Equality is a fundamental concept in mathematics that refers to the idea that two quantities are the same or have the same value. In mathematical expressions, the symbol "=" is used to denote equality. When two quantities are equal, they can be replaced with each other in an equation without changing the truth of the equation.

There are several types of equality that are commonly used in mathematics:

**Numerical Equality:**This type of equality involves comparing two numerical values to determine if they are the same, such as 5 = 5.**Variable Equality:**In algebra, variables are often used to represent unknown quantities. Variable equality involves determining if two expressions involving variables are equivalent, such as x + 3 = 7.**Set Equality:**In set theory, equality is used to compare sets to see if they contain the same elements, such as {1, 2, 3} = {3, 2, 1}.

When working with equality, it's important to keep in mind the following properties:

**Reflexive Property:**For any quantity a, a = a.**Symmetric Property:**If a = b, then b = a.**Transitive Property:**If a = b and b = c, then a = c.**Addition Property of Equality:**If a = b, then a + c = b + c.**Subtraction Property of Equality:**If a = b, then a - c = b - c.**Multiplication Property of Equality:**If a = b, then a * c = b * c (where c is not 0).**Division Property of Equality:**If a = b, then a / c = b / c (where c is not 0).

When solving equations, the goal is to find the value of the variable that makes the equation true. Here are some examples:

- Solve for x: 2x + 5 = 11
- Solve for y: 3y - 7 = 10
- Solve for z: 4z/2 = 6

To master the concept of equality, it's important to practice solving equations and understanding the properties of equality. Here are some key steps to follow:

- Understand the meaning of equality and the different types of equality used in mathematics.
- Familiarize yourself with the properties of equality and how they can be used to manipulate equations.
- Practice solving equations involving numerical and variable equality.
- Work on problems involving set equality and understand how to determine if two sets are equal.
- Review the steps for solving equations and apply them to different types of problems.

By following these steps and practicing regularly, you can gain a solid understanding of equality and become proficient in solving equations.

Study GuideSimilarity and scale Worksheet/Answer key

Similarity and scale Worksheet/Answer key

Similarity and scale Worksheet/Answer key

Similarity and scale Worksheet/Answer keyUsing Similar Polygons Worksheet/Answer keySimilar Polygons Worksheet/Answer keyUsing Similar Polygons Worksheet/Answer keySimilar Polygons

Number and Operations (NCTM)

Understand numbers, ways of representing numbers, relationships among numbers, and number systems.

Understand and use ratios and proportions to represent quantitative relationships.

Compute fluently and make reasonable estimates.

Develop, analyze, and explain methods for solving problems involving proportions, such as scaling and finding equivalent ratios.

Geometry (NCTM)

Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships.

Understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects.

Create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship.

Apply transformations and use symmetry to analyze mathematical situations.

Describe sizes, positions, and orientations of shapes under informal transformations such as flips, turns, slides, and scaling.

Measurement (NCTM)

Apply appropriate techniques, tools, and formulas to determine measurements.

Solve problems involving scale factors, using ratio and proportion.

Grade 8 Curriculum Focal Points (NCTM)

Geometry and Measurement: Analyzing two- and three-dimensional space and figures by using distance and angle

Students use fundamental facts about distance and angles to describe and analyze figures and situations in two- and three-dimensional space and to solve problems, including those with multiple steps. They prove that particular configurations of lines give rise to similar triangles because of the congruent angles created when a transversal cuts parallel lines. Students apply this reasoning about similar triangles to solve a variety of problems, including those that ask them to find heights and distances. They use facts about the angles that are created when a transversal cuts parallel lines to explain why the sum of the measures of the angles in a triangle is 180 degrees, and they apply this fact about triangles to find unknown measures of angles. Students explain why the Pythagorean Theorem is valid by using a variety of methods - for example, by decomposing a square in two different ways. They apply the Pythagorean Theorem to find distances between points in the Cartesian coordinate plane to measure lengths and analyze polygons and polyhedra.