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.

M3G1 - Geometry in Primary Grades (3)

M3G1. Students will further develop their understanding of geometric figures by drawing them. They will also state and explain their properties.

In Kindergarten, students are expected to "recognize" certain geometric figures. In Grade 1, students are expected to classify geometric figures by comparing their structures. In M3G1, the GPS mentions "properties" of geometric figures for the first time. By the end of Grade 3, students are expected to state and explain properties of geometric figures. The GPS does not clearly spell out what properties are to be explored. However, based on the types of figures students have previously explored, properties such as the equality of the base angles of isosceles triangles are within the reach for 3rd graders. Of course, we need to keep in mind that equality of angles at this point is based on the fact that two angles can be made to overlap each other completely as measuring angles is a Grade 4 expectation.

The general flow of the geometry standards in the GPS is very much consistent with the developmental levels of geometric thinking identified by Dina and Pierre van Hiele. According to the van Hiele's model, children can only treat geometric figures as wholes. Their thinking is based on how figures appear. In the second stage, children develop the ability to identify and use various components of geometric figures as objects of their thought. It is in this stage, they can start identifying figures based on relationships among their component parts. As children move into the 3rd level, typically called Informal Deduction, they can now start focusing on those relationships as the objects of their thought. One characteristic of students in this stage is that they can now start using some logical statements, like if ... then .... In order for students to be successful a typical high school geometry class, students must be at this stage at the beginning of the course. Since geometry is discussed throughout Math 1 ~ 4 in the GPS, and since geometry proof happens in Math 1, this means students must be at this stage by Grade 9 the latest.

As children begin to explore properties of geometric figures in Grade 3, it is essential that teachers distinguish properties from definitions. The van Hiele theory suggests that children in the second stage can identify relationships among various components of a given figure. However, as children move toward the third stage, they can identify the minimum amount of relationships that will be sufficient to define a shape. Those relationships become the definition of the shape, while other relationships are now treated as properties. For example, children in the second stage might be able to identify the following relationships in isosceles triangles:

• two sides are equal in length
• two angles are equal
• has a line of symmetry (you can fold it in half - symmetry is a Grade 6 topic)
• two angles that are equal are both acute angles
• third angles can be acute, right or obtuse
• etc.

As children move to the third stage, they can go from this laundry list of relationships to the realization that only the first relationship is necessary to define an isosceles triangle. A useful activity to help children go from a laundry list to the minimum set of defining characteristics is to have children actually draw shapes. As children are asked to draw isosceles triangle, they will realize that they don't need to use all of those characteristics to draw one - indeed if you draw a triangle with two equal length sides that's enough.

Although most 3rd graders are still in the second van Hiele stage, it is important that teachers' communication (with students and with parents) clearly distinguish definitions and properties. The parent letter for the third grade geometry unit have the following "terminology":

• Parallelogram: A quadrilateral with opposite sides that are parallel and of equal length and with opposite angles that are of equal measure.
• Rectangle: A quadrilateral with four right angles and two pairs of opposite, equal parallel sides.
• Rhombus: A parallelogram with four equal sides and equal opposite angles.

All of these are "laundry lists" and not definitions. Although at times it is perfectly fine to be less rigorous about our language use, when we do use language loosely, we should do so with clear understanding that we are indeed being less rigorous.