Math & History: Decimals: Multiplication

I. Standard Decimal Multiplication Algorithm

To multiply decimals:
1) Place the numbers in vertical form lining up the numbers on the right. Do not align the decimal points.

2) Multiply as you would when multiplying whole numbers. Ignore the decimals at this point.

3) Count the number of digits after the decimals in the numbers being multiplied and move the decimal point over that many spaces from the right in the answer.

Example: Multiply 12.26 x 2.5

Place the numbers in vertical form lining them up on the right.

12.26
x 2.5

Multiply as you would normally multiply whole numbers. At this point ignore the decimals.

12.26
x 2.5
6130
+ 24520
30650

Count the number of digits after the decimals in the numbers being multiplied and move the decimal point over that many spaces from the right of the answer.

12.26 (2 decimal places)
x 2.5 (1 decimal place)
6130
+ 24520
30.650 (3 decimal places starting from the right)

II. Conceptual Understanding

1. Whole number times decimal number

Number Line

Example

4 x 0.3

Thinking in terms of repeated addition we are asking "what is the total of 4 groups of 0.3?"

4 x 0.3 = 1.2

Khan: Multiplying decimals and whole numbers visually

2. Decimal number times whole number
Number Line

Example

0.3 x 4

Thinking of this problem in terms of scaling we are asking what is three tenths of four. To solve using the number line, we first divide the number line into tenths. 4 ÷ 10 = 0.4 so each tenth of the number 4 is 0.4 which is how the number line is partitioned. Finally we count the parts. We need three of the tenth sized parts (0.3 of the whole)

0.3 x 4 = 1.2

3. Decimal number times decimal number
Number Line

Example

0.4 x 0.3

We can think of this as a scaling problem. We are asking, 'What is four-tenths of three-tenths?'

First, we locate 0.3 on the number line. To find 'four-tenths' of that distance, we first need to know what one-tenth of it is. Since our scaling factor is in tenths, we divide 0.3 by 10, which gives us 0.03 (three-hundredths).

To visualize this, we divide each 'tenth' space on our number line into ten smaller equal intervals, which represent hundredths. Now, we can take 4 jumps of 0.03 starting from zero to land on our answer: 0.12