I had previously mentioned that my children are very picky and slow eaters. This creates much frustration for me, but at the same time has been on multiple occasions inspiration for interesting math conversations.
Yesterday, Katie was eating a cutlet for dinner, and I, as usual, was trying to make her hurry up a bit.
K: I’m almost done with it.
I look at the cutlet and notice that there is still almost half of it left.
Me: What fraction of the cutlet do you think must be remaining for you to be “almost done with it”?
K: I think a third.
Me: Why a third?
K: Because a third feels so much smaller than a half.
Me: Well, you definitely have more than a third left.
K: (taking a big bite) Not any more.
She went back to chewing, very slowly of course, and I thought the conversation was over. But about 5 minutes later, when she was still not finished with the cutlet, Katie suddenly decided to resume the conversation.
K: Mom, if someone could eat the whole cutlet in just one bite, could they say that they’re almost done with it before they take the bite?
Me: I don’t know. Can you be almost done with something before you even start?
K: Because if they can’t say it before they take the bite, they can never say it, because after taking the bite they will be fully done with the cutlet!
I couldn’t argue with her there.
So what do you think, does “being almost done with a task” depend on how quickly you do it or just on what fraction of it you have already done?
Musings of a Mathematical Mom 
The tortoise and the hare might get Katie’s brain exercised !
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Oh, good idea. Don’t know how I managed not to tell her that one yet.
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I think she outfoxed you.
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She ate the whole cutlet, which was definitely a winning outcome for me.
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What a great conversation! I love both your question (Can you be almost done with something before you start?) and Katie’s thinking that if you ate it in one bite, you could never say you’re almost done. It all comes down to context. Almost is a fascinating word, as I learned earlier this year:
http://exit10a.blogspot.com/2017/03/it-depends-on-meaning-of-almost.html
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What a great lesson, thanks for sharing! I love it questions where the answer is “it depends” because those lead to the best discussions. It’s also wonderful when students find ambiguity where the teacher may not have even seen it. This happened to me recently in a lesson on logic https://aofradkin.wordpress.com/2017/04/25/logical-fun-part-i/
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Depends on how the rate of consumption changes with the proportion remaining!
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So how do we figure out the threshold?
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Think of the following situation. If a procrastinator (any of us :-)) has a project due tomorrow but he did not start it yet. The boss calls, and the guy says “I am almost done”. So psychologically it is a typical situation.
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So you’re saying that it doesn’t depend on how big the project is?
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