Download Pre-Calculus Learning Objectives

[Mathematics: Pre-Calculus]
Grade Level: High School
Proficiency Scale – Semester 1.1
Standard:
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, cosine,
and tangent for x, π + x, and 2π – x in terms of their values for x, where x is any real number.
G.SRT.7: Explain and use the relationship between the sine and cosine of complementary angles.
G.SRT.9: (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
G.SRT.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
G.SRT.11: (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant
forces).
Essential Learning Objective: Students will solve any triangle through a variety of means to find missing sides, angles ,and area (law of sines, identities, law of cosines).
A proficiency scale includes statements of what students need to know and be able to do for a standard that sets out a logical progression of learning over time. Student
performance is represented by a proficiency level as determined by learning outcomes.
Correlating student performance to a proficiency scale:
• Level 4 – An example of application that is in-depth and goes beyond instruction of the standard
• Level 3 – Learning target/standard as stated in the common core
• Level 2 – Prerequisite skills and knowledge required to meet the learning target/standard
• Level 1 – Partial understanding of the simpler ideas and processes (Ex: English Language Learners and students with IEPs or 504s)
• Level 0 – Alternative curriculum required
Level 4
In addition to Level 3, in-depth inferences and applications that goes beyond instruction to the
Example Activities
standard. The student will:
• Students will analyze a real world application and come up with conclusions not discussed
in class.
• Example activity: Students will research a real-world application of the law of sines (or law
of cosines) and create a word problem based on the application. Students will solve their
problem afterwards.
• Example activity: Students will write a proof of the law of cosines.
3.5
Level 3
In addition to Level 3 performance, in-depth inferences and applications with partial success.
Students will solve any triangle through a variety of means to find missing sides, angles, and area
(law of sines, identities, law of cosines).
The student exhibits no major errors or omissions.
2.5
No major errors or omissions regarding 2.0 content and partial knowledge of the 3.0 content.
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[Mathematics: Pre-Calculus]
Level 2
Grade Level: High School
There are no major errors or omissions regarding the simpler details and processes as the student
will be able to:
•
Recognize and define terms, such as: sine, cosine, tangent, cosecant, secant, cotangent,
law of sines, law of cosines, area, degree measure, right triangle, oblique triangle, radian,
degree, measure of an angle, standard position, linear speed, angular speed, hypotenuse,
opposite side, adjacent side, 30-60-90 triangle, 45-45-90 triangle, solve a right triangle,
angle of elevation, angle of depression, reference angle, arcsine, arccosine, arctangent,
inverse trigonometric functions, bearings, degree, minute, second
•
Perform processes, such as:
- identify the quadrant where the terminal side of an angle lies given radian or degree angle
measure
- Use 45-45-90 special right triangles to find the missing hypotenuse or leg
- Use 30-60-90 special right triangles to find the missing hypotenuse or leg
- convert radians to degrees
- convert degrees to radians
- graph a given angle, in radians or degrees, in standard position
-find the circumference of a circle
- identify if a problem uses linear or angular speed.
- convert degrees or radians to degrees, minutes, seconds
- draw an appropriate diagram to represent a word problem with appropriate labels
- correctly solve a proportion
- use inverse functions to solve an equation to find the missing angle
- know the repiprocal identies of trig functions
- use pythagorean theorem to solve for a missing side of a right triangle
- identify whether an angle is that of elevation or depression
- determine whether a given triangle uses right triangle trigonometry, law of sines, or law
of cosines.
-determine how many solutions a triangle will produce
However, the student exhibits major errors or omissions regarding the more complex ideas and
processes.
1.5
Partial knowledge of the 2.0 content but major errors or omissions regarding the 3.0 content.
Level 1
With help, a partial understanding of some of the simpler details and processes and some of the more complex ideas and
processes.
Level 0
Even with help, no understanding or skill demonstrated.
.5
With help, a partial understanding of the 2.0 content but not the 3.0 content.
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[Mathematics: Pre-Calculus]
Grade Level: High School
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[Mathematics: Pre-Calculus]
Grade Level: High School
Proficiency Scale – Semester 1.2
Standard:
F.TF.4: (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
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.
Essential Learning Objective: Students will analyze key features of trigonometric functions graphically, numerically, analytically, and in words.
A proficiency scale includes statements of what students need to know and be able to do for a standard that sets out a logical progression of learning over time. Student
performance is represented by a proficiency level as determined by learning outcomes.
Correlating student performance to a proficiency scale:
• Level 4 – An example of application that is in-depth and goes beyond instruction of the standard
• Level 3 – Learning target/standard as stated in the common core
• Level 2 – Prerequisite skills and knowledge required to meet the learning target/standard
• Level 1 – Partial understanding of the simpler ideas and processes (Ex: English Language Learners and students with IEPs or 504s)
• Level 0 – Alternative curriculum required
Level 4
In addition to Level 3, in-depth inferences and applications that goes beyond instruction to the
Example Activities
standard. The student will:
•
3.5
Level 3
Represent trigonometric functions symbolically, in real-life scenarios, graphically, with a
verbal description, as a sequence and with input/output pairs to solve mathematical and
contextual problems.
In addition to Level 3 performance, in-depth inferences and applications with partial success.
Students will analyze key features of trigonometric functions graphically, numerically,
analytically, and in the written word.
The student exhibits no major errors or omissions.
2.5
Level 2
No major errors or omissions regarding 2.0 content and partial knowledge of the 3.0 content.
There are no major errors or omissions regarding the simpler details and processes as the student
will:
•
Recognize and define terms, such as: Sine, Cosine, Tangent, Cosecant, Secant, Cotangent,
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[Mathematics: Pre-Calculus]
•
Grade Level: High School
Amplitude, Period, Intercept, Maximum, Minimum, Translation, Domain, Range,
Asymptote, Reciprocal Function, Inverse Trigonometric Function, Phase Shift, Vertical Shift,
Perform processes, such as:
- Graph a basic sine, cosine, and tangent function
- Identify original period, intercepts, maxima, minima, and amplitude of a sine, cosine, and
tangent function
- Identify reflections, amplitude, and vertical translation given an equation
-Given a graph, tell whether the function is sinusoidal
-Identify range and domain of sine, cosine, and tangent functions
-Identify vertical asymptotes of the original tangent function
However, the student exhibits major errors or omissions regarding the more complex ideas and
processes.
Level 1
1.5
Partial knowledge of the 2.0 content but major errors or omissions regarding the 3.0 content.
With help, a partial understanding of some of the simpler details and processes and some of the more
complex ideas and processes.
.5
With help, a partial understanding of the 2.0 content but not the 3.0 content.
Level 0
Even with help, no understanding or skill demonstrated.
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[Mathematics: Pre-Calculus]
Grade Level: High School
Proficiency Scale – Semester 1.3
Standard:
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.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.
2
2
F.TF.8: Prove the Pythagorean identity sin (θ) + cos (θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
G.C.5: Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant
of proportionality; derive the formula for the area of a sector.
N.CN.4: Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar
forms of a given complex number represent the same number.
N.CN.5: Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties
N.CN.6: Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its
endpoints.
N.VM.1: Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors
and their magnitudes (e.g., v,|v|, ||v||, v).
N.VM.2: Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
N.VM.3: Solve problems involving velocity and other quantities that can be represented by vectors.
N.VM.4A: Add and subtract vectors.--Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is
typically not the sum of the magnitudes.
N.VM.4B: Add and subtract vectors.--Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
N.VM.4C: Add and subtract vectors.--Understand vector subtraction v – w as v + (–w), where –w is the additive inverse of w, with the same magnitude as w and pointing in the
opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.
N.VM.5A: Multiply a vector by a scalar.--Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication
component-wise, e.g., as c(v x, v y)=(cv x, cv y).
N.VM.5B: Multiply a vector by a scalar.--Compute the magnitude of a scalar multiple cv using ||cv|| =|c|v. Compute the direction of cv knowing that when|c|v ≠ 0, the
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[Mathematics: Pre-Calculus]
Grade Level: High School
direction of cv is either along v (for c > 0) or against v (for c < 0).
Essential Learning Objective: Students will utilize the unit circle, radians, and trigonometric functions to model and evaluate real-world and mathematical problems.
A proficiency scale includes statements of what students need to know and be able to do for a standard that sets out a logical progression of learning over time. Student
performance is represented by a proficiency level as determined by learning outcomes.
Correlating student performance to a proficiency scale:
• Level 4 – An example of application that is in-depth and goes beyond instruction of the standard
• Level 3 – Learning target/standard as stated in the common core
• Level 2 – Prerequisite skills and knowledge required to meet the learning target/standard
• Level 1 – Partial understanding of the simpler ideas and processes (Ex: English Language Learners and students with IEPs or 504s)
• Level 0 – Alternative curriculum required
Example Activities
Level 4
In addition to Level 3, in-depth inferences and applications that goes beyond instruction to the
standard. The student will:
•
•
3.5
Level 3
Students will analyze a real world application and come up with conclusions not discussed
in class.
Example Activity: Students will consider and calculate the best angle at which a picture
should be viewed on a wall (or the angle of vision when sitting in a movie theater).
Students will consider the viewing distance and the size of what they are viewing.
In addition to Level 3 performance, in-depth inferences and applications with partial success.
The student will utilize the unit circle, radians, vectors, and trigonometric functions to model and
evaluate real-world and mathematical problems.
The student exhibits no major errors or omissions.
2.5
Level 2
No major errors or omissions regarding 2.0 content and partial knowledge of the 3.0 content.
There are no major errors or omissions regarding the simpler details and processes as the student
will:
•
Recognize and define terms, such as: sine, cosine, tangent, cosecant, secant, cotangent,
law of sines, law of cosines, area, degree measure, right triangle, oblique triangle, radian,
degree, measure of an angle, central angle, trigonometry, angle, initial side, terminal side,
vertex, standard position, positive angle, negative angle, coterminal, complementary,
supplementary, linear speed, angular speed, unit circle, periodic, period, even function,
odd function, hypotenuse, opposite side, adjacent side, theta, 30-60-90 triangle, 45-45-90
triangle, reciprocal identity, quotient identity, pythagorean identity, solve a right triangle,
angle of elevation, angle of depression, reference angle, standard position of a vector,
component form of a vector, magnitude, unit vector, zero vector, directed line, initial
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[Mathematics: Pre-Calculus]
Grade Level: High School
point, terminal point, length, vector in the plane, scalar multiplication, vecor addition,
parallelogram law, resultant, scalar multiple, negative vector, difference of vectors, unit
vector in the direction of v, standard unit vectors, horizontal component of v, horizontal
component of v, linear combination, direction angle, dot product of vectors, angle
between vectors, orthogonal vectors, vector components, projection of u onto v, work,
absolute value of a complex number, trigonometric form of a complex number, modulus,
argument, DeMoivre’s Theorem, nth root of a complex number, nth roots of unity
•
Perform processes, such as:
- identify the quadrant where the terminal side of an angle lies given radian or degree angle
measure
- Use 45-45-90 special right triangles to find the missing hypotenuse or leg
- Use 30-60-90 special right triangles to find the missing hypotenuse or leg
-identify a positive and negative coterminal angle given an angle in radians or degrees
-identify a supplementary angle given an angle in radians or degrees
- convert radians to degrees
- convert degrees to radians
- graph a given angle, in radians or degrees, in standard position
- identify the quadrantal angles
- identify whether an angle in degrees or radians is acute, right, or obtuse
-find the circumference of a circle
- identify if a problem uses linear or angular speed.
- convert degrees or radians to degrees, minutes, seconds
- draw the unit circle
- identify that cosine and secant are odd functions
- identify that sine, cosecant, tangent, and cotangent are even function
- identify whether or not a function is periodic given a graph
- identify co-functions
- find the magnitude of a vector given the endpoints
- find the direction of a vector given the endpoints
- find the component form of a vector given the magnitude and direction
- graph a vector given a magnitude and an angle
However, the student exhibits major errors or omissions regarding the more complex ideas and
processes.
1.5
Partial knowledge of the 2.0 content but major errors or omissions regarding the 3.0 content.
Level 1
With help, a partial understanding of some of the simpler details and processes and some of the more complex ideas and
processes.
Level 0
Even with help, no understanding or skill demonstrated.
.5
With help, a partial understanding of the 2.0 content but not the 3.0 content.
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[Mathematics: Pre-Calculus]
Grade Level: High School
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[Mathematics: Pre-Calculus]
Grade Level: High School
Proficiency Scale – Semester 1.4
Standard:
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.9: (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.
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, cosine, and tangent for x, π + x, and 2π – x in terms of their values for x, where x is any real number.
Essential Learning Objective: Students will verify, simplify, apply, and solve trigonometric functions or equations.
A proficiency scale includes statements of what students need to know and be able to do for a standard that sets out a logical progression of learning over time. Student
performance is represented by a proficiency level as determined by learning outcomes.
Correlating student performance to a proficiency scale:
• Level 4 – An example of application that is in-depth and goes beyond instruction of the standard
• Level 3 – Learning target/standard as stated in the common core
• Level 2 – Prerequisite skills and knowledge required to meet the learning target/standard
• Level 1 – Partial understanding of the simpler ideas and processes (Ex: English Language Learners and students with IEPs or 504s)
• Level 0 – Alternative curriculum required
Level 4
In addition to Level 3, in-depth inferences and applications that goes beyond instruction to the
Example Activities
standard. The student will:
• Students will analyze a real world application and come up with conclusions not discussed
in class.
• Example Activity: Students will write proofs of trigonometric identities that are not
completed in class.
3.5
Level 3
In addition to Level 3 performance, in-depth inferences and applications with partial success.
The student will: Students will verify, simplify, apply, and solve trigonometric functions or
equations.
The student exhibits no major errors or omissions.
2.5
Level 2
No major errors or omissions regarding 2.0 content and partial knowledge of the 3.0 content.
There are no major errors or omissions regarding the simpler details and processes as the student
will:
•
Recognize and define terms, such as: Sine, Cosine, Tangent, Cosecant, Secant, Cotangent,
Sum and Difference Formulas, Reduction formulas, Double Angle formulas, Pythagorean
Identities, Quotient Identites, Reciprocal Identities, Half Angle formulas, Power-Reducing
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[Mathematics: Pre-Calculus]
Grade Level: High School
formulas, Product-to-sum formulas, Sum-to-product formulas
•
Perform processes, such as:
- Know Reciprocal Identies
- Know Quotient Identities
- Know Pythagorean Identities
- Know Cofunction Identities
- Know Even/Odd Identities
- Identity which quadrant an angle lies
- Determine the sign of the ratio according to which quadrant the terminal side lies
- Factor a trig expression
- Find a common denominator between two rational expression
- Know when cancelling terms is appropriate in rational expresssions
-Determine the conjugate of an expression
-Square a binomial correctly
- Correctly add fractions
- Rationalize and un-rationalize ratios
- Re-write an expression using the sum and difference formulas
- Re-write an expression using the double or half angle formulas
- Re-write an expresion using the power reducint formulas
- Re-write an expression using the product to sum formulas or sum to product formulas
However, the student exhibits major errors or omissions regarding the more complex ideas and
processes.
1.5
Level 1
.5
Level 0
Partial knowledge of the 2.0 content but major errors or omissions regarding the 3.0 content.
With help, a partial understanding of some of the simpler details and processes and some of the more complex ideas and
processes.
With help, a partial understanding of the 2.0 content but not the 3.0 content.
Even with help, no understanding or skill demonstrated.
11
[Mathematics: Pre-Calculus]
Grade Level: High School
Proficiency Scale Semester 2.1
Standard:
A.SSE.1A - Interpret expressions that represent a quantity in terms of its context.--Interpret parts of an expression, such as terms, factors, and coefficients.
A.SSE.3A - Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.--Factor a quadratic
expression to reveal the zeros of the function it defines.
A.SSE.3B - Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.--Complete the square in a
quadratic expression to reveal the maximum or minimum value of the function it defines.
A.APR.1- Understand that polynomials form a system analogous to the integers; namely, they are closed under the operations of addition, subtraction, and multiplication; add,
subtract, and multiply polynomials.
A.APR.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).
A.APR.3 - Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.
A.APR.6 - Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x)
less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.
A.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.
A.CED.3 - Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a
modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
A.REI.2 - Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
F.BF.1A - Write a function that describes a relationship between two quantities.--Determine an explicit expression, a recursive process, or steps for calculation from a context.
A.REI.4a - Solve quadratic equations in one variable.--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.
F.IF.1 - Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If
f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
F.IF.4 - For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key
features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative
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[Mathematics: Pre-Calculus]
Grade Level: High School
maximums and minimums; symmetries; end behavior; and periodicity.
F.BF.3 - Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the
graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.
F.BF.4a - Find inverse functions.--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) =2 x3
or f(x) = (x+1)/(x–1) for x ≠1.
F.BF.4b+ - Find inverse functions.--Verify by composition that one function is the inverse of another.
F.BF.4c+ - Find inverse functions.--Read values of an inverse function from a graph or a table, given that the function has an inverse.
F.BF.4d+ - Find inverse functions.--Produce an invertible function from a non-invertible function by restricting the domain.
G.GPE.5 - Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a
given line that passes through a given point).
N.CN.1.1 – Know there is a complex number i such that i2 = –1, and every complex number has the form a + bi with a and b real.
N.CN.1.2 – Use the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract,and multiply complex numbers.
N.CN.1.3 – Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. Represent complex numbers and their operations on the
complex plane.
N.CN.3.3- Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
Essential Learning Objective: Students will analyze key features of families of functions (linear, quadratic, cubic, square roots, absolute value, and rational) graphically,
numerically, analytically, and in words.
A proficiency scale includes statements of what students need to know and be able to do for a standard that sets out a logical progression of learning over time. Student
performance is represented by a proficiency level as determined by learning outcomes.
Correlating student performance to a proficiency scale:
• Level 4 – An example of application that is in-depth and goes beyond instruction of the standard
• Level 3 – Learning target/standard as stated in the common core
• Level 2 – Prerequisite skills and knowledge required to meet the learning target/standard
• Level 1 – Partial understanding of the simpler ideas and processes (Ex: English Language Learners and students with IEPs or 504s)
• Level 0 – Alternative curriculum required
Level 4
In addition to Level 3, in-depth inferences and applications that goes beyond instruction to the
Example Activities
standard. The student will:
•
Given multiple functions, in different forms, write multiple equivalent versions of the
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[Mathematics: Pre-Calculus]
Grade Level: High School
function and identify and compare key features. Determine how the change of a parameter
in each function impacts their other representations. (Use of dynamic graphers)
•
3.5
Level 3
Example Activity: Students will change the parameters of a function to simulate regression
analysis using appropriate technology.
In addition to Level 3 performance, in-depth inferences and applications with partial success.
The student will analyze key features of families of functions (linear, quadratic, cubic, square
roots, absolute value, and rational) graphically, numerically, analytically, and in the written word.
Students will analyze key features of families of functions (linear, quadratic, cubic, square
roots, absolute value, and rational) graphically, numerically, algebraically, and in written
format.
The student exhibits no major errors or omissions.
2.5
Level 2
No major errors or omissions regarding 2.0 content and partial knowledge of the 3.0 content.
There are no major errors or omissions regarding the simpler details and processes as the student
will:
• Recognize and define terms, such as: slope, slope-intercept form, point-slope form,
parabola, asymptote, translation, transformation, composition function, inverse
function, one-to-one function, polynomial function, quadratic function, minimum
value, maximum value, zeros/roots of a function, multiplicity, rational zero test,
remainder theorem, factor theorem, imaginary number, complex number,
fundamental theorem of algebra, vertical asymptote, horizontal asymptote, and slant
asymptote
•
Perform processes, such as: find the slope of a line, write the equation of a line using
both slope-intercept form and point-slope form, graph lines, graph parabolas, sketch
the graph of a polynomial function, identify translations and transformations that have
occurred from a graph and from an equation, add and subtract functions, create a
composition function, find the minimum/maximum value of a parabola, find and graph
the inverse of a function, perform the leading coefficient test on a polynomial
function, find the zeros/roots of a polynomial function, perform long division and
synthetic division of polynomials, use the rational zero test, add/subtract/multiply
complex numbers, plot complex numbers, identify asymptotes of rational functions,
complete application problems involving the terms and processes mentioned above
However, the student exhibits major errors or omissions regarding the more complex ideas and
processes.
1.5
Level 1
Partial knowledge of the 2.0 content but major errors or omissions regarding the 3.0 content.
With help, a partial understanding of some of the simpler details and processes and some of the more complex ideas and
processes.
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[Mathematics: Pre-Calculus]
.5
Level 0
Grade Level: High School
With help, a partial understanding of the 2.0 content but not the 3.0 content.
Even with help, no understanding or skill demonstrated.
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[Mathematics: Pre-Calculus]
Grade Level: High School
Proficiency Scale – Semester 2.2
Standard:
t
1/12 12t
A.SSE.3C: Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15 can be rewritten as (1.15
reveal the approximate equivalent monthly interest rate if the annual rate is 15%.
)
12t
≈ 1.012
to
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.
F.IF.8B: Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y =
(0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.
F.BF.1B: Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant
function to a decaying exponential, and relate these functions to the model.
F.BF.5: (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
ct
F.LE.4: For exponential models, express as a logarithm the solution to ab = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using
technology.
N.RN.1: Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for
1/3
1/3 3
(1/3)3
1/3 3
radicals in terms of rational exponents. For example, we define 5 to be the cube root of 5 because we want (5 ) = 5
to hold, so (5 ) must equal 5.
N.RN.2: Rewrite expressions involving radicals and rational exponents using the properties of exponents.
Essential Learning Objective: Students will analyze exponential and logarithmic equations graphically, numerically, analytically, and in words.
A proficiency scale includes statements of what students need to know and be able to do for a standard that sets out a logical progression of learning over time. Student
performance is represented by a proficiency level as determined by learning outcomes.
Correlating student performance to a proficiency scale:
• Level 4 – An example of application that is in-depth and goes beyond instruction of the standard
• Level 3 – Learning target/standard as stated in the common core
• Level 2 – Prerequisite skills and knowledge required to meet the learning target/standard
• Level 1 – Partial understanding of the simpler ideas and processes (Ex: English Language Learners and students with IEPs or 504s)
• Level 0 – Alternative curriculum required
Level 4
In addition to Level 3, in-depth inferences and applications that goes beyond instruction to the
Example Activities
standard. The student will:
•
Students will analyze a real world application and come up with conclusions not discussed
16
[Mathematics: Pre-Calculus]
•
•
3.5
Level 3
Grade Level: High School
in class.
Example Activity: Students will analyze real-world problems using continuous compound
interest.
Example Activity: Students will analyze logarithmic applications of social interactions, such
as society growth, industrialization, and purchasing trends.
In addition to Level 3 performance, in-depth inferences and applications with partial success.
The student will: Students will analyze exponential and logarithmic equations graphically,
numerically, analytically, and in words.
Students will analyze exponential and logarithmic equations graphically, numerically,
analytically, and in the written word.
The student exhibits no major errors or omissions.
2.5
Level 2
No major errors or omissions regarding 2.0 content and partial knowledge of the 3.0 content.
There are no major errors or omissions regarding the simpler details and processes as the student
will:
•
Recognize and define terms, such as: exponent, exponential function, logarithm,
logarithmic function, natural logarithm, natural logarithmic function, transformations
of functions, translations of functions, base, e, compound interest, continuously
compound interest,
•
Perform processes, such as:
However, the student exhibits major errors or omissions regarding the more complex ideas and
processes.
1.5
Partial knowledge of the 2.0 content but major errors or omissions regarding the 3.0 content.
Level 1
With help, a partial understanding of some of the simpler details and processes and some of the more complex ideas and
processes.
Level 0
Even with help, no understanding or skill demonstrated.
.5
With help, a partial understanding of the 2.0 content but not the 3.0 content.
17
[Mathematics: Pre-Calculus]
Grade Level: High School
Proficiency Scale – Semester 2.3
Standard:
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.
F.IF.3: Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined
recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.
F.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.
Essential Learning Objective: Students will solve problems by selection and evaluation of the correct sequence or series.
A proficiency scale includes statements of what students need to know and be able to do for a standard that sets out a logical progression of learning over time. Student
performance is represented by a proficiency level as determined by learning outcomes.
Correlating student performance to a proficiency scale:
• Level 4 – An example of application that is in-depth and goes beyond instruction of the standard
• Level 3 – Learning target/standard as stated in the common core
• Level 2 – Prerequisite skills and knowledge required to meet the learning target/standard
• Level 1 – Partial understanding of the simpler ideas and processes (Ex: English Language Learners and students with IEPs or 504s)
• Level 0 – Alternative curriculum required
Level 4
In addition to Level 3, in-depth inferences and applications that goes beyond instruction to the
Example Activities
standard. The student will:
•
•
•
3.5
Level 3
Students will analyze a real world application and come up with conclusions not discussed
in class.
Example Activity: Students will prove the formula for the summation of a geometric series.
Example Activity: Students will analyze real-world problems using the Taylor series.
In addition to Level 3 performance, in-depth inferences and applications with partial success.
The student will:
Students will solve problems by selection and evaluation of the correct sequence or series.
The student exhibits no major errors or omissions.
2.5
Level 2
No major errors or omissions regarding 2.0 content and partial knowledge of the 3.0 content.
There are no major errors or omissions regarding the simpler details and processes as the student
will:
18
[Mathematics: Pre-Calculus]
Grade Level: High School
However, the student exhibits major errors or omissions regarding the more complex ideas and
processes.
1.5
Partial knowledge of the 2.0 content but major errors or omissions regarding the 3.0 content.
Level 1
With help, a partial understanding of some of the simpler details and processes and some of the more complex ideas and
processes.
Level 0
Even with help, no understanding or skill demonstrated.
.5
With help, a partial understanding of the 2.0 content but not the 3.0 content.
19