Question

We will walk through the reasoning for why the standard algorithm for multiplication works.

a. Use both the standard/common algorithm and the partial products method to calculate 34×27. Show your work.
b. Draw a rectangle to represent an array for 34×27 (your rectangle need not be to scale, but should be labeled appropriately). Subdivide the rectangle in a natural way so that the parts of the rectangle correspond to the steps in the partial products algorithm. Indicate the correspondence between the parts of the rectangle and the steps in the algorithm. (For example, you could use a colored pencil and say "The pink part of the rectangle corresponds to...")
c. Use the distributive property to calculate 34×27. Use equations to demonstrate your work.
d. Explain how the partial products method, the rectangle you drew in (b), and the equations you made in part (c) all relate. Use them to why we place a 0 as a place holder in the second line of the common algorithm.

Answer 1

To calculate 34 x 27, you can use the standard/common algorithm or the partial products method. The rectangle representing the array for 34 x 27 can be divided into sections corresponding to the steps in the partial products algorithm. The partial products method, the rectangle, and the distributive property equations all relate by breaking down the multiplication problem into smaller parts.

Step-by-step explanation:

To calculate 34 x 27 using the standard/common algorithm, you would multiply each digit in the ones place of the bottom number (7) by each digit in the top number (34), and then multiply each digit in the tens place of the bottom number (2) by each digit in the top number (34), shifting the products one place to the left. Then, you would add the resulting products together to get the final answer.

To calculate 34 x 27 using the partial products method, you would multiply each digit in the ones place of the bottom number (7) by each digit in the top number (34) and write the products underneath each other. Then, you would multiply each digit in the tens place of the bottom number (2) by each digit in the top number (34) and write the products underneath each other, shifting each product one place to the left. Finally, you would add the resulting products and get the final answer.

The rectangle representing the array for 34 x 27 can be divided into four sections corresponding to the steps in the partial products algorithm. The first section represents the product of 4 x 7, the second section represents the product of 4 x 20, the third section represents the product of 30 x 7, and the fourth section represents the product of 30 x 20.

Using the distributive property, you can calculate 34 x 27 by breaking down the problem into smaller parts. You can multiply 30 x 20, 30 x 7, 4 x 20, and 4 x 7 separately, and then add the products together to get the final answer.

The partial products method, the rectangle representing the array, and the equations from the distributive property all relate because they all break down the multiplication problem into smaller parts. Placing a 0 as a placeholder in the second line of the standard algorithm helps to align the digits correctly and maintain the place value when adding the partial products together.