Download Math 105 - Department of Mathematics

SJSU Mathematics Department Course Syllabus
Fall 2005
Course Title: Concepts in Mathematics, Probability, and Statistics
SJSU Number: Math 105
Prerequisites: Math 12 with a grade of C- or better, two years of high school algebra, one year
of high school geometry, satisfaction of ELM requirement
Course Description
Mathematics 105 is the second course in a three-course sequence designed for prospective
elementary and middle school teachers. Students explore and develop understanding of
mathematical concepts and processes taught at those levels. In particular, students study
problem-solving techniques, functions and algebraic reasoning, ratio and proportions,
probability, data, graphs, and statistics. Throughout the course, students experience mathematics
learning in the way that we want their future students to experience mathematics learning. In
addition, students analyze their learning experiences from the perspective of a future teacher.
Technology is integrated throughout the course.
Note: This is the second course in a three-course sequence of mathematics courses for future
elementary and middle school teachers. For an overview of the three-course sequence, see p. 11.
Bibliography – Knowledge Base
Textbook
Musser, Burger, & Peterson’s Mathematics for Elementary School Teachers, 7th edition. (Note
that Math 12, Number Systems, and Math 106 Intuitive Geometry uses the same textbook.)
Required Topics and Suggested Schedule
Chapter 1
(1 week)
Chapter 7
(2 weeks)
Chapter 2
(0.5 week)
Chapter 9 &
Chapter 15
(3.5 weeks)
Introduction to Problem Solving (review from Math 12 with different emphases)
 Section 1.1: The Problem Solving Process and Strategies (Emphasize the
strategy use a variable. The strategies guess and test and draw a picture were
emphasized in Math 12.)
 Section 1.2: Three Additional Strategies (Emphasize the strategy look for a
pattern. The strategies make a list and solve a simpler problem were
emphasized in Math 12.)
Decimals, Ratio, Proportion, and Percent
 Section 7.3 Ratio and Proportion
 Section 7.4 Percents
Sets, Whole Numbers, and Numeration (only section 2.4)
 Section 2.4: Relations and Functions
Rational Numbers and Real Numbers with an Introduction to Algebra (last subsection
of Section 9.2 and section 9.3), & Geometry Using Coordinates (first two sections)
 Section 9.2 The Real Numbers (Review the first two-thirds of the section,
which were covered in Math 12, and teach the last subsection, Introduction to
Algebra.)
 Section 9.3 Functions and Their Graphs (first two subsections: The Cartesian
Coordinate System, Graphs of Linear Functions)
 Section 15.1 Distance and Slope in the Coordinate Plane
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
Section 15.2 Equations and Coordinates (omit the last subsection Equations of
Circles)
 Section 9.3 Functions and Their Graphs (remaining subsections: Graphs of
Quadratic Functions, Graphs of Exponential Functions, Graphs of Other
Common Functions. Of these three subsections, emphasize the quadratic
functions.)
Chapter 11
Probability
(3 weeks)
 Section 11.1 Probability and Simple Experiments
 Section 11.2 Probability and Complex Experiments
 Section 11.3 Simulation, Expected Value, Odds, and Conditional Probability
 Section 11.4 Additional Counting Techniques
Chapter 10
Statistics
(3 weeks)
 Section 10.1 Organizing and Picturing Information
 Section 10.2 Analyzing Data
 Section 10.3 Misleading Graphs and Statistics
Miscellaneous Review, quizzes, exams (2 weeks)
Total Time
15 weeks
Allocation
Journal Articles
1. Bay-Williams, Jennifer M., “What is Algebra in Elementary School?” Teaching Children
Mathematics 8 (December 2001): 196-200.
2. Stevens, Anthony C., Sharp, Janet M., Nelson, Becky, “The Intersection of Two Unlikely
Worlds: Ratios and Drums”, Teaching Children Mathematics 8 (February 2001): 376383.
3. Telese, James A., Abete Jr., Jesse, “Diet, Ratios, Proportions: A Healthy Mix”, Mathematics
Teaching in the Middle School 8 (September 2002): 8-13.
4. Tarr, James E., “Providing Opportunities to Learn Probability Concepts”, Teaching Children
Mathematics 8 (April 2002): 482-487.
Instructors’ References
Principles and Standards for School Mathematics, National Council of Teachers of
Mathematics, 2000.
Mathematics Framework for California Public Schools, California Department of Education,
1999.
Data in Depth, Exploring Mathematics with Fathom, Tim Erickson, Key Curriculum Press, 2001
Supplementary Activities Packet, SJSU Math Education Committee
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Goals and Objectives—Mathematical Content*
1. Functions and Algebraic Reasoning – Understand patterns, relations, and functions.
Represent and analyze mathematical situations and structures using algebraic symbols. Use
mathematical models to represent and understand quantitative relationships. Analyze change in
various contexts.
 Represent patterns, including relations and functions, through tables, graphs, verbal
rules, and symbolic rules.
 Explain the meaning of symbolic expressions (e.g., relating an expression to a situation
and vice versa).
 Recognize and create equivalent numerical and algebraic expressions (e.g., 2(a + 3) =
2a + 6) and identify the property or properties being used. Multiply, divide, and factor
polynomials.
 Represent, describe, and analyze situations, relationships, and problems, using patterns,
tables, expressions, verbal descriptions, equations, inequalities, variables, or graphs,
interchanging among these, as needed.
 Recognize that situations, relationships, and problems can be represented, described,
analyzed, and solved using functions.
 Give examples and non-examples of real-world quantities that are in a functional
relationship.
 Use functional relationships to analyze how change in one quantity results in change in
another. Determine the class or classes of functions that might model a quantitative
relationship.
 Find equivalent expressions for equalities and inequalities, and identify the property or
properties being used. Solve equations and inequalities, using and naming properties
being used.
 Understand linear functions and their properties (e.g., slope). Solve linear equations and
inequalities, and represent solutions on graphs. Create and interpret graphs of linear
equations and inequalities. Interpret graphs to get solutions to systems of equations.
 Understand quadratic functions and their properties (e.g. symmetry). Solve quadratic
equations through factoring and completing the square, and represent solutions on
graphs. Create and interpret graphs of quadratic equations and inequalities.
2. Proportional Reasoning – Understand and use ratios and proportions to represent
quantitative relationships. Develop meaning for percents, including percents greater than 100
and less than 1.
 Use proportional reasoning such as ratios, equivalent fractions, and similar triangles to
solve numerical, algebraic, and geometric problems.
 Recognize and give examples of different types of real-world ratios (e.g., part-part
comparison, part-whole comparison, rate). Represent ratios in different ways (e.g.,
fraction, decimal). Explain the relationship between ratio and the slope of a linear
function in the context of real-world problem situations.
*
Goals and objections in italic are directly quoted from the mathematics content specifications described in The
California Commission on Teacher Credentialing document Standards of Program Quality and Effectiveness for the
Subject Matter Requirement for the Multiple Subject Teaching Credential. Goals and objectives in italic must be
covered in the course in order for the course to remain compliant with this document. See pp. 9-11 for the full text
of these mathematics content specifications.
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
Represent a proportion as two equal ratios. Determine whether two quantities vary
proportionally. Write proportional relationships using different but equivalent equations.
Find the constant of variation given a set of equal ratios.
 Understand and model situations involving percent. Estimate and find exact answers to
real-world problems involving percent. Convert among percents, fractions, and decimals.
3. Probability – Understand and apply basic concepts of probability.
 Define the concept of probability in terms of a sample space of equally likely outcomes.
Express probabilities in a variety of ways, including ratios, proportions, decimals, and
percents. Give examples of situations with probabilities of 0 or 1.
 Use counting techniques to identify and count outcomes of an experiment. For example,
use tree diagrams, box arrays, the multiplication property, permutations, or combinations,
as appropriate
 Explain through words and examples the difference between theoretical and experimental
probability; find and compare experimental and theoretical probabilities; and make
predictions using experimental and theoretical probabilities.
 Use understanding of complementary, mutually exclusive, dependent, and independent
events to calculate probabilities of simple events. Find conditional probability in a given
situation.
 Construct and carry out a simulation to approximate probability.
 Find the odds in favor of or against an event.
4. Statistics – Formulate questions that can be addressed with data and collect, organize, and
display relevant data to answer them. Select and use appropriate statistical methods to analyze
data. Develop and evaluate inferences and predictions.
 Learn that patterns in data can be seen when data are represented using graphs, tables,
and charts. They will construct, read, and interpret a graph, table, or chart representing
a data set. In particular, students will use frequency tables, line plots, stem-and-leaf
plots, pictographs, histographs, bar graphs, box-and-whiskers plots, and line graphs.
Evaluate the reasonableness of particular data representations.
 Understand, compare and use measures of central tendency (mean, median, and mode) of
a collection of data. Give examples of situations where each measure might be used
appropriately.
 Understand, compare, and use measures of variability (range, standard deviation,
variance) of a collection of data. Give examples of situations where each measure might
be used appropriately. Give an intuitive explanation of variance and standard deviation.
Use formulas and/or technology to find variance and standard deviation and meaningfully
connect these measures to the normal curve
 Have a basic understanding of the design of surveys, such as the role of a random
sample. Evaluate and develop appropriate sampling techniques. Evaluate data collection
procedures and interpretations.
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Goals and Objectives – Mathematical Processes
One of the ways in which we weave together the three mathematics courses for prospective
teachers is to have well-defined, long-term goals that guide the implementation of the syllabi
throughout the sequence. These are goals that go above and beyond the learning of specific
content topics, and which take students longer than one semester to achieve. These goals
correspond roughly to the National Council of Teachers of Mathematics’ process standards –
they “highlight ways of acquiring and using content knowledge” (NCTM, 2000, p. 29). Each
course builds upon these goals in a progressive fashion.
1. Understand understanding
 Recognize the validity of different approaches
 Recognize the equivalence of different answers
 Analyze errors to identify misunderstandings
 Analyze levels of understanding
 Explain multiple ways of understanding the same idea
 Recognize when language use is ambiguous, well-defined, or meaningless
 Recognize examples and non-examples
2. Utilize representations and connections
 Identify situations that can be modeled using mathematics
 Represent situations appropriately using mathematics
 Translate from one representation to one another
 Explain how representations are connected to one another
3. Develop new reasoning and problem-solving skills
 Experiment, conjecture, verify
 Recognize patterns, recognize recurring ideas in different settings
 Reason by analogy, infer in situations of uncertainty
 Synthesize, deduce
4. Communicate mathematical ideas
 Use mathematical terminology, notation, and language effectively and accurately
 Express ideas logically and clearly
 Model English with mathematics, interpret mathematics into English
 Illustrate and support ideas graphically, numerically, symbolically, or verbally as
needed
5. Develop positive attitudes and beliefs about mathematics
 Learn and articulate how mathematics is useful outside of school
 Identify potential sources of negative attitudes and beliefs
 Model helpful attitudes and beliefs while working with fellow students
6. Use technology appropriately
 Appreciate the role of technology as a tool for learning and problem solving
 Recognize the capabilities and limitations of computational aids
 See how elementary mathematics and technology interact
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California Commission on Teacher Credentialing Standards
The California Commission on Teacher Credentialing requires all approved subject matter
programs to meet certain standards of program quality and effectiveness, as described in the
September 2001 document Standards of Program Quality and Effectiveness for the Subject
Matter Requirement for the Multiple Subject Teaching Credential. Here, we describe how
SJSU’s Math 105 contributes to meeting the content specifications in mathematics. (See pp. 810 for the mathematics content specifications.)
Part I: Content Domains for Subject Matter Understanding and Skill in Mathematics
Course Topics
Math 105
Functions &
Algebraic
Reasoning
Proportional
Reasoning
Probability
Data, Graphs, &
Statistics
Domain 1
Number
Sense
Domain 2
Algebra &
Functions
1.1
1.2
2.1
2.2
x
x
x
x
x
Domains
Domain 3
Measurement &
Geometry
3.1
3.2
3.3
Domain 4
Statistics, Data
Analysis, &
Probability
4.1
4.2
4.3
x
x
x
x
x
x
Domain 1.2 Computational Tools, Procedures, and Strategies
The elements of this domain are incorporated throughout the second and third courses of our
three-semester sequence of courses for future elementary and middle school teachers, with
particular emphasis placed on the use of technology.
Domain 2.1 Patterns and Functional Relationships
The exploration of patterns through tables, graphs, verbal rules, and symbolic rules sets the stage
for an investigation of relations and functions. Both manipulatives and graphing calculators aid
students in their explorations of patterns.
Domain 2.2 Linear and Quadratic Equations and Inequalities
These special kinds of equations and inequalities are investigated from the viewpoint of linear
and quadratic functions and their properties.
Domain 3.3 Techniques, Tools, and Formulas for Determining Measurements
One element of this domain, namely the relationship of proportional reasoning to the
construction of scale models or drawings is included in our unit on proportional reasoning.
Domain 4.1 Collection, Organization, and Representation of Data
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The representation of data through tables, graphs, and charts is a theme that begins with first unit
(functions and algebraic reasoning) and carries through all subsequent units. Students study
measures of central tendency and spread in our unit on statistics.
Domain 4.2 Inferences, Predictions, and Arguments Based on Data
Interpretation and sense-making of multiple representations of data is another theme that runs
throughout this course. This includes interpretation of technology-generated information.
Domain 4.3 Basic Notions of Chance and Probability
Students study both experimental and theoretical probabilities. In particular, they conduct
simulations using technology, and compare the simulation results with a theoretical analysis.
Part II: Subject Matter Skills and Abilities Applicable to the Content Domain in
Mathematics
Subject matter skills and abilities in mathematics are developed in a progressive fashion over the
three-semester sequence of mathematics courses for future elementary school teachers. Broadly,
our goals are for our students to (1) understand understanding, (2) utilize representations and
connections, (3) develop new reasoning and problem-solving skills, (4) communicate
mathematical ideas, (5) develop positive attitudes and beliefs about mathematics, and (6) use
technology appropriately. We progress towards these goals in Math 105 by integrating
technology into each unit in multiple ways.
The unit on functions and algebraic reasoning in Math 105 requires students to explore functions
numerically, graphically, and symbolically, using graphing calculators. Connections between
different representations of functions can be discovered by exploring patterns exhibited by
families of functions (i.e. linear functions, quadratic functions, exponential functions). This sets
the stage of exploring other mathematical ideas (proportional reasoning, probability, statistics)
via technology. The accessibility and ease of multiple representations via the graphing calculator
is made clear through several collaborative activities. At the same time, the limitations of the
calculators are brought to light, and alternatives to the calculators (Excel and Fathom) are
brought into the picture, allowing students to create and print data displays, as needed.
Subject Matter Competency in Mathematics for Multiple-Subjects Credential Candidates
All future multiple-subjects credential candidates will have to pass the CSET exam in multiple
subjects prior to entering a credential program in California. Subtest II covers mathematics and
science. Sample questions can be viewed at the web site http://www.cset.nesinc.com/.
Instructors might wish to go over sample mathematics questions from the CSET on occasion.
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Content Specifications in Mathematics*
Part I: Content Domains for
Subject Matter Understanding and Skill in
Mathematics
Domain 1: Number Sense
1.1
Numbers, Relationships Among Numbers, and Number Systems. Candidates for
Multiple Subject Teaching Credentials understand base ten place value, number theory
concepts (e.g., greatest common factor), and the structure of the whole, integer, rational,
and real number systems. They order integers, mixed numbers, rational numbers
(including fractions, decimals, and percents) and real numbers. They represent numbers in
exponential and scientific notation. They describe the relationships between the algorithms
for addition, subtraction, multiplication, and division. They understand properties of
number systems and their relationship to the algorithms, [e.g., 1 is the multiplicative
identity; 27 + 34 = 2 X 10 + 7 + 3 X 10 + 4 = (2 + 3) X 10 + (7 + 4)]. Candidates perform
operations with positive, negative, and fractional exponents, as they apply to whole
numbers and fractions.
1.2
Computational Tools, Procedures, and Strategies. Candidates demonstrate fluency in
standard algorithms for computation and evaluate the correctness of nonstandard
algorithms. They demonstrate an understanding of the order of operations. They round
numbers, estimate the results of calculations, and place numbers accurately on a number
line. They demonstrate the ability to use technology, such as calculators or software, for
complex calculations.
Domain 2: Algebra and Functions
2.1
Patterns and Functional Relationships. Candidates represent patterns, including
relations and functions, through tables, graphs, verbal rules, or symbolic rules. They use
proportional reasoning such as ratios, equivalent fractions, and similar triangles, to solve
numerical, algebraic, and geometric problems.
2.2
Linear and Quadratic Equations and Inequalities. Candidates are able to find
equivalent expressions for equalities and inequalities, explain the meaning of symbolic
expressions (e.g., relating an expression to a situation and vice versa), find the solutions,
and represent them on graphs. They recognize and create equivalent algebraic expressions
(e.g., 2(a+3) = 2a + 6), and represent geometric problems algebraically (e.g., the area of a
triangle). Candidates have a basic understanding of linear equations and their properties
(e.g., slope, perpendicularity); the multiplication, division, and factoring of polynomials;
and graphing and solving quadratic equations through factoring and completing the square.
*
Reprinted from California State Standards of Program Quality and Effectiveness for the Subject Matter Requirement for the Multiple Subject
Teaching Credential, September, 2001.
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They interpret graphs of linear and quadratic equations and inequalities, including solutions
to systems of equations.
Domain 3: Measurement and Geometry
3.1
Two- and Three-dimensional Geometric Objects. Candidates for Multiple Subject
Teaching Credentials understand characteristics of common two- and three-dimensional
figures, such as triangles (e.g., isosceles and right triangles), quadrilaterals, and spheres.
They are able to draw conclusions based on the congruence, similarity, or lack thereof, of
two figures. They identify different forms of symmetry, translations, rotations, and
reflections. They understand the Pythagorean theorem and its converse. They are able to
work with properties of parallel lines.
3.2
Representational Systems, Including Concrete Models, Drawings, and Coordinate
Geometry. Candidates use concrete representations, such as manipulatives, drawings, and
coordinate geometry to represent geometric objects. They construct basic geometric
figures using a compass and straightedge, and represent three-dimensional objects through
two-dimensional drawings. They combine and dissect two- and three-dimensional figures
into familiar shapes, such as dissecting a parallelogram and rearranging the pieces to form a
rectangle of equal area.
3.3
Techniques, Tools, and Formulas for Determining Measurements. Candidates estimate
and measure time, length, angles, perimeter, area, surface area, volume, weight/mass, and
temperature through appropriate units and scales. They identify relationships between
different measures within the metric or customary systems of measurements and estimate
an equivalent measurement across the two systems. They calculate perimeters and areas of
two-dimensional objects and surface areas and volumes of three-dimensional objects. They
relate proportional reasoning to the construction of scale drawings or models. They use
measures such as miles per hour to analyze and solve problems.
Domain 4: Statistics, Data Analysis, and Probability
4.1
Collection, Organization, and Representation of Data. Candidates represent a collection
of data through graphs, tables, or charts. They understand the mean, median, mode, and
range of a collection of data. They have a basic understanding of the design of surveys,
such as the role of a random sample.
4.2
Inferences, Predictions, and Arguments Based on Data. Candidates interpret a graph,
table, or chart representing a data set. They draw conclusions about a population from a
random sample, and identify potential sources and effects of bias.
4.3
Basic Notions of Chance and Probability. Candidates can define the concept of
probability in terms of a sample space of equally likely outcomes. They use their
understanding of complementary, mutually exclusive, dependent, and independent events
to calculate probabilities of simple events. They can express probabilities in a variety of
ways, including ratios, proportions, decimals, and percents.
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Part II: Subject Matter Skills and Abilities
Applicable to the Content Domains in Mathematics
Candidates for Multiple Subject Teaching Credentials identify and prioritize relevant and
missing information in mathematical problems. They analyze complex problems to identify
similar simple problems that might suggest solution strategies. They represent a problem in
alternate ways, such as words, symbols, concrete models, and diagrams, to gain greater insight.
They consider examples and patterns as means to formulating a conjecture.
Candidates apply logical reasoning and techniques from arithmetic, algebra, geometry, and
probability/statistics to solve mathematical problems. They analyze problems to identify
alternative solution strategies. They evaluate the truth of mathematical statements (i.e., whether
a given statement is always, sometimes, or never true). They apply different solution strategies
(e.g., estimation) to check the reasonableness of a solution. They demonstrate that a solution is
correct.
Candidates explain their mathematical reasoning through a variety of methods, such as words,
numbers, symbols, charts, graphs, tables, diagrams, and concrete models. They use appropriate
mathematical notation with clear and accurate language. They explain how to derive a result
based on previously developed ideas, and explain how a result is related to other ideas.
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Mathematics Course Sequence at SJSU for Prospective Elementary and Middle School Teachers
Math 12, Math 105, Math 106
The San Jose State Mathematics Department offers a three-semester sequence of courses designed for prospective
elementary and middle school teachers. In these courses, students explore and develop understanding of
mathematical concepts and processes taught at those levels. Throughout the three-course sequence, students
experience mathematics learning in the way that we want their future students to experience mathematics learning,
using technology, as appropriate. In addition, students analyze their own learning experiences from the perspective
of a future teacher. Students are expected to grow in mathematical sophistication, scholarly responsibility, and
pedagogical perspectives over the three-course sequence.
Math 12 Number Systems
In Math 12 is the first course in the three-course sequence. Several local community colleges offer equivalent
courses. In this course, students study problem solving techniques, numeration systems, the structure of the real
number system, and elementary number theory.
Course Prerequisites (reprinted from the SJSU 2004-2006 catalog, p. 303)
Two years of high school algebra, one year of high school geometry, satisfaction of ELM requirement
Math 105 Concepts in Mathematics, Probability, & Statistics
Math 105 is the second course in the three-course sequence. This is an upper division class that cannot be taken at a
local community college. Students study problem-solving techniques, functions and algebraic reasoning, ratio and
proportions, probability, data, graphs, and statistics.
Course Prerequisites (reprinted from the SJSU 2004-2006 catalog, p. 304)
Two years of high school algebra, one year of high school geometry, SJSU’s Math 12 with a C- or better
Math 106 Intuitive Geometry
Mathematics 106 is the third course in the three-course sequence. This is an upper division class that cannot be
taken at a local community college. Students analyze characteristics and properties of two- and three-dimensional
geometric shapes; develop mathematical arguments about geometric relationships; apply transformations and use
symmetry to analyze mathematical situations; represent geometric objects using representational systems such as
concrete models, drawings, and coordinate geometry; and use techniques, tools, and formulas for determining
measurements. In general, students are encouraged to think about geometry as the study of objects in a plane or in
space. They are asked to investigate situations involving geometric objects, state conjectures, and provide
explanations that support their conjectures. Technology is integrated extensively. In particular, students will use a
dynamic geometry system, such as Geometer’s Sketchpad or Cabri, as a tool for visual investigations.
Course Prerequisites (reprinted from the SJSU 2004-2006 catalog, p. 304)
Two years of high school algebra, one year of high school geometry, SJSU’s Math 12 and Math 105 with grades of
C- or better



General Notes
There are no exceptions to the prerequisites. This means that students must take the three courses
sequentially. (The level of mathematical sophistication, extent of scholarly expectations, and breadth of
pedagogical perspectives increase substantially from one course to the next in this sequence. The
prerequisites are designed to provide students with the greatest opportunity of success in the three-course
sequence as well as the best possible preparation for teaching mathematics at the elementary and middle
school levels.)
Grades of C or better MAY be required by some students’ majors in order for them to graduate. Students
are expected to determine this in consultation with their major advisors.
The topics in the three-course sequence include all of the topics covered in the mathematics portion of
Subtest II of the California Subject Examination for Teachers: Multiple Subjects. Students preparing for
teaching careers in California are strongly encouraged to take this three-course mathematics sequence in
preparation for this exam, even if their majors do not require all or any of the courses.
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