Summary of Equality: Same Operation on Both Sides

INTRODUCTION TO THE EQUALITY THEME: SAME OPERATION ON BOTH SIDES

Relevance of the Theme

• Mathematics Fundamentals: Understanding equality is the basis for solving equations and problems.
• Algebra Pillar: Knowing and applying the equality property prepares for future studies in Algebra.
• Tool for Life: Knowing that the balance of equality must always be maintained is a lesson that applies to numbers as well as to justice and balance in everyday situations.

Contextualization

• In the World of Numbers: We learn to add, subtract, multiply, and divide. Now it's time to understand that what we do on one side, we must do on the other to keep everything balanced.
• Relation to Other Disciplines: Just like in Science, where action causes reaction, in Mathematics, any action on one side of the equality demands an equal reaction on the opposite side.
• Essential Skill: We already know that 1 + 1 = 2. What if we want to add more? If we add 1 more to each side, the equality will still hold true: 2 + 1 = 2 + 1.
• Logical Development: Practicing equality helps strengthen logical reasoning – a very important skill for all types of learning.

Reminder: Always maintain balance, whether in numbers or in life!

THEORETICAL DEVELOPMENT: EQUALITY AND OPERATIONS

Components

• Equality ( = ): Sign that shows that the value on one side is exactly the same as on the other side.

  • Equality Balance: Imagining a balance can help understand. If both sides weigh the same, the balance is in equilibrium.
  • Breaking Equality: If we do something different on one side, the balance is lost, and the equality is broken.
  • Restoration of Equality: To keep the balance, the same operation must be done on both sides.

• Basic Operations:

  • Addition (+): Adding the same quantity to both sides maintains equality. Ex: If 3=3, then 3+2=3+2.
  • Subtraction (-): Subtracting the same quantity from both sides maintains equality. Ex: If 4=4, then 4-1=4-1.
  • Multiplication (×): Multiplying by the same number on both sides maintains equality. Ex: If 5=5, then 5×2=5×2.
  • Division (÷): Dividing by the same number on both sides maintains equality. Ex: If 6=6, then 6÷3=6÷3.

Key Terms

• Equation: A mathematical sentence that shows two expressions are equivalent.

  • Expression: Combination of numbers and operations without an equality sign.
  • Operand(s): Number(s) with which we perform the mathematical operation.

• Equality Property: Principle that states we can perform the same operation on both sides of the equality without changing the truth of the equation.

Examples and Cases

• Addition Example:

  • We have the equality 7=7.
  • If we add 2 to both sides, we get 7+2 on one side and 7+2 on the other, resulting in 9=9.
  • The equality remains true after the same operation on both sides.

• Subtraction Example:

  • Starting with 10=10.
  • Subtracting 3 from each side, we have 10-3 on one side and 10-3 on the other, giving us 7=7.
  • The equality still holds after the operation.

• Multiplication Example:

  • We begin with 2=2.
  • Multiplying both sides by 4, we get 2×4 on one side and 2×4 on the other, resulting in 8=8.
  • The equality is preserved even after multiplying both sides by the same number.

• Division Example:

  • Initially, we have 12=12.
  • Dividing both sides by 3, then 12÷3 on one side and 12÷3 on the other leads us to 4=4.
  • The equality remains true even after dividing by the same number on both sides.

Catch Phrase: Keep the balance! Do the same thing on both sides!

DETAILED SUMMARY

Key Points

• Equality Concept: Equality means that the value on one side is the same as on the other.
  • Visualization as a balance to understand the need for equilibrium.
• Impact of Operations on Equality: Performing different operations on each side breaks the equality.
  • Concrete examples demonstrate the maintenance of balance after the same operation on both sides.
• Mathematical Operations and Equality: Addition, subtraction, multiplication, and division applied equally maintain the truth of the equality.
  • Mathematical expressions as representations of actions performed on numbers.

Conclusions

• Preserved Equality: The same operation on both sides does not alter the truth of the equality.
• Basic Algebra: Introductory principles of algebra are established through the understanding of equality.
• Reasoning Skill: Practice with equalities develops logical thinking and problem-solving skills.

Exercises

1. Addition Exercise:
   • If we have the equality 8=8, what happens if we add 5 to both sides?
     • Expected answer: 8+5=8+5, resulting in 13=13.
2. Subtraction Exercise:
   • Given that 15=15, what will be the new equality if we subtract 4 from each side?
     • Expected answer: 15-4=15-4, giving us 11=11.
3. Combined Exercise:
   • We have the equality 20=20. If we multiply both sides by 2 and then subtract 10, what is the final result?
     • Expected answer: First, we multiply: 20×2=20×2, obtaining 40=40. Then, we subtract: 40-10=40-10, resulting in 30=30.

Important Reminder: Practice keeping the balance of numbers always in equilibrium. Remember, what you do on one side, repeat on the other!