State of State Standards & the Common Core in 2010

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Math Content-Specific Criteria

Arithmetic

Arithmetic should include standards for number sense as well as developmental and precursor standards for the essential capstone expectations, such as counting, comparing, place value, and common denominators.

In elementary and early middle school (high-priority grade levels for arithmetic), students must demonstrate:

• Clear understanding and instant recall of the single-digit addition and multiplication facts and the corresponding subtraction and division facts.
• Clear understanding of the properties of arithmetic, such as the inverse nature of addition and subtraction as well as the inverse nature of multiplication and division.
• Clear understanding of fractions as parts of a set, parts of a whole, and as numbers (e.g., the number line).

While we do not hold K-12 standards accountable to coverage at particular grade levels, we do expect all of them to include the following capstone standards (though not necessarily verbatim):

Students must understand and be fluent with the standard algorithms for whole-number addition, subtraction, multiplication, and division, and students must understand and be fluent with the standard procedures for the four arithmetic operations with fractions and decimals.

Other topics that should be covered, most frequently in late-middle and high school, include negative numbers, rational exponents, scientific notation, estimation, radicals, rational numbers as repeating decimals, and the arithmetic of complex numbers.

Measurement

In elementary or middle school, students should:

• Be able to measure lengths in centimeters and inches.
• Know and understand the formulas for the area of a rectangle and a triangle.
• Know how to convert within and between measurement systems.
• Be familiar with other types of measurement such as time, perimeter, angles, weight, volume, etc.

Ratios

Students should understand and be able to use rates, ratios, proportions, and percentages.

Algebra

In order to ensure college- and career-readiness, rigorous K-12 standards must include algebra standards that cover the following essential content.

Standards covering linear equations should ensure that students:

• Are able to solve equations and inequalities that are linear or involve the absolute value and know how to graph them.
• Know about slope and the various forms of linear equations and be able to write equations given different types of information, such as for a line through a given point with a given slope, a line through two points, or a line through a given point that is perpendicular to a given line.
• Are able to solve a system of two linear equations in two unknowns. Students should be fluent with the four arithmetic operations with polynomials and elementary factoring.
• Standards covering quadratic equations should ensure that students:
• Are able to graph quadratic equations and solve them by factoring, completing the square, and using the quadratic formula, including complex solutions.
• Are able to transform a quadratic equation into vertex form, find its vertex, its maximum or minimum, and its line of symmetry.

Finally, students should understand logarithmic and exponential functions as well as basic trigonometry and trigonometric functions.

Geometry

Geometry should be given a solid logical foundation that is made clear in the standards; for example, a Euclid-style axiomatic approach. As part of the study of high school geometry, students should understand:

Proofs of standard results about angles of triangles and angles associated with lines crossing parallel lines, including perpendicular lines.

• Proofs of the standard theorems about congruence and similarity of triangles as well as deeper results on triangles, including the Pythagorean Theorem and its converse.
• Proofs of the standard theorems about circles, chords, tangents, and angles.
• How to do standard geometric constructions. In addition, students should be introduced to various aspects of geometry in K-8, especially basic vocabulary, as long as it does not interfere with the important K-8 arithmetic priorities.

Data Analysis, Statistics, and Probability (DASP)

Students should be able to read, analyze, and make various kinds of graphs and tables, and they should know basic statistics and probability, particularly the counting arguments involving combinations and permutations. More DASP material is acceptable as long as it does not disturb the priorities of arithmetic, algebra, and geometry.

STEM-Ready Standards

The material needed by students pursuing mathematics-intensive (STEM) majors in college should be outlined in K-12 standards documents (though not every student should be required to take advanced math to graduate from high school). While most such content—such as arithmetic, algebra, and geometry—is already included in typical collegeready standards, some advanced topics might not be included.

These advanced standards include:

• The binomial theorem, geometric series, polar coordinates, and the arithmetic operations on rational expressions.
• More trigonometry, including the inverse trigonometric functions, the laws of sines and cosines, and angle sum identities.

Problem Solving

Across all grade levels, students should be able to use the essential material outlined in these criteria to solve complex multi-step exercises and word problems appropriate for each grade level.

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