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1 0:00:00.919,0:00: this is 0:00:05.630,0:00: common core state standards support video for mathematics 0:00:09.259,0:00: standard five n f 0:00:11.019,0:00: four a this standard reads 0:00:13.349,0:00: uh... apply and extend previous understandings of multiplication 0:00:16.720,0:00: to multiply a fraction or a whole number by a fraction 0:00:19.710,0:00: part a 0:00:21.230,0:00: interpret the product 0:00:22.580,0:00: a over b times q 0:00:24.279,0:00: as a parts of a partition of q into b equal parts 0:00:29.079,0:00: equivalently 0:00:30.179,0:00: as the result of operations a times q divided by b now this isn't a very long 0:00:36.000,0:00: standard compared to some others 0:00:38.670,0:00: but there's a whole lot to it

2 0:00:40.680,0:00: first of all it's about fractions 0:00:43.190,0:00: there's a whole lot of variables in here 0:00:46.210,0:00: and the reality is there's a lot more under the surface to this 0:00:50.590,0:00: then you would think at first 0:00:52.300,0:00: so let's take the first part this whole idea of 0:00:55.460,0:00: this product a over b times q 0:00:58.350,0:00: being 0:00:59.500,0:01: a parts 0:01:00.649,0:01: of a partition of q 0:01:02.820,0:01: into b equal parts 0:01:05.180,0:01: again it's a little bit confusing because of the variables 0:01:07.880,0:01: so let's start with our quantity of q and let's take the easiest possible scenario 0:01:12.940,0:01: let's let q be one 0:01:14.779,0:01:17.750

3 and so now we want to take 0:01:17.750,0:01: our one 0:01:19.170,0:01: and partition it into b equal parts and let's let the b be uh... be four 0:01:25.000,0:01: so we cut it up into four equal parts 0:01:27.340,0:01: and now we have some quantity a of those equal parts 0:01:31.890,0:01: and let's let a be three 0:01:34.050,0:01: so basically 0:01:35.509,0:01: what we have here from our a over b times q is 0:01:38.890,0:01: again we started off with one 0:01:41.350,0:01: we split it up into four parts 0:01:43.920,0:01: and we wanted three of them so we end up with three-fourths times one 0:01:48.189,0:01: which a lot of time we'd think of as 0:01:50.220,0:01: three-fourths 0:01:51.009,0:01: of one in plain english 0:01:53.480,0:01:57.180

4 and again this is the simplest scenario because the whole was one 0:01:57.180,0:01: now let's try a different scenario 0:01:59.210,0:02: what if the q itself was a proper fraction 0:02:02.229,0:02: in other words q is a fraction between zero and one so let's take this example 0:02:07.300,0:02: we want two-fifths times two-thirds 0:02:10.129,0:02: or in plain english two-fifths of two-thirds 0:02:13.659,0:02: so we first need to partition the two-thirds into five equal parts again 0:02:18.040,0:02: the two-thirds is the q 0:02:20.199,0:02: and 0:02:20.819,0:02: five is the b 0:02:24.249,0:02: so let's look at this visually so it can make a little bit more sense 0:02:27.829,0:02: so we're starting off with two-thirds 0:02:30.209,0:02: which is this here 0:02:31.949,0:02: but we need to split that up into five

5 equal parts 0:02:35.319,0:02: so let's do that 0:02:36.520,0:02: okay so there's 0:02:37.859,0:02: uh... one fifth 0:02:39.079,0:02: two fifths three-fifths 0:02:41.089,0:02: four-fifths which also takes care of the...fifth fifth 0:02:45.299,0:02: now our task is 0:02:47.099,0:02: well we need two 0:02:48.929,0:02: out of those parts 0:02:51.959,0:02: so what we need to do then okay well here's 0:02:55.649,0:02: one of those 0:02:56.959,0:02: parts here is the second of those parts 0:02:59.629,0:03: the third and so forth but we just need two of those 0:03:03.769,0:03: so here they are 0:03:05.319,0:03:06.489

6 so this is one 0:03:06.489,0:03: one part 0:03:07.629,0:03: and this is another part 0:03:10.049,0:03: now how do we determine the solution 0:03:12.309,0:03: well in a scenario like this we can actually just get the answer by counting 0:03:16.849,0:03: notice that uh... we extended these blue lines all the way over 0:03:22.239,0:03: to make a little bit more sense out of it and we can tell now that we have one two 0:03:25.969,0:03: three this way one two three four five this way 0:03:29.509,0:03: so we have a total of fifteen parts that we cut it into 0:03:33.699,0:03: and 0:03:34.530,0:03: we only want one two three four of these smaller rectangles four parts 0:03:40.249,0:03: so our solution is four-fifteenths 0:03:42.599,0:03: now it might be difficult for students to do 0:03:45.979,0:03:49.769

7 all those sub-divisions especially trying to get them into equal parts and 0:03:49.769,0:03: everything 0:03:50.729,0:03: so here's something that you can do to make it a lot simpler you can make some 0:03:54.219,0:03: fraction manipulatives 0:03:56.449,0:03: you know to find the solutions so 0:03:59.650,0:04: take some ah 0:04:00.770,0:04: some clear plastic some clear sheets 0:04:03.789,0:04: and what you want to do is make congruent squares and that's important 0:04:07.239,0:04: in other words all you squares need to be the same size 0:04:10.969,0:04: so then of course you need to have one for one-half you need another set for your thirds 0:04:15.209,0:04: you would have your one third and two thirds 0:04:18.449,0:04: you have a set for your fourths 0:04:20.110,0:04: you would have your one-fourth two-fourths and three-fourths 0:04:23.449,0:04: and of course you would continue on

8 0:04:25.580,0:04: and make the rest for your 0:04:28.469,0:04: fifths your sixths your sevenths and so forth 0:04:30.910,0:04: probably up to about uh... the tenths 0:04:33.470,0:04: so let's see how this works 0:04:35.530,0:04: let's say we wanted two-thirds times three-fourths 0:04:38.470,0:04: so this means that our q is three-fourths we're starting off with three-fourths 0:04:42.740,0:04: then 0:04:43.510,0:04: we find 0:04:44.830,0:04: our other manipulative that's two-thirds 0:04:48.349,0:04: so now what we need to do is 0:04:52.740,0:04: take the two-thirds and rotate it around to where this one is horizontal 0:04:57.120,0:05: compared to the original the other one being vertical 0:05:00.030,0:05: then what we need to do is 0:05:01.800,0:05: just slide it over

9 0:05:03.330,0:05: and just put one over the other so notice 0:05:06.409,0:05: the result now is 0:05:10.649,0:05: this situation 0:05:12.330,0:05: and so our solution is going to be 0:05:14.879,0:05: where the shaded areas overlap 0:05:17.729,0:05: and notice again here was our three-fourths but we only want two-thirds of that which is 0:05:22.599,0:05: indicated 0:05:23.770,0:05: this way 0:05:25.260,0:05: so two-thirds of three-fourths 0:05:27.760,0:05: would be right here 0:05:29.300,0:05: and again how do we get our solution well again we can do it just by 0:05:32.219,0:05: counting 0:05:33.380,0:05: we had let's see one two three four one two three that's twelve 0:05:37.249,0:05: so we have twelve total parts

10 0:05:39.560,0:05: and our solution the common shaded areas one two three four five six 0:05:44.249,0:05: so out solution is six over twelve now one thing that's important to notice 0:05:48.050,0:05: here 0:05:48.940,0:05: note that in this process when we put one manipulative over the other 0:05:53.739,0:05: we now just have one square manipulative 0:05:57.839,0:06: so notice that now we just have one 0:06:00.729,0:06: as our whole we really have six-twelfths of 0:06:03.410,0:06: uh... one 0:06:04.830,0:06: and again that was one of the main reasons the ah... really the critical reason 0:06:09.229,0:06: to make sure that your manipulatives these 0:06:12.360,0:06: uh... sheets 0:06:14.169,0:06: with the shaded areas 0:06:15.860,0:06: are congruent squares are the same size squares so that when we

11 0:06:19.720,0:06: rotate one it'll fit exactly over the other 0:06:22.739,0:06: and when you do that when one is over the other 0:06:25.650,0:06: again your whole is now one because you just have one square 0:06:30.629,0:06: now if the students keep doing these types of 0:06:33.419,0:06: problems over and over and they're going to start seeing a pattern 0:06:37.359,0:06: they're going to start seeing the pattern that 0:06:39.949,0:06: well like for this one 0:06:42.289,0:06: well look it looks like we multiplied the twos to get the four 0:06:45.689,0:06: and in the 0:06:46.540,0:06: denominator it looks like we multiplied the 0:06:48.759,0:06: five and the three to get the fifteen 'cause the same thing worked here on this other 0:06:52.689,0:06: example two times three is six 0:06:55.099,0:06: and on the bottom on the denominator three times four is twelve

12 0:06:58.710,0:07: so then they're going to see 0:07:00.440,0:07: in general that they multiply the numerators and they multiply the 0:07:04.300,0:07: denominators to get their solution 0:07:06.769,0:07: which of course is your standard algorithm for multiplying two fractions 0:07:11.439,0:07: so they'll have that down 0:07:13.219,0:07: by using these manipulatives and by figuring it out on their own 0:07:16.819,0:07: now there's one scenario that we haven't covered yet 0:07:19.910,0:07: what happens if your q 0:07:22.719,0:07: is something 0:07:23.650,0:07: bigger 0:07:24.340,0:07: than one we've covered one we've covered 0:07:27.139,0:07: uh... the situation where q was a fraction but we haven't covered this one 0:07:30.389,0:07: yet 0:07:30.970,0:07:34.470

13 if q is something bigger than one it might be easier to think of q as a 0:07:34.470,0:07: set instead of a whole it's 0:07:36.939,0:07: easier to think of a set of three 0:07:38.520,0:07: instead of a whole of three but 0:07:41.199,0:07: they are synonymous though your 0:07:44.059,0:07: your set here is your whole so three is the whole 0:07:47.130,0:07: we began with this quantity q which was three 0:07:50.400,0:07: and let say we need to cut that 0:07:52.180,0:07: uh... quantity of three up into twelve parts that's our b 0:07:55.859,0:07: so we've done that here 0:07:58.209,0:08: but noticed that we have to do it by cutting up each individual circle into 0:08:01.859,0:08: four part to get 0:08:03.499,0:08: twelve 0:08:04.610,0:08: for the total

14 0:08:05.919,0:08: and now we need some quantity a of those equal parts 0:08:09.380,0:08: let's let that be five 0:08:11.479,0:08: so here's 0:08:12.870,0:08: our situation so we had uh... 0:08:16.389,0:08: q was three 0:08:18.089,0:08: we cut that up into twelve parts 0:08:20.729,0:08: and we wanted five of them so we have five-twelfths times three 0:08:24.889,0:08: and 0:08:26.150,0:08: we've already figured out the standard algorithm 0:08:29.300,0:08: so we convert that three to be three over one to make it ah fit 0:08:33.460,0:08: and so we multiply the numerators multiply the denominators and we get fifteen 0:08:37.360,0:08: over twelve there's a little bit of a problem here 0:08:40.710,0:08: i don't see 0:08:41.849,0:08:43.450

15 fifteen anywhere 0:08:43.450,0:08: but if we simplify the fifteen over twelve to five over four okay i can see 0:08:47.400,0:08: that 0:08:48.390,0:08: i see 0:08:49.770,0:08: one two three four five-fourths over here 0:08:53.210,0:08: now let's check out what happened we started off with five-twelfths times three 0:08:57.340,0:08: five-twelfths of three 0:08:59.320,0:09: and the whole was our set of three 0:09:02.280,0:09: but then 0:09:03.840,0:09: we multiplied it we got fifteen over twelve 0:09:06.930,0:09: which was equivalent to five over four 0:09:12.140,0:09: here is the understanding here is the heart of the matter 0:09:15.030,0:09: the heart and soul of this 0:09:17.160,0:09: this idea of anything

16 0:09:20.000,0:09: can be expressed as something times one now the situation has changed to where 0:09:24.920,0:09: when we're dealing with our product our solution 0:09:28.110,0:09: the whole is one 0:09:29.579,0:09: it's not three anymore so let's review this one more time we started off with 0:09:34.439,0:09: five-twelfths times three 0:09:35.890,0:09: and our whole was a 0:09:36.910,0:09: set of three 0:09:39.620,0:09: then when we got our solution 0:09:42.000,0:09: five-fourths 0:09:43.850,0:09: that's really five-fourths times one so that changed to where our whole 0:09:48.670,0:09: for that is one 0:09:50.930,0:09: which would look 0:09:51.829,0:09: something like this 0:09:56.090,0:10: but looking at it this way our answer of five over four makes sense because again here's

17 0:10:00.280,0:10: four of them there's one more so there's five-fourths 0:10:04.640,0:10: so again what happened was 0:10:07.420,0:10: the whole for the resulting product is one 0:10:09.889,0:10: not what you started off with 0:10:12.080,0:10: so if we look at the examples that we've done 0:10:15.130,0:10: uh... 0:10:16.010,0:10: two-thirds 0:10:17.640,0:10: times three-fourths being six-twelfths and five-twelfths times three being 0:10:21.430,0:10: fifteen-twelfths 0:10:23.580,0:10: but again we ended up with 0:10:26.870,0:10: a whole of one 0:10:29.060,0:10: so in essence multiplication that involves a fraction results in changing 0:10:33.970,0:10: the original expression or context 0:10:36.470,0:10:39.610

18 to one where the resulting product will be an expression where the whole is 0:10:39.610,0:10: one 0:10:41.470,0:10: in other words let's say that you had two-thirds of three-fourths of a gallon 0:10:47.110,0:10: a student could say 0:10:48.630,0:10: that's how much you have you have two-thirds of three-fourths of a gallon 0:10:52.400,0:10: but that doesn't 0:10:54.920,0:10: make sense at least it's hard 0:10:56.829,0:10: to see that 0:10:58.060,0:11: but if we convert when we would do the multiplication now it makes it to where 0:11:01.910,0:11: i can visualize it 0:11:03.570,0:11: with one gallon as the whole so i would have six-twelfths of a gallon which of course is 0:11:07.689,0:11: half of a gallon 0:11:10.220,0:11: so that's what happens when converting this expression 0:11:13.720,0:11: to something with respect to one being

19 a whole 0:11:17.050,0:11: now let's look at another scenario 0:11:19.320,0:11: where our q is something bigger than one 0:11:22.970,0:11: ah let's let uh... 0:11:25.170,0:11: q be a four 0:11:27.320,0:11: and let's say we wanted two-thirds of that we have two-thirds times four 0:11:32.650,0:11: which of course we know is eight-thirds knowing our standard algorithm 0:11:36.710,0:11: but there's a lot more to this 0:11:38.710,0:11: so let's look at our standard and follow what it says so we have to begin with 0:11:41.960,0:11: the quality of four 0:11:43.450,0:11: and we need to partition 0:11:45.380,0:11: that 0:11:46.019,0:11: four into three equal parts 0:11:48.860,0:11: now we have a little bit of a problem 0:11:51.250,0:11:56.570

20 it's very difficult to look at this and say well 0:11:56.570,0:11: there's four rectangles here 0:11:58.290,0:12: but where in the world 0:12:00.030,0:12: do i draw my lines or or whatever 0:12:04.240,0:12: to split this up into three equal parts 0:12:06.510,0:12: i don't see it 0:12:07.590,0:12: well there's three equal parts 0:12:09.320,0:12: well what would happen if we take 0:12:11.730,0:12: each of those and cut them up into three equal parts 0:12:14.770,0:12: uh... maybe now there is something we can deal with here 0:12:17.350,0:12: so now i've got twelve of those smaller parts 0:12:20.900,0:12: and i want to split them up 0:12:22.940,0:12: three ways 0:12:24.310,0:12: and of course twelve divided by three is four

21 0:12:26.570,0:12: ah so now i can see that 0:12:29.480,0:12: what i can do then is 0:12:31.710,0:12: section this off in chunks of four 0:12:34.690,0:12: alright 0:12:38.300,0:12: so that's where i would draw my lines to split it up into three parts 0:12:42.380,0:12: and if we were to do some shading 0:12:44.520,0:12: this is what it would look like 0:12:46.290,0:12: where here is 0:12:47.360,0:12: the first of the three parts 0:12:49.109,0:12: there's the second of the there parts 0:12:51.060,0:12: and here is the third of the three parts 0:12:54.860,0:12: then of course now with this 0:12:56.139,0:13: now with the shaded drawing i can figure out my solution i have a whole 0:13:00.220,0:13: one here another whole one there and two thirds of another here because this is where i had 0:13:04.250,0:13:05.880

22 to stop right here 0:13:05.880,0:13: uh... so i've got two and two-thirds or eight over three like we figured out earlier 0:13:10.160,0:13: with the standard algorithm 0:13:11.850,0:13: but we still have this problem of 0:13:14.090,0:13: what happens when we have a set and it's very difficult to split it up into 0:13:18.700,0:13: the parts 0:13:19.760,0:13: that they're asking for 0:13:21.220,0:13: well let's try this 0:13:23.740,0:13: what would happen if we were to take the... what the standards says...the commutative 0:13:28.060,0:13: property 0:13:29.500,0:13: and reverse it 0:13:30.880,0:13: look at in terms of q times a over b 0:13:34.260,0:13: so we take the example that we were just working with the two-thirds times four 0:13:38.360,0:13: let's reverse it let's make it four times two-thirds

23 0:13:41.830,0:13: now keep in mind that for the two-thirds the whole is one it's just a regular 0:13:45.590,0:13: two-thirds like we are used to thinking about it 0:13:47.820,0:13: so 0:13:48.620,0:13: this would be one two-thirds 0:13:51.160,0:13: but in this scenario we've got four of them 0:13:53.450,0:13: so there's our four so we have four two-thirds 0:13:56.600,0:13: and to get the solution 0:13:58.840,0:14: really all we have to do is do a little bit of a 0:14:01.640,0:14: imaginative moving 0:14:03.730,0:14: and just uh... slide this over here take my purple parts and move them over here 0:14:09.080,0:14: my salmon-colored and move them over here 0:14:11.680,0:14: and then my last two-thirds and move it over here 0:14:15.040,0:14: and viola i can get my solution i've got a whole one here a whole one there and

24 0:14:19.260,0:14: two-thirds of another so iv'e got two and two-thirds or eight-thirds 0:14:22.990,0:14: and notice that i don't need this one anymore so i'll just x that out 0:14:27.210,0:14: what about the other example that we did the five-twelfths time three with the 0:14:31.120,0:14: circles 0:14:32.230,0:14: uh... what happens when we reverse that 0:14:34.870,0:14: and think of it as three five-twelfths 0:14:38.990,0:14: well here's what it look like visually 0:14:41.130,0:14: i got uh... one five-twelfths another five-twelfths here and another 0:14:45.010,0:14: five-twelfths over here 0:14:46.940,0:14: and just like we did with the rectangles it just takes a little bit of 0:14:49.910,0:14: imaginative moving 0:14:51.660,0:14: and so we take a this five-twelfths 0:14:54.139,0:14: and move it here 0:14:55.640,0:14: take the blue-shaded five-twelfths and move them

25 over here 0:14:59.440,0:15: and it's a little tougher with his last five-twelfth but it's ok we take it and 0:15:03.690,0:15: we move ah two of them here and the other three there 0:15:08.800,0:15: so there we have it we have a whole 0:15:11.540,0:15: one here and three-twelfths of another one 0:15:15.200,0:15: and we get our one and three-twelfths which would be one two three and 0:15:19.090,0:15: twelve and three more that's fifteen-twelfths 0:15:22.210,0:15: and again we don't need this one anymore 0:15:24.580,0:15: okay we can't neglect this last part of the standard 0:15:28.550,0:15: the equivalency as a result of operations a times q divided by b 0:15:35.660,0:15: so 0:15:36.509,0:15: a over b times q 0:15:38.350,0:15: now what happens here as a result of operations

26 0:15:42.420,0:15: a times q all over b notice that we just made this into a fraction bar 0:15:47.720,0:15: instead of the division sign to make it uh... equivalent to what we're dealing 0:15:51.970,0:15: with as far as the representation 0:15:55.020,0:15: so what happens here what's the difference 0:15:58.100,0:16: well let's take that example we just worked with the one with a circles 0:16:01.890,0:16: uh... five-twelfths times three 0:16:04.200,0:16: uh... equals 0:16:05.209,0:16: five times three all over twelve okay to go back and review this is what 0:16:10.009,0:16: happened 0:16:11.110,0:16: okay we start off with three 0:16:13.840,0:16: and we have to split up into twelve equal parts okay so we've done that 0:16:19.530,0:16: and then we want uh... five of those parts 0:16:23.480,0:16: uh... so here's one two three four so there's

27 0:16:25.749,0:16: five of those parts out of that set of three 0:16:28.380,0:16: and of course this diagram applies to the left side of the equation so now let's 0:16:33.220,0:16: look at the right hand side of the equation the original equation 0:16:36.790,0:16: five times three all over twelve 0:16:39.430,0:16: now if we do the standard algorithm 0:16:41.470,0:16: uh... we get fifteen over twelve 0:16:43.670,0:16: now that fifteen over twelve you know the numerator indicates how many 0:16:46.930,0:16: of this 0:16:48.910,0:16: bottom number that we have so it indicates that we have fifteen of these 0:16:53.810,0:16: the problem i'm having is that 0:16:55.580,0:16: where is the fifteen 0:16:56.920,0:16: i don't see it plus 0:16:59.220,0:17: unless otherwise indicated the whole for any fraction is a over b is one

28 0:17:04.010,0:17: and i'm dealing with three circles here not one so 0:17:07.300,0:17: there's some confusion in the problem here 0:17:10.829,0:17: so this diagram doesn't 0:17:12.940,0:17: fit here it doesn't work 0:17:15.680,0:17: so let's go back and look at our standard 0:17:18.269,0:17: and lets redo this side 0:17:20.330,0:17: to fit what the standard says as far as interpretation so now 0:17:24.580,0:17: this says 0:17:26.270,0:17: that i've got a whole of one 0:17:28.770,0:17: that must be partitioned into twelve parts 0:17:31.980,0:17: so we do that 0:17:34.220,0:17: we've partitioned it and now we need fifteen of those parts 0:17:39.070,0:17: so oh wait a minute there is only twelve that i can get out of that 0:17:43.120,0:17:46.890

29 so i need another circle and so there's 0:17:46.890,0:17: three more 0:17:48.250,0:17: and so that gives us our total of fifteen and connecting back to what we had 0:17:52.680,0:17: originally 0:17:54.150,0:17: again we don't need this third circle because we don't have a set of three 0:17:58.470,0:18: anymore we have a set of one so even though this was a short standard as far 0:18:04.600,0:18: the length 0:18:05.730,0:18: there was a lot to it there was a lot under the surface but hopefully this 0:18:09.040,0:18: will clarify this 0:18:10.720,0:18: and we try to dig a little bit deeper and go way beyond just the computation 0:18:15.179,0:18: because the students need that 0:18:16.870,0:18: uh... because again they need to really understand what's going on 0:18:20.500,0:18: with the models 0:18:22.510,0:18:23.740

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