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# Model Academic Standards for Mathematics

## A. MATHEMATICAL PROCESSES

Content Standard
Students in Wisconsin will draw on a broad body of mathematical knowledge and apply a
variety of mathematical skills and strategies, including reasoning, oral and written
communication, and the use of appropriate technology, when solving mathematical, realworld*
and nonroutine* problems.

Rationale:

In order to participate fully as a citizen and a worker in our contemporary world, a person
should be mathematically powerful. Mathematical power is the ability to explore, to
conjecture, to reason logically, and to apply a wide repertoire of methods to solve problems.
Because no one lives and works in isolation, it is also important to have the ability to
communicate mathematical ideas clearly and effectively.

## PERFORMANCE STANDARDS

By the end of grade 4 students will:

A.4.1 Use reasoning abilities to
• perceive patterns
• identify relationships
• formulate questions for further exploration
• justify strategies
• test reasonableness of results

A.4.2 Communicate mathematical ideas in a variety of ways, including words, numbers,
symbols, pictures, charts, graphs, tables, diagrams, and models*

A.4.3 Connect mathematical learning with other subjects, personal experiences, current
events, and personal interests
• see relationships between various kinds of problems and actual events
• use mathematics as a way to understand other areas of the curriculum (e.g.,
measurement in science, map skills in social studies)

A.4.4 Use appropriate mathematical vocabulary, symbols, and notation with
understanding based on prior conceptual work

A.4.5 Explain solutions to problems clearly and logically in oral and written work and
support solutions with evidence

By the end of grade 8 students will:

A.8.1 Use reasoning abilities to
• evaluate information
• perceive patterns
• identify relationships
• formulate questions for further exploration
• evaluate strategies
• justify statements
• test reasonableness of results
• defend work

A.8.2 Communicate logical arguments clearly to show why a result makes sense

A.8.3 Analyze nonroutine* problems by modeling*, illustrating, guessing, simplifying,
generalizing, shifting to another point of view, etc.

A.8.4 Develop effective oral and written presentations that include
• appropriate use of technology
• the conventions of mathematical discourse (e.g., symbols, definitions, labeled
drawings)
• mathematical language
• clear organization of ideas and procedures
• understanding of purpose and audience

A.8.5 Explain mathematical concepts, procedures, and ideas to others who may not be
familiar with them

A.8.6 Read and understand mathematical texts and other instructional materials and
recognize mathematical ideas as they appear in other contexts

By the end of grade 12 students will:

A.12.1 Use reason and logic to
• evaluate information
• perceive patterns
• identify relationships
• formulate questions, pose problems, and make and test conjectures
• pursue ideas that lead to further understanding and deeper insight

A.12.2 Communicate logical arguments and clearly show
• why a result does or does not make sense
• why the reasoning is or is not valid
• an understanding of the difference between examples that support a conjecture
and a proof of the conjecture

A.12.3 Analyze nonroutine* problems and arrive at solutions by various means, including
models* and simulations, often starting with provisional conjectures and
progressing, directly or indirectly, to a solution, justification, or counter-example

A.12.4 Develop effective oral and written presentations employing correct mathematical
terminology, notation, symbols, and conventions for mathematical arguments and
display of data

A.12.5 Organize work and present mathematical procedures and results clearly,
systematically, succinctly, and correctly

• mathematical texts and other instructional materials
• writing about mathematics (e.g., articles in journals)
• mathematical ideas as they are used in other contexts

## B. NUMBER OPERATIONS AND RELATIONSHIPS

Content Standard
Students in Wisconsin will use numbers effectively for various purposes, such as counting,
measuring, estimating, and problem solving.

Rationale:

People use numbers to quantify, describe, and label things in the world around them. It is
important to know the many uses of numbers and various ways of representing them.
Number sense is a matter of necessity, not only in one’s occupation but also in the conduct
of daily life, such as shopping, cooking, planning a budget, or analyzing information
reported in the media. When computing, an educated person needs to know which
operations (e.g., addition, multiplication), which procedures (e.g., mental techniques,
algorithms*), or which technological aids (e.g., calculator, spreadsheet) are appropriate.

## PERFORMANCE STANDARDS

By the end of grade 4 students will:

B.4.1 Represent and explain whole numbers*, decimals, and fractions with
• physical materials
• number lines and other pictorial models*
• verbal descriptions
• place-value concepts and notation
• symbolic renaming (e.g., 43 = 40+3 = 30+13).

B.4.2 Determine the number of things in a set by
• grouping and counting (e.g., by threes, fives, hundreds)
• combining and arranging (e.g., all possible coin combinations amounting to thirty
cents)
• estimation, including rounding

B.4.3 Read, write, and order whole numbers*, simple fractions (e.g., halves, fourths,
tenths, unit fractions*) and commonly-used decimals (monetary units)

B.4.4 Identify and represent equivalent fractions for halves, fourths, eighths, tenths,
sixteenths

B.4.5 In problem-solving situations involving whole numbers, select and efficiently use
appropriate computational procedures such as
• recalling the basic facts of addition, subtraction, multiplication, and division
• using mental math (e.g., 37 + 25, 40 x 7)
• estimation
• selecting and applying algorithms* for addition, subtraction, multiplication, and
division
• using a calculator

B.4.6 Add and subtract fractions with like denominators

B.4.7 In problem-solving situations involving money, add and subtract decimals

By the end of grade 8 students will:

B.8.1 Read, represent, and interpret various rational numbers* (whole numbers*,
integers*, decimals, fractions, and percents) with verbal descriptions, geometric
models*, and mathematical notation (e.g., expanded*, scientific*, exponential*)

B.8.2 Perform and explain operations on rational* numbers (add, subtract, multiply,
divide, raise to a power, extract a root, take opposites and reciprocals, determine
absolute value)

B.8.3 Generate and explain equivalencies among fractions, decimals, and percents

B.8.4 Express order relationships among rational numbers using appropriate symbols
(>, <, ≥, ≤, ≠)

B.8.5 Apply proportional thinking in a variety of problem situations that include, but are
not limited to
• ratios and proportions (e.g., rates, scale drawings*, similarity*)
• percents, including those greater than 100 and less than one (e.g., discounts, rate
of increase or decrease, sales tax)

B.8.6 Model* and solve problems involving number-theory concepts such as
• prime* and composite numbers
• divisibility and remainders
• greatest common factors
• least common multiples

B.8.7 In problem-solving situations, select and use appropriate computational procedures
with rational numbers such as
• calculating mentally
• estimating
• creating, using, and explaining algorithms* using technology (e.g., scientific

By the end of grade 12 students will:

B.12.1 Use complex counting procedures such as union and intersection of sets and
arrangements (permutations* and combinations*) to solve problems.

B.12.2 Compare real numbers using
• order relations (>, <) and transitivity*
• ordinal scales including logarithmic (e.g., Richter, pH rating)
• arithmetic differences
• ratios, proportions, percents, rates of change

B.12.3 Perform and explain operations on real numbers (add, subtract, multiply, divide,
raise to a power, extract a root, take opposites and reciprocals, determine absolute
value)

B.12.4 In problem-solving situations involving the application of different number systems
(natural, integers, rational*, real*) select and use appropriate
• computational procedures
• properties (e.g., commutativity*, associativity*, inverses*)
• modes of representation (e.g., rationals as repeating decimals, indicated roots as
fractional exponents)

B.12.5 Create and critically evaluate numerical arguments presented in a variety of
classroom and real-world situations (e.g., political, economic, scientific, social)

B.12.6 Routinely assess the acceptable limits of error when
• evaluating strategies
• testing the reasonableness of results
• using technology to carry out computations

## C. GEOMETRY

Content Standard
Students in Wisconsin will be able to use geometric concepts, relationships and procedures
to interpret, represent, and solve problems.

Note: Familiar mathematical content dealing with measurement of geometric objects (e.g.,
length, area, volume) is presented in “D. Measurement.”

Rationale:

Geometry and its study of shapes and relationships is an effort to understand the nature
and beauty of the world. While the need to understand our environment is still with us, the
rapid advance of technology has created another need—to understand ideas communicated
visually through electronic media. For these reasons, educated people in the 21st century
need a well-developed sense of spatial order to visualize and model real world* problem
situations.

## PERFORMANCE STANDARDS

By the end of grade 4 students will:

C.4.1 Describe two-and three-dimensional figures (e.g., circles, polygons, trapezoids,
prisms, spheres) by
• naming them
• comparing, sorting, and classifying them
• drawing and constructing physical models to specifications
• identifying their properties (e.g., number of sides or faces, two- or threedimensionality,
equal sides, number of right angles)
• predicting the results of combining or subdividing two-dimensional figures
• explaining how these figures are related to objects in the environment

C.4.2 Use physical materials and motion geometry (such as slides, flips, and turns) to
identify properties and relationships, including but not limited to
• symmetry*
• congruence*
• similarity*

C.4.3 Identify and use relationships among figures, including but not limited to
• location (e.g., between, adjacent to, interior of)
• position (e.g., parallel, perpendicular)
• intersection (of two-dimensional figures)

C.4.4 Use simple two-dimensional coordinate systems to find locations on maps and to
represent points and simple figures

By the end of grade 8 students will:

C.8.1 Describe special and complex two- and three-dimensional figures (e.g., rhombus,
polyhedron, cylinder) and their component parts (e.g., base, altitude, and slant
height) by
• naming, defining, and giving examples
• comparing, sorting, and classifying them
• identifying and contrasting their properties (e.g., symmetrical*, isosceles,
regular)
• drawing and constructing physical models to specifications
• explaining how these figures are related to objects in the environment

C.8.2 Identify and use relationships among the component parts of special and complex
two- and three-dimensional figures (e.g., parallel sides, congruent* faces)

C.8.3 Identify three-dimensional shapes from two-dimensional perspectives and draw two-dimensional
sketches of three-dimensional objects preserving their significant
features

C.8.4 Perform transformations* on two-dimensional figures and describe and analyze the
effects of the transformations on the figures

C.8.5 Locate objects using the rectangular coordinate system*

By the end of grade 12 students will:

C.12.1 Identify, describe, and analyze properties of figures, relationships among figures,
and relationships among their parts by
• constructing physical models
• drawing precisely with paper and pencil, hand calculators, and computer
software
• using appropriate transformations* (e.g., translations, rotations, reflections,
enlargements)
• using reason and logic

C.12.2 Use geometric models* to solve mathematical and real-world problems

C.12.3 Present convincing arguments by means of demonstration, informal proof, counterexamples,
or any other logical means to show the truth of
• statements (e.g., “these two triangles are not congruent”)
• generalizations (e.g., “the Pythagorean* theorem holds for all right triangles”)

C.12.4 Use the two-dimensional rectangular coordinate system* and algebraic procedures
to describe and characterize geometric properties and relationships such as slope*,
intercepts*, parallelism, and perpendicularity

C.12.5 Identify and demonstrate an understanding of the three ratios used in right-triangle
trigonometry (sine, cosine, tangent)

## D. MEASUREMENT

Content Standard
Students in Wisconsin will select and use appropriate tools (including technology) and
techniques to measure things to a specified degree of accuracy. They will use measurements
in problem-solving situations.

Rationale:

Measurement is the foundation upon which much technological, scientific, economic, and
social inquiry rests. Before things can be analyzed and subjected to scientific investigation
or mathematical modeling*, they must first be quantified by appropriate measurement
principles. Measurable attributes* include such diverse concepts as voting preferences,
consumer price indices, speed and acceleration, length, monetary value, duration of an
Olympic race, or probability of contracting a fatal disease.