Models and Partial Products

Key Concepts

• Use of numerical expressions for partial products
• Area model word problems with partial products

Introduction: 

• Student will learn about the area model and numerical expression 
• Understand partial products 
• Use of numerical expressions for partial products 
• Area model word problems with partial products 

What is Numerical Expression? 

The term numerical expression is made up of two words, numerical meaning numbers and expression meaning phrase. Thus, it is a phrase involving numbers.  

A numerical expression in mathematics can be a combination of numbers and integers combined using at least one mathematical operation such as addition, subtraction, multiplication, or division. 

Examples of Numerical Expressions 

10 + 5 

250 – 75 

60 × 5 + 10  

72 ÷ 8 × 5 – 4 + 1  

82 + 4 – 10 

Partial Products: 

The product of one term of a multiplicand and one term of its multiplier. 

3.4.1 Use of numerical expressions for partial products 

Find partial products for 5 × 3778 

Solution: 

Step 1: Just like traditional long multiplication, we multiply the ones digit of the second factor first. 

5 × 8 = 40 

Step 2: We multiply the 5 by the tens digit of the first factor.  

5 × 70 = 350 

We write the entire 350 beneath 40. 

Step 3: We multiply the 5 by the hundreds digit of the first factor. 

5 × 700 = 3500 

We write the entire 3500 beneath 350. 

Step 4: We multiply the 5 by the thousands digit of the first factor. 

5 × 3000 = 15000 

We write the entire 15000 beneath 3500. 

Step 5: Now that we have taken care of the multiplication, we just need to add our products up in order to make the final product. 

Example :1 

Find the partial product for 6 × 4563 

4 ,5 6 3 

               × 5 

 _______________________ 

                 1   5 5 × 3 

              3   0   0  5 × 6 0 

         2  5   0   0 5 × 5 0 0 

 + 2  0  0   0   0 5 × 4 0 0 0  

_______________________ 

     2  2 ,  8   1    5 

3.4.2 Area model word problems with partial products 

What is Area Model? 

An area model is a rectangular diagram or model used for multiplication and division problems, in which the factors or the quotient and divisor define the length and width of the rectangle. 

We have to break one large area of the rectangle into several smaller boxes, using number bonds to make the calculation easier. Then we add to get the area of the whole, which is the product or quotient. 

Example:1 

A rectangular auditorium has a length of 3654 m and a breadth of 9 m. Find the total area of the auditorium. 

Estimate 9 × 3654 is about 9 × 4000 = 36,000 

Step 1: Write the multiplicand and the multiplier using expanded forms. 

            3654 = 3000 + 600 + 50 + 4 

Step 2: Find the areas of the smaller rectangles.      

Step 3: Add the partial sums to get the total area. 

2 7 0 0 0 

  5 4 0 0 

   4 5 0 

  + 3  6

________________ 

  3 2 ,8 8  6       

 ________________ 

Thus, the area of the auditorium is 32,886 Sq. m. 

32,886 is reasonable because it is close to the estimate of 36,000. 

Example:2 

There are 1,352 nuts in each package. There are 8 packages. How many nuts are there in all? 

Solution: 

Estimate:  8 × 1,352 is about 8 × 1000= 8000 

Step 1: Write the multiplicand and the multiplier using expanded forms.  

              1,352 =1000 + 300 + 50 + 2 

Step 2: Find the areas of the smaller rectangles.  

Step 3: Add the partial sums to get the total area. 

 8 0 0 0 

 2 4 0 0 

    4 0 0 

  + 1  6

________________ 

 1 0, 8 1  6     

________________ 

There are 10,816 nuts in the package. 

10,816 is reasonable because it is close to 8000. 

Exercise:

1. Solve the following problems using partial products
   • 4 × 3482
   • 7 × 3286
   • 8 × 4489
2. Write the numerical expression for the following word problem
   Mary wants to see how many calories she has consumed so far today. She had an apple, which was 95 calories, a banana, which was 105 calories, a bowl of oatmeal, which was 150 calories and a glass of orange juice, which was 110 calories.
3. In a rectangular cinema hall, there are 554 rows of chairs with 6 chairs in each row.
   Find total how many chairs are there?
4. Find 3 × 4,356 using an area model and partial product model.
5. The area of a rectangular fence is 500500 square feet. If the width of the fence is 2020 feet, then find its length.
6. There are 1,467 candles in each package. There are 8 packages. How many candles are there in all?
7. Solve 5 × 2,656 using partial products and an area model. Match each partial product to its area on the model.
8. What are the partial products for 2 × 3,677 using the expanded notation for multiplication?
9. Find the partial products for 6 × 7898.
10. Match the numbers which have an answer of 3,456.
    908                      3
    735                      8
    864                      6
    432                      4

Concept map:

What have we learned:

• Student will learn about area model and numerical expression
• About  Partial products
• Use of numerical expressions for partial products
• Area model word problems with partial products