The general goal in this discipline is for students to
learn the techniques of matrix manipulation
so that they can solve systems of linear equations in any number of variables.
Linear
algebra is most often combined with another subject, such as trigonometry ,
mathematical
analysis, or precalculus.
Note: The sample
problems illustrate
the standards and
are written to help
clarify them. Some
problems are written
in a form that can be
used directly with
students; others will
need to be modified
before they are
used with students. |
1.0 Students solve linear equations in any number of
variables by using
Gauss- Jordan elimination . |
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2.0 Students interpret linear systems as coefficient matrices and the
Gauss-Jordan
method as row operations on the coefficient matrix.
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3.0 Students reduce rectangular matrices to row echelon form.
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4.0 Students perform addition on matrices and vectors.
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5.0 Students perform matrix multiplication and multiply vectors by
matrices
and by scalars.
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6.0 Students demonstrate an understanding that linear systems are
inconsistent
(have no solutions), have exactly one solution, or have infinitely many
solutions.
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7.0 Students demonstrate an understanding of the geometric
interpretation of
vectors and vector addition (by means of paralle lograms ) in the plane
and in
three-dimensional space.
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8.0 Students interpret geometrically the solution sets of systems of
equations. For
example, the solution set of a single linear equation in two variables
is interpreted
as a line in the plane, and the solution set of a two-by-two system is
interpreted as the intersection of a pair of lines in the plane.
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9.0 Students demonstrate an understanding of the notion of the inverse
to a square
matrix and apply that concept to solve systems of linear equations.
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10.0 Students compute the de terminants of 2 × 2 and 3 × 3 matrices and
are
familiar with their geometric interpretations as the area and volume of
the
parallelepipeds spanned by the images under the matrices of the standard
basis vectors in two-dimensional and three-dimensional spaces.
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11.0 Students know that a square matrix is invertible if, and only if,
its determinant
is nonzero. They can compute the inverse to 2 × 2 and 3 × 3 matrices
using
row reduction methods or Cramer’s rule.
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12.0 Students compute the scalar (dot) product of two vectors in
n-dimensional
space and know that perpendicular vectors have zero dot product. |
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Probability
and Statistics Mathematics Content Standards
This discipline is an introduction to the study of
probability, interpretation of data,
and fundamental statistical problem solving. Mastery of this academic content
will
provide students with a solid foundation in probability and facility in
processing
statistical information.
Note: The sample
problems illustrate
the standards and
are written to help
clarify them. Some
problems are written
in a form that can be
used directly with
students; others will
need to be modified
before they are
used with students. |
1.0 Students know the definition of the notion of
independent events and can use
the rules for addition , multiplication, and complementation to solve for
probabilities of particular events in finite sample spaces. |
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2.0 Students know the definition of conditional probability and
use it to solve for
probabilities in finite sample spaces.
A whole number between 1 and 30 is chosen at random. If the digits of
the number that is chosen add up to 8, what is the probability that the
number is greater than 12?
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3.0 Students demonstrate an understanding of the notion of discrete
random
variables by using them to solve for the probabilities of outcomes,
such as
the probability of the occurrence of five heads in 14 coin tosses.
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4.0 Students are familiar with the standard distributions (normal,
binomial , and
exponential) and can use them to solve for events in problems in which
the
distribution belongs to those families.
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5.0 Students determine the mean and the standard deviation of a normally
distributed
random variable.
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6.0 Students know the definitions of the mean, median, and
mode of a distribution
of data and can compute each in particular situations.
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7.0 Students compute the variance and the standard deviation of a
distribution
of data.
Find the mean and standard deviation of the following seven numbers:
Make up another list of seven numbers with the
same mean and a smaller
standard deviation. Make up another list of seven numbers with the same
mean and a larger standard deviation. (ICAS 1997)
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8.0 Students organize and describe distributions of data by using a
number of
different methods , including frequency tables, histograms, standard line
and
bar graphs, stem-and-leaf displays, scatterplots, and box-and- whisker
plots . |
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