multiplication and division with decimal fractions and use them.
In the last post, I discussed how the idea of relative size can be used to think about multiplying and dividing decimal numbers by whole numbers - M4N5(d). In this post, I want to continue to the next step, multiplying and dividing by decimal numbers. As I discussed in October, 2007, when the multiplier and the divisor is something other than a whole number, we must extend the meaning of division from an equal-group perspective to a more proportional one. Let's look at the two problems I left as "homework" last time.
Problem 3
One meter of wire weighs 2.4 grams. How much will 1.8 meters of the same wire weigh?
Problem 4
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?
Clearly, in Problem 3, we must multiply 2.4 by 1.8, while in Problem 4, we must divide 3.6 by 2.4. Since these situations involve a decimal multiplier and a decimal divisor, we can no longer use the equal group interpretation of multiplication and division - what does 1.5 or 2.4 groups mean? Rather, we must look at these situations more proportionally. In Problem 3, we are asking, if 2.4 is to 1, how much is to 1.8, and in Problem 4, if 3.6 is to 2.4, what is to 1? Alternately, if you use multiple comparison idea, Problem 3 asks how much is 1.5 times as much as 2.4, while Problem 4 asks 3.6 is 2.4 times as much as what?
Let's now think about how students can solve these problems using only what they have learned so far, which does not include how to multiply or divide by decimal numbers.
Problem 3
One meter of wire weighs 2.4 grams. How much will 1.8 meters of the same wire weigh?
One possible idea that students might use is to consider the multiplier, 1.8, in terms of the decimal unit using the idea of relative size. That is, 1.8 means there are 18 pieces of 0.1's. But what does that mean? A diagram might be helpful. Using a double number line (November, 2007), we can represent the problem like this:
When we say 1.8 is made up of 18 pieces of 0.1's, the diagram may look like this:
In other words, 1.8 meters can be thought of as a collection of 18 0.1 meter pieces. But, how does that help us find the missing number. We are not multiplying 2.4 by 18 - we don't have 18 groups of 2.4. What do we have 18 groups of on the top number line?
From this diagram, we can tell that what we have 18 of on the top number line is actually the weight of 0.1 meter wire. In other words, if we know how much a 0.1-meter wire weighs, then, we can find the answer. But, it's easy to see that the weight of a 0.1-meter wire can be determined by simply dividing 2.4 by 10, which is what students learned in Grade 4. Once we determine the weight of a 0.1-meter wire, i.e., 0.24 grams, then, we can multiply that by 18, which is also a Grade 4 idea. 0.24 x 18 = 4.32, so the weight of a 1.8-meter wire is 4.32 grams.
Here is another idea that students might come up with. Although we are looking for the weight of a 1.8-meter wire, let's first think about the weight of 18-meter wire, which is easy enough - simply multiply 2.4 by 18, a Grade 4 idea. However, since a 18-meter wire is 10 times as long a 1.8-meter wire, it should also weigh 10 times as much, too. So, in order to determine the weight of a 1.8-meter wire, we can simply divide that by 10 to find its weight. Since we already know how to divide decimal numbers by whole numbers, this last step should not be a problem. This line of reasoning may be represented on a number line like this:
Different students will feel more comfortable with different approaches. However, this second approach may be more useful to generalize into a written computation algorithm. In general, what we do in the first step is to make the multiplier into a whole number by multiplying it by an appropriate power of 10. Now, if the multiplicand is a decimal number, we end up multiplying it by a power of 10 to make it into a whole number as well (that's another way of thinking about the use of relative size). Now that we have two whole numbers, we can multiply them easily. However, this product is too big, and it must be divided by those powers of 10. For example,
Since multiplying by 10 means that the decimal point will move to the right one place while dividing by 10 means moving the decimal point to the left one place, we can describe what happened above this way: when we think of 3.7x4.26 as 37x426, we moved the decimal point 3 places to the right altogether, therefore, we have to move the decimal point to the left 3 places in the product of 37x246 to get the product for 3.7x4.26. And, this is (to us) the familiar multiplication algorithm for decimal numbers, isn't it?
Well, this has gotten a bit too long - of course, with actual 5th graders, you may need several lessons to get this much discussion done. Anyway, I think I must postpone the discussion of dividing by decimal number until next time. However, if you can think about how we solved Problem 3, you may find that Problem 4 can be solved in similar ways.
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