Northern California Math Department SLOs
Program Level
SLOs PreAlgebra
Beginning Algebra
Intermediate Algebra
Geometry
Statistics
Liberal Arts Math
PreCalculus/College Algebra
Trigonometry
Calculus I
Calculus II Calculus III
Linear Algebra
Differential Equations
Others
Program Level SLOs
College of San
Mateo
Developmental
1. Identify
and know when to apply basic arithmetic concepts.
2. Solve problems by application of arithmetic principles.
3. Represent problems
in written language, in symbolic form, and in graphical form.
4. Select and apply
appropriate formulas.
5. Organize work in a
logical, clearly stated order, correctly using mathematical symbols and
language.
6. Use calculators
effectively and appropriately.
7. State solutions to
application problems in the context of the problem and recognize inappropriate
and/or impossible answers.
8. Follow and
demonstrate understanding of mathematical exposition [text readings, handouts,
and lecture.]
9. Recognize the
usefulness of elementary mathematics
General
1. Follow mathematical
exposition, including descriptions of algorithms and derivations of formulas,
presented either orally or in writing.
2. Determine whether a
theorem or definition applies in a given situation, and use it appropriately if
it applies.
3. Use the language and
notation of differential and integral calculus correctly, and use appropriate
style and format in written work.
4. Demonstrate good
problemsolving habits, including:
a.
estimating
solutions and recognizing unreasonable results.
b.
considering a
variety of approaches to a given problem, and selecting one that is
appropriate.
c.
rejecting the
temptation to rely on mechanical techniques (either pencilandpaper or
electronic) that they do not understand.
d.
interpreting
solutions correctly, and answering the questions that were actually asked.
5. Use technology
(especially calculators) effectively and appropriately.
Las Positas
College
1. Students will demonstrate the
ability to use symbolic, graphical, numerical, and written representations of
mathematical ideas.
2. Students will read, write, listen
to, and speak mathematics with understanding.
3. Students will use mathematical reasoning to solve problems and a
generalized problem solving process to work word problems.
4. Students will learn mathematics through modeling realworld situations.
5. Students will use appropriate technology to enhance their mathematical
thinking and understanding, solve mathematical problems, and judge the
reasonableness of their results.
Lake Tahoe Community
College
1. Produce, interpret,
and analyze data and graphs
2. Solve mathematical equations
3. Construct, manipulate, and utilize mathematical functions
4. Engage in logical and critical thinking
5. Apply mathematical techniques to solve problems that arise in the real
world
Los Medanos
Community College
Developmental Math Program
Students completing the Developmental Math Program will
demonstrate:
1. Problemsolving abilities: Students will use
mathematical reasoning to solve problems and a generalized problem solving
process to work word problems.
a. The student can apply standard problemsolving methods and
use relevant concepts to solve problems.
b. The student uses a generalized problemsolving rubric if
such a rubric is used in the class.
c. The student’s written work demonstrates a conceptual
understanding of course concepts.
d. The student’s written work supports his/her solution.
e. The student evaluates the reasonableness of his/her
answer.
2. Mathematical versatility: Students will use verbal,
graphical, numerical, and symbolic representations of mathematical ideas to
solve problems.
a. Students will use a variety of representations to
demonstrate their understanding of mathematical concepts.
b. Students will use a multiprong approach to problem
solving.
c. Students will use appropriate technology to solve
mathematical problems and judge the reasonableness of their results.
3. Communication skills: Students will read, write,
listen to, and speak mathematics with understanding.
a. Students will read and listen to mathematical
presentations and arguments with understanding.
b. Students will communicate both in speaking and in writing
their understanding of mathematical ideas and procedures using appropriate
mathematical vocabulary and notation.
c. Students will coherently communicate their own
mathematical thinking to others.
4. Preparation: Students will recognize and apply math
concepts in a variety of relevant settings and demonstrate the math skills and
knowledge necessary to succeed in subsequent courses.
5. Effective Learning Attributes: Students will
demonstrate the characteristics of an effective learner.
a. Student has the will to succeed and demonstrates the
characteristics of a successful student: motivation, responsibility, focus,
perseverance, the ability to cope with anxiety, a good attitude toward learning,
and time management skills.
b. Student has the skills to succeed. (S)he uses appropriate
resources to improve learning and reach goals.
c. Student selfmonitors and selfregulates. (S)he assesses
personal strengths and weaknesses in his/her learning process and then seeks and
implements a strategy for improving learning.
Transfer Math Program
Students completing transferlevel math courses at LMC
will demonstrate:
1. Preparation and Mathematical Maturity: Be prepared
for the mathematical reasoning required in upper division work in their major,
including the ability to generalize mathematical concepts and comprehend
increasing levels of mathematical abstraction.
2. Mathematical Literacy: Communicate using mathematics:
a. Read with comprehension documents having mathematical
content and participate cogently in discussions involving mathematics;
b. Clearly articulate mathematical information accurately and
effectively, using a form, structure and style that suit the purpose (including
written and facetoface presentation).
3. Problemsolving ability:
a. Reason with and apply mathematical concepts, principles
and methods to solve problems or analyze scenarios in realworld contexts
relevant to their major;
b. Use technology effectively to analyze situations and solve
problems;
c. Estimate and check answers to mathematical problems in
order to determine reasonableness, identify alternatives, and select optimal
results.
4. Modeling ability:
a. Construct and interpret mathematical models using
numerical, graphical, symbolic and verbal representations with the help of
technology where appropriate in order to draw conclusions or make predictions;
b. Recognize and describe the limits of mathematical and
statistical methods.
5. Effective Learning skills:
a. Independently acquire further mathematical knowledge
without guidance, take responsibility for their own learning, and function
effectively in different learning environments.
b. Succeed in different learning environments, particularly
in a group setting of working collaboratively with others.
Ohlone College 1. Students should improve their
attitude towards math.
2. Students should have problem
solving skills at an appropriate level.
3. Students should retain information from
course to course.
4. Students should be completing the Math certificates and degree.
Solano Community
College
Three Categories:1. Students completing advanced math core courses.
2. Students meeting the minimum requirements to transfer.
3. Students meeting the minimum requirements to graduate with Associates
Degree.
Student Learning Outcomes
Be able to:
 Solve a problem applying appropriate math concepts and ideas.
 Effectively communicate solution(s).
PreAlgebra
College of San
Mateo
1. Strengthen core entry skills, which are to perform
a. Operations with whole numbers
b. Operations with fractions
c. Operations with decimals
d. Operations with percentages
2. Perform operations on integers.
3. Simplify and evaluate variable expressions.
4. Solve a one variable first degree linear equation that models
situation.
5. Construct linear graphs.
6. Convert units of measure (includes American and Metric systems).
7. Perform operations with polynomial
Lake Tahoe Community
College
Part I
1. Perform arithmetic operations with whole numbers, fractions, and decimals.
2. Translate written language into mathematical statements.
3. Apply the concepts in the course to reallife situations.
Part II
1. Solve problems involving decimals, percents and beginning algebra
2. Translate written statements in to mathematical statements.
3. Apply the topics of Basic Arithmetic (Part II) to real life situations.
Ohlone College
The
student will:1. Solve numerous problems in order to gain mastery of the
arithmetic skills needed for everyday situations.
2. Demonstrate a systematic and logical approach to solving arithmetic problems.
3. Demonstrate the knowledge and skills required to select the correct
introductory formulas, procedures, and concepts from algebra and geometry and
use them to calculate and problem solve.
4. Solve word problems that require concept of central tendency and
interpretation of statistical graphs.
Skyline College
By completing MATH 811 course students will be able to:
 Correctly choose and apply the four basic arithmetic
operations with whole numbers, decimals, fractions and sign numbers to
estimate and solve application problems that are part of their daily lives.
(Number Sense)
 Apply “Proportional Reasoning” to solve related
problems including ratios, rates, proportion, percent and conversions of
units.
 Compute the area and perimeter of geometric figures.
 Learn to use the services available to improve study
skills, test taking skills, problem solving skills and attitude toward
learning mathematics
Solano Community
College
1. Perform
arithmetic
operations on signed values including integers, fractions, and decimals.
2. Simplify algebraic expressions, evaluate formulas, and solve basic
linear equations and application problems.
3. Students obtain sufficient math proficiency to be successful in a
subsequent elementary algebra course.
Beginning
Algebra
College of San
Mateo
1. Identify and apply basic algebraic concepts including
slope, absolute value, scientific notation, equivalent equations, laws of
exponents, intercepts, horizontal lines, and vertical lines.
2. Solve systems of linear equations in two unknowns using
graphing, elimination, and substitution.
3. Solve equations and inequalities in one variable.
4. Solve quadratic equations by factoring and by using the
quadratic formula.
5. Solve elementary radical equations.
6. Graph linear equations.
7. Solve problems by application of linear functions.
8. Apply the properties of and perform operations with
radicals.
9. Apply the properties of and perform operations with integer exponents.
Lake Tahoe Community
College
First Quarter
1. Solve linear equations and inequalities.
2. Define and employ terminology and arithmetic relating to polynomials in one
variable.
3. Determine the equation and graph a line given information about the line.
4. Manipulate expressions with integral exponents.
5. Apply course topics to realworld situations.
Second Quarter
1. Factor a polynomial.
2. Apply the four basic operations to rational and radical expressions.
3. Solve equations with rational and radical expressions.
4. Solve a 2 x 2 system of linear equations.
5. Solve quadratic equations.
6. Apply course topics to real world situations.
Ohlone College
The student will:
1. Demonstrate basic skills in algebra up through quadratics and problem
solving.
2. Set up stated problems algebraically and solve the resulting equations.
3. Solve problems presented via formulas or procedures.
4. Graph linear equations.
5. Solve systems of linear equations using graphing, substitution, and
elimination methods.
6. Simplify exponential expressions with integer exponents.
7. Identify polynomials and perform operations with polynomials.
8. Factor polynomials using grouping, FOIL, special products formulas, and trial
and error methods.
9. Solve quadratic equations using factoring and their applications.
10. Simplify rational expressions and complex fractions and solve applications
of rational equations.
Skyline College
Upon completion of the course:
• Through real world applications, students will be able to create, manipulate,
and interpret mathematical models of relationships defined by either a constant
rate of change or a constant relative rate of change.
• Students will recognize, apply, and interpret multiple representations
(graphic, symbolic, numerical/data, verbal/applied) of linear functions and
their applications.
• Students will develop skills and attitudes for effectively solving problems at
an introductory algebra level.
Solano Community
College
1. Distinguish between and give examples of
equations, solutions to equations, and algebraic expressions.
2. Solve
mathematical equations appropriate to the elementary algebra curriculum.
3. Formulate realworld problems quantitatively and interpret the results.
Intermediate Algebra
College of San
Mateo
1. Identify and apply basic algebraic concepts including
domain, range, slope, absolute value, scientific notation, equivalent equations,
laws of exponents, intercepts, parallel lines, perpendicular lines, horizontal
lines, and vertical lines;
2. Solve systems of linear equations in three unknowns using elimination and
substitution
3. Solve equations and inequalities in one or two variables and involving
absolute values
4. Solve quadratic equations by factoring, completing the square, and quadratic
formula;
5. Solve exponential and logarithmic equations
6. Solve equations involving radicals
7. Perform basic operations on complex numbers
8. Find complex roots of a quadratic equation;
9. Sketch the graphs of functions and relations:
a. algebraic, including polynomial
and rational
b. logarithmic
c. exponential
d. circles;
10. Find and sketch inverse functions;
11. Problem solve by application of linear and quadratic functions
12. Apply the concepts of logarithmic and exponential functions;
13. Apply the properties of and perform operations with radicals;
14. Apply the properties of and perform operations with rational exponents;
15. Apply linear and quadratic functions
16. Graph linear inequalities in two variables;
17. Find the distance between two points;
18. Find the midpoint of a line segment.
Lake Tahoe Community
College
1. Apply the course topics to realworld situations.
2. Sketch and interpret the graphs of functions and relations introduced in
intermediate algebra.
3. Simplify mathematical expressions into forms more amenable to analysis.
4. Provide solutions to equations using methods from intermediate algebra.
Ohlone College
The student will:
1. Solve problems involving the mathematical concepts of function and functional
inverse.
2. Show increased skill in setting up and solving applications.
3. Solve mathematical problems using concepts that may be useful for learning
statistics: logarithms, sigma notation, and the binomial theorem.
4. Solve mathematical problems in topics useful for trigonometry: functions and
inverses and their graphs, quadratic equations, and conic sections.
Skyline College
Upon completion of the course:
• Through real world applications, students will be able to create, manipulate,
and interpret mathematical models of relationships involving linear,
exponential, polynomial, radical, and rational functions.
• Students will recognize, apply, and interpret multiple representations
(graphic, symbolic, numerical/data, verbal/applied) of functions and their
applications.
• Students will develop skills and attitudes for effectively solving problems at
an intermediate algebra level.
Solano Community
College
1. Evaluate
functions and
utilize functions and their graphs appropriate to the intermediate algebra
curriculum to solve and interpret the solutions to realworld problems.
2. Recognize exponential relationships and apply logarithmic principles to
solve and interpret the solutions to realworld problems.
Geometry
College of San
Mateo
1. Demonstrate familiarity with geometric vocabulary
a. Classify angles as acute, obtuse,
straight, or right angles.
b. Identify and name specific
polygons
c. Identify and name simple conic
sections
d. Identify alternate interior angles
and alternate exterior angles
e. Identify the four parts of an
axiomatic system.
f. Identify the hypothesis and
conclusion in and the converse and contrapositive of a conditional statement.
g. State and apply the triangular
inequality
2. Apply the parallel postulate.
3. Identify congruent figures and apply theorems relative to congruent figures.
4. Perform basic geometric constructions using a compass and straight edge.
5. Prove geometric theorems
a. Use reflexive, symmetric, and
transitive properties
b. Prove theorems with respect to
lines are parallel.
c. Prove two triangles are congruent
or not congruent
d. Construct indirect proofs
6. Perform calculations requiring geometric knowledge and formula
a. Apply formulas and theorem to find
area and perimeter.
b. Find volume and surface area of
prisms and pyramids.
c. Apply the Pythagorean Theorem.
d. Use ratio and proportion to solve
similar figures.
e. Solve for perimeter, area, and
volume of similar figures
7. Apply theorems about the geometry of a circle.
8. Graph figures.
9. Recognize transformations.
Lake Tahoe Community
College
1. Prove geometric statements using classical axioms and theorems.
2. Perform ruler and compass constructions.
3. Make deductions using the rules of logic.
4. Solve problems involving parallel lines, triangles, and angles.
Ohlone College
The student will:
1. Master the terms, postulates, and theorems of geometry.
2. Construct both direct and indirect proofs in geometric problems.
3. Const ruct and rmal logical arguments and give counterexamples to disprove
statements.
4. Prove basic theorems involving congruence and similarity.
5. Solve practical problems such as finding lengths, areas, volumes and angles
of geometric figures.
Solano Community
College
1. Apply
deductive
reasoning to construct formal proofs of geometric theorems.
2. Apply relevant theorems to solve problems related to geometric figures.
Statistics
College of San
Mateo
1. See that statistical analyses are useful for "real
world" questions and problems.
2. Understand the kinds of problems for which statistical analysis can be of
help.
3. Become knowledgeable and critical users of descriptive and inferential
statistical analyses; specifically:
a. Assess whether a descriptive analysis is appropriate for the data and problem
posed.
b. Explain what a descriptive analysis says about the data it is describing.
c. Demonstrate understanding of the concepts of variable, variability, the
distribution of a variable, the center, spread and shape of a distribution of a
variable, and how these features of a distribution are appropriately measured
and modeled.
d. Describe in words the relationship between two variables from graphical or
numerical analyses in the context of the data, whether the variables be
quantitative or categorical.
e. Identify appropriate methods and models of analyzing the relationship between
variables whether the variables be quantitative or categorical.
f. Construct graphical presentations displaying data, measure features of
distributions, and interpret the results in writing for data that the student
helped to collect.
g. Explain, with a mixture of prose and mathematics, why a formula does what it
is said to do.
4. Demonstrate awareness that the methods used to produce data can be as
important as what is done with the data.
5. Apply the language and the logic of probability, not only in “probability
problems” but also in the context of inference.
6. Show familiarity with the idea of a theoretical (or mathematically) defined
probability distribution, as illustrated by uniform distribution, normal
distributions or binomial distributions.
Lake Tahoe Community
College
1. Design and implement an unbiased study that will produce sound statistical
results.
2. Generate and interpret statistics graphs from data that arise from surveys
and experiments.
3. Implement the rules of probability.
4. Apply confidence intervals and test hypotheses to make conclusions about data
that come from practical applications.
5. Perform regression analysis to make informed predictions about relationships
between quantitative variables.
Ohlone College
The student will:
1. Understand basic statistical concepts and vocabulary.
2. Understand basic probability concepts and vocabulary.
3. Use statistical formulas.
4. Choose correct statistical tool for analysis of word problems.
5. Use technology for statistical applications.
Skyline College
Upon completion of this course the student will be able to:
• Organize, analyze, and utilize appropriate methods to draw conclusions based
on sample data by using tables, graphs, measures of central tendency, and
measures of dispersion.
• Have sufficient command of the concepts and terminology of probability and
statistics.
• Collect data, interpret and communicate the results using statistical analyses
such as confidence intervals, hypothesis tests, and regression analysis.
• Successfully apply statistics to ones selected major at a transfer institution
Solano Community
College
1. Critically analyze statistical information
presented in media, journals, etc."
2. Convert
data to statistical evidence and interpret the evidence.
Liberal Arts Mathematics
College of San
Mateo
1. apply principles from algebra and elementary number
theory to current technology, i.e. encryption techniques.
2. identify similarities and differences between Euclidean and nonEuclidean
Geometries
3. explain certain relationships between geometry and topology
4. demonstrate familiarity with the Platonic Solids
5. apply problem solving techniques learned in one area to another area
6. measure uncertainty using basic probability techniques
7. count using permutations and combinations
8. identify contributions of selected mathematicians
Lake Tahoe Community
College
1. Apply combinatorics and the rules of probability to real life situations.
2. Analyze statistical information and the 'Normal' distribution to make
conclusions based on data.
3. Incorporate the mathematics of finance to be consumerwise.
4. Utilize trigonometric formulas to solve problems involving triangles.
5. Develop exponential growth and decay models.
Ohlone College
The student will:
1. Apply mathematical principles and techniques to solve problems in areas such
as systems of numeration, algebraic modeling, basic trigonometry, intuitive
calculus, and math of finance.
2. Use critical thinking to arrive at conclusions from Venn Diagrams,
syllogistic forms, and truth tables.
3. Demonstrate a knowledge of probability and statistics by solving a variety of
counting problems, by calculating the probability of games of chance, and by
analyzing statistical data.
4. Relate a knowledge of the people, history and uses of mathematics through
research papers, projects, presentations, and class discussions.
Solano Community
College
1. Apply
problem solving skills to solve unfamiliar problems related to the topics
studied.
PreCalculus / College Algebra
College of San
Mateo
1. Solve and apply equations and inequalities including
linear, quadratic, absolute value, polynomial, rational, radical, exponential,
logarithmic, and trigonometric equations.
2. Graph linear, quadratic, absolute value, polynomial, rational, radical,
exponential, logarithmic and trigonometric functions, and parametric equations.
3. Perform function operations including composition, transposition, and finding
inverse functions.
4. Apply techniques for finding zeros of polynomial functions.
5. Solve systems of equations by application of algebraic techniques and/or
matrix techniques.
6. Apply formulas from analytic geometry.
7. Define, recognize, and solve for terms of arithmetic and geometric series.
Lake Tahoe Community
College
First Quarter
1. Produce and interpret graphs of functions and relations.
2. Apply techniques to solve polynomial and rational equations and inequalities.
3. Model real life situations using algebraic methods.
4. Simplify algebraic expressions using skills obtained in the course.
Second Quarter
1. Prove and derive mathematical statements using various methods including
induction.
2. Employ matrices and their properties to solve systems of equations.
3. Construct and interpret graphs of conic sections and transcendental
functions.
4. Apply the topics of the course to real world situations.
Ohlone College
The student will:
1. Solve equations involving algebraic and transcendental functions, and systems
of equations.2. Solve realworld applications of the above.
3. Graph algebraic and transcendental functions.
4. Apply a graphing calculator to all of the above.
5. Solve introductory sequence and series problems.
Skyline College
• Apply Algebraic concepts to equation and problem solving.
• Apply modeling techniques to solve real world problems.
• Apply graphing methods to be able to synthesize graphical concepts to check
algebraic solutions, as well as, to find solutions where algebraic ones are not
possible.
• Apply their functional awareness to be successful in Calculus
Solano Community
College 1. Demonstrate the ability to use functions as a mathematical tool to model the
conceptual ideas of algebra and trigonometry.
2. Construct, derive, and graph basic
trigonometric relationships.
Trigonometry
College of San
Mateo
1.
State and apply
correctly the definitions (unit circle, right triangle, and xyr), values for
key angles, properties (e.g. periodicity and domain and range), and basic
identities, for the six trig functions.
2.
Work with and
apply the algebraic relationships among the six trig functions: use algebra and
identities to derive other identities, verify identities, simplify expressions,
and solve trigonometric equations.
3.
Solve right
triangles using right triangle definitions of trig functions, and oblique
triangles using the laws of sines and cosines
4.
Solve applied trigonometry problems involving triangles or periodic
behaviors.
5.
Produce and
interpret graphs of sine and cosine functions, with correct amplitude, period,
phase shift, and vertical shift.
6.
Demonstrate
understanding of inverse trig functions and their applications.
7.
Model periodic
phenomena using sine and cosine functions.
8.
State solutions
to application problems in context and recognize inappropriate or impossible
answers.
9.
Follow and
demonstrate understanding of mathematical exposition [e.g. text readings,
handouts, and online resources].
10.
Organize work in
a logical, clearly stated order, correctly using mathematical symbols and
language.
Lake Tahoe Community
College
1. Provide and analyze graphs of trigonometric functions.
2. Apply trigonometric techniques to solve problems in real world contexts.
3. Derive and prove trigonometric properties and identities.
4. Produce solutions to equations using skills developed in trigonometry.
Ohlone College
The student will:
1. Identify six trigonometric functions and express them as the ratio of the
sides of a right triangle.
2. Solve right triangle problems.
3. Convert angles from degrees to radians and from radians to degrees.
5. Define the domain and range of trigonometric functions.
6. Graph trigonometric functions using amplitude, period, phase shift and/or
vertical translation.
7. Define domain and range of inverse trig functions.
8. Verify trigonometric identities using fundamental identities, sum and
difference formulas, double angle formulas and half angle formulas.
9. Solve trigonometric equations on either a restricted domain or a general
domain.
10. Applications.
11. Use Law of Sines and Law of Cosines to solve problems using oblique
triangles.
Skyline College
Upon completion of the course:
• Through real world applications, students will be able to create, manipulate,
and interpret mathematical models of periodic relationships.
• Students will recognize, apply, and interpret multiple representations
(graphic, symbolic, numerical/data, verbal/applied) of periodic functions and
their applications.
• Students will recognize, apply, and interpret proportional reasoning in the
context of right triangles and circles by analyzing a variety of problems, then
applying the trigonometric definitions, choosing appropriate ideas, and applying
these ideas to the solution of the problems.
• Students will develop skills and attitudes for effectively solving problems at
a transfer math course level.
Solano Community
College
1. Solve application
problems involving trigonometric functions and graphs.
Calculus I
College of San
Mateo
1. Calculate limits when they exist, and explain why when
they do not.
2. Determine where a function is continuous and/or differentiable, and explain
why.
3. Compute derivatives of polynomial, rational, algebraic, exponential,
logarithmic, and trigonometric functions.
4. Use techniques of differentiation, including the product, quotient, and chain
rules, and implicit differentiation.
5. Apply differentiation to the study of functions and their graphs, to
optimization and related rate problems, and to applications from science and
economics.
6. Compute antiderivatives of polynomial, rational, algebraic, exponential,
logarithmic, and trigonometric functions.
7. Interpret Riemann sums as definite integrals, relate definite integrals to
areas, and evaluate definite integrals using the fundamental theorem of
calculus.
Lake Tahoe Community
College
First Quarter
1. Differentiate functions of a single variable using the basic rules of
differentiation.
2. Apply the derivative to describe phenomena arising from reallife situations.
3. Sketch and analyze graphs using the first and second derivatives.
4. Prove corollaries and derive equations using the theorems that relate to
differential calculus.
5. Determine limits and continuity using graphical, analytical, and tabular
techniques.
Second Quarter
1. Employ integrals to applications from physics.
2. Apply the Fundamental Theorem of Calculus in determining indefinite
integrals.
3. Compute geometric quantities using integrals.
4. Solve differential equations.
5. Determine integrals and derivatives of transcendental functions.
Ohlone College
The student will:
1. Compute limits using numerical, graphical, and algebraic methods.
2. Differentiate algebraic, trigonometric, logarithmic, exponential, and inverse
trig functions.
3. Apply differentiation to problems in the areas of geometry, physics,
engineering, and business, including slopes of tangent lines and rates of
change.
4. Integrate algebraic, trigonometric, and exponential functions using
introductory techniques.
5. Apply integration to finding the area under a curve.
6. Demonstrate logical thinking, correct use of notation, and mathematical
precision in formulating and solving problems in the above areas.
7. Apply the appropriate use of a graphing calculator to each of the above
areas.
Skyline College
• Students will be able to explain and apply the techniques
of differential calculus to construct derivatives graphically, numerically and
analytically.
• Students will be able to translate problems from the physical, life and social
sciences into workable mathematical form, suitable for graphical, numerical,
analytical or verbal solutions.
• Students will be able to use technology where appropriate, in particular, the
use of a graphing calculator to
a. produce the graph of any function in an arbitrary viewing window
b. estimate the value of a numerical derivative of a function at a point and
c. estimate the value of a definite integral to a given degree of accuracy
Solano Community
College
1. Define
and apply the
concepts of limits, continuity, derivatives and antiderivatives to solve a
variety of word problems (both familiar and unfamiliar) and corroborate their
solutions with practical reasoning.
2. Demonstrate understanding of the geometric relationship between a
function, its first and second derivatives and its antiderivatives.
3. Interpret and analyze information to develop strategies for solving
problems involving related rates, optimization, and approximation by linear
models.
Calculus II
College of San
Mateo
1. Evaluate definite integrals using the fundamental
theorem of calculus.
2. Analyze geometric and physical situations to obtain Riemann sums, and
interpret and evaluate them as definite integrals.
3. Use numerical methods to estimate the value of definite integrals.
4. Use techniques of integration, including algebraic and trig substitutions,
integration by parts, and partial fractions, to evaluate definite and indefinite
integrals.
5. Find limits of sequences, or show that the limit does not exist
6. Determine whether series diverge, converge conditionally, or converge
absolutely, and find or estimate sums of series.
7. Find intervals of convergence of power series.
8. Find Taylor and Maclaurin series of functions.
9. Interpret and solve certain types of differential equations, including
separable and first order linear.
Lake Tahoe Community
College
1. Test series for convergence.
2. Relate analytic functions to their power series.
3. Apply calculus to functions of several variables.
4. Model real life applications using threedimensional constructs.
5. Perform arithmetic on vectors using both component and geometric forms.
Ohlone College
The student will:
1. Apply the use of integrals to problems involving volumes of solids, arc
length, surface area, and physics applications.
3. Solve basic differential equations.
4. Determine the convergence or divergence of infinite sequences and series by
using appropriate tests.
5. Determine polynomial representations of mathematical functions by using power
series.
6. Analyze mathematical relationships given in parametric and polar forms.
7. Graph conic sections and determine information about the conic from its
algebraic equation.
8. Demonstrate logical thinking, correct use of notation, and mathematical
precision in formulating and solving problems in the above areas.
9. Apply the use of a graphing calculator to each of the above areas.
Skyline College
• Students will be able to explain and apply the techniques
of integral calculus to construct antiderivatives graphically, numerically and
analytically.
• Students will be able to translate problems from the physical, life and social
sciences into workable mathematical form, suitable for graphical, numerical,
analytical or verbal solutions.
• Students will be able to use technology where appropriate, in particular, the
use of a graphing calculator to
o produce the graph of any function in an arbitrary viewing
window
o estimate the value of a numerical derivative of a function
at a point and
o estimate the value of a definite integral to a given degree
of accuracy
• Students will be able to solve differential equations with an emphasis on
qualitative solutions, modeling and interpretation.
• Students will be able to analyze the convergence or divergence of sequences
and series of constants with a variety of techniques.
Solano Community
College
1. Apply
numerical methods
to approximate definite integrals, improper integrals that converge, and
infinite series that converge and bound the errors.
2. Find antiderivatives using a variety of techniques of integration.
3. Interpret analyze information to develop strategies for finding area,
volume, and arc length.
Calculus III
College of San
Mateo
1. Compute limits of functions of several variables, or
show that the limits do not exist.
2. Determine where functions of several variables are continuous.
3. Compute partial derivatives and directional derivatives.
4. Solve extremum problems, using a) partial derivatives and b) Lagrange
multipliers.
5. Set up and evaluate multiple integrals, and use them in geometric and
physical applications.
6. Compute dot and cross products of vectors, and use them to find equations of
lines and planes in R3.
7. Find arc length, curvature, tangent and normal vectors to space curves.
8. Compute line and surface integrals.
9. Use cylindrical and spherical coordinates, and parametric equations, to study
functions of several variables and surfaces.
10. Compute and apply the gradient of scalar functions, and the divergence and
curl of vector fields.
Lake Tahoe Community
College
1. Perform calculus on vector value functions.
2. Evaluate double and triple integrals.
3. Integrate vector value functions.
4. Relate types of single and multiple integrals using the major theorems of
vector calculus.
5. Apply multivariable calculus to problems arising from physics.
Ohlone College
The student will:
1. Use vector methods to solve problems in three dimensional analytic geometry.
2. Analyze problems involving vector functions of a single variable. Topics
include two dimensional normal and tangential acceleration and curvature.
3. Determine the extreme value(s) of a multidimensional function, the tangent
plane to a three dimensional function, the directional derivative and gradient
of a function by using partial derivatives.
4. Use double and triple integrals to determine the areas and volumes bounded by
curves and surfaces, determine the surface area and center of mass of a solid.
Use polar, cylindrical and spherical coordinates for solving these types of
problems.
5. Evaluate line and surface integrals by using Green's Theorem, the Divergence
Theorem, and Stokes' Theorem
6. Demonstrate using a graphing calculator.
Skyline College
• Students will be able to explain and apply the techniques
of multivariable calculus to solve problems graphically, numerically and
analytically.
• Students will be able to translate problems from the physical, life and social
sciences into workable mathematical form, suitable for graphical, numerical,
analytical or verbal solutions.
• Students will be able to calculate derivatives and integrals, using vectors
and other tools fundamental to multivariable calculus.
• Students will be able to solve multivariable problems graphically and
analytically, by examining level sets, partial derivatives, finding extrema,
etc.
Solano Community
College
1. Define,
understand and
apply concepts of three dimensional analytical geometry, vector calculus, and
functions of two or more variables.
2. Make connections between the theorems and applications in vector
calculus.
Linear Algebra
College of San
Mateo
1. Parametrically construct the solution space of a linear
system using Gaussian elimination.
2. Use elementary row operations to reduce a matrix to row echelon form.
3. Successfully employ all of the standard operations with matrices and vectors.
4. Judge when a function is or is not a linear transformation.
5. Given a linear transformation,
a. Recognize the standard matrix (if
the domain and range are of the form Rn).
b. Construct the null space as a span
of vectors.
c. Construct the range space as a
span of vectors.
d. Assess if it is invertible, and,
if it is, construct the inverse.
e. Assess if it is one to one and/or
onto.
6. Construct elementary matrices corresponding to elementary row operations and
can use both to construct the inverse of an invertible square matrix.
7. Judge if a set of vectors is linearly independent or dependent.
8. Construct a basis for a given vector space.
9. Calculate the dimension of a given vector space.
10. Judge if a subset of a given vector space is a subspace.
11. Assess whether a given set and field with addition and scalar multiplication
is or is not a vector space.
12. Calculate the new representations of a vector or standard matrix under a
change of basis.
13. Calculate the determinant of a square matrix and use it to judge the linear
independence of row or columns and judge invertibility.
14. Apply the basic properties of the determinant.
15. Calculate a determinant by expansion by majors.
16. Apply the concept of matrix similarity.
17. Calculate the eigenvalues and construct a basis for the eigenspaces of a
matrix or linear transformation.
18. Construct the diagonal decomposition of a square matrix or else explain why
the matrix can not be diagonalized.
19. Judge when a given (vector, vector) to scalar operation is or is not an
inner product.
20. Express vectors and the representation of a linear transformation in terms
of a new basis.
21. Construct orthonormal bases of R^{n}.
Lake Tahoe Community
College
1. Apply the theory and techniques of linear algebra in applications from
physics, operations research and other scientific disciplines.
2. Solve linear systems, including under and overdetermined systems.
3. Prove lemmas and corollaries in linear algebra.
4. Relate linear transformations to their matrices with respect to given bases.
5. Describe linear transformations as functions mapping an ndimensional space
to an mdimensional space.
Ohlone College
The student will:
1. Use elementary row operations to reduce a matrix to reduced row echelon form
and apply this technique to solving systems of linear equations.
2. Perform the arithmetic of vectors and matrices.
3. Identify definitions and properties of the determinant of a matrix.
4. Identify definitions and properties of vector spaces, including subspaces,
basis and dimension.
5. Perform linear transformations and find the range, kernel, nullity and rank
of a transformation.
6. Expand their levels of thinking to more abstract mathematics and apply this
abstract thinking to proofs.
7. Apply the above material to current applications.
Skyline College
Upon completion of the course:
• Students will be able to formulate linear systems as mathematical models.
• Students will be able to represent any linear system with a suitable matrix
equation.
• Students will be able to compute the general solution to any linear system.
• Students will recognize inherent geometric and analytical properties of a
given matrix.
Solano Community
College
1. Compute
matrix row
operations, determinants, and inverses by hand and using technology.
2. Define, understand and apply concepts of linear systems, invertible
matrices, vector spaces, a basis, linear transformations, range and kernel
change of basis, eigenvalues and eigenvectors, matrix, inner product and
orthogonality.
3. Write simple proofs and give explanations that demonstrate
understanding of the concepts of linear systems, invertible matrices, vector
spaces, a basis, linear transformations, range and kernel change of basis,
eigenvalues and eigenvectors, matrix, inner product and orthogonality.
Differential Equations
College of San
Mateo
1. Recognize the types of first order and second order
ODE's.
2. Appropriately employ qualitative methods to analyze first order ODE's.
3. Solve separable first order ODE's.
4. Solve first order ODE's with homogeneous coefficients.
5. Explain the meaning of existence and uniqueness.
6. Recognize when a first order ODE must or must not have a unique solution.
7. Calculate approximate solutions to first order ODE's using selected numerical
methods.
8. Solve second order homogeneous linear ODE's with constant coefficients.
9. Solve second order linear ODE's with constant coefficients and a forcing
function.
10. Explain the meaning of linear independence.
11. Evaluate when given functions are linearly independent or not.
12. Employ the Wronskian to assess the linear independence of given functions.
13. Employ reduction of order to find the general solution of a second order ODE
given one solution.
14. Employ variation of parameters to solve second order ODE's.
15. Calculate the Laplace transform of a given function.
16. Calculate the inverse Laplace transform of a given function.
17. Calculate the Laplace transform for piecewise defined functions.
18. Employ convolution to calculate the Laplace transform of a given function.
19. Solve first and second order ODE's by employing Laplace transforms.
20. Solve first and second order ODE's using power series in the case of
nonsingular and singular points.
21. Construct the first order system equivalent to a given second order equation
and vice versa.
22. Employ both solution trajectories and phase plane analysis to explain
solutions to first order 2 x 2 systems of ODE's.
23. Solve first order, linear, autonomous first order systems of ODE's.
24. Categorize equilibrium points of systems of first order ODE's.
25. Calculate the Fourier series of given functions.
26. Solve given partial differential equations using separation of variables and
Fourier series.
Lake Tahoe Community
College
1. Apply ordinary differential equations to problems from physics, biology, and
other scientific disciplines.
2. Employ the technique of transformations in finding solutions to ordinary
differential equations.
3. Prove results from the field of differential equations.
4. Sketch direction fields for firstorder ordinary differential equations.
5. Solve differential equations using sequences, series, and matrices.
Ohlone College
The student will:
1. Solve first and second order linear differential equations using techniques
including separation of variables, reduction of order, constant coefficients,
and undetermined coefficients.
2. Solve an ordinary differential equation through the use of infinite series.
3. Solve an ordinary differential equation through the use of Laplace
Transforms.
4. Convert a higher order ordinary differential equation into a system of first
order differential equations.
5. Solve a partial differential equation by the method of separation of
variables.
6. Apply the methods of solving differential equations to problems in
springmass systems, electrical circuits, exponential growth and decay, mixing,
and Newton's Law of Cooling.
7. Use computer software to draw direction fields and solutions to differential
equations.
Skyline College
Upon completion of the course:
• Through real world applications, students will be able to create, manipulate,
solve, and interpret mathematical models describing various physical phenomena
using scalar or vector differential equations.
• Students will be able to apply analytical, numerical, and qualitative methods
to achieve solutions to a wide class of differential equations.
• Students will develop skills for effectively solving problems at an
introductory level of differential equations study.
Solano Community
College
1. Analyze
a differential
equation and select and implement an appropriate method to solve the equation.
2. Construct and solve a differential equation that models a given
situation.
3. Use numerical methods to approximate the solution of a differential
equation.
Others
Ohlone College
Introduction to MATLAB
The student will:
1. Use Matlab in an interactive mode.
2. Use Matlab in a programming mode.
3. Integrate Matlab with Microsoft Word.
4. Create and edit two and three dimensional graphs.
5. Solve a variety of mathematical problems including root finding, numerical
differentiation, numerical integration, and regression.
Discrete Math
The student will:
1. To define and describe those mathematical tools which are considered to be a
part of discrete mathematics.
2. To use these mathematical tools for problem solving in advanced courses in CS
and other related areas.
3. To apply these tools for modeling problems, CS and other areas.
4. To employ discrete mathematics in studying specific applications, such as: a)
data base management systems b) knowledgebased systems c) sorting techniques d)
language syntax e) communications networks f) logical circuit design g) Turing
machines, and other areas from artificial intelligence.
Math for the Associate of Arts Degree
The student will:
1. Demonstrate problem solving skills by applying mathematical principles and
techniques in real world areas.
2. Demonstrate critical thinking by examining and solving mathematical puzzles.
3. Analyze games of chance using probability theory and formulas.
4. Examine statistical principles used to display, interpret and analyze data.
5. Investigate how math is used in various professions such as sports,
carpentry, nursing, music, cooking, etc.
Skyline College
Math for Elementary School Teachers
Upon completion of the course:
• Students will be able to apply a variety of strategies to solve multi–step
problems; including making a table, chart or list, drawing pictures, making a
model, using patterns, working backward, guessing and checking, and comparing
with previous experience.
• Students will communicate mathematical thoughts, ideas, and solutions clearly
and concisely to others in the oral and written forms.
• Students will be able to describe the nature of a problem and the
problem–solving process
Applied Calculus
Upon completion of the course:
• Students will be able to solve real world applications using the concepts of
an average or instantaneous rate of change.
• Students will recognize, apply, and interpret multiple representations
(graphic, symbolic, numerical/data, verbal/applied) of the derivative and its
applications.
• Students will develop skills and attitudes for effectively solving problems at
an applied calculus level.
