Appendix A
Mathematics Benchmarks/NCTM Standards Comparisons
Number Concepts
AAAS Benchmarks
 The expression a/b can mean different things: a parts of size 1/b
each, a divided by b, or a compared to b. (9A grades
68, #5)
NCTM Standards (1989)
 Understand how basic arithmetic operations are related to one another. (Standard 6,
grades 58, #4)
NCTM Standards 2000 (Draft, October 1998)
 Develop meaning for fractions as parts of a unit whole, as a part of a collection, as
numbers, and as a division of whole numbers. (Standard 1, grades 35)
Number Skills
AAAS Benchmarks
 Use, interpret, and compare numbers in several equivalent forms such as integers,
fractions, decimals, and percents. (12 B, grades 68, #2)
NCTM Standards (1989)
 Understand, represent, and use numbers in a variety of equivalent forms (integer,
fraction, decimal, percent, exponential, and scientific notation) in realworld and
mathematical problem situations. (Standard 5, grades 58, #1)
NCTM Standards 2000 (Draft, October 1998)
 Work flexibly with equivalent fractions, decimals, percents; compare and order these
numbers efficiently and accurately, find their approximate locations on a number line; and
choose appropriate and convenient forms of these numbers for solving problems. (Standard
1, grades 68)
 Develop meaning for integers and be able to represent, compare, and order them.
(Standard 1, grades 68)
Geometry Concepts
AAAS Benchmarks
 Some shapes have special properties: Triangular shapes tend to make structures rigid,
and round shapes give the least possible boundary for a given amount of interior area.
Shapes can match exactly or have the same shape in different sizes. (9C, grades 68, #1)
NCTM Standards (1989)
 Identify, describe, compare, and classify geometric figures. (Standard 12, grades 58,
#1)
NCTM Standards 2000 (Draft, October 1998)
 Precisely describe, classify, and compare types of plane and solid figures (e.g. angles,
triangles, quadrilaterals, cylinders, cones, etc.) according to their main features.
(Standard 3, grades 68)
 Understand the concepts of congruence and similarity using transformations. (Standard 3,
grades 68)
Geometry Skills
AAAS Benchmarks
 Calculate the circumferences and areas of rectangles, triangles, and circles, and the
volumes of rectangular solids. (12 B, grades 68, #3)
NCTM Standards (1989)
 Extend their understanding of the concepts of perimeter, area, volume, angle measure,
capacity, and weight and mass. (Standard 13, grades 58, #5)
NCTM Standards 2000 (Draft, October 1998)
 Develop, understand, and use formulas to find the perimeter and area of rectangles.
(Standard 4, grades 35)
 Develop and use formulas for the perimeter and area of parallelograms, trapezoids,
circles, and simple composite figures. (Standard 4, grades 68)
 Develop and use formulas for the surface area and volume of prisms, pyramids, and
cylinders. (Standard 4, grades 68)
Algebra Graph Concepts
AAAS Benchmarks
 Graphs can show a variety of possible relationships between two variables. As one
variable increases uniformly, the other may do one of the following: increase or decrease
steadily, increase or decrease faster and faster, get closer and closer to some limiting
value, reach some intermediate maximum or minimum, alternately increase and decrease
indefinitely, increase or decrease in steps, or do something different from any of these.
(9B, grades 68, #3)
NCTM Standards (1989)
 Analyze functional relationships to explain how a change in one quantity results in a
change in another. (Standard 8, grades 58, #3)
 Analyze tables and graphs to identify properties and relationships. (Standard 93,
grades 58, #3)
NCTM Standards 2000 (Draft, October 1998)
 Represent a variety of relations and functions with tables, graphs, verbal rules, and,
when possible, symbolic rules. (Standard 2, grades 68)
 Model and solve contexualized problems using various representations, such as graphs and
tables, to understand the purpose and utility of each representation. (Standard 2, grades
68)
Algebra Equation Concepts
AAAS Benchmarks
 Symbolic equations can be used to summarize how the quantity of something changes over
time or in response to other changes. (11C, grades 68, #4)
NCTM Standards (1989)
 Apply algebraic methods to solve a variety of realworld and mathematical problems.
(Standard 96, grades 58, #6)
NCTM Standards 2000 (Draft, October 1998)
 Represent and investigate how a change in one variable relates to the change in a second
variable. (Standard 2, grades 35)
 Develop a sound conceptual understanding of equation and of variable. (Standard 2,
grades 68)
 Represent a variety of relations and functions with tables, graphs, verbal rules, and,
when possible, symbolic rules. (Standard 2, grades 68)
 Use symbolic algebra to represent situations and to solve problems, especially those
that involve linear relationships. (Standard 2, grades 68)
