# Model Academic Standards for Mathematics

## PERFORMANCE STANDARDS

**By the end of grade 4 students will:**

D.4.1 Recognize and describe measurable attributes*, such
as length, liquid capacity, time,

weight (mass), temperature, volume, monetary value, and angle size, and identify

the appropriate units to measure them

D.4.2 Demonstrate understanding of basic facts,
principles, and techniques of

measurement, including

• appropriate use of arbitrary* and standard units (metric and US Customary)

• appropriate use and conversion of units within a system (such as yards, feet,
and

inches; kilograms and grams; gallons, quarts, pints, and cups)

• judging the reasonableness of an obtained measurement as it relates to prior

experience and familiar benchmarks

D.4.3 Read and interpret measuring instruments (e.g., rulers, clocks, thermometers)

D.4.4 Determine measurements directly* by using standard
tools to these suggested

degrees of accuracy

• length to the nearest half-inch or nearest centimeter

• weight (mass) to the nearest ounce or nearest 5 grams

• temperature to the nearest 5°

• time to the nearest minute

• monetary value to dollars and cents• liquid capacity to the nearest fluid
ounce

D.4.5 Determine measurements by using basic relationships
(such as perimeter and area)

and approximate measurements by using estimation techniques

**By the end of grade 8 students will:**

D.8.1 Identify and describe attributes* in situations
where they are not directly* or easily

measurable (e.g., distance, area of an irregular figure, likelihood of
occurrence)

D.8.2 Demonstrate understanding of basic measurement
facts, principles, and techniques

including the following

• approximate comparisons between metric and US Customary units (e.g., a liter

and a quart are about the same; a kilometer is about six-tenths of a mile)

• knowledge that direct measurement* produces approximate, not exact, measures

• the use of smaller units to produce more precise measures

D.8.3 Determine measurement directly* using standard units
(metric and US Customary)

with these suggested degrees of accuracy

• lengths to the nearest mm or 1/16 of an inch

• weight (mass) to the nearest 0.1 g or 0.5 ounce

• liquid capacity to the nearest millileter

• angles to the nearest degree

• temperature to the nearest C° or F°

• elapsed time to the nearest second

D.8.4 Determine measurements indirectly* using

• estimation

• conversion of units within a system (e.g., quarts to cups, millimeters to

centimeters)

• ratio and proportion (e.g., similarity*, scale drawings*)

• geometric formulas to derive lengths, areas, volumes of common figures (e.g.,

perimeter, circumference, surface area)

• the Pythagorean* relationship

• geometric relationships and properties for angle size (e.g., parallel lines
and

transversals; sum of angles of a triangle; vertical angles*)

**By the end of grade 12 students will:**

D.12.1 Identify, describe, and use derived attributes*
(e.g., density, speed, acceleration,

pressure) to represent and solve problem situations

D.12.2 Select and use tools with appropriate degree of
precision to determine

measurements directly* within specified degrees of accuracy and error
(tolerance)

D.12.3 Determine measurements indirectly*, using

• estimation

• proportional reasoning, including those involving squaring and cubing (e.g.,

reasoning that areas of circles are proportional to the squares of their radii)

• techniques of algebra, geometry, and right triangle trigonometry

• formulas in applications (e.g., for compound interest, distance formula)

• geometric formulas to derive lengths, areas, or volumes of shapes and objects

(e.g., cones, parallelograms, cylinders, pyramids)

• geometric relationships and properties of circles and polygons (e.g., size of

central angles, area of a sector of a circle)

• conversion constants to relate measures in one system to another (e.g., meters
to

feet, dollars to Deutschmarks)

## E. STATISTICS AND PROBABILITY

**Content Standard**

Students in Wisconsin will use data collection and analysis, statistics and
probability in

problem-solving situations, employing technology where appropriate.

**Rationale:**

Dramatic advances in technology have launched the world
into the Information Age, when

data are used to describe past events or predict future events. Whether in the
business

place or in the home, as producers or consumers of information, citizens need to
be well

versed in the concepts and procedures of data analysis in order to make informed
decisions.

## PERFORMANCE STANDARDS

**By the end of grade 4 students will:**

E.4.1 Work with data in the context of real -world
situations by

• formulating questions that lead to data collection and analysis

• determining what data to collect and when and how to collect them

• collecting, organizing, and displaying data

• drawing reasonable conclusions based on data

E.4.2 Describe a set of data using

• high and low values , and range*

• most frequent value (mode*)

• middle value of a set of ordered data (median*)

E.4.3 In problem-solving situations, read, extract, and
use information presented in

graphs, tables, or charts.

E.4.4 Determine if the occurrence of future events are
more, less, or equally likely,

impossible, or certain

E.4.5 Predict outcomes of future events and test
predictions using data from a variety of

sources

**By the end of grade 8 students will:**

E.8.1 Work with data in the context of real-world
situations by

• formulating questions that lead to data collection and analysis

• designing and conducting a statistical investigation

• using technology to generate displays, summary statistics*, and presentations

E.8.2 Organize and display data from statistical
investigations using

• appropriate tables, graphs, and/or charts (e.g., circle, bar, or line for
multiple sets

of data)

• appropriate plots (e.g., line*, stem-and-leaf*, box*, scatter*)

E.8.3 Extract, interpret, and analyze information from
organized and displayed data by

using

• frequency and distribution, including mode* and range*

• central tendencies* of data (mean* and median*)

• indicators of dispersion (e.g., outliers*)

E.8.4 Use the results of data analysis to

• make predictions

• develop convincing arguments

• draw conclusions

E.8.5 Compare several sets of data to generate, test, and,
as the data dictate, confirm or

deny hypotheses

E.8.6 Evaluate presentations and statistical analyses from
a variety of sources for

• credibility of the source

• techniques of collection, organization, and presentation of data

• missing or incorrect data

• inferences

• possible sources of bias

E.8.7 Determine the likelihood of occurrence of simple
events by

• using a variety of strategies to identify possible outcomes (e.g., lists,
tables, tree

diagrams*)

• conducting an experiment

• designing and conducting simulations*

• applying theoretical notions of probability (e.g., that four equally likely
events

have a 25 percent chance of happening)

**By the end of grade 12 students will:**

E.12.1 Work with data in the context of real-world
situations by

• formulating hypotheses that lead to collection and analysis of one- and
twovariable

data

• designing a data collection plan that considers random sampling, control
groups ,

the role of assumptions, etc.

• conducting an investigation based on that plan

• using technology to generate displays, summary statistics*, and presentations

E.12.2 Organize and display data from statistical
investigations using

• frequency distributions

• percentiles*, quartiles, deciles

• line of best fit* (estimated regression line)

• matrices

E.12.3 Interpret and analyze information from organized
and displayed data when given

• measures of dispersion*, including standard deviation and variance

• measures of reliability

• measures of correlation*

E.12.4 Analyze, evaluate, and critique the methods and
conclusions of statistical

experiments reported in journals, magazines, news media, advertising, etc.

E.12.5 Determine the likelihood of occurrence of complex
events by

• using a variety of strategies (e.g., combinations*) to identify possible
outcomes

• conducting an experiment

• designing and conducting simulations*

• applying theoretical probability

## F. ALGEBRAIC RELATIONSHIPS

**Content Standard**

Students in Wisconsin will discover, describe, and generalize simple and complex
patterns

and relationships. In the context of real-world problem situations, the student
will use

algebraic techniques to define and describe the problem to determine and justify

appropriate solutions.

**Rationale:**

Algebra is the language of mathematics. Much of the
observable world can be characterized

as having patterned regularity where a change in one quantity results in changes
in other

quantities. Through algebra and the use of variables* and functions*,
mathematical

models* can be built which are essential to personal, scientific, economic,
social, medical,

artistic, and civic fields of inquiry.

## PERFORMANCE STANDARDS

**By the end of grade 4 students will:**

F.4.1 Use letters, boxes, or other symbols to stand for
any number, measured quantity, or

object in simple situations (e.g., N + 0 = N is true for any number)

F.4.2 Use the vocabulary, symbols, and notation of algebra
accurately (e.g., correct use of

the symbol “=”, effective use of the associative property of multiplication)

F.4.3 Work with simple linear patterns and relationships
in a variety of ways, including

• recognizing and extending number patterns

• describing them verbally

• representing them with pictures, tables, charts, graphs

• recognizing that different models * can represent the same pattern or

relationship

• using them to describe real-world phenomena

F.4.4 Recognize variability in simple functional*
relationships by describing how a change

in one quantity can produce a change in another (e.g., number of bicycles and
the

total number of wheels)

F.4.5 Use simple equations and inequalities in a variety
of ways, including

• using them to represent problem situations

• solving them by different methods (e.g., use of manipulatives, guess and check

strategies, recall of number facts)

• recording and describing solution strategies

F.4.6 Recognize and use generalized properties and
relationships of arithmetic (e.g.,

commutativity* of addition, inverse relationship of multiplication and division)

**By the end of grade 8 students will:**

F.8.1 Work with algebraic expressions in a variety of
ways, including

• using appropriate symbolism, including exponents* and variables*

• evaluating expressions through numerical substitution

• generating equivalent expressions

• adding and subtracting expressions

F.8.2 Work with linear and nonlinear patterns* and
relationships in a variety of ways,

including

• representing them with tables, with graphs, and with algebraic expressions,

equations, and inequalities

• describing and interpreting their graphical representations (e.g., slope*,
rate of

change, intercepts *)

• using them as models of real-world phenomena

• describing a real-world phenomenon that a given graph might represent

F.8.3 Recognize, describe, and analyze functional
relationships* by generalizing a rule

that characterizes the pattern of change among variables. These functional

relationships include exponential growth and decay (e.g., cell division,
depreciation)

F.8.4 Use linear equations and inequalities in a variety
of ways, including

• writing them to represent problem situations and to express generalizations

• solving them by different methods (e.g., informally, graphically, with formal

properties, with technology)

• writing and evaluating formulas (including solving for a specified variable)

• using them to record and describe solution strategies

F.8.5 Recognize and use generalized properties and
relations, including

• additive and multiplicative property of equations and inequalities

• commutativity* and associativity* of addition and multiplication

• distributive* property

• inverses* and identities* for addition and multiplication

• transitive* property

**By the end of grade 12 students will:**

F.12.1 Analyze and generalize patterns of change (e.g.,
direct and inverse variation) and

numerical sequences, and then represent them with algebraic expressions and

equations

F.12.2 Use mathematical functions* (e.g., linear*,
exponential*, quadratic*, power) in a

variety of ways, including

• recognizing that a variety of mathematical and real-world phenomena can be

modeled* by the same type of function

• translating different forms of representing them (e.g., tables, graphs,
functional

notation*, formulas)

• describing the relationships among variable quantities in a problem

• using appropriate technology to interpret properties of their graphical

representations (e.g., intercepts, slopes, rates of change, changes in rates of
change,

maximum*, minimum*)

F.12.3 Solve linear and quadratic equations, linear
inequalities, and systems of linear

equations and inequalities

• numerically

• graphically, including use of appropriate technology

• symbolically, including use of the quadratic formula

F.12.4 Model and solve a variety of mathematical and
real-world problems by using

algebraic expressions, equations, and inequalities

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