0:00:07.150,0:00: :00:08.880,0:00: this is common core state standards support video in mathematics
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- Imogen Stewart
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1 0:00:07.150,0:00: :00:08.880,0:00: this is common core state standards support video in mathematics 0:00:12.679,0:00: the standard is three O A point nine 0:00:15.990,0:00: this standard reads identify arithmetic patterns including patterns in the 0:00:20.289,0:00: addition table or multiplication table 0:00:23.119,0:00: and explain them using properties of operations 0:00:26.599,0:00: since the standard specifically mentions 0:00:29.919,0:00: the addition and multiplication tables 0:00:32.240,0:00: let's go ahead and start with that 0:00:34.840,0:00: let's look at the addition table first 0:00:38.240,0:00: one thing the students might notice is that across these diagonals 0:00:41.580,0:00: the numbers are the same 0:00:44.540,0:00: so for example 0:00:46.080,0:00: for the 0:00:47.140,0:00: fives 0:00:48.920,0:00: if we read from left to right 0:00:52.370,0:00: then this five comes from one plus four
2 0:00:56.390,0:00: this five comes from two 0:00:59.180,0:01: plus three 0:01:00.510,0:01: uh... this five comes from three plus two 0:01:03.610,0:01: and then this last five comes from four plus one 0:01:06.880,0:01: if we were to take ah... 0:01:08.560,0:01: this situation 0:01:10.020,0:01: these different combinations and use some manipulatives 0:01:13.180,0:01: if we were to take one from this group 0:01:16.080,0:01: and move it over to here 0:01:18.080,0:01: we have changed the one plus four to two plus three 0:01:21.800,0:01: now mathematically something else happened here and it's a little bit 0:01:25.010,0:01: abstract 0:01:26.170,0:01: what happened here we started off with 0:01:28.970,0:01: four we took one away from 0:01:30.610,0:01: this group over here 0:01:32.670,0:01: and moved it over here so in essence
3 0:01:35.650,0:01: we added one to this side 0:01:38.020,0:01: now notice that 0:01:40.310,0:01: when we subtract one and add one the net result is zero 0:01:44.979,0:01: mathematically 0:01:46.590,0:01: this is our additive inverse that anything plus its opposite is zero 0:01:51.630,0:01: so there's a net 0:01:52.700,0:01: change of zero so our sum is still five 0:01:55.810,0:01: but now we have two plus three instead of one plus four 0:01:58.590,0:02: next if we look at the diagonal with the nines we have pretty much the same pattern 0:02:03.250,0:02: uh... we start off with one plus eight and then two plus seven and so forth 0:02:08.269,0:02: and if we do the same thing 0:02:10.259,0:02: uh... what happens again is that 0:02:12.779,0:02: if i were to take one from this group and move it over here 0:02:16.159,0:02: i've changed the one plus eight to two plus seven 0:02:19.330,0:02: now when we move one 0:02:21.709,0:02: from one group to the other
4 0:02:23.359,0:02: we did basically the same thing as we did before 0:02:26.559,0:02: we subtracted one 0:02:28.199,0:02: from the right 0:02:29.609,0:02: side group 0:02:30.969,0:02: and we added it 0:02:33.779,0:02: to the left group 0:02:35.269,0:02: and so our result is two plus seven instead of one plus eight 0:02:38.419,0:02: let's look at the other diagonals 0:02:40.709,0:02: the pattern here seems to be that 0:02:44.509,0:02: each one down the diagonal is 0:02:46.299,0:02: is two larger than the previous one 0:02:50.040,0:02: so let's see the five comes from four plus one 0:02:53.309,0:02: the seven comes from five plus two 0:02:55.849,0:02: the nine comes from six plus three 0:02:58.349,0:03: and so forth 0:03:00.189,0:03: now let's see what ah... 0:03:01.439,0:03: happened here
5 0:03:02.629,0:03: to get from four to five 0:03:04.809,0:03: we had uh... 0:03:05.730,0:03: add one 0:03:06.939,0:03: and to go from the one to the two 0:03:09.719,0:03: we had to add one 0:03:12.459,0:03: now notice 0:03:14.679,0:03: that the increase in the sum was two 0:03:17.689,0:03: which is exactly 0:03:19.389,0:03: what we have 0:03:23.349,0:03: notice that what we're doing 0:03:26.629,0:03: we're establishing the foundation for the basic idea that what you do to one side 0:03:30.999,0:03: of an equation you add to the other 0:03:32.899,0:03: and that's exactly what we did here 0:03:35.069,0:03: we added two to the right hand side 0:03:37.809,0:03: and we added two to the left hand side but we did it 0:03:41.020,0:03: in increments 0:03:42.179,0:03: of one and one
6 0:03:45.219,0:03: and the process will repeat itself 0:03:47.859,0:03: for the next pairings 0:03:49.540,0:03: where again to get from the 0:03:51.139,0:03: five to the six we have to add one 0:03:53.789,0:03: and to get from the two to the three you had to 0:03:57.039,0:04: add one and that resulted in a net change of two 0:04:00.389,0:04: let's look at the addition table one more time 0:04:03.379,0:04: now let's look diagonally 0:04:05.409,0:04: and let's say this pairing and that pairing 0:04:07.739,0:04: so that is 0:04:08.989,0:04: six plus six 0:04:10.480,0:04: and the other one is five plus seven 0:04:12.729,0:04: we can use that same basic idea to get from the six 0:04:16.329,0:04: to the five 0:04:17.649,0:04: we had to 0:04:19.359,0:04: subtract one 0:04:20.699,0:04: and to get from the six to the seven 0:04:23.199,0:04:24.860
7 we had to add one 0:04:24.860,0:04: and notice again that's a net change of zero 0:04:27.509,0:04: and that's why 0:04:28.780,0:04: the sum in both cases is still twelve 0:04:31.139,0:04: what about other pairings 0:04:33.389,0:04: what about this one here six and ten 0:04:35.599,0:04: and eight and eight 0:04:37.449,0:04: same basic pattern 0:04:38.910,0:04: uh... they're both sixteen 0:04:40.759,0:04: and again 0:04:42.759,0:04: the change would result from adding two to the six 0:04:46.050,0:04: and subtracting two from the ten to get the eight 0:04:49.669,0:04: the plus two and the minus two subtracting two 0:04:53.569,0:04: uh... is a net change of zero resulting in the same sum of sixteen 0:04:57.479,0:05: let's look at our table one more time 0:05:01.070,0:05: what about this pairing here and that pairing 0:05:03.379,0:05: we have
8 0:05:04.569,0:05: five plus eight 0:05:06.009,0:05: and six plus seven 0:05:07.600,0:05: well let's see they're both thirteen 0:05:09.770,0:05: same basic idea we would have to add one to the five to get the six we would have to 0:05:14.020,0:05: subtract one from the eight to get to seven 0:05:16.569,0:05: and again we have a plus one and a minus one, net change of zero 0:05:21.269,0:05: so we still have the same sum of thirteen 0:05:23.520,0:05: a pattern that students might notice is that 0:05:25.819,0:05: some of these combinations are repeated in several places 0:05:29.289,0:05: in the addition table 0:05:30.529,0:05: what about other patterns what about what happens when we add even and odd 0:05:34.109,0:05: numbers 0:05:35.279,0:05: uh... even plus even what happens there if we add an even plus an odd number what happens 0:05:39.729,0:05: there 0:05:40.439,0:05: or an odd plus and odd 0:05:42.219,0:05: now of course the first thing we need to
9 make sure here is that uh... students 0:05:46.319,0:05: uh... understand the difference between even and odd numbers they need to 0:05:50.629,0:05: know the definitions 0:05:52.050,0:05: now students at this level are probably going to focus on examples to prove why a 0:05:56.410,0:05: pattern works 0:05:57.799,0:06: now mathematically that's not really a proof but then at this level that's 0:06:01.319,0:06: okay 0:06:02.259,0:06: but they still need to understand that 0:06:04.179,0:06: again that uh... 0:06:05.659,0:06: later on in pure mathematics that 0:06:08.059,0:06: just providing examples is not really a proof 0:06:10.860,0:06: but for now it's ok 0:06:12.409,0:06: so let's take even plus even 0:06:15.520,0:06: huh... it looks like um... 0:06:17.240,0:06: the sums are coming out even 0:06:19.180,0:06: now any even number 0:06:20.979,0:06: can be broken down
10 0:06:22.529,0:06: to a sum of twos because any even number is divisible by two 0:06:26.369,0:06: so we take this example here we can break the four to two plus two we can take the 0:06:30.580,0:06: ten 0:06:31.459,0:06: and break it down to the sum of these 0:06:33.340,0:06: five twos 0:06:35.299,0:06: and when we add them all together we have a bunch of twos that come out to be fourteen 0:06:40.549,0:06: if we take a more generic case 0:06:42.550,0:06: the first even number would be a bunch of twos 0:06:45.129,0:06: the second even number would be another bunch of twos 0:06:48.180,0:06: and the end result is a whole bunch of twos added together 0:06:51.319,0:06: which should still be an even number 0:06:53.179,0:06: what about even 0:06:54.539,0:06: plus odd 0:06:56.400,0:07: hmmm the pattern seems to be that the sums are coming out odd 0:07:01.819,0:07: if we take one example and break it down to see why 0:07:05.729,0:07:09.089
11 well let's see the even number four breaks down to two plus two 0:07:09.089,0:07: the odd number seven 0:07:10.799,0:07: breaks down to two plus two plus two but we have a plus one at the end 0:07:14.860,0:07: when we combine them all together we get eleven 0:07:18.280,0:07: so again we have that plus one at the end so it looks like the answer's again going to come out odd 0:07:24.870,0:07: let's take the generic case 0:07:26.930,0:07: our even number would be a bunch of twos 0:07:29.099,0:07: our odd number would be a bunch of twos 0:07:31.439,0:07: hmmm plus one at the and 0:07:33.449,0:07: and we combine them all together 0:07:35.430,0:07: we have all the twos from the even number 0:07:37.580,0:07: all of the twos from the odd number plus one at the end 0:07:41.039,0:07: which is still going to give us 0:07:42.530,0:07: an odd number for the sum 0:07:44.229,0:07: what about an odd number plus another odd number 0:07:47.470,0:07: the pattern seems to be 0:07:49.530,0:07:52.499
12 hmmm even 0:07:52.499,0:07: let's jump straight to the generic case the general case 0:07:56.219,0:08: uh... my first odd number would be so many twos plus a one 0:08:00.020,0:08: my second odd number 0:08:01.629,0:08: would be so many twos plus one at the end 0:08:04.759,0:08: now notice that we can take 0:08:06.979,0:08: uh... each of those ones and combine them together to be another two so 0:08:10.490,0:08: now we have 0:08:11.719,0:08: the sum of nothing but twos 0:08:15.050,0:08: which should result in an even number 0:08:17.509,0:08: so that's our pattern there an odd number plus an odd number will give us an even 0:08:21.460,0:08: result 0:08:22.039,0:08: so now let's look at multiplication and let's look at the 0:08:25.120,0:08: the table what if we look diagonally 0:08:27.900,0:08: three and eight and four and six 0:08:30.889,0:08: uh... we have 0:08:32.200,0:08:33.580
13 twenty-four for both 0:08:33.580,0:08: now look at this question 0:08:35.220,0:08: why do six times four eight times three result in the same product 0:08:39.430,0:08: now that's not really the same question is why are six times four and eight times three 0:08:43.320,0:08: equal we have already answered that one 0:08:45.880,0:08: they're equal because they both came out to be twenty-four 0:08:48.590,0:08: well let's look at it this way 0:08:50.550,0:08: the three times eight we can break this down 0:08:53.850,0:08: we can't bring down the three anymore but can break down the eight to 0:08:57.990,0:08: uh... four 0:08:58.930,0:09: times two 0:09:00.250,0:09: and then we can take the four and break it down to two times two 0:09:03.679,0:09: so here's all our factors 0:09:05.320,0:09: for three times eight 0:09:07.490,0:09: we can do the same thing with the four times six we can break the four down to 0:09:11.100,0:09: two times two
14 0:09:12.520,0:09: and we can break the six down to three times two 0:09:15.660,0:09: now notice what happened here in both cases we had a three for a factor 0:09:20.750,0:09: and then we can also match up all the different twos so again notice now 0:09:26.850,0:09: that in essence we have the same factors in both cases 0:09:29.690,0:09: we have a three 0:09:31.160,0:09: and we have three twos 0:09:32.540,0:09: for both of them so we're really multiplying the same factors in both 0:09:36.360,0:09: cases 0:09:38.580,0:09: if we look at another possibility like six times twelve and 0:09:42.310,0:09: uh... nine times eight 0:09:43.970,0:09: if we break down the six times twelve this is what we get 0:09:47.540,0:09: if we break down the nine times eight 0:09:49.870,0:09: this is what we get 0:09:51.240,0:09: and again notice the pattern 0:09:53.510,0:09: uh... we can match them all up so in essence we have the same factors 0:09:58.100,0:09:59.230
15 uh... for 0:09:59.230,0:10: both cases 0:10:01.040,0:10: and that explains why we get the same result 0:10:03.860,0:10: what about other possibilities what about this way 0:10:06.900,0:10: let's see we have two times twenty that's forty 0:10:09.550,0:10: and five times eight that's forty that works 0:10:12.660,0:10: and what about this 0:10:14.570,0:10: uh... six times forty is two hundred forty, sixteen times fifteen 0:10:18.050,0:10: if we multiply it out 0:10:19.080,0:10: that will come out to be two-forty 0:10:20.940,0:10: so this can be a lot of fun students looking for all these different 0:10:23.760,0:10: possibilities 0:10:25.150,0:10: where they can get uh... equal products 0:10:27.520,0:10: and this will give them a lot of practice with basic multiplication skills 0:10:32.130,0:10: now if go back to that question 0:10:34.190,0:10: why do six times four and eight times three result in the same product
16 0:10:37.590,0:10: well again the key is that 0:10:39.510,0:10: when you break it down 0:10:40.889,0:10: to all of the factors 0:10:42.720,0:10: and then the same thing here 0:10:45.850,0:10: we have the same factors 0:10:47.620,0:10: for both cases we're multiplying the same things 0:10:50.630,0:10: students can use the commutative and associative properties of multiplication to derive 0:10:55.000,0:10: all of the factor combinations for any given product 0:10:57.890,0:11: so let's take twenty-four for example if we break it all down 0:11:01.460,0:11: uh... notice what we can do here 0:11:03.750,0:11: the two times two times two 0:11:06.100,0:11: would be eight 0:11:08.350,0:11: times three 0:11:09.800,0:11: and then here if we group them this way then two times two would be four 0:11:14.240,0:11: and the two times three would be six 0:11:16.820,0:11: and over here 0:11:18.410,0:11:20.370
17 we would have two 0:11:20.370,0:11: times 0:11:21.470,0:11: uh... two times two is four times three is twelve 0:11:25.540,0:11: to get the other possibilities we would just change some of the orders around 0:11:29.340,0:11: and we would eventually get three times eight six times four and twelve times two 0:11:33.620,0:11: let's look at the multiplication 0:11:35.180,0:11: table again but from this perspective 0:11:37.490,0:11: notice that 0:11:38.649,0:11: this diagonal here is your perfect squares 0:11:42.150,0:11: four would be two times two nine would be three times three and so forth 0:11:46.550,0:11: now notice that we have 0:11:48.620,0:11: this part shaded 0:11:50.040,0:11: and this side not 0:11:52.470,0:11: but that's for a reason 0:11:54.250,0:11: notice that 0:11:56.530,0:11: students should be able to find numbers 0:11:59.270,0:12: on one side 0:12:00.519,0:12:01.890
18 that match up 0:12:01.890,0:12: with numbers on the other 0:12:03.339,0:12: for example here we have 0:12:05.040,0:12: twelve and twelve 0:12:06.780,0:12: uh... now this twelve here comes from multiplying two times six that's two groups of 0:12:11.690,0:12: six 0:12:12.720,0:12: and this other twelve is actually six groups of two 0:12:17.090,0:12: but they're both twelve 0:12:19.680,0:12: another combination 0:12:22.370,0:12: this thirty here would be five groups of six wheres this thirty is six groups of 0:12:27.010,0:12: five 0:12:29.240,0:12: so we can find a lot of these different pairings and notice that they occur 0:12:32.810,0:12: along diagonals 0:12:34.580,0:12: uh... in this case we have three groups of nine for this twenty seven and 0:12:38.140,0:12: nine groups of three 0:12:39.820,0:12: for this other twenty-seven 0:12:41.100,0:12: notice the pattern here
19 0:12:43.630,0:12: that we have these matching pairs of numbers the twenty-sevens the twenty- 0:12:47.280,0:12: fours the twenty-ones and so forth 0:12:49.390,0:12: so along this column 0:12:51.310,0:12: we have matching numbers 0:12:53.060,0:12: over here along this row 0:12:55.440,0:12: and notice that really what we have here is the commutative property 0:12:59.320,0:13: uh... all of these pairings that's the only difference is that you have 0:13:02.790,0:13: your factors in reverse order 0:13:04.920,0:13: so for example for this fifteen 0:13:07.330,0:13: this one is five groups of three 0:13:09.690,0:13: as opposed to this other fifteen that's three groups of five 0:13:13.120,0:13: what about uh... the multiplication patterns dealing with even 0:13:17.020,0:13: and odd numbers 0:13:18.450,0:13: again just using examples that if we take an even times an even 0:13:22.060,0:13: let's see that's twenty-four, sixteen, ten times six that's sixty 0:13:25.490,0:13:31.090
20 ah it looks like an even times an even is going to give us an even number 0:13:31.090,0:13: what about odd times even 0:13:32.600,0:13: well let's see that would be like five times four 0:13:34.920,0:13: that's twenty, seven times two is fourteen 0:13:37.100,0:13: and so far 0:13:38.610,0:13: ah so that pattern seems to be 0:13:40.750,0:13: an even result also 0:13:42.580,0:13: now we need to consider 0:13:44.270,0:13: the reversal of this which is just your commutative property an even times an odd 0:13:49.090,0:13: number well let's see forty-two, twenty-four fifty 0:13:53.120,0:13: looks like the pattern there is even 0:13:55.780,0:13: and in our last uh... combination 0:13:58.980,0:14: our last possibility is odd times odd 0:14:01.930,0:14: let's see five times three is fifteen seven times nine is sixty-three 0:14:06.170,0:14: ah looks like that pattern 0:14:08.100,0:14: is coming out to be odd 0:14:09.990,0:14: if we connect this back to what we were doing with addition
21 0:14:14.860,0:14: an even times an even number means that 0:14:17.810,0:14: our groups are all going to be composed of the same even number and we have an even 0:14:21.740,0:14: number of them 0:14:26.120,0:14: so when we do our combinations we're still going to end up with even numbers 0:14:30.110,0:14: and then the same thing uh... 0:14:33.210,0:14: here with an odd number of evens we still have the same even number repeated over 0:14:38.460,0:14: and over but we have an odd number of them 0:14:41.220,0:14: but that's okay 0:14:42.650,0:14: uh... if we... no matter how many iterations we were to go down at the end 0:14:47.600,0:14: we're gonna end up with an even plus an even 0:14:50.580,0:14: which will still give us 0:14:52.740,0:14: an even number for the result what about the reverse order even times odd 0:14:57.520,0:14: this time 0:14:59.110,0:15: each of our repeated numbers is an odd number 0:15:02.300,0:15: and we have an even number of those when
22 we combine 0:15:06.400,0:15: two odds that we already know that's going to give us an even sum 0:15:10.500,0:15: so if we do this we'll 0:15:12.490,0:15: see the pattern that 0:15:14.280,0:15: we'll still get 0:15:15.630,0:15: an even result and then last but not least an odd number times another odd number 0:15:21.960,0:15: so our groups are all going to be the same odd number but we have an odd number of those 0:15:26.620,0:15: if we ah... 0:15:28.290,0:15: keep combining odds with odds those will give us even 0:15:32.320,0:15: sums 0:15:33.700,0:15: but then we're going to have at the end no matter how many iterations you're going to end 0:15:37.760,0:15: up with 0:15:38.680,0:15: an even number plus an odd number which results in 0:15:42.310,0:15: another odd 0:15:43.870,0:15: if we wish to development little mathematicians it's vital that they not 0:15:47.370,0:15: only find the patterns
23 0:15:49.320,0:15: but also figure out why the pattern works and explain and justify the reasoning 0:15:55.550,0:15: why did it work? so we've seen that 0:15:57.050,0:16: patterns both from addition and multiplication 0:16:00.410,0:16: can really be powerful tools 0:16:02.700,0:16: because 0:16:03.900,0:16: it can do a lot of things first it can enhance 0:16:07.120,0:16: uh... the pattern recognition skills of the students 0:16:11.900,0:16: we can also use them to practice basic facts without all the drill and kill 0:16:16.390,0:16: in fact uh... 0:16:17.569,0:16: using those patterns in the tables and in other places was actually fun 0:16:21.610,0:16: students can experience the justification an explanation of the 0:16:25.210,0:16: reasoning 0:16:26.210,0:16: and if they're doing this in written format with doing journals that will 0:16:29.450,0:16: also help 0:16:30.339,0:16: working with patterns
24 0:16:31.820,0:16: will also help students learn their fundamental properties of mathematics especially 0:16:36.160,0:16: the commutative and associative properties 0:16:39.160,0:16: and last but not least 0:16:40.910,0:16: uh... you've seen that 0:16:42.700,0:16: we can also use these patterns to lay the foundation for more complex topics 0:16:47.720,0:16: uh... in this case uh... it was for example the... additive inverse 0:16:52.430,0:16: and the idea of prime factorization so done correctly...multiplication patterns 0:16:58.180,0:17: and addition patterns can really be useful and very powerful tools
Transcriber(s): Baldev, Prashant Verifier(s): DeLeon, Christina Date Transcribed: Spring 2008 Page: 1 of 5
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