Reasoning with Expressions, Equations, and Inequalities Overview

Introduction

In this module, you’ll be exploring expressions, equations, and inequalities. Yes, students were introduced to these very topics in 6th grade, but this unit will be going deeper. With expressions, students will learn to translate words into numbers, variables, and operations and apply the properties of operations to simplify numerical and algebraic expressions. This will flow right into solving equations containing integers, decimals, and fractions and identify the relationship between additive inverse in terms of change. Gradually students will transition to a deeper understanding of solving equations by exploring two step equations involving two operations, multiple operations, and variables on both sides of the equation. Finally, students will extend their knowledge of graphing, writing, and solving inequalities. All of these skills will be used in problem solving contexts and real-life situation Expressions Intro: Jeffery starts with 5 tickets. Andrew gives Jeffrey 2 more. How many tickets does Jeffrey end up with? 4+2=6

Essential Questions

Expressions

How can you write an expression from a given representation?
How can the properties of operations be used to simplify and expand linear expressions with rational coefficients?
How does rewriting an expression help clarify the problem?
How does rewriting an expression help show how the quantities are related?
How can factoring help us break down an expression?

Equations

How do you solve equations containing integers?
How do you solve equations containing decimals?
How do you solve equations containing fractions?
How do you solve equations with two operations?
How do you solve multi-step equations?
How do you solve equations with variables on both sides?

Inequalities

How do you solve inequalities that involve one-step operations?
How do you use multiplication or division to solve inequalities?
How do you combine these operations to solve multi-step inequalities?
How do you graph inequalities?

Key Terms

The following key terms will help you understand the content in this module.

Expressions

Expression - An expression is a mathematical phrase that contains operations and numbers. An algebraic expression contains one or more variables.

Coefficient - A coefficient is a number that is multiplied by a variable in an algebraic expression.

Evaluate - To evaluate an algebraic expression, substitute a number for the variable.

Factor - A number that is multiplied by another number to get a product ***to factor is the process of writing a number or an algebraic expression as a product.

Term - Terms are those parts of the expression between addition signs and subtraction signs. They can be a number, variable, or product of numbers and variables.

Like terms - Like terms are terms that have the same variable raised to the same power.

Variable - When a letter represents a number that can vary, it is called a variable.

Equations

Constant - The constant is a numerical term in an expression that does not change.

Coefficient - The coefficient is the numerical part of an algebraic term, for example, the 6 in 6x is the coefficient.

Equation - An equation is mathematical sentence relating expressions with an equal sign.

Evaluate - To evaluate, simply substitute a number for a variable in an algebraic expression.

Inverse Operation - An inverse operation is an operation that "undoes" another operation (e.g., subtraction is the inverse operation of addition division is the inverse operation of multiplication).

Reciprocal - Two numbers are reciprocals if their product is 1. Another name for reciprocal is "multiplicative inverse".

Solution - A solution is any value for a variable that makes an equation true.

Term - Each part of the expression separated by addition or subtraction signs is a term.

Variable - A variable is a symbol or letter that represents a number.

Inequalities

Addition property of inequality - You can add the same number to both sides of an inequality, and the inequality will still be true.

Algebraic Inequality - An Algebraic inequality is simply an inequality that contains at least one variable

Inequality - An inequality is a mathematical sentence that shows the relationship between quantities that are not equivalent.

Solution Set - A solution set is the set of values that make an inequality statement true.

Subtraction property of inequality - You can subtract the same number from both sides of an inequality, and the inequality will still be true.

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