Your dog has another neighborhood dog that she's friends with. You set up a time with your neighbor for the dogs to play together. Your dog was excited about it when you came. You're telling your neighbor about it:
She kind of perked up when I mentioned Hannah's name.
When someone suddenly sees or hears something that interests them, they "perk up": they open their eyes more and pay closer attention to what's happening.
You might "perk up" in a conversation when someone talks about a topic that you're interested in.
This means to talk about something briefly. Even if a person says only a few words about a topic, that would be considered "mentioning" that topic. Other words like "say", "talk about", etc. sound like they require a longer period of discussion.
You can use "mention" with a simple noun:
Did she mention me in the article?
Or you can use a dependent clause:
You can also use the structure "mention (doing something)"
He mentioned going to the gym earlier in the day.
I feel kind of obligated to go.
You look a little sick. Are you OK?
It's somewhat smaller than I thought it would be.
But only "kind of" sounds right before a verb, and only in casual conversation:
I kind of wanted to go with them.
People sometimes pronounce "kind of" in a way that sounds like "kinda".
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