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SPRINGFIELD PUBLIC SCHOOLS
DISCRETE MATHEMATICS II Discrete Mathematics II is designed for students who are planning a career in
computer science, business, education, the biological sciences, the social
sciences, or liberal arts. Calculators, "hands-on" activities, computer technology,
and visual media will be used to explore, develop, and solve problems dealing with
statistics, sets, probability, informal logic, and patterns. Optional topics that may be
covered are matrices and fractal geometry. This course will encourage the
modeling of real-world situations through finite methods. Prerequisite: C- or better
in Algebra II or successful completion of Algebra II Honors.
MATHEMATICAL PRACTICES
1.
Make sense of problems and persevere in solving them.
2.
Reason abstractly and quantitatively.
3.
Construct viable arguments and critique the reasoning of others.
4.
Model with mathematics.
5.
Use appropriate tools strategically.
6.
Attend to precision.
7.
Look for and make use of structure.
8.
Look for and express regularity in repeated reasoning.
Algebra
Creating Equations.
A.CED.1 Create equations and inequalities in one variable and use them to
solve problems. Include equations arising from linear and quadratic
functions, and simple rational and exponential functions.
A.CED.2 Create equations in two or more variables to represent
relationships between quantities; graph equations on coordinate
axes with labels and scales.
Functions
Analyze functions using different representations.
F.IF.7
Graph functions expressed symbolically and show key features of
the graph, by hand in simple cases and using technology for more
complicated cases.
a.
Graph linear and quadratic functions and show intercepts,
maxima, and minima.
b.
Graph exponential and logarithmic functions, showing
intercepts and end behavior, and trigonometric functions,
showing period, midline, and amplitude.
F.IF.8
Write a function defined by an expression in different but
equivalent forms to reveal and explain different properties of the
function.
a.
Use the properties of exponents to interpret expressions for
exponential functions. For example, identify percent rate of
change in functions such as y = (1.02)t, y = (0.97)t, y =
(1.01)12t, y = (1.2)t/10, and classify them as representing
exponential growth or decay.
F.IF.9
Compare properties of two functions each represented in a
different way (algebraically, graphically, numerically in tables, or by
verbal descriptions). For example, given a graph of one quadratic
function and an algebraic expression for another, say which has
the larger maximum.
Build a function that models a relationship between two quantities.
F.BF.1
Write a function that describes a relationship between two
quantities.
a.
Determine an explicit expression, a recursive process, or
steps for calculation from a context.
F.BF.2
Write arithmetic and geometric sequences both recursively and
with an explicit formula, use them to model situations, and translate
between the two forms.
Statistics and Probability
Summarize, represent, and interpret data on a single count or
measurement variable.
S.ID.1
Represent data with plots on the real number line (dot plots,
histograms, and box plots).
S.ID.2
Use statistics appropriate to the shape of the data distribution to
compare center (median, mean) and spread (interquartile range,
standard deviation) of two or more different data sets.
S.ID.3
Interpret differences in shape, center, and spread in the context of
the data sets, accounting for possible effects of extreme data
points (outliers).
Summarize, represent, and interpret data on two categorical and
quantitative variables.
S.ID.5
Summarize categorical data for two categories in two-way
frequency tables. Interpret relative frequencies in the context of the
data (including joint, marginal, and conditional relative
frequencies). Recognize possible associations and trends in the
data.
S.ID.6
Represent data on two quantitative variables on a scatter plot, and
describe how the variables are related.
a.
Fit a function to the data; use functions fitted to data to solve
problems in the context of the data. Use given functions or
choose a function suggested by the context. Emphasize
linear, quadratic, and exponential models.
b.
Fit a linear function for a scatter plot that suggests a linear
association.
Interpret linear models.
S.ID.7
Interpret the slope (rate of change) and the intercept (constant
term) of a linear model in the context of the data.
S.ID.8
Compute (using technology) and interpret the correlation
coefficient of a linear fit.
S.ID.9
Distinguish between correlation and causation.
SPRINGFIELD PUBLIC SCHOOLS
DISCRETE MATHEMATICS II Making Inferences and Justifying Conclusions
Understand and evaluate random processes underlying statistical
experiments.
S.IC.1
Understand statistics as a process for making inferences about
population parameters based on a random sample from that
population.
S.IC.2
Decide if a specified model is consistent with results from a
given data-generating process, e.g., using simulation. For
example, a model says a spinning coin falls heads up with
probability 0.5. Would a result of 5 tails in a row cause you to
question the model?
Make inferences and justify conclusions from sample surveys,
experiments, and observational studies.
S.IC.3
Recognize the purposes of and differences among sample
surveys, experiments, and observational studies; explain how
randomization relates to each.
S.IC.SPS.4 Apply the concepts of set theory, including Venn Diagrams, to
model relationships and solve problems.
S.IC.SPS.5 Apply the principles of logic and truth tables to critique
arguments and establish validity of conclusions.
Conditional Probability and Rules of Probability
Understand independence and conditional probability and use them to
interpret data.
S.CP.1
Describe events as subsets of a sample space (the set of
outcomes) using characteristics (or categories) of the outcomes,
or as unions, intersections, or complements of other events
(“or,” “and,” “not”).
S.CP.2
Understand that two events A and B are independent if the
probability of A and B occurring together is the product of their
probabilities, and use this characterization to determine if they
are independent.
Use the rules of probability to compute probabilities of compound
events in a uniform probability model.
S.CP.9
(+) Use permutations and combinations to compute probabilities
of compound events and solve problems.
Using Probability to Make Decisions
Use probability to evaluate outcomes of decisions.
S.MD.6
(+) Use probabilities to make fair decisions (e.g., drawing by
lots, using a random number generator).
S.MD.7
(+) Analyze decisions and strategies using probability concepts
(e.g., product testing, medical testing, pulling a hockey goalie at
the end of a game).