Adding and subtracting integers

Adding and subtracting integers involve operations that result in a positive, negative, or zero number. Integers are whole numbers, including positive numbers, negative numbers, and zero. They do not include decimals or fractions. 

• Positive integers: 1, 2, 3, 4…
   
• Negative integers: -1, -4, -8…
   
• Zero: It is neither positive nor negative.

Getting the hang of adding and subtracting integers really just comes down to watching the signs. When you are subtracting and adding integers, think of it like a battle: if the signs are different, they fight (subtract), and the bigger number decides the sign of the answer. It helps to practice adding and subtracting integers consistently to get a feel for the rhythm, especially if you use the "Keep-Change-Change" trick to convert complex subtraction problems into easier addition problems.

If you're still having trouble with negative numbers, downloading some printable adding and subtracting integers worksheets can be a huge help. Seeing the patterns on paper often helps the concept stick faster than memorizing rules. Once you've worked through a few issues, the logic becomes second nature.

Examples:

• \(3 + 4 = 7\)
   
• \(5 + (-2) = 3\)
   
• \(-6 - 3 = -9\)
   
• \(7 - (-1) = 8\)

Adding and subtracting integers follow specific rules that simplify calculations. Addition and subtraction are inverse operations. We can solve problems by using the following rules:

Note: Subtract absolute values. Keep the sign of the larger integer.

Examples:

• -4 + (-6)
  • Logic: You are combining debt. You owe $4, then you borrow $6 more.
  • Action: Add the numbers (4 + 6 = 10) and keep the negative sign.
  • Answer: -10

• -8 + 12
  • Logic: The signs are different, so they fight.
  • Action: Subtract the smaller number from the bigger number (12 - 8 = 4).
  • Sign: The 12 is positive, and it’s the bigger number, so it wins.
  • Answer: 4

• 5 - 9
  • Logic: You have 5, but you spend 9. You are going into debt.
  • Method (Keep-Change-Change):
    • Keep 5
    • Change - to +
    • Change 9 to -9
  • New Problem: 5 + (-9)
  • Answer: -4

• -3 - (-7)
  • Logic: Taking away debt is the same as gaining money.
  • Method (Keep-Change-Change):
    • Keep -3
    • Change - to +
    • Change -7 to positive 7
  • New Problem: -3 + 7
  • Answer: 4 (Because 7 is bigger than 3, the answer is positive).

• 15 + (-25)
  • Logic: You have 15 Positives and 25 Negatives.
  • Action: Subtract the values (25 - 15 = 10).
  • Sign: There are way more Negatives (25) than Positives (15), so the Negatives win.
  • Answer: -10

A number line is a horizontal straight line where integers are placed in equal intervals. To add and subtract integers on a number line, we use the following steps:
 

Addition of integers on a number line
 

• If we want to add a positive integer, we move to the right of the number line.
   
• If we want to add a negative integer, we move to the left of the number line.

Example:

• 4 + (-6) = -2
  • Start: At positive 4.
  • Move: The negative sign means move Left 6 spaces.
  • Land: You cross zero and land on -2.
     
• -3 + 5 = 2
  • Start: At negative -3.
  • Move: The positive sign means move Right 5 spaces.
  • Land: You cross zero and land on positive 2.

Subtraction of integers on a number line
 

• If we want to subtract a positive integer, we move to the left of the number line.
   
• To subtract a negative integer, we move to the right of the number line.

Example:

•  -2 - 3 = -5
  • Start: At negative -2.
  • Move: Subtraction moves Left 3 spaces.
  • Land: You end up deeper in the negatives at -5.
     
• 3 - 7 = -4
  • Start: At positive 3.
  • Move: Move Left 7 spaces.
  • Land: You pass zero and land on -4.

Once understanding the patterns and rules, adding and subtracting integers becomes simple and fun. Here are some quick and effective tips to help you master. 

• Use Concrete Manipulatives (Two-Color Counters): Start with physical objects before moving to abstract numbers. Use two-color counters (e.g., red for negative, yellow for positive) to demonstrate "Zero Pairs visually." When a student physically pairs a red and yellow chip to make them "disappear" (equal zero), they grasp why -1 + 1 = 0 far better than by memorizing a rule.
   
• Switch to Vertical Number Lines: Many students struggle with the left/right concept on a standard number line. Try using a vertical one instead (like a thermometer or a building with basements). It is often more intuitive for students to understand that adding and subtracting positive and negative integers involves going "up" (getting warmer/higher) or "down" (getting colder/lower).
   
• Contextualize with the "Money" Analogy: Money is the ultimate equalizer for math anxiety. Frame every problem in terms of cash and debt. Explain that addition is a transaction (getting money or incurring debt) and subtraction is removing something (forgiving a debt makes you richer). This real-world grounding helps the rules for adding and subtracting integers stick because students instinctively know that eliminating debt is a positive thing.
   
• Teach "Keep-Change-Change" for Subtraction: Subtraction is the biggest stumbling block. Teach students never to subtract integers directly. Instead, have them immediately rewrite every subtraction problem as an addition problem using Keep-Change-Change (Keep the first number, Change the sign of the second number, Change the operator from minus to plus). This unifies the process so they only need to focus on addition rules.
   
• Encourage Self-Checking with Tech: Once students have attempted the manual math, allow them to verify their answers using an adding and subtracting integers calculator. This shouldn't replace the work, but rather act as an instant feedback loop. If the calculator gives a different sign than they did, ask them to investigate why their "battle" logic (which number was bigger?) didn't match the result.
   
• Use Color-Coding for Signs: When writing problems on the board or paper, use distinct colors for positive and negative numbers (e.g., green for positive, red for negative). This visual distinction helps students quickly identify the "teams" in the equation. It reduces simple transcription errors and highlights the "Battle" concept of different signs fighting each other.
   
• Create a "Rule Cheat Sheet": Have students create a small reference card that summarizes the logic (e.g., "Same Signs = Add/Keep," "Different Signs = Subtract/Winner"). Having the rules for adding and subtracting integers visible reduces cognitive load, allowing them to focus on the calculation rather than panic-searching their memory for the correct rule during a test.

Addition and subtraction of integers are used in many real-life situations. Here are a few real-world applications where the addition and subtraction of integers are used:

• Banking: In bank accounts, transactions involve deposits where we add money to the account and withdrawals where money is subtracted from the account.
   
• Temperature changes: Meteorologists track temperature changes over time, including a rise (+) in temperature and a decrease (-) in temperature.
   
• Businesses: Businesses use integers to track profits (+) and losses (-) in financial statements. This is especially useful during accounting and budgeting.
   
• Speed and distance: To estimate the speed of a vehicle while driving, we add or subtract integers. If you start at a speed of +60 km/hr, then slowing down by 20 means, 60 - 20 = +40km/hr. 
   
• Game scores: While playing games or matches, the gained points of the player will be added(+), and the lost points will be subtracted (-).