Basically it's just that: your two core items are Rod of Ages and Atma's Impaler. You play on the synergy of these two items and a few other key items to build a hybrid champ that somewhat evenly raises his Survivability, AP, and Attack damage. The Atma's is basically your ticket to nearly free gold. To make it mathematically worth more than you payed for you need to recoup 825g from it. In 1125hp, you'll have a free B. F. Sword.

The RoA is going to give 12.6 AD. That's more than a Long Sword which is worth 450 or it is nearly half of the health required to be as good as a BF Sword. Only a couple more items with HP on them to go for that free BF sword.

Key items outside of the Rod of Ages and the Atmas:

Rylai's Crystal Scepter: This is a good 3rd item typically. It gives health and AP. It's also going to give 10 AD due to the health. This is equal to a free Long Sword. A 450 gold savings. Theoretically this makes up the value in Long Swords; congratulations you are well on your way to that B. F. Sword.

Frozen Mallet: This is for Champs who are mostly AoE or rely more on auto-attacks. 14 extra AD off this puppy.

Banshee's Veil: 7 bonus AD, a great passive and some extra mana.

Trinity Force: Doubling your base AD is nice and as most of these champs are spammers you should proc it often. Tons of good bonuses for the money. You're going to sneak 5 bonus AD off of this.

Spirit Visage: All of these champs have some sort of inbuilt sustain that this will pump up, give MR, and it's cheap. Only 5 bonus AD though, but it's cheap. Skarner's Ixtal's Impact, Kayle's Divine Blessing, Sion's Cannibalism, and Cho'Gath's Carnivore and Feast both benefit.

B-list:
Sunfire Aegis. Costs too much, does too little damage. Plus it's almost a sure thing that either the Tanky DPS or Tank has one. 9 Bonus attack damage.

Warmog's Armor. If you need a last item boost or are the primary tank. If you're just getting blown up as Skarner and the enemy keeps leaving lanes free for you to farm. 27 AD off of this maxed.

Haunting Guise. A cheap item that gives AP, Pen, and HP. What's not to love? Well, the fact that you're going to have to replace it later, which makes it less of a cost saving. If only this built into something better in the end. 4 Bonus Attack Damage.

C List:
Heart of Gold and Randuins. We don't want the Heart of Gold because we want to set up our Rod of Ages as fast as possible. 7 bonus Attack Damage. I love the Warden's Mail on tanks for its insane price and ability, but we want more damage. A big maybe for Cho'Gath.

Aegis of the Legion. Leave it for the tank or support. A big maybe for Cho'Gath.

Shurelya's Battlesong and Soul Shroud. Maybe maybe maybe on Kayle if you don't have a support and are just having a **** game, and need your teammates to win it for you. I'd pick the shroud over the Shurelyas. 6 ad and 10 ad respectively.

1) Rod of Ages gives champs early game stats that trickle into late game. All around with these champs you get the best mix of early survivability (health), sustainability (mana), and damage (AP). The Rod of Ages can typically be farmed in under 15, 20 if you need an item before it (like a Sheen), if you don't have it by 30 then you're having huge problems.

For these champs, I believe that the RoA is better than the Warmog's Armor (but similar) for a couple of reasons. First its two parts are more useful than the Warmog's pieces. Second it gives damage on its own, meaning you can farm it first without much worry about lagging in the early/laning game.

2) Atma's Impaler gives more defense and converts you into a hybrid character for the midgame. This also allows each of these champions a few extra seconds of tanking turrets and ad's.

3) The rest of the build is going to allow you a good mix of damage and health. Being hard to kill, and able to easily kill opponents is great.

4) Late game your AD will probably give you the edge in most fights, as these champions just don't bring the burst that going full AP will get you, and most of these champions can't cut it as raw AD.

5) This is to hybrids what Atmogs is to Tanky DPS. It is a percentage play strategy. You can go raw AP or raw AD or Tank, but each of these has its easy set counter. Some games you're going to rock as AP Skarner or AD Kayle, but others just crash spectacularly. This is good at balancing you out. Keeping your KDR positive and making sure that in every fight you're as useful as possible. You will be very hard to counter like a hybrid, but even harder to kill.

Skarner

Why is this better than:

Raw AP.
You will not find yourself dead all the time. Your damage will not be so dissimilar from Raw AP as you'll be scaling with AD and AP. You'll be able to kill more champs than just the squishes. Madred's Razor + Sheen is hyper fast jungling.

Top Lane Skarner.
Blue buff and red buff. Jungle creeps keep your Ult CD short.

Tanky DPS.
This is Tanky and DPS and can spike some damage too. If you just buy items like Wit's End and Sunfire Aegis, you will not scale your abilities. Sunfire is overpriced imo. Wits is great, especially early, but I think this is better.

Cho'Gath

The reason I like the Atmas is because your CD's are long. your abilities are dodgeable, and you can reach the end of your mana supply quickly midgame, so I tend to spend a lot of time auto attacking. AP is affecting Vorpal Spikes, but in order to make that a serious consideration you'll have to go full AP which wastes the potential of your ult and doesn't let you do much tanking at all. You kind of have to hope to be ignored if you're AP. This build adds to your HP and takes full advantage of your HP in every way while fixing some of your mana issues.

Kayle

This build focuses on making Kayle someone that the enemy doesn't want to kill first all the while dealing out as much debuffing and damage as possible. Her new passive, Holy Fervor and the items I've suggested makes her presence cause the enemy team to melt quickly. It's especially nice when you have a support buying buffing items. A Soraka and a Kayle can make a team fight very lopsided.

Sion

Sion's changes to ability cost make it hard to argue with AP Sion. The argument I'm going to deliver here is that AD Sion is one of a handful of champs with no upward bound on damage as long as he has an Atma's. This build allows you some of that midgame AP dominance mixed with a punishing AD finish. The longer the game goes the more dangerous you get, and you were already dangerous to begin with.

The key problem with AD Sion is that farming up that AD with the weaker shield damage sucks. You're also placing the bet that you can keep from dying with a weaker champ early for later game dominance.

Once again this rides the fine line between the two and gives a smoother transition from early to late game.

Jungle Sion's build:
Cloth Armor Madred's Razors Mercury's Treads Wriggle's Lantern Sheen Rod of Ages Atma's Impaler Trinity Force Last Whisper

Volibear

Tried this out. Works great. Makes your ult and E feel definitely worth it. Puts you in the same roll as Skarner. I built him with vastly more MR than Armor, because he gets hit with tons of anti-initiation moves. You also need another tank or another tanky dps on the team.

Works best in Dominion. Since you're guaranteed to level in the first fight, the catalyst is a nice chunk of hanging around and punching stuff. Also doable in summoners if you're short on AP and beefyness.

Leona

To quote:End Quote

I used Leona mainly as point back up in dominion. I ran around being the difference maker for a half health defender or pusher on my team. Definitely noticed a difference pushing minions with the W.

Gragas

Gragas becomes an angry dps who gets in the fray more than AP and sticks out of position opponents in place. His ult lets you force them out of position. E becomes stronger than Q which makes for more fun in team fights.

Nidalee - would do weird things to the Javelin Toss / Takedown, but may make Cougar Form stronger.
Jax - It'd take the squishy out the typical Guinsoo's Rageblade + Hextech Gunblade builds.
Maokai
Blitzcrank

11/10
Build Created, formatting added. Tweaked Sion's Skill Sequence with advice from Mastajdog.

11/30
Masteries changed. Volibear added.

12/8
Leona (Dominion) and Gragas added.