Add and subtract integers using counters

Key Notes:

🔢 What Are Integers?
  • Integers are positive and negative whole numbers, including zero.
    Examples: -5, -1, 0, 3, 7
🔴🔵 What Are Counters?

Counters are visual tools to help understand positive and negative integers.We use:

  • 🔵 Blue counters to represent positive numbers (+1)
  • 🔴 Red counters to represent negative numbers (–1)
➕ Adding Integers Using Counters
Same Signs – Add and Keep the Sign

Add the number of counters and keep the same sign.

Example:

  • (+3) + (+2)
  • 🔵🔵🔵 + 🔵🔵 = 🔵🔵🔵🔵🔵 → +5

Example:

  • (–4) + (–2)
  • 🔴🔴🔴🔴 + 🔴🔴 = 🔴🔴🔴🔴🔴🔴 → –6
Different Signs – Subtract and Use the Sign of the Bigger Number

Example:

  • (+5) + (–3)
  • 🔵🔵🔵🔵🔵 + 🔴🔴🔴
  • → Pair and cancel: 🔵🔴 🔵🔴 🔵🔴
  • Remaining: 🔵🔵 → +2

Example:

  • (–6) + (+2)
  • 🔴🔴🔴🔴🔴🔴 + 🔵🔵
  • → Pair and cancel: 🔴🔵 🔴🔵
  • Remaining: 🔴🔴🔴🔴 → –4
➖ Subtracting Integers Using Counters

Rule: Change Subtraction to Addition

a – b = a + (–b)

Examples:

Example 1:

  • (+4) – (+2) → Change to: (+4) + (–2)
  • 🔵🔵🔵🔵 + 🔴🔴 → Cancel pairs: 🔵🔴 🔵🔴
  • Remaining: 🔵🔵 → +2

Example 2:

  • (–5) – (–3) → Change to: (–5) + (+3)
  • 🔴🔴🔴🔴🔴 + 🔵🔵🔵 → Cancel pairs: 🔴🔵 🔴🔵 🔴🔵
  • Remaining: 🔴🔴 → –2
📝 Practice:
🔹 Part A: Represent with Counters

Draw 🔵 and 🔴 counters to represent the problems and solve.

  1. (+4) + (–3) = ______
  2. (–5) + (–2) = ______
  3. (+6) + (+2) = ______
  4. (–3) + (+5) = ______
  5. (+3) – (+1) = ______
  6. (–6) – (–4) = ______
  7. (+5) – (–2) = ______
  8. (–7) + (+3) = ______
  9. (–4) – (+2) = ______
  10. (+2) – (–5) = ______

✅ Answer Key

Part A:

  1. +1
  2. –7
  3. +8
  4. +2
  5. +2
  6. –2
  7. +7
  8. –4
  9. –6
  10. +7

🔹 Part B: Word Problems

Use counters (draw if needed) to solve:

  1. You gained 6 points in a game and then lost 4 points. What is your score now?
  2. A submarine is at –10 meters. It rises 6 meters. Where is it now?
  3. You owe 5 rupees and borrow 3 more. What is your total debt?

✅ Answer Key

Part B:

  1. +2
  2. –4 meters
  3. –8 rupees

💡 Tips:

  • Practice with real or drawn counters!negative side.
  • Cancel out 1 🔵 and 1 🔴 when together (they make 0).
  • Change subtraction to addition by adding the opposite.

Learn with an example

➡️ Use counters to add 2 + 3.

You start with:

You add:

Which picture shows the sum?

You are trying to calculate 2 + 3.

Start out with 2 positive counters, which represent 2:

Now add 3 positive counters:

Since the counters are all positive, just count them. There are 5 positive counters, so 2 + 3 = 5.

🔥Use counters to subtract 3 − (-1).

You start with:

You subtract :

Which picture shows the difference?

You are trying to calculate 3 − (-1).

Start out with 3 positive counters, which represent 3:

You would like to take away 1 negative counter. However, it’s not possible to take away 1 negative counter, because there aren’t any negative counters in the first group. You can fix this problem by adding 1 pair of positive and negative counters to the first group, so that you have the 1 negative counter that is needed:

Now take away 1 negative counter:

4 positive counters remain. So 3 − (-1) = 4.

🔥Use counters to subtract -1 − 2.

You start with:

You subtract:

Which picture shows the difference?

You are trying to calculate -1 − 2.

Start out with 1 negative counter, which represents

-1:

You would like to take away 2 positive counters. However, it’s not possible to take away 2 positive counters, because there aren’t any positive counters in the first group. You can fix this problem by adding 2 pairs of positive and negative counters to the first group, so that you have the 2 positive counters that are needed:

Now take away 2 positive counters:

Try some practice problems!