, discrete, or continuous.

K3. There are numerical values used to summarize both location and spread of the data.

S3. To identify or compute descriptive data values: mean, median, mode, range, minimum, maximum, quartiles, percentiles, interquartile range, variance, and standard deviation.

S4. To describe data sets using numerical summaries: mean, median, mode, range, quartiles, percentiles, interquartile range, variance, and standard deviation.

S5. To display a five number summary as a box-and-whisker plot.


Assessment Evidence



T1-T4 U1-U5 Q1-Q5




Summative:

K1-K5 S1-S8








Title of Unit Chances, Probabilities, and Odds Grade Level 11,12 Curriculum Area Discrete Math / Statistics Time Frame 5 weeks Developed By: A. Contella and C. Eccles


Established Goals (G): Common Core State Standards/Technology Standards Statistics HSS-CP.A.1 Describe events as subsets of a sample space. HSS-CP.A.2 Use probabilities to determine if events are independent. HSS-CP.A.3 Understand conditional probabilities. HSS-CP.A.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday

situations. HSS-CP.B.6 Find conditional probabilities and interpret in terms of the model. HSS-CP.B.9 Use permutations and combinations to compute probabilities of compound events and solve problems. HSS-MD.B.5 Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values. HSS-MD.B.6 Use probabilities to make fair decisions. HSS-MD.B.7 Analyze decisions and strategies using probability concepts.










Students will understand that… Students will keep considering… U1. There are mathematical ways to determine the amount of

uncertainty in a given situation. Q1. How can the level of uncertainty of a situation be determined?

U2. There are different ways in which uncertainty can be quantified and expressed.

Q2. How can the level of uncertainty be expressed quantitatively?

U3. Knowing the level of uncertainty can assist in decision making.

Q3. How can determining the level of uncertainty lead to decision making?



K1. The concepts and terms used to express uncertainty. S1. To understand concepts of basic framework for discussing probabilities: random experiment, sample space, probability spaces, basic rules for counting, and permutations & combinations.

K2. There are different ways to express uncertainty. S2. To determine and interpret permutations and combinations.

K3. How to interpret expressions of uncertainty. S3. To understand the differences of probabilities, chances, and odds.

K4. How to apply determination of uncertainty to making decisions.

S4. To determine and interpret probabilities, chances, and odds.

S5. To apply permutations and combinations to finding probabilities, chances, and odds.


Assessment Evidence



T1-T4 U1-U5 Q1-Q5




Summative:

K1-K5 S1-S8









Title of Unit Normal Distributions Grade Level 11,12 Curriculum Area Discrete Math / Statistics Time Frame 4-5 weeks Developed By: A. Contella and C. Eccles


Established Goals (G): Common Core State Standards/Technology Standards Statistics HSS-ID.A.2 Use statistics appropriate to the shape of the data distribution to compare center and spread of two or more different sets of

data. HSS-ID.A.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data

points. HSS-ID.A.4 Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages.

Understand that there are data sets for which this is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the curve.










Students will understand that… Students will keep considering… U1. there is an ideal distribution of data called “normal” and what

this means mathematically. Q1. What is the connection between normal curves and real-world

data? U2. distribution of data can be understood more fully by

comparing it to a normal distribution, even if the data does not fit a normal distribution.

Q2. What is the relationship of data distributions to statistical inference?

U3. the distribution of data and can be used for decision making. Q3. What is the relationship of data distributions to decision making?



K1. That displays of data distributions and comparison to normal curve can help understand and interpret data sets.

S1. To use bar graphs to display data distribution.

K2. That data sets can be standardized to make comparisons. S2. To compare displays of data distribution to the normal distribution.

K3. That data distributions can be used for estimation purposes. S3. To compare mean, median, and standard deviation of a data set to those of a normal distribution.

K4. That reliability of estimations made using data distributions can be quantified and expressed using confidence intervals.

S4. To standardize data using z-values.

S5. To understand and apply the 68-95-99.7 Rule for a normal curve distribution.

S6. To understand and apply the “honest-coin” and “dishonest-coin” principles to statistical inference.

S7. To determine standard error for a data distribution and express confidence intervals.


Assessment Evidence



T1-T4 U1-U5 Q1-Q5




Summative:

K1-K5 S1-S8




Belleville Board of Education - … Belleville Board of Education Discrete Mathematics and...

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