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Scaling as Multiplication
Scaling a jpg file
Most of us download “jpg” files onto our computer desktops.
Then we change their size by “dragging” with a mouse. We can either “expand” or “contract” the picture. If we are not careful the picture gets distorted.
We are doing the process of “scaling” a diagram up or down.
Scaling is executed by the computer software, using the basic principle of multiplication.
Scaling is another life situation which uses the process of multiplication.
Scaling a measuring ruler
We are all familiar with 15 cm rulers.
Imagine a 15 cm ruler made of elastic.
What happens if we stretch the elastic ruler to 4 times its original length?
The length of the ruler and the length of all its divisions would become 4 times their original size.
The 1cm mark would be 4 cm from the 0 mark.
The 5 cm mark would be 20 cm from the 0 mark.
The 15 cm mark would be 60 cm from the 0 mark.
The original “line segments” would get multiplied by 4!
We can say that the ruler has been “expanded” 4 times.
We can say that the ruler has been “scaled up” by a factor of 4.
We say that the ruler has been expanded by a ‘scale factor” of 4.
By convention, a scale factor is always used as “multiplier”
If we are reducing the size of a picture then the scale factor would be a fraction less than 1.
If a picture is reduced to half the size, the scale factor would be ½.
The Scale Factor is just a number without any dimensions
“Scaling” is a real-life situation which implies a multiplication.
In the example of the ruler, we can visualise the scale factor as a number which can be continuously varied – a whole number, a fraction, a decimal and even an irrational number!
Hence a scaling model can portray a situation where the multiplier can be any kind of real number!
Mathematicians are of the opinion that “scaling” is a better representation of multiplication than “repeated addition”
When the “scale factor” is a whole number, the life situation reduces to that of a “repeated addition”
Scaling is used widely in real life.
Maps are drawn to such a scale that an entire country can be represented as a map on a sheet of paper. 1 cm on the paper map could represent 100 km of actual distance. The scale factor would be a huge number.
An architect presents, a model of the house that he has designed, to the prospective buyer. If 1 cm in the model represents 1 m in the actual size, the scale factor will be 100.
A science teacher demonstrates the atomic structure of hydrogen using a bigger” scaled up” model.
Two triangles which are similar, are related by a specific scale factor.
The only way we can visualise microscopic & macroscopic events is to see them through scale models.
In Geography, most students have a totally wrong idea of the relative sizes & distances of the planetary system as the textbooks cannot show these diagrams to scale.
Here is a link to a beautiful video of the entire planetary system being created in a desert area in Nevada in the US, to scale, both with respect to their sized & distances. It is then that you realise the mindboggling emptiness of space!
https://www.theatlantic.com/video/index/417309/our-place-in-the-universe/
The “scaling model” of multiplication is a powerful way of representing change in the dimensions.
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