WEBVTT 00:00:05.036 --> 00:00:07.976 (Clare Heidema): Here are some problems, five of them. 00:00:08.386 --> 00:00:11.396 There may be other ways you would think about solving them. 00:00:12.006 --> 00:00:19.806 But if you were going to solve them using either one and three-fourths times one-half or one 00:00:19.806 --> 00:00:23.906 and three-fourths divided by one-half, which one would you use? 00:00:24.696 --> 00:00:28.036 (Teacher 1): It feels like multiplication, but now, I'm thinking... 00:00:28.796 --> 00:00:35.806 (Teacher 2): That's multiplication. 00:00:35.856 --> 00:00:36.686 (Teacher 1): Right? 00:00:36.766 --> 00:00:38.846 But does that solve the problem? 00:00:39.026 --> 00:00:40.456 (Teacher 2): Yep. 00:00:40.566 --> 00:00:42.446 (Teacher 1): Three-quarters of a half... 00:00:43.876 --> 00:00:47.956 (Teacher 2): Because it means that right here, of this half, right to here, 00:00:48.426 --> 00:00:53.666 there is one and three-quarters pounds of birdseed. 00:00:54.276 --> 00:00:57.426 Right? No... 00:01:05.406 --> 00:01:08.216 no. One and three-fourths pounds of half a bird feeder. 00:01:09.326 --> 00:01:10.086 Can you put it like that? 00:01:10.086 --> 00:01:14.196 (Teacher 1): But what I want to do is one and three-fourths times two. 00:01:15.316 --> 00:01:20.486 So maybe it is divide because that would be the reciprocal, right? 00:01:20.736 --> 00:01:27.056 (Heidema): Before we talk about all of them, let me just ask, what was an easy one? 00:01:28.736 --> 00:01:29.556 Number four? 00:01:31.026 --> 00:01:32.206 Why was four easy? 00:01:33.186 --> 00:01:34.026 Because it was...? 00:01:34.026 --> 00:01:34.496 (Teacher 3): Area. 00:01:34.646 --> 00:01:37.006 (Heidema): Area, and so what would you do? 00:01:37.006 --> 00:01:39.366 Which one of those calculations would you do for that problem? 00:01:39.366 --> 00:01:39.906 (Teacher 3): Multiply. 00:01:39.906 --> 00:01:41.926 (Heidema): Multiply one and three-fourths times one-half. 00:01:42.446 --> 00:01:45.006 Was there anything else that was fairly easy to think about? 00:01:45.476 --> 00:01:48.156 (Teacher 1): Number five. 00:01:48.156 --> 00:01:49.446 (Heidema): Number five? 00:01:49.696 --> 00:01:51.086 And tell me about number five. 00:01:51.816 --> 00:01:56.826 (Teacher 1): It was, basically, how many groups of half a cup are in one and three-quarter cups. 00:01:57.136 --> 00:02:01.896 (Heidema): So it was fairly easy when it was the lemonade that you had one 00:02:01.896 --> 00:02:07.526 and three-fourths cups of and you were trying to take out servings of size one-half 00:02:07.716 --> 00:02:10.776 that you could think about how many halves there were in. 00:02:11.066 --> 00:02:16.966 And I think, how many times did we hear that as-when you do one and three-fourths divided 00:02:16.966 --> 00:02:20.976 by one-half, it means how many halves are in one and three-fourths? 00:02:22.166 --> 00:02:23.046 What about number one? 00:02:24.036 --> 00:02:28.926 (Teacher 2): I thought that one was fairly simple just because it's telling you to take half 00:02:28.926 --> 00:02:33.406 of a whole already, because it says how many cups of sugar do you need to make half a batch 00:02:33.406 --> 00:02:36.846 of cookies if the full batch takes one and three-fourths cups. 00:02:36.916 --> 00:02:40.166 So I already knew that for the whole thing I needed one and three-fourths, 00:02:40.166 --> 00:02:42.446 so half I would have to split that in half. 00:02:43.156 --> 00:02:46.146 (Heidema): And so which of these two calculations did you use? 00:02:46.736 --> 00:02:48.156 (Teacher 2): The division. 00:02:49.356 --> 00:02:49.796 I divided. 00:02:49.796 --> 00:02:53.536 (Heidema): You divided, one and three-fourths divided by one-half? 00:02:53.786 --> 00:02:56.926 (Teacher): I think you divide it by two but you multiply it by half. 00:02:56.926 --> 00:03:00.956 (Heidema): And so what you have been talking about all along 00:03:00.956 --> 00:03:06.076 with fractions has been a measurement notion of division. 00:03:06.696 --> 00:03:09.586 But there is also with fractions, just like with whole numbers, 00:03:09.836 --> 00:03:13.166 there is what is called a partitive notion. 00:03:13.576 --> 00:03:18.356 It's parts, putting it into-in the case of dividing by two-into two parts. 00:03:18.876 --> 00:03:22.576 If we were dividing by one-half, it just sounds different 00:03:22.576 --> 00:03:24.716 because you can't think so easily about half parts. 00:03:25.826 --> 00:03:33.536 Okay? But that's the kind of thing that you are doing here is a partitive division problem. 00:03:33.536 --> 00:03:39.906 I would like you to write a story problem that's not a "How many two-thirds are there in...," 00:03:40.476 --> 00:03:45.786 not the measurement story type problem, but a partitive problem. 00:03:46.586 --> 00:03:49.826 Four and one-half divided by two-thirds. 00:03:50.476 --> 00:03:53.896 Partly because it was so difficult, that problem, 00:03:53.896 --> 00:03:59.056 in terms of just thinking about it that I thought maybe if we have a collection of them 00:03:59.056 --> 00:04:04.086 that you have thought about that it will sound a little easier, if you like, 00:04:04.086 --> 00:04:08.256 to think about it as a division-dividing by a fraction problem. 00:04:08.656 --> 00:04:09.956 You want to start us off here? 00:04:10.276 --> 00:04:17.116 (Teacher 2): Four and a half cups of flour is enough for two-thirds' batch of cookies. 00:04:17.786 --> 00:04:22.026 How much flour is needed for the full batch? 00:04:24.466 --> 00:04:31.276 (Teacher 4): How many gallons of gas fill a gas can if four 00:04:31.276 --> 00:04:36.436 and half gallons fills a gas can two-thirds full? 00:04:43.916 --> 00:04:48.166 (Heidema): We have something that, as you said, fills a part of it, 00:04:48.396 --> 00:04:49.676 but what do we need for the whole? 00:04:50.026 --> 00:04:55.516 And the part is the two-thirds part; we are going to divide by two-thirds. 00:04:56.266 --> 00:05:03.676 And there is lots of context, lots of things like that, real kind of problem sometimes, right? 00:05:03.676 --> 00:05:08.076 I don't know how real this is, but you could imagine. 00:05:08.136 --> 00:05:13.256 I have got four and a half yards, that's only going to get me two-thirds of the way, 00:05:13.756 --> 00:05:22.666 or if that four and a half gallons only filled up two-thirds of. 00:05:24.036 --> 00:05:24.866 That's a real problem. 00:05:25.056 --> 00:05:29.466 I have gone four and a half miles, and I have only gone two-thirds of the way. 00:05:31.866 --> 00:05:38.426 I have got four and a half cups but that's only going to make two-thirds of my batch. 00:05:39.696 --> 00:05:45.606 So they can be very real-sounding problems.