Add and subtract integers using counters

Key Notes:

1. Introduction to Integers

  • Integers include positive numbers, negative numbers, and zero.
  • For example: {−3,−2,−1,0,1,2,3}.

2. Using Counters for Integers

  • Counters are objects used to represent numbers visually. They help in understanding addition and subtraction of integers.
  • Two types of counters are used:
    • Positive counters (+1): Represent positive integers.
    • Negative counters (-1): Represent negative integers.

3. Addition of Integers Using Counters

  • Same Signs (Adding Positive or Negative Integers):
    • When adding two positive integers, place that many positive counters together.
    • When adding two negative integers, place that many negative counters together.
    • Example: 3+2=+5 (three positive counters and two more positive counters give a total of five positive counters).
    • Example: −3+(−2)=−5 (three negative counters and two more negative counters give a total of five negative counters).
  • Different Signs (Adding Positive and Negative Integers):
    • When adding a positive integer to a negative integer, combine them and the result will depend on the greater magnitude.
    • Example: +4+(−2):
      • Place four positive counters and two negative counters.
      • The result is +2 (since there are two more positive counters than negative ones).

4. Subtraction of Integers Using Counters

Example: −4−2=−6 (add two more negative counters to the four negative counters).

Subtracting Positive Integers:

To subtract a positive integer, remove that many positive counters.

Example: 5−3=2 (start with five positive counters, remove three, and two remain).

Subtracting Negative Integers:

To subtract a negative integer, it is the same as adding its positive counterpart.

Example: 5−(−3)=5+3=8 (removing negative counters is the same as adding positive counters).

Subtracting a Positive Integer from a Negative Integer:

When subtracting a positive integer from a negative integer, add the counters to the negative side.

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!