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From the point of view of a grade $4$ student - why and how is prime factorization of a number better in some sense (if at all)?

Visually, why is representing $30$ in the form of $2 times 3 times 5$, like this

(Source: mathlesstraveled.com)

might be better than say in the form $5 times 6$ like this?

If it is not, then why would we need it?

I know the fact that they are the building blocks of every number, but how could we make the kids appreciate this fact?

I'm also familiar with some 'contrived' problems which force prime-factorization, and with the application of primes to the field of cryptography. But the latter might not make much sense at the middle school level.

Edit: middle school = grade 4-7 = ages 8-12 = generally when prime factorization would be introduced.

I wouldn't say it's better. It's different and allows a different set of thought to emerge in the spectator's mind. If the real question underlying this is something like: Why should we teach prime factorisation, I have two answers (and others probably have more):

1. It's unique, allowing us to avoid discussions about whether $2times 15$ or $3times 10$ is the better factorisation of $30$.




2. It can be useful to think in terms of prime factors when doing other things, like calculating least common multiple/greatest common divisor (those should be understandable at the level you talk about, but coming from a different school system than you, your "middle school level" means absolutely nothing to me, please don't use references like that on international sites).

This isn't exactly addressing your question, but I would suggest that the factorization $30 = 2 times 3times 5$ is better represented as a 3-dimensional stack of boxes like this:

(but with the correct numbers in each direction), rather than as polyogon-like shapes.

I think this visualization may help with understanding how prime factorizations give a more "efficient" and "neat" way of understanding the number $30$ compared to other factorizations (in a vague non-technical sense).

It is not better in general. However, for various number-theoretical questions one needs the prime factorization, rather than another one. For example, the Fermat problem, asking which integers $nge 1$ are the sum of two squares, needs the prime factorization. We see that $n=30$ is not the sum of two squares, because in the prime factorization, not all factors $qequiv 3bmod 4$ occur with an even exponent. Indeed, $3$ appears with an odd exponent $1$.

I might explain to kids that you need it to find out if the two numbers are co-prime or not.

One application of co-prime (which kids might understand) is to make the gears life longer. If the number of teeth is co-prime between the meshing gears, the gears will wear evenly. Therefore, mechanical engineers try to design the number of teeth of the gear system in co-prime.

Graphical representations of algebraic relations are sometimes more enlightning than the algebraic formula itself (see only the online catalogues of funny function-curves, of fractals or many others), so I think there is no general rule that this or that representation of numbers is more "intuitive" or "nicer"...

Example: if you would consequently work with numbers in terms of their primefactorizations, then any multiplicative problem is extremely easy - but any additive problem is extremely difficult: if $z=p_1^{e_1}p_2^{e_2}$ then first the derivation and then the final expression of $z+1$ in that notational scheme is extremely difficult!

We have for cultural reasons the preference for the positional decimal-digit-system which makes addition extremely simple - and multiplication a complicated operation needing computations of each single digit with each other and which also needs observing carries and what not.

For the graphical representation of functions of numbers there is, if I recall the name correctly, for instance the "Moser-representation" which give hyperoperations (in the sense of a family/sequence of binary operations "+","*","^",...) a graphical representation and which arrives at a very concise notation-scheme for extremely huge numbers (see wikipedia)

P.s.: note that the given graphic in your question is not consistent: the pentagon should be rotated such that its sides are neighbored to the triangles and not the edges as it is shown there, compare to this corrected image:

$ qquad qquad $

I don't understand the top picture at all, so I wouldn't say that's simpler, because it clearly isn't.

I would say that a prime factorization is better in some sense because it's easiest to deal with something once you've broken it into the simplest possible pieces. This is a general principle really: If you have a complex problem to tackle, try to break the problem in tiny pieces and deal with those.

If you display your brick like pictures for $2times 15$, $3times 10$ and $6times 5$, you introduce the inverse function $y=frac{30}x$ as a bonus and state that all rectangles drawn under share the same area...

A prime factorization of a number $n$ is certainly useful if you are interested in all the ways of arranging $n$ units into a rectangular array.

The units could be square meters.

The units could be discrete 'dots', or people.

Say you have $30$ students in a class. You can arrange the classroom desks in a $6 times 5$ rectangular pattern. You train the students to leave the classroom in a $1 times 30$ pattern. If you are leading them out of the classroom, you train them to form a $2 times 15$ formation. In the school auditorium, the students are trained to sit in a designated $10 times 3$ section.