- Integer counter is a way to describe and model the integers using counters.

Example: Consider the figure shown. In this we use two different coloured counters to represent positive and negative integers.

- Integers are positive and negative.
- Similarly, counters are also considered to be positive and negative.
- We will consider blue as positive counters and orange as negative counters.

- If a blue counter represents +1 and an orange counter represents –1 then we can represent the integers using counters as:

- Inverse identity of integers says, when two same numbers having opposite signs are being added, the result will be zero.

Example: \(a+(-a)=0\)

where \(a=\) integer

- Similarly, in integer counters, when the same number of positive and negative counters are placed together, they are called zero pairs.

For example:

\(1+(-1)=0\) , \(2+(-2)=0\)

- Subtraction of integers can be shown by using integer counters.
- Subtraction means taking away some quantity from another.

Consider an example to understand it.

Here, 1 blue counter represents a positive integer (+1).

In the given figure, the first model has 3 blue counters but we have to remove 4 counters, so we add enough zero pairs in the first model to perform subtraction.

Here, 1 orange counter represents a negative integer (–1).

Now we can remove 4 blue counters from the above framed model.

Thus, the answer will be the remaining part.

Consider an another example to understand it better.

Here, we have 2 orange counters in the first model but want to remove 4 blue counters, so we add enough zero pairs in first model to perform subtraction.

Now, we can remove 4 blue counters from the above framed model. So, we are left with 6 orange counters.

Thus, the answer will be the remaining part.

- The counter models are built on the thought of multiplication as repeated addition or repeated subtraction.
- Multiplication is expressed through the expression,

\(\underbrace{A}_{\text{Number of groups}}×\underbrace{B}_{\text{Number of counters}}\)

"\(A\) groups of \(B\) counters"

- Join the counters when \(A\) has positive quantity.
- Take away the counters when \(A\) has negative quantity.

**Case 1:** When \(A\) has positive quantity

Consider an example:

\(3×4\)

The expression has

\(A=3\\B=4\)

That means join \(3\) groups of \(4\) positive counters.

- A group of \(4\) positive counters is shown.

- So, on joining \(3\) groups of \(4\) positive counters we obtain the figure shown.

**Case 2:** When \(A\) has negative quantity

Consider an example:

\(-2×4\)

The expression has

\(A=-2\\ B=4\)

That means take away \(2\) groups of \(4\) positive counters.

- In this case we start with no counters or zero.
- The expression tells to take away the counters, but it is not possible with no counters, so we use the property of additive inverse, i.e.

\(a+(-a)=0\)

**Step-1** Start with an empty box or no counters.

**Step-2** There are no counters to take out, so put in zero pairs until there are enough counters to take out.

So, we put \(8\) zero pairs in the empty box.

**Step-3** Now, take away \(A\) groups of \(B\) counters.

Taking away \(2\) sets of \(4\) positive counters.

Now, the left part is the result.

- Addition of integers can be shown using integer counters.
- Addition of counter models means joining the models together.
- Two cases arise when we add integer counters.

**Case-I** When same colored integer counters are being added

When integer counters being added are of the same color (positive or negative), we can add them by counting the total number of counters.

For example: If a blue counter represents a positive integer (+1) then find the sum of the given model.

- To calculate the sum of same colored counters, we will count the total number of counters.

Total number of counters = 6

**Case-II **When integer counters being added are of different colors

Consider an example to add the integer counters which are of two different colors (positive and negative).

**Step 1:** Draw all these counters together.

**Step 2:** Remove the zero pairs.

Thus, the resulting model represents the result.

- Factors are the different possible arrangements of an integer counter.
- If we arrange an integer counter in different ways, these all arrangements are the factors of that integer counter.
- Consider an example shown in the figure.

We can arrange it in different ways as:

Factors of an integer counter have the same number of counters.

\(\to\) Division of integers can be shown by using integer counter models.

\(\to\) In division of integers, \((A\div B),\;A\) represents the number of counters we have and \(B\) represents the number of counters each group contains.

\(\to\) The answer will show the number of \(B\) counters.

OR

\(\to\) If \(B\) represents the number of groups in which counters to be divided, the answer will show the number of counters in each group.

\(\to\) (i) If B has positive sign, the answer is the number of counters in each group.

(ii) If B has negative sign, the answer is the number of counters in each group to cancel out all the A counters.

**Case-1** When \(B\) has positive quantity

\(\to\) Consider an example-

\((-12)\div4\)

Here,

\(A\;\text{(Number of total counters) = –12 (12 orange counters)}\\ B\;\text{(Number of groups)}=4\)

We have to separate \(12\) negative counters into \(4\) groups.

\(\to\) Since each group contains \(3\) negative counters, so the quotient will be \(-3\).

**Case-2** When \(B\) has negative quantity

Consider an example-

\((-14)\div(-7)\)

Here,

A (Number of total counters) = –14 (14 orange counters)

B (Number of groups in which the total counters will be distributed for elimination) =7

\(\to\) Here, \(14\) counters are to distributed into \(7\) groups. So we have to create zero pairs.

\(\to\) Start with \(14\) negative integer counters.

\(\to\) Create \(7\) groups of \(14\) counters.

\(\to\) Here, \(7\) groups of positive counters each are needed to eliminate all groups.

\(\to\) In this case, the answer is the number of counters in each group to cancel out all the counters we had in starting.

Thus, the answer is 2 (positive counters).

\(\to\) On integer counters, multiple operations can be performed by taking one at a time, but the order of operations is taken into account.

\(\to\) In solving multiple operations we follow PEMDAS rule, i.e. we first perform the operations of multiplication/division and then addition/subtraction, in the order from left to right.

\(\to\) Consider an example-

\(2×(-3)+3\)

\(\to\) To solve the above expression using integer counters, we first solve the multiplication operation.

\(2×(-3)\)

where

\(A=2\\ B=-3\)

The expression says, join \(2\) groups of \(3\) negative counters.

\(\to\) After joining the groups, we have:

\(\to\) After solving the multiplication, the original expression becomes-

\(-6+3\)

\(\to\) Now, we solve the addition operation, which is performed by joining the counters together.

\(\to\) Here, we have \(6\) negative integer counters and \(3\) positive integer counters, to join together.

\(\to\) The model creates \(3\) zero pairs which can be removed.

\(\to\) So, the remaining part is the result, i.e. \(-3\).