M5N3. Students will further develop their understanding of the meaning of multiplication and division with decimal fractions and use them.
I have discussed these standards previously (October, 2007). However, I had an interesting conversation with a colleague from another university in Georgia recently. She also teaches mathematics courses for prospective elementary school teachers at her school. We were discussing how I incorporate Japanese curriculum materials to discuss decimal multiplication and division. She then shared with me that, when she teaches this topic, she relates it to fraction multiplication and division. Her approach is perfectly valid and viable; however, in the GPS, decimal multiplication and division are discussed before fraction multiplication and division. So, I feel it is important that our future elementary school teachers experience how decimal multiplication and division may be developed WITHOUT the knowledge of fraction multiplication and division. So, I want to discuss how multiplication and division of decimal numbers can be approached using the knowledge of whole numbers only.
Let's start with multiplying and dividing decimal numbers by whole numbers.
- One meter of wire weighs 5.7 grams. How much will 3 meters of the same wire weigh?
In the last post, I discussed the idea of relative size. If you use that idea, 5.7 grams can be considered as a collection of 57 0.1 g pieces. Thus, in 3 meters, we have 3 sets of 57 0.1 g's. That means we can use the calculation 57x3 to find out the total number of 0.1 g's in 3 meters. If there are 171 pieces of 0.1 g's, again using the idea of relative size, we know that is the same as 17.1 grams.
In general, when you are multiplying a decimal number by a whole number, you can just consider the given decimal number in terms of its smallest decimal place value as the unit. For example, 0.37 is 37 0.01's, 0.824 is 824 0.001's, etc.. You will then have a whole number as the multiplicand (in terms of a decimal unit). Since the multiplier is a whole number, we can use multiplication to find the total number of pieces of the decimal unit. We can then convert the final result into a decimal number by using the idea of relative size. Using the familiar paper-and-pencil algorithm, it basically means that the decimal point for the multiplicand (the number on top) and the product are in the same place.
Now, let's look at division of decimal numbers by whole numbers.
- A wire that is 3 meters long weighs 5.7 grams. How much will the same wire weigh if it is 1 meter long?
Again, using the idea of relative size, we can think of 5.7 grams as a collection of 57 pieces of 0.1 gram. Since 3 meters of this wire include these 57 pieces, and you want to know how much 1 meter will weigh, you simply need to divide 57 by 3, which is just a whole number division problem. The quotient, 19, tells us the number of 0.1 gram pieces in each group. Again, using the idea of relative size, we can conclude that 1 meter of this wire will weigh 1.9 grams.
In general, just as in the case of multiplication of decimal numbers by whole numbers, when you are dividing a decimal number by a whole number, you can consider the decimal dividend in terms of its smallest decimal place value as the unit. Then, we can simply use whole number division to find out how many pieces of the decimal unit will be in each group. The final quotient can be found by using the idea of relative size. Using the familiar long division notation, this suggests that the decimal point for the quotient and the dividend will be in the same place, i.e., the place values for the dividend and the quotient should line up, just as they did with whole numbers.
One point of complication we need to pay attention is what if the dividend (whole numbers pieces of a decimal unit) is not evenly divisible by the divisor. So, for example, what can we do if the weight of the 3-meter wire was 5.8 grams. When we divide 58 by 3, we have the remainder of 1. But, this is also the number of 0.1 gram piece. So, we can say that the answer to the division of 5.8 by 3 is 1.9 with the remainder of 0.1. In other words, the remainder is the number of the decimal unit. Thus, in the long division notation, the decimal point of the dividend and the decimal point for the remainder must also line up.
Of course, another option is to divide on - the remainder of 1 can be thought of as a collection of 10 0.1's (actually, the remainder is 0.1, so we are really talking about 10 0.01's), then we can keep dividing. Sometimes, this will result in a terminating decimal, while in other cases, you will have a repeating decimal. With this understanding of dividing on, when students learn about the quotient meaning of fractions they can then understand that every fraction can be re-written as a decimal number by simply dividing its numerator by the denominator.
OK, this post is already rather long. So, I will have to wait till the next time to discuss multiplication and division by decimal numbers. In the meantime, I encourage you to think about how students can reason about to solve the following problems, which involve multiplication and division by decimal numbers. Keep in mind that they have not learned how to multiply or divide by decimal numbers or fractions.
- One meter of wire weighs 2.4 grams. How much will 1.8 meters of the same wire weigh?
- A wire that is 2.4 meters long weighs 3.6 grams. How much will the same wire weigh if it is 1 meter long?