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M
ONTANA
S
TANDARDS
FOR
M
ATHEMATICS
Mathematics is intended to give students an ability to solve problems, to communicate their ideas and
strategies, and to apply their skills in other disciplines. Students are expected to understand and investigate
mathematical concepts, to use mathematics in real-world situations, and to select and use appropriate
technology to model and study mathematical processes.
Students will use mathematical methods to learn about six strands: Quantity (number), Algebraic
Representation, Shape (geometry), Measurement, Chance and the Use of Data, and Mathematical Patterns.
In every strand, it is important for all students to have a conceptual framework, a knowledge of procedures,
a sense of reasonable results, and a confidence to apply their skills.
Content Standards indicate what all students should know, understand, and be able to do in a specific content area.
Benchmarks define our expectations for students' knowledge, skills, and abilities along a developmental continuum in
each content area. That continuum is focused at three points-the end of grade 4, grade 8, and grade 12.
Content Standard 1 - Students engage in the mathematical processes of problem solv-
ing and reasoning, estimation, communication, connections and applications, and
using appropriate technology.
Content Standard 2 - Students demonstrate understanding of and an ability to use
numbers and operations.
Content Standard 3 - Students use algebraic concepts, processes, and language to
model and solve a variety of real-world and mathematical problems.
Content Standard 4 - Students demonstrate understanding of shape and an ability to
use geometry.
Content Standard 5 - Students demonstrate understanding of measurable attributes
and an ability to use measurement processes.
Content Standard 6 - The students demonstrate understanding of an ability to use
data analysis, probability, and statistics.
Content Standard 7 - Students demonstrate understanding of and an ability to use
patterns, relations and functions.
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Mathematics Content Standard 1
Students engage in the mathematical processes of problem solving and reasoning,
estimation, communication, connections and applications, and using appropriate
technology.
Rationale
These processes are essential to all mathematics and must be incorporated in all other mathematics standards.
Benchmarks
Students will:
End of Grade 4
End of Grade 8
Upon Graduation-End of Grade 12
1. solve problems from many contexts
1. formulate and solve multi-step and
1. recognize and formulate problems
using a variety of strategies (e.g.,
nonroutine problems using a variety
from situations within and outside
estimate, make a table, look for a
of strategies. Generalize methods to
mathematics and apply solution
pattern, and simplify the problem). Explain
new problem situations.
strategies to those problems.
the methods for solving these problems.
2. apply estimation strategies throughout
2. select and apply appropriate estimation
2. select, apply, and evaluate appropri-
the problem-solving process.
strategies throughout the problem-
ate estimation strategies throughout
solving process.
the problem-solving process.
3. communicate mathematical ideas in a
3. interpret and communicate mathematical
3. formulate definitions, make and
variety of ways (e.g., written, verbal,
ideas and logical arguments using correct
justify inferences, express
concrete, pictorial, graphical, algebraic).
mathematical terms and notations.
generalizations, and communicate
mathematical ideas and relationships.
4. recognize and investigate the relevance
4. recognize and investigate the relevance
4. apply and translate among different
and usefulness of mathematics through
and usefulness of mathematics through
representations of the same problem
applications, both in and out of school.
applications, both in and out of school.
situation or of the same mathematical
concept. Model connections between
problem situations that arise in
disciplines other than mathematics.
5. select and use appropriate technology
5. select and use appropriate technology to
5. select and use appropriate technology
to enhance mathematical understanding.
enhance mathematical understanding.
to enhance mathematical understand-
Appropriate technology may include,
Appropriate technology may include,
ing. Appropriate technology may
but is not limited to, paper and pencil,
but is not limited to, paper and pencil,
include, but is not limited to, paper
calculator, and computer.
calculator, computer, and data collection
and pencil, calculator, computer, and
devices.
data collection devices.
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Mathematics Content Standard 2
Students demonstrate understanding of and an ability to use numbers and operations.
Rationale
An understanding of numbers and how they are used is necessary in the everyday world. Computational
skills and procedures should be developed in context so the learner perceives them as tools for solving prob-
lems.
Benchmarks
Students will:
End of Grade 4
End of Grade 8
Upon Graduation-End of Grade 12
1. exhibit connections between the concrete
1. use the four basic operations with whole
1. use and understand the real number
and symbolic representation of a problem
numbers, fractions, decimals, and
system, its operations, notations,
or concept.
integers.
and the various subsystems.
2. use the number system by counting,
2. use mental mathematics and number sense 2. use definitions and basic operations
grouping and applying place value
in using order of operations, and order
of the complex number system.
concepts.
relations for whole numbers, fractions,
decimals, and integers.
3. model, explain, and use basic facts,
3. use the relationships and applications
the operations of addition and
of ratio, proportion, percent, and
subtraction of whole numbers,
scientific notation.
and mental mathematics.
4. model and explain multiplication and
4. develop and apply number theory concepts
division of whole numbers.
(e.g., primes, factors and multiples) in real-
world and mathematical problem situations.
5. model and explain part/whole
relationships in everyday situations.
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Mathematics Content Standard 3
Students use algebraic concepts, processes, and language to model and solve a variety of
real-world and mathematical problems.
Rationale
Algebra is the language of mathematics and science. Through the use of variables and operations, algebra
allows students to form abstract models from contextual information.
Benchmarks
Students will:
End of Grade 4
End of Grade 8
Upon Graduation-End of Grade 12
1. use symbols (e.g., boxes or letters) to
1. understand the concepts of variable,
1. use algebra to represent patterns of
represent numbers in simple situations.
expression and equation.
change.
2. explore the use of variables and open
2. represent situations and number patterns
2. use basic operations with algebraic
sentences to express relationships
using tables, graphs, verbal rules,
expressions.
(e.g., missing addend).
equations, and models.
3. use inverse operations and other
3. recognize and use the general properties
3. solve algebraic equations and in-
strategies to solve number sentences.
of operations (e.g., the distributive
equalities: linear, quadratic,
property).
exponential, logarithmic, and power.
4. solve linear equations using concrete,
4. solve systems of algebraic equations
numerical and algebraic methods.
and inequalities, including use of
matrices.
5. investigate inequalities and nonlinear
5. use algebraic models to solve
relationships informally.
mathematical and real-world
problems.
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Mathematics Content Standard 4
Students demonstrate understanding of shape and an ability to use geometry.
Rationale
The study of geometry helps students represent and make sense of the world by discovering relationships and
developing spatial sense.
Benchmarks
Students will:
End of Grade 4
End of Grade 8
Upon Graduation-End of Grade 12
1. describe, model, and classify two- and
1. identify, describe, construct, and com-
1. construct, interpret, and draw three-
three-dimensional shapes.
pare plane and solid geometric figures.
dimensional objects.
2. investigate and predict results of
2. understand and apply geometric
2. classify figures in terms of congru-
combining, subdividing, and changing
properties and relationships (e.g.,
ence and similarity and apply these
shapes.
the Pythagorean Theorem).
relationships.
3. identify lines of symmetry, congruent
3. represent geometric figures on a
3. translate between synthetic and
and similar shapes, and positional
coordinate grid.
coordinate representations.
relationships.
4. explore properties and transformations
4. deduce properties of figures using
of geometric figures.
transformations, coordinates, and
vectors in problem solving.
5. use geometry as a means of describing
5. apply trigonometric ratios (sine,
the physical world.
cosine and tangent) to problem
situations involving triangles.
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Mathematics Content Standard 5
Students demonstrate understanding of measurable attributes and an ability to use
measurement processes.
Rationale
The first step in scientific investigation is understanding the measurable attributes of objects.
Benchmarks
Students will:
End of Grade 4
End of Grade 8
Upon Graduation-End of Grade 12
1. estimate, measure, and investigate length,
1. estimate, make, and use measurements
1. apply concepts of indirect measure-
capacity, weight, mass, area,
to describe, compare, and/or contrast
ments (e.g., using similar triangles to
volume, time, and temperature.
objects in real-world situations.
calculate a distance).
2. develop the process of measuring and
2. select and use appropriate units and
2. use dimensional analysis to check
concepts related to units of measurement,
tools to measure to a level of accuracy
reasonableness of procedures.
including standard units (English and
required in a particular setting.
metric) and nonstandard units.
3. apply measurement skills to everyday
3. apply the concepts of perimeter, area,
3. investigate systems of derived
situations.
volume and capacity, weight and mass,
measures (e.g., km/sec, g/cm
3
).
angle measure, time, and temperature.
4. select and use appropriate tools and
4. demonstrate understanding of the structure 4. apply the appropriate concepts of
techniques.
and use of systems of measurement,
estimates in measurement, error in
including English and metric.
measurement, tolerance, and
precision.
5. use the concepts of rates and other
derived and indirect measurements.
6. demonstrate relationships between
formulas and procedures for determining
area and volume.
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Mathematics Content Standard 6
The students demonstrate understanding of and an ability to use data analysis, probability,
and statistics.
Rationale
With society's expanding use of data for prediction and decision making, it is important that students develop
an understanding of the concepts and processes used in analyzing data.
Benchmarks
Students will:
End of Grade 4
End of Grade 8
Upon Graduation-End of Grade 12
1. collect, organize, and display data.
1. systematically collect, organize, and
1. use curve fitting to make predictions
describe data.
from data.
2. construct, read, and interpret displays
2. construct, read, and interpret tables,
2. apply measures of central tendency
of data, including graphs.
charts, and graphs.
and demonstrate understanding of the
concepts of variability and
correlation.
3. formulate and solve problems that
3. draw inferences, construct, and evaluate
3. select an appropriate sampling
involve collecting and analyzing data.
arguments based on data analysis and
method for a given statistical analysis.
measures of central tendency.
4. demonstrate basic concepts of chance
4. construct sample spaces and determine
4. use experimental probability,
(e.g., equally likely events, simple
the theoretical and experimental proba-
theoretical probability, and simulation
probabilities).
bilities of events.
methods to represent and solve
problems, including expected values.
5. make predictions based on experimental
5. design a statistical experiment to
results or probabilities.
study a problem and communicate
the outcomes.
6. describe, in general terms, the normal
curve and use its properties to answer
questions about sets of data that are
assumed to be normally distributed.
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Mathematics Content Standard 7
Students demonstrate understanding of and an ability to use patterns, relations and
functions.
Rationale
One of the central themes of mathematics is the study of patterns, relations, and functions. Exploring pat-
terns helps students develop mathematical power and instills in them an appreciation for the beauty of math-
ematics.
Benchmarks
Students will:
End of Grade 4
End of Grade 8
Upon Graduation-End of Grade 12
1. recognize, describe, extend, and create
1. describe, extend, analyze, and create a
1. describe functions and their inverses
a variety of patterns.
variety of patterns and functions.
using graphical, numerical, physical,
algebraic, and verbal mathematical
models or representations.
2. represent and describe mathematical and
2. describe and represent relationships with
2. analyze the graphs of the families of
real-world relationships.
tables, graphs, and rules.
polynomial, rational, power, exponen-
tial, logarithmic, and periodic
functions.
3. analyze functional relationships to explain 3. analyze the effects of parameter
how a change in one quantity results in a
changes on the graphs of functions
change in another.
and relations, including translations.
4. use patterns and functions to represent
4. model real-world phenomena with a
and solve problems.
variety of functions.
5. describe functions using graphical,
5. use graphing for parametric
numerical, physical, algebraic, and verbal
equations, three-dimensional
models or representations.
equations, and recursive relations.
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Mathematics Performance Standards: A Profile of Four Levels
The Mathematics Performance Standards describe students' knowledge, skills, and abilities in the
mathematics content area on a continuum from kindergarten through grade twelve. These descriptions
provide a picture or profile of student achievement at the four performance levels: advanced, proficient,
nearing proficiency, and novice.
Advanced:
This level denotes superior performance.
Proficient:
This level denotes solid academic performance for each benchmark. Students reaching
this level have demonstrated competency over challenging subject matter, including
subject-matter knowledge, application of such knowledge to real-world situations, and
analytical skills appropriate to the subject matter.
Nearing
This level denotes that the student has partial mastery or prerequisite knowledge and
Proficiency:
skills fundamental for proficient work at each benchmark.
Novice:
This level denotes that the student is beginning to attain the prerequisite knowledge and
skills that are fundamental for work at each benchmark.
Grade 4 Mathematics
Advanced
A fourth-grade student at the advanced level in mathematics demonstrates superior performance. He/she:
(a) demonstrates self-motivation and emerging independence as a learner;
(b) accurately selects and uses problem-solving strategies;
(c) presents well-organized solutions and communicates in ways that exceed requirements;
(d) uses whole numbers accurately and fluently to estimate, compute, and determine whether results are accurate and
reasonable;
(e) effectively applies basic algebraic concepts and clearly communicates representations in a variety of ways;
(f)
examines relationships of shapes in the physical world and makes generalizations;
(g) selects and accurately uses appropriate tools for measurement;
(h) accurately predicts and makes reasonable decisions based on data; and
(i)
articulately and fluently communicates representations, analyzes patterns, and clearly describes relationships, and
applies them to varied situations.
Proficient
A fourth-grade student at the proficient level in mathematics demonstrates solid academic performance. He/she:
(a) selects and effectively uses appropriate problem-solving strategies;
(b) consistently presents organized solutions;
(c) uses whole numbers to estimate, compute, and determine whether results are accurate;
(d) applies basic algebra concepts and consistently communicates representations in a variety of ways;
(e) consistently examines and accurately uses relationships of shapes in the physical world;
(f)
determines measurable attributes of objects and selects appropriate tools for measurement;
(g) consistently predicts and makes reasonable decisions based on data; and
(h) consistently uses a variety of patterns and describes their relationships.
Nearing Proficiency
A fourth-grade student at the nearing proficiency level in mathematics demonstrates partial mastery of the
prerequisite knowledge and skills fundamental for proficient-level mathematics. He/she:
(a) sometimes selects and uses appropriate problem-solving strategies;
(b) sometimes presents organized solutions, but often with limited supporting information;
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(c) uses whole numbers to estimate and compute, and results are usually reasonable;
(d) sometimes applies basic algebraic concepts, but seldom communicates representations;
(e) examines some shapes in the physical world, and sometimes sees relationships;
(f)
determines measurable attributes of objects, but does not always select appropriate tools for measurement;
(g) often makes inconsistent predictions and inaccurate decisions based on data; and
(h) uses a limited range of patterns, and sometimes describes relationships within those patterns.
Novice
A fourth-grade student at the novice level in mathematics is beginning to attain the prerequisite knowledge and skills that
are fundamental at each benchmark in mathematics. He/she:
(a) selects and uses only a few problem-solving strategies;
(b) often presents poorly organized solutions, often without supporting information or explanation;
(c) lacks clarity and coherence when communicating mathematical concepts;
(d) uses whole numbers to estimate and compute, but is frequently inaccurate;
(e) sometimes determines whether results are reasonable;
(f)
demonstrates a basic algebraic understanding of concrete and symbolic representations, but often misconceptions are
present;
(g) describes, models, and classifies some shapes;
(h) determines some measurable attributes of objects, but often does not select appropriate tools for measurement;
(i)
sometimes predicts, but often makes inaccurate decisions based on data; and
(j)
recognizes and represents a limited range of patterns and describes relationships within those patterns, but is frequently
inaccurate.
Grade 8 Mathematics
Advanced
An eighth-grade student at the advanced level in mathematics demonstrates superior performance. He/she:
(a) demonstrates self-motivation and independence as a learner;
(b) is accurate and fluent when applying mathematical processes;
(c) effectively uses multiple strategies and extends concepts to new situations;
(d) explores hypothetical questions and articulates valid arguments;
(e) applies and extends rational numbers, proportionality, and algebraic concepts to solve real and theoretical problems;
(f)
applies complex measurement and geometric relationships to hypothetical situations;
(g) consistently makes accurate predictions and decisions based on basic probability and statistics; and
(h) recognizes interconnections within and outside mathematics.
Proficient
An eighth-grade student at the proficient level in mathematics demonstrates solid academic performance. He/she:
(a) effectively applies mathematical processes correctly to solve a variety of problems;
(b) applies mathematics in a variety of contexts;
(c) uses rational numbers, proportionality, and algebraic concepts to represent and accurately solve mathematical prob-
lems;
(d) consistently and accurately uses complex measurement, geometric relationships, and properties to describe the physi-
cal world;
(e) formulates logical arguments using appropriate mathematical ideas; and
(f)
consistently makes reasonable predictions and decisions based on basic probability and statistics.
Nearing Proficiency
An eighth-grade student at the nearing proficiency level in mathematics demonstrates partial mastery of
the prerequisite knowledge and skills fundamental for proficient-level mathematics. He/she:
(a) often uses incomplete and incorrect mathematical processes to solve problems, often inaccurately;
(b) communicates mathematical ideas, but often inaccurately;
(c) makes connections, but does not generalize and often his/her arguments lack appropriate supporting mathematical
ideas;
(d) sometimes understands and correctly uses numbers, operations, patterns, relations, and functions;
(e) sometimes uses inaccurate or incomplete representations of rational numbers, proportionality, and algebraic concepts
to solve mathematical problems;
(f)
sometimes has difficulty recognizing complex measurement and geometric relationships and properties which result in
inaccurate solutions; and
(g) makes simple predictions and decisions based on basic probability and statistics.
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Novice
An eighth-grade student at the novice level in mathematics is beginning to attain the prerequisite knowledge and skills that
are fundamental to each benchmark in mathematics. He/she:
(a) demonstrates limited and incomplete use of mathematical processes;
(b) communicates mathematical ideas, but they are often limited and incomplete;
(c) sometimes uses numbers, operations, patterns, relations, and functions accurately;
(d) makes only immediate, concrete, mathematical connections;
(e) seldom uses algebraic concepts to solve problems; and
(f)
makes simple and inconsistent predictions and decisions, often inaccurately, based on data, and seldom recognizes
complex measurement, geometric relationships, or properties.
Upon Graduation Mathematics
Advanced
A graduating student at the advanced level in mathematics demonstrates superior performance. He/she:
(a) is self-motivated, an independent learner, and extends and connects ideas;
(b) is accurate, articulate, and effective when applying mathematical processes;
(c) effectively uses multiple strategies, extends concepts to new situations, and skillfully communicates the results;
(d) explores hypothetical questions, uses complex reasoning to articulate valid arguments, and constructs proofs;
(e) uses appropriate technology to apply functions, graphs, and algebraic concepts to solve real and theoretical problems;
(f)
applies complex measurement and geometric and algebraic relationships to model a variety of problems and situations;
(g) consistently makes accurate and reasonable predictions and decisions based on data, probability, and statistics; and
(h) recognizes interconnections within and outside mathematics.
Proficient
A graduating student at the proficient level in mathematics demonstrates solid academic performance. He/she:
(a) consistently applies mathematical processes correctly to solve a variety of problems and communicate the results;
(b) applies mathematics in a variety of contexts;
(c) consistently uses appropriate technology to apply functions, graphs, and algebraic concepts to solve real and theoreti-
cal problems;
(d) uses complex reasoning to formulate logical arguments and proofs using appropriate mathematical ideas;
(e) consistently applies complex measurement and geometric and algebraic relationships to model a variety of problems
and situations;
(f)
makes reasonable predictions and decisions based on data, probability, and statistics; and
(g) recognizes interconnections within and outside mathematics.
Nearing Proficiency
A graduating student at the nearing proficiency level in mathematics demonstrates partial mastery of the
prerequisite knowledge and skills fundamental for proficient-level mathematics. He/she:
(a) applies incomplete and incorrect mathematical processes to solve problems, often inaccurately;
(b) communicates mathematical ideas and sometimes extends them, but often inaccurately;
(c) sometimes understands and uses appropriate technology to apply functions, graphs, and algebraic concepts to solve
real and theoretical problems;
(d) sometimes demonstrates difficulty recognizing complex measurement and geometric and algebraic relationships which
result in inaccuracies;
(e) sometimes makes predictions and decisions based on data, probability, and statistics, often inaccurately; and
(f)
makes connections, but does not generalize or prove them and often his/her arguments lack appropriate supporting
mathematical ideas and careful reasoning.
Novice
A graduating student at the novice level in mathematics is beginning to attain the prerequisite knowledge and skills that
are fundamental at each benchmark in mathematics. He/she:
(a) demonstrates limited and incomplete use of mathematical processes and problem-solving strategies;
(b) often uses limited and incomplete reasoning to formulate logical arguments and communicate mathematical ideas;
(c) makes only concrete, mathematical connections;
(d) seldom uses appropriate technology to apply functions, graphs, and algebraic concepts to solve problems;
(e) recognizes, on a limited basis, complex measurement, geometric relationships, and properties; and
(f)
makes some predictions and decisions, on a limited basis, based on data, but seldom recognizes statistical or probabil-
ity concepts.