M5N3c - Multiplying and dividing by numbers less than one

M5N3. Students will further develop their understanding of the meaning of multiplication and division with decimals and use them. c. Multiply and divide with decimal fractions including decimal fractions less than one and greater than one.
Consider a problem like the following:
A ribbon costs $ 1.80 for one meter. If you want to buy 0.8 meter of the ribbon, how much will it cost?
When children are asked what operation they would use to solve this problem, many will pick division. Some might think those children simply do not understand multiplication and division. However, that is not the case. Those children pick division as the operation because they know that "multiplication make bigger and division makes smaller." Research has shown that many children, and adults, hold this misconception.

Actually, calling it a "misconception" may be inappropriate. Rather, it is an overgeneralization children make based on their experiences. While students are working only with whole numbers, the only exception to this generalization is multiplication by 0 and 1. However, once they go beyond the "basic facts" stage of multiplication learning, practically all experiences involve multiplying by a number greater than one. The same can be said of division. The only time division does not result in a number less than the dividend is when it is divided by one. However, once again, practically all children's experiences before this point are division by a number greater than one.

Once the range of numbers is expanded to include decimal numbers and fractions, however, there are many cases where we do multiply or divide by numbers less than one. Therefore, it is an important goal of mathematics teaching that our students overcome this overgeneralization. A potentially powerful tool for this purpose is double number lines. If we represent the ribbon problem, it will look like this:

From this diagram, you can easily see that if the multiplier (represented on the bottom number line) is less than 1, the product (? mark) will be on the left of the multiplicand (1.80, i.e., amount corresponding to 1). On the other hand, if the multiplier is greater than 1, the product will be to the right of the multiplicand. Therefore, we can generalize:
If multiplier is greater than 1, multiplicand < product.
If multiplier is less than 1, product < multiplicand.
Similarly, you can use double number lines to contrast the situations when the divisors are less than 1 and those cases where the divisors are greater than 1.

However, the most difficult part for students (thus for teachers) is to help them understand that these situations are indeed situations where multiplication is the appropriate operation. For some students, double number line may not be sufficient. Another possible tool is to write mathematical expressions using words to describe the relationship among the quantities involved. In the ribbon problem, there are three quantities: cost of 1 meter of ribbon, total length of ribbon, and the price. The relationship among these three quantities can be expressed as
Price = [Cost of 1 meter] x [Total Length].Thus, for this problem, ? = 1.80 x 0.8.

An implicit, yet very important, goal of teaching multiplication and division of fractions and decimal numbers is to expand students' understanding of these operations. In early elementary grades, these operations are considered in equal group situations. Thus, when the multiplier or the divisor (in the case of fair-sharing division, the quotient in the case of measurement division) becomes something other than whole numbers, students have difficulty interpreting what it means. Through teaching of multiplication and division of decimal numbers and fractions, we want students to develop more proportional understanding of these operations. For example, A x B = C, should be interpreted as "A is to 1, C is to B," or "C is B times as much as A." Although this is not an explicitly stated goal in the GPS, it is something all teachers must keep in mind.

MKN1 h, i, j - Coins

MKN1. Students will connect numerals to the quantities they represent.
h. Identify coins by name and value (penny, nickel, dime, and quarter).

i. Count out pennies to buy items that together cost less than 30 cents.

j. Make fair trades using combinations involving pennies and nickels and pennies and dimes.

Quite frankly, I really don't understand why money and clock reading are in the mathematics curriculum. Those ideas should be learned in everyday contexts in which they mean something. The names and the values of each coin aren't mathematical concepts. However, whether or not I like these topics in the mathematics curriculum really matters much as teachers are expected to teach them. I heard many teachers say that money is a difficult idea for children. There may be a number of reasons for children to have difficulty with money. For one thing, being able to exchange a merchandise with a piece of metal or paper cannot be really "natural" to children. Another major source of difficulty for young children is the notion of exchanging coins. It's not quite logical why 5 shiny pennies can be exchanged with one, slightly larger coin, for example. Neither of these is mathematical ideas, and I'm not really sure how you can help young children with these ideas.

However, I do want to say something about exchanging coins. A part of the difficulty for young children is, I believe, because they have yet to develop a very sophisticated understanding of numbers. Kindergarten teachers are familiar with an exchange like the following:
Teacher: How many blue counters do you see?

Child: (counting to herself) One, two, three, four. Four !

Teacher: How many red counters do you see?

Child: (counting to herself) One, two, three. Three!

Teacher: So, how many counters are there altogether?

Child: One, two, three,...

Teacher: (interrupting). Wait a second. How many blue counters?

Child: One, two, three, four. Four.

Teacher: How many red counters?

Child: One, two, three. Three.

Teacher: So, how many counters altogether?

Child: One, two, three, four, five, six, seven. Seven!
Many adults are simply puzzled why a child will have to count all the counters when they know that there are four blue counters and three more red ones. However, for many young children, numbers exist only after they count - "four" cannot exist by itself. Furthermore, for many of them, four simply means four ones. Children must develop an understanding that four can be considered as an entity, or a unit, in itself before they can count on. In the previous post, I discussed the idea of five and ten as a benchmark. Children cannot think of five as a benchmark unless they can think of five as five ones and, simultaneously, one five. For adults, this is so obvious, and it is difficult to even fathom anyone (including children) not understanding it. However, research clearly shows that children don't automatically understand this idea.

So, if a child is still "counting all" stage, it is probably not reasonable to expect him/her to be able to make an exchange of five pennies with one nickel with understanding. One way to help children overcome difficulty with money is to help them develop good number sense - the ability to see numbers flexibly. Without number sense, money is much more difficulty to make sense.

MKN1f - Five and ten as benchmarks

MKN1. Students will connect numerals to the quantities they represent.
f. Estimate quantities using five and ten as a benchmark. (e.g. 9 is one five and four more. It is closer to 10, which can be represented as one ten or two fives, than it is to five.)

Although this indicator includes the words "estimate," what it is talking about isn't really estimation in the sense of "about how big" a number is. Rather, it is more about looking at a number from different perspectives. Thus, 8 isn't just eight ones, but rather, it is three more than 5 and two less than 10 as well. From that perspective, this standard relates very closely to MKN2 b, "Build number combinations up to 10 (e.g., 4 and 1, 2 and 3, 3 and 2, 4 and 1 for five) and for doubles to 10 (3 and 3 for six)." Using five and ten as a benchmark is in a way a special case of this indicator. Furthermore, being able to look at numbers from multiple perspectives is something that is continuously emphasized in the elementary GPS. According to Elementary School Teaching Guide for the Japanese Course of Study, the ability to see a number as a sum, a difference, a product, or a quotient of other numbers is an important foundation for algebraic thinking.

One common tool that is often used to help students develop this idea of five and ten as a benchmark is a ten frame:

It is just a table with 2 rows of 5 cells. Different numbers can be represented by filling up these cells with counters. However, when you represent numbers 6, 7, and 8, it is important that a row is filled up completely so that those numbers are represented as 5 and some more,

not like

The latter representation is useful if we want children to develop the understanding that 8 can be represented as 4 and 4 (MKN2b). Ten frames are very versatile tools, but that means we, as teachers, must be very intentional about how we use them to help students develop a specific understanding.