The Common Core content standards for mathematics are all part of specific domains, or categories, that are often relevant to multiple grade levels. The idea behind this approach is that certain concepts in math progress in complexity over several years.

## Counting and cardinality

Counting and cardinality involves getting comfortable with what numbers represent and how they’re used. Students count numbers 1 to 100, work on writing numbers 1 to 20, and solidify their understanding of numbers as representative of the total quantity of objects in a group.

You may also see this domain referenced as developing “number sense.” Counting and cardinality is the first step in a conceptual staircase of mathematics that students climb over the course of their school years.

## Operations and algebraic thinking

The emphasis in the operations and algebraic thinking domain (Grades K-5) is to develop comfort and fluency when using numbers to add, subtract, multiply, and divide. Numbers emerge as tools that students can use to identify and represent quantities, relationships, and patterns.

Starting with understanding how to take apart and put together numbers within 10 and understanding relationships between parts and wholes, students build upon these skills until they’re comfortable with multiplying and dividing up to 100, adding and subtracting decimals fluently, and multiplying and dividing decimals to the hundredths place.

This leads to the application of these skills in finding missing parts, solving problems with multiple steps, finding the solution to numerical expressions, and using these concepts to determine patterns and represent data on a coordinate plane.

## Number and operations in base ten

This domain (Grades K-5) stresses place value — the value of a digit according to its position in a number (for example, in the number 346, the 4 represents 40). Over the course of these grades, students gain an understanding of the use and function of numbers, place value with whole numbers, and eventually place value with multi-digit numbers and decimals.

## Number and operations — fractions

Fractions (Grades 3-5) can be challenging for many students. The standards in this domain are designed to help students see fractions as numbers, or more specifically as parts of a whole, instead of as foreign symbols that are difficult to comprehend.

Students explore the relationship of parts to wholes; the notation involved in expressing fractions; the ordering of fractions based on their values; and the use of fractions in addition, in subtraction, and in limited applications of multiplication and division.

## Measurement and data

In early grades (Grades K-5), students explore measurement and data by measuring objects and quantities and telling time as a means of collecting data. This undergirds the central understanding that numbers may be used as a means of classification based on quantity.

Students then begin to perform other operations, such as addition and subtraction, using gathered data. Geometric concepts and shapes are also introduced as a part of this domain. High school students pursue the use of data more extensively with an emphasis on statistics and probability.

## Geometry

Starting with a basic understanding of shapes in kindergarten, students explore properties of shapes and other geometric concepts in later grades. As students proceed through Grades 5-8 and into high school, they begin to focus on the application of geometry to real-world settings.

## Ratios and proportional relationships

In Grades 6-7, students strive to gain mastery of ratios and proportions. With a firm understanding of the basic tenets of multiplication and division, students explore relationships between sets of numbers and use mathematical terms to describe any relationships that exist. Significant emphasis is placed on real-world application.

## The number system

Students extend their abilities in addition, subtraction, multiplication, and division in Grades 6-8 as they apply these operations to multi-digit whole numbers and to fractions. The concept of rational and irrational numbers is also a component of these grades. Students continue to graph data and closed figures in a coordinate plane and use their results to make determinations for specific problems.

## Expressions and equations

In Grades 6-8, students apply their understanding of basic arithmetic and part/whole reasoning to solve algebraic expressions, discovering how to substitute letters for numbers and solve to find unknown values. Students then begin to compare various expressions to find out whether they’re equivalent.

Teachers introduce students to the idea of using exponents to denote significantly large or small numbers and show them how to solve linear equations and graph them in a coordinate plane.

## Functions

Understanding relationships between numbers and relationships between ratios is an essential aspect of Grade 8. Students must know what a function is: a rule, equation, or expression that produces only one output for every input.

For example, height is generally a function of age, because the older you are, the taller you get. If you measure your height on your birthday every year, your data set includes only one height for every birthday.

Students must also be able to display functions in various ways — with words, graphs, or numbers, for example. As they progress in their ability to recognize and use functions in Grade 8 and in high school, students are required to use functions to show the relationship between objects in real life.

## Statistics and probability

Students dive into statistics and probability in Grades 6-8 and again in high school. They explore basic concepts of variability first so they can get a handle on the fact that data varies from person to person. They also practice displaying sets of data in various forms.

After grasping the basics, students apply their understanding to make determinations about larger populations based on data samples. Students eventually use graphs, scatterplots, and tables to represent multiple forms of data that may include more than one variable.

## Domains in high school

The high-school content standards are grouped within six domains: number and quantity, algebra, functions, modeling, geometry, and statistics and probability. Some of these domains are extensions of domains that are addressed in lower grades. The rest are new domains that build on previously learned concepts.