Informative line

Understanding Integers Using Counters

Introduction and Representation of Integer Counters

Integer counters

  • 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.

Model the integers using counters
Model the integers using counters

Representation

  • 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.

Blue as positive counters and orange as negative counters
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:

Then we can represent the integers using counters as
Then we can represent the integers using counters as

Illustration Questions

Choose the correct option if

A

B

C

D

×

Given:

image

In option (A), we only have a blue counter and it represents a positive integer.

Also we have three blue counters in the figure, so it will represent +3.

+1 + (+1) + (+1) = +3

Hence, option (A) is incorrect.

image

In option (B), we only have an orange counter and it represents a negative integer.

Also we have three orange counters in the figure, so it will represent -3.

-1 + (-1) + (-1) = -3

Hence, option (B) is incorrect.

image

In option (C), we only have a blue counter and it represents a positive integer.

Also we have two blue counters in the figure, so it will represent +2.

+1 + (+1) = +2

Hence, option (C) is correct.

image

In option (D), we only have an orange counter and it represents a negative integer.

Also we have two orange counters in the figure, so it will represent -2.

-1 + (-1) = -2

Hence, option (D) is incorrect.

image

Choose the correct option if

image
A image
B image
C image
D image

Option C is Correct

Zero Pairs

Property:

  • 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:

\(1+(-1)=0\) ,      \(2+(-2)=0\)

Illustration Questions

Which one of the following options has 3 zero pairs? Given: 1 blue counter = a positive integer (+1) 1 orange counter = a negative integer (–1)

A

B

C

D

×

Given:

image

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

There should be 3 blue and 3 orange counters to make 3 zero pairs.

Only in option (D), there are three blue (positive) and three orange (negative) counters.

image

Hence, option (D) is correct.

Which one of the following options has 3 zero pairs? Given: 1 blue counter = a positive integer (+1) 1 orange counter = a negative integer (–1)

A image
B image
C image
D image

Option D is Correct

Subtraction of Integers Using Counters

  • 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).

Consider an example to understand it
Consider an example to understand it

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).

Add enough zero pairs in previous value to perform subtraction
Add enough zero pairs in previous value to perform subtraction

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

Thus, the answer will be the remaining part.

the answer will be remaining part
the answer will be remaining part

Consider an another example to understand it better.

Consider an another example to understand it better
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.

Add enough zero pairs in previous value to perform subtraction
Add enough zero pairs in previous value 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.

Thus, the answer will be remaining part
Thus, the answer will be remaining part

Illustration Questions

Choose the correct option which is equivalent to the given model. If 1 blue counter = a positive integer (+1) and 1 orange counter = a negative integer (–1)

A

B

C

D

×

Given:

image

Here,

image

In the given figure, the first model has 1 blue counter but want to remove 2 counters, so we add enough zero pairs in the first model to perform subtraction.

image

Now, we can remove 2 blue counters from the above framed model.

After taking away the 2 blue counters, the remaining part is,

image

So, we are left with 1 orange counter.

Thus,

image

Hence, option (A) is correct.

Choose the correct option which is equivalent to the given model. If 1 blue counter = a positive integer (+1) and 1 orange counter = a negative integer (–1)

image
A image
B image
C image
D image

Option A is Correct

Multiplication of Integers Using Counters

  • 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.

Illustration Questions

Choose the correct product of the given integer counter models. Consider 1 blue counter represents a positive integer (+1) and 1 orange counter represents a negative integer (–1).

A                              

B

C

D

×

Given models:

image

The model can be expressed as,

\(3×(-6)\)

The expression has

\(A=3\\ B=-6\)

That means join \(3\) groups of \(6\) negative counters.

A group of \(6\) negative counters is,

image

Joining \(3\) groups of \(6\) negative counters,

image

Hence, option (C) is correct.

Choose the correct product of the given integer counter models. Consider 1 blue counter represents a positive integer (+1) and 1 orange counter represents a negative integer (–1).

image
A

                             

.

image
B image
C image
D image

Option C is Correct

Addition of Integers Using Counters

  • 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.

Total number of counters = 6
Total number of counters = 6

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

Total number of counters = 6

Two different colors (positive and negative)
Two different colors (positive and negative)

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).

Draw all these counters together as
Draw all these counters together as

Step 1: Draw all these counters together.

Remove the zero pairs
Remove the zero pairs

Step 2: Remove the zero pairs.

Thus
Thus

Thus, the resulting model represents the result.

Illustration Questions

Find the sum, if 1 orange counter represents a negative integer (–1).

A

B

C

D

×

Given:

image

Here, all the counters are of same color, so we will count the total number of counters to calculate the sum.

Total number of counters = 4

\(\because\)  An orange counter = A negative integer (–1)

Thus,

image

Hence, option (B) is correct.

Find the sum, if 1 orange counter represents a negative integer (–1).

image
A image
B image
C image
D image

Option B is Correct

Factor

  • 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.

integer counter
integer counter

We can arrange it in different ways as:

We can arrange it in different ways as
We can arrange it in different ways as

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

Illustration Questions

Choose the factors of the given integer counter:

A Choose the factor of the given integer counter

B

C Choose the factor of the given integer counter

D Choose the factor of the given integer counter

×

Given:

image

Factor of an integer counter is the arrangement which has the same number of counters as that of integer counter model.

Option (A) has an arrangement of 6 counters.

\(\therefore\) It is a factor of the given integer counter.

Hence, option (A) is correct.

In options (B), (C) and (D), the number of counters are not same as the given integer counter.

Hence, options (A), (B) and (C) are incorrect.

Choose the factors of the given integer counter:

image
A image
B image
C image
D image

Option A is Correct

Division of Integers Using 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).

Illustration Questions

What will be the result, if we solve the give division using integer counters?   \(6\div2\)   \(\left[\begin {array}\ 1\text{ blue counter = A positive integer}\,(+1)\\ 1\text{ orange counter = A negative integer}\,(-1) \end {array}\right]\)

A

B

C

D

×

Given: \(6\div2\)

Here,

A (Number of total counters) = 6 (6 blue counters)

B (Number of groups in which total counters will be separated) = 2

\(\because\) B has positive quantity,

\(\therefore\) we have to separate \(6\) counters into \(2\) groups.

It can be shown as,

image

Since each group has \(3\) positive blue counters, so the answer is \(3.\)

Hence, option (A) is correct.

What will be the result, if we solve the give division using integer counters?   \(6\div2\)   \(\left[\begin {array}\ 1\text{ blue counter = A positive integer}\,(+1)\\ 1\text{ orange counter = A negative integer}\,(-1) \end {array}\right]\)

A image
B image
C image
D image

Option A is Correct

Multiple Operations on Integers Using 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. \(-3\).

Illustration Questions

Solve the given expression using integer counters.   \((-3)×3+5\)   \(\left[\begin {array}\ \text{1 blue counter = A positive integer}\;(+1)\\ \text{1 orange counter = A negative integer}\;(–1) \end {array}\right]\)

A

B

C

D

×

Given: \((-3)×3+5\)

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.

\((-3)×3\)

The expression says, "take away \(3\) groups of \(3\) positive integers".

We start with an empty box,

image

We put enough zero pairs to take away \(3\) groups of \(3\) positive integers.

So, we put \(9\) zero pairs in the empty box.

image

Now we can take away \(3\) sets of \(3\) positive integer counters.

So, the result of multiplication is \(-9\) .

 

image

The original expression becomes, \(-9+5\)

 

Solving addition operation by joining \(9\) negative integer counters and \(5\) positive integer counters.

image

The model has \(5\) zero pairs which can be removed.

The result is \(-4\)

image

Hence, option (B) is correct.

Solve the given expression using integer counters.   \((-3)×3+5\)   \(\left[\begin {array}\ \text{1 blue counter = A positive integer}\;(+1)\\ \text{1 orange counter = A negative integer}\;(–1) \end {array}\right]\)

A image
B image
C image
D image

Option B is Correct

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