5th Grade Patterns in Whole Numbers |Associative Distributive Property

In 5th grade patterns in whole numbers we will learn patterns in multiples of 3, patterns in the Associative property of whole numbers, patterns in the Distributive property of whole numbers, patterns in addition and subtraction of whole numbers, patterns in multiplication of whole numbers and different types of solve examples on patterns of whole numbers.

Patterns In Multiples of 3:

Consider some numbers which are multiples of 3:

12, 18, 27, 87, 99, 102, 111, 306, 615, 2463

Now, add the digits of these numbers. If the sum of the digits is a two-digit number, then further add its digits. What do you observe? 

The sum of the digits is 3, 6 or 9. Thus, we can say that the sum of the digits of any multiples of 3 is also a multiple of 3.

This property of whole numbers can be used to check whether a given number is a multiple of 3 or not without actual division.

Patterns in the Associative Property:

Associative property of addition and multiplication of whole numbers can be used to find the sum of numbers easily.

Consider the three numbers 89, 346 and 11.

To find the sum of these numbers, we can proceed as follows:

89 + 346 + 11 = (89 + 11) + 346

                      = 100 + 346

                      = 446

Similarly, 783 + 2945 + 217 = (783 + 217) + 2945

                                          = 1000 + 2945

                                          = 3945

To find the product of 125, 378 and 8, proceed as follows:

125 × 378 × 8 = (125 × 8) × 378

                       = 1000 × 378

                       = 378000

Similarly, 50 × 974 × 2 = (50 × 2) × 974

                                   = 100 × 974

                                   = 97400

Patterns in the Distributive Property:

Arithmetic calculation can be done easily using the distributive property of multiplication over addition.

Let us consider the following simplification:

897 × 13 + 87 × 897

Then, 897 × 13 + 87 × 897

      = 897(13 + 87)

      = 897 × 100 = 89700

Similarly, 723 × 956 + 44 × 723

           = 723(956 + 44)

          = 723 × 1000

          = 723000

Patterns in Addition and Subtraction:

We know that 10 – 2 = 9 – 1 = 8 and 93 – 4 = 92 – 3 = 89

This pattern can be used to subtract numbers easily.

For example:

1000 – 786 = 999 – 785 = 214

7900 – 2796 = 7899 – 2795 = 5104

Similarly, 2798 + 998 = 2798 + 1000 – 2

                                 = 3796

and 1234 – 99 = 1234 – 100 + 1

                      = 1135

Patterns in Multiplication:

Observe the following to explore the patterns in the multiplication:

(i) Product of a Number by another Number Ending with 5:

For Example:

482 × 5 = 482 × \(\frac{10}{2}\) = 2410

960 × 25 = 960 × \(\frac{100}{4}\) = 2400

72 × 15 = 72 × \(\frac{30}{2}\) = 36 × 30 = 1080

(ii) Product of a 2-digit or 3-digit Number Ending with 5 by Itself:

(ii) Simplification Involving Multiplication, Addition and Subtraction

(2 × 2) – (1 × 1) = 4 – 1 = 3 = 2 + 1

(3 × 3) – (2 × 2) = 9 – 4 = 5 = 3 + 2

(4 × 4) – (3 × 3) = 16 – 9 = 7 = 4 + 3

From this pattern, we get

(101 × 101) – (100 × 100) = 101 + 100 = 201 and

(999 × 999) – (998 × 998) =  999 + 998 = 1000 – 1 + 1000 – 2 = 2000 – 3 = 1997

5th Grade Math Problems 

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