Equivalent

In mathematics, the term "equivalent" refers to two or more expressions, equations, or values that have the same meaning, or represent the same quantity. When two or more mathematical expressions are equivalent, they are essentially equal to each other and can be interchanged without changing the meaning or value they represent.

Equivalent Expressions

Two algebraic expressions are equivalent if they have the same value for all the variables in the expressions. This means that the expressions may look different, but they represent the same quantity. For example, the expressions 2x + 3 and x + x + 3 are equivalent because they both represent the same quantity for any value of x.

Equivalent Equations

Equations are considered equivalent if they have the same solution. That is, when you solve each equation, you get the same value for the variable(s) in the equation. For instance, the equations 2x = 10 and x = 5 are equivalent because they both have the same solution, x = 5.

Equivalent Fractions

Fractions are equivalent if they represent the same portion of a whole. For example, 1/2 and 2/4 are equivalent fractions because they both represent half of a whole. To determine if two fractions are equivalent, you can simplify or reduce them to their simplest form and check if they are equal.

Study Guide

Here are some key points to remember when working with equivalent expressions, equations, and fractions:

Expressions are equivalent if they have the same value for all the variables in the expressions.
Equations are equivalent if they have the same solution when solved.
Fractions are equivalent if they represent the same portion of a whole, and can be determined by simplifying or reducing them to their simplest form.
When working with equivalent expressions, equations, or fractions, you can use properties and operations such as distribution, combining like terms, and simplifying to show their equivalence.

Understanding the concept of equivalence in mathematics is crucial for solving problems involving simplification, solving equations, and comparing quantities. Practice identifying equivalent expressions, equations, and fractions to strengthen your understanding of this fundamental concept.

◂Math Worksheets and Study Guides Eighth Grade. Displaying data

Organizing & Displaying Data

The resources above cover the following skills:

Data Analysis and Probability (NCTM)

Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them.

Select, create, and use appropriate graphical representations of data, including histograms, box plots, and scatterplots.

Select and use appropriate statistical methods to analyze data.

Discuss and understand the correspondence between data sets and their graphical representations, especially histograms, stem-and-leaf plots, box plots, and scatterplots.

Develop and evaluate inferences and predictions that are based on data.

Make conjectures about possible relationships between two characteristics of a sample on the basis of scatterplots of the data and approximate lines of fit.

Grade 8 Curriculum Focal Points (NCTM)

Data Analysis and Number and Operations and Algebra: Analyzing and summarizing data sets

Students use descriptive statistics, including mean, median, and range, to summarize and compare data sets, and they organize and display data to pose and answer questions. They compare the information provided by the mean and the median and investigate the different effects that changes in data values have on these measures of center. They understand that a measure of center alone does not thoroughly describe a data set because very different data sets can share the same measure of center. Students select the mean or the median as the appropriate measure of center for a given purpose.

Connections to the Grade 8 Focal Points (NCTM)

Data Analysis: Building on their work in previous grades to organize and display data to pose and answer questions, students now see numerical data as an aggregate, which they can often summarize with one or several numbers. In addition to the median, students determine the 25th and 75th percentiles (1st and 3rd quartiles) to obtain information about the spread of data. They may use box-and-whisker plots to convey this information. Students make scatterplots to display bivariate data, and they informally estimate lines of best fit to make and test conjectures.