Download Transition to College Math

Transition Syllabus – “Key Ideas”/ “Assessable Specifications”
Key Idea 1: Logical reasoning and mathematical knowledge (Chapter 1; Sections 1.1, 1.2 and 1.3)
COMPONENTS
Vocabulary: bar graphs, circle graphs, conjecture, counterexample, deductive reasoning, estimation, hypothesis, inductive reasoning, line graphs, Pascal’s triangle,
pie charts, sectors, square numbers, theorem, triangular numbers, unit price
Standards:
Assessable Specifications:
COMPONENTS
A.12.1 Use reason and logic to





evaluate information
perceive patterns
identify relationships
formulate questions, pose problems, and
make and test conjectures
pursue ideas that lead to further
understanding and deeper insight
A.12.2 Communicate logical arguments and clearly
show



why a result does or does not make sense
why the reasoning is or is not valid
an understanding of the difference
between examples that support a
conjecture and a proof of the conjecture
A.12.4 Develop effective oral and written
presentations employing correct mathematical
terminology, notation, symbols, and conventions for
mathematical arguments and display of data
MPS
Thinking
Levels
1.1 Students will be able to analyze a situation and describe
the problem(s) to be solved.
4
1.2 Students will be able to formulate a plan for solving the
problem.
3
1.3 Students will be able to use logical reasoning and
mathematical knowledge to obtain and justify correct
solutions.
4
Instructional Time
Sequence, i.e. 1st
Semester, 2nd
Semester
1st Semester
Key Idea 2: Interpret and communicate mathematical knowledge. (Chapter 3; Sections 3.1, 3.2, 3.3 and 3.4)
COMPONENTS
Vocabulary: antecedent, bi-conditional statement, component statement, compound statements, conditional statement, conjunction, connectives, consequent,
disjunction, equivalent, exclusive, implications, inclusive, negation, quantifiers, self-contradictions, simple statements, statement, tautology, truth table
Standards:
Assessable Specifications:
A.12.1 Use reason and logic to





evaluate information
perceive patterns
identify relationships
formulate questions, pose problems, and
make and test conjectures
pursue ideas that lead to further
understanding and deeper insight
A.12.3 Analyze non-routine* problems and arrive at
solutions by various means, including models* and
simulations, often starting with provisional
conjectures and progressing, directly or indirectly, to
a solution, justification, or counter-example
A.12.4 Develop effective oral and written
presentations employing correct mathematical
terminology, notation, symbols, and conventions for
mathematical arguments and display of data
C.12.3 Present convincing arguments by means of
demonstration, informal proof, counter-examples, or
any other logical means to show the truth of


statements (e.g., these two triangles are
not congruent)
generalizations (e.g., the Pythagorean*
theorem holds for all right triangles)
2.1 Students will be able to summarize and interpret
mathematical information from truth tables and from
other written formats.
2.2 Students will be able to use symbols, diagrams, graphs,
and words to clearly communicate mathematical ideas,
reasoning, and their implications; use correct
mathematical symbols, terminology and notation.
2.3 Students will be able to produce a variety of statements
and their negation, by translating them in both symbolic
form and everyday language, to support a position or
conclusion.
MPS
Thinking
Levels
Instructional Time
Sequence, i.e. 1st
Semester, 2nd
Semester
4
1st Semester
3
4
2.4 Students will be able to express conditional and
bi-conditional statements in every day language and
symbolic form, such as truth tables.
4
Key Idea 3: Perform appropriate operations. (Chapter 4; Sections 4.1, 4.2, and 4.4)
COMPONENTS
Vocabulary: additive system, base, binary system, digits, expanded form, exponent, exponential notation, hexadecimal system, Hindu-Arabic numerals, number,
numeral, numeration system, octal system, place value, positional, positional-value, powers, system of numeration
Standards:
Assessable Specifications:
B.12.3 Perform and explain operations on real
numbers (add, subtract, multiply, divide, raise to a
power, extract a root, take opposites and reciprocals,
determine absolute value)
3.1 Students will be able to accurately and efficiently
compute with real numbers in all forms, including rational
exponents and scientific notation.
B.12.4 In problem-solving situations involving the
application of different number systems (natural,
integers, rational*, real*) select and use appropriate

computational procedures

properties (e.g., commutativity*,
associativity*, inverses*)

modes of representation (e.g., rationals as
repeating decimals, indicated roots as
fractional exponents)
MPS
Thinking
Levels
Instructional Time
Sequence, i.e. 1st
Semester, 2nd
Semester
2
1st Semester
3.2 Students will be able to recognize and manipulate
different numeration systems; the Hindu-Arabic, Early
Positional, Bablyonian, Mayan, Egyptian, Roman, Ionic
Greek and the traditional Chinese system.
2
3.3 Students will be able to change numerals from one base
to other bases.
2
Key Idea 4: Number Theory and the Real Number System. (Chapter 5; Sections 5.1, 5.2, 5.3, 5.4, 5.5, 5.6 and 5.7)
COMPONENTS
Vocabulary: absolute value, additive inverses, arithmetic sequence, common denominator, common difference, common ratio, composite number, cube root,
decimal, denominator, difference, divisible, equivalent fractions, factors, factor tree, Fibonacci sequence, geometric sequence, imaginary number, improper fraction,
inequality symbols, integers, irrational number, least common multiple, lowest terms, mixed number, multiples, multiplicative inverses, natural numbers, negative
exponent, number line, number theory, numerator, order of operations, perfect square, prime factorization, prime number, principal square root, product, quotient,
radical, radical sign, radicand, rational numbers, real numbers, reciprocals, relatively prime, repeating decimal, scientific notation, simplified, subsets, sum, terms,
whole numbers
Standards:
Assessable Specifications:
B.12.2 Compare real numbers using

order relations (>,<) and transitivity*

ordinal scales including logarithmic (e.g.,
Richter, pH rating)

arithmetic differences

ratios, proportions, percents, rates of
change
4.1 Students will be able to solve problems using number
theory (prime and composite numbers, divisibility, prime
factoring, greatest common divisor, least common
multiple).
4.2 Students will be able to work with integers, absolute
value and the order of operations using a number line,
inequality symbols and real world examples.
MPS
Thinking
Levels
3/4
2
3/4
B.12.3 Perform and explain operations on real
numbers (add, subtract, multiply, divide, raise to a
power, extract a root, take opposites and reciprocals,
determine absolute value)
4.3 Students will be able to solve problems involving rational
numbers by being able to convert between various
formats, simplifying and performing basic operations.
2
B.12.4 In problem-solving situations involving the
application of different number systems (natural,
integers, rational*, real*) select and use appropriate
4.4 Students will be able to simplify and perform operations
with square roots including rationalizing the denominator.
3/4

computational procedures

properties (e.g., commutativity*,
associativity*, inverses*)

modes of representation (e.g., rationals as
repeating decimals, indicated roots as
fractional exponents)
F.12.1 Analyze and generalize patterns of change
(e.g., direct and inverse variation) and numerical
sequences, and then represent them with algebraic
expressions and equations.
4.5 Students will be able to recognize subsets and
properties of real numbers.
4.6 Students will be able to use properties of exponents,
convert between decimal and scientific notation and
perform computations and solve problems involving
scientific notation.
4.7 Students will be able to write terms and use the formulas
for arithmetic and geometric sequences.
1
2
Instructional Time
Sequence, i.e. 1st
Semester, 2nd
Semester
1st Semester
Key Idea 5: Algebra; Equations and Inequalities. (Chapter 6; Sections 6.1, 6.2, 6.3, 6.4, 6.5, and 6.6)
COMPONENTS
Vocabulary: algebraic expression, binomial, coefficient, constant term, equation, equivalent, evaluate, factoring, factors, formulas, like terms, linear equation, linear
inequality, mathematical models, model, prime, proportion, quadratic equations, quadratic formula, ratio, simplified, solution set, solutions, terms, trinomial, variables,
vary inversely, zero-product principle
Standards:
Assessable Specifications:
B.12.2 Compare real numbers using

order relations (>,<) and transitivity*

ordinal scales including logarithmic (e.g., Richter, pH
rating)

arithmetic differences

ratios, proportions, percents, rates of change
F.12.1 Analyze and generalize patterns of change (e.g., direct and
inverse variation) and numerical sequences, and then represent
them with algebraic expressions and equations
F.12.2 Use mathematical functions* (e.g., linear*, exponential*,
quadratic*, power) in a variety of ways, including




recognizing that a variety of mathematical and realworld phenomena can be modeled* by the same type
of function
translating different forms of representing them (e.g.,
tables, graphs, functional notation*, formulas)
describing the relationships among variable quantities
in a problem
using appropriate technology to interpret properties of
their graphical representations (e.g., intercepts,
slopes, rates of change, changes in rates of change,
maximum*, minimum*)
F.12.3 Solve linear and quadratic equations, linear inequalities,
and systems of linear equations and inequalities



numerically
graphically, including use of appropriate technology
symbolically, including use of the quadratic formula
F.12.4 Model and solve a variety of mathematical and real-world
problems by using algebraic expressions, equations, and
inequalities
MPS
Thinking
Levels
5.1 Students will be able to evaluate and simplify algebraic
expressions and formulas.
2
5.2 Students will be able to solve real world problems
involving linear equations and identify equations with
infinite or no solutions.
4
5.3 Students will be able to solve problems involving ratios
and proportions including direct and inverse variations.
3/4
5.4 Students will be able to solve real world problems
involving one variable linear inequalities and graph their
solutions.
3/4
5.5 Students will be able to multiply binomials, factor
trinomials and solve quadratic equations.
2
Instructional Time
Sequence, i.e. 1st
Semester, 2nd
Semester
2nd Semester
Key Idea 6: Algebra; Graphs, Function, and Linear Systems. (Chapter 7; Sections 7.1, 7.2, 7.3, 7.4, 7.5 and 7.6)
COMPONENTS
Vocabulary: axis of symmetry, Cartesian coordinate system, exponential function, function, linear function, linear system, origin, parabola, plot, quadratic function,
quadrants, rate of change, rectangular coordinate system, rise, run, slope, slope-intercept form, solution, vertex, vertical line test, x and y-axis, x and y-coordinates
Standards:
Assessable Specifications:
A.12.3: Analyze non-routine problems and arrive at
solutions by various means, including models and
simulations, often starting with provisional conjectures and
progressing, directly or indirectly, to a solution, justification,
or counter-example.
6.1 Students will be able to graph equations and express
functions using f (x) notation.
2
6.2 Students will be able to determine if an equation is a
function using the methods such as the vertical line test.
2
B.12.2: compare real number using; order relations (>,<)
and transitivity, ordinal scales including logarithmic,
arithmetic differences, rations, proportions, percents, rates
of change.
C.12.4: Use the two-dimensional rectangular coordinate
system and algebraic procedures ot describe and
characterize geometric properties and relationships such
as slope, intercepts, parallelism, and perpendicularity.
F.12.1: Analyze and generalize patterns of change and
numerical sequences, and then represent them with
algebraic expressions and equations.
F.12.2: Use mathematical functions (linear, exponential,
quadratic, power) in a variety of ways, including;
recognizing that a variety of mathematical and real-world
phenomena can be modeled by the same type of function,
translating different forms of representing them (tables,
graphs, functional notation, formulas), describing the
relationships among variable quantities in a problem,
using appropriate technology to interpret properties of their
graphical representations (intercepts, slopes, rates of
change, changes in a rates of change, maximum,
minimum)
F.12.3: Solve linear and quadratic equations, linear
inequalities, and systems of linear equations and
inequalities; numerically, graphically, including use of
appropriate technology, symbolically, including use of
quadratic formula.
F.12.4: Model and solve a variety of mathematical and
real-world problems using algebraic expressions,
equations, and inequalities.
MPS
Thinking
Levels
6.3 Students will be able to graph and interpret linear
equations using slopes and or intercepts.
2/3
6.4 Students will be able to graph and interpret quadratic
and exponential functions.
2/3
6.5 Students will be able to solve systems of equations and
interpret their results.
3
6.6 Students will be able to graph and interpret two variable
linear inequalities and a system of linear inequalities.
3
Instructional Time
Sequence, i.e. 1st
Semester, 2nd
Semester
2nd Semester
Key Idea 7: Consumer Mathematics and Financial Management. (Chapter 8; Sections 8.1, 8.2, 8.3, 8.4, 8.5 and 8.6)
COMPONENTS
Vocabulary: adjustable-rate mortgages, amortized, amount financed, annual percentage rate, average daily balance, banker’s rule, Bear market, bonds, Bull
market, discounted loan, dividends, capital gain, cash investment, closing price, compound interest, compounded – annually/semiannually/quarterly, compounding
period, continuous compounding, diversified portfolio, down payment, effective annual yield, effective rate, escrow amount, face value, finance charge, financial
portfolio, fixed installment loan, fixed-rate mortgages, future value, installment buying, installment loan, interest, itemized billing, lending money, loan amortization
schedule, mortgage, mutual fund, nominal rate, payoff amount, percent, percent increase/decrease, points, present value, previous balance method, principal, rate,
return, shareholder, simple interest, simple interest rate, stock exchange, stocks, trading, total installment price, unpaid balance method, variable-rate mortgages
Standards:
A.12.3 Analyze non-routine* problems and arrive at solutions by various
means, including models* and simulations, often starting with provisional
conjectures and progressing, directly or indirectly, to a solution, justification,
or counter-example
B.12.2 Compare real numbers using

order relations (>,<) and transitivity*

ordinal scales including logarithmic (e.g., Richter, pH rating)

arithmetic differences

ratios, proportions, percents, rates of change
B.12.5 Create and critically evaluate numerical arguments presented in a
variety of classroom and real-world situations (e.g., political, economic,
scientific, social)
D.12.3 Determine measurements indirectly*, using







estimation
proportional reasoning, including those involving squaring
and cubing (e.g., reasoning that areas of circles are
proportional to the squares of their radii)
techniques of algebra, geometry, and right triangle
trigonometry
formulas in applications (e.g., for compound interest, distance
formula)
geometric formulas to derive lengths, areas, or volumes of
shapes and objects (e.g., cones, parallelograms, cylinders,
pyramids)
geometric relationships and properties of circles and
polygons (e.g., size of central angles, area of a sector of a
circle)
conversion constants to relate measures in one system to
another (e.g., meters to feet, dollars to Deutschmarks
Assessable Specifications:
MPS
Thinking
Levels
7.1 Students will be able to convert between fractions,
decimals and percents.
2
7.2 Students will be able to solve real world problems
involving percent.
3/4
7.3 Students will be able to solve real world problems using
simple and compound interest.
2
7.4 Students will be able to solve real world problems
involving Installment Buying, Credit Cards, Mortgages
and Loans.
2/3
7.5 Students will be able to have a basic working ability of
stocks, bonds, mutual funds and investments.
3
Instructional Time
Sequence, i.e. 1st
Semester, 2nd
Semester
2nd Semester
Key Idea 8: Measurement. (Chapter 9; Sections 9.1, 9.2, and 9.3)
COMPONENTS
Vocabulary: area, capacity, dimensional analysis, Celsius, English system, Fahrenheit, length, linear measurement, linear units, mass, measure, Metric system,
square unit, unit fractions, weight
Standards:
Assessable Specifications:
D.12.1 Identify, describe, and use derived attributes*
(e.g., density, speed, acceleration, pressure) to
represent and solve problem situations
8.1 Students will be able to convert units within the metric
system.
2
8.2 Students will be able to convert between English and
metric systems.
2
8.3 Students will be able to express area, volume, mass,
weight and temperature in the appropriate English and
metric systems.
3
D.12.3 Determine measurements indirectly*, using







estimation
proportional reasoning, including those
involving squaring and cubing (e.g.,
reasoning that areas of circles are
proportional to the squares of their radii)
techniques of algebra, geometry, and right
triangle trigonometry
formulas in applications (e.g., for
compound interest, distance formula)
geometric formulas to derive lengths,
areas, or volumes of shapes and objects
(e.g., cones, parallelograms, cylinders,
pyramids)
geometric relationships and properties of
circles and polygons (e.g., size of central
angles, area of a sector of a circle)
conversion constants to relate measures in
one system to another (e.g., meters to
feet, dollars to Deutschmarks
MPS
Thinking
Levels
Instructional Time
Sequence, i.e. 1st
Semester, 2nd
Semester
2nd Semester
Key Idea 9: Geometry. (Chapter 10; Sections 10.1, 10.2, 10.3, 10.4, 10.5 and 10.6)
COMPONENTS
Vocabulary: acute angle, angle, angle of depression, angle of elevation, bases, center, circle, circumference, compass, complement, complementary angles,
corresponding angles, corresponding sides, degrees, diameter, hypotenuse, initial side, intersecting lines, inverse tangent, legs, line, obtuse angle, parallel lines,
perimeter, perpendicular lines, plane, point, polygon, polyhedron, protractor, pyramid, Pythagorean theorem, quadrilateral, radius, ray, rectangular solid, regular
polygon, right angle, right circular cone, right triangle, scale drawings, segment, side adjacent, side opposite, similar figures, sine, sphere, straight angle, supplement,
supplementary angles, surface area, tangent, terminal side, transversal, volume
Standards:
Assessable Specifications:
C.12.1 Identify, describe, and analyze properties of figures, relationships among figures, and
relationships among their parts by




constructing physical models
drawing precisely with paper-and-pencil, hand calculators, and computer
software
using appropriate transformations* (e.g., translations, rotations, reflections,
enlargements)
using reason and logic
C.12.3 Present convincing arguments by means of demonstration, informal proof, counterexamples, or any other logical means to show the truth of


statements (e.g., these two triangles are not congruent)
generalizations (e.g., the Pythagorean* theorem holds for all right triangles)
C.12.5 Identify and demonstrate an understanding of the three ratios used in right-triangle
trigonometry (sine, cosine, tangent)
D.12.2 Select and use tools with appropriate degree of precision to determine
measurements directly* within specified degrees of accuracy and error (tolerance)
D.12.3 Determine measurements indirectly*, using







estimation
proportional reasoning, including those involving squaring and cubing (e.g.,
reasoning that areas of circles are proportional to the squares of their radii)
techniques of algebra, geometry, and right triangle trigonometry
formulas in applications (e.g., for compound interest, distance formula)
geometric formulas to derive lengths, areas, or volumes of shapes and
objects (e.g., cones, parallelograms, cylinders, pyramids)
geometric relationships and properties of circles and polygons (e.g., size of
central angles, area of a sector of a circle)
conversion constants to relate measures in one system to another (e.g.,
meters to feet, dollars to Deutschmarks
MPS
Thinking
Levels
9.1 Students will be able to understand points, lines, rays,
segments and planes as the basis of geometry.
3/4
9.2 Students will be able to solve problems involving angle
measures.
3/4
9.3 Students will be able to solve problems involving
angles formed by parallel lines and transversals.
3/4
9.4 Students will be able to solve problems involving angle
relationships in triangles, similar triangles and the
Pythagorean Theorem.
9.5 Students will be able to classify polygons and
quadrilaterals by their characteristics.
9.6 Students will be able to solve problems involving
perimeter and the sums of the measurements of their
angles.
9.7 Students will be able to solve real world problems
using formulas to compute area, perimeter,
circumference, surface area and volume.
9.8 Students will be able to solve right triangle problems
using trigonometric ratios and/or parts of the triangle.
3/4
3/4
2
3/4
2
Instructional Time
Sequence, i.e. 1st
Semester, 2nd
Semester
2nd Semester
Key Idea 10: Counting Methods and Probability Theory. (Chapter 11; Sections 11.1, 11.2, 11.3, 11.4, and 11.5)
COMPONENTS
Vocabulary: combination, empirical probability, experiment, factorial, fundamental counting principle, permutation, theoretical probability, tree diagram
Standards:
Assessable Specifications:
B.12.1 Use complex counting procedures such as
union and intersection of sets and arrangements
(permutations* and combinations*) to solve problems
10.1 Students will be able to use the fundamental counting
principle to determine the number of possible outcomes
in a given situation.
MPS
Thinking
Levels
Instructional Time
Sequence, i.e. 1st
Semester, 2nd
Semester
2
2nd Semester
E.12.5 Determine the likelihood of occurrence of
complex events by




using a variety of strategies (e.g.,
combinations*) to identify possible
outcomes
conducting an experiment
designing and conducting simulations*
applying theoretical probability
10.2 Students will be able to solve problems involving
permutations and combinations
2
10.3 Students will be able to compute probability with
permutations and/or combinations.
2
10.4 Students will be able to compute theoretical and/or
empirical probability.
2
10.5 Students will be able to solve probability problems
using the fundamental counting principle, factorial
expressions, combinations, permutation formulas and
permutations of duplicate numbers.
3
Key Idea 11: Statistics. (Chapter 12; Sections 12.1 and 12.2)
COMPONENTS
Vocabulary: bimodal, classes, data, data item, data value, descriptive statistics, frequency, frequency distribution, frequency polygon, grouped frequency
distribution, histogram, inferential statistics, leaf, mean, measure of central tendency, median, midrange, mode, population, random sample, range, representative
sample, statistics, stem, stem-leaf plot, symbol of summation
Standards:
Assessable Specifications:
E.12.1 Work with data in the context of real-world
situations by




formulating hypotheses that lead to
collection and analysis of one- and twovariable data
designing a data collection plan that
considers random sampling, control
groups, the role of assumptions, etc.
conducting an investigation based on that
plan
using technology to generate displays,
summary statistics*, and presentations
E.12.2 Organize and display data from statistical
investigations using




11.1 Students will be able to identify appropriate populations
for real-world scenarios and select appropriate
sampling techniques for those populations.
MPS
Thinking
Levels
Instructional Time
Sequence, i.e. 1st
Semester, 2nd
Semester
3/4
2nd Semester
11.2 Students will be able to analyze and present sets of
data using appropriate graphical representation.
3
11.3 Students will be able to determine the mean, median,
mode and the midrange for a data set.
2
frequency distributions
percentiles*, quartiles, deciles
line of best fit* (estimated regression line)
matrices
E.12.3 Interpret and analyze information from
organized and displayed data when given



measures of dispersion*, including
standard deviation and variance
measures of reliability
measures of correlation*
E.12.4 Analyze, evaluate, and critique the methods
and conclusions of statistical experiments reported in
journals, magazines, news media, advertising, etc.
**All other Chapters and Sections may be used at the end or throughout the semesters if time permits. / ***Academic Standards were taken from Grade 12 Wisconsin State Standards.