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## 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 6-8, #5)

NCTM Standards (1989)

• Understand how basic arithmetic operations are related to one another. (Standard 6, grades 5-8, #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 3-5)

Number Skills

AAAS Benchmarks

• Use, interpret, and compare numbers in several equivalent forms such as integers, fractions, decimals, and percents. (12 B, grades 6-8, #2)

NCTM Standards (1989)

• Understand, represent, and use numbers in a variety of equivalent forms (integer, fraction, decimal, percent, exponential, and scientific notation) in real-world and mathematical problem situations. (Standard 5, grades 5-8, #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 6-8)
• Develop meaning for integers and be able to represent, compare, and order them. (Standard 1, grades 6-8)

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 6-8, #1)

NCTM Standards (1989)

• Identify, describe, compare, and classify geometric figures. (Standard 12, grades 5-8, #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 6-8)
• Understand the concepts of congruence and similarity using transformations. (Standard 3, grades 6-8)

Geometry Skills

AAAS Benchmarks

• Calculate the circumferences and areas of rectangles, triangles, and circles, and the volumes of rectangular solids. (12 B, grades 6-8, #3)

NCTM Standards (1989)

• Extend their understanding of the concepts of perimeter, area, volume, angle measure, capacity, and weight and mass. (Standard 13, grades 5-8, #5)

NCTM Standards 2000 (Draft, October 1998)

• Develop, understand, and use formulas to find the perimeter and area of rectangles. (Standard 4, grades 3-5)
• Develop and use formulas for the perimeter and area of parallelograms, trapezoids, circles, and simple composite figures. (Standard 4, grades 6-8)
• Develop and use formulas for the surface area and volume of prisms, pyramids, and cylinders. (Standard 4, grades 6-8)

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 6-8, #3)

NCTM Standards (1989)

• Analyze functional relationships to explain how a change in one quantity results in a change in another. (Standard 8, grades 5-8, #3)
• Analyze tables and graphs to identify properties and relationships. (Standard 9-3, grades 5-8, #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 6-8)
• 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 6-8)

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 6-8, #4)

NCTM Standards (1989)

• Apply algebraic methods to solve a variety of real-world and mathematical problems. (Standard 9-6, grades 5-8, #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 3-5)
• Develop a sound conceptual understanding of equation and of variable. (Standard 2, grades 6-8)
• Represent a variety of relations and functions with tables, graphs, verbal rules, and, when possible, symbolic rules. (Standard 2, grades 6-8)
• Use symbolic algebra to represent situations and to solve problems, especially those that involve linear relationships. (Standard 2, grades 6-8)