(This page modifed from original NAEP web page by Arthur Hu)

Chapter Three
1996 NAEP Mathematics Objectives

url: http://www.nagb.org/mchap3.html#content


Mathematical Content Areas and Assessment Strands

To conduct a meaningful assessment of mathematics proficiency, it is necessary to measure students' proficiencies in various content areas. As in the 1990 and 1992 assessments, five mathematical strands will be used to categorize mathematical content for the 1996 mathematics assessment. The strands are illustrated later in this chapter. Classification of topics into these strands cannot be exact, however, and inevitably involves some overlap. For example, some topics appearing under Data Analysis, Statistics, and Probability may be closely related to others that appear under Algebra and Functions. As assessment programs continue to be refined, it becomes less desirable to force every item into only one content description category. As described in the NCTM Standards, students are expected to solve problems that naturally involve more than one specific mathematical topic. Consequently, the assessment as a whole will address the topics and subtopics identified in this chapter, and every item will be categorized under primarily one topic and subtopic so that analysis of results may be somewhat specific. Ideally, however, the items will involve students in synthesizing knowledge across topics and subtopics, and occasionally it may be difficult to identify a unique topic for each item. In fact, it is desirable that at least half of the new items for the 1996 assessment should involve content from more than one topic, or even from more than one strand.

The following sections of this chapter provide a brief description of each content strand with a list of topics and subtopics illustrative of those to be included in the assessment. This level of specificity is needed to guide item writers and ensure adequate coverage of the content areas and abilities to be assessed. The five content strands are largely consistent with the strands used in the 1990 and 1992 assessments. The titles and emphases of the content areas have been modified to reflect more clearly the directions for curriculum and evaluation described in the NCTM Standards.

For each of grades 4, 8, and 12, the following symbols are used: a "Y" indicates that the subtopic could be assessed at that grade level, a "" indicates that the subtopic should not be assessed at that grade level, and a "#" indicates that the subtopic might be introduced in the assessment at a very simple level, probably using a manipulative or pictorial model. The test specifications include additional detail and descriptions of how item types, families, calculators, manipulatives, and special studies fit within and across topics and subtopics.

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Number Sense, Properties, and Operations

This strand focuses on students' understanding of numbers (whole numbers, fractions, decimals, integers, real numbers, and complex numbers), operations, and estimation, and their application to real-world situations. Students will be expected to demonstrate an understanding of numerical relationships as expressed in ratios, proportions, and percents. Students also will be expected to understand properties of numbers and operations, generalize from numerical patterns, and verify results.

Number sense includes items that address a student's understanding of relative size, equivalent forms of numbers, and his or her use of numbers to represent attributes of real-world objects and quantities. Items that call for students to complete open sentences involving basic number facts are considered part of this content area. Items that require some application of the definition of operations and related procedures are classified under the area of Algebra and Functions.

As in the NCTM Standards, the emphasis in computation is on understanding when to use an operation, knowing what the operation means, and being able to estimate and use mental techniques, in addition to performing calculations using computational algorithms. In terms of actual computation, students will be expected to demonstrate that they know how to perform basic algorithms and use calculators in appropriate ways, given more complex situations. While a few isolated computation items may be included, a priority will be placed on including items in which operations are used in meaningful contexts.

The grade 4 assessment will emphasize the development of number sense through the connection of a variety of models to their numerical representations, as well as emphasizing an understanding of the meaning of addition, subtraction, multiplication, and division. These concepts will be addressed for whole numbers, simple fractions, and decimals at this grade level, with continual emphasis on the use of models and their connection to the use of symbols.

The grade 8 assessment will include number sense extended to include both positive and negative numbers and will address properties and operations involving whole numbers, fractions, decimals, integers, and rational numbers. The use of ratios and proportional thinking to represent situations involving quantity is a major focus at this grade level, and students will be expected to know how to read, use, and apply scientific notation to represent large and small numbers.

At grade 12, the assessment will include both real and complex numbers and will allow students to demonstrate competency through approximately the precalculus or calculus level. Operations with powers and roots, as well as a variety of real and complex numbers, may be assessed. Including a broad range of items at this level will ensure that students who have had different types of high school mathematics courses will be able to demonstrate proficiency on some parts of this content area.

1996 NAEP Mathematics Content Strand 1

Y Subtropic can be assessed at this grade level.
Subtropic should not be assessed at this grade level.
# Subtropic may be introduced at a simple level
(e.g., using a manipulative or pictorial model).
Grade
Number Sense, Properties, and Operations 4 8 12

1. Relate counting, grouping, and place value
a. Use place value to model and describe whole numbers and decimals Y Y Y
b. Use scientific notation in meaningful contexts N Y Y
2. Represent numbers and operations in a variety of equivalent forms using models, diagrams, and symbols.
a. Model numbers using set models such as counters Y N N
b. Model numbers using number lines Y Y N
c. Use two- and three-dimensional region models to describe numbers Y Y Y
d. Use other models appropriate to a given situation (e.g., draw diagrams to represent a number or an operation; write a number sentence to fit a situation or describe a situation to fit a number sentence; interpret calculator or computer displays) Y Y Y
e. Read, write, rename, order, and compare numbers Y Y Y
3. Compute with numbers (i.e., add, subtract, multiply, divide)
a. Apply basic properties of operations Y Y Y
b. Describe effect of operations on size and order of numbers Y Y Y
c. Describe features of algorithms (e.g., regrouping with or without manipulatives, partial products) Y Y Y
d. Select appropriate computation method (e.g., pencil and paper, calculator, mental arithmetic) Y Y Y
4. Use computation and estimation in applications
a. Round whole numbers, decimals, and fractions in meaningful contexts Y Y Y
b. Make estimates appropriate to a given situation
i. Know when to estimate Y Y Y
ii. Select appropriate type of estimate (overestimate, underestimate, range of estimate) Y Y Y
iii. Describe order of magnitude (estimation related to place value; scientific notation) Y Y Y
c. Select appropriate method of estimation (e.g., front end, rounding) Y Y Y
d. Solve application problems involving numbers and operations, using exact answers or estimates, as appropriate Y Y Y
e. Interpret round-off errors using calculators/computers (i.e., truncating) N # Y
f. Verify solutions and determine the reasonableness of results
i. in real-world situations Y Y Y
ii. in abstract settings N N Y
5. Apply ratios and proportional thinking in a variety of situations
a. Use ratios to describe situations # Y Y
b. Use proportions to model problems N Y Y
c. Use proportional thinking to solve problems (including rates, scaling, and similarity) N Y Y
d. Understand the meaning of percent (including percents greater than 100 and less than 1) # Y Y
e. Solve problems involving percent N Y Y
6. Use elementary number theory
a. Describe odd and even numbers and their characteristics Y Y Y
b. Describe number patterns # Y Y
c. Use factors and multiples to model and solve problems N Y Y
d. Describe prime numbers N Y Y
e. Use divisibility and remainders in problem settings (including simple modular arithmetic) N # Y

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Measurement

The measurement strand focuses on an understanding of the process of measurement and on the use of numbers and measures to describe and compare mathematical and real-world objects. Students will be asked to identify attributes, select appropriate units and tools, apply measurement concepts, and communicate measurement-related ideas.

Students should understand and be able to use the measurement attributes of length, mass/weight, capacity, time, money, and temperature. Students will demonstrate their ability to extend basic concepts in applications involving, for example, perimeter, area, surface area, volume, and angle measure.

Students will use measuring instruments and apply measurement concepts to solve problems. Due to the inherent imprecision of measurement tools, it is important for students to recognize that measurement is an approximation.

When students use technology for calculations with imprecise measurements, errors are often carried or increased. Students need to be assessed on their judgments about such answers.

Of these measurement concepts, the focus at grade 4 is on time, money, temperature, length, perimeter, area, capacity, weight/mass, and angle measure. While assessment at grades 8 and 12 continues to include these measurement concepts, the focus shifts to more complex measurement problems that involve volume or surface area or that require students to combine shapes, translate, and apply measures. Students at these grade levels also should solve problems involving proportional thinking (such as scale drawing or map reading) and do applications that involve the use of complex measurement formulas. When appropriate and possible, measurement will be assessed with real measuring devices.

Items requiring straightforward computation with measures, especially those involving time and money, are included not as part of this content area but as a part of Number Sense, Properties, and Operations, instead.

Applications involving measurement provide a rich source for families of questions that illustrate the connections among number sense and operations, algebra, and geometry.

1996 NAEP Mathematics Content Strand 2

Y Subtropic can be assessed at this grade level.
N Subtropic should not be assessed at this grade level.
# Subtropic may be introduced at a simple level
(e.g., using a manipulative or pictorial model).
Grade
Measurement 4 8 12

1. Estimate the size of an object or compare objects with respect to a given attribute (e.g., length, area, capacity, volume, and weight/mass) Y Y Y
2. Select and use appropriate measurement instruments (e.g., manipulatives such as ruler, meter stick, protractor, thermometer, scales for weight or mass, and gauges) Y Y Y
3. Select and use appropriate units of measurement, according to two criteria:
a. Type of unit Y Y Y
b. Size of unit Y Y Y
4. Estimate, calculate (using basic principles or formulas), or compare perimeter, area, volume, and surface area in meaningful contexts to solve mathematical and real-world problems
a. Solve problems involving perimeter and area (e.g., triangles, quadrilaterals, other polygons, circles, and combined forms)
[Note: Grade 4 tasks done with manipulatives]
# Y Y
b. Solve problems involving volume and surface area (e.g., rectangular solids, cylinders, cones, pyramids, prisms, and combined forms)
[Note: Grades 4 and 8 use manipulatives]
# # Y
5. Apply given measurement formulas for perimeter, area, volume, and surface area in problem settings N Y Y
6. Convert from one measurement to another within the same system (customary or metric) N Y Y
7. Determine precision, accuracy, and error
a. Apply significant digits in meaningful contexts N Y Y
b. Determine appropriate size of unit of measurement in problem situation N Y Y
c. Apply concepts of accuracy of measurement in problem situations N Y Y
d. Apply absolute and relative error in problem situations N N Y
8. Make and read scale drawings N Y Y
9. Select appropriate methods of measurement (e.g., direct or indirect) Y Y Y
10. Apply the concept of rate to measurement situations N Y Y

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Geometry and Spatial Sense

As described in the NCTM Standards, spatial sense must be an integral component of the study and assessment of geometry. Understanding spatial relationships allows students to use the dynamic nature of geometry to connect mathematics to their world.

This content area is designed to extend well beyond low-level identification of geometric shapes into transformations and combinations of those shapes. Informal constructions and demonstrations (including drawing representations), along with their justifications, take precedence over more traditional types of compass-and-straightedge constructions and proofs. While reasoning is addressed throughout all of the content areas, this strand continues to lend itself to the demonstration of reasoning within both formal and informal settings. The extension of proportional thinking to similar figures and indirect measurement is an important connection here.

In grade 4, students are expected to model properties of shapes under simple combinations and transformations, and they are expected to use mathematical communication skills to draw figures given a verbal description. For grade 8, students are expected to have extended their understanding to include properties of angles and polygons and to apply reasoning skills to make and validate conjectures about transformations and combinations of shapes. At grade 12, students are expected to demonstrate proficiency with transformational geometry and to apply concepts of proportional thinking to a variety of geometric situations. They will have opportunities to demonstrate more sophisticated reasoning processes than at earlier grade levels, and they will be expected to demonstrate a variety of algebraic and geometric connections. The importance of these connections and their use in solving problems is indicated by the shifting emphasis in geometry toward coordinate geometry, as described in Chapter Four.

1996 NAEP Mathematics Content Strand 3

Y Subtropic can be assessed at this grade level.
N Subtropic should not be assessed at this grade level.
# Subtropic may be introduced at a simple level
(e.g., using a manipulative or pictorial model).
Grade
Geometry and Spatial Sense 4 8 12

1. Describe, visualize, draw, and construct geometric figures
a. Draw or sketch a figure given a verbal description [open-ended items] Y Y Y
b. Given a figure, write a verbal description of its geometric qualities N Y Y
2. Investigate and predict results of combining, subdividing, and changing shapes (e.g., paper folding, dissecting, tiling, and rearranging pieces of solids) Y Y Y
3. Identify the relationship (congruence, similarity) between a figure and its image under a transformation
a. Use motion geometry (informal: lines of symmetry, flips, turns, and slides) Y Y Y
b. Use transformations (translations, rotations, reflections, dilations, and symmetry)
i. Synthetic N # Y
ii. Algebraic N N Y
4. Describe the intersection of two or more geometric figures
a. Two dimensional N Y Y
b. Planar cross-section of a solid N Y Y
5. Classify figures in terms of congruence and similarity, and informally apply these relationships using proportional reasoning where appropriate N Y Y
6. Apply geometric properties and relationships in solving problems
a. Use concepts of 'between,' 'inside,' 'on,' and 'outside' Y Y N
b. Use the Pythagorean relationship to solve problems N Y Y
c. Apply properties of ratio and proportion with respect to similarity N # Y
e. Solve problems involving right triangle trigonometric applications N N Y
7. Establish and explain relationships involving geometric concepts
a. Make conjectures Y Y Y
b. Validate and justify conclusions and generalizations Y Y Y
c. Use informal induction and deduction Y Y #
8. Represent problem situations with geometric models and apply properties of figures in meaningful contexts to solve mathematical and real-world problems Y Y Y
9. Represent geometric figures and properties algebraically using coordinates and vectors
a. Use properties of lines (including distance, midpoint, slope, parallelism and perpendicularity) to describe figures algebraically N # Y
b. Algebraically describe conic sections and their properties N N Y
c. Use vectors in problem situations (addition, subtraction, scalar multiplication, dot product) N N Y

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Data Analysis, Statistics, and Probability

Because of its fundamental role in making sense of the world, this content area will receive increased emphasis. The important skills of collecting, organizing, reading, representing, and interpreting data will be assessed in a variety of contexts to reflect the pervasive use of these skills in dealing with information. Statistics and statistical concepts extend these basic skills to include analyzing and communicating increasingly sophisticated interpretations of data. Dealing with uncertainty and making predictions about outcomes require an understanding not only of the meaning of basic probability concepts but also the application of those concepts in problem-solving and decision-making situations.

Questions will emphasize appropriate methods for gathering data, the visual exploration of data, a variety of ways of representing data, and the development and evaluation of arguments based on data analysis. Students will be expected to apply these ideas in increasingly sophisticated situations that require increasingly comprehensive analysis and decision making.

For grade 4, students will be expected to apply their understanding of number and quantity by solving problems involving data, and they will use data analysis to broaden their number sense. They will be expected to be familiar with a variety of types of graphs. They will be asked to make predictions from data and explain their reasoning, and they will deal informally with measures of central tendency. Grade 4 students will also use the basic concept of chance in meaningful contexts not involving the computation of probabilities.

Probabilistic thinking and a variety of specialized graphs become increasingly important in grades 8 and 12. Students in grade 8 will be expected to analyze statistical claims and design experiments, and they may use simulations to model real-world situations. They should have some understanding of sampling, and they should be asked to make predictions based on experiments or data. They will begin to use some formal terminology related to probability, data analysis, and statistics. By grade 8, students should be comfortable with a variety of types of graphs to represent different types of data in different situations.

Students in grade 12 will be expected to use a wide variety of statistical techniques to model situations and solve problems. Students at this level should apply concepts of probability to explore dependent and independent events, and they should be somewhat knowledgeable about conditional probability. They should be able to use formulas and more formal terminology to describe a variety of situations. By this level, students should have a basic understanding of the use of mathematical equations and graphs to interpret data, including the use of curve fitting to match a set of data with an appropriate mathematical model.

1996 NAEP Mathematics Content Strand 4

Y Subtropic can be assessed at this grade level.
N Subtropic should not be assessed at this grade level.
# Subtropic may be introduced at a simple level
(e.g., using a manipulative or pictorial model).
Grade
Data Analysis, Statistics, and Probability 4 8 12

1. Read, interpret, and make predictions using tables and graphs
a. Read and interpret data Y Y Y
b. Solve problems by estimating and computing with data Y Y Y
c. Interpolate or extrapolate from data N Y Y
2. Organize and display data and make inferences
a. Use tables, histograms (bar graphs), pictograms, and line graphs Y Y Y
b. Use circle graphs and scattergrams N Y Y
c. Use stem-and-leaf plots and box-and-whisker plots N Y Y
d. Make decisions about outliers N Y Y
3. Understand and apply sampling, randomness, and bias in data collection
a. Given a situation, identify sources of sampling error N Y Y
b. Describe a procedure for selecting an unbiased sample N Y Y
c. Make generalizations based on sample results N Y Y
4. Describe measures of central tendency and dispersion in real-world situations # Y Y
5. Use measures of central tendency, correlation, dispersion, and shapes of distributions to describe statistical relationships
a. Use standard deviation and variance N N Y
b. Use the standard normal distribution N N Y
c. Make predictions and decisions involving correlation N N Y
6. Understand and reason about the use and misuse of statistics in our society
a. Given certain situations and reported results, identify faulty arguments or misleading presentations of the data # Y Y
b. Appropriately apply statistics to real-world situations # Y Y
7. Fit a line or curve to a set of data and use this line or curve to make predictions about the data, using frequency distributions where appropriate N N Y
8. Design a statistical experiment to study a problem and communicate the outcomes N Y Y
9. Use basic concepts, trees, and formulas for combinations, permutations, and other counting techniques to determine the number of ways an event can occur N Y Y
10. Determine the probability of a simple event
a. Estimate probabilities by use of simulations N Y Y
b. Use sample spaces and the definition of probability to describe events Y Y Y
c. Describe and make predictions about expected outcomes N Y Y
11. Apply the basic concept of probability to real-world situations
a. Informal use of probabilistic thinking Y Y Y
b. Use probability related to independent and dependent events N Y Y
c. Use probability related to simple and compound events N N Y
d. Use conditional probability N N Y

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Algebra and Functions

This strand extends from work with simple patterns at grade 4, to basic algebra concepts at grade 8, to sophisticated analysis at grade 12, and involves not only algebra but also precalculus and some topics from discrete mathematics. As described in the NCTM Standards, these algebraic concepts are developed throughout the grades with informal modeling done at the elementary level and with increased emphasis on functions at the secondary level. The nature of the algebraic concepts and procedures included in the assessment at all levels will reflect the NCTM Standards. Students will be expected to use algebraic notation and thinking in meaningful contexts to solve mathematical and real-world problems, specifically addressing an increasing understanding of the use of functions (including algebraic and geometric) as a representational tool.

The assessment at all levels will include the use of open sentences and equations as representational tools. Students will use the notion of equivalent representations to transform and solve number sentences and equations of increasing levels of complexity.

The grade 4 assessment will involve informal demonstration of students' abilities to generalize from patterns, including the justification of their generalizations. Students will be expected to translate between mathematical representations, to use simple equations, and to do basic graphing.

At grade 8, the assessment will include more algebraic notation, stressing the meaning of variable and an informal understanding of the use of symbolic representations in problem-solving contexts. Students at this level will be asked to use variables to represent a rule underlying a pattern. They should have a beginning understanding of equations as a modeling tool, and they should solve simple equations and inequalities by a variety of methods, including both graphical and basic algebraic methods. Students should begin to use basic concepts of functions as a way of describing relationships.

By grade 12, students will be expected to be adept at appropriately choosing and applying a rich set of representational tools in a variety of problem-solving situations. They should have an understanding of basic algebraic notation and terminology as they relate to representations of mathematical and real-world problem situations. Students should be able to use functions as a way of representing and describing relationships.

1996 NAEP Mathematics Content Strand 5

Y Subtropic can be assessed at this grade level.
N Subtropic should not be assessed at this grade level.
# Subtropic may be introduced at a simple level
(e.g., using a manipulative or pictorial model).
Grade
Algebra and Functions 4 8 12

1. Describe, extend, interpolate, transform, and create a wide variety of patterns and functional relationships
a. Recognize patterns and sequences Y Y Y
b. Extend a pattern or functional relationship Y Y Y
c. Given a verbal description, extend or interpolate with a pattern (complete a missing term) N Y Y
d. Translate patterns from one context to another # Y Y
e. Create an example of a pattern or functional relationship Y Y Y
f. Understand and apply the concept of a variable # Y Y
2. Use multiple representations for situations to translate among diagrams, models, and symbolic expressions Y Y Y
3. Use number lines and rectangular coordinate systems as representational tools
a. Identify or graph sets of points on a number line or in a rectangular coordinate system Y Y Y
b. Identify or graph sets of points in a polar coordinate system N Y Y
c. Work with applications using coordinates N Y Y
d. Transform the graph of a function N # Y
4. Represent and describe solutions to linear equations and inequalities to solve mathematical and real-world problems
a. Solution sets of whole numbers Y Y Y
b. Solution sets of real numbers # Y Y
5. Interpret contextual situations and perform algebraic operations on real numbers and algebraic expressions to solve mathematical and real-world problems
a. Perform basic operations, using appropriate tools, on real numbers in meaningful contexts (including grouping and order of multiple operations involving basic operations, exponents, and roots) N Y Y
b. Solve problems involving substitution in expressions and formulas N Y Y
c. Solve meaningful problems involving a formula with one variable N Y Y
d. Use equivalent forms to solve problems N Y Y
6. Solve systems of equations and inequalities using appropriate methods
a. Solve systems graphically N Y Y
b. Solve systems algebraically N N Y
c. Solve systems using matrices N N Y
7. Use mathematical reasoning
a. Make conjectures Y Y Y
b. Validate and justify conclusions and generalizations Y Y Y
c. Use informal induction and deduction # Y Y
8. Represent problem situations with discrete structures
a. Use finite graphs and matrices N # Y
b. Use sequences and series N N Y
c. Use recursive relations (including numerical and graphical iteration and finite differences) N N Y
9. Solve polynomial equations with real and complex roots using a variety of algebraic and graphical methods and using appropriate tools N N Y
10. Approximate solutions of equations (bisection, sign changes, and successive approximations) N # Y
11. Use appropriate notation and terminology to describe functions and their properties (including domain, range, function composition, and inverses) N N Y
12. Compare and apply the numerical, symbolic, and graphical properties of a variety of functions and families of functions, examining general parameters and their effect on curve shape N # Y
13. Apply function concepts to model and deal with real-world situations N # Y
14. Use trigonometry
a. Use triangle trigonometry to model problem situations N N Y
b. Use trigonometric and circular functions to model real-world phenomena N N Y
c. Apply concepts of trigonometry to solve real-world problems N N Y

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Chapter One | Chapter Two | Chapter Three | Chapter Four | Chapter Five
References | Appendix | Contents

NAEP Math Objectives Violated by the neew Wa test



1996 NAEP Mathematics Objectives
url: http://www.leconsulting.com/arthurhu/97/10/naep///www.nagb.org/mchap3.htmlScontent
The NAEP is a national test based on NTCM standards, the new-new
math. 


The new WA 1997 Assessment contains several questions which assesses
skills which the NEAP deems you should NOT assess in the 4th grade,
and are not commonly taught in most textbooks or curriculums. It is
claimed these skills are in the Essential Learnings guidelines, but
they are so general that these skills are NOT in fact mentioned in
detail. Therefore, this test is NOT a 4th grade level test, and it
was not propertly constructed with respect to measuring proficiency
of ESSENTIAL skills. Considering that only 20% of 4th graders passed
the "expected" standard, and the median performance was the lowest
score, reflecting little or no understanding of most of the problems,
this test is not appropriate, and it is a fraud to take what is 50th
percentile math skills equal to the national average, and the best
nations on the TIMSS, it is NOT appropriate to say that our students
are performing at "very low" levels.




Flagpole:
        Geometry and Spatial Sense                   4  8    12
        5. Apply ratios and proportional thinking
           in a variety of situations
           c. Use proportional thinking to solve     N  Y    Y
              problems (including rates, scaling,
              and similarity)


Bird Feeder:
        Measurement                                  4  8    12
        4.  Estimate, calculate (using basic
            principles or formulas), or compare
            perimeter, area, volume, and surface
            area in meaningful contexts to solve
            mathematical and real-world problems
            a. Solve problems involving perimeter    S  Y    Y
               and area (e.g., triangles,
               quadrilaterals, other polygons,
               circles, and combined forms)
               [Note: Grade 4 tasks done with
               manipulatives]


Bird feeder problem is a very complex task with reading charts, scale
drawings, computation, reading. It's not a straightforward test of
just one measurement concept.




Bird Feeder:


        Measurement                                  4  8    12
        6.  Convert from one measurement to another  N  Y    Y
            within the same system (customary or
            metric)


Requires conversion of 60 inches to feet without a calculator, does
not give conversion factor (which IS given on the 7th grade test)


Bird Feeder:
        Measurement
        8.  Make and read scale drawings             N  Y    Y


Bird feeder illustration is a scale drawing with hidden lines, you
have to interpret construction and placement of screws.


Flagpole:
        Geometry and Spatial Sense                   4  8    12
        6. Apply geometric properties and
           relationships in solving problems
           c. Apply properties of ratio and          N  S    Y
              proportion with respect to similarity


Battery - Frequency histogram is NOT a common bar graph type.
        Data Analysis, Statistics, and Probability   4  8    12
        2.  Organize and display data and make
            inferences
            a. Use tables, histograms (bar graphs),  Y  Y    Y
               pictograms, and line graphs


Battery - student was asked to find not the central, but 
graph with most items BELOW a certain point.


        Data Analysis, Statistics, and Probability   4  8    12
        4.  Describe measures of central tendency    S  Y    Y
            and dispersion in real-world situations