Measuring Areas: From Polygons to Land Maps

International Journal of Science and Mathematics …, 2010

The issue of the area of irregular shapes is absent from the modern mathematical textbooks in elementary education in Greece. However, there exists a collection of books written for educational purposes by famous Greek scholars dating from the eighteenth century, which propose certain techniques concerning the estimation of the area of such shapes. We claim that, when students deal for an adequate period with a succession of carefully designed tasks of the same conceptual basis-in our case that of the area of irregular shapes-then they "reinvent" problem-solving techniques for the estimation of their area, given that they have not been taught anything about these shapes. These techniques, in some cases, are almost the same as the abovementioned historical ones. In other cases, they could be considered to be an adaptation or extension of these.

2009

We present summarized strategies used by 5 th and 6 th graders estimating the area of irregular shapes and verifying their results, using the paper-and-pencil environment or the computational one. We also present our conclusions from the analysis of the collected data.

This paper concerns the results of the second stage of a two tier program designed to enhance students' technique usage in area measurement. The first stage involves 11 year old students; certain techniques were didactically introduced with the dual purpose of cementing the concept of area and area preservation, and of giving the students tools for explicit area measurement (either exact or estimates). The second stage deals with the development of the same techniques, but the focus is not now primarily on the direct enhancement of the central concept (area) but on the reassessing, re-examining and adapting of the techniques themselves. The paper reports on a case study concerning two 13-year old students' output analyzed from this latter context. Their work in particular shows several ways that they could refine the 'technique' of decomposition of plane figures.

One of the foundational students’ misconceptions of area measurement is conceiving area as the length of a line, instead the size of a surface. They do not see area as the number of measurement units covering a surface. Therefore, this study intends to develop learning activities and materials that support students in developing their understanding of area as the number of measurement units covering a surface. This research is conducted based on the view of the design research approach consisting of two cycles. The subjects of the study are taken from the third grade students of two primary schools in Surabaya. Meanwhile, the learning activities in this study are based on the view of PMRI (Indonesia realistic mathematics education). We found that student need to understand physically quantity of area and the measurement unit of area before they are introduced to the meaning of area as the number of measurement units covering a surface. Learning activity that provides students with the experience of covering activity and comparing area problems could lead to the discussion on the physically quantity of area, the measurement unit of area, and its properties (no gap, no overlapping, and unit consistency); meanwhile, learning activity that engaging students in comparing area of tiled floors can be used to develop students’ understanding of measuring area as the process of finding the number of measurement units.

2015

The perimeter and area are two important geometric concepts, which are taught through many years in schools. Although the curriculum and the textbooks in Hungary consist of both qualitative and quantitative approaches by teaching area and perimeter, the students' performance is low. The main goal of this research is to investigate students' ideas of the concepts area and perimeter from 5 th to 8 th grade. We identify typical solving strategies in order to understand students' imagination connected to these mathematical objects.

This study focuses on the role of tools, provided by a computer microworld, (C.AR.ME) on the strategies developed by 14-year-old pupils for the area measurement of a non-convex polygon. Pupils’ strategies on a transformation and a comparison task were interpreted and classified into categories in terms of the tools used for their development. The analysis of the data shows that an environment providing the pupils with the opportunity to select various tools and asking them to produce solutions ‘in any possible way’ can stimulate them to construct a plurality of solution strategies. The pupils selected tools appropriate for their cognitive development and expressed their own individual approaches regarding the concept of area measurement. The nature of tools used affected the nature of solution strategies that the pupils constructed. Moreover, all pupils were involved in the tasks and succeeded in completing them with more than one correct solution strategy thereby developing a broader view of the concept, although not all of them realized the same strategies. Three different approaches to area measurement emerged from the strategies which were constructed by the pupils in this microworld: automatic area measurement, provided by the environment, the operation of area measurement using spatial units and the use of area formulae. Almost all the pupils experienced qualitative aspects of area measurement through being involved in the process of covering areas using spatial units. Pupils also managed to use the area formulae meaningfully by studying it in relation to automatic area measurement and to area measurement using spatial units. Through these strategies, the concepts of conservation of area and its measurement as well as area formulae were viewed by the pupils as interrelated. Finally, some basic difficulties regarding area measurement were overcome in this computer environment.

The 28th International Scientific Conference “Educational Research and School Practice”, BOOK OF PROCEEDINGS "THE STATE PROBLEMS NEEDS MODERN EDUCATION COMMUNITY", Editors Jelena STEVANOVIĆ Dragana GUNDOGAN Branislav RANĐELOVIĆ, 2022

Learning area measurement in mathematics instruction implies certain phases that entail rhetorical and symbolical generalizations in terms of mathematical formulas (Smith III & Barrett, 2017; Zacharos, 2006; Zeljić & Ivančević, 2019). It is through mathematical formulas that geometry is represented in Serbian mathematics curricula and this fact is the starting point of our research. When pupils have to solve mathematical tasks, they use various strategies that differ in terms of the correctness of their solution, the time needed for completing the task, and task requirements and scope (Siegler, 1991). To measure the area of a rectangular and overcome the problems arising from pupils’ misunderstanding of the area formula, it is recommended to take a closer look at the structure of the rectangular array by covering the area of the rectangular with a mathematical manipulative in the form of a unit of measurement that pupils are intuitively familiar with from the onset (Đokić, 2014, 2017; Van de Walle, Karp & Bay-Williams, 2013). The concept of covering would enable pupils to conceptualize the relationship between the unit’s dimensions and the dimensions of the rectangular. After this phase, through length measurement and multiplication, pupils can solve the area measurement task using mathematical formulas with understanding. Apart from the covering strategy, the paper looks at the ways pupils conceptualize area (Outhred & Mitchelmore, 2000; Reynolds & Wheatley, 1996; Nunes, Light & Mason, 1993). Can an actual misunderstanding of the structure of a rectangular array be found in our mathematics curricula and, if so, can it be overcome by applying the idea of covering the area of a rectangular array? We conducted an empirical study on pupils’ strategic approaches to covering the area of a rectangular in order to understand how pupils calculate its area. We identified the strategies used by pupils and examined their stages of development.

Journal of Education and Culture Studies, 2019

The use of Virtual Globes and Pictometry continues to expand and develop in undergraduate spatial science education. Spatial science undergraduates measured the area of 30 rectangles on the earth’s surface and compared them to Pictometry hyperspectral imagery measurements within a web-based interface and the Google Earth interface compared to ArcGIS Explorer, Map Developers and ArcMap using the ArcMap 10.5.2 interface. An analysis of variance of the absolute mean area errors (p-value of 0.009271) concluded the accuracy of the five area measurements were statistically different at the 95% confidence interval. A Tukey pair-wise test found that the Pictometry and Google Earth methods were more accurate than the ArcGIS Explorer, Map Developers and ArcMap methods. The lowest standard deviation of errors (72.6 sq. ft.) for Pictometry was the most accurate and precise method for on-screen area measurement, followed by Google Earth (SD = 205.0 sq. ft.). The high variation of area measuremen...

International Group for the Psychology of Mathematics Education, 2019

As a component of a course on geometry for preservice elementary teachers (PSTs), we derive area formulas for a variety of polygons including triangles, quadrilaterals, and both regular and irregular shapes whose areas can be measured empirically using decomposition. Decomposing a circle to justify why its area can be measured using the standard formula is more challenging as it requires both empirical and deductive reasoning involving limits. In spite of the challenge, we expected decomposition strategies to transcend work with polygons and support PSTs when thinking about the area of circles. Results show that few PSTs utilized decomposition and instead focused on finding meaning in the symbolism of the formula. Concept images related to area will be discussed.

Learning and teaching measurement, 2003

1 Linear and Area Measurement in Prekindergarten to Grade 2 Michelle Stephan Douglas H. Clements Results from the NAEP international assessments indicate that students* understanding of measurement lags behind all other mathematics topics (National Center for Education ...

Journal of Humanities and Education Development

Mobile applications are rapidly growing in importance and can be used for various purposes. They are now widely used in education. One of the educational purposes for which mobile applications can be used is to calculate the exact result of surface area. This research aims to study the relationships between behavioral factors and the perceived usefulness of using the Measure mobile application in calculating polygonal surfaces. Also, the purpose of the research is to investigate the performance, satisfaction, and perceived behavior of the students while using this application. The methodology of the study is the observation method. The sample of the study includes 50 students from the primary and lower secondary school "Heronjtë e Lumës" Vërmica /Prizren, and from the primary and lower secondary school "Zef Lush Marku" Velezha/Prizren from the Republic of Kosovo. For data collection, the observation list with three sections of 10 indicators is the main instrument...

Kursor

The use of GPS is now widely used by various parties, especially in determining the positionof object location. GPS can be used to measure the land areas, on both wide and flat areas.It can also be used both at night and day, and even it can be used in bad weather conditions.This paper discusses the evaluation of the measurement result of land area using GPStechnology compare to the information containing on the land certificate document. Thegenerated GPS coordinates will be converted into units of meter so that length of each sidecan be known. Heron formula is used to calculate the area of a land plotted on the GoogleMap. From the series of measurement, GPS measurement fall above the land area expressedin the certificate and bellow the meter tape measurement. This measurement still need tostudy further to get a stable measurement. But it can be used by public community to showthe land location to facilitate the land trading. The limitation that must be considered inusing GPS is tha...

The present research was carried out with the participation of 106 students in their last grade in Elementary School and revealed certain problems that these students faced in understanding the concept of area measurement. The students in the sample persisted on using measurement strategies that often led to failure.

The 9th International Conference "Environmental Engineering 2014", 2014

At present, when digital maps are produced, areas are either calculated with the use of analytical-and-digital methods or they are directly acquired from databases. Besides the area of objects is a key attribute stored in cadastral registers and for other geodetic purposes. In the process of development and modernization or updating lands or buildings register database the knowledge of the certainty of analytical determination of areas is required. The knowledge of accurately determined areas is also indispensable for other activities, as e.g. in the case of geodetic maintenance of investments or in the process of control of direct subsidies for agriculture, which are performed within the Integrated Administration and Control System (IACS). The paper considers the case of the relative accuracy of calculation the area of the geometric figure (error-free assumption for the geodetic control network points) basing on mean square errors of X, Y coordinates of its vertices. A formula, which defines the accuracy of the calculation the area of the polygon, was determined from well known Gauss' equation on double area calculation. In analyses was stated permissible uncertainty of calculation of an area from coordinates of vertices' points for different methods of producing cadastral and other digital map data.

Mathematical Thinking and Learning, 2016

Research has found that elementary students face five main challenges in learning area measurement: (1) conserving area as a quantity, (2) understanding area units, (3) structuring rectangular space into composite units, (4) understanding area formulas, and (5) distinguishing area and perimeter. How well do elementary mathematics curricula address these challenges? A detailed analysis of three U.S. elementary textbook series revealed systematic deficits. Each presented area measurement in strongly procedural terms using a shared sequence of procedures across grades. Key conceptual principles were infrequently expressed and often well after related procedures were introduced. Particularly weak support was given for understanding how the multiplication of lengths produces area measures. The results suggest that the content of written curricula contributes to students' weak learning of area measurement. digitalcommons.unl.edu