exploring geometric components of circles

Exploring Geometric Components of CirclesChris Roberts

University of Maine at Farmington1/6/2016

EXPLORING GEOMETRIC COMPONENTS OF CIRCLES Roberts 2

As students progress through schooling systems and higher education, they

spend time analyzing, questioning and experimenting with geometric concepts both

inside and outside of the classroom. With the exploration of differing characteristics

and dimensions of shape, there is a large amount of material that can be processed

when researching these mathematical topics such as Euclidean and Spherical

geometry. Prior to exploring more complex and highly developed areas of geometry,

students in the United States must pass through years of assessment and

schoolwork that revolve around the Common Core State Standards found in

Kindergarten through twelfth grade classrooms. When researching this topic of set

standards for assessment, the United States was found to be one of the only

countries that have established levels of mathematical understanding that is met

and shared within an entire country. This aspect of individuality that revolves

around the United States mathematical curriculum is a component of Kindergarten

through twelfth grade education that is taught to both established and soon to be

educators around the country. Being established in 2009, the standards are still

fairly new and have begun to become incorporated into the new copies of

mathematical textbooks for schools such as Everyday Mathematics.

Due to the wide range of ages that are focused on in these Common Core

State Standards, this research assignment has been constructed in a manner that

splits the assessment narrative into two different concept maps to illustrate the

components that focus on circles. The first concept map, illustrated in Figure 1,

capture aspects of Kindergarten through sixth grade education. The mapping of


these standards focuses on the use of child-familiar language and all of the phrasing

was taken exactly from the Common Core narrative. Understanding the zone of

Figure 1: This concept map analyzes the components of the Common Core that connect to circles. This illustrates aspects of the standards created for students

between the grades of Kindergarten and sixth grade.

proximal understanding for the younger students in this grouping, it makes since to

have a presence of characteristics such as geometric vocabulary and the

understanding of the names of geometric illustrations. While the components of

mathematics that is present in this map can not be questioned, the aspects that are

absent can be analyzed by professionals and researchers. As the basic material of

geometry are illustrated as being an impact of early introduction to circles, there is

minimal narrations of what can come out of understanding of this topic. For these

ages, there is a strong focus on comparing the shape as an entity to other geometric

illustrations. There is no mention of gaining deeper understanding of this shape


besides the idea of using portions of the shape to illustrate fractions. This is much

different then the mentions of triangles and squares, which present a much deeper

analysis than circles in the Common Core for these ages.

The second grouping of ages to be analyzed focuses on the Common Core State

Standards for students between seventh and twelfth grade. The narratives set for

this age group can be found in Figure 2, and again this figure uses wording and

phrasing taken from the Common Core State Standards. The crucial addition found

Figure 2: This concept map analyzes the components of the Common Core that connect to circles. This illustrates aspects of the standards created for students

between the grades of seventh and twelfth grade.

within the narrative for this group is the mention of pi. With the addition of this

crucial component of circles, students are able to find calculations such as

circumference and area. The interesting part about this new concept is that the

standards to not elaborate on characteristics of area, circumference, or pi any

deeper than the idea of finding calculations by the use of formulas. In fact, formula

use is one of the only aspects of circle geometry mentioned for the students of


grades seventh and eighth. Arc lengths, radians, and theorems that relate to circles

in geometry are first mentioned for students between the grades of ninth and

twelfth. These are illustrated in bold near the bottom of Figure 2. The theorems that

are stated in the Common Core State Standards are also elaborated on in Figure 3.

Figure 3: These are the theorems and standards that are narrated in the high school Common Core State Standards for the geometry of circles.

As this area of geometry was explored, research was done to find what method of

mathematical instruction is being used in other countries around the world. One of

the countries that presented a very different method of mathematical education and

instruction was Australia. As similar components of shape and mathematics were

being taught, there was a much more abstract manner of presentation. While

looking through teaching material published by this country, a lesson plan was

found and compared to the construction of a teaching plan in the United States.

There was no mention or connection to standards that were established by the

country for the age group being taught by the classroom instructor. Relating the


plan back to the concept map for early education students, it is found that all of the

components are present. The focus of analysis with this document was the

additions found that differed from the focuses of United States mathematical

education. As mathematical vocabulary was used, such as diameter, circumference

and radius, the classroom investigation focused on how these characteristics

connected to pi. The reason for how this is different than the narrative of the

Common Core is that pi was explored and tested to deepen the students

understanding of what the irrational number stands for and represents. The

narrative of the Common Core found in the United States mentions all of this

vocabulary but the vocabulary is introduced as components of a geometric formula

for calculation.

With the exploration and research of how geometric instruction is conducted

in the United States and other countries such as Australia, a concept map can be

constructed that captures all of the components that both go into and come out of

the understanding and teaching of circles in geometry. As shown in Figure 4, there

are a lot of components that go into the understanding of circles that were

mentioned prior in the Common Core State Standards concept maps or the

Australian method of instruction. Additions are the components of deeper

understanding and knowledge of topics such as the base-ten system found in

mathematics. One other area of geometry that has been put into this concept math is

trigonometry. This is a connection to the component of geometric language and

captures an area of material and calculations that can be used to aid students

understand the ideas and functions of area and other aspects of circles.


Figure 4: This concept map is a thorough illustration of all aspects of circles in geometry found through research and intellectual conversations. Only the

components that go into the geometry of circles are narrated in this illustration.

Although the first portion of this concept map is not deeply expanded or more

elaborate than those of the Common Core State Standards mentioned earlier, the

aspects and intellectual ideas that result from the understanding of circles is much

more elaborate. With exploration of research and the communication of educated

components of both Euclidean and spherical geometry, a large amount of material is

found to come from characteristics and knowledge of circles in geometry. This


material is found in the concept map illustration found in Figure 5 below. Many

areas mentioned earlier are present in the outcome illustration of geometric

knowledge. Similar components are the use of formulas, recognition of geometric

aspects such as dilations and reflections, and the outcomes of using formulas.

Figure 5: This concept map is a thorough illustration of all aspects of circles in geometry found through research and intellectual conversations. Only the

components that result, or come out from, the understanding and familiarity of the geometry of circles are narrated in this illustration.


The two major areas of mathematics placed into this map is the components of

polygon inscription and the differing number of dimensions in geometry that lead to

both Euclidean and spherical geometry. In the time taken to research this topic,

these were two aspects not mentioned in classroom instruction. Polygon inscription

incorporates many factors mentioned throughout this paper, such as interior angles,

radians, and components of shape, but the idea begins to break down the circle and

not see the geometric illustration as a single entity. This concept of inscription

challenges ones knowledge of the characteristics and factors that make up circles.

One of the most intellectually challenging, and fascinating, area of geometry

explored with circles is the idea of three-dimensional geometry. As students become

introduced and educated about geometric concepts and ideas, it is common that

they are taught to think in a manner of a two-dimensional plain. With the abstract

idea challenging characteristics of Euclidean geometry with complex ideas of

spherical geometry, one can be more intellectually exposed to differing possibilities

of shape. These abstract areas of geometry need to be expanded on in classroom

instruction and explored to keep away from students presenting a lack of abstract

understanding and knowledge on the deep understanding of concepts of geometric

mathematics.


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