## Thursday, March 26, 2009

### M6G1 a - Symmetry

M6G1. Students will further develop their understanding of plane figures.
a. Determine and use lines of symmetry.

In the last entry, I mentioned that the topic of odd/even numbers is one of the topics some teachers are surprised to see discussed so much later than they used to. Another topic that some teachers have expressed their surprise because of the lateness of the treatment is the idea of symmetry. Many teachers of primary grades have children explore (reflective) symmetry through paper folding. They will have students make symmetrical shape by cutting a folded papers, or have them fold symmetric figures so that the two sides will coincide.

Clearly, young children can explore, and enjoy exploring, symmetries through such activities. However, as valuable as such informal experiences may be, they are still "informal" explorations. It is important for children to consider and understand symmetry as a mathematical idea, too. Such study of symmetry is the focus of this particular standard.

According to this standards, students are supposed to "further develop their understanding of plane figures" by studying symmetry. Thus, the purpose of studying symmetry isn't just about learning symmetry. Rather, using symmetry as a new perspective to review those shapes that have been previously studied. So, for example, what kinds of triangles are symmetric? From this perspective, isosceles triangles and equilateral triangles are in one group, symmetric triangles.

Students can also explore which types quadrilaterals have reflective (line) symmetry. Parallelograms, with the exception of those which are also rectangles, do not possess reflective symmetry. Many children (and adults) think that the line that are parallel and in between a pair of sides will serve as the line of symmetry. When they actually fold a parallelogram, they are surprised that the two sides do not match up. That experience, in turn, can help students understand that the line of reflection must be the perpendicular bisector of the segments connecting corresponding points. [The notion of "corresponding points" follows from their study of congruent figures in Grade 5 -- because the two sides of a symmetric figures are congruent, there are corresponding points. As a result, the formal study of symmetry must follow the study of congruence.] With that understanding, children can now determine the line of symmetry (M6G1a) without having to actually fold the paper, or simply eye balling it. This knowledge will also allow them to complete a figure when one side of the figure and the line of symmetry are given.

Many Japanese mathematics teachers consider the study of symmetry in Grade 6 as the culminating point of the study of geometry in elementary schools (in Japan, elementary schools cover grades 1 through 6). Children not only learn about symmetry, but they also learn to use symmetry as a perspective to re-analyze shapes they have learned. Most Grade 6 classrooms in Georgia are in middle schools. So, perhaps we can position the study of symmetry as an entry point into a more formal study of geometry in secondary schools.

## Tuesday, March 10, 2009

### M5N1 a - Even and odd numbers

M5N1. Students will further develop their understanding of whole numbers.
a. Classify the set of counting numbers into subsets with distinguishing characteristics (odd/even, prime/composite).
When teachers examine the GPS, there are (at least) a couple of topics they are surprised to see so much later in elementary schools than they are used to. The topic of odd/even numbers is one of those topics. Teachers are surprised that a topic that they used to discuss in the second grade (or even in the first grade) is now delayed until Grade 5. Some may be tempted to include this topic in an earlier grade. So, why does the GPS wait to discuss this topic until Grade 5? Although I cannot speak for the committee who developed the GPS, I can share with you the Japanese perspective.

There is no question that we can teach second grade children to distinguish odd/even numbers. We can connect to skip counting or simply tell students that all numbers in the sequence, "2, 4, 6, 8, 10, ..." are called even numbers and the rest are odd numbers. For larger numbers, they can simply use the rule, "if a number ends with a 0, 2, 4, 6, or 8, it is an even number." However, the point is NOT identifying even/odd numbers. Let's look at the GPS statement:
Classify the set of counting numbers into subsets with distinguishing characteristics (odd/even, prime/composite).
It is important to note that what we want students to understand is that "counting numbers" (whole numbers?) can be classified into different subsets by paying attention to various distinguishing characteristics. So, what is the distinguishing characteristics for even/odd numbers? It is the divisibility by 2. Even numbers are those numbers that are divisible by 2, while odd numbers are those that cannot be divided (with a whole number quotient) by 2. So, the emphasis is not about identifying even/odd numbers, but understanding ways to sort whole numbers. Even/odd numbers are just an example of one such classification schemes.

To help students focus more on ways to classify whole numbers, teachers may want to engage students with a task that require students to sort whole numbers in a similar way - by focusing on the remainder when divided by a number. You can watch a videotaped lesson from Japan, in which the teacher posed an interesting question involving the idea of classifying numbers by looking at the remainders when they are divided by 4.
http://tiny.cc/rkB1X