Stage I - Mathematics
Descriptors
6A - Students who meet the standard can demonstrate knowledge
and use of numbers and their many representations in a broad
range of theoretical and practical settings. (Representations)
- Illustrate the relationship between second and third roots
and powers of a number.
- Organize problem situations using matrices.
- Represent, order, and compare real numbers.
- Place real numbers on a number line.
6B - Students who meet the standard can investigate, represent,
and solve problems using number facts, operations and their
properties, algorithms, and relationships. (Operations
and properties)
- Compare and contrast the properties of numbers and number
systems, including the rational and the real numbers. **
- Determine an appropriate numerical representation of a
problem situation, including roots and powers, if applicable.
- Judge the effects of such operations as multiplication,
division, and computing powers and roots on the magnitudes
of quantities. *
- Solve problems using simple matrix operations (addition,
subtraction, scalar multiplication).
- Develop fluency in operations with real numbers using
mental computation or paper-and-pencil calculations for
simple cases and technology for more-complicated cases.
**
- Judge the reasonableness of numerical computations and
their results. *
6C - Students who meet the standard can compute and estimate
using mental mathematics, paper-and-pencil methods, calculators,
and computers. (Choice of method)
- Develop fluency in operations with real numbers and matrices
using mental computation or paper-and-pencil calculations
for simple cases and technology for more-complicated cases.
**
- Determine and explain whether exact values or approximations
are needed in a variety of situations.
- Determine an appropriate number of digits to represent
an outcome.
6D - Students who meet the standard can solve problems
using comparison of quantities, ratios, proportions, and percents.
- Explain how ratios and proportions can be used to solve
problems of percent, growth, and error tolerance.
- Set up and solve proportions for direct and inverse variation
of simple quantities.
7A - Students who meet the standard can measure and compare
quantities using appropriate units, instruments, and methods.
(Performance and conversion of measurements)
- Select units and scales that are appropriate for problem
situations involving measurement. **
- Convert between the U.S. customary and metric systems
given the conversion factor.
7B - Students who meet the standard can estimate measurements
and determine acceptable levels of accuracy. (Estimation)
- Estimate the magnitude and directions of physical quantities
(e.g., velocity, force, slope).
- Determine answers to an appropriate degree of accuracy
using significant digits.
7C - Students who meet the standard can select and use
appropriate technology, instruments, and formulas to solve
problems, interpret results, and communicate findings. (Progression
from selection of appropriate tools and methods to application
of measurements to solve problems)
- Solve problems using indirect measurement by choosing
appropriate technology, instruments, and/or formulas.
- Check measurement computations using unit analysis. **
- Describe the general trends of how the change in one measure
affects other measures in the same figure (e.g., length,
area, volume).
- Determine linear measures, perimeters, areas, surface
areas, and volumes of similar figures using the ratio of
similitude.
- Determine the ratio of similar figure perimeters, areas,
and volumes using the ratio of similitude.
- Calculate by an appropriate method the length, width,
height, perimeter, area, volume, surface area, angle measures,
or sums of angle measures of common geometric figures, or
combinations of common geometric figures.
- Solve problems involving multiple rates, measures, and
conversions.
8A - Students who meet the standard can describe numerical
relationships using variables and patterns. (Representations
and algebraic manipulations)
- Write equivalent forms of equations, inequalities, and
systems of equations. **
- Represent and explain mathematical relationships using
symbolic algebra. **
- Model and describe slope as a constant rate of change.
- Explain the difference between constant and non-constant
rate of change.
- Create an equation of a line of best fit from a set of
ordered pairs or set of data points.
- Simplify algebraic expressions using a variety of methods,
including factoring.
- Justify the results of symbol manipulations, including
those carried out by technology. **
- Identify essential quantitative relationships in a situation
and determine the class or classes of functions (e.g., linear,
quadratic) that might model the relationships. **
- Represent relationships arising from various contexts
using algebraic expression.
- Rewrite absolute value inequalities in terms of two separate
equivalent inequalities with the appropriate connecting
phrase of "AND" or "OR".
8B - Students who meet the standard can interpret and
describe numerical relationships using tables, graphs, and
symbols. (Connections of representations including the
rate of change)
- Describe the relationships of the independent and dependent
variables from a graph.
- Interpret the role of the coefficients and constants on
the graph of linear and quadratic functions given a set
of equations.
- Relate the effect of translations on linear graphs and
equations.
- Create and connect representations that are tabular, graphical,
numeric, and algebraic from a set of data.
- Recognize and describe the general shape and properties
of the graphs of linear, absolute value, and quadratic functions.
- Approximate and interpret rates of change from graphical
and numerical data. *
- Identify slope in an equation and from a table of values.
- Graph absolute values of linear functions on the Cartesian
plane.
- Recognize direct variation, inverse variation, linear,
and exponential curves from their graphs, a table of values,
or equations. **
- Interpret and use functions as a geometric representation
of linear and non-linear relationships.
8C - Students who meet the standard can solve problems
using systems of numbers and their properties. (Problem
solving)
- Describe and compare the properties of linear and quadratic
functions. **
- Solve problems by recognizing how an equation changes
when parameters change.
- Interpolate and extrapolate to solve problems using systems
of numbers.
- Solve problems using translations and dilations on basic
functions.
8D - Students who meet the standard can use algebraic
concepts and procedures to represent and solve problems. (Connection
of 8A, 8B, 8C to solve problems)
- Solve equivalent forms of equations, inequalities, and
systems of equations with fluency-mentally or with paper-and-pencil
in simple cases and using technology in all cases. **
- Create word problems that meet given conditions and represent
simple power or exponential relationships, or direct or
inverse variation situations.
- Solve simple quadratic equations using algebraic or graphical
representations.
- Solve problems of direct variation situations using a
variety of methods.
9A - Students who meet the standard can demonstrate and
apply geometric concepts involving points, lines, planes,
and space. (Properties of single figures, coordinate geometry
and constructions)
- Describe and apply properties of a polygon or a circle
in a problem-solving situation.
- Classify angle relationships for two or more parallel
lines crossed by a transversal.
- Analyze geometric situations using Cartesian coordinates.
**
- Represent transformations of an object in the plane using
sketches, coordinates, and vectors.
- Design a net that will create a given figure when folded.
- Solve problems using constructions.
- Gain insights into, and answer questions in, other areas
of mathematics using geometric models. **
- Calculate distance, midpoint coordinates, and slope using
coordinate geometry.
- Visualize a three-dimensional object from different perspectives
and describe their cross sections. **
- Identify and apply properties of medians, altitudes, angle
bisectors, perpendicular bisectors, and midlines of a triangle.
9B - Students who meet the standard can identify, describe,
classify and compare relationships using points, lines, planes,
and solids. (Connections between and among multiple geometric
figures)
- Solve problems using triangle congruence and similarity
of figures.
- Extend knowledge of plane figure relationships to relationships
within and between geometric solids.
- Identify relationships among circles, arcs, chords, tangents,
and secants.
- Solve problems in, and gain insights into, other disciplines
and other areas of interest such as art and architecture
using geometric ideas. **
- Analyze and describe the transformations that lead to
successful tessellations of one or more figures.
9C - Students who meet the standard can construct convincing
arguments and proofs to solve problems. (Justifications
of conjectures and conclusions)
- Create and critique arguments concerning geometric ideas
and relationships such as properties of circles, triangles
and quadrilaterals.
- Develop a formal proof for a given geometric situation
on the plane.
- Provide a counter-example to disprove a conjecture.
- Develop conjectures about geometric situations with and
without technology.
- Justify constructions using geometric properties.
- Describe the difference between an inductive argument
and a deductive argument.
9D - Students who meet the standard can use trigonometric
ratios and circular functions to solve problems.
- Determine distances and angle measures using indirect
measurement and properties of right triangles.
- Solve problems using 45o-45o-90o and 30o-60o-90o triangles.
10A - Students who meet the standard can organize, describe
and make predictions from existing data. (Data Analysis)
- Describe the meaning of measurement data and categorical
data, of univariate and bivariate data, and of the term
variable. **
- Display a scatter plot, describe its shape, and determine
regression coefficients, regression equations, and correlation
coefficients for bivariate measurement data using technological
tools.
- Evaluate published reports that are based on data by examining
the design of the study, the appropriateness of the data
analysis, and the validity of conclusions. *
- Analyze two-variable data for linear or quadratic fit.
- Make decisions based on data, including the relationships
of correlation and causation.
10B - Students who meet the standard can formulate questions,
design data collection methods, gather and analyze data, and
communicate findings. (Data Collection)
- Describe the characteristics of well-designed studies,
including the role of randomization in surveys and experiments.
**
- Discuss informally different populations and sampling
techniques.
- Decide if a survey was "successful" in gathering
intended data and justify the decision.
10C - Students who meet the standard can determine, describe
and apply the probabilities of events. (Probability, including
counting techniques)
- Determine geometric probability based on area.
- Calculate probability using Venn diagrams.
- Determine simple probabilities using frequency tables.
- Construct empirical probability distributions using simulations.
**
- Describe the concepts of conditional probability.
- Develop an understanding of permutations and combinations
as counting techniques. *
* National Council of Teachers of Mathematics. Principles
and Standards for School Mathematics. Reston, Va: National Council
of Teachers of Mathematics, 2000.
** Adapted from: National Council of Teachers of Mathematics.
Principles and Standards for School Mathematics. Reston, Va:
National Council of Teachers of Mathematics, 2000.
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