## Key Concepts

- Use arrays for partial products
- Use of area for partial products

**Introduction:**

- In this chapter, we will learn about the introduction of array.
- About partial products
- Use arrays in partial products
- Area of rectangle
- Area for partial products

**Use arrays for partial products**

#### Partial products:

A product obtained by multiplying a multiplicand by one digit of a multiplier having more than one digit.

**Example: 1 **

Find the product for 3 × 14.

In the below image indicate using arrays for partial products.

From the above image

** **The numbers 30 and 12 are called partial products.

42 is the product.

**Another way:**

14

× 3

______

12 3 × 4 ones

+ 30 3 × 1 tens

_______

** 42**

**______**

The product 42 is close to the estimate 40. The answer is reasonable.

We can use place value to break factors and distributive property to find partial products.

**Example: 2 **

Multiply 3 × 46

Break 46 into two parts: 40 and 6.

Then, multiply those two parts separately by 3:

3 × 40 is 120, and 3 × 6 is 18.

Then, add these two partial results: 120 + 18 = 138

138 is close to 130

**Note**: We can use place value to break factors and the distributive property to find the partial products.

Let us see another example:

Find 2 × 124

**Estimate:**

2 × 124 is about 2 × 100 = 200

**Open array:**

Here, we can use the distributive property

3 × 14 = (3 × 10) +(3 × 4)

**3.4 Use of area for partial products **

Area is the region bounded by the shape of an object. The space covered by the figure or any geometric shapes is the area of the shape.

Area of rectangle = Length × Breadth

A = lb

**Example1: **

A park is in the shape of a rectangle. It is 6 feet wide and 24 feet long. What is the area of the park?

6 × 24 is a numerical expression

We can use rectangular area model to show multiplication.

The product of 6 × 24 is the area of the rectangle.

Estimate 6 × 24 is about 6 × 20 = 120

120 + 24 = 144

The distributive property says that the sum of numbers is same as multiplying each part of the sum by that number and adding the partial products.

6 × 24 = (6 × 20) + (6 × 4)

**Another way:**

2 4

× 6

_________

2 4 6 × 4 ones

+ 1 2 0 6 × 2 tens

_________

1 4 4

The area of the park is 144 square feet.

The product, 144 is close to the estimate of 120. The answer is reasonable.

**Example 2: **

Find the area of 8 × 24

The picture illustrates the area of a rectangle with sides 8 and 24. It is also divided into two rectangles.

The area of the whole rectangle is 8 × 24 square units.

We can find that by calculating the areas of the two rectangles and adding.

The area of the first rectangle is 8 × 20 = 160 square units.

The area of the second rectangle is 8 × 4 = 32 square units.

So, the area of the whole rectangle is the sum 160 + 32 = 192 square units.

Another way

2 4

× 8

_______________

32 8 × 4

+ 160 8 × 20

_______________

1 9 2

### Exercise:

- Solve the following problems using partial products:

a) 87 × 4 b) 78 × 6 c) 32 × 5 - Solve using arrays.

148 × 3 b) 409 × 2 c) 306 × 4 - Explain how can you found the product of 3 and 57.
- Explore different ways to understand multiplying a three-digit number by a one-digit number.

What is the product of 3 and 254? - What will be the cost of gardening a 1 meter broad boundary around a rectangular plot having a perimeter of 340 meters at the rate of Rs. 8 per square meter?
- Draw a model to represent the product.

Then record the product.

a) 3 × 42 b) 8 × 34 c) 2 × 26 - What product does the model below represent?

8. A rectangular room has a length of 16 feet and a width of 8 feet. How much carpet is required to cover the entire room?

9.The length and width of a rectangular farm are 80 yards and 6 yards. Find the area of the farm.

10.Find the total distance around the rectangular field, if the length of the field is 50 meters and width is 8 meters. Also, find the area of the field.

### What have we learned:

- Introduction of array.
- Introduction of partial products
- Use arrays in partial products
- Area of rectangle
- Area for partial products