# Module 3

• Grade 7 Module 3: Expressions and Equations

This module consolidates and expands upon students’ understanding of equivalent expressions as they apply the properties of operations to write expressions in both standard form and in factored form. They use linear equations to solve unknown angle problems and other problems presented within context to understand that solving algebraic equations is all about the numbers. Students use the number line to understand the properties of inequality and recognize when to preserve the inequality and when to reverse the inequality when solving problems leading to inequalities. They interpret solutions within the context of problems. Students extend their sixth-grade study of geometric figures and the relationships between them as they apply their work with expressions and equations to solve problems involving area of a circle and composite area in the plane, as well as volume and surface area of right prisms.

(Excerpt taken from NYS Engage website)

NYS Engage Module 3 - link to all things related to Module 3, provided by NYS

Engage worksheets and Video Lessons for all lessons covered in Module

Important Vocabulary (taken from lesson 1 of the module):

Variable A variable is a symbol (such as a letter) that represents a number, i.e., it is a placeholder for a number.

Numerical Expression A numerical expression is a number, or it is any combination of sums, differences, products, or divisions of numbers that evaluates to a number.

Value of a Numerical Expression:  The value of a numerical expression is the number found by evaluating the expression.

Expression An expression is a numerical expression, or it is the result of replacing some (or all) of the numbers in a numerical expression with variables.

Equivalent Expressions Two expressions are equivalent if both expressions evaluate to the same number for every substitution of numbers into all the letters in both expressions.

An Expression in Expanded Form An expression that is written as sums (and/or differences) of products whose factors are numbers, variables, or variables raised to whole number powers is said to be in expanded form.  A single number, variable, or a single product of numbers and/or variables is also considered to be in expanded form.  Examples of expressions in expanded form include: 324, 3x, 5x + 3 - 40, etc.

Term Each summand of an expression in expanded form is called a term.  For example, the expression 2x + 3x + 5 consists of 3 terms:  2x, 3x, and 5.

Coefficient of the Term The number found by multiplying just the numbers in a term together.  For example, given the product 2 * x * 4, its equivalent term is 8x.  The number 8 is called the coefficient of the term 8x.

An Expression in Standard Form An expression in expanded form with all its like terms collected is said to be in standard form.  For example, 2x + 3x + 7 is an expression written in expanded form; however, to be written in standard form, the like terms 2x and 3x must be combined.  The equivalent expression 5x + 7 is written in standard form.

Important Vocabulary (taken from Lesson 7 of the module):

Equation:  An equation is a statement of equality between two expressions.

If A and B are two expressions in the variable x, then A = B is an equation in the variable x.

Students sometimes have trouble keeping track of what is an expression and what is an equation.  An expression never includes an equal sign (=) and can be thought of as part of a sentence.  The expression 3 + 4  read aloud is, “Three plus four,” which is only a phrase in a possible sentence.  Equations, on the other hand, always have an equal sign, which is a symbol for the verb “is.”  The equation 3 + 4 = 7 read aloud is, “Three plus four is seven,” which expresses a complete thought, i.e., a sentence.

Number sentences—equations with numbers only—are special among all equations.

Number Sentence:  A number sentence is a statement of equality (or inequality) between two numerical expressions.

A number sentence is by far the most concrete version of an equation.  It also has the very important property that it is always true or always false, and it is this property that distinguishes it from a generic equation.  Examples include 3 + 4 = 7 (true) and 3 + 3 = 7(false).  This important property guarantees the ability to check whether or not a number is a solution to an equation with a variable:  just substitute a number into the variable.  The resulting number sentence is either true or it is false.  If the number sentence is true, the number is a solution to the equation.  For that reason, number sentences are the first and most important type of equation that students need to understand.