Ugetsu

Right, most kanji have component parts, and if you can name and understand the component kanji, you can often guess at what the whole kanji means. This is very intentional and really the way the Chinese first came up with these characters. Here by the way is the raw for page one

Kodomo no Jikan 55 RAW Read Online

Sayo did not translate the big characters on the lower left:

夏　"SUMMER" - "natsu" or "ka" -- a head hanging down over "walking legs"
海　"SEA" - "umi" already mentioned
青木先生　"BLUE TREE BEFORE LIFE" -- you (and the Chinese reader) must be puzzled by that sequence but it will become clear in the Japanese pronunciation, "AOKI SENSEI"
実家　"REALITY HOUSE" "jitsu" "ie" -- in other words his real home.

Newnonimous

Yeah i'm getting this now little by little. I really really want to keep learning more of Kanji and especially Japanese but with work and other matters that i have i won't have the time to study them all, it's sad for me. you're a great teacher Ugetsu, i'll keep learning from you when i ever have the spare time, well i am on vacation right now and tomorrow so i'll still drop by ever few hours to see if you have anything else to teach.

Ugetsu

On the next page -- the top panel -- Rin is begging to go to Daisuke's home town, along with Kuro and Mimi, he says she has to get Reiji's permission, and she does. JUST the kanji:

海 SEA
見 SEE
言 SAY this is a wonderful kanji, "mouth" at the bottom and 4 "words" above.
This kanji is part of read, write, talk, etc., a "primitive."
黒 BLACK -- Kuro's name
美 BEAUTY -- Mimi's name
行　GOING (the verb "iku" to go is very common)
保護者 This phrase "Hogosha" means "Guardian" but breaking it up:
保 PROTECT
護 DELIBERATION
者 SOMEONE (this is a very common kanji)
出 EXIT -- this is used on exit signs in Japan. Also in the word "dashi"
BLUE TREE REALITY HOUSE -- see previous message, Aoki's home town
夏　SUMMER
休　REST -- a man leaning against a tree
仕事　Shigoto is the common word for "business" -- breaking it up:
仕　ATTEND
事 MATTER
出 EXIT

You can of course look at Sayo's version to get the whole translation but I think you can also see that, if one knew the kanji, it would be possible to run the eye over the page and get a rather clear view of what was going on.