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A visual approach to measurement concepts: Using mixed fractions to calculate the area of a rectangle

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Concepts of Measurement: Why are the basics so important for students of a truly global curriculum?

Measurement is one of the simplest, finite, and most absolute forms of admiration of nature. A good understanding of measurement concepts can and certainly will go a long way in developing the right amount of curiosity and perspective in young minds. Unfortunately, in most classroom scenarios (many of which now stand to be disrupted further in the wake of a global pandemic that demands physical distancing), the possibility of a child approaching a teacher for a one-on-one interaction to understand measurement with the level of attention that the subject deserves, has gone down drastically.

To make this process simpler for both children and their parents, the Cuemath team has set up a simple guide that can be referenced when trying to understand or explain the various concepts of measurement.

In this article, we will attempt to understand the concept of measuring a given rectangle area with maximum accuracy and aim to arrive at the same calculation by employing several different methods – numeric, fractional, arithmetic, and visual. This synergy of numeric calculations and visual verification is of paramount importance when imparting the knowledge of measurements to young children – because the ideas that shape up in their minds today will most definitely blossom into world-defying wonders in the years to come.

Ask any adult on the street about calculating the area of a rectangle, and they will be prompt to reply, “Just multiply the length with the breadth.” And sad as it may be, the lack of mathematical comprehension in the real world often forces us down a path of formulaic slavery. It will therefore be our sincere attempt to liberate young minds from such displays of formulaic apathy, and instead, create a climate of true curiosity and superior understanding. 

How is the area of a rectangle calculated?

Rectangle area (calculated) as length (multiplied by) breadth (the regular measurement model):

As all math enthusiasts are aware, the simplest way to calculate a rectangle area is to multiply its length and breadth (in unit measurements). The resultant number serves as the area of the rectangle (in unit-squared measurement).

For example, if a rectangle’s given length is 5 cm, and the breadth is 6 cm, its given area is calculated as 5 cm x 6 cm = 30 cm2.

Calculating the area of a rectangle as a geometric/visual representation:

At least at a primary instruction grade, these calculations are easy to demonstrate, calculate, and even visually represent. For example, the rectangle in question can ideally be represented as a composite of small, 1-by-1 squares. And when you reverse the calculation geometrically, you get 30 small squares, each of 1 square centimeter area.

When such a rectangle is plotted on a graph, with the length and breadth edges of the rectangle being marked out and separated centimeter by centimeter, the process of mapping the 30 “square centimeters” is actually pretty easy. This happens largely because, at such an instructional level, whole numbers are pretty easy to understand. In fact, another great way of arriving at an accurate area representation is possible, even with fractions instead of whole numbers.

For instance, if a rectangle is really slim, in the sense that it has a total length of 6 cms but a total breadth of half a centimeter, the resultant area (calculated both geometrically, plotted on a graph or just by multiplying the length and breadth edge units), will come out to be 3 square centimeters. And on the graph, this can be achieved by cutting (splitting) the rectangle halfway through the middle vertically and then shifting one of the “pieces” upwards, i.e., on top of the other piece, giving us a clean three square centimeter rectangle.

To get better visual representations regarding the various Concepts of Measurement, try out the measurement glossary by Cuemath.

Calculating the area of a rectangle in the absence of whole numbers: Mixed Fractions

What are mixed fractions?

The combination of a whole number followed by a partial number (or fraction) is called a mixed fraction or a mixed number. 

For example, 2 is a whole number.

¼ is a fraction.

And 2 ¼ is a mixed number (mixed fraction). Sometimes, improper numbers can also be represented as mixed fractions; for example, the improper fractional representation 9/5 can be represented as 1 ⅘ too. When we calculate the decimal value of either an improper fraction or a mixed number, the results are the same. For example, 9/5 is calculated as 1.8 in decimal number points, and so is 1 ⅘. 

Mixed fractions are used to represent non-whole measurements, non-ideal geometric conditions, and real-world measurement examples. 

For instance, it is never a guarantee that in the real world of physical planes and objects, a rectangle would end up possessing either whole number dimensions or fractional dimensions, individually. There may be a scenario in which a rectangle has a length of 1 ¾ cm and a total breadth of 2 ¼ cms. 

Calculating the area of a rectangle using mixed fractions: the decimal method (converting the mixed fractions into decimal point numbers and then multiplying the units):

The first approach towards calculating a rectangle area using mixed fractions is to convert each individual dimension (metric) as a decimal point number.

For example:

The length of the given rectangle in the scenario above is represented as “1 ¾” cms. It can also be represented in a decimal format, namely – 1.75 cms (¾ as a fraction corresponds to 0.75 as the decimal numeric value – for a simpler understanding of the Concepts of Measurement, please visit the super-glossary of measurement terms and indices as made available by Cuemath).

Similarly, the above problem’s breadth unit can be represented as 2 ¼ cms, or 2.25 cms, since ¼ as a fraction can be converted into decimal points as 0.25.

So now, we have two new dimension representations:

Length = 1.75 centimeters

Breadth = 2.25 centimeters

We can use the original area calculation method (area = length x breadth) and arrive at the result as: 

Area = length x breadth = 1.75 cms x 2.25 cms = 3.9375 square centimeters.

The only problem with this result is the decimal approximation, which now becomes necessary for us to be able to represent this area in an acceptable format. Ideally, this area figure would be estimated down to two decimal points (as per the original representation standards), and the final answer will be represented as 3.94 cm2.

Calculating the area of a rectangle using mixed fractions: the breakdown method (calculating the area projected by whole numbers and the area represented by fractions separately).

Another way of understanding the visual breakdown of the required area calculation in this process is to take the measurement units apart, from their given mixed number variant, into a whole number and a fraction. 

For example,

The length (1 ¾ cm) will be broken down as the whole number (1 cm) and the fractional part (¾ cms).

Similarly, we will break the breadth unit of the above problem (2 ¼ cms) as the whole number (2 cms) and the fractional part (¼ cms).

Now, the simplest way to arrive at the area of the complete rectangle as a result of our calculation is to understand that there are four rectangle areas which we will attempt to calculate and then piece together to find the total result.

This happens, because algebraically, 

(a+b)x(c+d) = ac + ad + bc + bd.

In our problem statement,

a = 1 cm

b = ¾ cms

c = 2 cms

d = ¼ cms

First, we calculate and set aside each area within the rectangle:

“Area one” = ac = 1 cm x 2 cm = 2 square centimeters.

“Area two” = ad = 1 cm x ¼ cm = ¼ square centimeters = 0.25 square centimeters

“Area three” = bc = ¾ cms x 2 cms = 3/2 square centimeters = 1 ½ square centimeters = 1.5 square centimeters

“Area four” = bd = ¾ cms x ¼ cms = ⅜ square centimeters = 0.375 square centimeters

Then, the total area of the rectangle will be calculated as a sum of all the four resultant areas:

ac + ad + bc + bd = 2 + ¼ + 1 ½ + ⅜ = 3.9375 square centimeters.

Conclusion

As we have observed through the many scopes and angles of examination – as we keep changing our approach to understand the concepts of measurement involved in calculating a rectangle area, we always arrive at the same result. However, the magic lies in our ability to adapt and celebrate the minutiae of each approach with the interest that they call for. 
We wish you (and your child) a fun time exploring and dissecting every mathematical concept with the same rigor and attention to detail. To better understand the unitary, fractional, and decimal Concepts of Measurement, please check out the simplified index of measurement, developed and distributed by Cuemath).

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