Arrays and Partial Products to Multiply

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:

1. Solve the following problems using partial products:
   a) 87 × 4                        b) 78 × 6                         c) 32 × 5
2. Solve using arrays.
   148 × 3                           b) 409 × 2                      c) 306 × 4
3. Explain how can you found the product of 3 and 57.
4. Explore different ways to understand multiplying a three-digit number by a one-digit number.
   What is the product of 3 and 254?
5. 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?
6. Draw a model to represent the product.
   Then record the product.
   a) 3 × 42                                      b) 8 × 34                         c) 2 × 26
7. 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

Concept map: