 # 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

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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