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Rows in crochet (almost) always start with a turning chain, or a number of chains required to get the yarn up to the proper height for that stitch or stitch pattern.
What is Standard
Single crochet rows start with a chain 1. The chain 1 does NOT count as the first stitch.
Half-double crochet rows start with a chain 2. The chain 2 does count as the first stitch.
Double crochet rows start with a chain 3. The chain 3 does count as the first stitch.
Exceptions to the Rule
The fact we call a way “standard” means that obviously there is another way. I frequently see variations such as a designer who counts the chain 1 in single crochet as the first stitch. I personally prefer to start my half-double crochet rows with a looser chain 1 that does not count as the first stitch- I think it gives a prettier edge.
Where is the Starting Chain in a Pattern
Different pattern writers have different preferences. I like to write the turning chain at the beginning of the row instructions.
Row 2: Ch 3 (counts as dc here and throughout), dc in each st across, turn.
Other designers prefer to write in the turning chain for a row at the end of the previous row.
Row 1: Sc in 2nd ch from the hook and in each ch across. Ch 3, turn.
It’s a matter of personal preference, whether that of the designer or publisher.