I'm grateful for you taking the time to teach me on your day off.
I appreciate that you have decided to teach me on your day off.
To be grateful for something or someone means that you are happy or thankful. It's just like saying thank you, but a little bit more formal. For example: I'm grateful for you taking the time to teach me on your day off. You can also use the word appreciate in a similar way. I appreciate that you have decided to teach me on your day off.
「grateful」は「うれしい(happy)」「感謝している(grateful)」という意味です。「Thank you」とかなり近いですが、「grateful」の方が若干フォーマルです。
例えば:
I'm grateful for you taking the time to teach me on your day off.
〔訳〕お休みに私のレッスンの時間を取って頂き、感謝しています
同様に「appreciate」も使えます:
I appreciate that you have decided to teach me on your day off.
〔訳〕休日にレッスンをして頂き、感謝しています
For this situation, you can show your gratitude by using words like "thank you" and "appreciate". The word "appreciate" shows that you understand they made a sacrifice to teach you, and you really thank them for it.
"free time" is another way of saying their "day off" or "non-working time". It is time they would have to freely do what they want outside of work.
Both of these phrases have a similar meaning. If we have to work on our day off, we usually say that we are giving up our day off. Meaning, we will not longer have our day off due to work. If someone gives up their day off for you, it is usually quite a big gesture so it is polite to thank them.
どちらの表現もよく似た意味があります。休日に働かなければいけないなら、giving up our day offと言います。仕事のせいで、休日がなくなったという意味です。休日出勤はかなり大きなことなので、感謝を表すのは大事です。
It's good of you to give up some of your time off just for me.
If the person was not supposed to work that day, it can be called a rest day. Since they are teaching you on this day, they are working and therefore gave up their rest day for you. They are also helping you, so you can also thank them for that and call it a day off. Lastly, you can express your appreciation for their sacrifice of their time off for your sake.
仕事をしない「休日」のことはrest dayと言えます。休みの日に教えてくれているわけなので、先生はあなたのために休日を返上している(giving up your rest day for me)ということになります。
また、先生はあなたを助けてくれているわけなので、それについて感謝することもできます(helping me out on your day off)。休日はday offとも言います。
最後に、先生があなたのために時間を犠牲にしてくれたことに対して感謝の気持ちを表してもいいです(give up some of your time off just for me)。
Thank you for … で「〜ありがとう」と英語で伝えることができます。
例えば、Thank you for helping なら「手伝ってくれてありがとう」になります。
Thank you for teaching me on your day off.
休みの日に教えてくれてありがとうございます。
Thank you for teaching me on Saturday.
土曜日に教えてくれてありがとうございます。