M4N5(d) & M5N3 - Multiplication & Division of Decimal Numbers

M4N5. Students will further develop their understanding of the meaning of decimal fractions and use them in computations.
d. Model multiplication and division of decimal fractions by whole numbers.
M5N3. Students will further develop their understanding of the meaning of multiplication and division with decimal fractions and use them.

So, what is the difference between these two statements (M4N5d and M5N3)? There are two aspects about these standards I would like to discuss, but I will focus on one in this post. Then, in the next post, I will address the other issue.

The primary difference between M4N5d and M5N3 is that, in Grade 4, students are to deal with problem situations where decimal numbers are used as the multiplicand or the dividend, but not as the multiplier or the divisor. As I discussed earlier (in a June 2007 post on M2N3a), the multiplier is the number of groups when there are equal sized groups, while the multiplicand is the number in a (or each) group. Thus, according to M4N5d, in Grade 4, students should be dealing with problems like:
With 1 gallon of gasoline, Steve’s car can travel 17.4 miles. How far can Steve’s car go with 8 gallons of gasoline?
But not problems like:
With 1 gallon of gasoline, Steve’s car can travel 18 miles. How far can Steve’s car go with 6.5 gallons of gasoline?

Some may ask, what’s the difference, they both involve multiplying a whole number and a decimal number. The big (mathematically) difference is that in the first problem we can think of it as 17.4+17.4+17.4+17.4+17.4+17.4+17.4+17.4, but not in the second problem. The second problem is not 6.5+6.5+6.5+6.5+…+6.5 (6.5 18 times). Why not? Think about what 6.5 represents in this problem. 6.5 is the amount of gasoline. So, why do we have 6.5 gallons 18 times? Actually, the problem involves 18+18+…+18, but we have to use 18 6.5 times? What does that mean? Helping students to deal with that situation, therefore, is the major focus of Grade 5 – and that’s why the GPS says, “the meaning of multiplication and division.”

So, what about division? In division, the dividend is the total amount. The divisor, however, may be the amount in one group or the number of groups. If the divisor is the amount in one group, then the quotient will tell us the number of groups (this is called measurement division), while, if the divisor is the number of groups, the quotient tells us the amount in each group (this is usually called fair sharing division). When the divisor is a decimal number, it may be a bit easier to conceive of the division situation as the measurement division. How many groups of 3.6 can you make with 18.7, for example. The fair sharing division situation will be something like this:
With 3.6 pints of paint, we can paint 18.7 square feet of board. How much area can we paint with 1 pint?
In this question, we are asked to determine the per-one unit amount. Again, this requires an expansion of the meaning of division.

Some may argue that, if it is easier to perceive division by a decimal number as the measurement division, why not focus on the measurement division. If we use the measurement division and if there is no whole number quotient, we will have to help students interpret the meaning of the quotient. Thus, either way, we have to help students expand their understanding of the meaning of division.

As Grades 4 and 5 teachers deal with multiplication and division of decimal numbers, they must pay very close attention to these points. In Grade 4, we are still using the same meaning of multiplication and division, and the focus is on helping students develop procedures for multiplying or dividing decimal numbers by whole numbers. In Grade 5, the first focus is helping students expand the meaning of multiplication and division, then we must help students develop computational procedures when the multiplier and the divisor become decimal numbers.