Algebraic Equation: Solving Equations by Applying Properties of Equality
This presentation covers the topic of solving simple equations using properties of equality for Grade 7 students. Through this lesson, students will learn to illustrate the properties of equality and solve problems involving algebraic expressions and formulas. By the end of the quarter, students will be able to solve simple equations, applying the knowledge gained through this mathematical journey.
Course Information
School Details
Name of School:
Quarter: 4th Quarter
Grade Level & Section: Grade 7
Subject Information
Subject: MATHEMATICS
Week: Week 4 Day 1
Topic: Algebraic Equation (Week 4)
2.2 Solving Equations by Applying Properties of Equality
Teacher & Schedule
Teacher:
Date and Time:
Content Standards and Learning Competencies
Content Standards
The learners should have knowledge and understanding of the solution of simple equations.
Performance Standards
By the end of the quarter, the learners are able to solve simple equations.
Learning Competencies
The learners...
1. Illustrate the properties of equality.
2. Solve problems involving algebraic expressions and formulas.
Learning Resources
1
Department of Education
Department of Education. (2020). Mathematics quarter 2 – module 4: algebraic expressions. (1). k-to-12-grade-7-learning-material-in-mathematics-quarter-3.pdf - Google Drive
2
CK-12 Foundation
CK-12 Foundation. (2024, January 11). Evaluating algebraic expressions and equations. https://flexbooks.ck12.org/cbook/ck-12-algebra-ii-with-trigonometry-concepts/section/1.4/primary/lesson/evaluating-algebraic-expressions-and-equations-alg-ii/
3
CueMath
CueMath. (2024, January 15). Properties of equality. Cuemath. https://www.cuemath.com/algebra/properties-of-equality/
4
National Repository of Online Courses
National Repository of Online Courses (2023, December 17). Solving one-step equations using properties of equality. LibreTexts Mathematics. https://math.libretexts.org/Bookshelves/Applied_Mathematics/Developmental_Math_(NROC)/10%3A_Solving_Equations_and_Inequalities/10.01%3A_Solving_Equations/10.1.01%3A_Solving_One-Step_Equations_Using_Properties_of_Equality
Pre-Lesson: Activating Prior Knowledge
Review Activity
A short review to activate prior knowledge before beginning the main lesson
1
Individual/Pair Work
Can be performed individually or in pairs with bar modeling
2
Partner Finding
For pair activity: find partners matching problems with answers
3
This section of the review focuses on solving word problems using the bar method. The review can be performed individually or in pairs. For individuals, the learners will prepare their bar modeling to solve the problems. For pair activity: The teacher will give each learner a strip of paper where one set is a problem and the other set is for the answer using bar modeling. The learners will be roamed around to find their partner.
Short Review Problems
Problem 1
Alexa has 24 photo cards. She gives 4 photo cards to each of her friends. How many friends receive the photo cards?
Problem 2
Sonny has 6 packs of shuttlecocks. There are 8 shuttlecocks in each pack. How many shuttlecocks are there?
Problem 3
Eldan has 4 boxes of pens. Each box has 10 pens. He gives the pen to 8 of his friends. How many pens does each of his friends receive?
Problem 4
Henry has 24 one-piece stickers in his collection. Ted has 6 times as many stickers as Henry. How many stickers does Ted have? How many stickers do they have altogether?
Problem 5
If John has 26 cards and Tim has 12 more than John's cards, how many cards does Tim have? How many cards are there altogether?
Lesson Purpose and Intention
Math problems have different ways of solving them, just like how we can share our knowledge and help those who are in need. Translating verbal phrases to algebraic equations is also a tool that will help facilitate the use of different methods in solving unknown values in an algebraic equation.
One way of solving a problem that involves algebraic equations is through the use of applying the properties of equalities and some formulas to solve the problems.
In this part, the teacher will introduce the lesson and its application to daily life.
Unlocking Content Vocabulary
Algebraic Expression
An expression that is made up of variables and constants along with algebraic operations (addition, subtraction, multiplication, and division).
Algebraic Equations
Can be defined as a mathematical statement in which two expressions are set equal to each other.
Bar model
Is one such tool that helps us visualize the given math problem using rectangles or bars.
Variable
Is a letter or symbol that represents an unknown number.
Equation
Is a mathematical statement that two expressions are equal.
Expressions
Are made up of terms, and the number of terms in each expression in an equation may vary.
Solving Equations by Applying Properties of Equality
Translating Mathematical Concepts
Translating, writing, and solving equations are crucial aspects of mathematics. Algebraic equations enable teachers and learners to solve problems involving unknown quantities.
Balance Principle
A key principle in working with algebraic equations is that you can add or subtract the same quantity from both sides of an equation to maintain its balance.
Justice Analogy
This concept is analogous to a balance scale used in the justice system, where equal weight must be given to both sides to ensure fairness and uphold justice for everyone.
Properties of Equality
1
Addition Property of Equality (APE)
For all real numbers a, b, and c, a = b if and only if a + c = b + c
2
Addition Property Explained
If we add the same number to both sides of the equal sign, then the two sides remain equal.
3
Addition Example
Example: 10 + 5 = 15 is true if and only if 10 + 5 + 8 = 15 + 8
The teacher will explain why there is no subtraction and division property of equality. Even though subtracting or dividing the same number from both sides of an equation preserves equality, these cases are already covered by APE and MPE. Subtracting the same number from both sides of an equation is the same as adding a negative number to both sides of an equation.
Multiplication Property of Equality
1
Multiplication Property of Equality (MPE)
For all real numbers a, b, and c, where c≠ 0, a = b if and only if ac = bc
2
Multiplication Property Explained
If we multiply the same number to both sides of the equation, then the two sides remain equal.
3
Multiplication Example
Example: 3(5) = 15 is true if and only if (3)(5) 2 = 15 (2)
Also, dividing the same number from both sides of an equation is the same as multiplying the reciprocal of the number to both sides of an equation.
Finding Solutions Using Properties of Equality
Identify the Problem
Finding the solutions to equations using properties of equality means finding the unknown so that the equation becomes true.
Example Problem
In the given equation x- 8 = 15, what value of x will make the expression equal?
Solution Process
x – 8 = 15 Given
x - 8 + 8 = 15 + 8 APE (Adding 8 to both sides)
x + 0 = 23 Simplify
x = 23 Simplify
Verification
Checking:
x – 8 = 15
23 – 8 = 15
15 = 15 True
Another Example Using Multiplication Property
Example Problem
In the equation 3x = 42, what is x?
Solution Process
3x = 42 Given
3x/3 = 42/3 MPE (Multiply 1/3 on both sides)
x = 14 Simplify
Verification
Checking:
3x = 42
3(14) = 42
42 = 42 True
Worked Example 1: Solving for x
Step 1: Identify the equation
x - 28 = 46
Given
Step 2: Apply Addition Property
x – 28 + 28 = 46 + 28
APE (Add 28 on both sides)
Step 3: Simplify
x = 74
Step 4: Verify
x – 28 = 46
74 – 28 = 46
46 = 46 True
Worked Example 2: Solving Negative Values
1
Step 1: Identify the equation
x + 15 = - 44
Given
2
Step 2: Apply Addition Property
x + 15 - 15 = - 44 – 15
APE (Add -15 on both sides)
3
Step 3: Simplify
x = - 59
4
Step 4: Verify
x + 15 = -44
-59 + 15 = -44
-44 = -44 True
Worked Example 3: Using Multiplication Property
Given Equation
4x = 128
Apply MPE
4x/4 = 128/4
MPE (Multiply 1/4 on both sides)
Simplify
x = 32
Verify
4x = 128
4(32) = 128
128 = 128 True
Worked Example 4: Equations with Variables on Both Sides
1
Solve
2x + 15 = x - 3
2
Collect Variables
2x - x + 15 = x - x - 3
3
Simplify
x + 15 = -3
4
Isolate Variable
x = -18
Solution steps in detail:
2x + 15 = x – 3 (Given)
2x - x + 15 = x - x -3 (APE: Add -x on both sides)
x + 15 = -3 (Simplify)
x + 15 – 15 = -3 -15 (APE: Add -15 on both sides)
x = - 18 (Simplify)
Checking: 2x + 15 = x – 3
2(-18) + 15 = -18 – 3
-36 + 15 = -21
-21 = -21 True
Worked Example 5: Two-Step Equation
5x
Left side
Variable term and constant
3x
Right side
Variable term and constant
2x
After collecting terms
Simplified form
8
Solution
Value of x
Solution for 5x – 14 = 3x + 12:
5x -3x – 14 = 3x -3x + 12 (APE: Add -3x on both sides)
2x – 14 = 12 (Simplify)
2x -14 + 14 = 12 + 14 (APE: Add 14 on both sides)
2x = 26 (Simplify)
x = 13 (MPE: Multiply 1⁄2 both sides)
Checking: 5x -14 = 3x + 12
5(8) – 14 = 3(8) + 12
40 – 14 = 24 + 12
26 = 36
Worked Example 6: Equation with Parentheses
Solution for 6 = 2(x -4):
6 = 2(x -4) (Given)
6 = 2(x) – 2(4) (Multiply 2 by the terms inside the parenthesis)
6 = 2x – 8 (Simplify)
6 + 8 = 2x – 8 + 8 (APE: Add 8 on both sides)
14 = 2x (Simplify)
14/2 = 2x/2 (MPE: Multiply 1/2 on both sides)
7 = x (Simplify)
Checking: 6 = 2(x – 4)
6 = 2(7-4)
6 = 2(3)
6 = 6 True
Lesson Activity: Quality Time
Activity Instructions
Paste all the equations with a solution in the boxes below. Arrange the equations in order by their solutions from least to greatest. Then write the letters in order on the lines below to form the word hidden.
Working with Equations
This activity helps students practice solving equations while also creating a fun word puzzle based on the ordered solutions.
_____ _____ _____ _____ _____ _____ _____ _____
Lesson Activity Solution
Equation
Solution
Letter
S = 2x + 4 = 18
x = 7
S
E = 3x + 2 = 20
x = 6
E
N = 3x - 2 = 13
x = 5
N
A = 2x + 1 = 3
x = 1
A
S = 4x + 5 = 3x + 15
x = 10
S
F = 12x = -144
x = -12
F
R = 6x + 5 = 4x + 13
x = 4
R
I = 6 = 3(x -1)
x = 3
I
Lesson Activity: Ordered Solution
F = -12
12x = -144
A = 1
2x + 1 = 3
I = 3
6 = 3(x -1)
R = 4
6x + 5 = 4x + 13
N = 5
3x - 2 = 13
E = 6
3x + 2 = 20
S = 7
2x + 4 = 18
S = 10
4x + 5 = 3x + 15
Making Generalizations and Abstractions
Properties of Equality
In solving equations, we apply the Properties of Equality to maintain balance and find the unknown value.
1
Addition Property
The Addition Property of Equality (APE) allows us to add the same number to both sides.
2
Multiplication Property
The Multiplication Property of Equality (MPE) allows us to multiply both sides by the same nonzero number.
3
Balance Principle
By following these properties, we ensure that the equation remains true, just like a balance scale in justice.
4
Problem Solving
This principle helps us systematically solve problems involving unknown quantities in algebra.
5
Evaluating Learning: Practice Problems
Problem 1
X – 2 = 13
Find the value of X that makes this equation true.
Problem 2
X + 9 = 21
Solve for X using the appropriate property of equality.
Problem 3
1/2(x) – 3 = 4
Determine the value of x in this equation.
Solutions to Practice Problems
Solution to Problem 1
X – 2 = 13
X – 2 + 2 = 13 + 2 (APE: Add 2 to both sides)
X = 15
Solution to Problem 2
X + 9 = 21
X + 9 – 9 = 21 - 9 (APE: Add -9 to both sides)
X = 12
Solution to Problem 3
1/2(x) – 3 = 4
1/2(x) – 3 + 3 = 4 + 3 (APE: Add 3 to both sides)
1/2(x) = 7
1/2(x) × 2 = 7 × 2 (MPE: Multiply by 2 on both sides)
x = 14
Balance Scale Analogy for Equations
Equality as Balance
The balance scale represents how both sides of an equation must remain equal. Whatever operation we perform on one side must be done to the other side as well.
Addition Property Visualized
Adding the same value to both sides maintains the balance, just as adding equal weights to both sides of a scale keeps it balanced.
Multiplication Property Visualized
Multiplying both sides by the same non-zero value maintains the balance, similar to multiplying all weights on both sides by the same factor.
Justice and Mathematics: A Connection
Balance in Justice
In our justice system, a balance scale represents the fair and equal treatment of all parties. Both sides must receive equal consideration for justice to be served.
Balance in Mathematics
Similarly, in mathematics, the properties of equality ensure that both sides of an equation remain balanced throughout the solution process, upholding the mathematical "justice" of the equation.
This concept is analogous to a balance scale used in the justice system, where equal weight must be given to both sides to ensure fairness and uphold justice for everyone.
Real-World Applications of Algebraic Equations
Shopping and Budgeting
Determining how many items you can buy with a set budget, or calculating discounts and sales tax.
Cooking and Recipes
Adjusting recipe ingredients proportionally when serving more or fewer people.
Construction and Design
Calculating materials needed based on area or volume formulas.
Travel Planning
Determining travel times, distances, or fuel requirements using rate equations.
Properties of Equality: Why No Subtraction or Division Properties?
1
Addition Covers Subtraction
Subtracting the same number from both sides of an equation is mathematically equivalent to adding the negative of that number to both sides.
2
Example of "Subtraction" as Addition
x + 5 = 12
x + 5 - 5 = 12 - 5 is the same as x + 5 + (-5) = 12 + (-5)
Therefore, subtraction is covered by the Addition Property of Equality (APE).
3
Multiplication Covers Division
Dividing both sides of an equation by a non-zero number is mathematically equivalent to multiplying both sides by the reciprocal of that number.
4
Example of "Division" as Multiplication
3x = 15
3x ÷ 3 = 15 ÷ 3 is the same as 3x × (1/3) = 15 × (1/3)
Therefore, division is covered by the Multiplication Property of Equality (MPE).
Bar Modeling for Algebraic Equations
Bar modeling is one such tool that helps us visualize the given math problem using rectangles or bars. This visual representation can help students understand the relationships between known and unknown values in an equation, making it easier to apply the properties of equality to find the solution.
Step-by-Step Approach to Solving Equations
1
Step 1: Identify the Equation
Clearly write down the given equation and identify the variable to be found.
2
Step 2: Simplify Both Sides
Combine like terms on each side of the equation (if necessary).
3
Step 3: Isolate the Variable
Use properties of equality to move all terms with the variable to one side and all constant terms to the other side.
4
Step 4: Apply Properties of Equality
Use the Addition Property of Equality (APE) or Multiplication Property of Equality (MPE) to isolate the variable.
5
Step 5: Verify Your Solution
Substitute your answer back into the original equation to check if both sides are equal.
Common Misconceptions in Solving Equations
Improper Use of Properties
Applying addition property to one side only or multiplying different values on each side breaks the balance of the equation.
Sign Errors
Mistakes in handling negative numbers during addition or multiplication can lead to incorrect solutions.
Not Checking Solutions
Failing to verify answers by substituting back into the original equation can result in accepting incorrect solutions.
Distributing Incorrectly
When dealing with expressions in parentheses, errors in applying the distributive property can occur.
Summary of Algebraic Equation Concepts
1
Application
Solve real-world problems
2
Verification
Check solutions for accuracy
3
Properties of Equality
APE and MPE to isolate variables
4
Equations
Mathematical statements with equal expressions
5
Algebraic Expressions
Variables, constants, and operations
Reflection on Learning
Students can use this chart to reflect on their understanding of different aspects of algebraic equations. As you progress through your learning journey, aim to improve your understanding of each concept. Remember that mathematics builds on foundational concepts, so strengthening your understanding of properties of equality will help with more complex equation types.
Prepared by the Mathematics Department
Prepared by:
___________________
Subject Teacher
Reviewed by:
___________________
Master Teacher/Head Teacher
For Additional Support:
Additional activities for application or remediation are available upon request.
Please contact the mathematics department for more resources.