Download learning trajectory display of the common core state standards for

LEARNING TRAJECTORY DISPLAY OF THE COMMON CORE STATE STANDARDS FOR MATHEMATICS
ALGEBRAIC REASONING - EXPONENTS, ROOTS AND SCIENTIFIC NOTATION, EXPRESSIONS, ONE VARIABLE EQUATIONS, INEQUALITIES AND WORD PROBLEMS, SIMULTANEOUS LINEAR
EQUATIONS AND INTRODUCTION TO FUNCTIONS AND LINEAR FUNCTIONS (GRADES 6 – 8)
ALGEBRA – DEFINITIONS AND PROOFS, EXPRESSIONS AND FORMULAS, GRAPHING, LINEAR, POLYNOMIAL AND RATIONAL FUNCTIONS, EXPONETIAL FUNCTIONS AND SYSTEMS OF EQUATIONS
AND INEQUALITIES; FUNCTIONS – FUNCTION DEFINITION, OPERATIONS, COMPOSITION, AND INVERSE, SEQUENCES, RATE OF CHANGE AND CHARACTERISTICS OF GRAPHS, QUADRATIC
FUNCTIONS, TRIGONOMETRIC FUNCTIONS, TRANSFORMATIONS AND MODELING WITH FUNCTIONS (HIGH SCHOOL)
GRADE 6
GRADE 7
GRADE 8
HIGH SCHOOL
EXPONENTS, ROOTS AND SCIENTIFIC NOTATION
6.EE.1 Write and evaluate
numerical expressions
involving whole number
exponents.
N/A
8.EE.1 Know and apply the properties
of integer exponents to generate
equivalent numerical expressions. For
example, 32 x 3-5 = 3-3 = (1/3)3 = 1/27
8.EE.3 Use numbers expressed in the
form of a single digit multiplied by an
integer power of 10 to estimate very
large or very small quantities, and to
express how many times larger one is
than the other. For example, estimate
the population of the United States as
3 x 108 and the population of the world
as 7 x 109, and determine that the
world population is more than 20 times
larger.
8.EE.4 Perform operations with
numbers expressed in scientific
notation, including problems where
both decimal and scientific notation
are used. Use scientific notation and
choose units of appropriate size for
measurements of very large or very
small quantities. (e.g., use millimeters
per year for seafloor spreading).
Interpret scientific notation that has
been generated by technology.
8.EE.2 Use square root and cube root
symbols to represent solutions to
equations of the form x2 = p and x3 = p
where p is a positive rational number.
Evaluate square roots of small perfect
squares and cube roots of small perfect
cubes. Know that √2 is irrational.
EXPONENTIAL FUNCTIONS
LEVEL 1
A-CED.1.ii Create exponential equations and inequalities in one variable, including equations with
coefficients represented by letters.
LEVEL 2
A-SSE.3.C.iv Use the properties of exponents to transform expressions for exponential functions. For
example, the expression 1.15t can be rewritten as (1.151/12 )12t ≈ 1.01212t to reveal the approximate
equivalent monthly interest rate if the annual rate is 15%.
LEVEL 3
N/A
LEVEL 4
N/A
LEVEL 5
N/A
LEVEL 6
N/A
LEVEL 7
N/A
LEVEL 8
N/A
(+) indicates topics students should learn in order to take advanced mathematics courses
LEARNING TRAJECTORY POSTERS (Drs. Jere Confrey, Alan Maloney and Kenny Nguyen) CAN BE PURCHASED AT http://www.wirelessgeneration.com/posters
The levels indicate increasing levels of complexity or sophistication of reasoning. Levels 4 and above are mostly STEM intending content.
LEARNING TRAJECTORY DISPLAY OF THE COMMON CORE STATE STANDARDS FOR MATHEMATICS
ALGEBRAIC REASONING - EXPONENTS, ROOTS AND SCIENTIFIC NOTATION, EXPRESSIONS, ONE VARIABLE EQUATIONS, INEQUALITIES AND WORD PROBLEMS, SIMULTANEOUS LINEAR
EQUATIONS AND INTRODUCTION TO FUNCTIONS AND LINEAR FUNCTIONS (GRADES 6 – 8)
ALGEBRA – DEFINITIONS AND PROOFS, EXPRESSIONS AND FORMULAS, GRAPHING, LINEAR, POLYNOMIAL AND RATIONAL FUNCTIONS, EXPONETIAL FUNCTIONS AND SYSTEMS OF EQUATIONS
AND INEQUALITIES; FUNCTIONS – FUNCTION DEFINITION, OPERATIONS, COMPOSITION, AND INVERSE, SEQUENCES, RATE OF CHANGE AND CHARACTERISTICS OF GRAPHS, QUADRATIC
FUNCTIONS, TRIGONOMETRIC FUNCTIONS, TRANSFORMATIONS AND MODELING WITH FUNCTIONS (HIGH SCHOOL)
GRADE 6
GRADE 7
GRADE 8
6.EE.2.ab.i Write, read and evaluate expressions in which
letters stand for numbers. a. Write expressions that record
operations with numbers and with letters standing for
numbers. For example, express the calculation, “subtract y
from 5” as 5 – y. b. Identify parts of an expression using
mathematical terms (sum, term, product, factor, quotient,
coefficient); view one or more parts of an expression as a
single entity. For example, describe the expression 2(8 + 7) as a
product of two factors; view (8 + 7) as both a single entity and
a sum of two terms.
7.EE.1 Apply
properties of
operations as
strategies to add,
subtract, factor, and
expand linear
expressions with
rational
coefficients.
N/A
6.EE.2.c.ii Evaluate expressions at specific values of their
variables. Include expressions embedded in formulas or
equations from real-world problems. Perform arithmetic
operations, including those involving whole number
exponents, in the conventional order when there are no
parentheses to specify a particular order (Order of Operations).
For example, use the formulas V = s3 and A = 6s2 to find the
value of the expression equal to the volume and surface area
of a cube with sides of length s = ½.
7.EE.2 Understand
that rewriting an
expression in
different forms in a
problem context
can shed light on
the problem and
how the quantities
in it are related. For
example, a + 0.05a
= 1.05a means that
“increase by 5%” is
the same as
“multiply by 1.05”.
HIGH SCHOOL
EXPRESSIONS
6.EE.3 Apply the properties of operations to generate
equivalent expressions. For example, apply the distributive
property to the expression 3(2 + x) to produce the equivalent
expression 6 + 3x; apply the distributive property to the
expression 24x + 18y to produce the equivalent expression
6(4x + 3y); apply properties of operations to y + y + y to
produce the equivalent expression 3y.
6.EE.4 Identify when two expressions are equivalent (i.e., when
the two expressions name the same number regardless of
which value is substituted into them). For example, the
expressions y + y + y and 3y are equivalent because they name
the same number regardless of which number y stands for.
EXPRESSIONS AND FORMULAS
LEVEL 1
A-SSE.2 Use the structure of an expression to identify ways to rewrite it.
For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference
of squares that can be factored as (x2 – y2)(x2 + y2).
LEVEL 2
A-SSE.3.i Choose and produce an equivalent form of an expression to
reveal and explain properties of the quantity represented by the
expression. * (REFERS TO MODELING)
LEVEL 3
A-CED.4 Rearrange formulas to highlight a quantity of interest, using the
same reasoning as in solving equations. For example, rearrange Ohm’s
Law V = IR to highlight resistance R.
LEVEL 4
A-SSE.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. For
example, calculate mortgage payments.
LEVEL 5
N/A
LEVEL 6
N/A
LEVEL 7
A-APR.7 (+) Understand that rational expressions form a system
analogous to the rational numbers, closed under addition, subtraction,
multiplication, and division by a nonzero rational expression; add,
subtract, multiply, and divide rational expressions.
LEVEL 8
N/A
6.EE.6 Use variables to represent numbers and write
expressions when solving a real-world or mathematical
problem; understand that a variable can represent an
unknown number, or, depending on the purpose at hand, any
number in a specified set.
(+) indicates topics students should learn in order to take advanced mathematics courses
LEARNING TRAJECTORY POSTERS (Drs. Jere Confrey, Alan Maloney and Kenny Nguyen) CAN BE PURCHASED AT http://www.wirelessgeneration.com/posters
The levels indicate increasing levels of complexity or sophistication of reasoning. Levels 4 and above are mostly STEM intending content.
LEARNING TRAJECTORY DISPLAY OF THE COMMON CORE STATE STANDARDS FOR MATHEMATICS
ALGEBRAIC REASONING - EXPONENTS, ROOTS AND SCIENTIFIC NOTATION, EXPRESSIONS, ONE VARIABLE EQUATIONS, INEQUALITIES AND WORD PROBLEMS, SIMULTANEOUS LINEAR
EQUATIONS AND INTRODUCTION TO FUNCTIONS AND LINEAR FUNCTIONS (GRADES 6 – 8)
ALGEBRA – DEFINITIONS AND PROOFS, EXPRESSIONS AND FORMULAS, GRAPHING, LINEAR, POLYNOMIAL AND RATIONAL FUNCTIONS, EXPONETIAL FUNCTIONS AND SYSTEMS OF EQUATIONS
AND INEQUALITIES; FUNCTIONS – FUNCTION DEFINITION, OPERATIONS, COMPOSITION, AND INVERSE, SEQUENCES, RATE OF CHANGE AND CHARACTERISTICS OF GRAPHS, QUADRATIC
FUNCTIONS, TRIGONOMETRIC FUNCTIONS, TRANSFORMATIONS AND MODELING WITH FUNCTIONS (HIGH SCHOOL)
GRADE 6
GRADE 7
GRADE 8
HIGH SCHOOL
ONE VARIABLE EQUATIONS, INEQUALITIES, AND
WORD PROBLEMS
6.EE.5 Understand
solving an equation or
inequality as a process of
answering a question:
which values from a
specified set, in any,
make the equation or
inequality true? Use
substitution to determine
whether a given number
in a specified set makes
an equation or inequality
true.
6.EE.7 Solve real-world
and mathematical
problems by writing and
solving equations of the
form x + p = q and px = q
for cases in which p, q
and x are all nonnegative
rational numbers.
6.EE.8 Write an inequality
of the form x > c or x < c
to represent a constraint
or condition in a realworld or mathematical
problem. Recognize that
inequalities of the form x
> c or x < c have infinitely
many solutions;
represent solutions of
such inequalities on
number line diagrams.
7.EE.4.a.i Use variables to represent
quantities in a real-world or
mathematical problem, and construct
simple equations to solve problems
by reasoning about the quantities. a.
Solve word problems leading to
equations of the form px + q = r and
p(x + q) = r, where p, q and r are
specific rational numbers. Solve
equations of these forms fluently.
Compare an algebraic solution to an
arithmetic solution, identifying the
sequence of the operations used in
each approach. For example, the
perimeter of a rectangle is 54 cm. Its
length is 6 cm. What is its width?
7.EE.4.b.ii Use variables to represent
quantities in a real-world or
mathematical problem, and construct
simple inequalities to solve problems
by reasoning about the quantities. b.
Solve word problems leading to
inequalities of the form px + q > r or
px + q < r, where p, q and r are
specific rational numbers. Graph the
solution set of the inequality and
interpret it in the context of the
problem. For example, As a
salesperson, you are paid $50 per
week plus $3 per sale. This week you
want your pay to be at least $100.
Write an inequality for the number of
sales you need to make, and describe
the solutions.
8.EE.7.ab Solve linear
equations in one
variable. a. Give
examples of linear
equations in one
variable with one
solution, infinitely
many solutions, or no
solution. Show which of
these possibilities is the
case by successively
transforming the given
equation into simpler
forms, until an
equivalent equation of
the form x = a, a = a, or
a = b results (where a
and b are different
numbers). b. Solve
linear equations with
rational number
coefficients, including
equations whose
solutions require
expanding expressions
using the distributive
property and collecting
like terms.
SYSTEMS OF EQUATIONS AND INEQUALITIES
LEVEL 1
N/A
LEVEL 2
A-REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs),
focusing on pairs of linear equations in two variables.
LEVEL 3
A-REI.7 Solve a simple system consisting of a linear equation and a quadratic equation in two
variables algebraically and graphically. For example, find the points of intersection between
the line y = -3x and the circle x2 + y2 = 3.
LEVEL 4
A-REI.8 (+) Represent a system of linear equations as a single matrix equation in a vector
variable.
LEVEL 5
A-REI.9 (+) Find the inverse of a matrix if it exists and use it to solve systems of linear
equations (using technology for matrices of dimension 3 x 3 or greater).
LEVEL 6
N/A
LEVEL 7
N/A
LEVEL 8
N/A
(+) indicates topics students should learn in order to take advanced mathematics courses
LEARNING TRAJECTORY POSTERS (Drs. Jere Confrey, Alan Maloney and Kenny Nguyen) CAN BE PURCHASED AT http://www.wirelessgeneration.com/posters
The levels indicate increasing levels of complexity or sophistication of reasoning. Levels 4 and above are mostly STEM intending content.
LEARNING TRAJECTORY DISPLAY OF THE COMMON CORE STATE STANDARDS FOR MATHEMATICS
ALGEBRAIC REASONING - EXPONENTS, ROOTS AND SCIENTIFIC NOTATION, EXPRESSIONS, ONE VARIABLE EQUATIONS, INEQUALITIES AND WORD PROBLEMS, SIMULTANEOUS LINEAR
EQUATIONS AND INTRODUCTION TO FUNCTIONS AND LINEAR FUNCTIONS (GRADES 6 – 8)
ALGEBRA – DEFINITIONS AND PROOFS, EXPRESSIONS AND FORMULAS, GRAPHING, LINEAR, POLYNOMIAL AND RATIONAL FUNCTIONS, EXPONETIAL FUNCTIONS AND SYSTEMS OF EQUATIONS
AND INEQUALITIES; FUNCTIONS – FUNCTION DEFINITION, OPERATIONS, COMPOSITION, AND INVERSE, SEQUENCES, RATE OF CHANGE AND CHARACTERISTICS OF GRAPHS, QUADRATIC
FUNCTIONS, TRIGONOMETRIC FUNCTIONS, TRANSFORMATIONS AND MODELING WITH FUNCTIONS (HIGH SCHOOL)
GRADE 6
GRADE 7
GRADE 8
HIGH SCHOOL
SIMULTANEOUS LINEAR EQUATIONS
N/A
N/A
8.EE.8.abc Analyze and solve pairs of
simultaneous linear equations. a.
Understand that solutions to a system of
two linear equations in two variables
correspond to points of intersection of
their graphs, because points of intersection
satisfy both equations simultaneously. b.
Solve systems of two linear equations in
two variables algebraically, and estimate
solutions by graphing the equations. Solve
simple cases by inspection. For example, 3x
+ 2y = 5 and 3x + 2y = 6 have no solution
because 3x + 2y cannot simultaneously by 5
and 6. c. Solve real-world and
mathematical problems leading to two
linear equations in two variables. For
example, given coordinates for two pairs of
points, determine whether the line through
the first pair of points intersects the line
through the second pair.
DEFINITIONS AND PROOFS
LEVEL 1
A-SSE.1.ab Interpret expressions that represent a quantity in terms of its context. * (REFERS
TO MODELING) 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. For
example, interpret P(1 + r)n as the product of P and a factor not depending on P.
LEVEL 2
A-REI.1 Understand solving equations as a process of reasoning and explain the reasoning.
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.
LEVEL 3
A-REI.5 Prove that, given a system of two equations in two variables, replacing one equation
by the sum of that equation and a multiple of the other produces a system with the same
solution.
LEVEL 4
N/A
LEVEL 5
N/A
LEVEL 6
F-TF.8 Prove the Pythagorean identity sin2(□) + cos2(□) = 1 and use it to find sin(□), cos(□), or
tan(□), given sin(□), cos(□), or tan(□) and the quadrant of the angle.
LEVEL 7
F-TF.9 (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use
them to solve problems.
LEVEL 8
N/A
(+) indicates topics students should learn in order to take advanced mathematics courses
LEARNING TRAJECTORY POSTERS (Drs. Jere Confrey, Alan Maloney and Kenny Nguyen) CAN BE PURCHASED AT http://www.wirelessgeneration.com/posters
The levels indicate increasing levels of complexity or sophistication of reasoning. Levels 4 and above are mostly STEM intending content.
LEARNING TRAJECTORY DISPLAY OF THE COMMON CORE STATE STANDARDS FOR MATHEMATICS
ALGEBRAIC REASONING - EXPONENTS, ROOTS AND SCIENTIFIC NOTATION, EXPRESSIONS, ONE VARIABLE EQUATIONS, INEQUALITIES AND WORD PROBLEMS, SIMULTANEOUS LINEAR
EQUATIONS AND INTRODUCTION TO FUNCTIONS AND LINEAR FUNCTIONS (GRADES 6 – 8)
ALGEBRA – DEFINITIONS AND PROOFS, EXPRESSIONS AND FORMULAS, GRAPHING, LINEAR, POLYNOMIAL AND RATIONAL FUNCTIONS, EXPONETIAL FUNCTIONS AND SYSTEMS OF EQUATIONS
AND INEQUALITIES; FUNCTIONS – FUNCTION DEFINITION, OPERATIONS, COMPOSITION, AND INVERSE, SEQUENCES, RATE OF CHANGE AND CHARACTERISTICS OF GRAPHS, QUADRATIC
FUNCTIONS, TRIGONOMETRIC FUNCTIONS, TRANSFORMATIONS AND MODELING WITH FUNCTIONS (HIGH SCHOOL)
GRADE 6
GRADE 7
GRADE 8
HIGH SCHOOL
INTRODUCTION TO FUNCTIONS AND LINEAR
FUNCTIONS
6.EE.9 Use variables to
represent two quantities in a
real-world problem that
change in relationship to one
another; write an equation to
express one quantity, thought
of as the dependent variable,
in terms of the other quantity,
thought of as the independent
variable. Analyze the
relationship between the
dependent and independent
variables using graphs and
tables, and relate these to the
equation. For example, in a
problem involving motion at
constant speed, list and graph
ordered pairs of distances and
times, and write the equation
d = 65t to represent between
distance and time.
N/A
8.F.1 Understand that a function is a rule
that assigns to each input exactly one
output. The graph of a function is the set
of ordered pairs consisting of an input and
the corresponding output.
8.F.5 Describe qualitatively the functional
relationship between two quantities by
analyzing a graph (e.g., where the function
is increasing or decreasing, linear or
nonlinear). Sketch a graph that exhibits the
qualitative features of a function that has
been described verbally.
8.F.2 Compare properties of two functions
each represented in a different way
(algebraically, graphically, numerically in
tables, or by verbal descriptions). For
example, given a linear function
represented by a function has the greater
rate of change.
LINEAR, POLYNOMIAL, AND RATIONAL FUNCTIONS
LEVEL 1
A-CED.1.i Create linear equations and inequalities in one variable and use them to solve problems.
A-REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented
by letters.
LEVEL 2
A-CED.1.iii Create quadratic equations and inequalities in one variable and use them to solve problems.
LEVEL 3
A-REI-4.ab (Level 2 and Level 3) 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.
A-REI-2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous
solutions may arise.
A-APR.1 Perform arithmetic operations on polynomials. Understand that polynomials form a system analogous
to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add,
subtract, and multiply polynomials.
A-APR.2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, 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).
LEVEL 4
LEVEL 5
LEVEL 6
A-APR.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a
rough graph of the function defined by the polynomial.
LEVEL 7
A-APR.4 Prove polynomial identities and use them to describe numerical relationships. For example, the
polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate Pythagorean triples.
LEVEL 8
A-APR.5 (+) Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a
positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle,
(the Binomial Theorem can be proved by mathematical induction or by a combinatorial argument).
(+) indicates topics students should learn in order to take advanced mathematics courses
LEARNING TRAJECTORY POSTERS (Drs. Jere Confrey, Alan Maloney and Kenny Nguyen) CAN BE PURCHASED AT http://www.wirelessgeneration.com/posters
The levels indicate increasing levels of complexity or sophistication of reasoning. Levels 4 and above are mostly STEM intending content.
LEARNING TRAJECTORY DISPLAY OF THE COMMON CORE STATE STANDARDS FOR MATHEMATICS
HIGH SCHOOL ONLY TOPICS
FUNCTIONS – FUNCTION DEFINITION, OPERATIONS, COMPOSITION, AND INVERSE, SEQUENCES, RATE OF CHANGE AND CHARACTERISTICS OF GRAPHS, QUADRATIC FUNCTIONS,
TRIGONOMETRIC FUNCTIONS, TRANSFORMATIONS AND MODELING WITH FUNCTIONS
HIGH SCHOOL
HIGH SCHOOL
FUNCTION DEFINITION, OPERATIONS, COMPOSITION AND INVERSE
LEVEL 1
SEQUENCES
F-IF.1 Understand that a function from one set (called the domain) to another set (called
the range) assigns to each element of the domain exactly one element of the range. If f is a
function and x is an element of its domain, the f(x) denotes the output of f corresponding
to the input of x. The graph of f is the graph of the equation y = f(x).
F-IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret
statements that use function notation in terms of a context.
F-IF.5 Relate the domain of a function to its graph and, where applicable, to the
quantitative relationship it describes. For example, if the function h(n) gives the number of
person-hours it takes to assemble n engines in a factory, then the positive integers would
be an appropriate domain for the function. * (REFERS TO MODELING)
F-LE.5 Interpret expressions for functions in terms of the situation they model. Interpret
the parameters in a linear or exponential function in terms of a context
F-BF.1.abc (Level 3 and Level 4) Write a function that describes a relationship between
two quantities. * (REFERS TO MODELING) a. Determine an explicit expression, a recursive
process, or steps for calculation from a context. b. Combine standard function types using
arithmetic operations. For example, build a function that models the temperature of a
cooling body by adding a constant function to a decaying exponential, and relate these
functions to the model. c. (+) Compose functions. For example, it T(y) is the temperature in
the atmosphere as a function of height, and h(t) is the height of a weather balloon as a
function of time, then T(h(t)) is the temperature at the location of the weather balloon as a
function of time.
F-IF.8.b.ii Write a function defined by an expression in different but equivalent forms to
reveal and explain different properties of the function. b. Use the properties of exponents
to interpret expressions for exponential functions. For example, identify percent rate of
change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)t, y = (1.01)12t y = (1.2)t/10, and
classify them as representing exponential growth and decay.
SEE ABOVE F-BF.1.abc
LEVEL 1
F-BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula,
use them to model situations, and translate between the two forms. * (REFERS TO MODELING)
LEVEL 2
F-IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is
a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1)
= 1, f(n + 1) = f(n) + f((n – 1) for n ≥ 1.
LEVEL 3
N/A
LEVEL 4
N/A
LEVEL 5
N/A
LEVEL 6
F-BF.4.abcd 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 example, f(x) =
2x3 or f(x) = (x + 1)/(x – 1) for 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. (+) from a non-invertible function by restricting the
domain.
F-BF.5 (+) Understand the inverse relationship between exponents and logarithms and use
this relationship to solve problems involving logarithms and exponents.
N/A
LEVEL 6
N/A
LEVEL 7
N/A
LEVEL 7
N/A
LEVEL 8
N/A
LEVEL 8
N/A
LEVEL 2
LEVEL 3
LEVEL 4
LEVEL 5
(+) indicates topics students should learn in order to take advanced mathematics courses
LEARNING TRAJECTORY POSTERS (Drs. Jere Confrey, Alan Maloney and Kenny Nguyen) CAN BE PURCHASED AT http://www.wirelessgeneration.com/posters
The levels indicate increasing levels of complexity or sophistication of reasoning. Levels 4 and above are mostly STEM intending content.
LEARNING TRAJECTORY DISPLAY OF THE COMMON CORE STATE STANDARDS FOR MATHEMATICS
HIGH SCHOOL ONLY TOPICS
FUNCTIONS – FUNCTION DEFINITION, OPERATIONS, COMPOSITION, AND INVERSE, SEQUENCES, RATE OF CHANGE AND CHARACTERISTICS OF GRAPHS, QUADRATIC FUNCTIONS,
TRIGONOMETRIC FUNCTIONS, TRANSFORMATIONS AND MODELING WITH FUNCTIONS
HIGH SCHOOL
HIGH SCHOOL
RATE OF CHANGE AND CHARACTERISTICS OF GRAPHS
QUADRATIC FUNCTIONS
LEVEL 1
F-IF.6 Calculate and interpret the average rate of change of a function (presented
symbolically or as a table) over a specified interval. Estimate the rate of change from a
graph. * (REFERS TO MODELING)
F-IF.7abcd (Level 1, Level 2 and Level 3) Graph functions expressed symbolically and show
key features of the graph, by hand in simple cases and using technology for more
complicated cases. * (REFERS TO MODELING) a. Graph linear and quadratic functions and
show intercepts, maxima, and minima b. Graph square root, root cube, and piecewisedefined functions, including step functions and absolute value functions. c. Graph
polynomial functions, identifying zeros when suitable factorizations are available, and
showing end behavior. d. (+) Graph rational functions, identifying zeros and asymptotes
when suitable factorizations are available, and showing end behavior. e. Graph exponential
and logarithmic functions, showing intercepts and end behavior, and trigonometric
functions, showing intercepts and end behavior, and trigonometric functions, showing
period, midline and amplitude.
LEVEL 1
N/A
LEVEL 2
LEVEL 2
A-SSE.3.a.ii Factor a quadratic expression to reveal the zeros of the function it defines.
LEVEL 3
F-IF.4 (Level 2, Level 3, Level 4 and Level 5) For a function that models a relationship
between two quantities, interpret key features of graphs and tables in terms of the
quantities, and sketch graphs showing key features given a verbal description of the
relationship. Key features [could] include: intercepts; intervals where the function is
increasing, decreasing, positive or negative; relative maximums and minimums;
symmetries; end behavior; and periodicity. * (REFERS TO MODELING)
SEE ABOVE F-IF.4
LEVEL 3
LEVEL 4
SEE ABOVE F-IF.4
LEVEL 4
LEVEL 5
LEVEL 6
SEE ABOVE F-IF.4
N/A
LEVEL 5
LEVEL 6
A-SSE.3.b.iii Complete the square in a quadratic expression to reveal the maximum or minimum
value of the function it defines.
F-IF.8.a.i Write a function defined by an expression in different but equivalent forms to reveal
and explain different properties of the function. a. Use the process of factoring and completing
the square in a quadratic function to show zeros, extreme values, and symmetry of the graph,
and interpret these in terms of a context.
N/A
N/A
LEVEL 7
N/A
LEVEL 7
N/A
LEVEL 8
N/A
LEVEL 8
N/A
(+) indicates topics students should learn in order to take advanced mathematics courses
LEARNING TRAJECTORY POSTERS (Drs. Jere Confrey, Alan Maloney and Kenny Nguyen) CAN BE PURCHASED AT http://www.wirelessgeneration.com/posters
The levels indicate increasing levels of complexity or sophistication of reasoning. Levels 4 and above are mostly STEM intending content.
LEARNING TRAJECTORY DISPLAY OF THE COMMON CORE STATE STANDARDS FOR MATHEMATICS
HIGH SCHOOL ONLY TOPICS
FUNCTIONS – FUNCTION DEFINITION, OPERATIONS, COMPOSITION, AND INVERSE, SEQUENCES, RATE OF CHANGE AND CHARACTERISTICS OF GRAPHS, QUADRATIC FUNCTIONS,
TRIGONOMETRIC FUNCTIONS, TRANSFORMATIONS AND MODELING WITH FUNCTIONS
HIGH SCHOOL
HIGH SCHOOL
TRIGONOMETRIC FUNCTIONS
TRANSFORMATIONS
LEVEL 1
N/A
LEVEL 1
N/A
LEVEL 2
N/A
LEVEL 2
F-IF.9 Compare properties of two functions each represented in a different way (algebraically,
graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one
quadratic function and an algebraic expression for another, say which has the larger maximum.
LEVEL 3
N/A
LEVEL 3
LEVEL 4
N/A
LEVEL 4
F-BF.3 (Level 3, Level 4 and Level 5) Identify the effect on the graph of replacing f(x) by f(x) + k,
kf(x), f(kx) and f(x + k) for specific values of k (both positive and negative); find the value of k
given the graphs. Experiment with cases and illustrate an explanation of the effects on the
graphs using technology. Include recognizing even and odd functions from their graphs and
algebraic expressions for them.
SEE ABOVE F-BF.3
LEVEL 5
SEE ABOVE F-BF.3
LEVEL 5
LEVEL 6
LEVEL 7
LEVEL 8
F-TF.1 Understand radian measure of an angle as the length of the arc on the unit circle
subtended by the angle.
F-TF.3(+) Use special triangles to determine geometrically the values of sine, cosine,
tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosines,
and tangent for x, π + x, and 2π – x in terms of their values for x, where x is any real
number.
F-TF.2 Explain how the unit circle in the coordinate plane enables the extension of
trigonometric functions to all real numbers, interpreted as radian measures of angles
traversed counterclockwise around the unit circle.
F-TF.4. (+) Use the unit circle to explain symmetry (odd and even) and periodicity of
trigonometric functions.
LEVEL 6
N/A
LEVEL 7
N/A
LEVEL 8
N/A
(+) indicates topics students should learn in order to take advanced mathematics courses
LEARNING TRAJECTORY POSTERS (Drs. Jere Confrey, Alan Maloney and Kenny Nguyen) CAN BE PURCHASED AT http://www.wirelessgeneration.com/posters
The levels indicate increasing levels of complexity or sophistication of reasoning. Levels 4 and above are mostly STEM intending content.
LEARNING TRAJECTORY DISPLAY OF THE COMMON CORE STATE STANDARDS FOR MATHEMATICS
HIGH SCHOOL ONLY TOPICS
FUNCTIONS – FUNCTION DEFINITION, OPERATIONS, COMPOSITION, AND INVERSE, SEQUENCES, RATE OF CHANGE AND
CHARACTERISTICS OF GRAPHS, QUADRATIC FUNCTIONS, TRIGONOMETRIC FUNCTIONS, TRANSFORMATIONS AND
MODELING WITH FUNCTIONS
HIGH SCHOOL
MODELING WITH FUNCTIONS
LEVEL 1
F-LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a
description of a relationship, or two input-output pairs (include reading these from a table).
LEVEL 2
F-LE.1abc 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.
F-LE.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing
linearly, quadratically, or (more generally) as a polynomial function.
LEVEL 3
LEVEL 4
N/A
LEVEL 5
F-LE.4 For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b
is 2, 10, or e; evaluate the logarithm using technology. * (REFERS TO MODELING)
F-TF.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
F-TF.6. (+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always
decreasing allows its inverse to be constructed.
F-TF.7. (+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions
using technology, and interpret them in terms of the context. * (REFERS TO MODELING)
LEVEL 6
LEVEL 7
LEVEL 8
(+) indicates topics students should learn in order to take advanced mathematics courses
LEARNING TRAJECTORY POSTERS (Drs. Jere Confrey, Alan Maloney and Kenny Nguyen) CAN BE PURCHASED AT http://www.wirelessgeneration.com/posters
The levels indicate increasing levels of complexity or sophistication of reasoning. Levels 4 and above are mostly STEM intending content.