Math SpringBoard Curriculum Map
Precalculus
Duration ____ Days
Unit One: Sequences and Bivariate Data
Unit Overview
Essential Questions
Academic Vocabulary
In prior courses, students have learned to analyze patterns and
How are recursive
 infinite sequence
investigate linear and nonlinear functions. This unit continues
relationships used to model
 converge
the study of patterns and functions by introducing students to
and investigate long term
 interpolation
arithmetic and geometric sequences and series, recursive
behavior involving sequential
 sigma notation
relationships, and best-fit linear models.
change?
 diverge
 extrapolation
 correlation coefficient
 regression line
How are median fit and least
 residual
squares regression fit lines
used to develop models for
bivariate data?
SpringBoard Activities
Content Focus
1.1 Arithmetic Sequences
DVD Promotions
• Arithmetic sequences and
series
• Sigma notation
• Proof by mathematical
induction
• Geometric sequences and
1.2 Geometric Sequences
Focus Standards and
Learning Objectives
AP/College Readiness
Unit 1 continues to develop the
algebra and graphing of functions. It
extends students’ understanding of
the properties and language of lines
of best fit and sequences and series
by:
• Providing contextual situations
where arithmetic and geometric
sequences and series can be applied.
• Allowing students the opportunity
to work with sequences and series in
a variety of ways: graphical,
analytical, numerical, and verbal.
• Establishing an introductory
understanding of convergence and
divergence of sequences and series
as listed in the course description for
AP Calculus.
• Using technology to explore,
interpret results, and support
conclusions.
• Giving students the opportunity to
collect, analyze, and draw
conclusions from data as described
in the topic list for AP Statistics.
Comments
She Sells Sea Shells
1.3 Modeling Recursive Relationships
Money Market Accounts
EA 1: Sequences
The Old Square Craft
1.4 Median Fit Line
Homework Help Line
1.5 Least-Squares Regression Lines
The Mile Run
1.6 Linear Relationships
Get Fit!
EA 2: Linear Regressions and Predictions
The Ice Cream Shoppe
Assessment and Performance Opportunities
Embedded Assessment 1
Embedded Assessment 2
Check Your Understanding
series
• Infinite geometric series
• Sequences expressed
recursively
• Sequences expressed
explicitly
• Sequences expressed
recursively
• Sequences expressed
explicitly
• Median-median line
• Estimate a trend line for
scatter plot data
• Construct a linear function to
model data
• Calculate sum of squared
residuals
• Regression line minimizes
the sum of the squared
residuals
• Interpolate and extrapolate
with linear models
• Correlation coefficient
• Appropriate use of linear
regression
• Residual plot
• Bivariate outliers
• Regression lines (medianmedian and least squares)
• Sum of squared residuals
• Outlier and the effect on the
correlation coefficient
Additional Resources
Teacher Reflection
Unit Practice
Unit Reflection
SB Online Unit Assessment
In class questioning of students
Monitoring of collaborative groups
Unit Overview
In prior units, students have studied arithmetic and geometric
sequences and series, and modeling functions. In this unit
students will review polynomial, rational, logarithmic, and
exponential functions, and will work with functions and their
transformations.
Math SpringBoard Curriculum Map
Precalculus
Duration ____ Days
Unit Two: Functions and their Graphs
Essential Questions
Academic Vocabulary
How are zeros and end
 relative maximum
behavior of polynomial
 multiple root
functions and their graphs
 relative minimum
related to the degree and the
 even function
factors of the polynomial?
 odd function
 oblique asymptote
 natural logarithm
How are logarithmic and
exponential functions used to
model real-world problems?
SpringBoard Activities
Content Focus
2.1 Polynomials
Sunspots
• Polynomial functions
• Relative maximum
• Relative minimum
• x- and y-intercepts
• End behavior
• Graphing polynomial
functions without a calculator
• Fundamental Theorem of
Algebra
2.2 Graphing Polynomial Functions
Graph It
Focus Standards and
Learning Objectives
AP/College Readiness
Unit 2 expands on students’
understanding of multiple functions
and makes connections between
them by:
• Providing contextual situations
dealing with polynomial, rational,
logarithmic, and exponential
functions, as required in the
prerequisite topic list for AP
Calculus.
• Giving students the opportunity to
look at these functions graphically,
numerically, algebraically, and
verbally, both in and out of
contextual situations.
• Further developing the
understanding of local extrema, and
methods and tools
that will be used at the calculus level
to find them.
• Beginning to look at the concepts
of a limit at infinity
Comments
2.3 Complex Polynomial Roots and Inequalities
Open Question
EA 1: Polynomial Functions
Coffee Time
2.4 Rational Functions
Playing Catch-Up
2.5 Rational Functions
Rationalizing Water Collection
EA 2: Rational Functions
Taneytown Reunion
2.6 Exponential Functions
Pennsylvania Lottery
• Linear Factorization
Theorem
• Rational Root Theorem
• Descartes’ Rule of Signs
• Factor Theorem
• Remainder Theorem
• Creating a polynomial from
its factors
• Complex Conjugate
Theorem
• Complex factors of a
polynomial
• Solving a polynomial
inequality
• Modeling polynomial
functions
• Key features of polynomial
functions
• Graphs of polynomial
functions
• Rational functions
R(x) =(ax+b)/(cx+d)
• Domain and range of a
rational function
• Evaluation of rational
functions using function
notation
• Vertical asymptote
• Horizontal asymptote
• Transformations of rational
functions
• Horizontal and vertical
asymptotes
• Holes
• Slant asymptotes
• Rational functions in context
• Graphing rational functions
• Compound interest
• Exponential functions
2.7 Logarithms
Power Trip
2.8 Transformations of Functions
I Doubt It
2.9 Effects of Transformations
Fore!
EA 3: Transformed Functions
Electric Wind
Assessment and Performance Opportunities
Embedded Assessment 1
Embedded Assessment 2
Embedded Assessment 3
Check Your Understanding
Unit Practice
Unit Reflection
SB Online Unit Assessment
In class questioning of students
Monitoring of collaborative groups
• Formulas and the effects of
parameters
•e
• Common logarithm
• Natural logarithm
• Properties of logarithms
• Change of Base Formula
• Expanding expressions
• Solving exponential and
logarithmic equations
• Transformations of functions
• Even and odd functions
• Operations on functions
• Composition of functions
• Characteristics of data sets
• Transforming bivariate data
• Straightening scatterplots
and linearizing data using
transformations
• Exponential functions
• Logarithmic functions
• Logarithmic re-expression
• Linear regression
• Solving logarithmic
equations
• Transformation of functions
Additional Resources
Teacher Reflection
Unit Overview
In this unit students extend their previous knowledge of right
triangle trigonometry in order to define the six trigonometric
functions in terms of a point on a terminal side of an angle in
standard position. The unit circle is developed along with the
graphs of the trigonometric functions. The inverse
trigonometric functions are defined and applied to the solving
of simple trigonometric equations.
Math SpringBoard Curriculum Map
Precalculus
Duration ____ Days
Unit Three: Trigonometric Functions
Essential Questions
Academic Vocabulary
What type of real-world
 amplitude
problems are modeled and
 angle in standard
solved using trigonometry?
position
 trigonometric function
 inverse trigonometric
 function
How are graphic
 periodic function
representations of
 radian
trigonometric functions
 unit circle
useful in understanding
real-life phenomena?
SpringBoard Activities
Content Focus
3.1 Angles and Angle Measure What's My Angle Measure?
• Angles in standard position
Focus Standards and
Learning Objectives
AP/College Readiness
Unit 3 continues to prepare students
for AP/College Readiness as they
explore trigonometric and inverse
trigonometric functions by:
• Providing a contextual situation
which develops the sine function.
• Providing opportunities to study
the six trigonometric functions and
their inverse functions, including
concepts such as: domain and range,
odd and even, and period, all of
which are included in the
prerequisites for AP Calculus.
• Giving students the opportunity to
learn the values of trigonometric
functions at the numbers 0, π/6,
π/4, π/3, , π/2, and their multiples, a
concept included in the prerequisites
for AP Calculus.
• Giving students opportunities to
work with functions graphically,
numerically, analytically, and
verbally.
• Providing opportunities for the
appropriate use of technology to
solve problems, experiment,
interpret results, and support
conclusions.
Comments
3.2 Sinusoidal Functions
Bicycle Wheels
3.3 Trigonometric Functions and the Unit Circle
Wheels Revisited
3.4 Graphs of the form y = A sin[B (x ? C)] + D
Trigonometric Graphs
3.5 Graphs of Other Trigonometric Functions
More Trig Graphs
EA 1: Angles, Unit Circle, and Trigonometric Graphs
Orbiting Spacecraft
• Coterminal angles
• Radian measure
• Converting between degree
and radian measure
• Reference triangles and right
triangle trigonometry review
• Introduction to sinusoidal
functions
• Periodic functions
• Amplitude
• Definitions of the
trigonometric functions in
terms of a point on the
terminal side of an angle in
standard position
• Unit circle
• Exact values of
trigonometric functions for
special angles
• Graphing functions of the
form
y = A sin [B(x - c)] + D and
y = A cos [B(x - c)] + D
• Writing equations for
sinusoidal curves
• Graphing cosecant, secant,
tangent, and cotangent
functions
• Sinusoidal models
• Graphing trigonometric
functions
• Coterminal angles
• Converting between degree
and radian measure
• Finding trigonometric values
for angles on the unit circle
• Solving for trigonometric
values using the definition of
the trigonometric functions in
3.6 Inverse Trigonometric Functions
Viewing Angle
3.7 Solving Simple Trigonometric Equations
Daylight Minutes
EA 2: Inverse Trig Functions and Equations
Show What You Know
Assessment and Performance Opportunities
Embedded Assessment 1
Embedded Assessment 2
Check Your Understanding
Unit Practice
Unit Reflection
SB Online Unit Assessment
In class questioning of students
Monitoring of collaborative groups
terms of a point on the
terminal side of an angle in
standard position
• Definitions of inverse
trigonometric functions
• Graphs of inverse
trigonometric functions
• Solving simple trigonometric
equations on a given interval
• Finding general solutions for
simple trigonometric
equations
• Reference angles
• Graphing inverse
trigonometric functions
• Evaluating inverse
trigonometric functions
• Solving simple trigonometric
equations
Additional Resources
Teacher Reflection
Math SpringBoard Curriculum Map
Precalculus
Duration ____ Days
Unit Four: Analytic Trigonometry and Trigonometric Applications
Unit Overview
Essential Questions
Academic Vocabulary
Unit 4 focuses on analytic trigonometry. Students learn
How are algebraic and
 trigonometric identity
fundamental identities, verify identities, evaluate expressions,
geometric concepts related
 Law of Sines
and solve equations. They solve a variety of problems using
to trigonometric identities
 angle of elevation /
right-triangle trigonometry, the Law of Cosines and the Law of and formulas?
 depression
Sines. Ultimately, students learn that trigonometric
 ambiguous case (SSA)
relationships are useful tools for performing calculations,
 Law of Cosines
solving problems, and modeling real-world phenomena.
How is trigonometry used to
solve real world problems
involving measure?
SpringBoard Activities
Content Focus
4.1 Trigonometric Identities
Imagine That
• Trigonometric Identities
• Pythagorean Identities
• Simplifying Expressions
• Verifying Identities
• Symmetry Identities
• Cofunction Identities
4.2 Identities and Equations
Triangle Measure
Focus Standards and
Learning Objectives
AP/College Readiness
Unit 4 builds upon what students
learned in the previous unit by
focusing on algebraic applications of
trigonometric functions, equations
and expressions. A thorough
knowledge of trigonometric
functions is a prerequisite skill for
Advanced Placement Calculus.
Unit 4 enhances this knowledge by:
• Building facility with the algebraic
manipulation of trigonometric
functions and equations.
• Preparing students to apply
fundamental trigonometric identities
in a variety of situations.
• Writing equations from verbal and
physical descriptions including the
limiting behavior of n-sided
polygons inscribed in a circle.
• Emphasizing written and verbal
communication by requiring students
to justify their thinking about
mathematical relationships.
• Using technology as a tool to verify
thinking and solve problems.
Comments
4.3 Multiple Angle Identities
Sounds Like Trigonometry
EA 1: Trigonometric Identities and Equations
Identify This, Solve That
4.4 Area of Triangles
Where Has Polly Gone?
4.5 Law of Cosines
The Chocolate Factory
4.6 The Law of Sines
Lost?
EA 2: Right and Oblique Triangles, Area
Tilting Towers and Triangles
Assessment and Performance Opportunities
Embedded Assessment 1
Embedded Assessment 2
• Trigonometric Equations
with Identities
• Sum and Difference
Identities
• Double Angle Identities
• Half Angle Identities
• Verifying Identities with
Multiple Angle Identities
• Solving Equations with
Multiple Angle Identities
• Trigonometric Expressions,
Identities and Equations
• Simplifying, Evaluating,
Verifying and Solving
• Right Triangle Trigonometry
Applications
• Area of Triangle Using
Trigonometry
• Heron’s Formula
• Modeling with
Trigonometric Functions
• Relationship Between
Circular and Linear Motion
• Unit circle definitions of sine
and cosine and identities
• The Law of Cosines
• Right Triangle Trigonometry
Applications
• Area of Triangle Using
Trigonometry
• Heron’s Formula
• Solving problems in context
• Solving right and oblique
triangles
• Finding the area of a triangle
Additional Resources
Teacher Reflection
Check Your Understanding
Unit Practice
Unit Reflection
SB Online Unit Assessment
In class questioning of students
Monitoring of collaborative groups
Math SpringBoard Curriculum Map
Precalculus
Duration ____ Days
Unit Five: Conic Sections, Polar Coordinates, and Complex Numbers
Unit Overview
Essential Questions
Academic Vocabulary
In prior units, students have learned to analyze patterns, graph
How are multiple
 polar coordinate
functions, verify identities and solve multi-step equations. This representations of conic
 conic section
unit continues the study of graphs by introducing students to
sections related and used to
 graph of a polar
conic sections and polar coordinates and graphs.
model real world situations?
equation
 polar form of a complex
number
How are polar and rectangular
graphing related?
SpringBoard Activities
Content Focus
5.1 Parabola Equations and Graphs
The Human Cannon Ball
• Standard form of a Parabola
• General form of a Parabola
• Focus
• Directrix
• Construction of an ellipse
from the locus definition
• Standard and general form of
the equation of an ellipse
• Construction of a hyperbola
5.2 Ellipses and Hyperbolas
Radio Navigation
Focus Standards and
Learning Objectives
AP/College Readiness
Unit 5 continues to develop the
concept of graphs and extends
students’ understanding of properties
and language of conic sections and
polar graphs by:
• Giving students the opportunity to
analyze the graphs of planar curves
as described in the course
description of AP Calculus.
• Providing contextual situations
where the graphs of conic sections
and polar curves can be applied to
solve problems.
• Allowing students to communicate
mathematics and explain solutions
both verbally and in written form.
• Modeling a written description of a
physical situation with a planar
curve.
• Using technology to explore,
interpret results, and support
conclusions.
Comments
5.3 Circles and Degenerate Conics
Crop Circles
EA 1: Equations of Conic Sections
Design of Conic Sections
5.4 Polar Graphs
Air Traffic Controller
5.5 Polar Curves and Polar Conics
Roses, Rings, and Hearts
from the locus definition
• Standard and general form of
the equation of a hyperbola
• Using a hyperbola to model a
real world situation
• Equation of a circle
• Degenerate conics
• General form of a conic
• Determining the equation of
an ellipse from given
information and determining
the foci.
• Rewriting the equation of a
parabola in standard form and
identifying the vertex,
directrix, focus and axis of
symmetry.
• Writing the equation of a
circle given the center and
radius.
• Determining the equation
of a hyperbola from given
information and determining
the foci.
• Points on the polar grid
• Relationships between
rectangular coordinates
and polar coordinates
• Graphs and rectangular form
for polar curves r = a cos(θ) or
r = a sin(θ)
• Relationships between
rectangular equations and
polar equations.
• Graphs of polar Rose,
Cardioid and Limaçon curves
with and without a graphing
calculator.
• Polar equations of conic
5.6 DeMoivre's Theorem
The Power of Complex Numbers
EA 2: Polar and Rectangular Coordinates
Totally Polar Bees
Assessment and Performance Opportunities
Embedded Assessment 1
Embedded Assessment 2
Check Your Understanding
Unit Practice
Unit Reflection
SB Online Unit Assessment
In class questioning of students
Monitoring of collaborative groups
sections.
• Complex numbers in
rectangular and polar forms
• Operations on complex
numbers, algebraically and
geometrically
• Absolute value of a complex
number
• Argument of a complex
number
• De Moivre’s Theorem
• Euler’s Identity and Euler’s
Formula
Additional Resources
Teacher Reflection
Math SpringBoard Curriculum Map
Precalculus
Duration ____ Days
Unit Six: Vectors, Parametric Equations, and Motion
Unit Overview
Essential Questions
Academic Vocabulary
Unit 6 focuses on motion and how to model it mathematically
How are parametric equations
 parametric equation
with both parametric equations and vector representations. This and vectors used to solve
 rectilinear/planar
unit builds on prior knowledge from all of the previous units as real world problems involving
motion
students create rich models and solve problems involving
motion?
 vector
linear, quadratic and circular motion. Vector concepts are
 cross product
applied in both two and three dimensions as students learn to
 dot product
model physical quantities that have both a magnitude and a
 three-dimensional
direction.
What are the advantages
 coordinate system
and disadvantages to using
Cartesian and parametric
coordinate systems for
graphing and solving
problems?
SpringBoard Activities
Content Focus
6.1 Parametric Equations
Ships in the Fog
• Parametric equations
• Relationship between
parametric and non-parametric
functions
• Linear and rectilinear Motion
• Parametric equations
6.2 Introduction to Vectors
Moving Walkways
Focus Standards and
Learning Objectives
AP/College Readiness
Simply stated, calculus is the study
of change and accumulation. This
unit focuses on motion and how to
represent it mathematically. In this
unit, students acquire critical
prerequisite skills by:
• Modeling motion using both
parametric equations and vectors as
specified in the Calculus BC
syllabus.
• Making the connections between
verbal, numerical, algebraic and
graphical representations of relations
and functions explicit to students.
• Writing equations for motion from
physical and verbal models and
representing motion in the
coordinate plane.
• Communicating their
understanding and justifying their
thinking verbally and in writing.
• Using technology to explore
relationships, test conjectures, and
refine algebraic models for solving
complicated problems.
Comments
6.3 Vectors in Two and Three Dimensions
The Robotic Arm
6.4 Parametric Equations Revisited
Keep Your Eye on the Ball
EA 1: Parametric Equations and Vectors
A Pirate's Life
Assessment and Performance Opportunities
Embedded Assessment 1
Check Your Understanding
Unit Practice
Unit Reflection
SB Online Unit Assessment
In class questioning of students
Monitoring of collaborative groups
• Vectors in 1- and 2dimensions
• Vector addition and scalar
multiplication
• Vectors in 2- and 3dimensions
• Vector operations
• Representing fi gures in
3-dimensions
• Dot and cross products of
vectors
• Modeling motion with
parametric equations
• Non-linear parametric
equations
• Using parametric equations
to model situations and solve
problems
• Using vectors to model
situations and solve problems
• Representing geometric
figures in a 3-dimensional
coordinate system
• Operations with vectors
Additional Resources
Teacher Reflection
Math SpringBoard Curriculum Map
Precalculus
Duration ____ Days
Unit Seven: Foundations for Calculus
Unit Overview
Essential Questions
Academic Vocabulary
In prior units, students have studied sequences and explored the How is the concept of limit
 limit
characteristics of many types of functions, including
applied in mathematics to
 continuous
trigonometric and rational functions. In this unit, students will
function discontinuities?
 discontinuous
extend their study of sequences to include their related
functions. In addition, students will continue to examine
characteristics of functions, specifi cally end behavior and
possible discontinuities, to develop an understanding of limits
Why are limits important in
and their connection to the continuity of a function.
the study of end behavior?
SpringBoard Activities
Content Focus
7.1 Modeling Functions
Bungee Jumping
• Trigonometric functions/
harmonic motion
• Exponential
functions/damped
harmonic motion,
• Limit of a function at a point
• One-sided limits
• Limit of a sequence
• Limits at infinity
• Continuity of a function
7.2 Limits
Close Enough
7.3 Limit of a Sequence and Limits at Infinity
Reached Your Limit?
7.4 Continuity
Focus Standards and
Learning Objectives
AP/College Readiness
Unit 7 continues to develop the
algebraic and graphical
representations of functions and
extends students’ understanding of
the properties, graphs, algebra, and
language of functions by:
• Giving students opportunities to
work with multiple representations
of functions and to make
connections between the
representations.
• Allowing students to communicate
mathematics and explain solutions
both verbally and in writing.
• Using technology to help solve
problems and support conclusions.
• Modeling written descriptions of
physical situations with functions.
• Developing an intuitive
understanding of the limiting process
and of continuity.
Comments
• Intermediate Value Theorem
• Extreme Value Theorem
Embedded Assessment 1: Limits and Continuity
• Finding limits of a function
Ready for Calculus
at a particular point or at
infinity by using graphs, tables
or algebraic methods
• Convergence and divergence
of a sequence
• Continuity of a function
Assessment and Performance Opportunities Additional Resources
Continuing On
Embedded Assessment 1
Check Your Understanding
Unit Practice
Unit Reflection
SB Online Unit Assessment
In class questioning of students
Monitoring of collaborative groups
Teacher Reflection