Constructing and Deconstructing Equations

Introduction

In this lesson plan, your learner will practice constructing and deconstructing equations. They will discover which operations are permissible when working with equations and define four of the properties of equality. They will also develop an understanding of inverse operations. These skills will prepare them for manipulating and solving algebraic equations.

Key Concepts for Constructing and Deconstructing Equations

Here are a few key concepts that are helpful to know for this lesson:

• Constructing Equations: Constructing equations involves applying different operations to an equation to manipulate it so that the variable is not alone on one side. This process often involves adding, subtracting, multiplying, or dividing both sides of the equation by specific values to rearrange the terms.
• Deconstructing Equations: Deconstructing equations refers to applying inverse operations to isolate the variable on one side of the equation. By performing the inverse operation (opposite operation) of addition, subtraction, multiplication, or division, the equation can be simplified to a form where the variable is isolated.
• Inverse Operations: Addition and subtraction are inverse operations of each other. Adding a value and then subtracting the same value will nullify each other. Similarly, multiplication and division are inverse operations. Multiplying a value and then dividing by the same value will cancel each other out.
• Equivalent Equations: Equivalent equations are equations that have the same solution or represent the same relationship between variables. This means that if you solve one equation to find the value of the variable, the same value will satisfy all equivalent equations. For example, 3x = 12 and x = 4 are equivalent equations because they both represent x being equal to 4.

Properties of Equality

In algebra, several properties are used to manipulate and solve equations. Here are the ones introduced in this lesson:

• Addition Property of Equality: This property states that if you add the same quantity to both sides of an equation, the equality is preserved.
• Subtraction Property of Equality: Similarly, subtracting the same quantity from both sides of an equation maintains equality.
• Multiplication Property of Equality: Multiplying both sides of an equation by the same value (other than zero) preserves equality.
• Division Property of Equality: Dividing both sides of an equation by the same value (other than zero) also maintains equality.

Teaching Plan

The following activities will help your learner become confident with constructing and deconstructing equations. Remember to go at a pace that is comfortable for your learner.

Examples and visuals to support the lesson:

1. Exploring Operations in Equations

In this activity, your learner will apply various operations to equations to determine which ones are "legal."

• Begin by asking your learner if 12 = 12 is a true equation and explain how they know. Remind them that an equation is true if both sides represent the same value.
• Next, have them add 4 to both sides of the equation and ask whether the resulting equation, 16 = 16, is true. Encourage them to continue applying several operations and determine the truth of the resulting equations.
• Let your learner reflect on which operations resulted in true equations and which ones resulted in false equations. Then, formalize the ideas by introducing academic vocabulary: Addition Property of Equality, Subtraction Property of Equality, Multiplication Property of Equality, and Division Property of Equality.

Skill Check
I can identify operations that result in true equations.

2. Constructing Equations

This activity will help your learner understand how operations transform equations.

• Provide your learner with an equation that has a variable on one side and a number on the other side, such as x = 5. Review and discuss what variables are in equations and what the equation x = 5 tells us about the value of x.
• Have your learner perform an operation, such as adding 3 to both sides of the equation, and write the resulting equation. Ask them whether the value of x changes as they transform the equation. For example, when adding 3 to both sides of x = 5, the transformed equation is x + 3 = 8, yet x is still equal to 5.

Skill Check
I can apply operations to both sides of an equation resulting in an equivalent equation.

3. Deconstructing Equations

In this activity, your learner will perform inverse operations to deconstruct equations.

• Have your learner review the equations they constructed in the previous activity. Ask them what action they need to perform to deconstruct the equation and get it back to its original form.
• Encourage your learner to reflect on the operations used to construct and deconstruct the equations. Explain that as they deconstructed the equations, they applied the properties of equality.
• Discuss the relationships between the operations used to construct and deconstruct each equation, emphasizing the connection between inverse properties: addition and subtraction are inverses, while multiplication and division are inverses.

Skill Check
I can use inverse operations to deconstruct an equation back to its original form.

4. Identifying Equivalent Equations

This activity can be used to check your learner's understanding of the properties of equality by identifying equivalent equations.

• Provide your learner with an equation, such as x = 8, along with a list of equations that have been constructed with various operations.
• Have your learner identify which equations are equivalent to x=8 and discuss their reasoning for each one.
• For equations that are equivalent, have your learner name the property of equality that was applied.

Skill Check
I can use what I know about constructing and deconstructing equations to identify equivalent equations.

Summary

While constructing and deconstruction equations, your learner has explored inverse operations and the properties of equality. The ability to construct and deconstruct equations provides a solid foundation for understanding how to manipulate expressions while maintaining balance. This understanding empowers your learner to understand and solve equations with confidence.

Teaching Plan adapted from Utah Middle School Math Project under CC BY 4.0.