Real-World Connections to Area and Perimeter

Introduction

In most cases the concepts of area and perimeter are often confused by students where sometimes they are likely to interchangeably use formulas for area and perimeter. Thus, there is always the need for the students to clearly understand these concepts because without building conceptual understanding, the concepts of area and perimeter become purely symbolic (Beckmann, 2013). Hence, it is highly important to developing connections between the concepts of area and perimeter by devising concrete models that relate area and perimeter as well as applying these concepts to the real world as an appropriate way of helping in the stimulating students’ thinking which is essential for making the concepts of area and perimeter which seem abstract more meaningful (Van de Walle et al., 2013). This application task involves solving a real-world mathematical task concerning the concepts of area and perimeter followed by creation and implementation of new task which is slightly similar.

Track and Field Problem

The track is as shown below:

| 100 meters |

Estimating the perimeter of the track in feet.

The perimeter is 400 meters equivalent to 40,000 centimeters because 1 meter = 100 centimeters

1 feet = 30 centimeters

Therefore, the perimeter in feet is obtained by:

Determining the needed rectangular fence to be placed around track (in feet).

In order to get the needed fencing, we must obtain the diameter and radius of the two curves of the track as shown below:

Perimeter of a rectangle = 2(sum of the two equal sides)

Sum of side A distance = 10 feet on each side of the track multiplied by 2 + diameter of track curve

Sum of side B distance = 10 feet on each side of the track multiplied by 2 + radius of each track curve + the distance of the straightaway side of the track

Therefore,

Sum of side A = 10 x 2 + 212 feet

= 232 feet

Sum of side B = 10 x 2 + 106 x 2 + 333.33

Perimeter of a rectangle = the needed fencing

Perimeter of the rectangle = 2(797.33) feet

= 1,594.66 feet

How the track and field problem was solved

The estimation of the track perimeter in feet was solved by converting its perimeter given in meters into feet. This was started by first converting the meters into centimeters and then the centimeters were then converted into feet by dividing the number of centimeters by thirty because there are 30 centimeters in feet. Solving for the needed fencing, we must obtain the diameter and radius of the two curves of the track in order to get the distance added to the 10 feet on each side of the track followed by cumulating of the diameter or radius where possible as well as the straightaway distance on sides of the track as shown in the above calculations.

Creating a drawing to show recommendations of where each of the 4 events should be located on the field and a brief explanation to support the decisions.

| 100 meters |

Boundary line

8 ft 35°

Boundary line

The reason why the four events have been located in the respective locations shown in the field is because upon considering the measurements for each of the four events’ locations, there must be considerations of how they fit effectively. For instance, the long jump and high jump have been located on the edges of the field because there respective measurements are effectively accommodated at the edges of the field. Moreover, the short put and high jump have been located in the middle of the field mainly because it accommodates the boundary lines appropriately as well as giving an opportunity for the thrown short put or discuss to effectively land within the field because of the additional space provided by the curve of the track.

How one or more Standards for Mathematical Practice were incorporated during the problem-solving process

In order to solve the given problem, incorporation of various Standards for Mathematical Practice was inevitable. For example, those incorporated included MP1 which is to make sense of problems and persevere in solving them; MP2 which is to Reason abstractly and quantitatively; as well as MP5 which is to use appropriate tools strategically (Van de Walle et al., 2013).

THE CREATED AND IMPLEMENTED TASK WITH A GROUP OF STUDENTS

The created activity will concern determining the dimensions of a paddock for zero grazed dairy cows through tethering. The activity will consider two zero grazed cows through tethering using a rope measuring 106 centimeters. It is assumed that while grazing the two cows should each be restricted within their respective areas determined by the length of their ropes which are equal. Therefore, the task should determine the following:

What is the area grazed by each cow?
What is the area grazed by the two cows?
What distance can be covered by each dairy cow when they have maximally stretched their ropes?
What are the dimensions of the rectangle that perfectly accommodates the two dairy cows?

Question one

This is given by the area of the circle

Radius = length of the rope

Question two

This is given by the sum of the areas of two circles

This is given by:

= 35,313.17 x 2 cm²

= 70,626.32 cm²

Question three

This is given by the circumference of the circle

Question four

The dimensions of the rectangle in given by the sum of the diameters of the circles where each cow grazes

Diameter = 212 centimeters (dimension of one side)

Dimension of the other side of the rectangle = 212 x 2 centimeters

= 424 centimeters

212 centimeters

424 centimeters

How well students were able to distinguish between the concepts of area and perimeter

During the learning activity, the students were able to make various distinctions between the concepts of area and perimeter. For instance they managed to explain the difference between the two and also a significant number of them were able to provide area and perimeter formulas for a number of figures such as rectangle, circles, semicircles and squares.

The visual representations used by students

The students managed to present there solutions visually particularly by drawing the images in order for them to easily solve the problem during the learning activity, but a considerable number of them were not able to perfectly position the two circles in which the cows were supposed to be grazing which hindered them to draw a rectangle in which the cows were supposed to graze.

One or more Standards for Mathematical Practice were demonstrated by the students during the learning activity

In the process of the learning activity the students were able to demonstrated a number of Standards for Mathematical Practice such as MP1 which is to make sense of problems and persevere in solving them; MP2 which is to reason abstractly and quantitatively; MP4 which is to model with mathematics; MP5 which is to use appropriate tools strategically; as well as MP6 which is to attend to precision. These demonstrations enabled a significant number of students to correctly solve the problem given during the learning activity (Beckmann, 2013).

The obtained insights

Through the analysis of the students’ data there were various insights obtained. For instance, I discovered that the students were conversant with the area formulas and perimeter formulas. However, the students had various challenges in the application part of the learning activity where there some mixture of the positioning and solving the learning activity problem in order to make decisions.

Conclusion

In conclusion, this task has perfectly covered the application part of the connections between the concepts of area and perimeter in order to emphasize on the importance of conceptual understanding amongst students. This has been achieved by solving a real-world problem application of the concepts of area and perimeter (track and field) as well as creation and implementation of the same concepts on a similar task (zero grazed cows). Also a number of Standards for Mathematical Practice were incorporated in solving the track and field problem; whereas the students also demonstrated a number of Standards for Mathematical Practice during the learning activity.

References

Beckmann, S. (2013). Mathematics for elementary teachers with activities, (4^th ed.). Boston, MA: Pearson.

Laureate Education, (2013). Area and perimeter [Video file]. Retrieved on January 24 2014 from https://class.waldenu.edu

Van de Walle, J.A., Karp, K.S., & Bay-Williams, J.M. (2013). Elementary and middle school mathematics: Teaching developmentally (8^th ed.). Upper Saddle River, NJ: Pearson.