## Friday, October 31, 2008

### M2N1(b) & M3N1(b): Relative Magnitudes & Relative Sizes

M2N1 Students will use multiple representation of numbers to connect symbols to quantities.
b. Understand the relative magnitudes of numbers using 10 as a unit, 100 as a unit, or 1000 as a unit. Represent 2-digit numbers with drawings of tens and ones and 3-digit numbers with drawings of hundreds, tens, and ones.

M3N1 Students will further develop their understanding of whole numbers and ways of representing them.
b. Understand the relative sizes of digits in place value notation (10 times, 100 times, 1/10 of a single digit whole number) and ways to represent them.

In these two standards, you see phrases, "relative magnitudes" and "relative sizes." These standards actually elaborate what these phrases mean further - using 10, 100 or 1000 as a unit, and 10, 100 times or 1/10 of a single digit whole number. These statements seem to suggest that these phrases may be related but different.

As you know, the GPS was heavily influenced by the 1989 Japanese Course of Study (COS). Interestingly, in the COS, they use the same words, which can be translated either "relative magnitude" or "relative size." The Japanese Ministry of Education produces a document that explains the COS, and in this document, they explain what they meant by "relative size/magnitude":
"To understand the relative size of numbers" mean to grasp numbers' size by units of tens and hundreds." (Grade 2)
"In this grade, broaden the range of numbers up to unit of ten-thousands, and help children deepen their understanding of the relative size of numbers." (Grade 3)

Thus, it appears that, in the original Japanese COS, the authors' focus was on the meaning that is consistent with the meaning suggested by M2N1(b). So, what does this mean? Let's look at an example, 38291. Teachers and students are familiar with the question, "What numeral is in the hundreds place?" However, the idea of "relative magnitude/size" suggests another question: How many hundreds does this number include? The answer is 382. We can also say that this number also include 3829 tens. With our numeration system, therefore, telling the relative size of numbers is rather easy. Whatever the unit you want to use to consider the given number, think of that place as the "ones" place and consider the number made up of the numerals to the left. So, in 15076821, there are 1507 ten-thousands, 150768 hundreds, etc.. Actually, the way we read number in English take advantage of this idea. The number 38291 is read as "38 thousands 291," not "3 ten-thousands 8 thousands, ...," and 15076821 is "15 millions..."

This idea can also be extended to decimal numbers (and Japanese textbooks emphasizes this way of looking at numbers). For example, consider the number 0.873. You can say this number has 8 0.1's, 87 0.01's, or 873 0.001's. You can even say this number includes 8730 0.0001's. Moreover, the idea of considering a number using units other than 1 is an important foundation for fraction learning as well. It is very useful to consider non-unit fractions as collections of unit fractions. For example, 3/4 is 3 one-fourth's. When you consider numbers from this perspective, 30+40, 300+400, 0.3+0.4, and 3/5+4/5 can all be related to "3+4." The only difference is the unit, 3 and 4 of what (tens, hundreds, 0.1's or one-fifth's) we are combining.

By the way, there is actually a Grade 3 standard in the 1989 Japanese COS that states, "(Students are) To know about the size of 10 times, 100 times, 1/100 of a whole number and how to represent them." The elaboration document goes on to explain this standard by saying:
When teaching 10 times bigger, 100 times bigger, or 1/10 of a whole number, it is necessary to help children pay attention to the fact that the order of numerals does not change and that the size of corresponding numerals is 10 times, 100 times, or 1/10 of the original numbers.

Finally, the elaboration document states that this idea is related to the idea of "relative size/magnitude" discussed in another Grade 3 standard. So, it appears that M3N1(b) is more about this standard. However, as I noted above, the way we read large numbers in English take advantage of this idea, it is nevertheless important that Grade 3 mathematics instruction revisits and extends this idea further.