Sup dude! I'm Glacem(writing from my bro's acount). I have mained Anivia for 2 years and she was the champion that just "spoke" to me. I would love to help you! I will list you some tips. But know that some(or most) of them may be very dum and straight forward. You have to keep the mentality that they are very important. After all to make a strong home you need strong fundations.
WARNING: I have writen a novel! It is alot, but it is useful! Please read it. K? Ty!
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Know Anivia! What I mean is that you must know her from the inside out. The basic Q+E combo. The speed of your Q. The speed of Anivia. the auto-attacks(how to CS with them).
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Mana managment! I could add this to the firts tip but it is so important that it is a rule of itself. In the early game you must be semi-conservative the usage of mana. If there are 2-3 minions and you can kill them with one Q, go for it! After level 6 you have to be very mindful and smart with you use of the ultimate. You can wave clear with it, but use only your ulti to do it if you are still in lane. There is an interesing interaction where the ulti can prock one more tick after it is canceled, so learn to use it. To learn the feeling of your mana usage try to play with no blue buff. It is a bit tricky, but it will help.
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Q+E combo! Anivia has one of the largest auto-attack ranges in the game of 600 the E(frost bite) has the same range. If you are against something very slippery it is best to use your Q(FlashFrost) when you are absolutely fucking dead sure that you will land it, if you feel like you won't land it an auto and one E will do the job. Of they come in your face stun and run. If they run throw E, if you are safe to use E that go for it(under tower, near your team, behind a wall). In lane this combo will hurt. Q is best to be trown alittle bit to the side where you think they will go.
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Deathfire Touch! Try it out. It work for me better then thunderlord's. After you try it out pick the one which feel better for you.
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Builds! Some things I do is to rush Seeker's Armguard
against AD like Zed(get
ASAP vs this guy), Yasuo and others. And
when there is a fed AP. Also not always is good to take
. It is situational.
is a must against a heavy tank team.
can be a good option when there is alot of burst or CC(Haven't tested it). I prefer to rush Rod of Ages. Take boots according to the situation.
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The Wall! This is the thing that makes the difference between a good Anivia and a GREAT Anivia. If you are not used to it then the best uses are to wall of the pasts when you are taking objectives and to use it for vision(can work like a ward). More advanced uses are to make the opponent to walk in your Q and to split the enemy team. Also you can use it for catching peaple out of possition. Also(this is important) when you are chasing someone the wall is a good idea, but you must keep in mind that even if the range shows that you can catch them there is the possibility that the wall will be placed in such a way that it will push the enemy forward and you will wall yourself. The wall is mastered with practice. Don't be affraid to use it, but think before you use it.
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Position! In team fight Anivia can be the force that can turn the tide. The cooldown on yor abilities is kinda long so you have to do the best with them. When there is danger for you to get blown up is best to utilize your Q and R to their full potencial. The basics are to set the ulti in a good place and to use your Q for stun or to snipe a carry out of possition(later in the game it may one-shot an ADC). The wall should be used with caution. And when it is dangerous to use you E, just don't. Q and R can do alot of damage on their own.
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Practice! Although that Anivia look very easy and straight forward, she requiers alot of decision making and recource management. Just play. Don't be affraid to make mistakes and just keep going. You will learn. When you can "feel" her then you will learn her.
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The EGG! Anivia's passive is very good in the early game and can save you, get you a kill and tilt your enemies. But do not think that it will save you all the times. Late game it is more or less useless if there is noone to defend you. Baiting with it is very dependant on your opponent. The Egg can tank many things in the early game. Don't to to aggresive and try to use it just to get your Heath back if there is nothing else that you can do(like to kill them used tower or to bait them into a gang).
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Early Game! There you can bully very well. Baiting or just nuking you lane opponent if a good start of a lead or tilting your enemy. After one or two succesful gangs and a good CS score you can do just fine without blue and withuot help. You can clear, kill, push and distract only with a
, boots, doran, tear(if you feel like you need it) and some posts. Try to keep a good CS score, don't get tilted if you are bullied, and remember that Anivia can be useful even when behind. It is better to give up on CS when there is a posibility for you die and feed.
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Mid Game! You damage will look like it is less the in the early game, but this is not true. If you went even with your laner you will still hurt like a truck. Anivia's objective control is very good, so you can go for a 4-man gang onbot and then go for the drake. You can defend the dragon pit extremly well. Personaly after we get the dragon of the enemy had taked the dragon, I find myself defending mid against two, three or even four people. This is no problem. You should have Rod of Ages, tier two boots and some parts of your next item. Rod gives you more then enought sustain to defend mid on your own for a very reasonable time.
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Late Game! Here is abit tricky. If you have tear you will scale very well in damage. If you don't have
it's no problem(in some situations it is better). Anivia scales very well. You still have your objective control and defending capabilities, but it is much harded to keep yourself alive. Unlike in the earlier part of the game you are not that tanky in the late game, you can dishout a massive amount of damage, but the other can do the same to you. When behind Anivia can stall the game so well that the only way to push against her is eather baron or killing her. Be safe, possition well, do your best to wall off people and to control the fight and do not let to be picked off. Tanks will recieve alot of damage from a Liandry powered Glacial Storm, and squishies can be insta-bursted. Do NOT get cocky.