Chapter Three: 2005 NAEP Mathematics Objectives


In order to describe the specific mathematics that should be assessed at each grade level, it is necessary to organize the domain of mathematics into component parts. This is accomplished by utilizing the five content areas, as described in chapter 2. Though such an organization brings with it the danger of fragmentation, the hope is that the objectives and the test items built on them will, in many cases, cross some of the boundaries of these content areas.

One of the goals of this framework is to provide more clarity and specificity in the objectives for each grade level. To accomplish this, a matrix was created that depicts the particular objectives that are appropriate for assessment under each subtopic. Within Number, for example, and the subtopic of Number Sense, specific objectives are listed for assessment at grade 4, grade 8, and grade 12. The same objective at different grade levels depicts a developmental sequence for that concept or skill. An empty cell in the matrix is used to convey the fact that a particular objective is not appropriate for assessment at that grade level.

In order to fully understand the objectives and their intent, please note the following:

  • Further clarification of some of these objectives, along with some sample items, may be found in the companion document, Assessment and Item Specifications for the 2005 NAEP Mathematics Assessment.

  • While all test items will be assigned a primary classification, some test items could potentially fall into more than one content area or more than one objective.

  • When the word “or” is used in an objective, it should be understood as inclusive; that is, an item may assess one or more of the concepts included. However, all concepts described should be measured across the full range of the assessment.

  • These objectives describe what is to be assessed on the 2005 NAEP. They should not be interpreted as a complete description of mathematics that should be taught at these grade levels.

Mathematical Content Areas

NUMBER PROPERTIES AND OPERATIONS

Numbers are our main tools for describing the world quantitatively. As such, they deserve a privileged place in the 2005 NAEP framework. With whole numbers, we can count collections of discrete objects of any type. We can also use numbers to describe fractional parts and even to describe continuous quantities such as length, area, volume, weight, and time, and more complicated derived quantities such as rates, speed, density, inflation, interest, and so forth. Thanks to Cartesian coordinates, we can use pairs of numbers to describe points in a plane or triples of numbers to label points in space. Numbers let us talk in a precise way about anything that can be counted, measured, or located in space.

Numbers are not simply labels for quantities. They form systems with their own internal structure. The arithmetic operations (addition and subtraction, multiplication and division) help us model basic real-world operations. For example, joining two collections, or laying two lengths end to end, can be described by addition, while the concept of rate depends on division. Multiplication and division of whole numbers lead to the beginnings of number theory, including concepts of factorization, remainder, and prime number. Besides the arithmetic operations, the other basic structure of the real numbers is ordering, as in which is greater and lesser. These reflect our intuitions about the relative size of quantities, and provide a basis for making sensible estimates.

The accessibility and usefulness of arithmetic are greatly enhanced by our efficient means for representing numbers: the Hindu-Arabic decimal place value system. In its full development, this remarkable system includes decimal fractions, which let us approximate any real number as closely as we wish. Decimal notation allows us to do arithmetic by means of simple, routine algorithms, and it also makes size comparisons and estimation easy. The decimal system achieves its efficiency through sophistication, as all the basic algebraic operations are implicitly used in writing decimal numbers. To represent ratios of two whole numbers exactly, we supplement decimal notation with fractions.

Comfort in dealing with numbers effectively is called number sense. It includes firm intuitions about what numbers tell us; an understanding of the ways to represent them symbolically (including facility with converting between different representations); the ability to calculate, either exactly or approximately, and by several means (mentally, with paper and pencil, or with calculator, as appropriate); and skill in estimation. The ability to deal with proportion, including percents, is another important part of number sense.

Number sense is a major expectation of the 2005 NAEP. At fourth grade, students are expected to have a solid grasp of whole numbers, as represented by the decimal system, and to have the beginnings of understanding fractions. By eighth grade, they should be comfortable with rational numbers, represented either as decimal fractions (including percents) or as common fractions. They should be able to use them to solve problems involving proportionality and rates. In middle school also, number should begin to coalesce with geometry via the idea of the number line. This should be connected with ideas of approximation and the use of scientific notation. Eighth graders should also have some acquaintance with naturally occurring irrational numbers, such as square roots and pi. By 12th grade, students should be comfortable dealing with all types of real numbers.

Number Properties and Operations

1) Number sense
GRADE 4 GRADE 8 GRADE 12
a) Identify the place value and actual value of digits in whole numbers. a) Use place value to model and describe integers and decimals.  
b) Represent numbers using models such as base 10 representations, number lines, and two-dimensional models. b) Model or describe rational numbers or numerical relationships using number lines and diagrams.  
c) Compose or decompose whole quantities by place value (e.g., write whole numbers in expanded notation using place value: 342 = 300 + 40 + 2).    
d) Write or rename whole numbers (e.g., 10: 5 + 5, 12 – 2, 2 x 5). d) Write or rename rational numbers. d) Write, rename, represent, or compare real numbers (e.g., pi, square root of 2, numerical relationships using number lines, models, or diagrams).
e) Connect model, number word, or number using various models and representations for whole numbers, fractions, and decimals. e) Recognize, translate between, or apply multiple representations of rational numbers (fractions, decimals, and percents) in meaningful contexts.  
  f) Express or interpret numbers using scientific notation from real-life contexts. f) Represent very large or very small numbers using scientific notation in meaningful contexts.
  g) Find or model absolute value or apply to problem situations. g) Find or model absolute value or apply to problem situations.
    h) Interpret calculator or computer displays of numbers given in scientific notation.
  i) Order or compare rational numbers (fractions, decimals, percents, or integers) using various models and representations (e.g., number line).  
j) Order or compare whole numbers, decimals, or fractions. j) Order or compare rational numbers including very large and small integers, and decimals and fractions close to zero. j) Order or compare real numbers, including very large or small real numbers.


2) Estimation
GRADE 4 GRADE 8 GRADE 12
a) Use benchmarks (well-known numbers used as meaningful points for comparison) for whole numbers, decimals, or fractions in contexts (e.g., 1/2 and .5 may be used as benchmarks for fractions and decimals between 0 and 1.00). a) Establish or apply benchmarks for rational numbers and common irrational numbers (e.g., ) in contexts. a) Establish or apply benchmarks for real numbers in contexts.
b) Make estimates appropriate to a given situation with whole numbers, fractions, or decimals by:
  • knowing when to estimate,
  • selecting the appropriate type of estimate, including over- estimate, underestimate, and range of estimate, or
  • selecting the appropriate method of estimation (e.g., rounding).

b) Make estimates appropriate to a given situation by:

  • identifying when estimation is appropriate,
  • determining the level of accuracy needed,
  • selecting the appropriate method of estimation, or
  • analyzing the effect of an estimation method on the accuracy of results.

b) Make estimates of very large or very small numbers appropriate to a given situation by:

  • identifying when estimation is appropriate or not,
  • determining the level of accuracy needed,
  • selecting the appropriate method of estimation, or
  • analyzing the effect of an estimation method on the accuracy of results.
c) Verify solutions or determine the reasonableness of results in meaningful contexts. c) Verify solutions or determine the reasonableness of results in a variety of situations including calculator and computer results. c) Verify solutions or determine the reasonableness of results in a variety of situations including scientific notation, calculator, and computer results.
  d) Estimate square or cube roots of numbers less than 1,000 between two whole numbers. d) Estimate square or cube roots of numbers less than 1,000 between two whole numbers.


3) Number operations
GRADE 4 GRADE 8 GRADE 12
a) Add and subtract:
  • whole numbers, or
  • fractions with like denominators, or
  • decimals through hundredths.
a) Perform computations with rational numbers. a) Perform computations with real numbers including common irrational numbers or the absolute value of numbers.
b) Multiply whole numbers:
  • no larger than two-digit by two-digit with paper and pencil computation, or
  • larger numbers with use of calculator.
   
c) Divide whole numbers:
  • up to three-digits by one-digit with paper and pencil computation, or
  • up to five-digits by two-digits with use of calculator.
   
d) Describe the effect of operations on size (whole numbers). d) Describe the effect of multiplying and dividing by numbers including the effect of multiplying or dividing a rational number by:
  • zero, or
  • a number less than zero, or
  • a number between zero and one,
  • one, or
  • a number greater than one.
d) Describe the effect of multiplying and dividing by numbers including the effect of multiplying or dividing a real number by:
  • zero, or
  • a number less than zero, or
  • a number between zero and one, or
  • one, or
  • a number greater than one.
  e) Provide a mathematical argument to explain operations with two or more fractions.  
f) Interpret whole number operations and the relationships between them. f) Interpret rational number operations and the relationships between them.  
g) Solve application problems involving numbers and operations. g) Solve application problems involving rational numbers and operations using exact answers or estimates as appropriate. g) Solve application problems involving numbers, including rational and common irrationals, using exact answers or estimates as appropriate.


4) Ratios and proportional reasoning
GRADE 4 GRADE 8 GRADE 12
a) Use simple ratios to describe problem situations. a) Use ratios to describe problem situations.  
  b) Use fractions to represent and express ratios and proportions. b) Use proportions to model problems.
  c) Use proportional reasoning to model and solve problems (including rates and scaling). c) Use proportional reasoning to solve problems (including rates).
  d) Solve problems involving percentages (including percent increase and decrease, interest rates, tax, discount, tips, or part/whole relationships). d) Solve problems involving percentages (including percent increase and decrease, interest rates, tax, discount, tips, or part/whole relationships).


5) Properties of number and operations
GRADE 4 GRADE 8 GRADE 12
a) Identify odd and even numbers. a) Describe odd and even integers and how they behave under different operations.  
b) Identify factors of whole numbers. b) Recognize, find, or use factors, multiples, or prime factorization. b) Solve problems involving factors, multiples, or prime factorization.
  c) Recognize or use prime and composite numbers to solve problems. c) Use prime or composite numbers to solve problems.
  d) Use divisibility or remainders in problem settings. d) Use divisibility or remainders in problem settings.
e) Apply basic properties of operations. e) Apply basic properties of operations. e) Apply basic properties of operations.
f) Explain or justify a mathematical concept or relationship (e.g., explain why 15 is an odd number or why 7–3 is not the same as 3–7). f) Explain or justify a mathematical concept or relationship (e.g., explain why 17 is prime). f) Provide a mathematical argument about a numerical property or relationship.


MEASUREMENT

Measuring is the process by which numbers are assigned in order to describe the world quantitatively. This process involves selecting the attribute of the object or event to be measured, comparing this attribute to a unit, and reporting the number of units. For example, in measuring a child, we may select the attribute of height and the inch as the unit for the comparison. In comparing the height to the inch, we may find that the child is about 42 inches. If considering only the domain of whole numbers, we would report that the child is 42 inches tall. However, since height is a continuous attribute, we may consider the domain of rational numbers and report that the child is 413/16 inches tall (to the nearest 16th of an inch). Measurement also allows us to model positive and negative numbers as well as irrational numbers.

This connection between measuring and number makes measuring a vital part of the school curriculum. Measurement models are often used when students are learning about number and operations. For example, area and volume models can help students understand multiplication and the properties of multiplication. Length models, especially the number line, can help students understand ordering and rounding numbers. Measurement also has a strong connection to other areas of school mathematics and to the other subjects in the school curriculum. Problems in algebra are often drawn from measurement situations. One can also consider measurement to be a function or a mapping of the attribute to a set of numbers. Much of school geometry focuses on the measurement aspect of geometric figures. Statistics also provides ways to measure and to compare sets of data. These are some of the ways in which measurement is intertwined with the other four content areas.

In this NAEP Mathematics Framework, attributes such as capacity, weight/mass, time, and temperature are included, as well as the geometric attributes of length, area, and volume. Although many of these attributes are included in the grade 4 framework, the emphasis is on length, including perimeter, distance, and height. More emphasis is placed on area and angle in grade 8. By grade 12, volumes and rates constructed from other attributes, such as speed, are emphasized.

Units involved in items on the NAEP assessment include nonstandard, customary, and metric units. At grade 4, common customary units such as inch, quart, pound, and hour and the common metric units such as centimeter, liter, and gram are emphasized. Grades 8 and 12 include the use of both square and cubic units for measuring area, surface area, and volume; degrees for measuring angles; and constructed units such as miles per hour. Converting from one unit in a system to another (such as from minutes to hours) is an important aspect of measurement included in problem situations. Understanding and using the many conversions available is an important skill. There are a limited number of common, everyday equivalencies that students are expected to know (see the Assessment and Item Specifications document for more detail).

Items classified in this content area depend on some knowledge of measurement. For example, an item that asks the difference between a 3-inch and a 13/4-inch line segment is a number item, while an item comparing a 2-foot segment with an 8-inch line segment is a measurement item. In many secondary schools, measurement becomes an integral part of geometry; this is reflected in the proportion of items recommended for these two areas.



Measurement

1) Measuring physical attributes
GRADE 4 GRADE 8 GRADE 12
a) Identify the attribute that is appropriate to measure in a given situation.    
b) Compare objects with respect to a given attribute, such as length, area, volume, time, or temperature. b) Compare objects with respect to length, area, volume, angle measurement, weight, or mass.  
c) Estimate the size of an object with respect to a given measurement attribute (e.g., length, perimeter, or area using a grid). c) Estimate the size of an object with respect to a given measurement attribute (e.g., area). c) Estimate or compare perimeters or areas of two-dimensional geometric figures.
    d) Estimate or compare volume or surface area of three-dimensional figures.
    e) Solve problems involving the coordinate plane such as the distance between two points, the midpoint of a segment, or slopes of perpendicular or parallel lines.
    f) Solve problems of angle measure, including those involving triangles or other polygons or parallel lines cut by a transversal.
g) Select or use appropriate measurement instruments such as ruler, meter stick, clock, thermometer, or other scaled instruments. g) Select or use appropriate measurement instrument to determine or create a given length, area, volume, angle, weight, or mass.  
h) Solve problems involving perimeter of plane figures. h) Solve mathematical or real-world problems involving perimeter or area of plane figures such as triangles, rectangles, circles, or composite figures. h) Solve mathematical or real-world problems involving perimeter or area of plane figures such as polygons, circles, or composite figures.
i) Solve problems involving area of squares and rectangles.    
  j) Solve problems involving volume or surface area of rectangular solids, cylinders, prisms, or composite shapes. j) Solve problems involving volume or surface area of rectangular solids, cylinders, cones, pyramids, prisms, spheres, or composite shapes.
  k) Solve problems involving indirect measurement such as finding the height of a building by comparing its shadow with the height and shadow of a known object. k) Solve problems involving indirect measurement such as finding the height of a building by finding the distance to the base of the building and the angle of elevation to the top.
  l) Solve problems involving rates such as speed or population density. l) Solve problems involving rates such as speed, density, population density, or flow rates.
    m) Use trigonometric relations in right triangles to solve problems.




2) Systems of measurement
GRADE 4 GRADE 8 GRADE 12
a) Select or use appropriate type of unit for the attribute being measured such as length, time, or temperature. a) Select or use appropriate type of unit for the attribute being measured such as length, area, angle, time, or volume. a) Select or use appropriate type of unit for the attribute being measured such as volume or surface area.
b) Solve problems involving conversions within the same measurement system such as conversions involving inches and feet or hours and minutes. b) Solve problems involving conversions within the same measurement system such as conversions involving square inches and square feet. b) Solve problems involving conversions within or between measurement systems, given the relationship between the units.
  c) Estimate the measure of an object in one system given the measure of that object in another system and the approximate conversion factor. For example:
  • Distance conversion: 1 kilometer is approximately 5/8 of a mile.
  • Money conversion: US dollar is approximately 1.5 Canadian dollars.
  • Temperature conversion: Fahrenheit to Celsius
 
d) Determine appropriate size of unit of measurement in problem situation involving such attributes as length, time, capacity, or weight. d) Determine appropriate size of unit of measurement in problem situation involving such attributes as length, area, or volume.  
e) Determine situations in which a highly accurate measurement is important. e) Determine appropriate accuracy of measurement in problem situations (e.g., the accuracy of each of several lengths needed to obtain a specified accuracy of a total length) and find the measure to that degree of accuracy. e) Determine appropriate accuracy of measurement in problem situations (e.g., the accuracy of measurement of the dimensions to obtain a specified accuracy of area) and find the measure to that degree of accuracy.
  f) Construct or solve problems (e.g., floor area of a room) involving scale drawings. f) Construct or solve problems (e.g., number of rolls needed for insulating a house) involving scale drawings.
    g) Compare lengths, areas, or volumes of similar figures using proportions.


GEOMETRY

Geometry began as a practical collection of rules for calculating lengths, areas, and volumes of common shapes. In classical times, the Greeks turned it into a subject for reasoning and proof, and Euclid organized their discoveries into a coherent collection of results, all deduced using logic from a small number of special assumptions called postulates. Euclid’s Elements stood as a pinnacle of human intellectual achievement for over 2000 years.

The 19th century saw a new flowering of geometric thought, going beyond Euclid, and leading to the idea that geometry is the study of the possible structures of space. This had its most striking application in Einstein’s theories of relativity, which describes the behavior of light, and also of gravity, in terms of a four-dimensional geometry, which combines the usual three dimensions of space with time as an additional dimension.

A major insight of the 19th century is that geometry is intimately related to ideas of symmetry and transformation. The symmetry of familiar shapes under simple transformations (that our bodies look more or less the same if reflected across the middle, or that a square looks the same if rotated by 90 degrees) is a matter of everyday experience. Many of the standard terms for triangles (scalene, isosceles, equilateral) and quadrilaterals (parallelogram, rectangle, rhombus, square) refer to symmetry properties. Also, the behavior of figures under changes of scale is an aspect of symmetry with myriad practical consequences. At a deeper level, the fundamental ideas of geometry itself (for example, congruence) depend on transformation and invariance. In the 20th century, symmetry ideas were seen to also underlie much of physics, not only Einstein’s relativity theories, but atomic physics and solid-state physics (the field that produced computer chips).

School geometry roughly mirrors the historical development through Greek times with some modern additions, most notably symmetry and transformations. By grade 4, students are expected to be familiar with a library of simple figures and their attributes, both in the plane (lines, circles, triangles, rectangles, and squares) and in space (cubes, spheres, and cylinders). In middle school, understanding of these shapes deepens, with the study of cross-sections of solids and the beginnings of an analytical understanding of properties of plane figures, especially parallelism, perpendicularity, and angle relations in polygons. Right angles and the Pythagorean theorem are introduced, and geometry becomes more and more mixed with measurement. The basis for analytic geometry is laid by study of the number line. In high school, attention is given to Euclid’s legacy and the power of rigorous thinking. Students are expected to make, test, and validate conjectures. Via analytic geometry, the key areas of geometry and algebra are merged into a powerful tool that provides a basis for calculus and the applications of mathematics that helped create the modern technological world in which we live.

Symmetry is an increasingly important component of geometry. Elementary students are expected to be familiar with the basic types of symmetry transformations of plane figures, including flips (reflection across lines), turns (rotations around points), and slides (translations). In middle school, this knowledge becomes more systematic and analytical, with each type of transformation being distinguished from other types by their qualitative effects. For example, a rigid motion of the plane that leaves at least two points fixed (but not all points) must be a reflection in a line. In high school, students are expected to be able to represent transformations algebraically. Some may also gain insight into systematic structure, such as the classification of rigid motions of the plane as reflections, rotations, translations, or glide reflections, and what happens when two or more isometries are performed in succession (composition).

Geometry

1) Dimension and shape
GRADE 4 GRADE 8 GRADE 12
a) Explore properties of paths between points. a) Draw or describe a path of shortest length between points to solve problems in context.  
b) Identify or describe (informally) real-world objects using simple plane figures (e.g., triangles, rectangles, squares, and circles) and simple solid figures (e.g., cubes, spheres, and cylinders). b) Identify a geometric object given a written description of its properties. b) Use two-dimensional representations of three-dimensional objects to visualize and solve problems involving surface area and volume.
c) Identify or draw angles and other geometric figures in the plane. c) Identify, define, or describe geometric shapes in the plane and in three-dimensional space given a visual representation. c) Give precise mathematical descriptions or definitions of geometric shapes in the plane and in three-dimensional space.
  d) Draw or sketch from a written description polygons, circles, or semicircles. d) Draw or sketch from a written description plane figures (e.g., isosceles triangles, regular polygons, curved figures) and planar images of three-dimensional figures (e.g., polyhedra, spheres, and hemispheres).
  e) Represent or describe a three-dimensional situation in a two-dimensional drawing from different views. e) Describe or analyze properties of spheres and hemispheres.
f) Describe attributes of two- and three-dimensional shapes. f) Demonstrate an understanding about the two- and three-dimensional shapes in our world through identifying, drawing, modeling, building, or taking apart.  


2) Transformation of shapes and preservation of properties
GRADE 4 GRADE 8 GRADE 12
a) Identify whether a figure is symmetrical, or draw lines of symmetry. a) Identify lines of symmetry in plane figures or recognize and classify types of symmetries of plane figures. a) Recognize or identify types of symmetries (e.g., point, line, rotational, self-congruences) of two- and three-dimensional figures.
    b) Give or recognize the precise mathematical relationship (e.g., congruence, similarity, orientation) between a figure and its image under a transformation.
c) Identify the images resulting from flips (reflections), slides (translations), or turns (rotations). c) Recognize or informally describe the effect of a transformation on two-dimensional geometric shapes (reflections across lines of symmetry, rotations, translations, magnifications, and contractions). c) Perform or describe the effect of a single transformation on two- and three-dimensional geometric shapes (reflections across lines of symmetry, rotations, translations, and dilations).
d) Recognize which attributes (such as shape and area) change or don’t change when plane figures are cut up or rearranged. d) Predict results of combining, subdividing, and changing shapes of plane figures and solids (e.g., paper folding, tiling, and cutting up and rearranging pieces). d) Describe the final outcome of successive transformations.
e) Match or draw congruent figures in a given collection. e) Justify relationships of congruence and similarity, and apply these relationships using scaling and proportional reasoning. e) Justify relationships of congruence and similarity, and apply these relationships using scaling and proportional reasoning.
  f) For similar figures, identify and use the relationships of conservation of angle and of proportionality of side length and perimeter.  


3) Relationships between geometric figures
GRADE 4 GRADE 8 GRADE 12
a) Analyze or describe patterns of geometric figures by increasing number of sides, changing size or orientation (e.g., polygons with more and more sides).    
b) Assemble simple plane shapes to construct a given shape. b) Apply geometric properties and relationships in solving simple problems in two and three dimensions. b) Apply geometric properties and relationships in solving multistep problems in two and three dimensions (including rigid and nonrigid figures).
c) Recognize two-dimensional faces of three-dimensional shapes. c) Represent problem situations with simple geometric models to solve mathematical or realworld problems. c) Represent problem situations with geometric models to solve mathematical or real-world problems.
  d) Use the Pythagorean theorem to solve problems. d) Use the Pythagorean theorem to solve problems in two- or three-dimensional situations.
    e) Describe and analyze properties of circles (e.g., perpendicularity of tangent and radius, angle inscribed in a semicircle).
f) Describe and compare properties of simple and compound figures composed of triangles, squares, and rectangles. f) Describe or analyze simple properties of, or relationships between, triangles, quadrilaterals, and other polygonal plane figures. f) Analyze properties or relationships of triangles, quadrilaterals, and other polygonal plane figures.
  g) Describe or analyze properties and relationships of parallel or intersecting lines. g) Describe or analyze properties and relationships of parallel, perpendicular, or intersecting lines, including the angle relationships that arise in these cases.


4) Position and direction
GRADE 4 GRADE 8 GRADE 12
a) Describe relative positions of points and lines using the geometric ideas of parallelism or perpendicularity. a) Describe relative positions of points and lines using the geometric ideas of midpoint, points on common line through a common point, parallelism, or perpendicularity.  
  b) Describe the intersection of two or more geometric figures in the plane (e.g., intersection of a circle and a line). b) Describe the intersections of lines in the plane and in space, intersections of a line and a plane, or of two planes in space.
  c) Visualize or describe the cross section of a solid. c) Describe or identify conic sections and other cross sections of solids.
d) Construct geometric figures with vertices at points on a coordinate grid. d) Represent geometric figures using rectangular coordinates on a plane. d) Represent two-dimensional figures algebraically using coordinates and/or equations.
    e) Use vectors to represent velocity and direction.


5) Mathematical reasoning
GRADE 4 GRADE 8 GRADE 12
a) Distinguish which objects in a collection satisfy a given geometric definition and explain choices. a) Make and test a geometric conjecture about regular polygons. a) Make, test, and validate geometric conjectures using a variety of methods including deductive reasoning and counterexamples.

DATA ANALYSIS AND PROBABILITY

Data analysis covers the entire process of collecting, organizing, summarizing, and interpreting data. This is the heart of the discipline called statistics; it is in evidence whenever quantitative information is used to determine a course of action. To emphasize the spirit of statistical thinking, data analysis should begin with a question to be answered, not with the data. Data should be collected only with a specific question (or questions) in mind and only after a plan (usually called a design) for collecting data relevant to the question is thought out. Beginning at an early age, students should grasp the fundamental principle that looking for questions in an existing data set is far different from the scientific method of collecting data to verify or refute a well-posed question. A pattern can be found in almost any data set if one looks hard enough, but a pattern discovered in this way is often meaningless, especially from the point of view of statistical inference.

In the context of data analysis, or statistics, probability can be thought of as the study of potential patterns in outcomes that have not yet been observed. We say that the probability of a balanced coin coming up heads when flipped is one-half because we believe that about half of the flips would turn out to be heads if we flipped the coin many times. Under random sampling, patterns for outcomes of designed studies can be anticipated and used as the basis for making decisions. If the coin actually turned up heads 80 percent of the time, we would suspect that it was not balanced. The whole probability distribution of all possible outcomes is important in most statistics problems because the key to decisionmaking is to decide whether a particular observed outcome is unusual (located in a tail of the probability distribution). For example, 4 as a grade point average is unusually high among most groups of students, 4 as the pound weight of a baby is unusually low, and 4 as the number of runs scored in a baseball game is not unusual in either direction.

By grade 4, students should be expected to apply their understanding of number and quantity to pose questions that can be answered by collecting appropriate data. They should be expected to organize data in a table or a plot and summarize the essential features of center, spread, and shape both verbally and with simple summary statistics. Simple comparisons can be made between two related data sets, but more formal inference based on randomness should come later. The basic concept of chance and statistical reasoning can be built into meaningful contexts, though, such as, “If I draw two names from among those of the students in the room, am I likely to get two girls?” Such problems can be addressed through simulation.

Building on the same definition of data analysis and the same principles of describing distributions of data through center, spread, and shape, grade 8 students will be expected to use a wider variety of organizing and summarizing techniques. They can also begin to analyze statistical claims through designed surveys and experiments that involve randomization, with simulation being the main tool for making simple statistical inferences. They will begin to use more formal terminology related to probability and data analysis.

Students in grade 12 will be expected to use a wide variety of statistical techniques for all phases of the data analysis process, including a more formal understanding of statistical inference (but still with simulation as the main inferential analysis tool). In addition to comparing univariate data sets, students at this level should be able to recognize and describe possible associations between two variables by looking at two-way tables for categorical variables or scatterplots for measurement variables. Association between variables is related to the concepts of independence and dependence, and an understanding of these ideas requires knowledge of conditional probability. These students should be able to use statistical models (linear and nonlinear equations) to describe possible associations between measurement variables and should be familiar with techniques for fitting models to data.

Data Analysis and Probability

1) Data representation
GRADE 4 GRADE 8 GRADE 12
The following representations of data are indicated for each grade level. Objectives in which only a subset of these representations is applicable are indicated in the parenthesis associated with the objective.
Pictographs, bar graphs, circle graphs, line graphs, line plots, tables, and tallies. Histograms, line graphs, scatterplots, box plots, circle graphs, stem and leaf plots, frequency distributions, tables, and bar graphs. Histograms, line graphs, scatterplots, box plots, circle graphs, stem and leaf plots, frequency distributions, and tables.
a) Read or interpret a single set of data. a) Read or interpret data, including interpolating or extrapolating from data. a) Read or interpret data, including interpolating or extrapolating from data.
b) For a given set of data, complete a graph (limits of time make it difficult to construct graphs completely). b) For a given set of data, complete a graph and then solve a problem using the data in the graph (histograms, line graphs, scatterplots, circle graphs, and bar graphs). b) For a given set of data, complete a graph and then solve a problem using the data in the graph (histograms, scatterplots, line graphs).
c) Solve problems by estimating and computing within a single set of data. c) Solve problems by estimating and computing with data from a single set or across sets of data. c) Solve problems by estimating and computing with univariate or bivariate data (including scatterplots and two-way tables).
  d) Given a graph or a set of data, determine whether information is represented effectively and appropriately (histograms, line graphs, scatterplots, circle graphs, and bar graphs). d) Given a graph or a set of data, determine whether information is represented effectively and appropriately (bar graphs, box plots, histograms, scatterplots, line graphs).
  e) Compare and contrast the effectiveness of different representations of the same data. e) Compare and contrast the effectiveness of different representations of the same data.


2) Characteristics of data sets
GRADE 4 GRADE 8 GRADE 12
  a) Calculate, use, or interpret mean, median, mode, or range. a) Calculate, interpret, or use mean, median, mode, range, interquartile range, or standard deviation.
b) Given a set of data or a graph, describe the distribution of the data using median, range, or mode. b) Describe how mean, median, mode, range, or interquartile ranges relate to the shape of the distribution. b) Recognize how linear transformations of one-variable data affect mean, median, mode, and range (e.g., effect on the mean by adding a constant to each data point).
  c) Identify outliers and determine their effect on mean, median, mode, or range. c) Determine the effect of outliers on mean, median, mode, range, interquartile range, or standard deviation.
d) Compare two sets of related data. d) Using appropriate statistical measures, compare two or more data sets describing the same characteristic for two different populations or subsets of the same population. d) Compare two or more data sets using mean, median, mode, range, interquartile range, or standard deviation describing the same characteristic for two different populations or subsets of the same population.
  e) Visually choose the line that best fits given a scatterplot and informally explain the meaning of the line. Use the line to make predictions. e) Given a set of data or a scatterplot, visually choose the line of best fit and explain the meaning of the line. Use the line to make predictions.
    f) Use or interpret a normal distribution as a mathematical model appropriate for summarizing certain sets of data.
    g) Given a scatterplot, make decisions or predictions involving a line or curve of best fit.
    h) Given a scatterplot, estimate the correlation coefficient (e.g., Given a scatterplot, is the correlation closer to 0, .5, or 1.0? Is it a positive or negative correlation?).


3) Experiments and samples
GRADE 4 GRADE 8 GRADE 12
  a) Given a sample, identify possible sources of bias in sampling. a) Identify possible sources of bias in data collection methods and describe how such bias can be controlled and reduced.
  b) Distinguish between a random and nonrandom sample. b) Recognize and describe a method to select a simple random sample.
    c) Make inferences from sample results.
  d) Evaluate the design of an experiment. d) Identify or evaluate the characteristics of a good survey or of a well-designed experiment.


4) Probability
GRADE 4 GRADE 8 GRADE 12
a) Use informal probabilistic thinking to describe chance events (i.e., likely and unlikely, certain and impossible). a) Analyze a situation that involves probability of an independent event. a) Analyze a situation that involves probability of independent or dependent events.
b) Determine a simple probability from a context that includes a picture. b) Determine the theoretical probability of simple and compound events in familiar contexts. b) Determine the theoretical probability of simple and compound events in familiar or unfamiliar contexts.
  c) Estimate the probability of simple and compound events through experimentation or simulation. c) Given the results of an experiment or simulation, estimate the probability of simple or compound events in familiar or unfamiliar contexts.
  d) Use theoretical probability to evaluate or predict experimental outcomes. d) Use theoretical probability to evaluate or predict experimental outcomes.
e) List all possible outcomes of a given situation or event. e) Determine the sample space for a given situation. e) Determine the number of ways an event can occur using tree diagrams, formulas for combinations and permutations, or other counting techniques.
  f) Use a sample space to determine the probability of the possible outcomes of an event. f) Determine the probability of the possible outcomes of an event.
g) Represent the probability of a given outcome using a picture or other graphic. g) Represent probability of a given outcome using fractions, decimals, and percents.  
  h) Determine the probability of independent and dependent events. (Dependent events should be limited to linear functions with a small sample size.) h) Determine the probability of independent and dependent events.
    i) Determine conditional probability using two-way tables.
  j) Interpret probabilities within a given context. j) Interpret probabilities within a given context.


ALGEBRA

Algebra was pioneered in the Middle Ages by mathematicians in the Middle East and Asia as a method of solving equations easily and efficiently by manipulation of symbols, rather than by the earlier geometric methods of the Greeks. The two approaches were eventually united in the analytic geometry of René Descartes. Modern symbolic notation, developed in the Renaissance, greatly enhanced the power of the algebraic method; from the 17th century forward, algebra in turn promoted advances in all branches of mathematics and science.

The widening use of algebra led to the study of its formal structure. Out of this were gradually distilled the “rules of algebra,” a compact summary of the principles behind algebraic manipulation. A parallel line of thought produced a simple but flexible concept of function and also led to the development of set theory as a comprehensive background for mathematics. When it is taken liberally to include these ideas, algebra reaches from the foundations of mathematics to the frontiers of current research.

These two aspects of algebra, a powerful representational tool and a vehicle for comprehensive concepts such as function, form the basis for the expectations throughout the grades. By grade 4, students are expected to be able to recognize and extend simple numeric patterns as one foundation for a later understanding of function. They can begin to understand the meaning of equality and some of its properties, as well as the idea of an unknown quantity as a precursor to the concept of variable.

As students move into middle school, the ideas of function and variable become more important. Representation of functions as patterns, via tables, verbal descriptions, symbolic descriptions, and graphs, can combine to promote a flexible grasp of the idea of function. Linear functions receive special attention. They connect to the ideas of proportionality and rate, forming a bridge that will eventually link arithmetic to calculus. Symbolic manipulation in the relatively simple context of linear equations is reinforced by other means of finding solutions, including graphing by hand or with calculators.

In high school, students should become comfortable in manipulating and interpreting more complex expressions. The rules of algebra should come to be appreciated as a basis for reasoning.

Nonlinear functions, especially quadratic functions, and also power and exponential functions, are introduced to solve real-world problems. Students should become accomplished at translating verbal descriptions of problem situations into symbolic form. Expressions involving several variables, systems of linear equations, and the solutions to inequalities are encountered by grade 12.

Algebra

1) Patterns, relations, and functions
GRADE 4 GRADE 8 GRADE 12
a) Recognize, describe, or extend numerical patterns. a) Recognize, describe, or extend numerical and geometric patterns using tables, graphs, words, or symbols. a) Recognize, describe, or extend arithmetic, geometric progressions, or patterns using words or symbols.
b) Given a pattern or sequence, construct or explain a rule that can generate the terms of the pattern or sequence. b) Generalize a pattern appearing in a numerical sequence or table or graph using words or symbols. b) Express the function in general terms (either recursively or explicitly), given a table, verbal description, or some terms of a sequence.
c) Given a description, extend or find a missing term in a pattern or sequence. c) Analyze or create patterns, sequences, or linear functions given a rule.  
d) Create a different representation of a pattern or sequence given a verbal description.    
e) Recognize or describe a relationship in which quantities change proportionally. e) Identify functions as linear or nonlinear or contrast distinguishing properties of functions from tables, graphs, or equations. e) Identify or analyze distinguishing properties of linear, quadratic, inverse (y = k/x) or exponential functions from tables, graphs, or equations.
  f) Interpret the meaning of slope or intercepts in linear functions.  
    g) Determine the domain and range of functions given various contexts.
    h) Recognize and analyze the general forms of linear, quadratic, inverse, or exponential functions (e.g., in y = ax + b, recognize the roles of a and b).
    i) Express linear and exponential functions in recursive and explicit form given a table or verbal description.


2) Algebraic representations
GRADE 4 GRADE 8 GRADE 12
a) Translate between the different forms of representations (symbolic, numerical, verbal, or pictorial) of whole number relationships (such as from a written description to an equation or from a function table to a written description). a) Translate between different representations of linear expressions using symbols, graphs, tables, diagrams, or written descriptions. a) Translate between different representations of algebraic expressions using symbols, graphs, tables, diagrams, or written descriptions.
  b) Analyze or interpret linear relationships expressed in symbols, graphs, tables, diagrams, or written descriptions. b) Analyze or interpret relationships expressed in symbols, graphs, tables, diagrams, or written descriptions.
c) Graph or interpret points with whole number or letter coordinates on grids or in the first quadrant of the coordinate plane. c) Graph or interpret points that are represented by ordered pairs of numbers on a rectangular coordinate system. c) Graph or interpret points that are represented by one or more ordered pairs of numbers on a rectangular coordinate system.
  d) Solve problems involving coordinate pairs on the rectangular coordinate system. d) Perform or interpret transformations on the graphs of linear and quadratic functions.
e) Verify a conclusion using algebraic properties. e) Make, validate, and justify conclusions and generalizations about linear relationships. e) Use algebraic properties to develop a valid mathematical argument.
    f) Use an algebraic model of a situation to make inferences or predictions.
  g) Identify or represent functional relationships in meaningful contexts including proportional, linear, and common nonlinear (e.g., compound interest, bacterial growth) in tables, graphs, words, or symbols. g) Given a real-world situation, determine if a linear, quadratic, inverse, or exponential function fits the situation (e.g., half-life bacterial growth).
    h) Solve problems involving exponential growth and decay.


3) Variables, expressions, and operations
GRADE 4 GRADE 8 GRADE 12
a) Use letters and symbols to represent an unknown quantity in a simple mathematical expression.    
b) Express simple mathematical relationships using number sentences. b) Write algebraic expressions, equations, or inequalities to represent a situation. b) Write algebraic expressions, equations, or inequalities to represent a situation.
  c) Perform basic operations, using appropriate tools, on linear algebraic expressions (including grouping and order of multiple operations involving basic operations, exponents, roots, simplifying, and expanding). c) Perform basic operations, using appropriate tools, on algebraic expressions (including grouping and order of multiple operations involving basic operations, exponents, roots, simplifying, and expanding).
    d) Write equivalent forms of algebraic expressions, equations, or inequalities to represent and explain mathematical relationships.


4) Equations and inequalities
GRADE 4 GRADE 8 GRADE 12
a) Find the value of the unknown in a whole number sentence. a) Solve linear equations or inequalities (e.g., ax + b = c or ax + b = cx + d or ax + b > c). a) Solve linear, rational, or quadratic equations or inequalities.
  b) Interpret "=" as an equivalence between two expressions and use this interpretation to solve problems.  
  c) Analyze situations or solve problems using linear equations and inequalities with rational coefficients symbolically or graphically (e.g., ax + b = c or ax + b = cx + d). c) Analyze situations or solve problems using linear or quadratic equations or inequalities symbolically or graphically.
  d) Interpret relationships between symbolic linear expressions and graphs of lines by identifying and computing slope and intercepts (e.g., know in y = ax + b, that a is the rate of change and b is the vertical intercept of the graph). d) Recognize the relationship between the solution of a system of linear equations and its graph.
  e) Use and evaluate common formulas [e.g., relationship between a circle’s circumference and diameter (C = pi d), distance and time under constant speed]. e) Solve problems involving more advanced formulas [e.g., the volumes and surface areas of three dimensional solids; or such formulas as: A = P(1 + r)t, A = Pert].
    f) Given a familiar formula, solve for one of the variables.
    g) Solve or interpret systems of equations or inequalities.


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Mathematics Framework for the 2005 National Assessment of Educational Progress