Real-World Applications
Introduction
Geometric measurements connect geometry and numbers which are two critical domains in mathematics as each provide each other conceptual support. Measurements are central to mathematics, to sciences subjects such as physics, chemistry, and even in real life application situations. Mathematics curriculum cannot be complete without including measurements as one of the core components. Measurement involves assigning a number to a magnitude of an attribute between objects such as a width relative to a unit of measurement such as yard, inch, feet etc. Measurement is a continuous attribute shared by objects. For example length can be increased by adding other lengths or subdivided to lesser lengths. Attributes between objects can be compared. For example the length between an object and another can be compared with the length between two other separate objects. Measurements enable students to indirectly compare between objects attributes such as length or width and infer meaning that can be applied to solve problems in real life situations (http://www.corestandards.org/Math).
A Community Recreation Center (CRC) owns a rectangular shaped parking lot which measures 50yards in length and 40 yards in width. The parking lot has 36 parking spaces and can therefore park 36 vehicles at a time. Measurements of each parking space have not been provided however each parking space is rectangular in shape. According to the current plot design, 24 parking spaces are designed close to each other whereas there is a drive in aisle between the next 12 parking spaces. Measurements of the existing driving aisle have also not been provided. The parking lot is shown in the diagram below;
The current design of the parking lot provides enough space for easy parking and pulling out of the parking lot. I have been hired by the Community Recreation Centre (CRC) to determine whether it is possible to re-design the parking lot to accommodate 15 additional rectangular shaped parking spaces to make a total of 51 parking spaces in the parking lot. The task of redesigning the parking lot will require that the parking lot is painted and new parking spaces each measuring 9feet in width and 18feet in length are created. The main constraint in the task is that the parking lot measurements cannot be increased beyond the current 50 yards by 40 yards. The other challenge is that the parking lot will need to have adequate driving aisles to enable 51 vehicles to pull in and out of the parking lot with ease; without causing traffic congestion and inconveniencing other users of the parking lot.
The parking lot in its current design is measured in yards e.g. the width and the length of the parking lot are all measured in yards as the preferred unit of measurement. The new parking spaces are to be measured in feet measuring 9feet in width by 18 feet in length. To compare the two units of measurements it is prudent to either convert the yards into feet or the feet unit of measurement into yards. I could convert the units of measurements of feet into yards but this would not be prudent since it would result in units of measurements stated in a fraction form. For example converting feet into yard will result in decimal points since 1 foot is equal to 1/3 yards.
In this exercise I will convert yards into feet to deal with whole numbers which are easier to manipulate, understand and read to infer meaning of different attributes since 1 yard is equivalent to 3 feet (http://ime.math.arizona.edu/progressions/).
In this task I will have to design new parking spaces without reference to the measurements of the existing parking spaces since they have not been provided. The next challenge is that I must lay out suitable driving aisles to ensure vehicles are able to park without causing traffic jams or inconveniencing other motorists intending to use the parking lot. The starting point in undertaking this task of redesigning the parking lot is by using mathematical calculations to obtain the maximum number of parking spaces that the parking lot can hold. I will use this information to advise the management of the Community Recreation Centre on whether the parking lot can accommodate further additional parking spaces should the need arise in future.
One of the Community Recreation Centre’s instructions is that the new parking spaces must be rectangular in shape and must not block each other. The manager would need to advise park attendants on vehicle measurements to ensure only vehicles which comply with those measurements are allowed to access the parking lot. This would ensure congestion does not occur and increase mobility within the parking lot. The current design of the parking lot is provided below;
Strategies to increase the parking slots from 36 to 51 by adding additional 15 parking slots
To determine whether it is possible to increase the number of parking spaces from the current 36 parking spaces to 51 by adding 15 new spaces I must first determine the total area of the parking lot. The formula for determining the area of a rectangular plot is obtained by multiplying the width of the plot by its length i.e. Area (in square yards) =length in yards multiplied by width in yards. For the parking lot therefore the area is; 50yards*40yards=2000 square yards. However I must determine the total area of the parking lot in square feet to enable me compare measurements of the parking spaces since they are provided in feet. To convert these measurements I must determine the equivalent of yards to feet which is that 1 yard is equivalent to 3 feet. The length of the parking lot of 50yards is -50yd*3ft/1yd=150 feet whereas the width of the parking lot given as 40yards is 40yd*3ft/1yd=120feet.The area of the parking lot in square feet therefore is=>150feet*120feet=18,000square feet.
The total number of parking spaces that the parking lot can possibly hold can be obtained by dividing the total area of the parking lot by the area of a single rectangular parking space. The area of a single parking space is given by multiplying the length by the width of the single parking space in focus. The area therefore is obtained by multiplying the length and width of a parking space that is 9feet*18feet=162square feet. The maximum number of parking spaces that the parking lot can hold can be obtained by dividing the area of the entire parking lot by the area of a single rectangular parking space. Thus 18000square fleet/162square feet* I parking space=111.11parking spaces. This is to determine in a glance whether the parking lot can accommodate 51 parking spaces. In this case it can since it can actually hold 111 parking spaces in total which is more than two times the total number that is targeted except this does not allow for the driving lanes. This is because it would not be possible for 111 parking spaces to fit into the parking lot and still make the parking lot serviceable (http://ime.math.arizona.edu/progressions/).
I must set aside space for vehicles to drive in and out (driving aisles) to avoid congestion and ensure the vehicles are able to smoothly drive in and out of the parking lot. In this case I have set aside space to accommodate two, one-way driving aisles and one, two-way driving aisle. I must therefore calculate the total area “in square feet” that will be taken up by the driving aisles. The area in square feet of one-way driving aisle is obtained by multiplying the width of the one-way driving aisle, which is 12 feet, by the length of the parking lot, which is 150 feet; hence the total area of a single one way drive in aisle is 12 ft * 150ft= 1800square feet. I am multiplying the width of one, one-way driving aisle by the length of the parking lot because one, one-way driving aisle will start from one end of the parking lot and terminate at the end of the parking lot hence covering the entire length of the parking lot. This will enable vehicles to enter from one end and exit the parking lot from the exit directly at the end of the driving aisle. The total area to be occupied by the two (2), one-way driving aisles therefore will be obtained by multiplying the area occupied by one, one-way driving aisle which is 1800square feet by two (2) which is the number of one-way driving aisles in the new redesigned parking lot; hence 1800 square feet* 2 one-way driving aisle =3600 square feet. The next step is determining the total area “in square feet” that will be occupied by the one (1) two-way driving aisle that will be in the newly redesigned parking lot. This similarly will be obtained by multiplying the width of the one-two way driving aisle with the length of the parking lot since the driving aisle will start from one end and terminate at the other end directly opposite it. Therefore the area is obtained by multiplying its width which is 24feet by 150 feet which is the length of the parking lot that is 24feet*150feet=3600square feet.
The total area to be occupied or taken up by the three driving aisles will be obtained by adding up the total area in square feet of the two (2) one-way driving aisles plus the total area of the one (1) two-way driving aisle. The total area to be occupied by the two (2) one-way driving aisles will be 3600 square feet whereas the total area to be occupied by the one (1) two-way driving aisle will be 3600 square feet which when added up equals 7,200 square feet.
The next step is to obtain the area that will be occupied by the parking lines in the new parking lot. Each parking space will have two parallel parking lines each measuring 9 feet in length. The parking lines will demarcate adjacent parking spaces. Since the parking lines will be shared by adjacent parking spaces i shall count one single line per each parking space. Parking spaces 1-10 will be demarcated by 11 parking lines each measuring 9 feet which will be the width of a single parking space. Parking space no 1 and no 10 will be demarcated on one side by a parking line and on the next by a parking line which will form part of the perimeter wall of the parking lot. Parking spaces 11-51 will each have 46 parking lines each measuring 9 feet which is the width of a single parking space. The total parking lines measuring 9 feet will be 11+46=57 parking lines.
The length of the parking spaces each measuring 18 feet will be obtained by counting the space between the parking spaces. It is important to note at this point that half of the parking spaces numbers 11-50 are each sharing parking lines that measure the length of each single parking space. For example parking space number 11 is directly opposite parking space number 21. This means the parking line measuring the length of parking space number 11 is the same parking line measuring the length of parking space number 21. In this case I will count the parking line once for number 11 and 21. Between parking spaces numbers 1-10 there are 10 parking lines each measuring 18 feet. This is because each parking line will be rectangular with a width of 9 feet and a length of 18feet. The rectangle length of each parking space is however open at one end; which is the end which a vehicle will pull into the parking space. Between parking space numbers 11 to 51 there will be 20 parking lines each measuring 18 feet and each will measure the length of each parking space.
To determine the total area to be occupied by the parking lines we must determine the area of the 57 parking lines each measuring 9 feet and the 31 parking lines each measuring 18 feet. The width of one parking line is given as 4inches which when converted to feet gives 1.33feet since 1 inch is equivalent to 1/3 feet. This implies that the area of one parking line measuring 9 feet will be given by multiplying 9feet*1.33 feet=11.97square feet. The total area of the 57 parking lines therefore will be 57*11.97square feet=682.3square feet. The area of one parking line measuring 18 feet in length will be given by multiplying 18feet*1.33 feet=23.94square feet. The total area of the 31 parking lines will be 31*23.94square feet=742.14square feet. The total space in square feet to be occupied by the parking lines is 682.3square feet +742.14square feet=1,424.44 square feet.
The total space to be occupied by the driving aisles and the parking lines will be 7,200 square feet + 1,424.44square feet=8,624.44square feet. The area that is available to create parking spaces is 18000squarefeet minus 8,624.44 = 9,375.56 square feet. This translates to 9,375.56 square feet divided by 162square feet=57parking spaces. The number of parking spaces that can be created will be 57 parking spaces.
The redesigned parking lot will be as shown in the diagram below;
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
| 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |
| 40 Yds | |||||||||
| 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 |
| 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 |
| 51 | |||||||||
| 50 Yds | |||||||||
Note: 1-51 represent the additional parking spaces in rectangular shape each measuring 9feet in width by 18 feet in length.
From the fore going analysis therefore i can create an additional 21 more parking spaces on top of the existing 36 parking spaces to have a total of 57 parking spaces each measuring 9feet in width by 18feet in length. But for my case I am required to add only 15 additional parking spaces to total 51 additional parking spaces.
Standards of mathematical practice employed in the exercise include the formula to determine area of a rectangle which is obtained by multiplying its length by its width and also conversion of distance. In the case above I converted yards into feet in which I was able to determine that 1 yard is equivalent to 3 feet and inches to feet (http://www.corestandards.org/Math).
An idea for a cognitively demanding mathematical task
An idea for a cognitively demanding mathematical task that could be used to help students learn how to estimate measurements and convert measurements from one unit to another in the context of a real-world problem in grades 5–8 could be formulated as follows. In a parking lot measuring 400meters in length by 200 centimeters in length redesign the parking lot to accommodate ¾ of the parking spaces to measure 18feet in length by 9 feet in width and the remaining ¼ of the total parking spaces to accommodate 36 feet in length by 18 feet in width. Each parking line will measure 4inches in width. Parking spaces to be parallelogram in design
Summary
In summary it is important for a student to learn computational estimation to be able to develop their abilities to solve cognitively-demanding mathematical tasks. Student should be able to logically understand the how to add, subtract and divide whole numbers to obtain a logical solution to a cognitively demanding mathematical task. Teachers should continuously instruct their pupils on computational estimation to help them develop computational estimation skills which then help them to solve cognitively –demanding mathematical tasks (Tsao & Pan, 2013). For a student to be competent he must be able to understand conversion of measurement units from one unit of measurement to the next for example converting yards to feet as in our case above. Real life application of mathematical concepts should also be used to assist students to understand application of mathematical concepts to be able to solve cognitively- demanding mathematical tasks. Understanding of what terms of measurement actually mean in real life is critical. For example the distance in real life of a certain unit of measurement is important. Students must be able to develop numbers sense by being able to understand the meaning of numbers , know the relative size of numbers, comprehend how arithmetic operations affect results and be able to develop multiple relationships among the numbers to be able to solve cognitively- demanding mathematical tasks (Tsao & Lin, 2011; Berry & Kim, 2008).
References
Berry, R. A. W., & Kim, N. (2008). Exploring teacher talk during mathematics instruction in an
inclusion classroom. The Journal of Educational Research, 101(6), 363-377,384. Retrieved from http://search.proquest.com/docview/204189609?accountid=45049
http://www.corestandards.org/Math
http://ime.math.arizona.edu/progressions/
Tsao, Y., & Pan, T. (2013).The computational estimation and instructional perspectives of elementary school teachers. Journal of Instructional Pedagogies, 11, 1-15. Retrieved from http://search.proquest.com/docview/1440862387?accountid=45049
Tsao, Y., & Lin, Y. (2011). The study of number sense and teaching practice. Journal of Case Studies in Education, 2, 1-14. Retrieved from http://search.proquest.com/docview/887907244?accountid=45049
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