"I have six children."
As far as I can tell, 人 is the counter for individual people and not groups of people.
子供 can be either child or children
子供たち means children
This is how I break it down in my head when having to decide what I'm trying to say :
-です (desu) for to be/is
-います (imasu) to affirm someone's existence. (living)
-あります (arimasu) for something's existence (non living)
彼らは私の兄弟です。(karera wa watashi no kyoudai desu) - They are my siblings.
私は兄弟います。(watashi wa kyoudai imasu) - I have siblings.
Hope that trick helps!
I'm no expert, but my understanding is that 子供が六人います can mean either "There are six kids" or "I have six kids."
According to this and other sources, が is the default particle you usually use with います.
I'm not sure that I fully understand the differences between が and は because there are entire books written on the subject. However, I think that は would be used for emphasis in this case. "I have six kids" rather than six dogs or six goats or six of something else.
As there is a counter, I'm not sure if は could be used to emphasize the number. Might 子供が六人います mean "I have six kids" if for example someone thought you had two kids or four kids?
based on my understanding (I'm also a learner so take it with a grain of salt) this wouldn't be correct because 子供 is not the main subject of the sentence but the implied subject 私 (わたし). は is usually used for the primary subject of the sentence. In your sentence the children become the main subject so the sentence is more accurately translated "there are six children" (that's also how google translates it, although it also translates the original sentence from the exercise the same)
私 is background information that would be indicated by the topic particle. I'm sure that's what you meant, but unless there's a conjunction, there wouldn't be two "subjects" in a sentence.
You can think of it as "As for (x)" when translating a phrase "x は." "As for the children, there are six of them" vs. "As for me, there are (I have) six children."
Is adding 私は (or わたしは) to the beginning of the sentence correct here? It's been my understanding that 私 is the implied subject here, but it can be added explicitly if desired. I tried adding it in the website interface but I get a mistake. Wanted to flag "my answer should have been accepted" but I don't really know if I'm right.
Short answer: Because the words are completely disarranged.
Long answer: Maybe you would translate it like "My people six children are." - makes no real sense. In Japanese "(living being[s]) が (number) います" means (besides other meanings) "there are (number) (living being[s])". Number is optional. So "子どもがいます" could mean "I have a child" / "I have children" or just "There is a child" / "There are children". Here a more clear translation would be "私は子どもが六人います" ("I have six children"). But it depends on situation. If someone asks you about your children, it would be clear that you talk about them, so "子どもが六人います" would be a clear answer to that question. Hope I cleared things up somehow :-)
Answered in a comment above. It seems to be that -ta chi makes child into 'a group of children'. Japanese doesn't have plurals in the same we do, so 'a group of children' is the closest thing when Duo looks for a Japanese translation for the English word children. In this context though, using -ta chi would make it more like "I have six groups of children".