Stage D - 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)
- Represent, order, and compare decimals to demonstrate
understanding of the place-value structure in the base-ten
number system **
- Identify prime numbers through 100.
- Recognize equivalent representations for decimals and
generate them by composing and decomposing numbers (e.g.,
0.15 = 0.1 +0.005 )
- Represent fractions as parts of unit wholes, as parts
of a set, as locations on a number line, and as divisions
of whole numbers **
- Explore numbers less than zero by extending a number line
and through familiar applications. *
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)
- Describe classes of numbers according to characteristics
such as factors and multiples.
- Solve addition or subtraction number sentences and word
problems using fractions with like denominators.
- Solve multi-step number sentences and word problems using
whole numbers and the four basic operations.
- Select and use one of various algorithms to multiply and
divide.
6C - Students who meet the standard can compute and estimate
using mental mathematics, paper-and-pencil methods, calculators,
and computers. (Choice of method)
- Develop and use strategies (e.g., compatible numbers,
front-end estimation) to estimate the results of wholenumber
computations and to judge the reasonableness of such results.
**
- Estimate the sum or difference of a number sentence containing
decimals using a variety of strategies.
6D - Students who meet the standard can solve problems
using comparison of quantities, ratios, proportions, and percents.
- Determine 50% and 100% of a given group in context.
7A - Students who meet the standard can measure and compare
quantities using appropriate units, instruments, and methods.
(Performance and conversion of measurements)
- Measure angles using a protractor or angle ruler.
- Measure with a greater degree of accuracy.
- Convert U.S. customary measurements into larger or smaller
units with the help of conversion charts.
- Convert linear metric measurements into larger or smaller
units with the help of a conversion chart.
7B - Students who meet the standard can estimate measurements
and determine acceptable levels of accuracy. (Estimation)
- Develop and discuss strategies for estimating the perimeters,
areas, and volumes of regular and nonregular shapes. **
- Develop and use common referents for volume, weight/mass,
capacity, area, and angle measures to make comparisons and
estimates.
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)
- Select and apply appropriate standard units and tools
to measure the size of angles.**
- Determine the volume of a cube or rectangular prism using
concrete materials.
- Create an accurate representation of a polygon with a
given perimeter or area.
8A - Students who meet the standard can describe numerical
relationships using variables and patterns. (Representations
and algebraic manipulations)
- Identify a number pattern, both increasing and decreasing,
and extend the number sequence.
- Determine the missing number(s) in a complex repeating
pattern.
- Construct and solve simple number sentences using a symbol
for a variable.
- Make generalizations given a specific pattern.
- Create, describe, and extend patterns.
- Describe a pattern with one operation, verbally and symbolically,
given a table of input/output numbers.
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)
- Create a table that describes a function rule for a single
operation.
- Demonstrate, in simple situations, how a change in one
quantity results in a change in another quantity (e.g.,
increase the measure of the side of a square and the perimeter
increases).
- Identify situations with varying rates of change using
words, tables, and graphs e.g., growth of a plant). **
8C - Students who meet the standard can solve problems
using systems of numbers and their properties. (Problem
solving; number systems, systems of equations, inequalities,
algebraic functions)
- Solve problems with whole numbers using appropriate field
properties.
8D - Students who meet the standard can use algebraic
concepts and procedures to represent and solve problems. (Connection
of 8A, 8B, and 8C to solve problems)
- Solve one-step linear equations with one missing value
in isolation and in problem solving situations.
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)
- Identify, draw, and label lines, line segments, rays,
parallel lines, intersecting lines, perpendicular lines,
acute angles, obtuse angles, right angles, and acute, obtuse,
right, scalene, isosceles, and equilateral triangles.
- Identify, draw, and build regular, irregular, convex,
and concave polygons.
- Read and plot ordered pairs of numbers in the positive
quadrant of the Cartesian plane.
- Describe paths and movement using coordinate systems.
- Differentiate between polygons and non-polygons.
- Identify and label radius, diameter, chord, and circumference
of a circle.
- Explore and describe rotational symmetry of two- and three-dimensional
shapes.**
- Construct a circle with a specified radius or diameter
using a compass.
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)
- Determine congruence and similarity of given shapes. **
- Explore polyhedra using concrete models.
9C - Students who meet the standard can construct convincing
arguments and proofs to solve problems. (Justifications of
conjectures and conclusions)
- Make and test conjectures about mathematical properties
and relationships and justify the conclusions. **
10A - Students who meet the standard can organize, describe
and make predictions from existing data. (Data analysis)
- Represent data using tables and graphs such as line plots
and line graphs. **
- Describe the shape and important features of a set of
data and compare related data sets. **
- Arrange given data in order, least to greatest or greatest
to least, and determine minimum value, maximum value, range,
mode, and median for an odd number of data points.
- Compare different representations of the same data and
evaluate how well each representation shows important aspects
of the data. *
- Propose and justify conclusions and predictions that are
based on data. **
10B - Students who meet the standard can formulate questions,
design data collection methods, gather and analyze data and
communicate findings. (Data Collection)
- Collect data using observations and experiments. **
- Propose a further investigation to verify or refute a
prediction. **
10C - Students who meet the standard can determine, describe
and apply the probabilities of events. (Probability including
counting techniques)
- List all possible outcomes of a single event and tell
whether an outcome is certain, impossible, likely, or unlikely.
- Describe the probability of an event using terminology
such as “5 chances out of 8."
* 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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