**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.

**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

A.12.6 Read and understand

• mathematical texts and other instructional materials

• writing about mathematics (e.g., articles in journals)

• mathematical ideas as they are used in other contexts

**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.

**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

calculators, spreadsheets)

**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

**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.

**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)

**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.