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Strategy to Multiply

Key Concepts

  • Estimation, Standard algorithm, partial products, and word problems

Estimation, standard algorithm, partial products, and word problems 

Estimation: 

Estimation means finding a number that is close enough to the right answer. 

Example: 

Estimate the product of 7 × 489 by rounding to the nearest hundred. 

7 × 489 

489 –>        500 

Multiply 7 × 500 = 3500 

Multiply 7 × 489 = 3,423 

3,423 is close to 3,500. It is reasonable. 

What is meant by a standard algorithm? 

A standard algorithm or method is a specific method of computation that is conventionally taught for solving particular mathematical problems.  

Standard algorithm is a way of doing multiplication by using partial products or multiplying in parts. 

What is meant by partial products? 

A model that breaks numbers down into their factors or place values to make multiplication easier. 

Example 1:

Multiply 215 × 9 

Step1: Multiply 200 × 9 = 1800 

Step2: Multiply 10 × 9 = 90 

Step3: Multiply 5 × 9 = 45 

Step4: Add 1800 + 90 + 45 =1935 

Example2:  

Each T-Shirts’ cost is $485. Find the cost of 6 T-Shirts?  

Each frock’s cost is $4,480. Find the cost of 3 frocks?  

Also, find the total cost for T-Shirts and Frocks. 

Solution: 

6 × 485 = c 

Estimate: 6 × 485 is about 6 × 480= 2,880 

Break apart 485 using place value and the distributive property. 

6 × 485 = 6 × (400 + 85) 

= 6 × 400 + 6 × 85 

= 2400 + 510 

= 2,910 

The total cost of T-Shirts is $ 2,910. 

Find the cost of the dress 3 × 4,480 = y 

Estimate 3 × 4,480 is about 3 × 4,500 =13,440 

Use an area model partial products. 

Example 2: 

Find the product of 3 × (7+4) using the distributive property 

Using the distributive law, we: 

  • Multiply, or distribute, the outer term to the inner terms. 
  • Combine like terms. 
  • Solve the equation. 

3(7+4) = 3(7) + 3(4) 

= 21 + 12 

= 33 

Example 3: 

Let us understand this concept with distributive property examples 

For example, 3(2 + 5) = 3 (7) = 21 

Or 

By distributive law 

3(2 + 5) = 3 × 2 + 3 × 5 

= 6 + 15 

= 21 

Here we are distributing the process of multiplying 3 evenly between 2 and 5. We observe that whether we follow the order of the operation or distributive law or not, the result is the same. 

Example 4:

Multiply 9 × 4860 using the standard algorithm model. 

Solution: 

Steps: 

  1. Multiply the ones   
  1. Multiply the tens   
  1. Multiply the hundreds, add any extra hundreds that we were carried over. 
  1. Multiply the thousands, add any extra thousands. 

Example 5: 

Use partial products to multiply 7 × $332. 

Solution:  

Step 1: Estimate the product.                          

332 rounds to 300; 7 × $300 = 2100 

Step 2: Multiply the 3 hundreds, or 300, by 7. 

7 × 300=$2100 

Step 3: Multiply the 3 tens, or 30, by 7.  

7 × 30 =$ 210 

Step 4: Multiply the 2 ones, or 2, by 7. 

7 × 2=$14 

Step 5: Add the partial products. 

So, 7 × $332 = $2,324. Since $2,324 is close to the estimate of $2,100 

Example 6:  

Use partial products to multiply 4 × $534.  

Step 1: Estimate the product.  

4 × $534 is about estimate 4 × $ 500 = $2000        

Step 2: Multiply the 5 hundreds, or 500, by 4. 

4 × 500 = 2000 

Step 3: Multiply the 3 tens, or 30, by 4  

4 × 30 =120 

Step 4: Multiply the 4 ones, or 4, by 4. 

4 × 4 = 16 

Step 5: Add the partial products. 

So, 4 × $ 534 = $2,136. Since $2,136 is close to the estimate of $2,000. 

Exercise:

1.       Solve the following using partial products:

(a) 78 × 8                                                 (b)  6 × 765                                     (c)   4 × 456

2.       Solve the following using distributive property:

   (a)  4 × 564                                             (b)  2 × 465                                     (c)   3 × 590

3.       Find each product by choosing an appropriate strategy.

   (a)  3 × 45                                                (b) 6 × 876                                      (c)   5 × 25

4.       Estimate the sum by rounding the number to the nearest thousand and then multiply.

          (a)  3 x 3,953                                           (b)  5 × 5,458

5.       Solve the following using standard algorithm:

          (a)  5 × 34                                                (b)  6 × 345                                     (c) 7 × 543

6.        A theatre can seat 460 people at one time. If a movie in the theatre fills every seat, every  day for 3 weeks, how many tickets were sold?

7.        A banana contains 105 calories. Last week, Nate and Haley ate a total of 14 bananas. How many calories did they eat?

An office needs to buy 20 new printers and 100 packages of paper. Each printer costs $300. Each package of paper costs $25. What is the cost of the printers and paper combined?

9.        A company sells jerseys for $50 each. In October, they see 32 jerseys. In April, they sold 55 jerseys. What is the total amount of money they earned by selling jerseys in October and April combined?

10.      A toy shop sells 4 toys that cost $200 each. Find the cost of 4 toys?

Concept Map:

What have we learned:

  • Understand estimation
  • Understand  standard algorithm
  • Understand  partial products
  • Understand word problems

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