Most native English speakers are unaware of R-Controlled Vowels. However, these can be troublesome for Newcomers (ELLs). I always teach R-Vowels as they occur in class. For example, when mother, father, sister and brother are taught, I also teach 'er' says /er/. Likewise, when shirt and skirt are part of a clothing lesson, I teach 'ir' says /er/. A common sentence used to teach five different ways to say /er/ is - Her first nurse works early.
A good warmup sentence to introduce the R-controlled vowels is - The storm hurt her first car. You can draw a car with a tree limb on top of it. I underline all of the combinations and review the pronunciations.
This chart shows the five basic spellings and sounds. I find that students often like to learn each separately, in a systematic manner.
The letter 'w' changes the R-controlled vowel sounds. Students are often familiar with the words: war, word, and work. Since many of my students are refugees, the word 'war' is common in their vocabulary.
So what about the words like 'hear' or 'fire'? 'Hear' has a long vowel team followed by 'r'. Also, 'care' has a silent e long vowel followed by a 'r'. I contrast these by talking about hear vs her and care vs car.
Another little known English rule is when there is a double r - the vowel is usually long. This includes words such as carry, marry, berry, and cherry.