Download Wolfram: Normal Approximation to a Binomial Random Variable

Curriculum and Instruction – Mathematics
Quarter 2
Finite Math
Introduction
In 2014, the Shelby County Schools Board of Education adopted a set of ambitious, yet attainable goals for school and student performance. The District
is committed to these goals, as further described in our strategic plan, Destination2025. By 2025,
 80% of our students will graduate from high school college or career ready
 90% of students will graduate on time
 100% of our students who graduate college or career ready will enroll in a post-secondary opportunity
In order to achieve these ambitious goals, we must collectively work to provide our students with high quality, College and Career Ready standardsaligned instruction. The Tennessee State Standards provide a common set of expectations for what students will know and be able to do at the end of a
grade. College and Career Ready Standards are rooted in the knowledge and skills students need to succeed in post-secondary study or careers. The TN
State Standards represent three fundamental shifts in mathematics instruction: focus, coherence and rigor.
Focus
•
The TN Standards call for a greater focus in
mathematics. Rather than racing to cover topics
in a mile-wide, inch-deep curriculum, the
Standards require us to significantly narrow and
deepen the way time and energy is spent in the
math classroom. We focus deeply on the major
concepts of each subject so that students can
gain strong foundations: solid conceptual
understanding, a high degree of procedural skill
and fluency, and the ability to apply the math
they know to solve problems inside and outside
the math classroom.
Coherence
Rigor
Thinking across grades:
•
The TN Standards are designed around coherent
progressions from grade to grade. Learning is
carefully connected across grades so that
students can build new understanding onto
foundations built in previous years. Each
standard is not a new event, but an extension of
previous learning.
Conceptual understanding:
•
The TN Standards call for conceptual
understanding of key concepts, such as place
value and ratios. Students must be able to access
concepts from a number of perspectives so that
they are able to see math as more than a set of
mnemonics or discrete procedures.
Procedural skill and fluency:
•
The Standards call for speed and accuracy in
calculation. While the high school standards for
math do not list high school fluencies, there are
suggested fluency standards for algebra 1,
geometry and algebra 2.
Linking to major topics:
•
Instead of allowing additional or supporting
topics to detract from course, these concepts
serve the course focus. For example, instead of
data displays as an end in themselves, they are
an opportunity to do grade-level word
problems.
Application:
•
The Standards call for students to use math
flexibly for applications in problem-solving
contexts. In content areas outside of math,
particularly science, students are given the
opportunity to use math to make meaning of and
access content.
Shelby County Schools 2016/2017
Revised 8/31/16
1 of 11
Curriculum and Instruction – Mathematics
Quarter 2
8. Look for and
express regularity
in repeated
reasoning
7. Look for and
make use of
structure
1. Make sense of
problems and
persevere in
solving them
2. Reason
abstractly and
quatitatively
Mathematical
Practices(MP)
6. Attend to
precision
3. Construct viable
arguments and
crituqe the
reasoning of
others
4. Model with
mathematics
5. Use appropriate
tools strategically
Finite Math
The Standards for Mathematical Practice describe varieties of expertise, habits of minds and
productive dispositions that mathematics educators at all levels should seek to develop in
their students. These practices rest on important National Council of Teachers of
Mathematics (NCTM) “processes and proficiencies” with longstanding importance in
mathematics education. Throughout the year, students should continue to develop
proficiency with the eight Standards for Mathematical Practice.
This curriculum map is designed to help teachers make effective decisions about what
mathematical content to teach so that, ultimately our students, can reach Destination 2025.
To reach our collective student achievement goals, we know that teachers must change their
practice so that it is in alignment with the three mathematics instructional shifts.
Throughout this curriculum map, you will see resources as well as links to tasks that will
support you in ensuring that students are able to reach the demands of the standards in your
classroom. In addition to the resources embedded in the map, there are some high-leverage
resources around the content standards and mathematical practice standards that teachers
should consistently access:
The TN Mathematics Standards
The Tennessee Mathematics Standards:
Teachers can access the Tennessee State standards, which are featured
https://www.tn.gov/education/article/mathematics-standards
throughout this curriculum map and represent college and career ready
learning at reach respective grade level.
Standards for Mathematical Practice
Mathematical Practice Standards
Teachers can access the Mathematical Practice Standards, which are
https://drive.google.com/file/d/0B926oAMrdzI4RUpMd1pGdEJTYkE/view featured throughout this curriculum map. This link contains more a more
detailed explanation of each practice along with implications for instructions.
Shelby County Schools 2016/2017
Revised 8/31/16
2 of 11
Curriculum and Instruction – Mathematics
Quarter 2
Finite Math
Purpose of the Mathematics Curriculum Maps
This curriculum framework or map is meant to help teachers and their support providers (e.g., coaches, leaders) on their path to effective, college and career ready
(CCR) aligned instruction and our pursuit of Destination 2025. It is a resource for organizing instruction around the TN State Standards, which define what to teach
and what students need to learn at each grade level. The framework is designed to reinforce the grade/course-specific standards and content—the major work of the
grade (scope)—and provides a suggested sequencing and pacing and time frames, aligned resources—including sample questions, tasks and other planning tools.
Our hope is that by curating and organizing a variety of standards-aligned resources, teachers will be able to spend less time wondering what to teach and searching
for quality materials (though they may both select from and/or supplement those included here) and have more time to plan, teach, assess, and reflect with
colleagues to continuously improve practice and best meet the needs of their students.
The map is meant to support effective planning and instruction to rigorous standards; it is not meant to replace teacher planning or prescribe pacing or instructional
practice. In fact, our goal is not to merely “cover the curriculum,” but rather to “uncover” it by developing students’ deep understanding of the content and mastery of
the standards. Teachers who are knowledgeable about and intentionally align the learning target (standards and objectives), topic, task, and needs (and
assessment) of the learners are best-positioned to make decisions about how to support student learning toward such mastery. Teachers are therefore expected-with the support of their colleagues, coaches, leaders, and other support providers--to exercise their professional judgement aligned to our shared vision of effective
instruction, the Teacher Effectiveness Measure (TEM) and related best practices. However, while the framework allows for flexibility and encourages each
teacher/teacher team to make it their own, our expectations for student learning are non-negotiable. We must ensure all of our children have access to rigor—highquality teaching and learning to grade-level specific standards, including purposeful support of literacy and language learning across the content areas.
Additional Instructional Support
Shelby County Schools adopted our current math textbooks for grades 6-8 in 2010-2011. The textbook adoption process at that time followed the requirements set
forth by the Tennessee Department of Education and took into consideration all texts approved by the TDOE as appropriate. We now have new standards; therefore,
the textbook(s) have been vetted using the Instructional Materials Evaluation Tool (IMET). This tool was developed in partnership with Achieve, the Council of Chief
State Officers (CCSSO) and the Council of Great City Schools. The review revealed some gaps in the content, scope, sequencing, and rigor (including the balance of
conceptual knowledge development and application of these concepts), of our current materials.
The additional materials purposefully address the identified gaps in alignment to meet the expectations of the CCR standards and related instructional shifts while still
incorporating the current materials to which schools have access. Materials selected for inclusion in the Curriculum Maps, both those from the textbooks and
external/supplemental resources (e.g., EngageNY), have been evaluated by district staff to ensure that they meet the IMET criteria.
Shelby County Schools 2016/2017
Revised 8/31/16
3 of 11
Curriculum and Instruction – Mathematics
Quarter 2
Finite Math
How to Use the Mathematics Curriculum Maps
Overview
An overview is provided for each quarter. The information given is intended to aid teachers, coaches and administrators develop an understanding of the content the
students will learn in the quarter, how the content addresses prior knowledge and future learning, and may provide some non-summative assessment items.
Tennessee State Standards
The TN State Standards are located in the left column. Each content standard is identified as the following: Major Work, Supporting Content or Additional Content.; a
key can be found at the bottom of the map. The major work of the grade should comprise 65-85% of your instructional time. Supporting Content are standards that
supports student’s learning of the major work. Therefore, you will see supporting and additional standards taught in conjunction with major work. It is the teacher’s
responsibility to examine the standards and skills needed in order to ensure student mastery of the indicated standard.
Content
Teachers are expected to carefully craft weekly and daily learning objectives/ based on their knowledge of TEM Teach 1. In addition, teachers should include related
best practices based upon the TN State Standards, related shifts, and knowledge of students from a variety of sources (e.g., student work samples, MAP, etc.).
Support for the development of these lesson objectives can be found under the column titled ‘Content’. The enduring understandings will help clarify the “big picture”
of the standard. The essential questions break that picture down into smaller questions and the objectives provide specific outcomes for that standard(s). Best
practices tell us that clearly communicating and making objectives measureable leads to greater student mastery.
Instructional Support and Resources
District and web-based resources have been provided in the Instructional Resources column. Throughout the map you will find instructional/performance tasks, iReady lessons and additional resources that align with the standards in that module. The additional resources provided are supplementary and should be used as
needed for content support and differentiation.
Shelby County Schools 2016/2017
Revised 8/31/16
4 of 11
Curriculum and Instruction – Mathematics
Quarter 2
Finite Math
Topics Addressed in Quarter



Random Variables, Averages, and Statistics
Logic for Finite Math
Systems of Linear Equations
Overview
In quarter two students begin to bridge the study of probability and the application of probability theory to the field of statistics. Students solve problems and use the concepts of random
variable, mean, and standard deviation. The normal random variable and the binomial random variable is also studied. Student study a few concepts involving logic including truth tables,
statements and deduction. Students conclude the quarter by reviewing systems of linear equations and applying these concepts to analyze and solve real-world problems involving linear
systems.
Fluency
The high school standards do not set explicit expectations for fluency, but fluency is important in high school mathematics. Fluency in algebra can help students get past the need to manage
computational and algebraic manipulation details so that they can observe structure and patterns in problems. Such fluency can also allow for smooth progress toward readiness for further
study/careers in science, technology, engineering, and mathematics (STEM) fields. These fluencies are highlighted to stress the need to provide sufficient supports and opportunities for practice to
help students gain fluency. Fluency is not meant to come at the expense of conceptual understanding. Rather, it should be an outcome resulting from a progression of learning and thoughtful
practice. It is important to provide the conceptual building blocks that develop understanding along with skill toward developing fluency.
The fluency recommendations for Algebra I listed below should be incorporated throughout your instruction over the course of the school year.



A/G
A-APR.A.1
A-SSE.A.1b
Solving characteristic problems involving the analytic geometry of lines
Fluency in adding, subtracting, and multiplying polynomials
Fluency in transforming expressions and seeing parts of an expression as a single object
References:



http://www.corestandards.org/
http://www.nctm.org/
http://achievethecore.org/
Shelby County Schools 2016/2017
Revised 8/31/16
5 of 11
Curriculum and Instruction – Mathematics
Quarter 2
TN STATE STANDARDS
Conceptual Category: Data Analysis,
Statistics, and Probability
Domain: Organize and Interpret data
D-ID.1 Organize data for problem solving.
D-ID.2 Use a variety of counting methods to
organize information, determine probabilities,
and solve problems.
Finite Math
CONTENT
INSTRUCTIONAL SUPPORT & RESOURCES
Chapter 4 Random Variables, Averages, and Statistics
(Allow approximately 3 weeks for instruction, review, and assessment)
Enduring Understanding(s):
Tennessee Finite Math Textbook
4.1 Random Variables and Probability Density
 Interpretation of data is dependent
Functions
upon the graphical displays and
numerical summaries.
 The question to be answered
Additional Resources
determines the data to be collected and
Khan Academy Video – Probability Density
how best to collect it.
Functions
 The normal distribution is a
Khan Academy Video – Random Variables
fundamental component of statistical
inference.
FiniteHelp Practice Problems
 Density curves are used to mimic
FiniteHelp Notation Guide
probability.
Finite Help
 The normal distribution is used to
Finite Math Student Resources
model the spread of data.
 Probability models are useful tools for
making decisions and predictions.
 Probability is the basis of statistical
inference.
 The notion and behavior of a random
variable is foundational to
understanding probability distributions.
 Probability models are useful tools for
making decisions and predictions.
Essential Question(s):
Important Terms & Concepts (Chapter 4)
Approximation method, binomial random
variable, density function, expected value,
expected value of a binomial random variable,
mean, normal approximation to a binomial
random variable, normal random variable,
random variable, standard deviation, standard
normal curve, standard normal random
variable, variance, variance of a binomial
random variable
Writing in Math
Explain the difference between positively
skewed, negatively skewed, and normally
distributed sets of data and describe an
example of each.
Find a real-world data set that appears to
represent a normal distribution and one that
does not. Describe the characteristics of each
distribution. Create a visual representation of
each set of data.

What is a density function?
Glencoe Reading & Writing in the Mathematics
Classroom

How do density functions relate to
probability?
Graphic Organizers (9-12)

How are measures of central tendency
relevant to density functions?
Graphic Organizers (dgelman)

How can density functions be used to
express relative standing?
Literacy Skills and Strategies for Content Area
Teachers
Shelby County Schools 2016/2017
Revised 8/31/16
6 of 11
Curriculum and Instruction – Mathematics
Quarter 2
TN STATE STANDARDS
Finite Math
CONTENT

What is a normal distribution?

How does one assess normality?
INSTRUCTIONAL SUPPORT & RESOURCES

What does a normal distribution imply
about the spread of data?
Objectives:
Students will:

Conceptual Category: Data Analysis,
Statistics, and Probability
Domain: Organize and Interpret data
D-ID.6 Calculate expected value, e.g., to
determine the fair price of an investment.
Conceptual Category: Data Analysis,
Statistics, and Probability
Domain: Organize and Interpret data
D-ID.4 Calculate and interpret statistical
problem using measures of central tendency
and graphs.
D-ID.6 Calculate expected value, e.g., to
determine the fair price of an investment.
Construct a probability distribution for
random variables.
Objectives:
Students will:

Determine the expected value of a
random variable.

Determine the standard deviation of a
random variable.

Use the expected value to determine
the average payoff or loss in a game
of chance.
Objectives:
Students will:

Determine the probabilities for a normally
distributed variable using a z-score.

Determine the binominal probabilities for a
normally distributed variable.
Tennessee Finite Math Textbook
4.2 Expected Values and Standard
Deviations of Random Variables
Additional Resources
Finite Math Student Resources
FiniteHelp Practice Problems
TI Activity: It’s To Be Expected
Tennessee Finite Math Textbook
4.3 Normal Random Variables and the
Normal Approximation to the Binomial
Additional Resources
Wolfram: Normal Approximation to a Binomial
Random Variable
Shelby County Schools 2016/2017
Revised 8/31/16
7 of 11
Curriculum and Instruction – Mathematics
Quarter 2
Finite Math
TN STATE STANDARDS
CONTENT
INSTRUCTIONAL SUPPORT & RESOURCES
Chapter 11 Logic for Finite Math
(Allow approximately 3 weeks for instruction, review, and assessment)
Conceptual Category: Geometry and
Measurement
Domain: Investigate logic
G-L.1 Define the order of operations for the
logical operators.
G-L.2 Define conjunction, disjunction, negation,
conditional, and biconditional.
Essential Question(s):

Can you describe the method used in
mathematics to assign a truth value to a
compound statement?

What is the importance of the concepts
implication and proof?
Tennessee Finite Math Textbook
11.1 Statements, Connectives, and Negation
Additional Resources
Wolfram: Logic
Objectives:
Students will:

Define the order of operations for the
logical operators.

Define conjunction, disjunction, negation,
conditional, and biconditional.
Important Terms & Concepts (Chapter 11)
Biconditional, Conditional, Conjunction,
Counterexample, Deduction, Disjunction,
Equivalence, Implication, Logical connective
“and”, Logical connective “not”, Logical
connective ”or”, Negative, Truth table, Valid
argument
Writing in Math
Glencoe Reading & Writing in the Mathematics
Classroom
Graphic Organizers (9-12)
Graphic Organizers (dgelman)
Literacy Skills and Strategies for Content Area
Teachers
Conceptual Category: Geometry and
Measurement
Domain: Investigate logic
G-L.4 Construct and use a truth table to draw
conclusions about a statement.
G-L.7 Analyze arguments with quantifiers
through the use of Venn diagrams.
Objectives:
Students will:
Conceptual Category: Geometry and
Measurement
Objectives:
Students will:

Construct and use a truth table to draw
conclusions about a statement.

Analyze arguments with quantifiers
through the use of Venn diagrams
Tennessee Finite Math Textbook
11.2 Truth Tables
Additional Resources
Truth Tables Video
Tennessee Finite Math Textbook
11.3 Equivalence, Implication, and Deduction
Writing in Math
Explain the difference between inductive and
deductive reasoning and give an example of
Shelby County Schools 2016/2017
Revised 8/31/16
8 of 11
Curriculum and Instruction – Mathematics
Quarter 2
TN STATE STANDARDS
Domain: Investigate logic
G-L.2 Construct and use a truth table to draw
conclusions about a statement.
G-L.3 Solve a variety of logic puzzles.
G-L.5 Apply the laws of logic to judge
the validity of arguments.
G-L.6 Give counterexamples to disprove
statements.
Finite Math
CONTENT

Construct and use a truth table to draw
conclusions about a statement.

Solve a variety of logic puzzles
and apply the laws of logic to
judge the validity of arguments.

Give counterexamples to disprove
statements.
INSTRUCTIONAL SUPPORT & RESOURCES
Additional Resources
Logic Puzzles
each.
Logic is used in electrical engineering in
designing circuits. Find another example of
the use of logic in the real world and write a
brief report about your findings.
Chapter 5 Systems of Linear Equations
(Allow approximately 3 weeks for instruction, review, and assessment)
HSF.IF.C.7.A
Graph linear functions.
HSF.LE.A.1.B
Recognize situations in which one quantity
changes at a constant rate per unit interval
relative to another.
HSF.LE.A.2
Construct linear functions given a graph, a
description of a relationship, or two inputoutput pairs (include reading these from a
table).
Enduring Understanding(s):

A mathematical model consists of both
symbolic notations and relations among
symbols.

Many situations in which mathematical
concepts and methods are applied in real
world situations involve quantities related
to each other through one or more
equations.
Essential Question(s):

How can you model a simulation to
represent a real life situation?
Objectives:
Students will learn:
 Create and graph linear functions in realworld problems.
 Recognize situations in which one quantity
changes at a constant rate per unit interval
relative to another.
Tennessee Finite Math Textbook
5.1 Review of Equations and Graphs of Lines
Additional Resources
FiniteHelp Video- 5.1
Finite Math Student Resources
Important Terms & Concepts (Chapter 5)
Algorithm for solving a system of linear
equations, augmented matrix, coefficient
matrix, consistent system, coordinates in the
plane and in the three-dimensional space,
Echelon method, equation of a plane, function,
function notation, Gaussian elimination,
Gauss-Jordan elimination, general equation of
a line, inconsistent system, intercept, line,
linear extrapolation, linear interpolation, linear
model, reduced row-echelon form, reduction
method, row-echelon form, slope, slopeintercept equation of a line, solution of a
system of equations, Theorem on the solution
of a system of linear equations
Writing in Math
Explain how the rate of change and slope are
related and how to find the slope of a line.
Shelby County Schools 2016/2017
Revised 8/31/16
9 of 11
Curriculum and Instruction – Mathematics
Quarter 2
TN STATE STANDARDS
HAS.REI.C.6
Solve systems of linear equations exactly
and approximately (e.g., with graphs),
focusing on pairs of linear equations in
two variables.
Conceptual Category: Algebra
Domain: Linear systems, matrices, and their
applications
A-LM.5 Identify and write the general solution to
a system of linear equations; in the case of an
infinite solution set, select various particular
solutions given specific properties.
Conceptual Category: Algebra
Domain: Linear systems, matrices, and their
applications
2. A-LM.5 Identify and write the general
solution to a system of linear equations;
in the case of an infinite solution set,
select various particular solutions given
specific properties.
Finite Math
CONTENT
Objectives:
Students will:
 Solve systems of linear equations exactly
and approximately (e.g., with graphs),
focusing on pairs of linear equations in two
variables.
INSTRUCTIONAL SUPPORT & RESOURCES
Tennessee Finite Math Textbook
5.2 Formulation and Solution of Systems of
Linear Equations in Two Variables
Additional Resources
FiniteHelp Video- 5.2
Writing in Math
Explain why you would solve an equation like
r = ax + b by solving the system of equations
y = r and y = ax + b.
Tennessee Finite Math Textbook
5.3 Formulation and Solution of Systems of
Linear Equations in Three or More Variables
Additional Resources
FiniteHelp Video- 5.3
Writing in Math
Describe the advantages and disadvantages
of using an augmented matrix to solve a
system of equations.
 Identify and write the general solution to a
system of linear equations; in the case of an
infinite solution set, select various particular
solutions given specific properties.
Objectives:
Students will:
 Identify and write the general solution to a
system of linear equations; in the case of an
infinite solution set, select various particular
solutions given specific properties.
Finite Math Student Resources
Shelby County Schools 2016/2017
Revised 8/31/16
10 of 11
Curriculum and Instruction – Mathematics
Quarter 2
Finite Math
RESOURCE TOOLBOX
Textbook Resources
Tennessee Finite Math
by Dan Maki and Maynard Thompson
Published by McGraw Hill 2011
CCSS/PARCC
Common Core Standards - Mathematics
Common Core Standards - Mathematics Appendix A
The Mathematics Common Core Toolbox
State Academic Standards (Finite Math)
TN Department of Education Math Standards
Edutoolbox (formerly TNCore)
Videos
Khan Academy
Illuminations (NCTM)
Discovery Education
The Futures Channel
The Teaching Channel
Teachertube.com
FiniteHelp Lecture Videos
Calculator
Texas Instruments Education
TI-Nspired
http://www.atomiclearning.com/ti_84
TICommonCore.com
http://www.casioeducation.com/educators
Interactive Manipulatives
Rossmanchance.com
Additional Sites
NCTM Math Illuminations
Core Math Tools
Math is Fun
Wolfram Math World
Nrich
STatistics Education Web
Online Algebra and Trigonometry Tutorial
Literacy
Glencoe Reading & Writing in the Mathematics Classroom
ACT
Finite Help
Graphic Organizers (9-12)
Graphic Organizers (dgelman)
Literacy Skills and Strategies for Content Area Teachers
TN ACT Information & Resources
ACT College & Career Readiness Mathematics Standards
Tasks/Lessons
UT Dana Center
Mars Tasks
Inside Math Tasks
Math Vision Project Tasks
Better Lesson
Edutoolbox (formerly TNCore)
Shelby County Schools 2016/2017
Revised 8/31/16
11 of 11