This post has a lot of pictures and is too long for some email clients; you can read it on the Substack website or in the app.

This is part of a new series about how to help yourself or your kids improve in mathematics. It’s based partly on my own experience of having to learn how to learn mathematics, partly on extensive tutoring experience, and partly on many conversations with homeschooling parents, as well as parents struggling to understand Common Core mathematics. Many future editions are already planned, but feel free to leave suggestions for future editions in the comments (open for paid subscribers) or by email to hollymathnerd at gmail dot com.

I may put future editions behind the paywall; I’m not sure yet. If you want to subscribe to be sure you get them all, this link will give you 10% off. If you can’t afford a paid subscription, email me and I’ll give you a free one.

Previous posts in the series:

Part 1: Addition and Subtraction

Part 2: Multiplication, Division, and Fractions

## Part 3: Why Prime Numbers Are Everything

#### aka The Fundamental Theorem of Arithmetic

A Theorem is a mathematical proposition that is proved by reliance on already known facts.

The Pythagorean Theorem is the one that most people have heard of and can recite. To refresh your memory:

In the Pythagorean Theorem, we don’t prove the meaning of operations like squared or addition or the definition of a right triangle. Those are accepted facts with already established definitions.

There are hundreds of proofs of the Pythagorean Theorem, including an original one by a US President!

## Two Reminders, First

This post is about the Fundamental Theorem of Arithmetic. Once you understand this, you will understand the root of mathematics as we use it, and mathematical fluency will become much easier to obtain.

But there are two small things I need to remind you of to be sure you’ll understand.

#### Reminder One, Exponents

The first is exponents. Exponents are a mathematical shorthand for how many copies of a number are multiplied.

Just like every whole number is part of a fraction with a denominator of 1, but we just don’t usually write it, every number with no exponent has an exponent of 1 that we usually don’t write. This is true of all numbers, but here are a few examples:

#### Reminder Two, Prime Numbers

Every number greater than 1 is either prime or composite. Prime numbers are only divisible by themselves and 1. Composite numbers have more than just 1 and themselves as factors.

Example: 13 is prime because 13 can only be evenly divided (without a remainder) by 1 or 13.

10 is composite because 10 can be evenly divided by 1, 2, 5, or 10.

Prime numbers are wicked awesome and play important roles in cryptography and other code systems, among other important applications.

Here’s a chart that shows all the numbers from 1 to 100, with the primes highlighted in red:

# The Key To Becoming Fluent in Math

Understanding the Fundamental Theorem of Arithmetic (FTOA) is a game changer for mathematical fluency.

The FTOA tells us that, up to the order, every integer has a unique combination of prime factors.

Let’s take the number 360 for an example.

There are many ways to use multiplication to get 360. If you can list them all, you’ve got a list of the factors of 360.

Notice that many of those ways can be broken down further.

There is only one way to get 360 by multiplication that cannot be broken down any further. That’s the prime decomposition of 360.

Every number greater than 1 either is prime or has a unique (up to the order) prime decomposition. “Up to the order” just means that re-arranging the order of the prime factors doesn’t count as a different prime decomposition.

Learning to recognize prime factors as the building blocks of numbers is a game-changer.

It turns factoring, which is a crucial part of success in algebra and calculus, into a fun game.

Prime factors enable you to find all the other factors, too. There are lots of ways to do this, but one of the most fun ones is a factoring tree. When a number is even, start by dividing it by 2. Keep going until you have an odd number to work with. Then try 3, 5, 7, and other odd numbers until you get another even split. When you’re down to all prime factors at the branches of the trees, you will have both the prime decomposition and all the other factors!

Notice that all the other factors of 360 can be found from the prime decomposition tree. Take any one of them — for example, 45 — and you can find it from the prime decomposition. See the 5, 3, and 3 branches? They form 45. (3 x 3 x 5 = 45).

#### How To Get Good At Prime Decomposition

Memorization of some of the baseline powers really helps. You start to learn to spot those factors in other numbers, and they make you fluent in using the exponent rules, too.

Factoring trees are a great way to practice. Here are three more examples.

Here is something I coded that does prime decompositions. (It will work best on a computer or tablet, not a phone.)

It’s better to use it just to check yourself until you’ve done some for practice, but it can be a real time-saver for a large number. Google will give you a scary warning when you click the play button to run it, but that’s just them covering their backsides — the only code in it is what you see, and it does nothing other than prime decomposition. When you put in a number, it’ll give you back the prime decomposition. For example, it’ll return this for 360:

#### How Understanding the FTOA Promotes Mathematical Success

The approaches I suggest in parts 1 and 2 of this series will help kids get good at arithmetic, which is the most important key to success in algebra. When the arithmetic is second nature, all the focus, thought, cognitive energy, and effort can go into thinking through the algebra.

The ability to do factoring — to think through numbers and representations of numbers (which is what x, y, and other variables used in algebra are) in terms of factoring — is a superpower for algebra and becomes crucial for calculus. Teaching kids to understand the FTOA and to find the prime decompositions of bigger numbers is giving them a* massive *leg up for all mathematical tracks yet to come.

Understanding the FTOA changed everything for me, and I’ve always suspected that if kids could learn it at the right time — way earlier than I did, as a junior in college — it would pay massive dividends.

If working with the FTOA helps you (or your kids) please drop me an email or comment and let me know about your experience. I consult (locally) and correspond with a number of homeschooling parents and am always interested in expanding my base of understanding and experience in how kids learn math.

In part 4, I show some Algebra 1 problems from Khan Academy and demonstrate how the approach I’ve described makes Algebra much easier and more doable than it would be otherwise.

In a future post: the distributive and commutative properties, along with the sign rules (why negative times negative is positive and all the rest).

Thanks for putting this together Holly. So much of my math education never came close to this kind of framing and explanation. It was all zoomed in on one specific type of calculation with no context. Wish I had this when I was in school, but I'm saving all of it for my kids.

This isn't particularly essential to your explanation of TFToA, but I want to ask because it always comes up when I go over exponents with my students: How you would convince someone that a^0=1? I've got my standby explanation, but I'm curious how you would go about it.