Subtracting integers can be tricky, but it's a crucial skill in math. We'll explore methods like using number lines and counters to visualize subtraction, and learn how to simplify expressions by converting subtraction to addition of opposites.

We'll also dive into real-world applications, like temperature changes and financial transactions. By mastering these concepts, you'll be better equipped to handle more complex math problems and everyday calculations involving negative numbers.

Subtraction of Integers

Subtraction with number lines

• Number line method involves starting at the minuend (first number) on the number line
  • Move left if the subtrahend (second number) is positive
  • Move right if the subtrahend is negative
  • The endpoint represents the difference (result)
• Counter method uses two colors of counters to represent positive and negative integers
  • Start with counters equal to the absolute value of the minuend (positive counters for positive minuend, negative counters for negative minuend)
  • Remove positive counters if the subtrahend is positive
  • Add negative counters if the subtrahend is negative
  • Remaining counters represent the difference

Simplification of integer expressions

• Subtracting a positive integer is equivalent to adding its opposite (negative)
  • $a - b = a + (-b)$ (subtracting 5 is the same as adding -5)
• Subtracting a negative integer is equivalent to adding its opposite (positive)
  • $a - (-b) = a + b$ (subtracting -3 is the same as adding 3)
• Subtract integers with the same sign by finding the absolute difference and using the sign of the larger absolute value
  • $-a - (-b) = -a + b = b - a$ when $|b| > |a|$ (-7 - (-3) = -7 + 3 = 3 - 7 = -4)
  • $a - b$ when $|a| > |b|$ (8 - 5 = 3)
• Subtract integers with different signs by adding their absolute values and using the sign of the larger absolute value
  • $a - (-b) = a + b$ (4 - (-6) = 4 + 6 = 10)
  • $-a - b = -(a + b)$ (-9 - 2 = -(9 + 2) = -11)

Evaluation of variable expressions (algebraic expressions)

• Substitute given values for variables in the expression
• Follow order of operations (PEMDAS)
  1. Parentheses
  2. Exponents
  3. Multiplication and Division (left to right)
  4. Addition and Subtraction (left to right)
• Simplify the expression using rules for subtracting integers
  • If $x = 3$ and $y = -5$, evaluate $2x - 3y$
    1. $2(3) - 3(-5)$
    2. $6 - (-15)$
    3. $6 + 15 = 21$

Word problems to algebraic expressions

• Identify unknown quantity and assign a variable
• Translate word problem into algebraic expression
  • Positive integers for quantities added or gained
  • Negative integers for quantities subtracted or lost
• Solve algebraic expression to find value of unknown quantity
  • "John has $10. He spends $7 on lunch and earns $5 for helping a friend. How much money does he have now?"
    • Let $x$ represent John's final amount
    • $x = 10 - 7 + 5$
    • $x = 8$, so John has $8

Real-world applications of subtraction

• Temperature changes
  • Subtracting positive temperature change decreases temperature (subtracting 5°C means temperature drops by 5°C)
  • Subtracting negative temperature change increases temperature (subtracting -3°C means temperature rises by 3°C)
• Elevation changes
  • Subtracting positive elevation change decreases elevation (moving downward)
  • Subtracting negative elevation change increases elevation (moving upward)
• Financial transactions
  • Subtracting positive amount decreases balance (withdrawal or expense)
  • Subtracting negative amount increases balance (deposit or income)

Key Concepts in Integer Subtraction

• Integer: A whole number that can be positive, negative, or zero
• Additive inverse: Two numbers that sum to zero (e.g., 5 and -5)
• Inverse operation: Subtraction is the inverse operation of addition
  • This relationship is used to solve equations involving subtraction