The elementary school standards
are set at a level of performance approximately equivalent to the
end of fourth grade. The middle school standards are set at a level
of performance approximately equivalent to the end of eighth grade.
The high school standards are set at a level of performance approximately
equivalent to the end of tenth grade or the end of the common core.
It is expected that some students might achieve these levels earlier
and others later than these grades. 
Arithmetic
and Number Concepts/Number and Operation Concepts 
Elementary School
The student produces evidence that demonstrates understanding of
arithmetic and number concepts; that is, the student:
a
Adds, subtracts, multiplies, and divides whole numbers, with and
without calculators; that is:
• adds, i.e., joins things together, increases;
• subtracts, i.e., takes away, compares, finds the difference;
• multiplies, i.e., uses repeated addition, counts by multiples,
combines things that come in groups, makes arrays, uses area models,
computes simple scales, uses simple rates;
• divides, i.e., puts things into groups, shares equally; calculates
simple rates;
• analyzes problem situations and contexts in order to figure
out when to add, subtract, multiply, or divide;
• solves arithmetic problems by relating addition, subtraction,
multiplication, and division to one another;
• computes answers mentally, e.g., 27 + 45, 30 x 4;
• uses simple concepts of negative numbers, e.g., on a number
line, in counting, in temperature, “owing.”
b
Demonstrates understanding of the base ten place value system and
uses this knowledge to solve arithmetic tasks; that is:
• counts 1, 10, 100, or 1,000 more than or less than, e.g.,
1 less than 10,000, 10 more than 380, 1,000 more than 23,000, 100
less than 9,000;
• uses knowledge about ones, tens, hundreds, and thousands
to figure out answers to multiplication and division tasks, e.g.,
36 x 10, 18 x 100, 7 x 1,000, 4,000 ÷ 4.
c
Estimates, approximates, rounds off, uses landmark numbers, or uses
exact numbers, as appropriate, in calculations.
d
Describes and compares quantities by using concrete and real world
models of simple fractions; that is:
• finds simple parts of wholes;
• recognizes simple fractions as instructions to divide, e.g.,
¼ of something is the same as dividing something by 4;
• recognizes the place of fractions on number lines, e.g.,
in measurement;
• uses drawings, diagrams, or models to show what the numerator
and denominator mean, including when adding like fractions, e.g.,
1/8 + 5/8,
or when showing that 3/4
is more than 3/8;
• uses beginning proportional reasoning and simple ratios,
e.g., “about half of the people.”
e
Describes and compares quantities by using simple decimals; that
is:
• adds, subtracts, multiplies, and divides money amounts;
• recognizes relationships among simple fractions, decimals,
and percents, i.e., that
is the same as 0.5, and
is the same as 50%, with concrete materials, diagrams, and in real
world situations, e.g., when discovering the chance of a coin landing
on heads or tails.
f
Describes and compares quantities by using whole numbers up to 10,000;
that is:
• connects ideas of quantities to the real world, e.g., how
many people fit in the school’s cafeteria; how far away is
a kilometer;
• finds, identifies, and sorts numbers by their properties,
e.g., odd, even, multiple, square.

Middle School
The student produces evidence that demonstrates understanding of
number and operation concepts; that is, the student:
a
Consistently and accurately adds, subtracts, multiplies, and divides
rational numbers using appropriate methods (e.g., the student can
add
+
mentally or on paper but may opt to add
+
on a calculator) and raises rational numbers to whole number powers.
(Students should have facility with the different kinds and forms
of rational numbers, i.e., integers, both whole numbers and negative
integers; and other positive and negative rationals, written as
decimals, as percents, or as proper, improper, or mixed fractions.
Irrational numbers, i.e., those that cannot be written as a ratio
of two integers, are not required content but are suitable for introduction,
especially since the student should be familiar with the irrational
number .)
b
Uses and understands the inverse relationships between addition
and subtraction, multiplication and division, and exponentiation
and rootextraction (e.g., squares and square roots, cubes and cube
roots); uses the inverse operation to determine unknown quantities
in equations.
c
Consistently and accurately applies and converts the different kinds
and forms of rational numbers.
d
Is familiar with characteristics of numbers (e.g., divisibility,
prime factorization) and with properties of operations (e.g., commutativity
and associativity), short of formal statements.
e
Interprets percent as part of 100 and as a means of comparing quantities
of different sizes or changing sizes.
f
Uses ratios and rates to express “parttopart” and “wholetowhole”
relationships, and reasons proportionally to solve problems involving
equivalent fractions, equal ratios, or constant rates, recognizing
the multiplicative nature of these problems in the constant factor
of change.
g
Orders numbers with the > and < relationships and by location
on a number line; estimates and compares rational numbers using
sense of the magnitudes and relative magnitudes of numbers and of
baseten place values (e.g., recognizes relationships to “benchmark”
numbers 1/2 and 1 to
conclude that the sum 1/2
+ 5/6 must be between
1 and 1 1/2 (likewise,
13/24 + ¼)).

High School
The student produces evidence that demonstrates understanding of
number and operation concepts; that is, the student:
a
Uses addition, subtraction, multiplication, division, exponentiation,
and rootextraction in forming and working with numerical and algebraic
expressions.
b
Understands and uses operations such as opposite, reciprocal, raising
to a power, taking a root, and taking a logarithm.
c
Has facility with the mechanics of operations as well as understanding
of their typical meaning and uses in applications.
d
Understands and uses number systems: natural, integer, rational,
and real.
e
Represents numbers in decimal or fraction form and in scientific
notation, and graphs numbers on the number line and number pairs
in the coordinate plane.
f
Compares numbers using order relations, differences, ratios, proportions,
percents, and proportional change.
g
Carries out proportional reasoning in cases involving partwhole
relationships and in cases involving expansions and contractions.
h
Understands dimensionless numbers, such as proportions, percents,
and multiplicative factors, as well as numbers with specific units
of measure, such as numbers with length, time, and rate units.
i
Carries out counting procedures such as those involving sets (unions
and intersections) and arrangements (permutations and combinations).
j
Uses concepts such as prime, relatively prime, factor, divisor,
multiple, and divisibility in solving problems involving integers.
k
Uses a scientific calculator effectively and efficiently in carrying
out complex calculations.
l
Recognizes and represents basic number patterns, such as patterns
involving multiples, squares, or cubes.

Geometry and Measurement Concepts 
Elementary School
The student produces evidence that demonstrates understanding of
geometry and measurement concepts; that is, the student:
a
Gives and responds to directions about location, e.g., by using
words such as “in front of,” “right,” and “above.”
b
Visualizes and represents two dimensional views of simple rectangular
three dimensional shapes, e.g., by showing the front view and side
view of a building made of cubes.
c
Uses simple two dimensional coordinate systems to find locations
on a map and to represent points and simple figures.
d
Uses many types of figures (angles, triangles, squares, rectangles,
rhombi, parallelograms, quadrilaterals, polygons, prisms, pyramids,
cubes, circles, and spheres) and identifies the figures by their
properties, e.g., symmetry, number of faces, two or threedimensionality,
no right angles.
e
Solves problems by showing relationships between and among figures,
e.g., using congruence and similarity, and using transformations
including flips, slides, and rotations.
f
Extends and creates geometric patterns using concrete and pictorial
models.
g
Uses basic ways of estimating and measuring the size of figures
and objects in the real world, including length, width, perimeter,
and area.
h
Uses models to reason about the relationship between the perimeter
and area of rectangles in simple situations.
i
Selects and uses units, both formal and informal as appropriate,
for estimating and measuring quantities such as weight, length,
area, volume, and time.
j
Carries out simple unit conversions, such as between cm and m, and
between hours and minutes.
k
Uses scales in maps, and uses, measures, and creates scales for
rectangular scale drawings based on work with concrete models and
graph paper.

Middle School
The student produces evidence that demonstrates understanding of
geometry and measurement concepts in the following areas; that is,
the student:
a
Is familiar with assorted two and threedimensional objects, including
squares, triangles, other polygons, circles, cubes, rectangular
prisms, pyramids, spheres, and cylinders.
b
Identifies similar and congruent shapes and uses transformations
in the coordinate plane, i.e., translations, rotations, and reflections.
c
Identifies three dimensional shapes from two dimensional perspectives;
draws two dimensional sketches of three dimensional objects that
preserve significant features.
d
Determines and understands length, area, and volume (as well as
the differences among these measurements), including perimeter and
surface area; uses units, square units, and cubic units of measure
correctly; computes areas of rectangles, triangles, and circles;
computes volumes of prisms.
e
Recognizes similarity and rotational and bilateral symmetry in two
and threedimensional figures.
f
Analyzes and generalizes geometric patterns, such as tessellations
and sequences of shapes.
g
Measures angles, weights, capacities, times, and temperatures using
appropriate units.
h
Chooses appropriate units of measure and converts with ease between
like units, e.g., inches and miles, within a customary or metric
system. (Conversions between customary and metric are not required.)
i
Reasons proportionally in situations with similar figures.
j
Reasons proportionally with measurements to interpret maps and to
make smaller and larger scale drawings.
k
Models situations geometrically to formulate and solve problems.

High School
The student produces evidence that demonstrates understanding of
geometry and measurement concepts; that is, the student:
a
Models situations geometrically to formulate and solve problems.
b
Works with two and threedimensional figures and their properties,
including polygons and circles, cubes and pyramids, and cylinders,
cones, and spheres.
c
Uses congruence and similarity in describing relationships between
figures.
d
Visualizes objects, paths, and regions in space, including intersections
and cross sections of three dimensional figures, and describes these
using geometric language.
e
Knows, uses, and derives formulas for perimeter, circumference,
area, surface area, and volume of many types of figures.
f
Uses the Pythagorean Theorem in many types of situations, and works
through more than one proof of this theorem.
g
Works with similar triangles, and extends the ideas to include simple
uses of the three basic trigonometric functions.
h
Analyzes figures in terms of their symmetries using, for example,
concepts of reflection, rotation, and translation.
i
Compares slope (rise over run) and angle of elevation as measures
of steepness.
j
Investigates geometric patterns, including sequences of growing
shapes.
k
Works with geometric measures of length, area, volume, and angle;
and nongeometric measures such as weight and time.
l
Uses quotient measures, such as speed and density, that give “per
unit” amounts; and uses product measures, such as personhours.
m
Understands the structure of standard measurement systems, both
SI and customary, including unit conversions and dimensional analysis.
n
Solves problems involving scale, such as in maps and diagrams.
o
Represents geometric curves and graphs of functions in standard
coordinate systems.
p
Analyzes geometric figures and proves simple things about them using
deductive methods.
q
Explores geometry using computer programs such as CAD software,
Sketchpad programs, or LOGO.

Function and Algebra Concepts 
Elementary School
The student produces evidence that demonstrates understanding of
function and algebra concepts; that is, the student:
a
Uses linear patterns to solve problems; that is:
• shows how one quantity determines another in a linear (“repeating”)
pattern, i.e., describes, extends, and recognizes the linear pattern
by its rule, such as, the total number of legs on a given number
of horses can be calculated by counting by fours;
• shows how one quantity determines another quantity in a functional
relationship based on a linear pattern, e.g., for the “number
of people and total number of eyes,” figure out how many eyes
100 people have all together.
b
Builds iterations of simple nonlinear patterns, including multiplicative
and squaring patterns (e.g., “growing” patterns) with
concrete materials, and recognizes that these patterns are not linear.
c
Uses the understanding that an equality relationship between two
quantities remains the same as long as the same change is made to
both quantities.
d
Uses letters, boxes, or other symbols to stand for any number, measured
quantity, or object in simple situations with concrete materials,
i.e., demonstrates understanding and use of a beginning concept
of a variable.

Middle School
The student produces evidence that demonstrates understanding of
function and algebra concepts; that is, the student:
a
Discovers, describes, and generalizes patterns, including linear,
exponential, and simple quadratic relationships, i.e., those of
the form f(n)=n² or f(n)=cn²,
for constant c, including A=r²,
and represents them with variables and expressions.
b
Represents relationships with tables, graphs in the coordinate plane,
and verbal or symbolic rules.
c
Analyzes tables, graphs, and rules to determine functional relationships.
d
Finds solutions for unknown quantities in linear equations and in
simple equations and inequalities.

High School
The student produces evidence that demonstrates understanding of
function and algebra concepts; that is, the student:
a
Models given situations with formulas and functions, and interprets
given formulas and functions in terms of situations.
b
Describes, generalizes, and uses basic types of functions: linear,
exponential, power, rational, square and square root, and cube and
cube root.
c
Utilizes the concepts of slope, evaluation, and inverse in working
with functions.
d
Works with rates of many kinds, expressed numerically, symbolically,
and graphically.
e
Represents constant rates as the slope of a straight line graph,
and interprets slope as the amount of one quantity (y) per unit
amount of another (x).
f
Understands and uses linear functions as a mathematical representation
of proportional relationships.
g
Uses arithmetic sequences and geometric sequences and their sums,
and sees these as the discrete forms of linear and exponential functions,
respectively.
h
Defines, uses, and manipulates expressions involving variables,
parameters, constants, and unknowns in work with formulas, functions,
equations, and inequalities.
i
Represents functional relationships in formulas, tables, and graphs,
and translates between pairs of these.
j
Solves equations symbolically, graphically, and numerically, especially
linear, quadratic, and exponential equations; and knows how to use
the quadratic formula for solving quadratic equations.
k
Makes predictions by interpolating or extrapolating from given data
or a given graph.
l
Understands the basic algebraic structure of number systems.
m
Uses equations to represent curves such as lines, circles, and parabolas.
n
Uses technology such as graphics calculators to represent and analyze
functions and their graphs.
o
Uses functions to analyze patterns and represent their structure.

Statistics and Probability Concepts 
Elementary School
The student produces evidence that demonstrates understanding of
statistics and probability concepts in the following areas; that
is, the student:
a
Collects and organizes data to answer a question or test a hypothesis
by comparing sets of data.
b
Displays data in line plots, graphs, tables, and charts.
c
Makes statements and draws simple conclusions based on data; that
is:
• reads data in line plots, graphs, tables, and charts;
• compares data in order to make true statements, e.g., “seven
plants grew at least 5 cm”;
• identifies and uses the mode necessary for making true statements,
e.g., “more people chose red”;
• makes true statements based on a simple concept of average
(median and mean), for a small sample size and where the situation
is made evident with concrete materials or clear representations;
• interprets data to determine the reasonableness of statements
about the data, e.g., “twice as often,” “three times
faster”;
• uses data, including statements about the data, to make a
simple concluding statement about a situation, e.g., “This
kind of plant grows better near sunlight because the seven plants
that were near the window grew at least 5 cm.”
d
Gathers data about an entire group or by sampling group members
to understand the concept of sample, i.e., that a large sample leads
to more reliable information, e.g., when flipping coins.
e
Predicts results, analyzes data, and finds out why some results
are more likely, less likely, or equally likely.
f
Finds all possible combinations and arrangements within certain
constraints involving a limited number of variables.

Middle School
The student produces evidence that demonstrates understanding of
statistics and probability concepts; that is, the student:
a
Collects data, organizes data, and displays data with tables, charts,
and graphs that are appropriate, i.e., consistent with the nature
of the data.
b
Analyzes data with respect to characteristics of frequency and distribution,
including mode and range.
c
Analyzes appropriately central tendencies of data by considering
mean and median.
d
Makes conclusions and recommendations based on data analysis.
e
Critiques the conclusions and recommendations of others’ statistics.
f
Considers the effects of missing or incorrect information.
g
Formulates hypotheses to answer a question and uses data to test
hypotheses.
h
Represents and determines probability as a fraction of a set of
equally likely outcomes; recognizes equally likely outcomes, and
constructs sample spaces (including those described by numerical
combinations and permutations).
i
Makes predictions based on experimental or theoretical probabilities.
j
Predicts the result of a series of trials once the probability for
one trial is known.

High School
The student demonstrates understanding of statistics and probability
concepts; that is, the student:
a
Organizes, analyzes, and displays singlevariable data, choosing
appropriate frequency distributions, circle graphs, line plots,
histograms, and summary statistics.
b
Organizes, analyzes, and displays twovariable data using scatter
plots, estimated regression lines, and computer generated regression
lines and correlation coefficients.
c
Uses sampling techniques to draw inferences about large populations.
d
Understands that making an inference about a population from a sample
always involves uncertainty and that the role of statistics is to
estimate the size of that uncertainty.
e
Formulates hypotheses to answer a question and uses data to test
hypotheses.
f
Interprets representations of data, compares distributions of data,
and critiques conclusions and the use of statistics, both in school
materials and in public documents.
g
Explores questions of experimental design, use of control groups,
and reliability.
h
Creates and uses models of probabilistic situations and understands
the role of assumptions in this process.
i
Uses concepts such as equally likely, sample space, outcome, and
event in analyzing situations involving chance.
j
Constructs appropriate sample spaces, and applies the addition and
multiplication principles for probabilities.
k
Uses the concept of a probability distribution to discuss whether
an event is rare or reasonably likely.
l
Chooses an appropriate probability model and uses it to arrive at
a theoretical probability for a chance event.
m
Uses relative frequencies based on empirical data to arrive at an
experimental probability for a chance event.
n
Designs simulations including Monte Carlo simulations to estimate
probabilities.
o
Works with the normal distribution in some of its basic applications.

Problem Solving and Mathematical Reasoning 
Elementary School
The student demonstrates logical reasoning throughout work in mathematics,
i.e., concepts and skills, problem solving, and projects; demonstrates
problem solving by using mathematical concepts and skills to solve
nonroutine problems that do not lay out specific and detailed steps
to follow; and solves problems that make demands on all three aspects
of the solution process—formulation, implementation, and conclusion.
Formulation
a
Given the basic statement of a problem situation, the student:
• makes the important decisions about the approach, materials,
and strategies to use, i.e., does not merely fill in a given chart,
use a prespecified manipulative, or go through a predetermined
set of steps;
• uses previously learned strategies, skills, knowledge, and
concepts to make decisions;
• uses strategies, such as using manipulatives or drawing sketches,
to model problems.
Implementation
b
The student makes the basic choices involved in planning and carrying
out a solution; that is, the student:
• makes up and uses a variety of strategies and approaches
to solving problems and uses or learns approaches that other people
use, as appropriate;
• makes connections among concepts in order to solve problems;
• solves problems in ways that make sense and explains why
these ways make sense, e.g., defends the reasoning, explains the
solution.
Conclusion
c
The student moves beyond a particular problem by making connections,
extensions, and/or generalizations; for example, the student:
• explains a pattern that can be used in similar situations;
• explains how the problem is similar to other problems he
or she has solved;
• explains how the mathematics used in the problem is like
other concepts in mathematics;
• explains how the problem solution can be applied to other
school subjects and in real world situations;
• makes the solution into a general rule that applies to other
circumstances.

Middle School
The student demonstrates problem solving by using mathematical
concepts and skills to solve nonroutine problems that do not lay
out specific and detailed steps to follow, and solves problems that
make demands on all three aspects of the solution process—formulation,
implementation, and conclusion.
Formulation
a
The student participates in the formulation of problems; that is,
given the basic statement of a problem situation, the student:
• formulates and solves a variety of meaningful problems;
• extracts pertinent information from situations and figures
out what additional information is needed.
Implementation
b
The student makes the basic choices involved in planning and carrying
out a solution; that is, the student:
• uses and invents a variety of approaches and understands
and evaluates those of others;
• invokes problem solving strategies, such as illustrating
with sensemaking sketches to clarify situations or organizing information
in a table;
• determines, where helpful, how to break a problem into simpler
parts;
• solves for unknown or undecided quantities using algebra,
graphing, sound reasoning, and other strategies;
• integrates concepts and techniques from different areas of
mathematics;
• works effectively in teams when the nature of the task or
the allotted time makes this an appropriate strategy.
Conclusion
c
The student provides closure to the solution process through summary
statements and general conclusions; that is, the student:
• verifies and interprets results with respect to the original
problem situation;
• generalizes solutions and strategies to new problem situations.
Mathematical reasoning
d
The student demonstrates mathematical reasoning by generalizing
patterns, making conjectures and explaining why they seem true,
and by making sensible, justifiable statements; that is, the student:
• formulates conjectures and argues why they must be or seem
true;
• makes sensible, reasonable estimates;
• makes justified, logical statements.

High School
The student demonstrates problem solving by using mathematical
concepts and skills to solve nonroutine problems that do not lay
out specific and detailed steps to follow, and solves problems that
make demands on all three aspects of the solution process—formulation,
implementation, and conclusion.
Formulation
a
The student participates in the formulation of problems; that is,
given the statement of a problem situation, the student:
• fills out the formulation of a definite problem that is to
be solved;
• extracts pertinent information from the situation as a basis
for working on the problem;
• asks and answers a series of appropriate questions in pursuit
of a solution and does so with minimal “scaffolding” in
the form of detailed guiding questions.
Implementation
b
The student makes the basic choices involved in planning and carrying
out a solution; that is, the student:
• chooses and employs effective problem solving strategies
in dealing with nonroutine and multistep problems;
• selects appropriate mathematical concepts and techniques
from different areas of mathematics and applies them to the solution
of the problem;
• applies mathematical concepts to new situations within mathematics
and uses mathematics to model real world situations involving basic
applications of mathematics in the physical and biological sciences,
the social sciences, and business.
Conclusion
c
The student provides closure to the solution process through summary
statements and general conclusions; that is, the student:
• concludes a solution process with a useful summary of results;
• evaluates the degree to which the results obtained represent
a good response to the initial problem;
• formulates generalizations of the results obtained;
• carries out extensions of the given problem to related problems.
Mathematical reasoning
d
The student demonstrates mathematical reasoning by using logic to
prove specific conjectures, by explaining the logic inherent in
a solution process, by making generalizations and showing that they
are valid, and by revealing mathematical patterns inherent in a
situation. The student not only makes observations and states results
but also justifies or proves why the results hold in general; that
is, the student:
• employs forms of mathematical reasoning and proof appropriate
to the solution of the problem at hand, including deductive
and inductive reasoning, making and testing conjectures,
and using counterexamples and indirect proof;
• differentiates clearly between giving examples that support
a conjecture and giving proof of the conjecture.

Mathematical Skills and Tools 
Elementary School
The student demonstrates fluency with basic and important skills
by using these skills accurately and automatically, and demonstrates
practical competence and persistence with other skills by using
them effectively to accomplish a task, perhaps referring to notes,
books, or other students, perhaps working to reconstruct a method;
that is, the student:
a
Adds, subtracts, multiplies, and divides whole numbers correctly;
that is:
• knows single digit addition, subtraction, multiplication,
and division facts;
• adds and subtracts numbers with several digits;
• multiplies and divides numbers with one or two digits;
• multiplies and divides three digit numbers by one digit numbers.
b
Estimates numerically and spatially.
c
Measures length, area, perimeter, circumference, diameter, height,
weight, and volume accurately in both the customary and metric systems.
d
Computes time (in hours and minutes) and money (in dollars and cents).
e
Refers to geometric shapes and terms correctly with concrete objects
or drawings, including triangle, square, rectangle, side, edge,
face, cube, point, line, perimeter, area, and circle; and refers
with assistance to rhombus, parallelogram, quadrilateral, polygon,
polyhedron, angle, vertex, volume, diameter, circumference, sphere,
prism, and pyramid.
f
Uses +, , x, ÷, /,,
$, ¢, %, and . (decimal point) correctly in number sentences
and expressions.
g
Reads, creates, and represents data on line plots, charts, tables,
diagrams, bar graphs, simple circle graphs, and coordinate graphs.
h
Uses recall, mental computations, pencil and paper, measuring devices,
mathematics texts, manipulatives, calculators, computers, and advice
from peers, as appropriate, to achieve solutions; that is, uses
measuring devices, graded appropriately for given situations, such
as rulers (customary to the 1/8
inch; metric to the millimeter), graph paper (customary to the inch
or halfinch; metric to the centimeter), measuring cups (customary
to the ounce; metric to the milliliter), and scales (customary to
the pound or ounce; metric to the kilogram or gram).

Middle School
The student demonstrates fluency with basic and important skills
by using these skills accurately and automatically, and demonstrates
practical competence and persistence with other skills by using
them effectively to accomplish a task (perhaps referring to notes,
or books, perhaps working to reconstruct a method); that is, the
student:
a
Computes accurately with arithmetic operations on rational numbers.
b
Knows and uses the correct order of operations for arithmetic computations.
c
Estimates numerically and spatially.
d
Measures length, area, volume, weight, time, and temperature accurately.
e
Refers to geometric shapes and terms correctly.
f
Uses equations, formulas, and simple algebraic notation appropriately.
g
Reads and organizes data on charts and graphs, including scatter
plots, bar, line, and circle graphs, and Venn diagrams; calculates
mean and median.
h
Uses recall, mental computations, pencil and paper, measuring devices,
mathematics texts, manipulatives, calculators, computers, and advice
from peers, as appropriate, to achieve solutions.

High School
The student demonstrates fluency with basic and important skills
by using these skills accurately and automatically, and demonstrates
practical competence and persistence with other skills by using
them effectively to accomplish a task, perhaps referring to notes,
or books, perhaps working to reconstruct a method; that is, the
student:
a
Carries out numerical calculations and symbol manipulations effectively,
using mental computations, pencil and paper, or other technological
aids, as appropriate.
b
Uses a variety of methods to estimate the values, in appropriate
units, of quantities met in applications, and rounds numbers used
in applications to an appropriate degree of accuracy.
c
Evaluates and analyzes formulas and functions of many kinds, using
both pencil and paper and more advanced technology.
d
Uses basic geometric terminology accurately, and deduces information
about basic geometric figures in solving problems.
e
Makes and uses rough sketches, schematic diagrams, or precise scale
diagrams to enhance a solution.
f
Uses the number line and Cartesian coordinates in the plane and
in space.
g
Creates and interprets graphs of many kinds, such as function graphs,
circle graphs, scatter plots, regression lines, and histograms.
h
Sets up and solves equations symbolically (when possible) and graphically.
i
Knows how to use algorithms in mathematics, such as the Euclidean
Algorithm.
j
Uses technology to create graphs or spreadsheets that contribute
to the understanding of a problem.
k
Writes a simple computer program to carry out a computation or simulation
to be repeated many times.
l
Uses tools such as rulers, tapes, compasses, and protractors in
solving problems.
m
Knows standard methods to solve basic problems and uses these methods
in approaching more complex problems.

Mathematical Communication 
Elementary School
The student uses the language of mathematics, its symbols, notation,
graphs, and expressions, to communicate through reading, writing,
speaking, and listening, and communicates about mathematics by describing
mathematical ideas and concepts and explaining reasoning and results;
that is, the student:
a
Uses appropriate mathematical terms, vocabulary, and language, based
on prior conceptual work.
b
Shows mathematical ideas in a variety of ways, including words,
numbers, symbols, pictures, charts, graphs, tables, diagrams, and
models.
c
Explains solutions to problems clearly and logically, and supports
solutions with evidence, in both oral and written work.
d
Considers purpose and audience when communicating about mathematics.
e
Comprehends mathematics from reading assignments and from other
sources.

Middle School
The student uses the language of mathematics, its symbols, notation,
graphs, and expressions, to communicate through reading, writing,
speaking, and listening, and communicates about mathematics by describing
mathematical ideas and concepts and explaining reasoning and results;
that is, the student:
a
Uses mathematical language and representations with appropriate
accuracy, including numerical tables and equations, simple algebraic
equations and formulas, charts, graphs, and diagrams.
b
Organizes work, explains facets of a solution orally and in writing,
labels drawings, and uses other techniques to make meaning clear
to the audience.
c
Uses mathematical language to make complex situations easier to
understand.
d
Exhibits developing reasoning abilities by justifying statements
and defending work.
e
Shows understanding of concepts by explaining ideas not only to
teachers and assessors but to fellow students or younger children.
f
Comprehends mathematics from reading assignments and from other
sources.

High School
The student uses the language of mathematics, its symbols, notation,
graphs, and expressions, to communicate through reading, writing,
speaking, and listening, and communicates about mathematics by describing
mathematical ideas and concepts and explaining reasoning and results;
that is, the student:
a
Is familiar with basic mathematical terminology, standard notation
and use of symbols, common conventions for graphing, and general
features of effective mathematical communication styles.
b
Uses mathematical representations with appropriate accuracy, including
numerical tables, formulas, functions, equations, charts, graphs,
and diagrams.
c
Organizes work and presents mathematical procedures and results
clearly, systematically, succinctly, and correctly.
d
Communicates logical arguments clearly, showing why a result makes
sense and why the reasoning is valid.
e
Presents mathematical ideas effectively both orally and in writing.
f
Explains mathematical concepts clearly enough to be of assistance
to those who may be having difficulty with them.
g
Writes narrative accounts of the history and process of work on
a mathematical problem or extended project.
h
Writes succinct accounts of the mathematical results obtained in
a mathematical problem or extended project, with diagrams, graphs,
tables, and formulas integrated into the text.
i
Keeps narrative accounts of process separate from succinct accounts
of results, and realizes that doing so can enhance the effectiveness
of each.
j
Reads mathematics texts and other writing about mathematics with
understanding.

Putting Mathematics to Work 
Elementary School
The student conducts at least one large scale project each year,
beginning in fourth grade, drawn from the following kinds and, over
the course of elementary school, conducts projects drawn from at
least two of the kinds.
A single project may draw on more than one kind.
a
Data study, in which the student:
• develops a question and a hypothesis in a situation where
data could help make a decision or recommendation;
• decides on a group or groups to be sampled and makes predictions
of the results, with specific percents, fractions, or numbers;
• collects, represents, and displays data in order to help
make the decision or recommendation; compares the results with the
predictions;
• writes a report that includes recommendations supported by
diagrams, charts, and graphs, and acknowledges assistance received
from parents, peers, and teachers.
b
Science study, in which the student:
• decides on a specific science question to study and identifies
the mathematics that will be used, e.g., measurement;
• develops a prediction (a hypothesis) and develops procedures
to test the hypothesis;
• collects and records data, represents and displays data,
and compares results with predictions;
• writes a report that compares the results with the hypothesis;
supports the results with diagrams, charts, and graphs; acknowledges
assistance received from parents, peers, and teachers.
c
Design of a physical structure, in which the student:
• decides on a structure to design, the size and budget constraints,
and the scale of design;
• makes a first draft of the design, and revises and improves
the design in response to input from peers and teachers;
• makes a final draft and report of the design, drawn and written
so that another person could make the structure; acknowledges assistance
received from parents, peers, and teachers.
d
Management and planning, in which the student:
• decides on what to manage or plan, and the criteria to be
used to see if the plan worked;
• identifies unexpected events that could disrupt the plan
and further plans for such contingencies;
• identifies resources needed, e.g., materials, money, time,
space, and other people;
• writes a detailed plan and revises and improves the plan
in response to feedback from peers and teachers;
• carries out the plan (optional);
• writes a report on the plan that includes resources, budget,
and schedule, and acknowledges assistance received from parents,
peers, and teachers.
• writes a report that includes recommendations supported by
diagrams, charts, and graphs, and acknowledges assistance received
from parents, peers, and teachers.
e
Pure mathematics investigation, in which the student:
• decides on the area of mathematics to investigate, e.g.,
numbers, shapes, patterns;
• describes a question or concept to investigate;
• decides on representations that will be used, e.g., numbers,
symbols, diagrams, shapes, or physical models;
• carries out the investigation;
• writes a report that includes any generalizations drawn from
the investigation, and acknowledges assistance received from parents,
peers, and teachers.

Middle School
The student conducts at least one large scale investigation or
project each year drawn from the following kinds and, over the course
of middle school, conducts investigations or projects drawn from
three of the kinds.
A single investigation or project may draw on more than one kind.
a
Data study based on civic, economic, or social issues, in which
the student:
• selects an issue to investigate;
• makes a hypothesis on an expected finding, if appropriate;
• gathers data;
• analyzes the data using concepts from Standard 4, e.g., considering
mean and median, and the frequency and distribution of the data;
• shows how the study’s results compare with the hypothesis;
• uses pertinent statistics to summarize;
• prepares a presentation or report that includes the question
investigated, a detailed description of how the project was carried
out, and an explanation of the findings.
b
Mathematical model of physical phenomena, often used in science
studies, in which the student:
• carries out a study of a physical system using a mathematical
representation of the structure;
• uses understanding from Standard 3, particularly with respect
to the determination of the function governing behavior in the model;
• generalizes about the structure with a rule, i.e., a function,
that clearly applies to the phenomenon and goes beyond statistical
analysis of a pattern of numbers generated by the situation;
• prepares a presentation or report that includes the question
investigated, a detailed description of how the project was carried
out, and an explanation of the findings.
c
Design of a physical structure, in which the student:
• generates a plan to build something of value, not necessarily
monetary value;
• uses mathematics from Standard 2 to make the design realistic
or appropriate, e.g., areas and volumes in general and of specific
geometric shapes;
• summarizes the important features of the structure;
• prepares a presentation or report that includes the question
investigated, a detailed description of how the project was carried
out, and an explanation of the findings.
d
Management and planning, in which the student:
• determines the needs of the event to be managed or planned,
e.g., cost, supply, scheduling;
• notes any constraints that will affect the plan;
• determines a plan;
• uses concepts from any of Standards 1 to 4, depending on
the nature of the project;
• considers the possibility of a more efficient solution;
• prepares a presentation or report that includes the question
investigated, a detailed description of how the project was carried
out, and an explanation of the plan.
e
Pure mathematics investigation, in which the student:
• extends or “plays with,” as with mathematical puzzles,
some mathematical feature, e.g., properties and patterns in numbers;
• uses concepts from any of Standards 1 to 4, e.g., an investigation
of Pascal’s triangle would have roots in Standard 1 but could
tie in concepts from geometry, algebra, and probability; investigations
of derivations of geometric formulas would be rooted in Standard
2 but could require algebra;
• determines and expresses generalizations from patterns;
• makes conjectures on apparent properties and argues, short
of formal proof, why they seem true;
• prepares a presentation or report that includes the question
investigated, a detailed description of how the project was carried
out, and an explanation of the findings.

High School
The student conducts at least one large scale investigation or
project each year drawn from the following kinds and, over the course
of high school, conducts investigations or projects drawn from at
least three of the kinds.
A single investigation or project may draw on more than one kind.
a
Data study, in which the student:
• carries out a study of data relevant to current civic, economic,
scientific, health, or social issues;
• uses methods of statistical inference to generalize from
the data;
• prepares a report that explains the purpose of the project,
the organizational plan, and conclusions, and uses an appropriate
balance of different ways of presenting information.
b
Mathematical model of a physical system or phenomenon, in which
the student:
• carries out a study of a physical system or phenomenon by
constructing a mathematical model based on functions to make generalizations
about the structure of the system;
• uses structural analysis (a direct analysis of the structure
of the system) rather than numerical or statistical analysis (an
analysis of data about the system);
• prepares a report that explains the purpose of the project,
the organizational plan, and conclusions, and uses an appropriate
balance of different ways of presenting information.
c
Design of a physical structure, in which the student:
• creates a design for a physical structure;
• uses general mathematical ideas and techniques to discuss
specifications for building the structure;
• prepares a report that explains the purpose of the project,
the organizational plan, and conclusions, and uses an appropriate
balance of different ways of presenting information.
d
Management and planning analysis, in which the student:
• carries out a study of a business or public policy situation
involving issues such as optimization, costbenefit projections,
and risks;
• uses decision rules and strategies both to analyze options
and balance tradeoffs; and brings in mathematical ideas that serve
to generalize the analysis across different conditions;
• prepares a report that explains the purpose of the project,
the organizational plan, and conclusions, and uses an appropriate
balance of different ways of presenting information.
e
Pure mathematics investigation, in which the student:
• carries out a mathematical investigation of a phenomenon
or concept in pure mathematics;
• uses methods of mathematical reasoning and justification
to make generalizations about the phenomenon;
• prepares a report that explains the purpose of the project,
the organizational plan, and conclusions, and uses an appropriate
balance of different ways of presenting information.
f
History of a mathematical idea, in which the student:
• carries out a historical study tracing the development of
a mathematical concept and the people who contributed to it;
• includes a discussion of the actual mathematical content
and its place in the curriculum of the present day;
• prepares a report that explains the purpose of the project,
the organizational plan, and conclusions, and uses an appropriate
balance of different ways of presenting information.

