## Multiplication by Two-Digit Numbers: When Students Ask

**How do I use mental math to multiply by multiples of 10?**

First, students must know that because of the properties of operations and numbers, it is possible to break any number into a product of its factors and then change the order of those factors. For example, 8 x 30 can be written as 8 x (3 x 10). Then the factors can be reordered or regrouped as (8 x 3) x 10, so that you can multiply basic facts first. With that in mind, encourage students to use mental math to multiply by saying the following: “You know that 8 x 3 = 24. Multiplying 24 by 10 is the easiest part. Simply put one 0 behind 24 to get 240. If you were multiplying 8 x 300, you would put two 0s behind 24 to get 2,400. The only time you have to be really careful with the number of 0s you put behind the basic fact is when the basic fact itself ends in a 0. For example, 8 x 50 = (8 x 5) x 10. Since 8 x 5 = 40, you see one 0 already. But you still need to put another 0 behind 40 to get the answer to 8 x 50. So, the answer is 400.”**What does the Distributive Property say about multiplying by two-digit numbers?**

Here is the answer to the question in words: You multiply one of the numbers first by the tens digit of the second number and then by the ones digit of the second number. However, a number example may be more easily understood. Use 38 x 27 as an example. Here is what the Distributive Property permits you to do: 38 x 27 = (38 x 20) + (38 x 7). It may also be helpful to think of the Distributive Property as a*distribution manager*. The Distributive Property tells us that the first factor, 38, must get*distributed*(or passed out) to each of the digits of the second number, 2 and 7.**When would I multiply money amounts by two-digit numbers?**

Students probably have had to solve word problems or even real-life problems that involve money amounts. Brainstorm with your class some examples of situations in which money amounts are multiplied by two-digit numbers. You may even want to have students look through newspapers or magazines to find real-life examples of money amounts that could be multiplied by two-digit numbers, such as finding out how much money would be earned in 45 days if $125 was earned per day. Then have your class write a set of word problems based on the examples they find. For example, a long-distance phone company charges $0.07 per minute; determine how much you would pay for a 25-minute call.**Does the product of a number and a multiple of 10, 100, or 1,000 always end in a zero?**

The answer to this is “Yes, always.” In fact, the product of a number and a multiple of 10 will end with*at least*one zero; the product of a number and a multiple of 100 will end with*at least*two zeros; and the product of a number and a multiple of 1,000 will end with*at least*three zeros.**What do you mean when you say that the product of a number and a multiple of 10 will end with***at least*one zero?

The phrase*end with at least one zero*means that the product must end in one zero, but it could end in more than one zero. For example, 5 x 80 is 400. You can see that 400 ends in two zeros even though 80 is a multiple of 10. The reason for this is that 40, the answer to the basic fact 5 x 8, already ends in a zero. But, you still need to put another zero behind 40 to get the answer to 5 x 80.