MORRIS SCHOOL DISTRICT
MORRISTOWN, NJ
2012-2013
ALGEBRA II CURRICULUM
S U P ER IN TE N DE NT
Dr. Thomas J. Ficarra
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Board of Education: 2012-2013
Nancy Bangiola, President
Board Members:
Dr. Peter Gallerstein, Vice President
Alan Albm (Morris Plains Representative)
Fran Rossoff
Jeanette Thomas
Lisa Pollak
Teresa Murphy
Ann Rhines
Dr. Angela Rieck
Leonard Posey
Central Office:
Dr. Thomas J. Ficarra, Superintendent of Schools
Andy Williams, Director of Curriculum, Grades 6 – 12
Michael Amendola, Supervisor of Mathematics & Science
Morristown High School Administration
Linda D. Murphy, Principal
Curriculum Writers:
Kathleen Wood
Kerry FitzMaurice
Christina Bifulco
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Table of Contents
Part I
Rationale and Philosophy
Goals and Objectives
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6
Part II
Units of Study
Mastery Objectives
Teaching and Learning Activities
Assessment and Testing strategies
Text and Materials
Procedures for use of Supplemental Materials
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Part III
Curriculum Map
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Part IV: Appendix
References
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Rationale and Philosophy
In order to prepare for global competition and high expectations for all, Morris School District students must have increased opportunities for
mathematical experiences that extend critical thinking and reasoning. Specifically, access to higher mathematics is essential. Algebra II is a
course to build on their work with linear, quadratic, and exponential functions, students extend their repertoire of functions to include
polynomial, rational, and radical functions. Students work closely with the expressions that define the functions, and continue to expand and
hone their abilities to model situations and solve equations.
Key considerations:
 State and National Expectations
There has been great activity on the state and national level in terms of expectations for the skills, knowledge and expertise students
should master in mathematics to succeed in work and life in the 21 st century. NJ is currently moving forward with the “Framework for 21 st
Century Learning,” a partnership with business, education, and government to develop a collective vision to strengthen American
education (www.21stcenturyskills.org). The six components include: Core Content – Mathematics, 21st Century Content, Learning and
Thinking skills, Information and Communications Technology (ICT) Literacy, Life Skills – Real World Applications, and 21st Century
Assessments.
 Equity and Access to Higher Mathematics
The belief that all students, not just a select few, have access to mathematical learning environments that enable them to meet worldclass standards for both college and the world of work continues to be an essential goal of the Morris School District. The National Council
of Teachers of Mathematics initially reflected this important consideration in The Equity Principle whereby, “Excellence in mathematics
education requires equity – high expectations and strong support for all students (NCTM 2000).” Subsequent research has revealed that
“It is important that high schools do everything to promote success among all students – encouraging enrollment by students from all
demographics in advanced math courses.” NCTM’s recent release of the scientifically research-based, Focus in High School Mathematics:
Reasoning and Sense Making (2009), stressed the role of educators to help students with a wide range of backgrounds develop
connections between applications of new learning and their existing knowledge, increasing their likelihood of understanding and thereby
allowing increased options and entry into advanced mathematics.
 Building on Existing Partnerships
The Algebra II course fills a critical need for extensive study towards the development of abstract algebraic thinking. This goal is aligned
with the district’s vision of providing rich opportunities for all students to move forward on Blooms’ Knowledge Taxonomy continuum
from Knowledge and Awareness to Comprehension, Application, Analysis, Synthesis, and Evaluation. Specifically, scaffolding on a
foundation of algebra skills, this course moves students from the acquisition and assimilation of concepts towards increased application
and adaptation. Adaptation occurs when students have the competence to think in complex ways and to apply their knowledge and skills.
Building on the district’s existing partnerships, this course would provide a springboard for entrance into higher level mathematics courses
for a greater number of Morris School District students. [Rigor and Relevance Framework: International Center for Leadership in
Education, Bill Daggett and Ray McNulty, www.leadered.com]
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Goals and Objectives (outcomes):
Algebra is the study of patterns and functions. In teaching and learning Algebra II, it is important for teachers and students to comprehend the
following Big Ideas and Enduring Understandings and to establish connections and applications of the individual skills and concepts to these
broad principles as the critical goals and objectives of the course:
 Patterns and Functions
Algebra provides language through which we describe and communicate mathematical patterns that arise in both mathematical and nonmathematical situations, and in particular, when one quantity is a function of a second quantity or where the quantities change in
predictable ways. Ways of representing patterns and functions include tables, graphs, symbolic and verbal expressions, sequences, and
formulas.
 Equivalence
There are many different – but equivalent – forms of a number, expression, function, or equation, and these forms differ in their efficacy
and efficiency in interpreting or solving a problem, depending on the context. Algebra extends the properties of numbers to rules
involving symbols; when applied properly, these rules allow us to transform and expression, function, or equation into an equivalent form
and substitute equivalent forms for each other. Solving problems algebraically typically involves transforming one equation to another
equivalent equation until the solution becomes clear.
 Representation and Modeling with Variables
Quantities can be represented by variables, whether the quantities are unknown, changing over time, parameters, or probabilities.
Representing quantities by variables gives us the power to recognize and describe patterns, make generalizations, prove or explain
conclusions, and solve problems. Representing quantities with variables also enables us to model situations in all areas of human
endeavor and to represent them abstractly.
 Linearity
In many situations, the relationship between two quantities is linear so the graphical representation of the relationship is a geometric line.
Linear functions can be used to show a relationship between two variables that has a constant rate of change and to represent the
relationship between two quantities, which vary proportionately. Linear functions can also be used to model, describe, analyze, and
compare sets of data.
 Connections Between Algebra & Geometry
Geometric objects can be represented algebraically (for example, lines can be described using coordinates), and algebraic expressions can
be interpreted geometrically (for example, systems of equations and inequalities can be solved graphically).
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 Connections Between Algebra & Systematic Counting, Probability, and Statistics
Algebra provides a language and techniques for analyzing situations that involve chance and uncertainty, including systematic listing and
counting of all possible outcomes (as well as informal explorations of Pascal’s Triangle), the determination of their probabilities, the
calculation of probabilities of various events, predications based on experimental probabilities, and correlations between two variables.
 Applications of Algebraic Concepts
Using the model, Webb’s Depth-of-Knowledge Levels, Algebra II has a primary goal of increasing opportunities for students to reinforce
Levels 1 & 2 and to engage in Level 3 Strategic Thinking and Level 4 Extended Thinking:
Strategic Thinking requires “reasoning, planning, using evidence, and a higher level of thinking than the previous two levels. In most
instances, requiring students to explain their thinking is a Level 3. Activities that require students to make conjectures are also at this
level. The cognitive demands at Level 3 are complex and abstract. The complexity does not result from the fact that there are multiple
answers, a possibility for both Levels 1 and 2, but because the task requires more demanding reasoning. An activity, however, that has
more than one possible answer and requires students to justify the response they give would most likely be a Level 3. Other Level 3
activities include drawing conclusions from observations; citing evidence and developing a logical argument for concepts; explaining
phenomena in terms of concepts; and using concepts to solve problems.”
Extended Thinking “requires complex reasoning, planning, developing, and thinking most likely over an extended period of time. The
extended time period is not a distinguishing factor if the required work is only repetitive and does not require applying significant
conceptual understanding and higher-order thinking. For example, if a student has to take the water temperature from a river each
day for a month and then construct a graph, this would be classified as a Level 2. However, if the student is to conduct a river study
that requires taking into consideration a number of variables, this would be a Level 4. At Level 4, the cognitive demands of the task
should be high and the work should be very complex. Students should be required to make several connections—relate ideas within
the content area or among content areas—and have to select one approach among many alternatives on how the situation should be
solved, in order to be at this highest level. Level 4 activities include designing and conducting experiments; making connections
between a finding and related concepts and phenomena; combining and synthesizing ideas into new concepts; and critiquing
experimental designs.” (Wisconsin Center for Educational Research, www.facstaff.wcer.wisc.educ/normw)
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Units of Study:
(Note: emphasis on problem solving, applications, and modeling)
Unit 1: Linear Functions; Systems of Linear Equations and Inequalities
Topic 1: Linear Equations and Inequalities
Topic 2: Systems of Equations & Inequalities
Unit 2: Quadratic Functions
Topic 3: Radical Expressions, the Parabola, Factoring and Complex Numbers
Unit 3: Polynomial Functions
Topic 4: Graphing, Operations, and Solving
Unit 4: Radical Functions and Rational Exponents
Topic 5: Simplifying, Operations and Solving Radical Equations and Inequalities
Unit 5: Rational Functions
Topic 6: Operations, Solving, and Graphing Rational Equations and Inequalities
Topic 7: Functions, Inverses, Transformations
Unit 6: Exponential & Logarithmic Functions
Topic 8: Evaluate, Simplify and Solve
Unit 7: Trigonometric Functions
Topic 9: Periodic Functions, Radians, and Graphing
Unit 8: Probability & Statistics
Topic 10: Central Tendency, Standard Deviation, and Normal Distribution
Optional Units:
Unit 9: Matrices
Topic 11: Operations and Solving Systems with Matrix Equations
Unit 10: Conic Sections
Topic 12: Parabolas, Circles, Ellipses, Hyperbolas
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Mastery Objectives:
MASTERY OBJECTIVES
(NJCCCS)
Algebra II is correlated to the 2010 Common Core Curriculum Content Standards in Mathematics.
2010 Common Core Curriculum Content Standards in Mathematics
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Modeling Standards
Modeling is best interpreted not as a collection of isolated topics but rather in relation to other standards. Making mathematical models is a
Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★).
All course of study must include the following, which replace the Workplace readiness standards:
Career Education and Consumer, Family, and Life Skills
Career and Technical Education: All students will develop career awareness and planning, employability skills, and foundational knowledge
necessary for success in the workplace.
Consumer, Family, and Life Skills: All students will demonstrate critical life skills in order to be functional members of society.
Scans Workplace Competencies
Effective workers can productively use:
Resources: They know how to allocate time, money, materials, space and staff.
Interpersonal Skills: They can work on teams, teach others, serve customers, lead, negotiate, and work well with people from culturally diverse
backgrounds.
Information: They can acquire and evaluate data, organize and maintain files, interpret and communicate, and use computers to process
information.
Systems: They understand social, organizational, and technological systems; they can monitor and correct performance; and they can design or
improve systems.
Technology: They can select equipment and tools, apply technology to specific tasks, and maintain and troubleshoot equipment.
Scans Foundations Skills
Competent workers in the high-performance workplace need:
Basic Skills: reading, writing, arithmetic, and mathematics, speaking and listening.
Thinking Skills – the ability to learn, reason, think creatively, make decisions, and solve problems.
Personal Qualities – individual responsibility self-esteem and self-management, sociability, integrity, and honesty.
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Teaching/Learning Activities
The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their
students through teaching and learning activities. A secondary document [Curriculum Map: Algebra II], has been prepared to provide greater
detail as to specific teaching/learning activities for this course.
Teaching/Learning practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of
these are the NCTM Process Standards of problem solving, reasoning and proof, communication, representation, and connections.
The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning,
strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in
carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as
sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).
The teaching/learning activities in Algebra II are focused on developing the following mathematical proficiencies:
1. Make sense of problems and persevere in solving them.
Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They
analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution
pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the
original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students
might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get
the information they need. Mathematically proficient students can 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. Mathematically proficient
students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can
understand the approaches of others to solving complex problems and identify correspondences between different approaches.
2. Reason abstractly and quantitatively.
Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities
to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically
and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to
contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative
reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of
quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.
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3. Construct viable arguments and critique the reasoning of others.
Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing
arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze
situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others,
and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from
which the data arose. Mathematically proficient students are also able to 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. Elementary students can construct
arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though
they are not generalized or made formal until later grades. Later in algebra, students learn to determine domains to which an argument applies.
Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve
the arguments.
4. Model with mathematics.
Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In
early algebraic reasoning, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply
proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a
design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply
what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need
revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, twoway tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret
their 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.
5. Use appropriate tools strategically.
Mathematically proficient students 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. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of
these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school
students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using
estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the
results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade
levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve
problems. They are able to use technological tools to explore and deepen their understanding of concepts.
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6. Attend to precision.
Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their
own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful
about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and
efficiently, express numerical answers with a degree of precision appropriate for the problem context. Initially, students give carefully formulated
explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.
7. Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure. Students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3,
in preparation for learning about the distributive property. In the expression x2 + 9x + 14, algebra students can see the 14 as 2 × 7 and the 9 as 2 +
7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems.
They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single
objects or as being composed of several objects. For 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.
8. Look for and express regularity in repeated reasoning.
Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. By paying attention to
the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, algebra students might abstract the
equation (y – 2)/(x – 1) = 3. 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. As they work to solve a problem, mathematically proficient students
maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.
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Assessment and Testing Strategies
Sound and productive classroom assessments are built on a foundation of the following five key dimensions (Stiggins et al, 2006):
Key 1: Assessment serves a clear and appropriate purpose.
Did the teacher specify users and uses, and are these appropriate?
Key 2: Assessment reflects valued achievement targets.
Has the teacher clearly specified the achievement targets to be reflected in the exercises? Do these represent important learning
outcomes?
Key 3: Design.
Does the selection of the method make sense given the goals and purposes? Is there anything in the assessment that might lead to
misleading results?
Key 4: Communication.
Is it clear how this assessment helps communication with others about student achievement?
Key 5: Student Involvement.
Is it clear how students are involved in the assessment as a way to help them understand achievement targets, practice hitting those
targets, see themselves growing in their achievement, and communicate with others about their success as learners?
The Algebra II course will include a variety of assessment tools for the effective teaching and learning of mathematics. Indicators of Sound
Classroom Assessment Practice will consist of both formative and summative assessments that may include, but are not limited to:

Observation

Interviews

Portfolios (Project, Growth, Achievement, Competence, Celebration)

Paper-and-pencil tests/quizzes

Performance Tasks

Journals/Self-Reflection

Correct and Reflect on all assessments (optional for students)
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Texts and Materials
Student Text:
 Holt, Rinehart and Winston Algebra II (2003).
 Prentice Hall Algebra II Common Core (2012)
Teacher Materials and Resources:
o Blueprints for Success Gold Seal Lessons: Successful Practice Network. www.leadered.com/spn.html
o Greenes, C., & Rubenstein, R. (2008). Algebra and Algebraic Thinking in School Mathematics, Seventieth Yearbook. Reston, VA: National
Council of Teachers of Mathematics.
o National Council of Teachers of Mathematics. (2001). Navigating Through Algebra in Grades 9-12. Reston, VA: NCTM.
o National Council of Teachers of Mathematics. (2006). Navigating Through Mathematical Connections in Grades 9-12. Reston, VA: NCTM.
o National Council of Teachers of Mathematics. (2004). Navigating Through Probability in Grades 9-12. Reston, VA: NCTM.
o New Jersey Common Core Curriculum Standards 2010. www.njcccs.org.
o Prentice Hall teacher-provided materials/Digital path/SuccessNet. (2012). www.pearsonsuccessnet.com
Technology/Computer Software
o Wolframalpha.com
o TI-Nspire/TI-83/TI-84
o Geogebra www.geogebra.com
o Pearson SuccessNet www.pearsonsuccessnet.com
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Procedures for Use of Supplemental Instructional Materials
Instructional materials not approved by the Board of Education must be brought to the attention of the building principal or vice-principal before
use in any instructional area. Materials that are approved include all textbooks, videos, and other supplemental material acquired through
purchase orders, and/or other school funds. Resources from the County Education Media and Technology Center are also acceptable, with age
appropriateness reviewed.
All instructional materials not explicitly Board approved as outlined in above, which are intended for use in any instructional setting must be
approved by the building principal or vice- principal at least 5 schools days prior to use. The principal or vice-principal may request to review a copy
of the materials, video, etc, prior to use in the classroom.
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Curriculum Map
Unit 1 - Linear Functions; Systems of Linear Equations and Inequalities
Content/Objective
The learner will:
- solve linear and literal
equations and inequalities
- solve real-life word
problems
- create a linear equation
to model a situation
- define slope and yintercept to write a linear
equation, including parallel
and perpendicular lines
- write and graph a linear
equation
- write and solve
compound inequalities
- solve and graph absolute
value equations and
inequalities
Essential Questions/
Enduring Understandings
Essential Questions:
-How can we solve different
types of algebraic
equations?
-How can we write an
equation for a line that is
parallel or perpendicular to a
given line?
Enduring Understandings:
Slope is the rate of change
Suggested Activity/Appropriate Materials-Equipment
Instructional Focus:
- Create equations and inequalities in one variable and use them to
solve problems (A.CED.1)
Sample Assessments:
Supplement resources with equations that include operations with all
numbers. (A.CED.2)
Solve for x: (A.CED.3)
-(x+2) - 2x = -3(x+1)
c(x+2) - 5 = b(x+3)
x/a - 5 = b
4(n-2) - 6 > 18
- Develop through provoking questions using linear equations to
reflect real-life situations. (A.CED.3)
The park is opening a new swimming pool. You can pay a daily fee
of $3 or buy a membership for the 12-week summer for $82 and pay
only $1 per day to swim. How many days would you have to swim
to make the membership worthwhile?
- Which equation describes a line with slope 12 and y-intercept 4 ?
(A.CED.1)
a. y=12(x+4) b. y=12x+4
c. y=4x + 12
d. y = x + 3
- Three times the difference of twelve and a number is fifty. Find the
number. (A.CED.1)
- Six less than a number is greater than 54.(A.CED.1)
- The sum of four consecutive integers is 250. What is the greatest
of these integers? (A.CED.4)
- Connect with Geometry using formula's like A=lw, A = πr ^ 2
(A.CED.4)
- Absolute value equations can be analyzed using the definition of
absolute value.
- Calculate and interpret the average rate of change of a function
(presented symbolically or as a table) over a specified interval.
Estimate the rate of change from a graph (F.IF.6)
Suggested
Evaluation/Assessment
Pre-Assessment, Checkpoint
exercises, DoNows,
portfolios, oral questioning,
closure, self-reflection
journals, projects, tests,
technology-based
assessments, teacher
constructed quizzes and tests
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The learner will:
- solve and graph a system
of linear equations and
inequalities
- solve real-life word
problems
Essential Questions:
How does graphing a
system help you to solve?
How does writing equivalent
equations help you to solve
a system of equations?
Enrichment Extension:
- solve a system of linear
equations with three
variables
- write and graph a set of
constraints for a linearprogramming problem
- use linear programming
to find the maximum or
minimum value of an
objective function
Enduring Understanding:
The solution to a system of
equations is the point of
intersection
Instructional Focus:
-Systems can be solved using a variety of strategies, including
substitution, graphing, and elimination
-Represent constraints by equations or inequalities, and by systems
of equations... (A.CED.3)
- Create equations in two or more variables to represent
relationships... (A.CED.2)
Pre-Assessment, Checkpoint
exercises, Do Nows,
portfolios, oral questioning,
closure, self-reflection
journals, projects, tests,
technology-based
assessments, teacher
constructed quizzes and tests
Sample Assessments:
A chemist uses a solution containing 30% insecticide and another
solution containing 50% insecticide. How much of each solution
should
the chemist mix to get 200 L of a 43% insecticide?
Solve: 3x + 2y = 41
5x - 3y = 24
Solve by graphing: y > x + 4
y < 2x - 1
Instructional Strategies:
- Algebraic principles and techniques can be extended from familiar
linear equations to more complex situations
- Equations, inequalities, and systems of equations can be used to
model the constraints of a given situation.
- Systems can be solved using a variety of strategies, including
substitution, graphing, and elimination.
- Use a graphing calculator to solve a linear programming problem.
Graphing calculators can be used to find the feasible region and its
vertices. This can be a helpful tool when the coordinate of the
vertices are not integers.
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Unit 2 - Quadratic Functions
Content/Objective
The learner will:
- simplify radical
expression containing
numbers
- define, identify, and
graph a quadratic function,
including vertex, axis of
symmetry, and
maximum/minimum values
- write a quadratic in
standard form
- model data with quadratic
functions
- factor a quadratic
expression
- use factoring to solve a
quadratic equation and
find the zeros of a
quadratic function
- use completing the
square to solve a quadratic
equation
- use the vertex form of a
quadratic function to locate
the axis of symmetry of its
graph
- use the quadratic formula
to find the real roots of
quadratic equations
- determine the number of
solutions by using the
discriminant
- perform operations with
complex numbers
- find complex number
solutions of quadratic
equation
- solve and graph
quadratic inequalities
Essential Questions/
Enduring Understandings
Essential Questions:
What are the advantages of
a quadratic function in vertex
form? in standard form?
- What characteristics of
quadratic functions are
important?
Enduring Understanding:
Quadratic equations can be
solved in more than one way
For any quadratic function in
standard form the values of
a, b, and c provide key
information about its graphs
Suggested Activity/Appropriate Materials-Equipment
Instructional Focus:
- Solving quadratic equations and inequalities using factoring,
completing the square and the quadratic formula, including complex
solutions
- Graphing quadratic equations by hand and using a technology
graphing aid
- Describing how the discriminant indicates the nature of the
solutions of the equation .
- Create equations and inequalities in one variable and use them to
solve problems (A.CED.1)
Suggested
Evaluation/Assessment
Pre-Assessment, Checkpoint
exercises, DoNows,
portfolios, oral questioning,
closure, self-reflection
journals, projects, tests,
technology-based
assessments, teacher
constructed quizzes and tests
Sample Assessments:
- Let f(x) = x^2 + x + c.
Part A. For what values of c are the roots of f(x) not real? Show
or explain your work.
Part B. Using one of the values for c that you found in Part A,
determine the roots of f(x). Show or explain your work.
- Write a quadratic equation whose roots have a sum of 4 and a
product of -4, and find the roots of this equation.
- The area in square feet of a rectangular field is x^2-120x+3500.
The width, in feet, is x-50. What is the length is feet?
Instructional Strategies:
- Have students investigate the connections between the various
parts of the quadratic formula (-b/2a and b^2-4ac), the graph of the
function and the solution(s) (A.SSE.1)
- Give students several quadratic equations to solve and have them
complete a table showing the equation, the roots, the sum of the
roots and the product of the roots. Then have them use the
information in the table to make a conjecture about the relationship
between a quadratic equation and the sum and product of its roots,
writing their conjectures in a paragraph. Have them exchange
papers with another student, read the conjecture, and decide
whether it is valid. If it is valid, they should write questions that will
help prove the conjecture. If it is not valid, they should write
questions that will challenge it. Have students return the papers and
respond to the questions raised.
- Have students work with models of everyday physics situations
including trajectory and velocity-time
- Use geotechnical engineering and measurement of compaction of
24
soil as an additional application of quadratic functions
- Use technology to model a larger set of data with a quadratic
function
- Know there is a complex number i such that i^2=-1 and every
complex number has the form a + bi with a and b real (N.CN.1)
- Use the relation i^2=-1 and the commutative, associative, and
distributive properties to add, subtract, and multiple complex
numbers (N.CN.2)
- Solve quadratic equations with real coefficients that have complex
solutions (N.CN.7)
- Extend polynomial identities to the complex numbers (N.CN.8)
- Know the Fundamental Theorem of Algebra; show that it is true for
quadratic polynomials (N.CN.9)
25
Unit 3 - Polynomial Functions
Content/Objective
The learner will:
- identify and classify
polynomials
- graph polynomial
functions, include
describing the end
behavior and the
Fundamental Theorem of
Algebra
- perform operations with
polynomials, including long
division, synthetic division,
the Factor and Remainder
Theorems
- solve polynomial
equations by factoring,
graphing, and the Rational
Root Theorem
- analyze the factored form
of a polynomial
- write a polynomial from
its zeros
- expand a binomial using
Pascal's Triangle
- use the Binomial
Theorem
Essential Questions/
Enduring Understandings
Essential Questions:
-What characteristics of
functions are important?
- For a polynomial function,
how are factors, zeros,
roots, and x-intercepts
related?
Enduring Understandings:
- A quantity can be
represented numerically in
various ways.
- The degree of a polynomial
equation tells you how many
roots the equation has.
- Key characteristics include
domain and range,
intercepts, end behavior,
and degree.
- Finding the zeros of a
polynomial function will help
you factor the polynomial,
graph the function, and solve
the related polynomial
equation.
- If f(x) is a polynomial, and a
is a root of f, then (x-a) is a
factor of f(x). If (x-a) is a
repeated factor, that is (xa)^k is a factor of f(x), then a
is a root with multiplicity of k.
Suggested Activity/Appropriate Materials-Equipment
Instructional Focus:
- Identifying the type of symmetry and relation to odd/even
exponents
- Identifying or writing a polynomial function of a given degree
(A.SSE.1)
- Explain the relationship between the number of real (and complex)
solutions and the graph of a polynomial equation.
- Use the structure of an expression to identify ways to rewrite it
(A.SSE.2)
Suggested
Evaluation/Assessment
Pre-Assessment, Checkpoint
exercises, DoNows,
portfolios, oral questioning,
closure, self-reflection
journals, projects, tests,
technology-based
assessments, teacher
constructed quizzes and tests
Sample Assessments:
- Find the maximum volume for a box made from a sheet of 8" x 10"
paper if squares of varying sizes are cut from each corner.
(Geometric problems involving volume provide a context for cubic
functions.)
Instructional Strategies:
- Consider differences among subsequent entries in a table of
values with a constant increment in the x-values in order to
determine the degree of the polynomial
- Know the Fundamental Theorem of Algebra (N.CN.9)
- Understand that polynomials form a system of analogous to the
integers, namely, they are closed under the operation of addition,
subtraction, and multiplication (A.APR.1)
- Add, subtract, and multiply polynomials (A.APR.1)
- Find the areas and volumes of figures with sides given as a
polynomial length
- Understand and apply the Remainder Theorem (A.APR.2)
- Identify zeros of a polynomials when suitable factorizations are
available, and use the zeros to construct a rough graph of the
function defined by the polynomial (A.APR.3 and F.IF.4) ... show end
behavior (F.IF.7)
- Prove and use the polynomial identities (A.APR.4)
- Understand and apply the Binomial Theorem (A.APR.5)
- Relate the domain of a function to its graph and, where applicable,
to the quantitative relationship it describes (F.IF.5)
26
Unit 4 - Radical Functions and Rational Exponents
Content/Objective
The learner will:
- simplify expressions
involving positive and
negative exponents
- add, subtract, multiply,
divide, and simplify radical
expressions
- rationalize a
denominator, including the
conjugate
- simplify expressions with
rational exponents
- solve radical equations
and inequalities in one
variable showing how
extraneous solutions may
arise
Essential Questions/
Enduring Understandings
Essential Questions:
-What are some equivalent
ways to represent algebraic
expressions?
- To simplify the nth root of an
expression, what must be true
about the expression?
- When you square each side of
an equation, is the resulting
equation equivalent to the
original?
- When should you check for
extraneous solutions?
Enduring Understandings:
- A quantity can be represented
numerically in various ways.
- Convert between and among
radical and exponential forms
of numerical and algebraic
expressions.
- You can write a radical
expression in an equivalent
form using a fractional
(rational) exponent instead of a
radical symbol
- Solving a square root equation
may require that you square
each side of the equation. This
can introduce extraneous
solutions.
Suggested Activity/Appropriate Materials-Equipment
Instructional Focus:
- Solve radical equations numerically, algebraically, and graphically with
and without the use of technology
- Converting between expressions involving rational exponents and those
involving roots and integral powers
- Consider domain restrictions (undefined values) when finding solutions
Suggested
Evaluation/Assessment
Pre-Assessment, Checkpoint
exercises, DoNows, portfolios,
oral questioning, closure, selfreflection journals, projects,
tests, technology-based
assessments, teacher constructed
quizzes and tests
Sample Assessments:
- Shirley wants to use the [y^x] key on her calculator to find the
approximate value of the expression
. Help Shirley by writing the
expression in the form y^x.
- How are the processes of multiplying radical expressions and multiplying
polynomial expression alike? How are the processes different?
Instructional Strategies:
- Problems involving applications of the Pythagorean Theorem, perimeter,
or distance are useful here. For example, students might be asked to find
the perimeter of a rectangle whose sides are formed by the hypotenuse of
right triangles: the legs of the first triangle have length 1 and 2, while the
legs of the second have lengths 2 and 4.
- Solve simple ... radical equations/inequalities in one variable, and give
examples showing how extraneous solutions may arise. (A.REI.2/11)
- Stress that the laws of exponents are the same for rational exponents as
they were for whole number exponents in Algebra I.
- Create equations and inequalities in one variable and use them to solve
problems (A.CED.1)
- Graph functions expressed symbolically and show key features of the
graph, by hand in simple cases and using technology for more complicated
cases. Graph square root functions. (F.IF.7)
27
Unit 5 - Rational Functions
Content/Objective
The learner will:
- add, subtract, multiply,
divide, and simplify rational
expressions
- simplify complex fractions
- solve rational equations and
inequalities in one variable
showing how extraneous
solutions may arise
- graph rational functions
including vertical and
horizontal asymptotes
The learner will:
- determine whether a relation
is a function
- write a function in function
notation and evaluate it
Essential Questions/
Enduring Understandings
Essential Questions:
-What are the effects of varying
parameters in rational
functions?
- are a rational expression and
its simplified form equivalent?
Enduring Understanding:
- a rational function is a ratio of
polynomial functions. If a
rational function is in simplified
form and the polynomial in the
denominator is not constant, the
graph of the rational function
features asymptotic behavior. it
looks quite different from the
graphs of either of its
polynomial components
Essential Questions:
- What is the difference
between an equation and a
function?
-When does a transformation
Suggested Activity/Appropriate Materials-Equipment
Instructional Focus:
- Create equations and inequalities in one variable and use them to solve
problems (A.CED.1)
- Rewrite simple rational expressions in different forms;... using inspection
or for more complicated examples a computer algebra system (A.APR.6)
- Understand that rational expressions form a system analogous to the
rational numbers (A.ARP.7)
- Add, subtract, multiply, and divide rational expressions (A.ARP.7)
- Solve simple rational equations in one variable and give examples
showing how extraneous solutions may arise (A.REI.2)
- Graph functions expressed symbolically and show key features of the
graph, by hand in simple cases and using technology for more complicated
cases (F.IF.7)
Suggested
Evaluation/Assessment
Pre-Assessment, Checkpoint
exercises, DoNows, portfolios,
oral questioning, closure, selfreflection journals, projects,
tests, technology-based
assessments, teacher constructed
quizzes and tests
Sample Assessments:
- The force required to stretch a spring is directly proportional to the
distance d. A force of 6 lbs stretches a spring 1.2 ft. What force will be
needed to stretch the spring 3 ft?
(Answer: 15 lbs)
- Write three rational expressions that simplify to (x+1)/(x-1)
- One pump can fill a tank with oil in 4 hours. A second pump can fill the
same tank in 3 hours. If both pumps are used at the same time, how long
will they take to fill the tank?
Instructional Strategies:
- Focus on domain restrictions and how these domain restrictions will
impact the graph of the rational function
- A toy company is considering a cube or sphere-shaped container for
packaging a new product. The height of the cube would equal the diameter
of the sphere. Compare the ratios of the volumes to the surface areas of the
containers. Which packaging will be more efficient? For a sphere SA=
- Application: Total resistance of parallel circuits
- Explain the difference in the process of adding two rational expressions
using the lowest common denominator and adding them using a common
denominator that is not the LCD. Include an example in your explanation
- Students can use the dot-and-connected mode on a graphing calculator to
graph rational functions. Sometimes the dot mode is better since the
connected mode can join branches of a graph that should be separated.
Instructional Focus:
- Identify the effect on the graph of replacing f(x) by f(x) + k,... (F.BF.3)
- Find inverse functions (F.BF.4)
- Write a function that describes a relationship between two quantities
(F.BF.1)
Pre-Assessment, Checkpoint
exercises, DoNows, portfolios,
oral questioning, closure, selfreflection journals, projects,
tests, technology-based
28
- perform operations with
functions to write new
functions
- find the composition of two
functions
- find the inverse of a relation
or function
- determine whether the
inverse of a function is a
function
- analyze transformation of
functions, including
translations, reflections,
rotations, compressions, and
stretches
- write and graph a piecewise
and step functions
change the location of a graph
and when does it change the
shape of a graph?
- How are a function and its
inverse function related
(graphically, numerically and
algebraically)?
- How can you perform
computations with functions
and how do these combinations
effect the domain of the
resultant function?
- How can you use functions to
describe situations that change
drastically from one point in
time to the next?
Enduring Understanding:
There are sets of functions
called families in which each
function is a transformation of a
special function called the
parent
Functions can be added,
subtracted, multiplied, and
divided based on how these
operations are performed for
real numbers. One difference,
however, is that the domain of
each function must be
considered
- Graph... piecewise-defined functions, including step functions and
absolute value functions (F.IF.7)
assessments, teacher constructed
quizzes and tests
Sample Assessment:
- Liza was absent from school and emailed two of her friends to help her
understand how to decide if a relation is a function.
Mike said: Make a table, and see if you get two of the same y-values.
John said: Look at the graph. See if a vertical line crosses the graph in
more than one place. If it does, then we have a function.
Which student is correct? Why? Provide a counterexample to explain any
errors made by either Mike or John.
- If k(x)=
and m(x)=3x, what is the domain of k(m(x)) over the
set of real numbers?
- Describe and graph the transformation: y = - l x-1 I - 5
- Given a graph, determine the piecewise function. Special note should be
taken for what values of x are defined with what piece of the function
-Describe the conditions under which an inverse relation is a function
-If f(x)= 3/5 +2/5x and g(x) = 2 / (5x-3), determine whether or not f and g
are inverses and explain how you know
Instructional Strategies:
- Connect algebraic operations with functions to the graphical
representations of their sums, differences, products and quotients.
- The formula V=4/3πr^3 expresses the relationship between the volume V
and radius r of a sphere. A weather balloon is being inflated so that the
radius is changing with respect to time according to the equation r=t+1,
where t is the time in minutes and r is the radius in feet. Write a
composition function to represent the volume of the weather balloon after t
minutes.
- How can finding the inverse of a function help you when you are given
values for the dependent values? --> find the inverse of formula for
converting Celsius to Fahrenheit temperatures
- Relate piecewise functions to social studies topics such as minimum wage
or the cost of mailing a first class letter.
- Research average wages in different countries over time, creating graphs
and piecewise-defined functions for each country.
- Discuss different functions and compare their properties, similarities and
differences (F.IF.9)
- You can graph inverses of functions on a graphing calculator by using the
“Drawinv” feature without knowing the equation of the inverse. This can
be used to view the shape of the graph and find particular points on the
graph. It will not, however, provide the equation that defines the inverse.
- Use an excel spreadsheet to perform function operations.
29
Unit 6 - Exponential and Logarithmic Functions
Content/Objective
The learner will:
- write and evaluate
exponential expressions to
model growth and decay
situations
- classify an exponential
function as representing
exponential growth or
exponential decay
- calculate the growth of
investments under various
conditions
- graph exponential functions
including those that have a
base of e
- write equivalent forms for
exponential and logarithmic
expressions and use to solve
- simplify and evaluate
expressions involving the
properties of logarithms
- graph logarithmic functions
in terms of transformations
and change of base formula
- evaluate, simplify and solve
natural logarithmic
expressions equations
- model and solve real-world
problems involving
exponential and logarithmic
relationships
Essential Questions/
Enduring Understandings
Essential Questions:
- What are the key
characteristics of an
Exponential function?
- Without graphing, how can
you tell When an Exponential
function represents Exponential
growth or Exponential decay?
- What kind of functions are
used to describe patterns that
grow through multiplication or
division?
- how do you model a quantity
that changes regularly over
time by the same percentage?
- how are exponents and
Logarithms related?
Enduring Understandings:
-Exponential notation is a
powerful way to express
repeated products of the same
number.
- Exponential functions have
numerous applications in the
natural world, in social science,
and in business.
- Distinguish between and
graph functions that are growth
or decay functions
- Translate from Exponential to
logarithmic form and vice versa
- Logarithms provide an
efficient method For solving
problems with variable
exponents
Suggested Activity/Appropriate Materials-Equipment
Instructional Focus:
- Explain or illustrate the effect that changes in a parameter (a or c) or the
base (b) have on the graph of the exponential function f(x)=ab^x+c.
- Write a function defined by an expression in different but equivalent forms
to reveal and explain different properties of the function (F.IF.8)
Sample Assessment:
- The equation c=523430(1.193)^t models the pounds of U.S. copper
produced in the period from 1987 to 1992. Which statement best interprets
the coefficient and base of this equation?
A. The copper production in 1987 was 523,430 pounds, and it
increased at a rate of 1.93% per year during that period.
**B. The copper production in 1987 was 523,430 pounds, and it
increased at a rate of 19.3% per year during that period.
C. The copper production increased by a factor of 523,430 x 1.193
pounds per year during that period.
D. The copper production at the beginning of 1987 was at 1.193
pounds, and it increased by a factor of 523,430 pounds per year during that
period.
- A company offers its employees a choice of two salary schemes (A or B)
over a period of ten years. Scheme A offers a starting salary of $33,000 the
first year and an annual increase of $1200 per year. Scheme B offers a
starting salary of $30,000 the first year and an annual increase of 7% of the
previous year's pay. An employee about to join this company hires you to
examine the two schemes and decide which one is better for her. Prepare a
variety of ways to present the information to the client, such as tables,
graphs, and equations.
- If a culture had 500 cells at noon and 600 cells at 1:00pm, what is the
approximate doubling time of the cell population? Approximate how many
cells will there be at 4:00pm?
- A car has an orginal value of $20,000. The value decreased at a rate of
18% each year. (A.SSE.1)
Part A: Write a function where f(x) represents the value of the car in
dollars and x represents years. ( f(x)=20000(0.82)^x )
Part B: After how many years will the car be worth less than 1/2 of the
original value? Show or explain your work. ( approximately 3.493 years)
- Bonnie decided to invest her $600 tax refund rather than spending it. She
found a bank that would pay her 4% interest, compounded quarterly. If she
deposits the entire $600 and does not deposit or withdraw any other amount,
how long will it take her to double her money in the account? Explain or
Suggested
Evaluation/Assessment
Pre-Assessment, Checkpoint
exercises, DoNows, portfolios,
oral questioning, closure, selfreflection journals, projects,
tests, technology-based
assessments, teacher
constructed quizzes and tests
30
show your work. (A.SSE.4)
Instructional Strategies:
- Have students work with a partner on the Web to find three situations that
can be modeled by exponential equations. Students should write a
description of each situation, write and solve one problem based on each of
the three situation, and prepare to present the solution to the problem.
- Create equations and inequalities in one variable and use them to solve
problems (A.CED.1)
- Investigate population models for endangered species or models for
disappearance of the rainforest as examples of exponenital functions.
- Graph exponential and logarithmic functions, showing intercepts and end
behavior (F.IF.7)
- Combine standard function types using arithmetic operations. For
example, build a function that models the temperature of a cooling body by
adding a constant function to a decaying exponential, and related these
functions to the model. (F.BF.1)
- For exponential models, express as a logarithm ... evaluate the logarithm
using technology (F.LE.4)
- Culminating Activity: Students can connect data points to determine the
type of function that models the data. Then, use the “Stat Plot” feature on a
calculator to graph a table of data and regression features to find the bestfitting function. The purpose of fitting a curve to data is to approximate a
function that has the points on its graph and draw conclusions. For example,
have students predict the cost of attending a specific college in 5, 10, 15
years.
- Determine whether an exponential function is a good model for the values
in the table by entering the table into a graphing calculator. Then, transform
the y-values into log y. An exponential model of the original data is
appropriate if the points (x, log y) are linear.
31
Unit 7 - Trigonometric Functions
Content/Objective
The Learner will:
- identify cycles and periods
of periodic functions
- find the amplitude of
periodic functions
- convert from degree
measure to radian measure
and vice versa
- find coterminal and
reference angles
- find the trigonometric
function values of angles in
standard form
- find the length of an arc of a
circle
- identify properties of the
sine and cosine functions in
terms of the unit circle
- graph transformations of
sine and cosine functions
- verify trigonometric
identities
Essential Questions/
Enduring Understandings
Essential Questions:
How can periodic functions be
used to model real phenomena?
How do you convert between
degrees and radians?
How do you find coordinate
points on the unit circle?
How do you find coterminal
angles?
How can you find a reference
angle?
How do you find the length of
an arc?
How do you find the sine and
cosine for angles that are
multiples of 30 degrees, 45
degrees ?
How do you graph the sine and
cosine functions?
How do you transform the
graphs of sine and cosine?
How do you verify
trigonometric identities?
Enduring Understanding:
Period behavior repeats over
intervals of constant length.
Suggested Activity/Appropriate Materials-Equipment
Instructional Focus:
- Choose trigonometric functions to model periodic phenomena with
specified amplitude, frequency, and midline (F.TF.5)
- Prove the Pythagorean identify
(F.TF.8)
- Understand radian measure of an angle as the length of the arc on the unit
circle subtended by the angle (F.TF.1)
Suggested
Evaluation/Assessment
Pre-Assessment, Checkpoint
exercises, DoNows, portfolios,
oral questioning, closure, selfreflection journals, projects,
tests, technology-based
assessments, teacher
constructed quizzes and tests
Sample Assessments:
- What do all periodic functions have in common? (F.TF.5)
-Sketch a graph of a periodic function that has a period of 3 and amplitude of
2. (F.TF.5)
-Which angles are coterminal?
a. -315
b. 45
c. 315
d. 405 (F.TF.2)
-What are the sine and cosine of a 60 degree angle? (F.TF.2)
- Use the formula for arc length to find how far a satellite travels in 1 hour if
it is making a circular orbit around the Earth and it completes one orbit
every 2 hours. (F.TF.2)
Instructional Strategies:
-Students may use the graphing calculators to explore graphs of
trigonometric functions. When graphing a periodic function, it is important
to view at least one cycle of a graph to determine domain, range, period, and
amplitude.
-Students may look for examples of periodic functions throughout the world,
from local temperature to heights of tides.
32
Unit 8 - Probability and Statistics
Content/Objective
The learner will:
- find the probability of an
event using theoretical,
experimental, empirical, and
simulation methods
- calculate measure of central
tendency including mean,
median, and mode
- find the standard deviation
and variance of a set of
values
- find the probability of an
event given that the data is
normally distributed
- identify sampling methods
- recognize bias in samples
and surveys
Enrichment Extension:
- use probabilities to make
fair decisions and analyze
decisions (S.MD.6/ S.MD.7)
Essential Questions/
Enduring Understandings
Essential Questions:
- How can the collection,
organization, interpretation, and
display of data be used to
answer questions?
- How can sampling techniques
affect data reliability?
- How are measures of central
tendency different from
standard deviation?
- What is the normal
distribution and how is it used?
Enduring Understandings:
- Standard deviation is a
measure of how far the
numbers in a data set deviate
from the mean.
- The normal distribution
allows us to predict the
probability that data is typical
or not.
- Some data comes from
surveys, some from
observational studies, and some
from randomized controlled
experiments
- Understand statistics as a
process for making inferences
about population parameters
based on a random sample from
that population (S.IC.1)
Suggested Activity/Appropriate Materials-Equipment
Instructional Focus:
- Distinguish between random sampling from a population in sample surveys
and random assignment of treatments of experimental units in an experiment
(S.IC.3)
- Discuss bias in conclusions drawn
- Interpret oral, written, graphic, pictorial, or multi-media reports on data
- Identify and explain misleading uses of data by considering the
completeness and source of the data, the design of the study, and the way the
data are analyzed and displayed.
- Decide if a specified model is consistent with results from a given datagenerating process (S.IC.2)
- Understand common examples that fit the normal distribution (height and
weight) and examples that do not (salaries, housing prices, size of cities) and
explain the distinguishing characteristics of each
Suggested
Evaluation/Assessment
Pre-Assessment, Checkpoint
exercises, DoNows, portfolios,
oral questioning, closure, selfreflection journals, projects,
tests, technology-based
assessments, teacher
constructed quizzes and tests
Sample Assessments:
- Randomization is an important characteristic of a well-designed survey.
Name two sources of bias that can be eliminated by randomization.
- A machine is used to fill plastic soda bottles. The amount of soda
dispensed into each bottle varies slightly. Suppose the amount of soda
dispensed into the bottles is normally distributed. If at least 99% of the
bottles must have between 585 and 595 milliliters of soda, find the greatest
standard deviation, to the nearest hundredth, that can be allowed. (answer:
1.57)
- The lifetime of a battery is normally distributed with a mean life of 40
hours and a standard deviation of 1.2 hours. Find the probability that a
randomly selected battery lasts longer than 42.4 hours. (answer: 2.5%)
Instructional Strategies:
- Use the mean and standard deviation of a data set to fit it to a normal
distribution, to estimate population percentages. ... [and] use calculators,
spreadsheets, and tables to estimate areas under the normal curve (S.ID.4)
- Use data from a sample survey to estimate a population mean or proportion
(S.IC.4)
- Use data from a randomized experiment to compare two treatments
(S.IC.5)
- Evaluate reports based on data (S.IC.6)
- Some possible real world applications may be clinical trials in medicine, an
opinion poll, or a report on the effect of smoking on health.
33
- Use real-world data sets from the Web, analyzing questions of interest and
share analysis.
- Use the Web to conduct a simple survey of students from around the world
(ex. on length of time spent of homework), analyze and post the results
Unit 9: Matrices
Content/Objective
The learner will:
- Find sums and differences
of matrices and the scalar
product of a number and a
matrix
- Use matrix multiplication to
solve mathematical and real
world problems
- Find and use the inverse and
determinant of a matrix if it
exists
- Use matrices to solve
systems of linear equations in
mathematical and real world
situations
Essential Questions/
Enduring Understandings
Essential Questions:
- How can use a matrix to
organize data?
- How can you use a matrix
equation to model a real world
situation?
Suggested Activity/Appropriate Materials-Equipment
Instructional Focus:
- Understand that the 0 and identity matrices, play a role in matrix addition
and multiplication similar to the role of the 0 and 1 (N.VM.10)
- Use matrices to represent and manipulate data (N.VM.6)
- Understand that unlike multiplication of numbers, matrix multiplication for
square matrices is not a commutative operation but still satisfies the
associative and distributive properties (N.VM.9)
Suggested
Evaluation/Assessment
Pre-Assessment, Checkpoint
exercises, DoNows, portfolios,
oral questioning, closure, selfreflection journals, projects,
tests, technology-based
assessments, teacher
constructed quizzes and tests
Sample Assessments:
- A financial manager wants to invest $50,000 for a client by putting some of
the money in a low-risk investment that earns 5% per year and some of the
money in a high-risk investment that earns 14% per year. How much money
should be invested at each interest rate to earn $5000 in interest per year?
- Write a matrix that has no inverse.
- Explain how to determine whether two matrices can be multiplied and
what the dimensions of the product matrix will be.
Instructional Strategies:
- Use matrices to represent and manipulate data (N.VM.6)
- Multiple matrices by scalars to produce new matrices (N.VM.7)
- Add, subtract, and multiply matrices of appropriate dimensions (N.VM.8)
34
Unit 10: Conics
Content/Objective
The learner will:
- Classify a conic section as
the intersection of a plane and
a double cone
- Define key terms, write and
graph the standard equation
of a parabola, circle, ellipse,
hyperbola
Essential Questions/
Enduring Understandings
Essential Questions:
- What is the intersection of a
cone and a plane parallel to a
line along the side of a cone?
- What is the difference
between the algebraic
representations of ellipses and
hyperbolas?
Enduring Understanding:
- There are 4 types of curves
known as conic sections:
parabolas, circles, ellipses, and
hyperbolas. Each curve has its
own distinct shape and
properties
Suggested Activity/Appropriate Materials-Equipment
Instructional Focus:
-Determine which conic section is represented by an equation based on the
relative size and sign of the parameters of the equation
- Exploring each conic graphically numerically, geometrically and
algebraically
Sample Assessments:
- The point (1,4) lies on the graph of y=x^2. The graph is then translated so
that the new equation is y=(x-2)^2 -3. What are the coordinates of the image
of the point (1,4)? (Answer: (3,1))
Suggested
Evaluation/Assessment
Pre-Assessment, Checkpoint
exercises, DoNows, portfolios,
oral questioning, closure, selfreflection journals, projects,
tests, technology-based
assessments, teacher
constructed quizzes and tests
Instructional Strategies:
- Solve and graph each conic section x^2+y^2=25, 4x^2+y^2=16, 9x^216y^2=144, y=(1/4)x^2
- Students should use computers or graphing calculators to investigate the
graphs of different types of conic sections
- Use real world problems of interest to students, possibly including
parabolic mirrors, satellite dish design, planet or satellite orbits, wooden
plaques, whispering galleries, tracking systems, telescopes, or long range
navigational systems
35
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Daggett, Bill and McNulty, Ray. Rigor and Relevance Framework: International Center for Leadership in Education (www.leadered.com).
Danielson, Charlotte (2007). Enhancing Professional Practice: A Framework for Teaching, 2nd Edition. Alexandria, VA: Association for Supervision
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Killion, Joellen P. (2008). Collaborative Professional Learning in School and Beyond: A Toolkit for New Jersey Educators. Trenton, NJ: New Jersey
Department of Education, the New Jersey Professional Teaching Standards Board, and the National Development Council.
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of Mathematics.
National Research Council (1999). A Guide for Using Mathematics and Science Education Standards. Washington, D.C.: National Academy Press.
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