Area Measurement: Nonsquare Units and New Connections

Area Measurement: Nonsquare Units and New Connections

Amanda L. Miller Illinois State University

Abstract

In this paper, I report the results of a dissertation study on area measurement. The study

utilized structured, task-based interviews conducted to explore how students think about area

measurement when using not only square units but also nonsquare, rectangular units and

triangular units The major design was that of a cross-sectional study, with five students from

each of four different grade groups: Grades 1, 3, 5, and 7 (ages 7, 9, 11, and 13). Overall, the

data in this study extends the literature on area measurement development to indicate

progressions in students’ covering, enumerating, subdividing, and spatial structuring

schemes, regardless of unit shape. Implications for instruction and research are discussed.

Although area measurement is one of the most commonly taught (and used) forms of

geometric measurement (Curry, Mitchelmore, & Outhred, 2006), research indicates that

measuring area is more than a procedure or skill quickly learned and retained. Instead,

students can perform rote procedures without understanding (Sarama & Clements, 2009). In

order to investigate why students struggle with area measurement, researchers have analyzed

textbooks and manipulative materials and attempted to identify common errors and

misconceptions. One of these common misconceptions stems from measuring a continuous

quantity (area) with discrete objects (e.g., foam squares). Cavanaugh (2008) found that,

“Many textbooks present regions which are already partitioned so that students need only

count the squares one by one to discover the area of a shape” (p. 55). Other researchers have

exhibited the prevalence of textbook exercises that present rectangular regions with

dimensions displayed with number labels on a pair of orthogonal sides, prompting students to

consider and operate on the displayed number labels only (e.g., Miller, Kara, & Eames,

2012). Such representations had a tendency to encourage students to limit the learning of area

conceptions to rote counting of units or formula usage. These narrowed approaches cause

some students to develop an incomplete conceptual understanding of area concepts and

principles, including identifying a unit, relating unit size to the dimensions, constructing of

arrays, and tessellating the plane (Battista, 2003; Cavanagh, 2008; Clements & Sarama,

2007).

Similar results were reported when area measurement experiences were focused on

covering with physical tiles and then counting those tiles. In their review of related literature,

Outhred and Mitchelmore (2000) argued that using concrete materials to tile a rectangular

figure “may conceal the very relations they are intended to illustrate” (p. 146). Some

manipulatives, such as foam squares or grid-overlays, pre-structure an array, allowing

students to determine correctly the area of the region without attending to the structure

(Lehrer, 2003; Outhred & Mitchelmore, 2000). In other words, students are often able to

create an array of square tiles or drawn units without conceptually understanding that a

square within an array is an individual unit, a component of a row, and a component of a

column (Sarama & Clements, 2009). As Kamii and Kysh (2006) argued, “Empirically

covering a surface with squares or a grid and counting them is one thing, and being able to

think about a square as a unit for area is quite another thing” (p. 113).

Other researchers have reported that students lack a conceptual understanding of the

rectangular area formula, and thus, are unable to recognize the unit of measurement for area.

Battista, Clements, Arnoff, Battista, and Borrow (1998) found that students had difficulties

visualizing the spatial structure of rectilinear regions or the row and column array of squares.

When shown an incomplete array (Figure 1), only 19% of second graders, 31% of fourth

graders, and 78% of fifth graders were able to determine correctly the number of squares in

the completed array.

Figure 1. Example of an incomplete array

While continuing to study students’ development of area measurement, Battista

(2003) determined, “many children simply learn to multiply lengths to generate areas without

understanding that these products are generating arrays” (p. 110).

Rationale for Studying Area Measurement with Nonsquare Area Units

Although researchers have identified several student misconceptions related to area

units, the field has yet to determine how to address them. Since the 1970s, researchers have

analyzed how children learn area unit concepts, such as selecting units (Heraud, 1987; Maher

& Beattys, 1986), constructing units (Reynolds & Wheatley, 1996), identifying units (Kamii

& Kysh, 2006), and relating area units (e.g., Carpenter & Lewis, 1976; Hiebert, 1981; Barrett

et al., 2011). However, children’s conceptions of area are usually investigated solely in terms

of square (standard) units (e.g., Battista, Clements, Arnoff, Battista, & Borrow, 1998; Kamii

& Kysh, 2006). From the literature, it is clear that providing students with area tasks

involving only square (standard) units is insufficient.

More attention needs to be paid to students’ development of area conceptions in terms

of nonsquare (nonstandard) units in order to provide researchers and teachers with a more

generalized characterization of the ways elementary and middle school students think about

area measurement. Such a characterization would integrate students’ behaviors and reasoning

processes with different types of area units, which would include but not be limited to square

units.

Multiple researchers have called for such a de-limitation. During the 1980s, Hiebert

(1981) argued, “Children can also benefit from working with nonsquare units to cover a

region” (p. 42), not just square units. During the 1990s, Simon (1995) and Simon and Blume

(1994) made similar assertions but in reference to pre-service teachers. Specifically, Simon

(1995) claimed,

Measuring with a nonsquare rectangle to determine the area encourages a level of

visualization that is not required when one uses a ruler to determine square units, that

is, they will have to take into account what they are counting, the unit of measure,

which is based on how they are laying the tiles on the table. (p. 133)

Despite the history of these supplications, such a generalized characterization of the ways

elementary and middle school students think about area measurement has yet to be described.

To initiate such a characterization, this descriptive study was conducted to explore the ways

elementary and middle school students think about area measurement when using not only

square units but also nonsquare, rectangular units and triangular units.

Prior to this study, it was my conjecture that (a) the use of nonsquare, rectangular

units would help elementary and middle school students develop an appreciation for the use

of square units as standard area units and (b) doing so would clarify conceptual, procedural,

and intuitive aspects of area measurement.

Research Questions

The following questions guided this study:

1. In what ways do students enumerate and structure two-dimensional space with a

variety of area units?

2. What are conceptual, procedural, and intuitive aspects of area measurement that relate

to students’ enumeration and structuring techniques when working with area units?

Theoretical Considerations

This study was informed by intuitionism and constructivism. As asserted by Fischbein

(1990), “mathematical activity is essentially a constructive process;” hence, researchers have

much to learn from “observing the child’s spontaneous behaviour when coping with

mathematical problems” (p. 7). He argued that such spontaneous behaviour cannot be

categorized as only conceptual or procedural in nature. He asserted that there are conceptual,

procedural, and intuitive aspects at every level of mathematical activity (Fischbein, 1990;

2002). These components can act in harmony or disharmony, creating a dynamic that should

be investigated. Although intuitivism and constructivism are not, generally, integrated, this

study required a theoretical approach that accounted for conceptual, procedural, and intuitive

aspects of mathematical thinking and learning related to the domain of area measurement.

Context and Methods

This research study incorporated qualitative methods to explore the ways elementary

and middle school students resolved area measurement tasks with square and nonsquare

units. The major design was that of a cross-sectional study, with students from four different

grade groups: Grades 1, 3, 5, and 7 (ages 7, 9, 11, and 13). In each age group, five students

were selected from two classes per grade within one school, providing a total of 20

participants.

Research Design

This study employed structured, task-based interviews (Goldin, 2000) aimed at

understanding the students’ current mathematical knowledge at a defined time. Individual,

structured, task-based interview design was selected due to its use in prior studies attempting

to model the relationship between internal and external representational systems (Goldin,

2000). This design has four components: a subject, an interviewer, a set of preplanned tasks,

and a carefully described theoretical framework. Individual, structured, task-based interviews

require that all participants receive the same structured mathematical tasks, yet allow for

further probing. At times, I chose to gather more information and pose unscripted follow up

questions, such as, “Tell me more,” or “How did you come up with your answer?”

Participants

I chose to work with elementary and middle school students from a Midwestern

public school district in the United States, hereafter referred to as Prairie School. The school

is located in a sub-urban community with a population of approximately 125,000 people in

2012. According to Prairie School documents, the school’s student population is reflective of

the community’s demographics. During the 2010 – 2011 school year (the year before this

study), students were 67.7% White/ nonHispanic, 9.3% Hispanic, 8.8% Black/ nonHispanic,

8.3% Asian, and 5.9% multi-ethnic. Prairie School uses the same curricular materials and

texts as nearby districts, Everyday Mathematics (e.g. Bell et al., 2004).

Data Sources

Participant selection was based on a seven item initial survey administered verbally

for first graders and in written form for third, fifth, and seventh graders. After five students

per grade were selected, 10 tasks were posed within two one-on-one interviews. For each

interview task, a child was presented with a rectilinear region and rectilinear unit. Linear

dimensions were displayed in a variety of ways (e.g., tickmarks, dots, grids), but not with

number labels (see Figure 2). The participant was then presented with tasks that required him

or her to operate on conceptual, figural, and physical area units (Steffe, von Glasersfeld,

Richards, & Cobb, 1983).

Figure 2. Ten interview tasks

The participants were posed the same tasks in the same order. Each study interview was

between 11 and 30 minutes in length and occurred during the normal school day. The total

interview time per child was between 31 and 51 minutes in length. All interviews were video

recorded and transcribed. Thus, the data collected as part of this study included transcripts

from the initial surveys and the study interviews, interviews scripts, video recordings, my

reflection after each interview, and copies of student work.

Analysis

After the 40 study interviews had been administered, I performed three phases of

analysis. In the first phase, I started naming and describing the strategies I observed students

using. In the second phase, I performed a comparative analysis (Corbin & Strauss, 2008) to

create initial concepts, or free nodes, to account for and make sense of the strategies I

observed. This comparative analysis consisted of constant comparisons, a process of first

sorting data bits, next organizing data bits, then creating categories of data bits, and finally

constructing, revising, and refining categories or themes in the data while repeatedly and

systematically searching the data (Corbin & Strauss, 2008). By constantly comparing units or

bits of data, I developed an analytical tool that was rooted in my own data. This is in contrast

to using an analytical tool developed by someone else based on a different sample, different

set of tasks, and different research questions.

With the constant comparative method, I interpreted my data to investigate the ways

students resolved area measurement tasks with a variety of area units. To make these constant

comparisons, I rewatched a subtask and attributed a code, or node, for a sentence, action, or

strategy. When I finished a subtask, I went on to the next subtask. When I finished an

interview, I rewatched the interview to see if created any new nodes near the end of the

interview that I would want to use on an earlier subtask. If a meaningful bit of data was coded

with a free node, it was also marked with additional nodes to mark the participant, the task,

and the sub-task to identify the data bit source. I repeated this process per subtask, task,

interview, child, and grade.

In the third phase, I made theoretical comparisons per child, per grade, per task, and

per unit type. According to Corbin and Strauss (2008), “theoretical comparisons are tools

designed to assist the analysis with arriving at a definition or understanding of some

phenomenon by looking at the property and dimensional level” (p. 75). To make theoretical

comparisons, I uploaded all of my transcripts and created nodes for all of my codes in the

qualitative research software Nvivo 8 developed by QSR International (2010). I ran

matrix-coding queries on my nodes to compare codes and groups of codes. In a

matrix-coding query, one free node, tree node, or attribute can be selected to be displayed in a

column (such as structuring with individual units one by one) and a different free node, tree

node, or attribute can be selected to be displayed in a row (such as grade).

After completing the three-phase process, I recruited a check coder. This individual

independently coded 10% of my data, at least one subtask per participant. Between one and

six codes were utilized per subtask. Interrater reliability among Rater 1 and Rater 2 was

computed to be R = 0.73, which was above Marques and McCalls’s (2005) recommended R

for interraters and dissertation researchers. During the post-coding discussion, we compared

codes and came to a consensus for each disagreement. Definitions and use provisions were

elaborated upon based on this conversation. A revised R after coding incongruities were

resolved was not computed.

Summary of Findings

In the section below, I share the results and analyses that support the answering of

Research Questions 1 and 2: In what ways do students enumerate and structure

two-dimensional space with a variety of area units; and what are conceptual, procedural, and

intuitive aspects of area measurement that relate to students’ enumeration and structuring of

two-dimensional space? To do so, I present descriptive accounts of the ways four groups of

students performed on the individual interview tasks. These four groups, or tiers, were

created based on student overall success indices. Student overall success indices were

determined by scaling raw counts for correct responses per sub-task, incorrect responses per

sub-task, and corrected incorrect responses per sub-task. Note that an uncorrected correct

responses was classified as an incorrect response. These raw counts were then scaled. Correct

responses were multiplied by three points, corrected incorrect responses by two points, and

incorrect responses by one point. Next, these scaled counts were summed and then divided by

the total number of subtasks that the individual participant was posed. Table 1 lists each

participant’s tier group number, overall success index, grade, and gender.

Table 1. Overall Success Index per Student

Fictional Name Tier Group Overall Success Index Grade Gender Charlie 1 92.47% 5 Male Jacob 1 89.66% 5 Male

Stephanie 1 89.58% 7 Female Irene 1 88.17 5 Female

Matthew 1 86.67% 7 Male Chris 2 86.46% 5 Male Abel 2 86.46% 7 Male

Dameon 2 83.87% 7 Male Nora 2 82.80% 7 Female Cathy 2 81.48% 1 Female

Eli 3 78.79% 3 Male Octavia 3 75.00% 3 Female Salena 3 73.81% 5 Female Jerome 3 65.69% 3 Male Rebecca 3 63.81% 3 Female

Sally 4 60.19% 1 Female Carter 4 60.19% 3 Male Olive 4 52.69% 1 Female Orrin 4 51.96% 1 Male

Brandon 4 48.48% 1 Male

Although trends per grade, gender, and handedness were also considered, more similarities

were found per tier group than the other categorizations. Individual profiles from Phase 1 of

analysis and matrix queries from Phase 3 of analysis informed the writing of tier group

synopses.

Tier Group 1 Results

The five students categorized into tier group 1 (TG1) had overall success indices

between 86.67% and 92.47% (see Table 1). TG1 students (spanning grades 5 and 7) exhibited

the least variability of the four tier groups in their enumeration and structuring techniques.

They exhibited a capacity to operate mentally and figurally on a unit of units (row and

nonrow), use perceptual support across subtasks, determine the number of units in a row or

column, a unit of units and iterate this unit of units repeatedly and exhaustively in one-to-one

correspondence with the elements of the orthogonal column to structure an array, and apply

the rectangular area formula with understanding. However, they not only exhibited this

success with square units but also nonsquare units (see Figure 3). In fact, TG1 students were

more successful on triangular unit tasks than square unit tasks.

Figure 3. Matthew’s drawing on Task 9

These results indicate that TG1 students had developed a set of complete and integrated

covering, enumerating, and spatial structuring schemes. Their abstraction of spatial

structuring techniques with square units allowed for transfer on novel tasks that included

nonsquare units and a variety of dimension demarcation displays.



between 81.48% and 86.46% (see Table 1). TG2 students (spanning grades 1 to 7) exhibited

more variability in their enumeration and structuring techniques than TG1 students. They

exhibited a capacity to operate mentally and figurally on a combination of a unit of units (row

and nonrow) and individual units, use perceptual support on some sub-tasks but not all, use

mixed drawing and counting actions by recognizing, drawing, or counting at least some rows

(or columns) as a unit of units, and apply arithmetic approaches with and without

understanding. TG2 students were more successful on square unit tasks than triangular or

rectangular unit tasks (see Figure 4).

Figure 4. Abel’s enumeration of physical tiles on Task 5

This indicates that TG2 students had developed a set of mental construct of unit of units that

had not yet been abstracted and thus integrated into a general concept of area unit, as it had

for TG1 students. That is, TG2 students may have integrated enumerating and covering

schemes without a complete set of spatial structuring schemes. Although TG2 students had

developed an expectation of congruity, they had not yet determined how to articulate or enact

that congruity across unit types and dimension demarcation displays.



between 63.81% and 78.79% (see Table 1). TG3 students (spanning grades 3 and 5) exhibited

the most variability in enumeration and structuring techniques of the four tier groups (see

Figure 5).

a. b. c. Figure 5. Octavia's a) Structuring with Individual Units on Task 1, b) Use of a Mixed

Drawing Strategy on Task 2, and c) Use of Parallel Row and Column Segments on Task 6

Although they attempted to structure with a unit of units (row and nonrow), TG3 students

were more successful when structuring with individual units. In addition, TG3 students were

more likely to use perceptual support incorrectly than to use it correctly. They exhibited a

propensity to count individual units one by one systematically, rather than use more advanced

enumeration strategies. These results indicate that TG3 students had made progress with

mixed drawing and counting actions by recognizing, drawing, or counting at least some rows

(or columns) as a unit of units but had not yet developed a mental construct of a unit of units,

as the TG2 students had. That is, TG3 students may have partially integrated enumerating

and covering schemes without a complete set of spatial structuring schemes. Furthermore,

TG3 students had also not yet developed an expectation of congruity among strategies, as

evidenced by their lack of consistency in strategies among sub-tasks, unit types, and

demarcation of dimension types.



between 48.48% and 60.19% (see Table 1). TG4 students (spanning grades 1 and 3) were

comparably invariant in their strategies. They only exhibited more variability in enumeration

and structuring techniques than TG1 students. They attempted to cover with individual

shapes that were sometimes approximations of squares, rectangles, or triangles that were not

constrained in size, shape, or location. Other times, TG4 students covered with individual

units that were somewhat constrained in size, shape, and/ or location. Due to their struggles

with unit concepts, TG4 students rarely operated on a unit of units, and when they did, the

units of units were usually row-based. Furthermore, TG4 students were less likely use

perceptual support than use it (correctly or incorrectly). These results indicate the TG4

students had not yet developed a mental construct of unit. Instead, they struggled to make

sense of experiences not only when they were asked to structure mentally or figurally with a

variety of area units, but also when they were asked to tile with physical manipulatives. That

is, TG4 students exhibited an inability to organize or coordinate the space with physical,

figural, and mental units (see Figure 6). They had neither integrated their enumerating and

covering schemes, nor developed spatial structuring schemes.

a. b. Figure 6. a) Carter’s tiling and b) Olive’s drawing on Task 9

Nontier Group Based Findings

By comparing strategies across tiers, I identified three collections of trends. These

three collections are conceptual aspects, procedural aspects, and intuitive aspects of area

measurement. Because these aspects influence one another, it is important to examine the role

each of them plays in area measurement.

Conceptual Aspects. The first collection of trends indicated conceptual aspects of

area measurement. Conceptual aspects are defined to be implicit or explicit principles or

features of a concept. These aspects are relational, flexible, and generalizable (Rittle-Johnson,

Sieger, & Alibali, 2001; Schneider & Stern, 2005). A student’s mental image of an area unit

and how he operates on it is a conceptual aspect of area measurement. This is because a unit

is a conceptual entity (cf. Barrett et al., 2011). I interpreted the strategy codes no anchoring,

anchoring, structuring with individual units, structuring with units of units as rows or

columns, structuring with units of units not as rows or columns, and structuring with parallel

row and column segments to be attending to conceptual aspects of area measurement.

Procedural Aspects. The second collection of trends indicated procedural aspects of

area measurement. Procedural aspects are defined to be conventional and routine sequences

of actions such that the objects operated on are symbols (cf., Hiebert & Lefevre, 1986).

Algorithms and other arithmetic-based computation strategies are examples of procedures.

Hence, I classified many of the enumeration strategy codes as procedural aspects of area

measurement.

Intuitive Aspects. Intuitive aspects of area measurement are defined to be

self-evident, immediate, and natural ideas or notions about area measurement concepts and

operations. Some intuitive aspects are based on everyday experiences, and others are based

on educational experiences (cf., Fischbein, 2002).

Two of the emergent strategies considered to be related to intuitive aspects of area

measurement are the one row, one column strategy and the corner strategy. The one row, one

column strategy was used by five students (all from TG2 and TG3 but spanning grades 3, 5,

and 7) when drawing on five tasks across unit types. Sometimes the student using he strategy

first drew individual units in one row and then one column before drawing individual units to

cover the interior of the region. Other times they used the strategy before curtailing to draw

row and column segments, indicating a mixed drawing strategy.

In contrast to the one row, one column strategy, the corner strategy is based on

(Western) everyday experiences. Young children, such as Orrin, placed four units in the four

corners and then considered the region covered (see Figure 7). Some of the participants first

covered the corners, then the remaining border, and finally the interior of the region with

figural or physical units. Others covered the corners before filling in the rest of the region

without focusing on the remaining border next. The corner strategy was used by five children

(from TG2 and TG3 but spanning all four grades) on seven tasks, spanning the unit types.

Figure 7. Orrin’s tiling on Task 9

Only one student gave an explanation for her strategy. Rebecca started in the four

corners six times on five different tasks. When asked about her strategy, Rebecca asserted, “I

like to do it because there’s a game named four corners.” Although I asked the students if

they liked puzzles, the corner strategy users did not explicitly make the connection.

Discussion of Findings

Overall, the data in this study support previous findings that area measurement is

developmental in nature (e.g., Battista et al., 1998; Sarama & Clements, 2009). I identified an

increasing pattern, generally based on grade, in children’s use of more sophisticated

strategies and a decreasing pattern in their use of less sophisticated strategies. This was

especially pronounced with transitions from structuring with individual units to structuring

with groups of units (as rows and nonrows) and then to structuring with parallel row and

column segments. These results extend the existing literature on area measurement

development to indicate that the construction of individual units, groups of units, and arrays

are developmental in nature, regardless of unit shape.

There were two notable emergent strategies: one row, one column strategy and corner

strategy. I consider the one row, one column strategy to be based on educational experiences

because of its potential connection to textbook displays of regions. Figure 8 is an area task

that imitates the format of area measurement tasks often printed in elementary and middle

school mathematics textbooks.

Figure 8. Textbook area task

In such a task, a child may be asked to determine the area of the rectangle, prompting the

child to multiply the length and the width of the rectangle. Hence, the child may be prompted

to acknowledge the importance of a pair of orthogonal sides. The one row, one column first

strategy is an extension of this acknowledgement that may or may not be used to quantify a

two-dimensional region correctly. In some cases, children that used this strategy then

multiplied the number of units in a row times the number of units in a column to quantify the

number of units covering the region. Within an array of squares, the number in a row times

the number in a column does produce a correct number but not a correct quantity. Referring

to the operation of n rows times m columns produces mn row-columns, which is an erroneous

abstraction of the rectangular area formula. Rather, one correct arithmetic approach would be

n rows times m square units per row, producing nm square units. Within an array of

nonsquare units, the number in a row times the number in a column does not necessarily

produce a correct number or a correct quantity.

The second notable strategy was the corner strategy. There were many variations of

this strategy. I consider the corner strategy to be based on everyday experiences because of

its connection to jigsaw puzzles. A common strategy for assembling a jigsaw puzzle is first

finding the corner pieces (pieces with two straight edges), then finding the border pieces

(pieces with one straight edge), and finally assembling the interior pieces (pieces with no

straight edges), assembling adjacent pieces by fit.

There are many parallels to the ways young children cover, figurally or physically, a

rectilinear region with area units and the ways individuals assemble a jigsaw puzzle. In both

situations individuals have a tendency to use similar strategies, including corner strategies,

border strategies, next to existing strategies, and trial and error strategies. Furthermore, when

covering an area and assembling a puzzle, it is important to eliminate gaps (large and small)

and overlaps in order to create a planar region completely covered (although in the assembly

of a puzzle, the "units" are not congruent). These connections have not been investigated in

the literature, neither the strategy correspondences nor the space-covering comparisons. Yet,

they have direct instructional implications. Such a comparison may help children recognize

and then eliminate their space-covering errors.

Educational Importance of the Research

These results indicate that the account of ways children develop spatial structuring

schemes is too narrow. Researchers have described how children progress from iterating

individual units to cover, to row-based, two-dimensional structuring schemes, and then to

array-based, two-dimensional structuring schemes (Battista et al., 1998; Sarama & Clements,

2009). However, such a progression does not account for nonrow, two-dimensional

structuring. The results from this study, especially from the triangular unit tasks, indicate that

nonrow, two-dimensional structuring should be considered. It is important for future research

to account for such a delimited notion of nonrow, two-dimensional structuring schemes and

the ways they support meaningful use of arrays, grids, and coordinate systems (Miller, 2013).

These results not only extend the existing literature on how children think about area

measurement, but also suggest multiple instructional implications. The emergent strategies of

one row, one column first and corners are worthy of further investigation. However, before

promoting the use of the one row, one column first strategy or discussing the use of jigsaw

puzzles in math lessons, more study is needed. Is the one row, one column first strategy a

transient strategy supporting the shift from around the border to array structuring? Does the

one row, one column first strategy propagate an erroneous abstraction of the rectangular area

formula? Does the connection of covering and area to assembling a jigsaw puzzle help

children recognize and then eliminate their space-covering errors? Are there other strategy

correspondences? Does the lack of congruence among jigsaw puzzle pieces detract from

these correspondences? These and other questions need answering before recommending that

jigsaw puzzles be used to support area measurement development.

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Area Measurement: Nonsquare Units and New Connections

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