You sure you’re on radicals? Radicals don’t have readings, per se, though most of them exist as kanji in their own right, which do.

Anyway, meet the two types of readings: kun’yomi and on’yomi. Basically, the Japanese writing system was imported from China by Buddhist scholars way back in the year dot, but prior to that, it already had its own spoken language. Basically, kanji were then used to write down the words that already existed, so the native Japanese reading was given to the kanji – this is called the kun’yomi. However, the kanji already had Chinese readings attached to them, and these were introduced as well – this is called the on’yomi.

When Koichi introduces the kanji, he calls the on’yomi the “reading” – the kun’yomi occasionally comes up in the “vocabulary” section, but it’s never explicitly indicated as such. I’ve not the foggiest idea why he does it like this. As a rough rule of thumb, when a kanji’s standing on its own, or has okurigana (= hiragana ending used to indicate verb/adjective conjugation) then it uses kun’yomi. When it’s in a compound word with other kanji, then it uses on’yomi. This is a VERY rough rule of thumb, though, and there’s about a thousand and one exceptions. You’ll get a feel for it after a while.

In this case, 八 has an on’yomi of はち and a kun’yomi of や. Numbers tend to use on’yomi most of the time, but when attached to the generic counter つ, they always use kun’yomi (though this is only ever used for the numbers one to ten – kun’yomi does exist for higher numbers, but it basically never gets used). For any other counter, you generally use on’yomi (notable exceptions are 人, which uses kun’yomi for 一人 (ひとり) and 二人 (ふたり) only, and on’yomi after that, and 日, which has the absolute freakiest readings which I won’t get into at present to avoid confusing you).

So:

一 (いち) -> 一つ (ひとつ)
二 (に) -> 二つ (ふたつ)
三 (さん) -> 三つ (みっつ)
四 (し／よん) -> 四つ (よっつ) (Because し = 死 = death, よん is often used instead)
五 (ご) -> 五つ (いつつ)
六 (ろく) -> 六つ (むっつ)
七 (しち／なな) -> 七つ (ななつ) (The readings for 七 tend to be fairly interchangeable, dunno why)
八 (はち) -> 八つ (やっつ)
九 (きゅう／く) -> 九つ (ここのつ) (く sometimes gets used because it’s easier to say)
十 (じゅう) -> 十 (とお) (No つ for this one. Dunno why)