How To Do Integers Grade 7

Integers are whole numbers that can be positive, negative, or zero. Understanding and mastering integers is a fundamental step in building a solid foundation in mathematics for 7th graders. This article will provide a comprehensive guide on how to perform various operations with integers, including addition, subtraction, multiplication, and division. We will break down the rules, provide examples, and offer helpful tips to make learning integers easier and more effective.

Introduction to Integers

Integers are a crucial part of the number system and are used extensively in algebra, geometry, and various real-world applications. Unlike whole numbers, which only include positive numbers and zero, integers include both positive and negative numbers, extending infinitely in both directions.

Key Concepts:

• Positive Integers: These are whole numbers greater than zero (e.g., 1, 2, 3, ...).
• Negative Integers: These are whole numbers less than zero (e.g., -1, -2, -3, ...).
• Zero: Zero is an integer that is neither positive nor negative.
• Number Line: A visual representation of integers, with zero at the center, positive integers to the right, and negative integers to the left.

Understanding these basic concepts is essential before delving into operations with integers.

Addition of Integers

Adding integers involves combining two or more integers to find their sum. The rules for adding integers depend on whether the integers have the same sign or different signs.

1. Adding Integers with the Same Sign:

When adding two integers with the same sign (both positive or both negative), add their absolute values and keep the common sign.

• Example 1: Add 3 + 5
  • Both numbers are positive.
  • Add their absolute values: |3| + |5| = 3 + 5 = 8
  • Keep the positive sign: +8
  • So, 3 + 5 = 8
• Example 2: Add -4 + (-6)
  • Both numbers are negative.
  • Add their absolute values: |-4| + |-6| = 4 + 6 = 10
  • Keep the negative sign: -10
  • So, -4 + (-6) = -10

2. Adding Integers with Different Signs:

When adding two integers with different signs (one positive and one negative), subtract the smaller absolute value from the larger absolute value. The result takes the sign of the integer with the larger absolute value.

• Example 1: Add -7 + 3
  • The numbers have different signs.
  • Find the absolute values: |-7| = 7 and |3| = 3
  • Subtract the smaller absolute value from the larger: 7 - 3 = 4
  • The larger absolute value is |-7| = 7, which has a negative sign.
  • So, -7 + 3 = -4
• Example 2: Add 9 + (-2)
  • The numbers have different signs.
  • Find the absolute values: |9| = 9 and |-2| = 2
  • Subtract the smaller absolute value from the larger: 9 - 2 = 7
  • The larger absolute value is |9| = 9, which has a positive sign.
  • So, 9 + (-2) = 7

Tips for Adding Integers:

• Visualize on a Number Line: Imagine moving along a number line. Adding a positive integer means moving to the right, and adding a negative integer means moving to the left.
• Use Counters: Use different colored counters (e.g., red for negative, blue for positive) to represent integers and physically combine them.
• Practice Regularly: Consistent practice will reinforce the rules and improve your speed and accuracy.

Subtraction of Integers

Subtracting integers can be thought of as adding the opposite. To subtract an integer, change the subtraction to addition and change the sign of the integer being subtracted. Then, follow the rules for addition.

Rule: a - b = a + (-b)

• Example 1: Subtract 5 - 8
  • Change the subtraction to addition and change the sign of 8: 5 + (-8)
  • Now, follow the addition rules for integers with different signs.
  • |-8| = 8 and |5| = 5
  • 8 - 5 = 3
  • Since |-8| is larger, the result is negative: -3
  • So, 5 - 8 = -3
• Example 2: Subtract -3 - 4
  • Change the subtraction to addition and change the sign of 4: -3 + (-4)
  • Now, follow the addition rules for integers with the same sign.
  • |-3| = 3 and |-4| = 4
  • 3 + 4 = 7
  • Keep the negative sign: -7
  • So, -3 - 4 = -7
• Example 3: Subtract 2 - (-6)
  • Change the subtraction to addition and change the sign of -6: 2 + 6
  • Now, follow the addition rules for integers with the same sign.
  • 2 + 6 = 8
  • So, 2 - (-6) = 8
• Example 4: Subtract -1 - (-5)
  • Change the subtraction to addition and change the sign of -5: -1 + 5
  • Now, follow the addition rules for integers with different signs.
  • |5| = 5 and |-1| = 1
  • 5 - 1 = 4
  • Since |5| is larger, the result is positive: 4
  • So, -1 - (-5) = 4

Tips for Subtracting Integers:

• Rewrite the Problem: Always rewrite the subtraction problem as an addition problem by changing the sign of the integer being subtracted.
• Use the Number Line: Visualize moving to the left when subtracting a positive integer and moving to the right when subtracting a negative integer.
• Practice with Different Scenarios: Work through various examples to become comfortable with the rule of "adding the opposite."

Multiplication of Integers

Multiplying integers involves finding the product of two or more integers. The rules for multiplying integers depend on the signs of the integers being multiplied.

1. Multiplying Integers with the Same Sign:

When multiplying two integers with the same sign (both positive or both negative), the product is always positive.

• Rule:
  • Positive x Positive = Positive
  • Negative x Negative = Positive
• Example 1: Multiply 4 x 6
  • Both numbers are positive.
  • 4 x 6 = 24
  • So, 4 x 6 = 24
• Example 2: Multiply -3 x -5
  • Both numbers are negative.
  • 3 x 5 = 15
  • The product is positive: 15
  • So, -3 x -5 = 15

2. Multiplying Integers with Different Signs:

When multiplying two integers with different signs (one positive and one negative), the product is always negative.

• Rule:
  • Positive x Negative = Negative
  • Negative x Positive = Negative
• Example 1: Multiply -2 x 7
  • The numbers have different signs.
  • 2 x 7 = 14
  • The product is negative: -14
  • So, -2 x 7 = -14
• Example 2: Multiply 8 x -4
  • The numbers have different signs.
  • 8 x 4 = 32
  • The product is negative: -32
  • So, 8 x -4 = -32

Tips for Multiplying Integers:

• Remember the Rules: A simple way to remember the rules is: "Same signs, positive answer; different signs, negative answer."
• Ignore Signs Initially: Multiply the absolute values of the integers first, then determine the sign of the product.
• Practice Mental Math: Try to memorize basic multiplication facts to improve your speed.

Division of Integers

Dividing integers involves finding the quotient of two integers. The rules for dividing integers are similar to those for multiplying integers.

1. Dividing Integers with the Same Sign:

When dividing two integers with the same sign (both positive or both negative), the quotient is always positive.

• Rule:
  • Positive ÷ Positive = Positive
  • Negative ÷ Negative = Positive
• Example 1: Divide 12 ÷ 3
  • Both numbers are positive.
  • 12 ÷ 3 = 4
  • So, 12 ÷ 3 = 4
• Example 2: Divide -15 ÷ -5
  • Both numbers are negative.
  • 15 ÷ 5 = 3
  • The quotient is positive: 3
  • So, -15 ÷ -5 = 3

2. Dividing Integers with Different Signs:

When dividing two integers with different signs (one positive and one negative), the quotient is always negative.

• Rule:
  • Positive ÷ Negative = Negative
  • Negative ÷ Positive = Negative
• Example 1: Divide -20 ÷ 4
  • The numbers have different signs.
  • 20 ÷ 4 = 5
  • The quotient is negative: -5
  • So, -20 ÷ 4 = -5
• Example 2: Divide 24 ÷ -6
  • The numbers have different signs.
  • 24 ÷ 6 = 4
  • The quotient is negative: -4
  • So, 24 ÷ -6 = -4

Tips for Dividing Integers:

• Remember the Rules: The same sign rules apply to both multiplication and division: "Same signs, positive answer; different signs, negative answer."
• Check Your Work: Multiply the quotient by the divisor to ensure it equals the dividend.
• Be Mindful of Zero: Division by zero is undefined. Always be careful when dealing with zero in division problems.

Order of Operations with Integers

When solving mathematical expressions involving integers, it is essential to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

PEMDAS:

1. Parentheses: Solve any expressions inside parentheses or brackets first.
2. Exponents: Evaluate any exponents.
3. Multiplication and Division: Perform multiplication and division from left to right.
4. Addition and Subtraction: Perform addition and subtraction from left to right.

Example 1: Solve 2 x (3 + (-5)) - 4 ÷ 2

1. Parentheses: 3 + (-5) = -2
   • The expression becomes: 2 x (-2) - 4 ÷ 2
2. Multiplication: 2 x (-2) = -4
   • The expression becomes: -4 - 4 ÷ 2
3. Division: 4 ÷ 2 = 2
   • The expression becomes: -4 - 2
4. Subtraction: -4 - 2 = -6
   • So, 2 x (3 + (-5)) - 4 ÷ 2 = -6

Example 2: Solve -3 + 4 x (-2) - (-10) ÷ 5

1. Multiplication: 4 x (-2) = -8
   • The expression becomes: -3 + (-8) - (-10) ÷ 5
2. Division: (-10) ÷ 5 = -2
   • The expression becomes: -3 + (-8) - (-2)
3. Addition: -3 + (-8) = -11
   • The expression becomes: -11 - (-2)
4. Subtraction: -11 - (-2) = -11 + 2 = -9
   • So, -3 + 4 x (-2) - (-10) ÷ 5 = -9

Tips for Order of Operations:

• Write Down Each Step: Break down the problem into smaller steps and write down each step to avoid errors.
• Follow PEMDAS Carefully: Ensure you are following the correct order of operations.
• Double-Check Your Work: Review each step to catch any mistakes.

Real-World Applications of Integers

Integers are not just abstract mathematical concepts; they are used in various real-world situations. Understanding how to apply integers can make them more relatable and easier to grasp.

1. Temperature:

Temperature is a common example of using integers. Temperatures above zero are represented by positive integers, while temperatures below zero are represented by negative integers.

• Example: If the temperature is -5°C in the morning and rises by 8°C in the afternoon, the new temperature is -5 + 8 = 3°C.

2. Finances:

Integers are used to represent gains and losses in finances. Positive integers represent income or profits, while negative integers represent expenses or debts.

• Example: If you have $50 in your account and spend $75, your new balance is 50 - 75 = -$25.

3. Altitude:

Altitude or elevation is another application of integers. Heights above sea level are represented by positive integers, while depths below sea level are represented by negative integers.

• Example: A submarine diving 300 feet below sea level is at an altitude of -300 feet. If it then rises 150 feet, its new altitude is -300 + 150 = -150 feet.

4. Sports:

In sports, integers can represent scores, yards gained or lost, and other performance metrics.

• Example: In football, if a team gains 15 yards on one play and loses 7 yards on the next, their net gain is 15 + (-7) = 8 yards.

5. Time Zones:

Time zones are often represented using integers to indicate the difference from Coordinated Universal Time (UTC).

• Example: New York City is UTC-5, meaning it is 5 hours behind UTC. If it is 3:00 PM UTC, it is 3:00 PM - 5 hours = 10:00 AM in New York City.

Tips for Real-World Applications:

• Identify the Context: Understand the situation and what the integers represent.
• Translate into Mathematical Expressions: Convert the real-world problem into a mathematical expression using integers.
• Solve and Interpret: Solve the expression and interpret the result in the context of the problem.

Common Mistakes to Avoid

When working with integers, there are several common mistakes that students often make. Being aware of these mistakes can help you avoid them and improve your accuracy.

1. Incorrectly Applying Addition Rules:

• Mistake: Adding integers with different signs by adding their absolute values instead of subtracting.
• Correct: Subtract the smaller absolute value from the larger absolute value and use the sign of the integer with the larger absolute value.

2. Forgetting to Change Subtraction to Addition:

• Mistake: Subtracting integers without changing the subtraction to addition and changing the sign of the integer being subtracted.
• Correct: Rewrite the subtraction problem as an addition problem (a - b = a + (-b)).

3. Mixing Up Multiplication and Division Rules:

• Mistake: Applying the wrong sign rules when multiplying or dividing integers.
• Correct: Remember: "Same signs, positive answer; different signs, negative answer."

4. Ignoring Order of Operations:

• Mistake: Performing operations in the wrong order (e.g., adding before multiplying).
• Correct: Follow PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

5. Misinterpreting Real-World Problems:

• Mistake: Failing to correctly translate a real-world problem into a mathematical expression.
• Correct: Carefully read and understand the context of the problem before translating it into an equation.

Tips to Avoid Mistakes:

• Practice Regularly: Consistent practice helps reinforce the rules and improve your skills.
• Show Your Work: Writing down each step can help you identify and correct errors.
• Double-Check Your Answers: Take the time to review your work and ensure your answers are correct.

Strategies for Mastering Integers

Mastering integers requires more than just memorizing rules; it involves understanding the concepts and developing effective problem-solving strategies. Here are some strategies to help you master integers:

1. Use Visual Aids:

• Number Line: Use a number line to visualize integers and their relationships.
• Counters: Use different colored counters to represent positive and negative integers.

2. Relate to Real-World Examples:

• Temperature: Think about how integers are used to represent temperatures above and below zero.
• Finances: Relate integers to gains and losses in finances.

3. Practice Regularly:

• Consistent Practice: Solve a variety of problems involving addition, subtraction, multiplication, and division of integers.
• Online Resources: Utilize online resources, such as websites and apps, to practice and test your knowledge.

4. Break Down Complex Problems:

• Simplify: Break down complex problems into smaller, more manageable steps.
• Follow PEMDAS: Always follow the order of operations to ensure accuracy.

5. Seek Help When Needed:

• Ask Questions: Don't hesitate to ask your teacher or classmates for help if you are struggling with a concept.
• Tutoring: Consider getting a tutor to provide personalized instruction and support.

Conclusion

Mastering integers is a critical step in building a strong foundation in mathematics. By understanding the rules for addition, subtraction, multiplication, and division, and by practicing regularly, 7th graders can develop the skills and confidence they need to succeed in more advanced math courses. Remember to use visual aids, relate to real-world examples, and break down complex problems into manageable steps. With consistent effort and the right strategies, you can master integers and unlock new possibilities in mathematics.