Download Algebra I - Hackettstown School District

Hackettstown
HACKETTSTOWN, NEW JERSEY
Algebra 1
Grade 8-9
CURRICULUM GUIDE
FINAL DRAFT
August 1, 2014
Mr. David C. Mango, Superintendent
Ms. Nadia Inskeep, Director of Curriculum & Instruction
Developed by:
Stevie Klie
Carl Robinson
Suzanne Sloan
This curriculum may be modified through varying techniques,
strategies and materials, as per an individual student’s
Individualized Education Plan (IEP).
Approved by the Hackettstown Board of Education
At the regular meeting held on
8/20/2014
And
Aligned with the Common Core Content Standards
Hackettstown
Table of Contents
Philosophy and Rationale: Page 3
Mission Statement: Page 3
Units:
Unit 1: Page 4
Unit 2: Page 7
Unit 3: Page 11
Unit 4: Page 13
Unit 5: Page 16
Unit 6: Page 19
Unit 7: Page 21
Unit 8: Page 24
NJ Content Standards: Page 27
21st Century Skills: Page 30
Hackettstown
Philosophy and Rationale
The mission of the Hackettstown Mathematics Department is to design and implement a mathematics
curriculum that stresses the key ideas identified in the Common Core State Standards. This goal is to be
achieved by continually returning to the organizing principles of mathematics and the laws of algebraic and
geometric structure to reinforce those concepts.
Through a variety of real world scenarios and applications, our students will be challenged to think
critically, apply, evaluate, and communicate the ideas and concepts presented in each course as it relates to the
field of mathematics. Students will also be challenged to analyze their performance in an effort to become
independent problem solvers.
Our students’ performance will be assessed using formative techniques such as homework, quizzes, and
tests. Additionally, our students will be assessed using state and national assessment tools based upon the
branch of mathematics being studied.
Mission Statement
Building on Tradition and success, the mission of the Hackettstown School District is to educate and inspire
students through school, family and community partnerships so that all become positive, contributing members
of a global society, with a life-long commitment to learning.
Hackettstown
Stage 1: Desired Results
Unit: 1
Topic: One Variable Expressions, Equations & Inequalities
Content Standards
A.SSE.1.a Interpret expressions that represent a quantity in terms of its context.*
a. Interpret parts of an expression, such as terms, factors, and coefficients.
A.SSE.1.b Interpret expressions that represent a quantity in terms of its context.*
b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For
example, interpret P(1+r)n as the product of P and a factor not depending on P.
A.CED.1
Create equations and inequalities in one variable and use them to solve problems. Include equations
arising from linear and quadratic functions, and simple rational and exponential functions.
Essential Questions
Enduring Understandings
1. Symbolic fluency; what is it and how do I get some? Students will understand that:
2. What are some ways to represent, describe, and
Variables are symbols that take the place
analyze patterns that occur in our world?
of numbers or a range of numbers; they
have different meanings depending on
how they are being used
Algebraic representations can be used to
generalize patterns and relationships
Rules of arithmetic and algebra can be
used together with the concept of
equivalence to transform equations so
solutions can be found to solve problems
In a proportion, the ratio of two quantities
remains constant
Knowledge and Skills:
Students will be instructed on:
1. Using dimensional analysis for unit conversion
2. Representing patterns in t-charts and by using algebraic expressions
3. Translating word expressions into algebraic expressions and vice versa
4. Simplifying and evaluating algebraic expressions
5. Interpreting parts of an expression, such as terms, factors, and coefficients
6. Combining like terms
7. Explaining each step in solving a simple equation as following from the equality of numbers asserted at
the previous step, starting from the assumption that the original equation has a solution
8. Constructing a viable argument to justify a solution method
9. Solving equations with one variable
10. Applying and solving problems using D = RT
11. Solving contextual problems and equations
12. Using proportions to find missing measures in similar polygons
13. Translating a word story into a qualitative graph and vice versa
14. Understanding and using the symbols >, <, >, <
15. Solving and graphing inequalities and absolute value equations on a number line
Hackettstown
Learning Expectations/Objectives
Stage 2: Evidence of Understanding
Hackettstown Benchmarks:
Students will:
1. Use dimensional analysis for unit conversion
2. Represent patterns in t-charts and by using algebraic expressions
3. Translate word expressions into algebraic expressions and vice versa
4. Simplify and evaluating algebraic expressions
5. Interpret parts of an expression, such as terms, factors, and coefficients
6. Combine like terms
7. Explain each step in solving a simple equation as following from the equality of numbers asserted at the
previous step, starting from the assumption that the original equation has a solution
8. Constructing a viable argument to justify a solution method
9. Solve equations with one variable
10. Apply and solve problems using D = RT
11. Solve contextual problems and equations
12. Use proportions to find missing measures in similar polygons
13. Translate a word story into a qualitative graph and vice versa
14. Understand and use the symbols >, <, >, <
15. Solve and graph inequalities and absolute value equations on a number line
Assessment Methods:
Formative:
Warm up
Sample problems in class
Homework
Quizzes
Unit Tests
Summative:
Performance based tasks
Essays
Other Evidence and Student Self-Assessment:
Homework groups
Evaluation of answers and determine feasibility of them in groups and pairs
Hackettstown
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Stage 3: Learning Plan
Assessing prior knowledge: Students will be given a pre-test to determine their knowledge of the
benchmarks of Unit 1.
Sample hook: Students will decide which Bed, Bath, and Beyond coupon they will apply to a $15
purchase. The purpose of this activity is to illustrate that some discounts may save more money, for
example, a $5 off coupon versus a 20% off coupon.
The lessons will incorporate solving or manipulating simple equations in order to come up with a
solution. We will expand on previously learned material by giving students contextual word problems
that cause students to write equations and apply their skills. Students will then be given an opportunity to
come up with their own stories and translate those stories into mathematical models. Once students have
mastered one variable equations, we will re-introduce the Bed, Bath, and Beyond problem that will
include two variables.
Integration of 21st Century Skills
Be Self-directed Learners
Go beyond basic mastery of skills and/or curriculum to explore and expand one’s own learning and
opportunities to gain expertise
Demonstrate initiative to advance skill levels towards a professional level
Demonstrate commitment to learning as a lifelong process
Reflect critically on past experiences in order to inform future progress
Activity: Students will compose a table for D = Rate Time word problems. They will write sentences
explaining the validity of their answers and applying the word problems to real life activities.
Produce Results
Demonstrate additional attributes associated with producing high quality products including the abilities
to:
- Work positively and ethically
- Manage time and projects effectively
- Multi-task
- Participate actively, as well as be reliable and punctual
- Present oneself professionally and with proper etiquette
- Collaborate and cooperate effectively with teams
- Respect and appreciate team diversity
- Be accountable for results
Sample activity: Students will complete a walk-around activity in pairs. Students will solve equations
and search the room for the answer. If students complete all of the equations correctly and check their
answers, they will arrive back at their original problem.
Time Allotment: 13 Class Periods
Resources:
Student Materials:
Textbook
Technology:
Graphing Calculator
Teaching Materials:
SmartBoard
Hackettstown
Stage 1: Desired Results
Unit: 2
Topic: Two Variable Expressions, Equations & Inequalities
Content Standards
N.Q.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and
interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays
A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph
equations on coordinate axes with labels and scales.
A.REI.5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that
equation and a multiple of the other produces a system with the same solutions.
A.REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of
linear equations in two variables.
A.REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the
coordinate plane, often forming a curve (which could be a line).
F.IF.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to
each element of the domain exactly one element of the range. If f is a function and x is an element of its domain,
then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
F.IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use
function notation in terms of a context.
F.IF.5 For a function that models a relationship between two quantities, interpret key features of graphs and
tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the
relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or
negative; relative maximums and minimums; symmetries; end behavior; and periodicity.*
F.IF.6 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.7a Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and
using technology for more complicated cases.* Graph linear and quadratic functions and show intercepts,
maxima, and minima.
F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically,
numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an
algebraic expression for another, say which has the larger maximum.
F.LE.1a Distinguish between situations that can be modeled with linear functions and with exponential
functions. Prove that linear functions grow by equal differences over equal intervals, and that exponential
functions grow by equal factors over equal intervals.
F.LE.1b Distinguish between situations that can be modeled with linear functions and with exponential
functions. Recognize situations in which one quantity changes at a constant rate per unit interval relative to
another.
F.LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph,
a description of a relationship, or two input-output pairs (include reading these from a table).
F.LE.5 Interpret the parameters in a linear or exponential function in terms of a context.
S.ID.6a Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given
functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.
S.ID.6c Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
Fit a linear function for a scatter plot that suggests a linear association.
S.ID.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the
data.
S.ID.8 Compute (using technology) and interpret the correlation coefficient of a linear fit.
Hackettstown
Essential Questions
1. How can we use mathematical models to describe
the change over time?
2. What makes a solution optimal?
3. What is the definition of the word absolute and how
does it pertain to mathematics?
Enduring Understandings
Students will understand that:
Relationships and functions can be used in
real context
In a functional relationship, one variable is
defined in terms of the other variable
Solving inequalities presents the
opportunity for multiple solutions
Absolute value refers to the magnitude of
quantities
Knowledge and Skills:
Students will be instructed on:
1. Identifying functional relationships, domain and range, independent and dependent quantities or variables
2. Relating the domain of a function to its graph and, where applicable, to the quantitative relationship it
describes
3. Graphing points and a line in a coordinate plane
4. Manipulating an equation
5. Graphing a line using a) table b) two points c) slope and a point d) x- and y- intercepts
6. Describing the effects of varying the parameter’s ‘m’ and ‘b’ linear functions of the form f(x) = mx + b
or y = mx + b
7. Using functional notation to represent equations
8. Writing the equation of a line using a) two points b) slope and one point on the line c) the graph of the
line d) x- and y- intercepts
9. Finding slope and use it to determine linear relationships
10. Solving and graphing a linear inequalities and absolute value equations
11. Providing examples of ordered pairs that are included in the solution set of a two-variable linear
inequality
12. Graphing a linear inequality in a coordinate plane
13. Graphing an absolute value equation on the coordinate plane
14. Differentiating between the solutions and graphs of |ax +b| = c, a|x + b| = c, |ax| + b = c
15. Using direct and inverse variation to model real-life situations and solving problems
Hackettstown
Learning Expectations/Objectives
Stage 2: Evidence of Understanding
Hackettstown Benchmarks:
Students will:
1. Identify functional relationships, domain and range, independent and dependent quantities or
variables
2. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it
describes
3. Graph points and a line in a coordinate plane
4. Manipulate an equation
5. Graph a line using a) table b) two points c) slope and a point d) x- and y- intercepts
6. Describe the effects of varying the parameter’s ‘m’ and ‘b’ linear functions of the form f(x) = mx + b
or y = mx + b
7. Use functional notation to represent equations
8. Write the equation of a line using a) two points b) slope and one point on the line c) the graph of the
line d) x- and y- intercepts
9. Find slope and use it to determine linear relationships
10. Solve and graph a linear inequalities and absolute value equations
11. Provide examples of ordered pairs that are included in the solution set of a two-variable linear
inequality
12. Graph a linear inequality in a coordinate plane
13. Graph an absolute value equation on the coordinate plane
14. Differentiate between the solutions and graphs of |ax +b| = c, a|x + b| = c, |ax| + b = c
15. Use direct and inverse variation to model real-life situations and solve problems
Assessment Methods:
Formative:
Warm up
Sample problems in class
Homework
Quizzes
Unit Tests
Summative:
Performance based tasks
Essays
Other Evidence and Student Self-Assessment:
Homework groups
Evaluation of answers and determine feasibility of them in groups and pairs
Hackettstown
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Stage 3: Learning Plan
Sample hook: Students will be given the same Bed, Bath, and Beyond problem, except they will use a
second variable for the purchased amount.
The lessons will incorporate graphing equations in order to show the possible inputs with their respective
outputs for that equation. Through word problems, we will help the students understand how the slope of
a line affects the equation’s graph. Once students grasp the main ideas of working with two variables,
students will be presented with higher order thinking questions. When students have understood two
variable equations, we will re-introduce the Bed, Bath, and Beyond problem that will incorporate
systems.
ICT Literacy
Apply Technology Effectively
Use technology as a tool to research, organize, evaluate and communicate information
Use digital technologies (computers, PDAs, media players, GPS, etc.), communication/networking tools
and social networks appropriately to access, manage, integrate, evaluate and create information to
successfully function in a knowledge economy
Activity: Students will investigate the slope and y intercept of a line. By using a calculator and other
forms of technology, students will compare various lines to discover how the slope and y intercept affect
the graph of a line.
Manage Goals and Time
Set goals with tangible and intangible success criteria
Balance tactical (short-term) and strategic (long-term) goals
Utilize time and manage workload efficiently
Activity: Surf’s Up - students will work together to solve a problem involving expenses and income for
a business.
Time Allotment: 20 Class Periods
Resources:
Student Materials:
Textbook
Technology:
Graphing Calculator
Teaching Materials:
SmartBoard
Hackettstown
Stage 1: Desired Results
Unit: 3 Topic: Systems of Equations, Inequalities & Absolute Values
Content Standards
A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and
interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities
describing nutritional and cost constraints on combinations of different foods.
A.REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x)
intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to
graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or
g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.*
A.REI.12 Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in
the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the
intersection of the corresponding half-planes.
F.IF.7b 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, cube root, and piecewise-defined functions,
including step functions and absolute value functions.
Essential Questions
Enduring Understandings
1. What is a solution set and how is it used to represent Students will understand that:
multiple solutions?
Linear inequalities are similar to linear
2. How do you represent multiple solutions?
equations, but the difference is the infinite
3. What does it mean for things to be unequal?
amount of solutions
Systems of equations can be used to solve
real-life situations
Knowledge and Skills:
Students will be instructed on:
1. Solving a system of linear inequalities using graphing and interpret the solution in the context of the
problem
2. Solving a system of equations by graphing, substitution, and linear combinations
3. Identifying systems with special cases (no solution, infinitely many solutions)
4. Translating between context, tables, graphs, and equations
5. When finished with this unit, put in unit on graphing calculator including liner regression and correlation
coefficient
Learning Expectations/Objectives
Stage 2: Evidence of Understanding
Hackettstown Benchmarks:
Students will:
1. Solve a system of linear inequalities using graphing and interpret the solution in the context of the
problem
2. Solve a system of equations by graphing, substitution, and linear combinations
3. Identify systems with special cases (no solution, infinitely many solutions)
4. Translate between context, tables, graphs, and equations
5. Input data using a graphing calculator, including liner regression and correlation coefficient
Hackettstown
Assessment Methods:
Formative:
Warm up
Sample problems in class
Homework
Quizzes
Unit Tests
Summative:
Performance based tasks
Essays
Other Evidence and Student Self-Assessment:
Homework groups
Evaluation of answers and determine feasibility of them in groups and pairs
Stage 3: Learning Plan
- Assessing prior knowledge: Students will be given a pre-test of their knowledge of the benchmarks of
Unit 3.
- Sample hook: Now students will be looking at the Bed, Bath and Beyond problem and applying two
different discounts separately to an unknown purchase amount and trying to figure out at what amount
spent, which discount would save more money.
Students will now use their knowledge from Unit 2 to work with multiple two variable equations,
inequalities, and absolute value equations. We will discover the possible solutions to systems and
understand why there may be one, none or many solutions. Again, we will include word problems to help
the students visualize real life scenarios.
Be Self-directed Learners
Go beyond basic mastery of skills and/or curriculum to explore and expand one’s own learning and
opportunities to gain expertise
Demonstrate initiative to advance skill levels towards a professional level
Demonstrate commitment to learning as a lifelong process
Reflect critically on past experiences in order to inform future progress
Activity: An analysis of heating system options – individual project on system of equations. Students
will take the role of a homeowner to determine which system of heating their house is cheaper over 40
years.
Create Media Products
Understand and utilize the most appropriate media creation tools, characteristics and conventions
Understand and effectively utilize the most appropriate expressions and interpretations in diverse, multicultural environments
Activity: Print shop – students will solve a non-routine problem using a variety of algebra skills and
present their solutions to the class using PowerPoint, smart software, or other various forms of
technology.
Time Allotment: 17 Class Periods
Resources:
Student Materials:
Textbook
Technology:
Graphing Calculator
Teaching Materials:
SmartBoard
Hackettstown
Stage 1: Desired Results
Unit: 4
Topic: Exponents, Roots, & Radicals
Content Standards
N.RN.1 Explain how the definition of the meaning of rational exponents follows from extending the properties
of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For
example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must
equal 5.
N.RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.
N.RN.3 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number
and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number
is irrational.
A.REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous
solutions may arise.
A.REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the
coordinate plane, often forming a curve (which could be a line).
A.REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x)
intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to
graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or
g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.*
F.IF.7b 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, cube root, and piecewise-defined functions,
including step functions and absolute value functions.
F.IF.8b Write a function defined by an expression in different but equivalent forms to reveal and explain
different properties of the function. Use the properties of exponents to interpret expressions for exponential
functions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t,
y = (1.2)t/10, and classify them as representing exponential growth or decay.
F.LE.1a Distinguish between situations that can be modeled with linear functions and with exponential
functions. Prove that linear functions grow by equal differences over equal intervals, and that exponential
functions grow by equal factors over equal intervals.
F.LE.1c Distinguish between situations that can be modeled with linear functions and with exponential
functions. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval
relative to another.
F.LE.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity
increasing linearly, quadratically, or (more generally) as a polynomial function.
F.LE.5 Interpret the parameters in a linear or exponential function in terms of a context.
Essential Questions
Enduring Understandings
1. How can we model situations using exponents?
Students will understand that:
2. How can we use mathematical language to describe nonMathematical models can be used to
linear change?
describe physical relationships; these
3. Does every mathematical operation have an inverse
relationships are often non-linear
operation?
Radicals are numbers too
4. How do I decide to use an exact answer and when to use
an estimate?
Hackettstown
Knowledge and Skills:
Students will be instructed on:
1. Understanding and applying the laws of exponents
2. Solving problems with radicals and evaluating radical expressions, discussing extraneous solutions
3. Analyzing, evaluating, and modeling exponential growth and decay
4. Graphing the square root and cube root functions
5. Understanding and using scientific notation
6. Determining the difference between rational vs. irrational radicals and numbers
7. Explaining why the sum or product of two rational numbers is rational; that the sum of a rational number
and an irrational number is irrational; and that the product of a nonzero rational number and an irrational
number is irrational
8. Differentiating between situations in which one quantity changes at a constant rate per unit intervals
relative to another and situations in which a quantity grows or decays by a constant percent rate per unit
interval relative to another
9. Observing using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity
increasing linearly and quadratically
Learning Expectations/Objectives
Stage 2: Evidence of Understanding
Hackettstown Benchmarks:
Students will:
1. Understand and apply the laws of exponents
2. Solve problems with radicals and evaluate radical expressions, discussing extraneous solutions
3. Analyze, evaluate, and model exponential growth and decay
4. Graph the square root and cube root functions
5. Understand and use scientific notation
6. Determine the difference between rational vs. irrational radicals and numbers
7. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number
and an irrational number is irrational; and that the product of a nonzero rational number and an irrational
number is irrational
8. Differentiate between situations in which one quantity changes at a constant rate per unit intervals
relative to another and situations in which a quantity grows or decays by a constant percent rate per unit
interval relative to another
9. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity
increasing linearly and quadratically
Assessment Methods:
Formative:
Warm up
Sample problems in class
Homework
Quizzes
Unit Tests
Summative:
Performance based tasks
Essays
Other Evidence and Student Self-Assessment:
Homework groups
Evaluation of answers and determine feasibility of them in groups and pairs
Hackettstown
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Stage 3: Learning Plan
Sample hook: Using the BBB problem, we will discuss buying a grill that costs $2500. Unfortunately at
the time the student will not have enough money to buy the grill, therefore they will have to finance the
grill. For example, let’s say BBB charges 5% interest and gives you 5 years to pay off the grill, the
students will use the compounded interest formula to discover how much they will need to pay each
month.
The students will discover the function of exponential growth through graphing and investigations, which
will ultimately lead them to compare the graphs of linear functions, from Units 2 and 3, and exponential
functions. Also, in this unit students will explore why the sum or product of two rational numbers is
rational; that the sum of a rational number and an irrational number is irrational; and that the product of a
nonzero rational number and an irrational number is irrational.
Work Independently
Monitor, define, prioritize and complete tasks without direct oversight
Activity: Population Growth – students analyze the population growth of a town and report to the city
planners about building highways.
Interact Effectively with Others
Know when it is appropriate to listen and when to speak
Conduct themselves in a respectable, professional manner
Activity: Students will engage in a jigsaw activity including operations with radicals. Teachers will
differentiate the problems based on student ability and students will use peer teaching to communicate
operations with radicals.
Time Allotment: 16 Class Periods
Resources:
Student Materials:
Textbook
Technology:
Graphing Calculator
Teaching Materials:
SmartBoard
Hackettstown
Stage 1: Desired Results
Unit: 5
Topic: Polynomials
Content Standards
A.SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2,
thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).
A.SSE.3a Choose and produce an equivalent form of an expression to reveal and explain properties of the
quantity represented by the expression.*
A.APR.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under
the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
A.APR.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct
a rough graph of the function defined by the polynomial.
F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and
tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the
relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or
negative; relative maximums and minimums; symmetries; end behavior; and periodicity.*
F.IF.7a Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and
using technology for more complicated cases.*
a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
F.IF.8a Write a function defined by an expression in different but equivalent forms to reveal and explain
different properties of the function.
a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values,
and symmetry of the graph, and interpret these in terms of a context
F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically,
numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an
algebraic expression for another, say which has the larger maximum.
Essential Questions
Enduring Understandings
1. How can we use mathematical language to describe Students will understand that:
non-linear change?
Real world situations involving quadratic
2. How can we use technology to analyze, graph,
relationships can be solved using multiple
solve, and apply quadratic functions?
representations
Mathematical models can be used to
describe physical relationships.
Graphs, tables, and equations are
alternative ways for depicting and
analyzing patterns of non-linear change
Knowledge and Skills:
Students will be able to:
1. Adding, subtracting, multiplying polynomials
2. Using special product patterns for the product of a sum and a difference, and for the square of a
binomial
3. Identifying the degree and classifying polynomials
Learning Expectations/Objectives
Stage 2: Evidence of Understanding
Hackettstown Benchmarks:
Students will:
1. Add, subtract, multiply polynomials
2. Use special product patterns for the product of a sum and a difference, and for the square of a
binomial
3. Identify the degree and classify polynomials
Hackettstown
Assessment Methods:
Formative:
Warm up
Sample problems in class
Homework
Quizzes
Unit Tests
Summative:
Performance based tasks
Essays
Other Evidence and Student Self-Assessment:
Homework groups
Evaluation of answers and determine feasibility of them in groups and pairs
Stage 3: Learning Plan
- Assessing prior knowledge: Students will be given a pre-test of their knowledge of the benchmarks of
Unit 5.
- Sample hook: Now halfway through the year we will be changing the constant problem. Instead, we will
be looking at the dimensions of a kitchen and an island in the middle of the kitchen. For example, we
want the width to be x feet and it’s length to be twice as long as the width (2x feet). Then, the island we
want it’s width to be 10 feet less than the width of the kitchen (x-10) and it’s length to be 5 feet less than
the kitchen’s width (x-5). We are looking to find the area of the kitchen without the area of the island
(2x*x) – (x-10)(x-5) = x2 + 15x -50. We can look at different values of x, the width, to see what happens
to the area of the kitchen floor.
Throughout this unit, we will be working to progress from the sample hook. We can make the
dimensions more and more complicated each problem we work on. Therefore we will be incorporating,
multiplying polynomials and adding or subtracting polynomials. The focus for this unit is to concentrate
on applying the topic to real world situations so the concept is concrete for the students and they can
make sense of the problem. Next we’ll be looking at quadratics, the students will be familiar with them
since most of our polynomial problems ended with a quadratic expression.
Guide and Lead Others
Use interpersonal and problem-solving skills to influence and guide others toward a goal
Leverage strengths of others to accomplish a common goal
Inspire others to reach their very best via example and selflessness
Demonstrate integrity and ethical behavior in using influence and power
Activity: Students will use algebra tiles to model a polynomial expression to their partner. The partner
will have to write the expression mathematically.
Adapt to Change
Work effectively in a climate of ambiguity and changing priorities
Activity: Students will be given the area of a room and expressions for each side. They will have to find
the dimensions of the room. Then students will create a problem using the same area but having different
dimensions.
Time Allotment: 13 Class Periods
Hackettstown
Resources:
Student Materials:
Textbook
Technology:
Graphing Calculator
Teaching Materials:
SmartBoard
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Stage 1: Desired Results
Unit: 6
Topic: Quadratics
Content Standards
A.SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2,
thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).
A.SSE.3a Choose and produce an equivalent form of an expression to reveal and explain properties of the
quantity represented by the expression.*
A.APR.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under
the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
A.APR.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct
a rough graph of the function defined by the polynomial.
F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and
tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the
relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or
negative; relative maximums and minimums; symmetries; end behavior; and periodicity.*
F.IF.7a Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and
using technology for more complicated cases.*
a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
F.IF.8a Write a function defined by an expression in different but equivalent forms to reveal and explain
different properties of the function.
a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values,
and symmetry of the graph, and interpret these in terms of a context
F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically,
numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and
an algebraic expression for another, say which has the larger maximum.
Essential Questions
Enduring Understandings
1. How can we use mathematical language to describe Students will understand that:
non-linear change?
Real world situations involving quadratic
2. How can we use technology to analyze, graph,
relationships can be solved using multiple
solve, and apply quadratic functions?
representations
Mathematical models can be used to
describe physical relationships
Graphs, tables, and equations are
alternative ways for depicting and
analyzing patterns of non-linear change
Knowledge and Skills:
Students will be able to:
1. Factoring and solving quadratic expressions using a variety of methods
2. Solving a quadratic equation in factor form
3. Solving a quadratic equation by finding the square roots or factoring
4. Finding the x and y intercepts, vertex, axis of symmetry of quadratic equations and using them to
graph an equation.
5. Using quadratics to model real-life settings
Learning Expectations/Objectives
Stage 2: Evidence of Understanding
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Hackettstown Benchmarks:
Students will:
1. Factor and solve quadratic expressions using a variety of methods
2. Solve a quadratic equation in factor form
3. Solve a quadratic equation by finding the square roots or factoring
4. Find the x and y intercepts, vertex, axis of symmetry of quadratic equations and use them to graph an
equation.
5. Use quadratics to model real-life settings
Assessment Methods:
Formative:
Warm up
Sample problems in class
Homework
Quizzes
Unit Tests
Summative:
Performance based tasks
Essays
Other Evidence and Student Self-Assessment:
Homework groups
Evaluation of answers and determine feasibility of them in groups and pairs
Stage 3: Learning Plan
- Sample hook: The students will look back at area of the island from the hook in Unit 5. The area started
out as (x-10)(x-5) and by double distribution became x2 - 15x + 50, which is a quadratic expression.
Now, the students need to look at the quadratic and figure out how we could work backwards for it to
become the factored form, (x-10)(x-5).
Again, we will be working with real life problems where we would need to use quadratics and factoring.
Students will also discover the graph of a quadratic through an investigation of plugging in x values,
noting the output, and plotting them on a graph. Ultimately, we will be able to put together all of the
concepts for quadratics and discuss the graph of one in a real life setting.
Work Independently
Monitor, define, prioritize and complete tasks without direct oversight
Activity: Sloppy student – students analyze and correct the mistakes made by another student who is
graphing a quadratic.
Be Responsible to Others
Act responsibly with the interests of the larger community in mind
Activity: Group of 3 activity – students create, factor, expand, compare, and draw quadratics.
Time Allotment: 20 Class Periods
Resources:
Student Materials:
Textbook
Technology:
Graphing Calculator
Teaching Materials:
SmartBoard
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Stage 1: Desired Results
Unit: 7
Topic: Rational Equations
Content Standards
N.Q.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and
interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
A.APR.6 Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x),
where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using
inspection, long division, or, for the more complicated examples, a computer algebra system.
A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph
equations on coordinate axes with labels and scales
A.CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving
equations. For example, rearrange Ohm’s law V =IR to highlight resistance R.
A.REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x)
intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to
graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or
g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.*
Essential Questions
Enduring Understandings
1. Can fractions have numerators and denominators
Students will understand that:
that are polynomials?
Formulas and expressions with units can
2. Where do rational expressions occur in real-life?
be written in different formats and still be
equivalent
Quantities may vary directly, inversely, or
neither way
Because the variables in rational
expressions represent numbers, the rules
for operations on rational expressions are
the same as the rules for operations of
numerical fractions
Knowledge and Skills:
Students will be able to:
1. Solving equations involving several variables for one variable in terms of the others
2. Identifying rational expressions
3. Using knowledge of simplifying fractions to simplify rational expressions
4. Recognizing when an expression is undefined
5. Solving and using rational expressions for geometric probability and work problems
6. Using direct and inverse variation to model real-life situations
7. Multiplying and Dividing rational expressions
8. Solving work problems
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Learning Expectations/Objectives
Stage 2: Evidence of Understanding
Hackettstown Benchmarks:
Students will:
1. Solve equations involving several variables for one variable in terms of the others
2. Identify rational expressions
3. Use knowledge of simplifying fractions to simplify rational expressions
4. Recognize when an expression is undefined
5. Solve and use rational expressions for geometric probability and work problems
6. Use direct and inverse variation to model real-life situations
7. Multiply and Divide rational expressions
8. Solve work problems
Assessment Methods:
Formative:
Warm up
Sample problems in class
Homework
Quizzes
Unit Tests
Summative:
Performance based tasks
Essays
Other Evidence and Student Self-Assessment:
Homework groups
Evaluation of answers and determine feasibility of them in groups and pairs
Stage 3: Learning Plan
- Assessing prior knowledge: Students will be given a pre-test of their knowledge of the benchmarks of
Unit 7.
- Sample hook: Now we would like to discuss flooring the kitchen the we have previously discussed.
Suppose Kenny can floor the kitchen in 6 hours and Bob can floor the kitchen in 8 hours, how long
would it take Kenny and Bob to floor the room together?
, x = 24/7 3.4 hours.
We will be looking at all different aspects of rational expressions and equations. We will begin with the
work problems like the sample hook and move onto more complicated rational problems. The students
will apply previous knowledge of writing out an equation from a word problem. Also, the prior
knowledge from Unit 6, factoring, will be helpful when the students need to simplify rational
expressions, which is included in all of the rational problems.
Be Flexible
- Incorporate feedback effectively
- Deal positively with praise, setbacks and criticism
- Understand, negotiate and balance diverse views and beliefs to reach workable solutions, particularly in
multi-cultural environments
- Activity: Delivery Truck – students will set up and solve a work problem to find out how long it will
take two people to unload a delivery truck. Students will report their findings in small group settings.
Work Independently
- Monitor, define, prioritize and complete tasks without direct oversight
- Activity: Students will take a non-routine rational equation and apply all of the previously learned
algebraic techniques.
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Time Allotment: 20 Class Periods
Resources:
Student Materials:
Textbook
Technology:
Graphing Calculator
Teaching Materials:
SmartBoard
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Stage 1: Desired Results
Unit: 8
Topic: Probability, Statistics, and Discrete Math
Content Standards
S.ID.1 Represent data with plots on the real number line (dot plots, histograms, and box plots).
S.ID.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and
spread (interquartile range, standard deviation) of two or more different data sets.
S.ID.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible
effects of extreme data points (outliers).
S.IC.1 Understand statistics as a process for making inferences about population parameters based on a random
sample from that population.
S.IC.2 Decide if a specified model is consistent with results from a given data-generating process, e.g., using
simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5
tails in a row cause you to question the model?
S.IC.6 Evaluate reports based on data.
S.CP.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of
the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).
S.CP.2 Understand that two events A and B are independent if the probability of A and B occurring together is
the product of their probabilities, and use this characterization to determine if they are independent.
S.CP.3 Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A
and B as saying that the conditional probability of A given B is the same as the probability of A, and the
conditional probability of B given A is the same as the probability of B.
S.CP.4 Construct and interpret two-way frequency tables of data when two categories are associated with each
object being classified. Use the two-way table as a sample space to decide if events are independent and to
approximate conditional probabilities. For example, collect data from a random sample of students in your
school on their favorite subject among math, science, and English. Estimate the probability that a randomly
selected student from your school will favor science given that the student is in tenth grade. Do the same for
other subjects and compare the results.
S.CP.5 Recognize and explain the concepts of conditional probability and independence in everyday language
and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the
chance of being a smoker if you have lung cancer.
S.CP.6 Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and
interpret the answer in terms of the model.
S.CP.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the
model.
Essential Questions
Enduring Understandings
1. How do people analyze information to make a good Students will understand that:
and fair decision? Can data representation influence
Tables, graphs, charts, tree diagrams, and
decisions?
symbols are alternative ways of
2. How can we determine how many ways something
representing data and relationships that can
can occur and does order matter?
be translated from one to another
Data sets can be analyzed in various ways
to provide a sense of the shape of the data
Data can be messy and not always easily
analyzed
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Knowledge and Skills:
Students will be instructed on:
1. Calculating mean, median, mode, range, and standard deviation for a set of data
2. Finding the interquartile range and the five number summary for a set of data
3. Representing data in tree diagrams, histograms, stem and leaf, two way frequency tables, and box-andwhisker plots and interpreting their meaning and implications
4. Comparing and contrasting a population and a sample and determine when bias might affect these
5. Finding the probability of events including conditional probability and independence of events
6. Solving problems involving the probabilities and odds of mutually exclusive events or complementary
events
7. Using the counting principle to determine combinations and permutations
8. Determining the effects of outliers on statistics
Learning Expectations/Objectives
Stage 2: Evidence of Understanding
Hackettstown Benchmarks:
Students will:
1. Calculate mean, median, mode, range, and standard deviation for a set of data
2. Find the interquartile range and the five number summary for a set of data
3. Represent data in tree diagrams, histograms, stem and leaf, two way frequency tables, and box-andwhisker plots and interpret their meaning and implications
4. Compare and contrast a population and a sample and determine when bias might affect these
5. Find the probability of events including conditional probability and independence of events
6. Solve problems involving the probabilities and odds of mutually exclusive events or complementary
events
7. Use the counting principle to determine combinations and permutations
8. Determine the effects of outliers on statistics
Assessment Methods:
Formative:
Warm up
Sample problems in class
Homework
Quizzes
Unit Tests
Summative:
Performance based tasks
Essays
Other Evidence and Student Self-Assessment:
Homework groups
Evaluation of answers and determine feasibility of them in groups and pairs
Hackettstown
-
Stage 3: Learning Plan
Sample hook: Students are given a simple probability problem, for example, given 4 green marbles and 4
red marbles, what is the probability that you choose a green marble? Then we will incorporate more
marbles, other colors and progress to a slightly more complicated probability problem.
This unit should have a very hands-on approach where the students are conducting their own
experiments. This will lead the students to verify the difference between theoretical and experimental
probabilities. Also, the students will take sets of data and find the measures of central tendencies (mean,
median, and mode) and the range. Students will take the information they have complied and use their
own judgment to decide which measure best represents the data.
Interact Effectively with Others
Know when it is appropriate to listen and when to speak
Conduct themselves in a respectable, professional manner
Activity: Students will work together to find the mean, median, mode, and range of a given set of data.
Each student will have a different role and responsibility and will communicate their individual task to
the rest of the group.
Information Literacy
Access and Evaluate Information
Access information efficiently (time) and effectively (sources)
Evaluate information critically and competently
Activity: Students will perform experiments to investigate the difference between theoretical and
experimental probability.
Time Allotment: 13 Class Periods
Resources:
Student Materials:
Textbook
Technology:
Graphing Calculator
Teaching Materials:
SmartBoard
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New Jersey Core Curriculum and Common Core Content Standards
N.Q.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and
interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays
N.Q.2 Define appropriate quantities for the purpose of descriptive modeling.
N.Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
A.SSE.1 Interpret expressions that represent a quantity in terms of its context.*
A.SSE.1.A Interpret parts of an expression, such as terms, factors, and coefficients.
A.SSE.1.B Interpret complicated expressions by viewing one or more of their parts as a single entity. For
example, interpret P(1+r)n as the product of P and a factor not depending on P.
A.SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 - y4 as (x2)2 (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 - y2)(x2 + y2).
A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the
quantity represented by the expression.*
A.SSE.3.A Factor a quadratic expression to reveal the zeros of the function it defines.
A.SSE.3.C Use the properties of exponents to transform expressions for exponential functions. For example the
expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly
interest rate if the annual rate is 15%.
A.APR.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under
the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
A.APR.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct
a rough graph of the function defined by the polynomial.
A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations
arising from linear and quadratic functions, and simple rational and exponential functions.
A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph
equations on coordinate axes with labels and scales.
A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities,
and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities
describing nutritional and cost constraints on combinations of different foods.
A.CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving
equations. For example, rearrange Ohm's law V = IR to highlight resistance R.
A.REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at
the previous step, starting from the assumption that the original equation has a solution. Construct a viable
argument to justify a solution method.
A.REI.2 Solve simple rational and radical equations in one variable, and give examples showing how
extraneous solutions may arise.
A.REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients
represented by letters.
A.REI.4 Solve quadratic equations in one variable.
A.REI.4.B Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the
square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when
the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.
A.REI.5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that
equation and a multiple of the other produces a system with the same solutions.
A.REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of
linear equations in two variables.
A.REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the
coordinate plane, often forming a curve (which could be a line).
A.REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x)
intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to
graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or
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g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.*
A.REI.12 Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in
the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the
intersection of the corresponding half-planes.
CCSS.MATH.CONTENT.HSF.IF.A.1
Understand that a function from one set (called the domain) to another set (called the range) assigns to each
element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then
f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
F.IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use
function notation in terms of a context.
F.IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the
integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1)
for n ≥ 1.
F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and
tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the
relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive,
or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.*
F.IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it
describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a
factory, then the positive integers would be an appropriate domain for the function.*
F.IF.6 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.7 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.A Graph linear and quadratic functions and show intercepts, maxima, and minima.
F.IF.7.B Graph square root, cube root, and piecewise-defined functions, including step functions and absolute
value functions.
F.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain
different properties of the function.
F.IF.8.A Use the process of factoring and completing the square in a quadratic function to show zeros, extreme
values, and symmetry of the graph, and interpret these in terms of a context.
F.IF.8.B Use the properties of exponents to interpret expressions for exponential functions. For example,
identify percent rate of change in functions such as y = (1.02)ᵗ, y = (0.97)ᵗ, y = (1.01)12ᵗ, y = (1.2)ᵗ/10, and
classify them as representing exponential growth or decay
F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically,
numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an
algebraic expression for another, say which has the larger maximum.
F.BF.1 Write a function that describes a relationship between two quantities.*
F.BF.1.B 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 relate
these functions to the model.
F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values
of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an
explanation of the effects on the graph using technology. Include recognizing even and odd functions from their
graphs and algebraic expressions for them.
F.LE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.
F.LE.1.A Prove that linear functions grow by equal differences over equal intervals, and that exponential
functions grow by equal factors over equal intervals.
F.LE.1.B Recognize situations in which one quantity changes at a constant rate per unit interval relative to
another.
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F.LE.1.C Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval
relative to another.
S.ID.1 Represent data with plots on the real number line (dot plots, histograms, and box plots).
S.ID.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and
spread (interquartile range, standard deviation) of two or more different data sets.
S.ID.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible
effects of extreme data points (outliers).
S.ID.4 Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate
population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use
calculators, spreadsheets, and tables to estimate areas under the normal curve.
Summarize, represent, and interpret data on two categorical and quantitative variables
S.ID.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative
frequencies in the context of the data (including joint, marginal, and conditional relative frequencies).
Recognize possible associations and trends in the data.
S.ID.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
S.ID.6.A Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use
given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential
models.
S.ID.6.B Informally assess the fit of a function by plotting and analyzing residuals.
S.ID.6.C Fit a linear function for a scatter plot that suggests a linear association.
S.ID.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of
the data.
S.ID.8 Compute (using technology) and interpret the correlation coefficient of a linear fit.
S.ID.9 Distinguish between correlation and causation.
S.CP.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories)
of the outcomes, or as unions, intersections, or complements of other events ("or," "and," "not").
S.CP.2 Understand that two events A and B are independent if the probability of A and B occurring together is
the product of their probabilities, and use this characterization to determine if they are independent.
S.CP.3 Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A
and B as saying that the conditional probability of A given B is the same as the probability of A, and the
conditional probability of B given A is the same as the probability of B.
S.CP.4 Construct and interpret two-way frequency tables of data when two categories are associated with each
object being classified. Use the two-way table as a sample space to decide if events are independent and to
approximate conditional probabilities. For example, collect data from a random sample of students in your
school on their favorite subject among math, science, and English. Estimate the probability that a randomly
selected student from your school will favor science given that the student is in tenth grade. Do the same for
other subjects and compare the results.
S.CP.5 Recognize and explain the concepts of conditional probability and independence in everyday language
and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the
chance of being a smoker if you have lung cancer.
S.CP.6 Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and
interpret the answer in terms of the model.
S.CP.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in terms of the
model.
S.CP.8 (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) =
P(B)P(A|B), and interpret the answer in terms of the model.
S.CP.9 (+) Use permutations and combinations to compute probabilities of compound events and solve
problems.
http://www.state.nj.us/education/cccs/
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Integration of 21st Century Theme(s)
The following websites are sources for the following 21st Century Themes and Skills:
http://www.nj.gov/education/code/current/title6a/chap8.pdf
http://www.p21.org/about-us/p21-framework .
http://www.state.nj.us/education/cccs/standards/9/index.html
21st Century Interdisciplinary Themes (into core subjects)
• Global Awareness
• Financial, Economic, Business and Entrepreneurial Literacy
• Civic Literacy
• Health Literacy
• Environmental Literacy
Learning and Innovation Skills
• Creativity and Innovation
• Critical Thinking and Problem Solving
• Communication and Collaboration
Information, Media and Technology Skills
• Information Literacy
• Media Literacy
• ICT (Information, Communications and Technology) Literacy
Life and Career Skills
• Flexibility and Adaptability
• Initiative and Self-Direction
• Social and Cross-Cultural Skills
• Productivity and Accountability
• Leadership and Responsibility
Integration of Digital Tools
Classroom computers/laptops
Technology Lab
FM system
Other software programs