Substitution in Mathematics

Substitution is a fundamental concept in mathematics, particularly in algebra. It involves replacing a variable in an expression with a specific value or another expression. This technique is often used to simplify expressions, solve equations, and evaluate functions.

Basic Substitution

When performing substitution, we replace a variable with a specific value. For example, in the expression 3x + 5, if we substitute x with 2, the expression becomes 3(2) + 5 = 6 + 5 = 11.

Substitution in Equations

In equations, substitution can be used to solve for the value of a variable. For example, in the equation 2x - 3 = 7, we can substitute a value for x (let's use 5) to solve for the variable.

Substitution in Functions

When dealing with functions, substitution involves replacing the input variable with a specific value and then evaluating the function. For instance, if we have the function f(x) = 2x + 3 and we want to find f(4), we substitute 4 for x to get f(4) = 2(4) + 3 = 8 + 3 = 11.

Study Guide

When working on problems involving substitution, it's important to remember the following key points:

Identify the variable you want to substitute and the specific value you will use.
Be careful with signs and operations when replacing variables with values.
Always double-check your work to ensure the substitution was done accurately.
Practice substitution in various contexts, including equations and functions, to build proficiency.

By mastering the concept of substitution, you'll be better equipped to handle algebraic manipulations, solve equations, and work with functions in mathematics.

◂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.