Measurement and Geometry
1. Students demonstrate understanding of the structure of standard
measurement systems, use derived units, unit conversions, dimensional analysis,
with appropriate accuracy.
1.1 use quotient measures (e.g., speed, density) relate them to slope
and "per unit" amounts, and product measures (e.g., person-days)
1.2 use dimensional analysis to check answers and determine the units
of a solution
1.3 use the concept of significant digits in giving answers to an appropriate
degree of accuracy
2. Students select and use appropriate units, tools, and degrees
of accuracy to solve problems involving geometric and non-geometric measures.
2.1 know, use, derive formulas for, and solve problems involving perimeter,
circumference, area, volume, lateral area, and surface area of common geometric
figures
2.2 describe how changes in the dimensions of an object affect the perimeter,
area, and volume (e.g., tripling the radius of a sphere multiplies its
volume by 27)
2.3 know, use, derive formulas for, and solve problems involving weight,
monetary, and time systems selecting appropriate units and degrees of accuracy
3. Students identify, find missing measures, and solve problems
involving angles, right triangles, other polygons, circles, planes and solid
geometric objects.
3.1 find and use measures of sides, interior and exterior angles of
triangles and polygons to classify figures (e.g., isosceles, obtuse, convex,
regular) and solve problems (e.g., determine the number of degrees in a
central angle of a regular polygon).
3.2 describe the relationships between vertical angles, angles that
are supplementary and complementary, and angles formed when parallel lines
are cut by a transversal express and use these to find missing angle measures
in such systems
3.3 use the Pythagorean Theorem, its converse, properties of special
right triangles (e.g., sides in the ratio 3-4-5, angles of 30-60-90 degrees),
and right triangle trigonometry to find missing information about triangles
3.4 develop and implement a plan for obtaining indirect measures using
similarity, proportional reasoning and trigonometric ratios
3.5 compare, contrast, classify, and solve problems involving quadrilaterals
(square, rhombus, rectangle, parallelogram, trapezoid, kite, cyclic) on
the basis of their definitions and properties (e.g., opposite sides, consecutive
angles, diagonals)
3.6 apply the properties of angles, arcs, chords, radii, tangents, and
secants to solve problems involving circles
3.7 apply the triangle inequality properties (given information concerning
the lengths of sides and/or measures of angles) to determine whether a
triangle exists and to order sides and angles
3.8 identify the structural parts (e.g., angles, shape of sides, orientation
of sides, circumference) and characteristics (e.g., symmetry, shape of
cross-sections) and use these to classify objects and answer questions
about them (e.g., a penny can be seen as a cylinder with small height so
its volume is V= [pi] r^2h).
4. Students demonstrate understanding of an axiomatic system,
and the nature of proof.
4.1 identify and give examples of undefined terms, axioms, theorems,
inductive, and deductive reasoning
4.2 construct and judge the validity of a logical argument including
giving counterexamples and understanding quantifiers
5. Students identify, use, and prove relationships between figures
involving congruence and similarity.
5.1 prove the Pythagorean Theorem using algebraic and geometric arguments
as well as deductive proof.
5.2 identify similar and congruent triangles and other polygons and
their corresponding parts
5.3 prove basic theorems about congruent triangles and other polygons
using deductive, algebraic, and transformational arguments
6. Students visualize and describe objects, paths, and regions
in space.
6.1 use geometric language to describe the intersection of two planes
or a plane and a solid object (the cross section it cuts)
6.2 use geometric language to describe how the conic sections are derived
as cross sections of a cone
6.3 construct a 3-dimensional figure from a 2-dimensional drawing and
a 2-dimensional representation of a 3-dimensional object (e.g., nets, prisms,
cones, pyramids)
Functions
1. Students give information about and perform algebraic operations
on standard (quadratic, exponential, rational, radical, absolute value,
and factorable polynomial) functions.
1.1 give the domain, range, zeros, and express the inverse (for 1-to-1
functions) of a standard function
1.2 represent a standard function as a table of values, an equation,
or a graph and translate among these representations.
1.3 determine the composition of two standard functions including necessary
restrictions on the domain
1.4 demonstrate and explain the effect that changing a parameter has
on the graph of a function (e.g., describe the effect on the graph of y = x^2 + 6x +
C as C grows from -5 to 5)
2. Students demonstrate an understanding of arithmetic and geometric
sequences and series.
2.1 identify, describe, extend, and find the nth term of arithmetic
and geometric sequences
2.2 find the sum of finite arithmetic and geometric sequences and infinite
geometric sequences with |r| < 1, use and interpret [sigma] notation
2.3 relate arithmetic and geometric sequences to linear and exponential
functions, and express them in explicit and recursive form
3. Students demonstrate understanding and use of basic types of
functions (linear, exponential, polynomial, rational, radical, absolute
value, step, and piecewise defined).
3.1 identify a function given as a table of values and equation or a
graph as linear, exponential, polynomial, rational, radical (square root
or cube root), absolute value, step, or piecewise and explain key characteristics
of the function.
3.2 determine which type of function best models a situation, write
an equation using technology where appropriate, and use this equation to
answer questions about the situation.
3.3 find and interpret the maximum or minimum value of a quadratic function
3.4 find the value of a function for a given element in its domain and
determine the natural domain of a given function
Statistics, Data Analysis, and Probability
1. Students collect, display, analyze, and interpret single and
bi-variate data and critique the conclusions and uses of statistics in both
school materials and public documents
1.1 create, analyze and, interpret data using graphs (frequency distributions,
histograms, control charts, scatter plots) and analysis tools (estimated
and computer generated regression lines, correlation coefficients, and
residuals.)
1.2 apply curve fitting techniques to data and assess the "goodness
of fit" of a regression line or curve and its usefulness as a model
for the data and make predictions by interpolating or extrapolating from
the data or its graph
1.3 explain and critique the process of a survey or experiment, how
that might have contributed to or influenced the results (e.g., reliability
of sampling procedures, bias, missing or incorrect information), and describe
misuses of statistical or numerical data
2. Students estimate relative frequency, compute probability,
and demonstrate understanding of ways to make predictions from samples and
experiments in situations involving uncertainty including dependent and
conditional events.
2.1 choose an appropriate probability model and use it to arrive at
a theoretical probability for a simple, compound, or conditional chance
event
2.2 graph and interpret probability distributions including the binomial
distributions and use them to discuss whether an event is rare or reasonably
likely
2.3 demonstrate understanding of a random variable and how it can be
used to make predictions about a population from a sample
Credits
The Draft Standards were prepared by:
The State of California Academic Standards Commission
The Commission for the Establishment of Academic Content and Performance Standards
Comments may be addressed to The Commission
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