Download Chapter 2 Probabilities, Counting, and Equally Likely Outcomes

Curriculum and Instruction – Mathematics
Quarter 1
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 6/29/16
1 of 13
Curriculum and Instruction – Mathematics
Quarter 1
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 6/29/16
2 of 13
Curriculum and Instruction – Mathematics
Quarter 1
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 6/29/16
3 of 13
Curriculum and Instruction – Mathematics
Quarter 1
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 6/29/16
4 of 13
Curriculum and Instruction – Mathematics
Quarter 1
Finite Math
Topics Addressed in Quarter



Sets, Partitions, and Tree Diagrams
Probabilities, Counting, and Equally Likely Outcomes
Probability
Overview
During this quarter students develop a common notation and terminology for sets and set notation which will be useful in understanding probability. Sets, ways of combining sets, set operations
and a technique of representing sets with diagrams is developed. Students are introduced to a special type of set needed for the work on probability, and three methods of counting the
elements in particular kinds of sets are developed: partitions, tree diagrams, and the multiplication principle. Concepts from set theory are used to develop probability and the counting
principles/tools to compute probabilities in special situations. Finally in this quarter students will see how the properties of probabilities which hold in general settings are related to those which
hold when outcomes are equally likely. Also, students will study experiments and the concept of tree diagrams to represent certain types of experiments as a way of representing experiments
and as an aid to solving problems.
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 6/29/16
5 of 13
Curriculum and Instruction – Mathematics
Quarter 1
TN STATE STANDARDS
Conceptual Category: Geometry and
Measurement
Domain: Set Theory
G-ST
1. Define sets, subsets, proper subsets, and
empty sets with correct notation.
G-ST
3. Perform set operations such as union,
intersection, complement, and set
difference.
Finite Math
CONTENT
INSTRUCTIONAL SUPPORT & RESOURCES
Chapter 1 Sets, Partitions, and Tree Diagrams
(Allow approximately 2 weeks for instruction, review, and assessment)
Enduring Understanding(s):
Tennessee Finite Math Textbook
1.1 Review of Sets and Set Operations
 Mathematical models can be used to
interpret and predict the behavior of real
world phenomena.
Additional Resources
 Recognizing the predictable patterns in
FiniteHelp Video - 1.1
mathematics allows the creation of
FiniteHelp Chapter 1 Notation Guide
functional relationships.
Finite Help
 Varieties of mathematical tools are used
Finite Math Student Resources
to analyze and solve problems and
explore concepts.

Clear and precise notation and
mathematical vocabulary enables
effective communication and
comprehension.

Using prior knowledge of mathematical
ideas can help discover more efficient
problem solving strategies.
Essential Question(s):

How do I use the language of math (i.e.
symbols, words) to make sense of/solve a
problem?

How does the math I am learning in the
classroom relate to the real-world?

How do I effectively communicate about
math with others in verbal form? In written
form?

How do I explain my thinking to others, in
written form? In verbal form?

How do I effectively represent quantities
and relationships through mathematical
Important Terms & Concepts (Chapter 1)
Cartesian product, complement, de Morgan’s
law, disjoint sets, element, empty(null) set,
experiment, intersection, Multiplication
Principle, number of elements in a set,
pairwise disjoint, partition, partition principle,
sample space, set, set equality, subset, tree
diagram, union, universal set, Venn Diagram
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
Shelby County Schools 2016/2017
Revised 6/29/16
6 of 13
Curriculum and Instruction – Mathematics
Quarter 1
TN STATE STANDARDS
Finite Math
CONTENT
INSTRUCTIONAL SUPPORT & RESOURCES
notation?
Objectives:
Students will:
Conceptual Category: Geometry and
Measurement
Domain: Set Theory
G-ST
2. Analyze and compare sets, including
using Venn Diagrams.

Develop a common notation and
terminology for sets and set operations.

Discuss the meaning of a universal set.
Objectives:
Students will:

Represent sets and relations
between sets using diagrams or
pictures.
Tennessee Finite Math Textbook
1.2 Venn Diagrams and Partitions
Additional Resources
FiniteHelp Video- 1.2
Finite Math Student Resources
Task
Q1_finiteresource_GreatExpectinProb
Conceptual Category: Geometry and
Measurement
Domain: Set Theory
G-ST
3. Perform set operations such as union,
intersection, complement, and set
difference.
Conceptual Category: Data Analysis.
Statistics, and Probability
Domain: Organize and Interpret data
D-ID
3. Analyze survey data using Venn
Diagrams
Objectives:
Students will:

Determine the number of subsets of a set.

Perform operations with sets.
Tennessee Finite Math Textbook
1.3 Sizes of Sets
Additional Resources
FiniteHelp Video- 1.3
Shelby County Schools 2016/2017
Revised 6/29/16
7 of 13
Curriculum and Instruction – Mathematics
Quarter 1
Finite Math
TN STATE STANDARDS
Conceptual Category: Geometry and
Measurement
Domain: Set Theory
G-ST
4. Operate with sets and use set theory to
solve problems.
CONTENT
Objectives:
Students will:

Use three methods to represent sets.
INSTRUCTIONAL SUPPORT & RESOURCES
Tennessee Finite Math Textbook
1.4 Sets of Outcomes and Trees
Additional Resources
FiniteHelp Video- 1.4
Chapter 2 Probabilities, Counting, and Equally Likely Outcomes
(Allow approximately 3 weeks for instruction, review, and assessment)
Conceptual Category: Data Analysis,
Statistics, and Probability
Domain: Organize and interpret data
D-ID
2. Use a variety of counting methods to
organize information, determine
probabilities, and solve problems.
Enduring Understanding(s):

How to distinguish between types of
probability

How to find the probability of two events
occurring in sequence

How to find the probability that at least
one of two events will occur

How to count the number of ways an
event can occur

How to represent and interpret binomial
probability distributions
Essential Question(s):

How can one differentiate between the
three types of probability?

What is conditional probability?

How can one determine if two events will
occur in sequence?

How can one determine if two events are
mutually exclusive?
Objectives:
Students will:

Tennessee Finite Math Textbook
2.1 Probabilities, Events, and Equally Likely
Outcomes
Additional Resources
FiniteHelp Video- 2.1
Finite Help
Finite Math Student Resources
Important Terms & Concepts (Chapter 2)
Assignment of probabilities, binomial
coefficients, combination principle, deductive
method of assigning probabilities, dependent
events, equally likely outcomes, equiprobable
measures, event, experiment, mutually
exclusive events, performance, permutation,
permutation principle, probability of an event,
probability of E, probability measure, Relative
frequency method of assigning probabilities,
simple event, weight
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
Use the Fundamental Counting Principle
Shelby County Schools 2016/2017
Revised 6/29/16
8 of 13
Curriculum and Instruction – Mathematics
Quarter 1
TN STATE STANDARDS
Finite Math
CONTENT
INSTRUCTIONAL SUPPORT & RESOURCES
to determine the number of possible
outcomes in a given situation.

Use the possible outcomes to determine
the probability of an event/experiment.
Conceptual Category: Data Analysis,
Statistics, and Probability
Domain: Develop concepts in probability
D-CP
2. Evaluate expressions indicating
permutations or combinations.
D-CP
3. Define the relationship between
permutations and the multiplication
principle.
Objectives:
Students will:
Conceptual Category: Data Analysis,
Statistics, and Probability
Domain: Develop concepts in probability
D-CP
2. Evaluate expressions indicating
permutations or combinations.
Objectives:
Students will:
Conceptual Category: Data Analysis,
Statistics, and Probability
Domain: Develop concepts in probability
D-CP
1. Differentiate between permutations and
combinations.
4. Use permutations and combinations to
compute probabilities of compound
events and solve problems.
Objectives:
Students will:


Use the permutations formula to find the
number of ways a group of objects can be
arranged in order.
Use the combinations formula to find
the number of ways to choose
several objects from a group without
regard to order.
 Distinguish between using permutations and
combinations to solve problems.
 Use counting principles (permutations,
combinations, and fundamental counting
principle) to find probabilities.
Tennessee Finite Math Textbook
2.2 Counting Arrangements: Permutations
Additional Resources
FiniteHelp Video- 2.2
Finite Math Student Resources
Tennessee Finite Math Textbook
2.3 Counting Partitions: Combinations
Additional Resources
FiniteHelp Video- 2.3
Tennessee Finite Math Textbook
2.4 Computing Probabilities by Using Equally
Likely Outcomes
Additional Resources
FiniteHelp Video- 2.4
Shelby County Schools 2016/2017
Revised 6/29/16
9 of 13
Curriculum and Instruction – Mathematics
Quarter 1
Finite Math
TN STATE STANDARDS
CONTENT
INSTRUCTIONAL SUPPORT & RESOURCES
Task(s)
Q1_finiteactivity_PermComb
(or see TI Activity: Permutations &
Combinations)
Chapter 3 Probability
(Allow approximately 4 weeks for instruction, review, and assessment)
Conceptual Category: Data Analysis,
Statistics, and Probability
Domain: Organize and interpret data
D-ID
2. Use a variety of counting methods to
organize information, determine
probabilities, and solve problems.
3. Analyze survey data using Venn
Diagrams
Enduring Understanding(s):
Probability is a tool for measuring long-term
behavior.
Essential Question(s):
• What is probability theory?
• How can probability be simulated?
• What is a probability distribution?
•
Tennessee Finite Math Textbook
3.1 Probability Measures: Axioms and
Properties
Additional Resources
FiniteHelp Video- 3.1
Finite Math Student Resources
How does one determine probability for a
given event?
Important Terms & Concepts (Chapter 3)
Axioms of a probability measure, Bayes
probabilities, Bayes’ Theorem, Bernoulli
process, Bernoulli trial, conditional probability,
equiprobable measure, independence,
probability measure, properties of a probability
measure, probabilities on trees, stochastic
process
Writing in Math
Glencoe Reading & Writing in the Mathematics
Classroom
Objectives:
Students will learn:
Graphic Organizers (9-12)
 Axioms for a Probability Measure
Graphic Organizers (dgelman)
 Properties for a Probability Measure
Literacy Skills and Strategies for Content Area
Teachers
Conceptual Category: Data Analysis,
Statistics, and Probability
Domain: Develop concepts in probability
D-CP
5. Understand and apply the relationship
between conditional probabilities and the
probabilities of individual events.
Objectives:
Students will:
 Distinguish between independent and
dependent events.
Tennessee Finite Math Textbook
3.2 Conditional Probability and Independence
Additional Resources
FiniteHelp Video- 3.2
 Compute conditional probabilities.
Task(s)
Q1_finitetask_Independence
Shelby County Schools 2016/2017
Revised 6/29/16
10 of 13
Curriculum and Instruction – Mathematics
Quarter 1
TN STATE STANDARDS
Finite Math
CONTENT
Conceptual Category: Data Analysis,
Statistics, and Probability
Domain: Organize and interpret data
D-ID
2. Use a variety of counting methods to
organize information, determine
probabilities, and solve problems.
D-ID
3. Analyze survey data using Venn
diagrams
Conceptual Category: Data Analysis,
Statistics, and Probability
Domain: Develop concepts in probability
D-CP
5. Understand and apply the relationship
between conditional probabilities and the
probabilities of individual events.
Conceptual Category: Data Analysis,
Statistics, and Probability
Domain: Organize and interpret data
D-ID
2. Use a variety of counting methods to
organize information, determine
probabilities, and solve problems.
Conceptual Category: Data Analysis,
Statistics, and Probability
Domain: Develop concepts in probability
D-CP
6. Calculate conditional probabilities using
Bayes’ Theorem.
INSTRUCTIONAL SUPPORT & RESOURCES
(or see TI Activity: Statistics: Independence is
the Word)
Objectives:
Students will:
 Use pictures, diagrams, and trees to find
conditional probabilities.
Tennessee Finite Math Textbook
3.3 Stochastic Process and Trees
Additional Resources
FiniteHelp Video- 3.3
Finite Math Student Resources
Objectives:
Students will:
 Find the probability of an event using Bayes’
Theorem.
Tennessee Finite Math Textbook
3.4 Bayes Probabilities
Additional Resources
FiniteHelp Video- 3.4
Shelby County Schools 2016/2017
Revised 6/29/16
11 of 13
Curriculum and Instruction – Mathematics
Quarter 1
TN STATE STANDARDS
Finite Math
CONTENT
Conceptual Category: Data Analysis,
Statistics, and Probability
Domain: Organize and interpret data
D-ID
2. Use a variety of counting methods to
organize information, determine
probabilities, and solve problems.
Conceptual Category: Data Analysis,
Statistics, and Probability
Domain: Develop concepts in probability
D-CP
5. Understand and apply the relationship
between conditional probabilities and the
probabilities of individual events.
Conceptual Category: Data Analysis,
Statistics, and Probability
Domain: Organize and interpret data
D-ID
2. Use a variety of counting methods to
organize information, determine
probabilities, and solve problems.
INSTRUCTIONAL SUPPORT & RESOURCES
Task(s)
Q1_finitetask_BayesTheorem
Objectives:
Students will:
Tennessee Finite Math Textbook
3.5 Bernoulli Trials
 Determine if an event is a Bernoulli trial.
 Compute the probability of an event that is a
Bernoulli trial.
Additional Resources
FiniteHelp Video- 3.4
Finite Math Student Resources
Shelby County Schools 2016/2017
Revised 6/29/16
12 of 13
Curriculum and Instruction – Mathematics
Quarter 1
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
Literacy
Glencoe Reading & Writing in the Mathematics Classroom
ACT
Finite Help
Graphic Organizers (9-12)
TN ACT Information & Resources
ACT College & Career Readiness Mathematics Standards
Graphic Organizers (dgelman)
Literacy Skills and Strategies for Content Area Teachers
Shelby County Schools 2016/2017
Revised 6/29/16
13 of 13