Chapter 4.6, Problem 13P

The following method for multiplying $21\times 23$ relies on repeatedly doubling numbers (by adding them to themselves):

$$

$\begin{array}{l}21=16+4+1\\ 368\\ \end{array}$

The answer, 483, to the multiplication problem $21\times 23,$ is shown on the bottom of the column on the right.

Notice that this method requires only addition; students do not need to know how to multiply.

a. Discuss and explain the method of repeated doubling. Address the following questions in your discussion:

i. What is the significance of the column of the numbers 2, 4, 8, 16?

ii. What is the significance of writing $21=16+4+1?$

iii. How are the numbers 368, 92, and 23 chosen for the column on the right? Why do we add those numbers?

b. Use the method of repeated doubling to calculate $26\times 35$ . Show your work.

c. Use the method of repeated doubling to calculate $37\times 51$ . Show your work.