Commutative Property

The commutative property is a fundamental property in mathematics that applies to binary operations, such as addition and multiplication. It states that the order of the numbers does not affect the result of the operation. In other words, when you add or multiply two numbers, you can change the order of the numbers and still get the same result.

Addition

The commutative property of addition states that for any two numbers a and b, the sum of a + b is equal to the sum of b + a. In mathematical terms, this can be expressed as:

a + b = b + a

Multiplication

Similarly, the commutative property of multiplication states that for any two numbers a and b, the product of a x b is equal to the product of b x a. In mathematical terms, this can be expressed as:

a x b = b x a

Examples

For example, let's consider the following addition and multiplication:

For addition: 3 + 5 = 5 + 3

For multiplication: 4 x 7 = 7 x 4

Study Guide

To better understand the commutative property, it's important to practice with various numbers and operations. Here are some study guide questions and examples to help reinforce your understanding:

What is the commutative property of addition?
What is the commutative property of multiplication?
Provide examples of addition and multiplication that demonstrate the commutative property.
Are there any operations that do not follow the commutative property? If so, provide examples.
How does the commutative property apply to real-life situations?

By practicing these questions and working through various examples, you can solidify your understanding of the commutative property and its applications in mathematics.

◂Math Worksheets and Study Guides Eighth Grade. Linear equations

The resources above cover the following skills:

Algebra (NCTM)

Represent and analyze mathematical situations and structures using algebraic symbols.

Explore relationships between symbolic expressions and graphs of lines, paying particular attention to the meaning of intercept and slope.

Grade 8 Curriculum Focal Points (NCTM)

Algebra: Analyzing and representing linear functions and solving linear equations and systems of linear equations

Students use linear functions, linear equations, and systems of linear equations to represent, analyze, and solve a variety of problems. They recognize a proportion (y/x = k, or y = kx) as a special case of a linear equation of the form y = mx + b, understanding that the constant of proportionality (k) is the slope and the resulting graph is a line through the origin. Students understand that the slope (m) of a line is a constant rate of change, so if the input, or x-coordinate, changes by a specific amount, a, the output, or y-coordinate, changes by the amount ma. Students translate among verbal, tabular, graphical, and algebraic representations of functions (recognizing that tabular and graphical representations are usually only partial representations), and they describe how such aspects of a function as slope and y-intercept appear in different representations. Students solve systems of two linear equations in two variables and relate the systems to pairs of lines that intersect, are parallel, or are the same line, in the plane. Students use linear equations, systems of linear equations, linear functions, and their understanding of the slope of a line to analyze situations and solve problems.

Connections to the Grade 8 Focal Points (NCTM)

Geometry: Given a line in a coordinate plane, students understand that all 'slope triangles' - triangles created by a vertical 'rise' line segment (showing the change in y), a horizontal 'run' line segment (showing the change in x), and a segment of the line itself - are similar. They also understand the relationship of these similar triangles to the constant slope of a line.