2014 - 2015
Algebra 1 /Algebra 1 Honors
Curriculum Map
Mathematics Florida Standards
Volusia County Curriculum Maps are revised annually and updated throughout the year.
The learning goals are a work in progress and may be modified as needed.
Florida Standards
Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them. (MAFS.K12.MP.1)
Solving a mathematical problem involves making sense of what is known and applying a thoughtful and logical process
which sometimes requires perseverance, flexibility, and a bit of ingenuity.
2. Reason abstractly and quantitatively. (MAFS.K12.MP.2)
The concrete and the abstract can complement each other in the development of mathematical understanding:
representing a concrete situation with symbols can make the solution process more efficient, while reverting to a concrete
context can help make sense of abstract symbols.
3. Construct viable arguments and critique the reasoning of others. (MAFS.K12.MP.3)
A well-crafted argument/critique requires a thoughtful and logical progression of mathematically sound statements and
supporting evidence.
4. Model with mathematics. (MAFS.K12.MP.4)
Many everyday problems can be solved by modeling the situation with mathematics.
5. Use appropriate tools strategically. (MAFS.K12.MP.5)
Strategic choice and use of tools can increase reliability and precision of results, enhance arguments, and deepen
mathematical understanding.
6. Attend to precision. (MAFS.K12.MP.6)
Attending to precise detail increases reliability of mathematical results and minimizes miscommunication of mathematical
explanations.
7. Look for and make use of structure. (MAFS.K12.MP.7)
Recognizing a structure or pattern can be the key to solving a problem or making sense of a mathematical idea.
8. Look for and express regularity in repeated reasoning. (MAFS.K12.MP.8)
Recognizing repetition or regularity in the course of solving a problem (or series of similar problems) can lead to results
more quickly and efficiently.
Algebra 1: Florida Standards
The fundamental purpose of this course is to formalize and extend the mathematics that students learned in the middle grades. The critical areas,
called units, deepen and extend understanding of linear and exponential relationships by contrasting them with each other and by applying linear
models to data that exhibit a linear trend, and students engage in methods for analyzing, solving, and using quadratic functions. The Mathematical
Practice Standards apply throughout each course and, together with the content standards, prescribe that students experience mathematics as a
coherent, useful, and logical subject that makes use of their ability to make sense of problem situations.
Relationships Between Quantities and Reasoning with Equations/Inequalities: By the end of eighth grade students have learned to solve
linear equations in one variable and have applied graphical and algebraic methods to analyze and solve systems of linear equations in two
variables. This unit builds on these earlier experiences by asking students to analyze and explain the process of solving an equation. Students
analyze and explain the process of solving an equation. Students develop fluency, writing, interpreting, and translating between various forms of
linear equations and inequalities, and using them to solve problems. They master the solution of linear equations and apply related solution
techniques and the laws of exponents to the creation and solution of simple exponential equations.
Linear/Exponential Relationships and Functions: In earlier grades, students define, evaluate, and compare functions, and use them to model
relationships between quantities. In this unit, students will learn function notation and develop the concepts of domain and range. They explore
many examples of functions, including sequences; they interpret functions given graphically, numerically, symbolically, and verbally, translate
between representations, and understand the limitations of various representations. Students build on and informally extend their understanding of
integer exponents to consider exponential functions. They compare and contrast linear and exponential functions, distinguishing between additive
and multiplicative change. Students explore systems of equations and inequalities, and they find and interpret their solutions. They interpret
arithmetic sequences as linear functions and geometric sequences as exponential functions. This unit also builds upon students’ prior experiences
with data, providing students with more formal means of assessing how a model fits data. Students use regression techniques to describe and
approximate linear relationships between quantities. They use graphical representations and knowledge of the context to make judgments about
the appropriateness of linear models. With linear models, they look at residuals to analyze the goodness of fit.
Expressions and Equations: In this unit, students build on their knowledge from the unit of Linear and Exponential Relationships, where they
extended the laws of exponents to rational exponents. Students apply this new understanding of number and strengthen their ability to see
structure in and create quadratic and exponential expressions. They create and solve equations, inequalities, and systems of equations involving
quadratic expressions.
Quadratic Functions and Modeling: In this unit, students consider quadratic functions, comparing the key characteristics of quadratic functions
to those of linear and exponential functions. They select from among these functions to model phenomena. Students learn to anticipate the graph
of a quadratic function by interpreting various forms of quadratic expressions. In particular, they identify the real solutions of a quadratic equation
as the zeroes of a related quadratic function. Students expand their experience with functions to include more specialized functions—absolute
value, step, and those that are piecewise-defined.
Algebra 1: Florida Standards At A Glance
First Quarter
SMT
Unit 1 – Equations and
Inequalities
MAFS.912.A-CED.1.1
MAFS.912.A-CED.1.2
MAFS.912.A-CED.1.3
MAFS.912.A-CED.1.4
MAFS.912.A-REI.1.1
MAFS.912.A-REI.1.2
MAFS.912.A-REI.2.3
MAFS.912.A-REI.4.10
MAFS.912.A-SSE.1.1
MAFS.912.N-Q.1.1
MAFS.912.N-Q.1.2
MAFS.912.N-Q.1.3
MAFS.912.F-BF.1.2
DIA 1
Unit 2 – Functions
MAFS.912.F-IF.1.1
MAFS.912.F-IF.1.2
MAFS.912.F-IF.2.4
MAFS.912.F-IF.2.5
MAFS.912.F-IF.2.6
MAFS.912.F-IF.3.9
MAFS.912.F-LE.1.2
MAFS.912.F-BF.2.4
Second Quarter
Unit 3 – Linear and Exponential
Relationships
MAFS.912.F-BF.1.1a, b
MAFS.912.F-BF.2.3
MAFS.912.F-IF.1.3
MAFS.912.F-IF.3.7a, e
MAFS.912.F-IF.3.8b
MAFS.912.F-LE.1.1
MAFS.912.F-LE.1.3
MAFS.912.A-SSE.2.3
MAFS.912.F-LE.2.5
MAFS.912.N-RN.1.1
MAFS.912.N-RN.1.2
MAFS.912.F-BF.1.2
MAFS.912.F-LE.1.2
MAFS.912.A-SSE.2.4
DIA 3
Unit 4 – Systems of Equations
and Inequalities
MAFS.912.A-CED.1.3
MAFS.912.A.-REI.3.5
MAFS.912.A.-REI.3.6
MAFS.912.A.-REI.4.12
MAFS.912.A-REI.4.11
SMT
DIA 2
*Highlighted standards are Algebra 1 Honors ONLY.*
Third Quarter
Fourth Quarter
Unit 5 – Analyzing Univariate
Data
MAFS.912.S-ID.1.1
MAFS.912.S-ID.1.2
MAFS.912.S-ID.1.3
MAFS.912.S-ID.1.4
Unit 9 – Graphing Quadratic
Equations
MAFS.912.A-CED.1.2
MAFS.912.F-BF.2.3
MAFS.912.F-IF.3.7a, c, d
MAFS.912.F-IF.3.8a
Unit 6 – Analyzing Bivariate Data
MAFS.912.S-ID.2.6
MAFS.912.S-ID.3.7
MAFS.912.S-ID.3.8
MAFS.912.S-ID.3.9
MAFS.912.S-ID.2.5
Unit 10 – Piecewise and Absolute
Value Functions
MAFS.912.A-REI.2.3
MAFS.912.A-REI.4.11
MAFS.912.F-IF.3.7b
MAFS.912.F-IF.3.9
DIA 4
DIA 6
Unit 7 – Polynomials
MAFS.912.A-APR.1.1
MAFS.912.A-SSE.1.1
MAFS.912.A-SSE.1.2
Unit 8 – Solving Quadratic
Equations
MAFS.912.A-CED.1.1
MAFS.912.A-CED.1.2
MAFS.912.A-APR.2.2
MAFS.912.A-APR.2.3
MAFS.912.A-APR.3.4
MAFS.912.A-APR.4.6
MAFS.912.A-REI.2.4
MAFS.912.A-SSE.2.3
MAFS.912.N-RN.2.3
DIA 5
Unit 11 – Using Graphs of
Functions
MAFS.912.F-BF.1.1b
MAFS.912.F-IF.2.4
MAFS.912.F-IF.2.5
MAFS.912.F-IF.3.7e
MAFS.912.F-IF.3.9
MAFS.912.F-LE.1.3
MAFS.912.A-REI.3.7
Diagnostic EOC
Fluency Recommendations
A/G- Algebra I students become fluent in solving characteristic problems involving the analytic geometry of lines, such as writing
down the equation of a line given a point and a slope. Such fluency can support them in solving less routine mathematical problems
involving linearity, as well as in modeling linear phenomena (including modeling using systems of linear inequalities in two
variables).
A-APR.1- Fluency in adding, subtracting, and multiplying polynomials supports students throughout their work in Algebra, as well as
in their symbolic work with functions. Manipulation can be more mindful when it is fluent.
A-SSE.1b- Fluency in transforming expressions and chunking (seeing parts of an expression as a single object) is essential in factoring,
completing the square, and other mindful algebraic calculations.
The following Mathematics and English Language Arts CCSS should be taught throughout the course:
MAFS.912.N-Q.1.1: 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.
MAFS.912.N-Q.1.2: Define appropriate quantities for the purpose of descriptive modeling.
MAFS.912.N-Q.1.3: Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
LACC.910.RST.1.3: Follow precisely a complex multistep procedure when carrying out experiments, taking measurements or
performing tasks, attending to special cases or exceptions defined in the text.
LACC.910.RST.2.4: Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in
context and topics.
LACC.910.RST.3.7: Translate quantitative or technical information expressed in words in a text into visual form and translate
information expressed visually or mathematically into words.
LACC.910.SL.1.1: Initiate and participate effectively in a range of collaborative discussions with diverse partners.
LACC.910.SL.1.2: Integrate multiple sources of information presented in diverse media or formats evaluating the credibility and
accuracy of each source.
LACC.910.SL.1.3: Evaluate a speaker’s point of view, reasoning, and use of evidence and rhetoric, identifying any fallacious reasoning
or exaggerated or distorted evidence.
LACC.910.SL.2.4: Present information, findings and supporting evidence clearly, concisely, and logically such that listeners can follow
the line of reasoning.
LACC.910.WHST.1.1: Write arguments focused on discipline-specific content.
LACC.910.WHST.2.4: Produce clear and coherent writing in which the development, organization, and style are appropriate to task,
purpose, and audience.
LACC.910.WHST.3.9: Draw evidence from informational texts to support analysis, reflection, and research.
Unit 1 – Solving and Applying Equations and Inequalities
Students will analyze and explain the process of evaluating, simplifying and writing expressions and solving equations and inequalities.
Students will develop fluency on writing, graphing, interpreting, translating between linear equations and inequalities, and use them to solve
problems.
Essential Question(s):
How are algebraic equations used in real life?
How can you use the properties of equality to support your solution to a linear equation?
How do you justify the solution to a linear inequality?
How do you solve literal equations and inequalities?
How do you rewrite formulas?
By which methods do we translate expressions, equations and inequalities into real world products?
How can you use linear equations and inequalities to model the result of real world?
What is the slope of a linear function and how can you use it to graph the function?
How can you represent a function symbolically from a graph, a verbal description, or a table of values?
Standard
Learning Goals
I can:
The students will:
MAFS.912A-CED.1.1.
• Understand variables, expressions, terms, factors,
Create equations and inequalities in one variable and use them to solve problems.
coefficients, and order of operations.
o define expression, term, factor, and coefficient.
o interpret the real-world meaning of the terms, factors,
MAFS.912.A-CED.1.2
and coefficients of an expression in terms of their units.
Create equations in two or more variables to represent relationships between quantities
o group the parts of an expression differently in order to
better interpret their meaning.
MAFS.912A.CED.1.3
Represent constraints by equations or inequalities, and by systems of equations
and/or inequalities, and interpret solutions as viable or non-viable options in a modeling
context.
MAFS.912A.CED.1.4
Rearrange formulas to highlight a quantity of interest, using the same reasoning as in
solving equations.
MAFS.912A.REI.1.1
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.
•
Write expressions, equations, and inequalities in one
variable.
o identify the variables and quantities represented in a
real-world problem.
o write the equation or inequality that best models the
problem.
•
Solve equations and inequalities in one variable.
o solve the linear equation or inequality.
o interpret the solution in the context of the problem.
o determine the best model for the real-world problem.
o apply order of operations and inverse operations to
solve equations.
o construct an argument to justify my solution process.
o interpret solutions in the context of the situation
modeled and decide if they are reasonable.
o solve linear equations in one variable including
MAFS.912.A-REI.1.2:
Solve simple rational and radical equations in one variable, and give examples showing
how extraneous solutions may arise.
o
o
MAFS.912A.REI.2.3
Solve linear equations and inequalities in one variable, including equations with
coefficients represented by letters.
MAFS.912.A-REI.4.10.
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).
MAFS.912.A-SSE.1.1:
Interpret expressions that represent a quantity in terms of its context.
a. Interpret parts of an expression, such as terms, factors and coefficients.
b. Interpret complicated expressions by viewing one or more of their parts as a single
entity.
equations with coefficients represented by letters.
solve linear inequalities in one variable including
equations with coefficients represented by letter.
graph linear inequalities in one variable on a number
line.
•
Solve literal equations.
o solve formulas for a specified variable.
•
Write linear equations and inequalities in two variables.
o understand slope- intercept form, point-slope form,
standard form.
o convert between the different forms of linear
equations.
o interpret the real-world meaning of the terms, factors,
and coefficients of an expression in terms of their units
•
Graph linear equations and inequalities in two variables.
o set up coordinate axes using an appropriate scale and
label the axes
o graph equations on coordinate axes with appropriate
labels and scales.
o choose an appropriate scale and origin for graphs and
data displays.
o interpret the scale and origin for graphs and data
displays.
o identify the variables or quantities of significance from
the data provided.
o identify or choose the appropriate unit of measure for
each variable or quantity.
o explain that every ordered pair on the graph of an
equation represents values that make the equation
true.
o verify that any point on a graph will result in a true
equation when their coordinates are substituted into
the equation.
•
Solve simple radical and rational equations in one variable.
o define extraneous solution.
o solve a rational equation in one variable.
o determine which numbers cannot be solutions of a
rational equation and explain why they cannot be
solutions.
o generate examples of rational equations with
extraneous solutions.
MAFS.912.N-Q.1.1
Use units as a way to understand problems and to guide the solution of multi-step
problems; choose and interpret units consistently in formulas.
MAFS.N-Q.1.2
Define appropriate quantities for the purpose of descriptive modeling.
MAFS.912.N-Q.1.3
Choose a level of accuracy appropriate to limitations on measurement when reporting
quantities.
MAFS.912.F-BF.1.2:
Write arithmetic and geometric sequences both recursively and with an explicit formula,
use them to model situations, and translate between the two forms.
o
o
o
solve a radical equation in one variable.
determine which numbers cannot be solutions a
radical equation and explain why they cannot be
solutions.
generate examples of radical equations with
extraneous solutions
•
Write arithmetic sequences recursively and explicitly.
o explain that recursive formula tells me how a
sequence starts and tells me how to use the previous
value(s) to generate the next element of the sequence.
o explain that an explicit formula allows me to find any
element of a sequence without knowing the element
before it.
o distinguish between explicit and recursive formulas for
sequences.
o define an arithmetic sequence as a sequence of
numbers that is formed so that the difference between
consecutive terms is always the same known as a
common difference.
o determine the common difference between two terms
in an arithmetic sequence.
o explain how to change a term of an arithmetic
sequence into the next term and write a recursive
formula for the sequence, 𝑎𝑛 = 𝑎𝑛−1 + 𝑑.
o write an explicit formula for an arithmetic sequence,
𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑑..
o decide when a real world problem models an
arithmetic sequence and write an equation to model
the situation.
•
Throughout Unit 1:
o label units through multiple steps of a problem.
o choose appropriate units for real world problems involving
formulas.
o use and interpret units when solving formulas.
o report measured quantities in a way that is reasonable for
the tool used to make the measurement.
o report calculated quantities using the same level of
accuracy as used in the problem statement.
Unit 1 – Solving and Applying Equations and Inequalities
Remarks
Resources
Use this opportunity to review operations with integers, rational numbers, number
sense, etc as you teach this standard.
http://insidemathematics.org/problems-of-the-month/pomonbalance.pdf
Students may believe that solving an equation such as 3x + 1 = 7 involves “only
removing the 1,” failing to realize that the equation 1 = 1 is being subtracted to produce
the next step.
http://insidemathematics.org/common-core-math-tasks/highschool/HS-A-2003%20Number%20Towers.pdf
When using Distributive Property, students often multiply the number (or variable)
outside the parentheses by the first term in the parentheses, but neglect to multiply that
same number by the other term(s) in the parentheses.
Regarding variables on both sides, students often will try to combine the terms as if
they are on the same side of the equation rather than eliminating one of the variables.
Students may confuse the rule of reversing the inequality when multiplying or dividing
by a negative number, with the need to reverse the inequality anytime a negative sign
shows up in solving the last step of the inequality. Example: 3x > -15 or x < - 5
(Rather than correctly using the rule: -3x >15 or x< -5)
Students may struggle to solve literal equations/ formulas due to not containing any
numbers, so reiterating that the same steps (inverse operations) are used whether
dealing with eliminating a variable or number may be helpful.
Refer to slope as rate of change to prepare students for Unit 2.
http://insidemathematics.org/common-core-math-tasks/highschool/HS-A-2003%20Number%20Towers.pdf
http://insidemathematics.org/common-core-math-tasks/highschool/HS-F-2008%20Functions.pdf
http://www.math-aids.com/Algebra/Pre-Algebra/Expressions/
http://teachers.henrico.k12.va.us/math/hcpsalgebra1/Documents/34/EqnsCardGame.doc
http://lzlomek.wordpress.com/2012/10/10/activities-for-solvingequations/
http://www.map.mathshell.org/materials/download.php?fileid=1261
http://www.livebinders.com/play/play?present=true&tab_layout=top&id
=330370
https://www.teachingchannel.org/videos/graphing-linear-equationslesson
http://www.nasa.gov/audience/foreducators/exploringmath/algebra1/Pr
ob_Exercise_detail.html
http://www.mathwarehouse.com/algebra/linear_equation/linearinequality.php
http://www.nasa.gov/audience/foreducators/exploringmath/algebra1/Pr
ob_Exercise_detail.html
http://www.nasa.gov/audience/foreducators/exploringmath/algebra1/Pr
ob_BoneDensity_detail.html
Unit 2 - Functions
Students will learn function notation and develop the concepts of domain and range. They explore many examples of functions, including
sequencing; they interpret functions given graphically, numerically, symbolically, and verbally, translate between representations, and
understand the limitations of various representations.
Essential Question(s):
In which ways do you interpret domain and range in multiple formats?
How can you represent a function symbolically from a graph, a verbal description, or a table of values?
How can you use operations to combine?
What is the inverse of a function, and how can you find the inverse of a linear function?
Standard
Learning Goals
I can:
The students will:
MAFS. 912.F-IF.1.1
• define and understand functions.
Understand that a function from one set (called the domain) to another set
o define relation, domain and range.
(called the range) assigns to each element of the domain exactly one
o define a function as a relation in which each input (domain)
element of the range. If f is a function and x is an element of its domain,
has exactly one output (range).
the f(x) denotes the output of f corresponding to the input x. The graph of f
o determine if a graph, table or set of ordered pairs represent
is the graph of the equation y=f(x).
a function.
o determine if stated rules (both numeric and non-numeric)
MAFS. 912.F-IF.1.2
produce ordered pairs that form a function.
Use function notation, evaluate functions for inputs in their domains, and
o explain that when ‘x’ is an element of the input of a function
interpret statements that use function notation in terms of a context.
f(x) represents the corresponding output.
o explain that function notation is not limited to f(x); other
MAFS. 912.F-IF.2.4
latters can also be used to we can tell different functions
For a function that models a relationship between two quantities, interpret
apart.
key features of graphs and tables in terms of the quantities, and sketch
o explain that the graph of ‘f’ is the graph of the equation
graphs showing key features given a verbal description of the relationship.
y=f(x).
o state the appropriate domain of a function that represents a
MAFS. 912.F-IF.2.5
problem situation, defend my choice, and explain why other
Relate the domain of a function to its graph and, where applicable, to the
numbers might be excluded from the domain.
quantitative relationship it describes.
o identify and eliminate the part(s) of a graph that cause it to
MAFS. 912.F-IF.2.6
fail the vertical line test.
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
• When a function is presented using a table or graph:
change from a graph.
o locate the information that explains what each quantity
represents.
MAFS.912.F-IF.3.9
o interpret the meaning of an ordered pair.
Compare properties of two functions each represented in a different way
o determine if negative inputs make sense in the problem
(algebraically, graphically, numerically in tables, or by verbal descriptions).
situation.
For example, given a graph of one quadratic function and an algebraic
o determine if negative outputs make sense in the problem
expression for another, say which has the larger maximum.
situation.
o identify the y-intercept.
MAFS.912.F-LE.1.2
o use the definition of function to explain why there can only be
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).
o
o
o
MAFS.912.F-BF.2.4
Find inverse functions.
a. Solve an equation of the form f(x) = c for a simple function f that
has an inverse and write an expression for the inverse. For
(𝑥+1)
example, 𝑓(𝑥) = 2 𝑥 3 𝑜𝑟 𝑓(𝑥) =
𝑓𝑜𝑟 𝑥 ≠ 1.
o
o
(𝑥−1)
b. Verify by composition that one function is the inverse of another.
c. Read values of an inverse function from a graph or a table, given
that the function has an inverse.
d. Produce an invertible function from a non-invertible function by
restricting the domain.
o
one y-intercept.
use the problem situation to explain what an x-intercept
means.
identify the x-intercept.
use the definition of function to explain why some functions
have more than one x-intercept.
use the problem situation to explain what an x-intercept
means.
explain how the domain of a function is represented in its
graph.
state, defend and explain the appropriate domain of a
function that represents a problem situation.
•
When a function is presented using a verbal or written description:
o locate the information that explains what each quantity
represents.
o decide which quantity should be used as the input.
o identify which parts of the description indicate, if applicable,
the function’s y-intercept, x-intercept(s).
o create a graph that matches the description and indicates all
of the key features of the function.
•
Write and evaluate functions using function notation.
o
o
o
o
o
•
decode function notation and explain how the output of a
function is matched to its input.
convert a table, graph, set of ordered pairs or description
into function notation by identifying the rule used to turn
inputs into outputs and writing the rule.
use order of operations to evaluate a function for a given
domain value.
identify the numbers that are not in the domain of a function.
choose and analyze inputs (and outputs) that make sense
based on the problem.
calculate and interpret the rate of change of a function.
o define and explain interval, rate of change and average rate
of change.
o calculate the average rate of change of a function,
represented either by function notation, a graph or a table
over a specific interval.
o compare the rates of change of two or more functions.
o interpret the meaning of the average rate of change in the
context of the problem.
•
write the inverse of a function.
o define inverse of a function
PART A
o write the inverse of a function by solving f(x) = c; for x
o explain that after f(x) = c; for x, c can be considered the input
and x the output
o write the inverse of a function in standard notation by
replacing the x in my inverse equation with y and replacing
the c in my inverse equation with x.
PART B
o use the composition of functions to verify that g(x) and f(x)
are inverses by showing that g(f(x))=f(g(x)) = 1.
PART C
o decide if a function has an inverse using the horizontal line
test
o use the definitions of functions, inverse functions, and 1:1
functions to explain why the horizontal line test works.
o list values of an inverse given a table or graph of a function
that has an inverse.
PART D
o identify and eliminate the part of the graph that caused it to
fail the vertical line test
o state the domain of a relation that has been altered in order
to pass the horizontal line test
o write the inverse of the invertible function in function notation
Unit 2 - Functions
Remarks
Resources
Using Function Notation I
http://www.illustrativemathematics.org/illustrations/598
Using Function Notation II
http://www.illustrativemathematics.org/illustrations/599
Domains http://www.illustrativemathematics.org/illustrations/635
Points on a Graph
http://www.illustrativemathematics.org/illustrations/630
Customers
http://www.illustrativemathematics.org/illustrations/624
Cell Phones
http://www.illustrativemathematics.org/illustrations/634
Oakland Coliseum
http://www.illustrativemathematics.org/illustrations/631
www.learnzillion.com
Choose Domain, Choose Standard (Site contains 895 lessons lasting
3-7 minutes on High School standards.)
https://www.khanacademy.org/commoncore/map
Under high school choose domain.
Standards are listed with information about available
lessons/exercises from which you may choose.
www.cpalms.org
You will find a minimum of 3 formative assessment tasks with
rubrics for each standard as well as many other resources.
The Algebra Nation project at the University of Florida
www.algebranation.com
Unit 3 - Linear and Exponential Relationships
Students build on and informally extend their understanding of integer exponents to consider exponential functions. They compare and
contrast linear and exponential functions, distinguishing between additive and multiplicative change. They interpret arithmetic sequences
as linear functions and geometric sequences as linear functions and geometric sequences as exponential functions.
Essential Question(s):
In what ways can you recognize, describe, input and compare linear and exponential functions?
What are discrete exponential functions and how can you represent them?
How do you write, graph, and interpret an exponential growth and decay function?
How can you solve problems modeled by equations involving variable exponents?
How can you recognize, describe, and compare linear and exponential functions?
How can you model real world problems using an exponential function?
Standard
Learning Goals
I can:
The students will:
MAFS.912.F-BF.1.1a
• Simplify expressions involving exponents
Determine an explicit expression, a recursive process, or steps for
o define a linear function and exponential function.
calculation from a context.
o evaluate and simplify expressions containing zero and
integer exponents.
MAFS.912.F-BF.1.1b
o multiply monomials.
Combine standard function types using arithmetic operations. For
o apply multiplication properties of exponents to evaluate and
example, build a function that models the temperature of a cooling body
simplify expressions.
by adding a constant function to a decaying exponential, and relate
o divide monomials.
these functions to the model.
o apply division properties of exponents to evaluate and simplify
expressions.
MAFS.912.F-BF.2.3
o apply properties of rational exponents to simplify expressions.
Identify the effect on the graph of replacing f(x) by f(x) + k, k (f(x),
o convert between radicals and rational exponents.
f(kx),and f(x+k) for specific values of k. ( positive and negative) ; find the
value of k given the graphs. Experiment with cases and illustrate an
• recognize and write exponential functions.
explanation of the effects on the graph using technology
o classify exponential functions in function notation as growth or
decay
MAFS.912.F-IF.3.7
o explain how can simple geometric transformations changes a
Graph function expressed symbolically and show key features of the
growth graph to a decay graph.
graph, by hand in simple cases and using technology for more
o demonstrate that an exponential function has a constant
complicated functions.
multiplier or equal intervals.
a. Graph linear and quadratic functions ; show intercepts, max and
o identify situations that display equal ratios of change over
min.
equal intervals and can be modeled with exponential functions.
e. Graph exponential and logarithmic functions, showing intercepts
o distinguish between situations modeled with linear functions
and end behavior, and trigonometric functions showing period,
and withexponential functions when presented with a realmidline and amplitude.
world problem.
o understand exponential functions and how they are used.
MAFS.912.F-IF.3.8b
o recognize differences between graphs of exponential
Write a function defined by an expression in different but equivalent
functions with different bases.
forms to reveal and explain different properties of the function.
o apply exponential functions to model applications that include
a. Use properties of exponents to interpret expressions for
growth and decay in different contexts.
exponential function.
o
o
MAFS.912.F-LE.1.1
Distinguish between situations that can be modeled with linear functions
and with exponential functions.
a. Prove that linear functions grow by equal differences over equal
intervals, and that exponential functions grow by equal factors over
equal intervals.
b. Recognize situations in which one quantity changes at a constant
rate per unit interval relative to another.
c. Recognize situations in which a quantity grows or decays by a
constant percent rate per unit interval relative to another.
MAFS.912.F-LE.1.3
Observe using graphs and tables that a quantity increasing
exponentially eventually exceeds a quantity increasing linearly,
quadratically, or (more generally) as a polynomial function.
o
o
o
o
•
graph exponential functions.
o explain the parent function for exponentials
o determine domain, range, and end behavior ( horizontal
asymptote) by inspection of the graph
o use technology to graph a polynomial and to find piece values
for the x-intercept(s) and the maximums and minimum (turning
points)
o substitute convenient values for x to generate a table and
graph of an exponential function.
o explain how a simple geometric transformation changes a
growth graph to a decay graph.
o interpret the components of an exponential function in the
context of a problem.
•
describe changes to the parent function given a function.
o explain why f(x) + k translates the original graph of f(x) up k
units and why f(x) –k translates the original graph of F(x) down
k units
o explain why f(x + k) translates the original graph of f(x) left k
units and why f(x-k) translates the original graph pf f(x) right k
units
o explain why kf(x) vertically stretches or shrinks the graphs of
f(x) by a factor of k and predict whether a given value of k will
cause a stretch or a shrink
o explain why f(kx) horizontally stretches or shrinks the graph of
1
f(x) by a factor of and predict whether a given value of k will
𝑘
cause a stretch or a shrink
o describe the transformation that change a graph of f(x) into a
different graph when given pictures of the pre-image and
image.
o determine the value of k given the graph of a transformed
function.
MAFS.912.A-SSE.2.3c:
Choose and produce an equivalent form of an expression to reveal and
explain properties of the quantity represented by the expression.
c. Use the properties of exponents to transform expressions for
exponential functions.
MAFS.912.F-LE.2.5:
Interpret the parameters in an exponential function in terms of a context.
MAFS.912.N-RN.1.1:
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
to be the cube root of 5 because we want
=
to hold, so
must equal 5.
MAFS.912.N-RN.1.2:
Rewrite expressions involving radicals and rational exponents using the
properties of exponents.
MAFS.912.F-BF.1.2:
Write arithmetic and geometric sequences both recursively and with an
explicit formula, use them to model situations, and translate between
the two forms.
define an exponential function f(x)=a𝑏𝑥 .
rewrite exponential functions using the properties of
exponents
identify the names and definitions of the parameters ‘a’, ‘b’
x
and ‘c’ in the exponential function f(x)=a(b )+c.
explain the meaning (using appropriate units) of the constant
a, b, c and other points of an exponential function when the
exponential function models a real-world relationship.
compose an original problem situation and construct an
exponential function to model it.
o
o
o
MAFS.912.F-LE.1.2:
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).
MAFS.912.A-SSE.2.4:
Derive the formula for the sum of a finite geometric series (when the
common ratio is not 1), and use the formula to solve problems.
MAFS.912.F-IF.1.3
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.
o
o
o
•
graph the listed transformations when given a graph of f(x)
and a value of k ( f(x) ± k, f(x±k) , kf(x) and f(kx).
use a graphing calculator to generate examples of functions
with different k values
analyze the similarities and differences between function with
different k values.
recognize from a graph if the function is even or odd.
explain that a function is even when f(-x) = f(x) and its graph
has y –axis symmetry
explain that a function is odd when f(-x) = -f(x) and its graph
has 180° rotational symmetry.
relate linear and exponential functions to arithmetic and
geometric sequences.
o identify the quantities being compared in a real-world
problem.
o write a function to describe a real-world problem.
o compose two or more functions.
o determine if a function is linear or exponential given a
sequence, a graph, a verbal description or a table.
o describe the algebraic process used to construct the linear
function and exponential function that passes through two
points.
o convert a list of numbers (a sequence) into a function by
making the whole numbers (0, 1, 2, etc.) the inputs and the
elements of the sequence the outputs.
o explain that a recursive formula tells me how a sequence
starts and tells me how to use the previous value(s) to
generate the next element of the sequence.
o explain that an explicit formula allows me to find any element
of a sequence without knowing the element before it.
o distinguish between explicit and recursive formulas for
sequences.
o explain why the recursive formula for an arithmetic sequence
uses addition and why the explicit formula uses multiplication
o define a geometric sequence as a sequence of numbers that
is formed so that the ratio of consecutive terms is always the
same known as the common ratio.
o distinguish between arithmetic and geometric sequences.
o determine the common ratio between two terms in a
geometric sequence.
o explain how to change a term of a geometric sequence into
the next term and write a recursive formula for the sequence,
𝑎𝑛 = 𝑟 ∙ 𝑎𝑛−1 .
o write an explicit formula for a geometric sequence, 𝑎𝑛 =
o
o
o
o
•
𝑎1 𝑟 𝑛−1 .
explain why the recursive formula for a geometric sequence
uses multiplication and why the explicit formula uses
exponentiation.
translate between the recursive and explicit forms of
geometric sequences.
decide when a real world problem models a geometric
sequence and write an equation to model the situation.
translate between the recursive and explicit forms of
arithmetic sequences.
evaluate finite geometric series.
o define a finite geometric series and common ratio.
o derive the formula for the sum of a finite geometric series,
𝑆𝑛 = 𝑎1 ((1 − 𝑟 𝑛)/(1−𝑟)).
o express the sum of a finite geometric series.
o calculate the sum of a finite geometric series.
o recognize real-world scenarios that are modeled by geometric
sequences.
o use the formula for the sum of a finite geometric series to
solve real-world problems.
Unit 3 - Linear and Exponential Relationships
Remarks
Resources
Evaluating a Special Exponential Expression
http://www.illustrativemathematics.org/HSN
www.learnzillion.com
Choose Domain, Choose Standard (Site contains 895 lessons lasting
3-7 minutes on High School standards.)
https://www.khanacademy.org/commoncore/map
Under high school choose domain.
Standards are listed with information about available
lessons/exercises from which you may choose.
www.cpalms.org
You will find a minimum of 3 formative assessment tasks with
rubrics for each standard as well as many other resources.
The Algebra Nation project at the University of Florida
www.algebranation.com
Unit 4 – Systems of Equations and Inequalities
Students explore systems of equations and inequalities, and they find and interpret their solutions.
Essential Question(s):
How can we model and find solutions in real world situations using systems of equations and inequalities?
How can you solve a system of equations or inequalities?
Can systems of equations model real world situations?
Standard
Learning Goals
I can:
The students will:
MAFS.912A.CED.1.3
• solve systems of equations by graphing.
Represent constraints by equations or inequalities, and by systems of
o graph a system of linear equations and determine the
equations and/or inequalities, and interpret solutions as viable or non-viable
approximate solution to the system of linear equations by
options in a modeling context.
estimating the point of intersection.
o graph the system on coordinate axes with appropriate
MAFS.912A-REI.3.5
labels and scales.
Prove that given a system of two equations in two variables, replacing one
o solve a system of two linear equations by graphing and
equation by the sum of that equation and a multiple of the other produces a
determining the point of intersection.
system with the same solutions.
o explain that a point of intersection on the graph of a system
of equations, y=f(x) and y=g(x), represents a solution to both
MAFS.912.A-REI.3.6
equations use a graphing calculator to determine the
Solve systems of linear equations exactly and approximately (with graphs),
approximate solutions to a system of equations, f(x) and
focusing on pairs of linear equations in two variables.
g(x).
o explain that a point of intersection on the graph of a system
MAFS.912.A-REI.4.12
of equations represents a solution to both equations.
Graph the solutions to a linear inequality in two variables as a half-plane
o use a graphing calculator to determine the approximate
(excluding the boundary in the case of a strict inequality), and graph the
solutions to a system of equations.
solution set to a system of linear inequalities in two variables as the
intersection of corresponding half-planes.
• solve systems of equations algebraically.
o write the system of equations and/or inequalities that best
MAFS.912.A-REI.4.11:
models the problem.
Explain why the x-coordinates of the points where the graphs of the
o solve a system of two linear equations algebraically using
substitution.
equations y=f(x) and y=g(x) intersect are the solutions of the
o solve a system of two linear equations algebraically using
equation f(x)=g(x); find the solutions approximately, e.g., using
elimination.
technology to graph the functions, make tables of values, or find
o explain why some linear systems have no solutions or
successive approximations. Include cases where f(x) and/or g(x) are
infinitely many solutions.
linear, polynomial, rational, absolute value, exponential, and
o solve a system of linear equations algebraically to find an
logarithmic functions.
exact solution.
o infer that the x-coordinate of the points of intersection for
o
o
o
•
y=f(x) and y=g(x) are also solutions for f(x)=g(x).
interpret solutions in the context of the situation modeled
and decide if they are reasonable.
infer that since y=f(x) and y=g(x), f(x)=g(x) by the
substitution property.
infer that the x-coordinate of the points of intersections are
solutions for f(x) = g(x).
solve systems of inequalities.
o solve and graph linear inequalities with two variables.
o solve and graph system of linear inequalities.
o explain that the solution set for a system of linear
inequalities is the intersection of the shaded regions of both
inequalities and check points in the shaded region to verify
solution.
Unit 4 – Systems of Equations and Inequalities
Remarks
Resources
Number Towers http://www.insidemathematics.org/commoncore-math-tasks/high-school/HS-A2003%20Number%20Towers.pdf
Graphs http://www.insidemathematics.org/common-core-mathtasks/high-school/HS-A-2006%20Graphs2006.pdf
Supply and Demand
http://illuminations.nctm.org/Lesson.aspx?id=2544
Road Rage
http://illuminations.nctm.org/Lesson.aspx?id=2808
www.learnzillion.com
Choose Domain, Choose Standard (Site contains 895 lessons lasting
3-7 minutes on High School standards.)
https://www.khanacademy.org/commoncore/map
Under high school choose domain.
Standards are listed with information about available
lessons/exercises from which you may choose.
www.cpalms.org
You will find a minimum of 3 formative assessment tasks with
rubrics for each standard as well as many other resources.
The Algebra Nation project at the University of Florida
www.algebranation.com
Unit 5 – Analyzing Univariate Data
Part 1: Students will use multiple representations and knowledge of the context to make judgments about the appropriateness of models.
Essential Questions:
How can you represent data?
Of the multiple methods available how do you chose the best method?
What statistics can you use to characterize and compare the center and spread of data sets and which are most affected by outliers?
How can you compare, estimate, and categorize data sets by histograms, box plots, and frequency tables?
Standard
Learning Goals
I can:
The students will:
MAFS.912.S-ID.1.1
• display data using the best representation.
Represent data with plots on the real number line (dot plots, histograms, and box
o choose the best representation (dot plot, histogram, box plot) for
plots).
a set of data.
o decide if a representation preserves all the data values or
MAFS.912.S-ID.1.2
presents only the general characteristics of a data set.
Use statistics appropriate to the shape of the data distribution to compare center
o construct a histogram for a set of data
(median, mean) and spread (interquartile range, standard deviation) of two or more
o construct a dot plot for a set of data and choose the appropriate
different data sets.
scale to represent data on a number line.
MAFS.912.S-ID.1.3
Interpret differences in shape, center, and spread in the context of the data sets,
accounting for possible effects of extreme data points (outliers).
MAFS.912.S-ID.1.4
Use the mean and standard deviation data set to fit it to a normal distribution and to
estimate population percentages. Recognize that they are data sets for which such a
procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate
area under the normal curve,
•
analyze and interpret sets of data given a display.
o describe the center of the data distribution (mean or median).
o choose the histogram with the largest mean when shown several
histograms.
o describe the spread of the data distribution (interquartile range
or standard deviation).
o choose the histogram with the greatest standard deviation when
shown several histograms.
o choose the box-and-whisker plot with the greatest interquartile
range when shown several box-and-whisker plots.
o compare the distributions of two or more data sets by examining
their shapes, centers, and spreads when drawn on the same
scale.
o interpret the differences in the shape, center, and spread of a
data set in the context of a problem.
o identify the outliers for the data set.
o predict the effect an outlier will have on the shape, center, and
spread of a data set.
o decide whether to include the outliers as part of the data set or to
remove them.
•
recognize and interpret a data set with a normal distribution.
o use mean and standard deviation of a set of data to fit the data to
a normal curve
o use the 68-95-99.7 Rule to estimate the percent of a normal
population that falls within 1, 2, or 3 standard deviations of the
mean.
o recognize that normal distributions are only appropriate for
unimodal and symmetric shapes
o estimate the area under a normal curve using a calculator, table,
or spreadsheet
Unit 5 – Analyzing Univariate Data
Remarks
Resources
The United States of Obesity
http://www.amstat.org/education/stew/pdfs/TheUnitedStatesofO
besity.pdf
Saga of Survival
http://www.amstat.org/education/stew/pdfs/SagaofSurvival.pdf
NFL Quarterback Salaries
http://www.amstat.org/education/stew/pdfs/NFLQuarterbackSalar
ies.pdf
An A-MAZE-ING Comparison
http://www.amstat.org/education/stew/pdfs/AnAmazingComparis
on.pdf
Haircut Costs
http://www.illustrativemathematics.org/illustrations/942
Unit 6 – Analyzing Bivariate Data
Part 2: Students will use regression techniques to describe and approximate linear relationships between quantities. Students will use the linear models
to look at residuals to analyze the goodness of fit.
Essential Questions:
How do you write an equation to model trends and data?
How do you write an equation to show trends in data?
How can you decide whether a correlation exists between paired numerical data?
How can you find a linear model for a set paired numerical data, and how do you evaluate the goodness of fit?
Standard
Learning Goals
I can:
The students will:
MAFS.912-S-ID.2.6
• model bivariate data using linear and exponential functions.
Represent data on two quantitative variables on a scatter plot, and describe how the
o identify the independent and dependent variable and describe
variables are related. Fit a function to the data; use functions fitted to data to solve
the relationship of the variables.
problems in the context of the data. Use given functions or choose a function
o construct a scatter plot with an appropriate scale.
suggested by the context. Emphasize linear, quadratic, and exponential models.
o identify any outliers on the scatter plot.
Informally assess the fit of a function by plotting and analyzing residuals. Fit a linear
o determine when linear, quadratic, and exponential models
function for a scatter plot that suggests a linear association.
should be used to represent a data set.
o determine whether linear and exponential models are increasing
MAFS.912.S-ID.3.7
and decreasing.
Interpret the slope (rate of change) and the intercept (constant term) of a linear model
o use technology to find the function of best fit for a scatter plot.
in the context of the data.
o use the function of best fit to make predictions.
o compute the residuals (observed value minus predicted value)
MAFS.912.S-ID.3.8
for the set of data and the function of best fit.
Compute (using technology) and interpret the correlation coefficient of a linear fit.
o construct a scatter plot of the residuals.
o analyze the residual plot to determine whether the function is an
MAFS.912.S-ID.3.9
appropriate fit.
Distinguish between correlation and causation.
o sketch the line of best fit on a scatter plot that appears linear.
o write the equation of the line of best fit (y=mx+b) using
MAFS.912.S-ID.2.5
technology or by using two points on the best fit line.
Summarize categorical data for two categories in two-way frequency tables. Interpret
o interpret the meaning of the slope in terms of the units stated in
relative frequencies in the context of the data (including joint, marginal, and
the data.
conditional relative frequencies). Recognize possible associations and trends in the
o interpret the meaning of the y-intercept in terms of the units
data.
stated in the data.
•
determine if a linear model is the line of best fit.
o explain the correlation coefficient applies only to quantitative
variables and linear models of best fit.
o explain that the correlation coefficient must be between -1 and 1
o
o
o
o
o
o
o
o
•
inclusive and explain what each of these values means.
explain the correlation coefficient as a measure of the “goodness
of a linear fit.”
compute the correlation coefficient (r) using a graphing calculator
or other appropriate technology.
use the correlation coefficient to interpret the linear model in
terms of its sign (i.e., direction) and its magnitude (i.e., strength).
use the correlation coefficient to determine if a linear model is a
good fit for the data (significance).
recognize that correlation does not imply causation and that
causation is not illustrated on a scatter plot.
choose two variables that could be correlated because one is the
cause of the other and defend my selection.
choose two variables that could be correlated even though
neither variable could reasonably be considered to be the cause
of the other and defend my selection.
determine if statements of causation seem reasonable or
unreasonable and defend my opinion.
display and interpret bivariate categorical data.
o read and interpret the data displayed in a two-way frequency
table.
o write clear summaries of data displayed in a two-way frequency
table.
o calculate percentages using the ratios in a two-way frequency
table to yield relative frequencies.
o calculate joint, marginal, and conditional relative frequencies.
o interpret and explain the meaning of relative frequencies in the
context of a problem.
o make appropriate displays of joint, marginal, and conditional
distributions.
o describe patterns observed in the data.
Unit 6 – Analyzing Bivariate Data
Remarks
Resources
Musical Preferences
http://www.illustrativemathematics.org/illustrations/123
Population
http://www.insidemathematics.org/common-core-mathtasks/high-school/HS-S-2004%20Population.pdf
Speed Trap
http://www.illustrativemathematics.org/illustrations/1027
Coffee and Crime
http://www.illustrativemathematics.org/illustrations/1307
Olympic Men’s 100-yard Dash
http://www.illustrativemathematics.org/illustrations/1554
Golf and Divorce
http://www.illustrativemathematics.org/illustrations/44
Exploring Linear Data
http://illuminations.nctm.org/Lesson.aspx?id=1189
Advanced Data Grapher
http://illuminations.nctm.org/Lesson.aspx?id=3081
Histogram Tool
http://illuminations.nctm.org/Activity.aspx?id=4152
Impact of a Superstar
http://illuminations.nctm.org/Lesson.aspx?id=2303
www.learnzillion.com
Choose Domain, Choose Standard (Site contains 895 lessons lasting
3-7 minutes on High School standards.)
https://www.khanacademy.org/commoncore/map
Under high school choose domain.
Standards are listed with information about available
lessons/exercises from which you may choose.
www.cpalms.org
You will find a minimum of 3 formative assessment tasks with
rubrics for each standard as well as many other resources.
The Algebra Nation project at the University of Florida
www.algebranation.com
Unit 7 – Polynomials
Part 1: Students will see structure in and will quadratic and polynomials expressions.
Essential Question:
In what ways can you recognize and create quadratic and polynomial expressions?
How are monomials and polynomials related?
Can two algebraic expressions that appear to be different be equivalent?
How are the properties of real numbers related to polynomials?
Standard
The students will:
Learning Goals
I can:
MAFS.912.A-APR.1.1
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.
•
identify and describe a polynomial.
o define expression, term, factor, and coefficient.
o interpret the real-world meaning of the terms, factors, and
coefficients of an expression in terms of their units.
MAFS. 912.A-SSE.1.1
Interpret expressions that represent a quantity in terms of its context.
a. Interpret parts of an expression, such as terms, factors, and coefficients.
b. Interpret complicated expressions by viewing one or more of their parts as a
n
single entity. For example, interpret P(1+r) as the product of P and a factor
not depending on P.
•
perform operations on polynomials.
o apply the definition of a polynomial to explain why adding,
subtracting, or multiplying two polynomials always produces
a polynomial.
o add, subtract, and multiply polynomials.
o explain why equivalent expressions are equivalent.
MAFS. 912.A-SSE.1.2
4
Use the structure of an expression to identify ways to rewrite it. For example, see x 4
2 2
2 2
y as (x ) -(y ) , thus recognizing it as a difference of squares that can be factored as
2 2
2
2
(x -y )(x +y ).
•
factor polynomials.
o look for and identify clues in the structure of expressions
(e.g., like terms, common factors, difference of squares,
perfect squares) in order to rewrite it another way.
o apply models for factoring and multiplying polynomials to
rewrite expressions.
Unit 7 – Polynomials
Remarks
Resources
www.learnzillion.com
Choose Domain, Choose Standard (Site contains 895 lessons lasting
3-7 minutes on High School standards.)
https://www.khanacademy.org/commoncore/map
Under high school choose domain.
Standards are listed with information about available
lessons/exercises from which you may choose.
www.cpalms.org
You will find a minimum of 3 formative assessment tasks with
rubrics for each standard as well as many other resources.
The Algebra Nation project at the University of Florida
www.algebranation.com
Unit 8 – Solving Quadratic Equations
Part 2: Students will create and solve equations, inequalities, and systems of equations involving quadratic expressions.
Essential Questions:
In what ways can you represent quadratic equations using real world data?
What are the characteristics of quadratic equation?
How can you use quadratic equations in real-world situations?
Standard
Learning Goals
I can:
The students will:
MAFS.912.A-CED.1.1:
• divide polynomials.
Create equations and inequalities in one variable and use them to solve problems.
o divide polynomials using long division and synthetic
Include equations arising from linear and quadratic functions, and simple rational and
division and apply the Remainder Theorem (when
exponential functions.
appropriate) to check the answer.
o apply the Remainder Theorem to determine if a divisor
MAFS.912.A-CED.1.2:
(x-a) is a factor of the polynomials p(x).
Create equations in two or more variables to represent relationships between quantities.
• solve quadratic equations by factoring, completing the
MAFS.912.A-APR.2.2:
square, quadratic formula, and square root method.
Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the
o factor a quadratic expression to find the zeros of the
remainder on division by x-a is p(a), so p(a)=0 if and only if (x-a) is a factor of p(x).
function it represents.
o identify zeros of factored quadratics.
o identify the multiplicity of the zeroes of a factored
MAFS.912.A-APR.2.3:
quadratic.
Identify zeroes of polynomials when suitable factorizations are available, and use the
o identify and factor a perfect square trinomial.
2
zeroes to construct a rough graph of the function defined by the polynomial.
o complete the square of ax +bx+c to write the quadratic
2
in the form (x-p) =q.
o derive the quadratic formula by completing the square
2
MAFS.912.A-APR.3.4:
of ax +bx+c.
Prove polynomial identities and use them to describe numerical relationships. For
o determine the best method to solve a quadratic
example, the polynomial identity (𝑥 2 + 𝑦 2 )2 = (𝑥 2 − 𝑦 2 )2 + (2𝑥𝑦)2 can be used to
equation in one variable.
generate Pythagorean Triples.
o solve quadratic equations by inspection, finding square
roots, completing the square, the quadratic formula, and
MAFS.912.A-APR.4.6:
factoring.
𝑎(𝑥)
o classify real numbers as rational or irrational according
�𝑏(𝑥) in the form
Rewrite simple rational expressions in different forms; write
to their definitions.
o
𝑟(𝑥)
o explain that complex solutions result when the radicand
�𝑏(𝑥), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x)
𝑞(𝑥) +
2
is negative in the quadratic formula (b -4ac<0).
o predict whether a quadratic will have a minimum or a
less than the degree of b(x) , using inspection, long division, or, for the more
maximum based on the value of a.
complicated examples, a graphing calculator.
o identify the maximum or minimum of a quadratic written
MAFS.912.A-REI.2.4:
Solve quadratic equations in one variable.
a. Use the method of completing the square to transform any quadratic equation in x
into an equation of the form (x – p)² = q that has the same solutions. Derive the
quadratic formula from this form.
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.
MAFS.912.A-SSE.2.3: (a-b)
Choose and produce an equivalent form of an expression to reveal and explain
properties of the quantity represented by the expression.
a. Factor a quadratic expression to reveal the zeroes of the function it defines.
b. Complete the square in a quadratic expression to reveal the maximum or
minimum value of the function it defines.
in vertex form.
•
interpret the graphical meaning of a solution to a quadratic
equation.
o explain how the multiplicity of the zeroes provides a
clue as to how the graph will behave when it
approaches and leaves the x-int.
o sketch a rough graph using the zeroes of a polynomial
and other easily identifiable points.
•
solve real world problems involving quadratic equations.
o identify the variables and quantities represented in a
real-world problem.
o determine the best models for a real-world problem.
o write the equation or inequality that best models the
problem.
o interpret the solution in the context of the problem.
o identify variables and quantities represented in a real
world problem
o determine the best model for the real world problem
(e.g. linear versus non-linear)
o write equation that best models the problem
•
simplify polynomial and rational expressions.
o verify polynomial identities (sums and differences of like
powers.
𝒙𝟐 − 𝒚𝟐 = (𝒙 − 𝒚)(𝒙 + 𝒚)
𝟑
𝒙 + 𝒚𝟑 = (𝒙 + 𝒚)(𝒙𝟐 − 𝒙𝒚 + 𝒚𝟐 )
𝒙𝟑 − 𝒚𝟑 = (𝒙 − 𝒚)(𝒙𝟐 + 𝒙𝒚 + 𝒚𝟐 )
o factor polynomials completely by applying the
polynomial identities.
o use polynomial identities to describe numerical
relationships.
o define rational expressions
o determine the best method of simplifying the given
rational expression (inspection, long division, computer
algebra system).
MAFS.912.N-RN.2.3
Explain why the sum or product of two rational numbers is rational; the at 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.
o
o
o
45𝑥²+9𝑥
simplify rational expressions by inspection (e.g.,
= 5x + 1)
simplify rational expressions using long division
simplify complicated rational expressions using a
graphing calculator.
9𝑥
o
write a rational expression
𝑎(𝑥)
𝑏(𝑥)
where a(x) is the
dividend and b(x) is the divisor in the form : q(x) +
𝑟(𝑥)
𝑏(𝑥)
where q(x) is the quotient and r(x) is the remainder
o
Remarks
(e.g.,
𝑥³−𝑥 2 +4𝑥−10
𝑥+2
= x² - 3x + 10 +
−30
𝑥+2
)
Resources
www.learnzillion.com
Choose Domain, Choose Standard (Site contains 895 lessons lasting
3-7 minutes on High School standards.)
https://www.khanacademy.org/commoncore/map
Under high school choose domain.
Standards are listed with information about available
lessons/exercises from which you may choose.
www.cpalms.org
You will find a minimum of 3 formative assessment tasks with
rubrics for each standard as well as many other resources.
The Algebra Nation project at the University of Florida
www.algebranation.com
Unit 9 – Graphing Quadratic Functions
Part 1: Students will learn to anticipate the graph of a quadratic function by interpreting various forms of quadratic expressions. In particular, they identify
the real solutions of a quadratic equation as the zeros of a related quadratic function.
Essential Question:
How can you represent various forms of quadratic functions and predict their solutions?
How can you solve a quadratic equation by graphing?
How can you describe key attributes of a graph by analyzing its equation?
Standard
Learning Goals
I can:
The students will:
MAFS.912.A-CED.1.2
• graph quadratic functions and recognize its features.
Create equations in two or more variables to represent relationships between
o explain that the parent function for quadratic functions is the
2
quantities; graph equations on coordinate axes with labels and scales.
parabola f(x)=x .
o explain that the minimum or maximum of a quadratic is called
MAFS.912.F-BF.2.3
the vertex.
Identify the effect on the graph of replacing f(x) by f(x) + k, k(f(x)), f(kx), and f(x+k) for
o identify whether the vertex of a quadratic will be a minimum or
specific values of k (both positive and negative); find the value of k given the graphs.
maximum by looking at the equation.
Experiment with cases and illustrate an explanation of the effects on the graph using
o find the y-intercept of a quadratic by substituting 0 for x and
technology. Include recognizing even and odd functions from their graphs and
evaluating.
algebraic expressions for them.
o estimate the vertex of a quadratic by evaluating different values
of x.
MAFS.912.F-IF.3.7a
o use calculated values while looking for a minimum or maximum
Graph linear and quadratic functions and show intercepts, maxima and minima.
to decide if the quadratic has x-intercepts.
o estimate the x-intercepts of a quadratic by evaluating different
MAFS.912.F-IF.3.7c
values of x.
Graph polynomial functions, identifying zeros when suitable factorizations are
o graph a quadratic using evaluated points.
available, and showing end behavior.
o use technology to graph a quadratic and to find precise values
for the x-intercept(s) and the maximum or minimum.
MAFS.912.F-IF.3.7d
o identify zeros of a quadratic by factoring.
Graph rational functions, identifying zeroes and asymptotes when suitable
o explain that there are three forms of quadratic functions:
2
2
factorization is available, and showing end behavior.
standard form (f(x)=ax +bx+c), vertex form (f(x)=a(x-h) +k), and
factored form (f(x)=a(x-x1)(x-x2) and that the graph of a quadratic
MAFS.912.F-IF.3.8a
function is a parabola.
Write a function defined by an expression in different but equivalent forms to reveal
o convert a standard form quadratic to factored form by factoring
and explain different properties of the function.
and to vertex form by completing the square.
a. Use the process of factoring and completing the square in a quadratic
o write the function that describes a parabola in all three forms
function to show zeros, extreme values, and symmetry of the graph, and
when given a graph with x-intercepts, y-intercept, and vertex
interpret these in terms of a context.
labeled.
•
use graph of quadratic functions to solve real world problems.
o identify variables and quantities represented in a real world
problem
o determine the best model for the real world problem (ex. linear,
quadraitc)
o write the equation that best models the problem
o set up coordinate axes using appropriate scales and label the
axis
o graph equations on coordinate axes with appropriate label and
scales.
•
describe transformations from the parent function given a quadratic
function.
o explain why f(x) + k translates the original graph of f(x) up k units
and why f(x) –k translates the original graph of F(x) down k units
o explain why f(x + k) translates the original graph of f(x) left k
units and why f(x-k) translates the original graph pf f(x) right k
units
o explain why kf(x) vertically stretches or shrinks the graphs of f(x)
by a factor of k and predict whether a given value of k will cause
a stretch or a shrink
o explain why f(kx) horizontally stretches or shrinks the graph of
1
f(x) by a factor of and predict whether a given value of k will
𝑘
cause a stretch or a shrink
o describe the transformation that change a graph of f(x) into a
different graph when given pictures of the pre-image and image.
o determine the value of k given the graph of a transformed
function.
o graph the listed transformations when given a graph of f(x) and a
value of k ( f(x) ± k, f(x±k) , kf(x) and f(kx).
o use a graphing calculator to generate examples of functions with
different k values
o analyze the similarities and differences between function with
different k values.
o recognize from a graph if the function is even or odd.
o explain that a function is even when f(-x) = f(x) and its graph has
y –axis symmetry
o explain that a function is odd when f(-x) = -f(x) and its graph has
180° rotational symmetry.
•
graph rational functions.
o define rational functions as the ratio of two polynomials
o
o
o
o
o
o
state end behavior of a rational function by inspection of the
graph
find y-intercept and x-intercept of the rational function
describe the end behavior
determine if a rational function has an horizontal asymptote and
find it
sketch the graph based on domain, intercepts and end behavior.
use technology to graph
Unit 9 – Graphing Quadratic Functions
Remarks
Resources
www.learnzillion.com
Choose Domain, Choose Standard (Site contains 895 lessons lasting
3-7 minutes on High School standards.)
https://www.khanacademy.org/commoncore/map
Under high school choose domain.
Standards are listed with information about available
lessons/exercises from which you may choose.
www.cpalms.org
You will find a minimum of 3 formative assessment tasks with
rubrics for each standard as well as many other resources.
The Algebra Nation project at the University of Florida
www.algebranation.com
Unit 10 – Piecewise and Absolute Value Functions
Students will expand their experience with functions to include more specialized functions—absolute value, step, and those that are piecewise-defined.
Essential Question:
How are piecewise functions and step functions different from other functions?
How can you use graphing to solve equations involving absolute value?
How can you use an absolute value function in a real-world situation?
Standard
The students will:
Learning Goals
I can:
MAFS.9.12.A-REI.2.3
•
solve absolute value equations and inequalities.
Solve linear equations and inequalities in one variable, including equations with
coefficients represented by letters.
•
identify and evaluate piecewise functions.
o define piecewise functions as functions that have different
rules for evaluation depending on the value of the input.
o identify which evaluation rule to use for a specific value of x.
o represent a piecewise function with a table or graph by
evaluating several values of x.
o compare properties of two functions when represented in
different ways (algebraically, graphically, numerically in
tables, or by verbal descriptions).
•
graph piecewise and absolute value functions.
o graph a step function by substituting values for x and plotting
the points.
o explain that the parent function for absolute value functions
is f(x)=|𝑥|.
o explain why absolute value is considered a composite
function.
o identify the two evaluation rules use for f(x)=|𝑥| and other
simple absolute values (example: use y=x to evaluate
f(x)=|𝑥| for positive x values and y=-x for negative x values.)
o know that the minimum or maximum of an absolute value is
called the vertex and identify whether the vertex will be a
maximum or a minimum by looking at the equation.
o estimate the x-intercepts of an absolute value by evaluating
different values of x.
o graph an absolute value using evaluated points.
o use technology to graph an absolute value and to find
precise values for the x-intercept(s) and the
MAFS.912.A-REI.4.11
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.
MAFS.912.F-IF.3.7b
Graph square root, cube root, and piecewise-defined functions, including
step functions and absolute value functions.
MAFS.912.F-IF.3.9
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 a larger maximum.
o
o
maximum/minimum.
infer that the x-coordinate of the points of intersection for
y=f(x) and y=g(x) are also solutions for f(x)=g(x).
use a graphing calculator to determine the approximate
solutions.
Unit 10 – Piecewise and Absolute Value Functions
Remarks
Resources
www.learnzillion.com
Choose Domain, Choose Standard (Site contains 895 lessons lasting
3-7 minutes on High School standards.)
https://www.khanacademy.org/commoncore/map
Under high school choose domain.
Standards are listed with information about available
lessons/exercises from which you may choose.
www.cpalms.org
You will find a minimum of 3 formative assessment tasks with
rubrics for each standard as well as many other resources.
The Algebra Nation project at the University of Florida
www.algebranation.com
Unit 11 – Using Graphs of Functions
Part 2: Students will compare key characteristics of quadratic functions to those of linear and exponential functions and model phenomena from among
these functions.
Essential Question:
How can you use quadratic functions in real world situations?
Standard
Learning Goals
I can:
The students will:
MAFS.912.F-BF.1.1b
• can compare quadratic functions with linear and exponential
Combine standard function types using arithmetic operations. For example, build a
functions.
function that models the temperature of a cooling body by adding a constant function
o recall the parent functions.
to a decaying exponential, and relate these functions to the model.
o apply transformations to equations of parent functions
o use graphs or tables to compare the output values of linear,
MAFS. 912.F-IF.2.4
quadratic, and exponential functions.
For a function that models a relationship between two quantities, interpret key
o estimate the intervals for which the output of one function is
features of graphs and tables in terms of the quantities, and sketch graphs showing
greater than the output of another function when given a
key features given a verbal description of the relationship. Key features include:
table or graph.
intercepts; intervals where the function is increasing, decreasing, positive, or
o use technology to find the point at which the graphs of two
negative; relative maximums and minimums; symmetries; end behavior; and
functions intersect.
periodicity.
o use the points of intersection to precisely list the intervals for
which the output of one function is greater than the output of
MAFS. 912.F-IF.2.5
another function.
Relate the domain of a function to its graph and, where applicable, to the quantitative
o use graphs or tables to compare the rates of change of
relationship it describes. For example, if the function h(n) gives the number of
linear, quadratic, polynomial, and exponential functions.
person-hours it takes to assemble n engines in a factor, then the positive integers
o explain why exponential functions eventually have greater
would be an appropriate domain for the function.
output values than linear or quadratic functions by comparing
single functions of each type.
MAFS. 912.F-IF.3.7e
o compare properties of two functions when represented in
Graph exponential and logarithmic functions, showing intercepts and end behavior,
different ways.
and trigonometric functions, showing period, midline, and amplitude.
o
•
MAFS.912.F-IF.3.9
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.
determine and use a function model to solve a real world problem.
o combine different parent functions (adding, subtracting,
multiplying and/or dividing) to write a function that describes
a real-world problem.
o locate the information that explains what each quantity
represents and interpret the meaning of an ordered pair.
o determine if negative inputs and outputs make sense in the
MAFS.912.F-LE.1.3
Observe using graphs and tables that a quantity increasing exponentially eventually
exceeds a quantity increasing linearly, quadratically, or (more generally) as a
polynomial function.
o
o
MAFS.912.A-REI.3.7
Solve simple system consisting of a linear equation and a quadratic equation in two
variables algebraically and graphically.
o
o
o
o
o
o
o
o
•
problem situation
identify the y-intercept and use the problem situation o
explain what the y-intercept means
identify the x-intercept(s) and use the problem situation to
explain what an x-intercept means.
use the function definition to explain why some functions
have more than one x-intercept and why there can only be
one y-intercept.
define and identify “increasing interval” and “decreasing
interval” as a set of function inputs for which the output
increases or decreases, respectively, as the input increases.
use the problem situation to explain where and why the
function is increasing or decreasing.
identify the relative maximums and relative minimums on a
table or graph and use the problem situation to explain the
meaning of each.
identify reflective and rotational symmetries in a table or
graph and use the problem situation to explain why the
function has symmetry.
define positive and negative end behavior as the trend of a
function’s outputs as the input grows increasingly positive or
increasingly negative, respectively and use the problem
situation to explain the end behavior of a function.
explain how the domain of a function is represented in its
graph
state the appropriate domain of a function that represents a
problem situation, defend my choice, and explain why other
numbers might be excluded from the domain.
solve systems of linear and quadratic equations.
o distinguish between equations that are linear and those that
are quadratic
o use substitution to solve a system of equations in which one
equation is linear and one equation is quadratic.
o graph a linear equation on a coordinate plane.
o graph quadratic equations on a coordinate plane.
o determine the approximate solution of a system of equations
in which one equation is linear and one equation is quadratic
by graphing and estimating the point(s) of intersection.
Unit 11 – Using Graphs of Functions
Remarks
Resources
www.learnzillion.com
Choose Domain, Choose Standard (Site contains 895 lessons lasting
3-7 minutes on High School standards.)
https://www.khanacademy.org/commoncore/map
Under high school choose domain.
Standards are listed with information about available
lessons/exercises from which you may choose.
www.cpalms.org
You will find a minimum of 3 formative assessment tasks with
rubrics for each standard as well as many other resources.
The Algebra Nation project at the University of Florida
www.algebranation.com