1 00:00:00,000 --> 00:00:00,740 2 00:00:00,740 --> 00:00:02,849 Let's review a little bit about what we know so 3 00:00:02,850 --> 00:00:04,820 far about subtraction. 4 00:00:04,820 --> 00:00:09,210 So if I say 5 minus 3, what does that mean? 5 00:00:09,210 --> 00:00:11,060 Well, there's a couple of ways to think about it. 6 00:00:11,060 --> 00:00:16,804 I could have 5-- let's say I had to 5 berries. 7 00:00:16,803 --> 00:00:21,929 So 1, 2, 3, 4, 5. 8 00:00:21,929 --> 00:00:25,519 So I could have 5 berries, and when I say minus 3 you're 9 00:00:25,519 --> 00:00:27,219 subtracting 3 from it. 10 00:00:27,219 --> 00:00:30,359 I can view that as saying I'm going to take away 11 00:00:30,359 --> 00:00:31,739 3 of these berries. 12 00:00:31,739 --> 00:00:35,219 So if I take away that berry, that berry, and that berry. 13 00:00:35,219 --> 00:00:38,109 So I took away 1, 2, 3 berries. 14 00:00:38,109 --> 00:00:40,219 How many berries do I have left? 15 00:00:40,219 --> 00:00:43,129 Well the only berries I have left are right here-- 1, 2. 16 00:00:43,130 --> 00:00:46,750 So I have 2 berries left just like that. 17 00:00:46,750 --> 00:00:50,060 Now the other way, the other way that I could visualize or 18 00:00:50,060 --> 00:00:53,710 think about 5 minus 3, I'll do it over here. 19 00:00:53,710 --> 00:00:57,250 5 minus 3-- is to think about what the difference 20 00:00:57,250 --> 00:00:59,670 between 5 and 3 is. 21 00:00:59,670 --> 00:01:00,609 So let me draw this. 22 00:01:00,609 --> 00:01:02,280 So let's say I have 5 berries. 23 00:01:02,280 --> 00:01:04,960 1, 2, 3, 4, 5. 24 00:01:04,959 --> 00:01:07,559 And let's say that you have 3 berries. 25 00:01:07,560 --> 00:01:09,600 Here's a slightly different color. 26 00:01:09,599 --> 00:01:11,959 You have 3 berries. 27 00:01:11,959 --> 00:01:16,109 So another way to think about 5 minus 3 is how many more 28 00:01:16,109 --> 00:01:18,980 berries do I have than you have? 29 00:01:18,980 --> 00:01:21,500 And if you look right here, well, you see, this berry 30 00:01:21,500 --> 00:01:24,109 is another-- you have also one berry there. 31 00:01:24,109 --> 00:01:27,040 We both have one berry there, we both have one berry there. 32 00:01:27,040 --> 00:01:29,970 But I've got 1, 2 berries that you don't have. 33 00:01:29,969 --> 00:01:33,200 So once again, I have 2 more berries than you have. 34 00:01:33,200 --> 00:01:35,219 Now we can also think of this from the number 35 00:01:35,219 --> 00:01:38,069 line point of view. 36 00:01:38,069 --> 00:01:42,299 So let me draw a number line just like that. 37 00:01:42,299 --> 00:01:43,269 It's my number line. 38 00:01:43,269 --> 00:01:44,849 We've learned on the addition videos we can 39 00:01:44,849 --> 00:01:46,619 keep going off forever. 40 00:01:46,620 --> 00:01:49,219 And actually, we could even go to the left of 0 and go into 41 00:01:49,219 --> 00:01:52,159 negative numbers, which we'll see in future videos. 42 00:01:52,159 --> 00:01:53,719 But I'll start at 0. 43 00:01:53,719 --> 00:02:02,500 0, 1, 2, 3, 4, 5-- I'll just go up to 7. 44 00:02:02,500 --> 00:02:07,329 So if we do 5 minus 3, if we view 3 as being taken away from 45 00:02:07,329 --> 00:02:10,889 5, 5 minus 3 means start at 5. 46 00:02:10,889 --> 00:02:14,889 If I did 5 plus 3 I would jump 3 spots to the right because 47 00:02:14,889 --> 00:02:16,939 that's increasing the number of things I have. 48 00:02:16,939 --> 00:02:20,329 But since I'm subtracting 3, I want to decrease by 3. 49 00:02:20,330 --> 00:02:29,010 So I decrease by 1, 2, 3 and I get to 2 just like that. 50 00:02:29,009 --> 00:02:31,769 Now if we visualize it from this point of view, let me 51 00:02:31,770 --> 00:02:33,159 draw another number line. 52 00:02:33,159 --> 00:02:33,819 I want to show you. 53 00:02:33,819 --> 00:02:37,280 I mean this is, I'm taking away 3 and here I'm saying, how 54 00:02:37,280 --> 00:02:38,909 many more is 5 than 3? 55 00:02:38,909 --> 00:02:41,689 Even though they're the exact same answer, but there are two 56 00:02:41,689 --> 00:02:43,939 different ways to think about it. 57 00:02:43,939 --> 00:02:45,479 Let me draw a number line here again. 58 00:02:45,479 --> 00:02:46,829 Let me draw the same number line. 59 00:02:46,830 --> 00:02:49,750 60 00:02:49,750 --> 00:02:57,629 I have 0, 1, 2, 3, 4, 5, 6, 7. 61 00:02:57,629 --> 00:03:01,329 So if I were to plot where 5 is on this number line, so 62 00:03:01,330 --> 00:03:02,770 this is the 5 right there. 63 00:03:02,770 --> 00:03:04,750 I'll put a little pink square around it. 64 00:03:04,750 --> 00:03:06,490 5 is right there. 65 00:03:06,490 --> 00:03:10,719 Now 3, let me do 3 in this yellow color. 66 00:03:10,719 --> 00:03:13,109 3 is right here on the number line. 67 00:03:13,110 --> 00:03:18,830 So in this way of thinking about 5 minus 3 you're saying, 68 00:03:18,830 --> 00:03:21,630 what is the difference-- let me write that down. 69 00:03:21,629 --> 00:03:30,840 Here we're saying, what is the difference between 5 and 3? 70 00:03:30,840 --> 00:03:36,870 71 00:03:36,870 --> 00:03:38,930 And to figure out the difference you actually have to 72 00:03:38,930 --> 00:03:43,129 say, how much do you have to add to 3 to get to 5? 73 00:03:43,129 --> 00:03:46,199 So the difference here, how different is 5 than 3? 74 00:03:46,199 --> 00:03:50,379 Well you have to go up 1 and then up 2 to get to 5. 75 00:03:50,379 --> 00:03:54,430 So the difference between 5, which is all the way over here, 76 00:03:54,430 --> 00:04:04,520 and 3, which is just that far, is 2, just like that. 77 00:04:04,520 --> 00:04:05,880 That right there is 2. 78 00:04:05,879 --> 00:04:07,250 Let me draw that in another box. 79 00:04:07,250 --> 00:04:08,270 So that's 2 right here. 80 00:04:08,270 --> 00:04:11,770 I want to make this difference between subtraction and 81 00:04:11,770 --> 00:04:14,060 difference-- I want to make it at least reasonably clear to 82 00:04:14,060 --> 00:04:17,600 you because these are two different ways of viewing 83 00:04:17,600 --> 00:04:20,340 subtraction, but it ends up being the exact same operation. 84 00:04:20,339 --> 00:04:22,649 You're going to get the same answer regardless of which 85 00:04:22,649 --> 00:04:24,529 way you think about it. 86 00:04:24,529 --> 00:04:26,759 Now, I could view-- let me do different numbers now. 87 00:04:26,759 --> 00:04:30,550 Let me do 7 minus 4. 88 00:04:30,550 --> 00:04:34,030 So I could view this as, maybe I have a 7 foot 89 00:04:34,029 --> 00:04:35,239 long piece of wood. 90 00:04:35,240 --> 00:04:39,060 91 00:04:39,060 --> 00:04:40,759 It's 7 feet long. 92 00:04:40,759 --> 00:04:43,709 If I put a ruler up against it I would have 0, 93 00:04:43,709 --> 00:04:50,479 1, 2, 3, 4, 5, 6, 7. 94 00:04:50,480 --> 00:04:52,689 So I have a 7 foot long piece of wood. 95 00:04:52,689 --> 00:04:55,930 And then I could saw off 4 of those feet. 96 00:04:55,930 --> 00:04:57,985 So if I were to saw off 4 of these feet-- so 97 00:04:57,985 --> 00:05:01,750 I saw off 1, 2, 3, 4. 98 00:05:01,750 --> 00:05:03,170 How much wood do I have left? 99 00:05:03,170 --> 00:05:06,040 So all of this stuff right here, I'm eliminating. 100 00:05:06,040 --> 00:05:07,660 I'm sawing it off. 101 00:05:07,660 --> 00:05:09,160 I'm sawing it off of the wood. 102 00:05:09,160 --> 00:05:10,820 Maybe I should do that in a darker color to show 103 00:05:10,819 --> 00:05:13,250 that I'm sawing it off. 104 00:05:13,250 --> 00:05:15,240 So all of this stuff is going to disappear. 105 00:05:15,240 --> 00:05:16,720 I'm grinding it away. 106 00:05:16,720 --> 00:05:17,870 I'm sawing that off. 107 00:05:17,870 --> 00:05:21,889 So I'm just left with-- after I saw the 4 inches or feet or 108 00:05:21,889 --> 00:05:27,879 whatever of the wood, I'm left with 1, 2, 3 inches of wood. 109 00:05:27,879 --> 00:05:28,600 So this is 3. 110 00:05:28,600 --> 00:05:33,700 So 7 minus 4 is equal to 3. 111 00:05:33,699 --> 00:05:36,409 This is viewing subtraction as literally taking away. 112 00:05:36,410 --> 00:05:40,380 I sawed off the wood, so I took away wood. 113 00:05:40,379 --> 00:05:45,069 Now I could think of it in a slightly different way of 114 00:05:45,069 --> 00:05:47,930 thinking about it, but give you the exact same answer. 115 00:05:47,930 --> 00:05:49,610 We could say 7 minus 4. 116 00:05:49,610 --> 00:05:53,500 So once again, I could have the 7 inch long piece 117 00:05:53,500 --> 00:05:56,149 of wood like that. 118 00:05:56,149 --> 00:06:05,279 So if I put a ruler here that's 1, 2, 3, 4, 5, 6, 7. 119 00:06:05,279 --> 00:06:07,939 So once again, a 7 inch long piece of wood. 120 00:06:07,939 --> 00:06:11,079 And now instead of taking 4 away of it I'm comparing it-- 121 00:06:11,079 --> 00:06:13,740 so that's a 7-- I'm comparing it to a 4 inch long 122 00:06:13,740 --> 00:06:14,280 piece of wood. 123 00:06:14,279 --> 00:06:18,829 So I have another 4 inch long piece of wood right there. 124 00:06:18,829 --> 00:06:22,750 That's my 4 inch long piece of wood. that's 7, this is 4. 125 00:06:22,750 --> 00:06:25,769 You could view 7 minus 4 as taking 4 inches away from 126 00:06:25,769 --> 00:06:26,579 the long piece of wood. 127 00:06:26,579 --> 00:06:30,609 Or you could view seven minus 4 as the difference between the 4 128 00:06:30,610 --> 00:06:33,530 inch piece of wood and the 7 inch piece of wood. 129 00:06:33,529 --> 00:06:35,059 So in this case, what's the difference? 130 00:06:35,060 --> 00:06:38,310 To go from the 4 inch piece of wood to the 7 inch piece of 131 00:06:38,310 --> 00:06:45,030 wood I would have to grow by 3 inches, or I would have to add 132 00:06:45,029 --> 00:06:48,369 a 3 inch piece of wood somehow. 133 00:06:48,370 --> 00:06:50,600 Or the wood would somehow have to grow by 3 inches in 134 00:06:50,600 --> 00:06:52,170 order to become 7 inches. 135 00:06:52,170 --> 00:06:55,480 So these are 2 completely equivalent ways to 136 00:06:55,480 --> 00:06:56,430 view subtraction. 137 00:06:56,430 --> 00:06:59,370 That's all a little bit of a review from the last video. 138 00:06:59,370 --> 00:07:02,290 Now what I also want to do in this video is start tackling 139 00:07:02,290 --> 00:07:03,379 slightly larger problems. 140 00:07:03,379 --> 00:07:06,170 But you'll see that really, the number line applies just 141 00:07:06,170 --> 00:07:09,069 equally as well as to kind of the simpler problems 142 00:07:09,069 --> 00:07:12,259 that we've done before. 143 00:07:12,259 --> 00:07:16,379 Let's do 17 minus 9. 144 00:07:16,379 --> 00:07:18,279 So just like everything else, there's two ways 145 00:07:18,279 --> 00:07:18,989 we could've done it. 146 00:07:18,990 --> 00:07:24,280 You know, the more slow way is you could draw 17 objects. 147 00:07:24,279 --> 00:07:26,739 Let's say I have 17 chips. 148 00:07:26,740 --> 00:07:35,790 1, 2, 3, 4, 5, 6, 7, 8, 9 10, 11, 12, 13, 14, 15, 16, 17. 149 00:07:35,790 --> 00:07:37,750 And I'm going to take away 9 of them. 150 00:07:37,750 --> 00:07:44,889 So I'm going to take away 1, 2, 3, 4, 5, 6, 7, 8, 9. 151 00:07:44,889 --> 00:07:46,810 How many am I left with? 152 00:07:46,810 --> 00:07:52,449 I'm left with 1, 2, 3, 4 5, 6, 7, 8. 153 00:07:52,449 --> 00:07:56,300 So 17 minus 9 is equal to 8. 154 00:07:56,300 --> 00:07:58,389 But that took a long time and you could imagine, if this 155 00:07:58,389 --> 00:08:00,509 number was a lot bigger it would've taken me forever to 156 00:08:00,509 --> 00:08:02,569 draw all of these circles and then scratch out things. 157 00:08:02,569 --> 00:08:04,750 And it would've wasted paper and time. 158 00:08:04,750 --> 00:08:07,160 And we have other things to do. 159 00:08:07,160 --> 00:08:10,050 So another way you could do it, and maybe this would be easier 160 00:08:10,050 --> 00:08:11,930 for you to visualize, is to draw the number line. 161 00:08:11,930 --> 00:08:14,259 You always don't have to start at 0. 162 00:08:14,259 --> 00:08:20,480 So if we draw the number line, if we say that's 18, 17, 16, 163 00:08:20,480 --> 00:08:32,210 15, 14, 13, 12, 11, 10, 9, 8, 7-- you could imagine, I could 164 00:08:32,210 --> 00:08:34,550 keep going to the left all the way to 0. 165 00:08:34,549 --> 00:08:36,789 But I start at 17. 166 00:08:36,789 --> 00:08:40,459 I could start at 17 and take away 9 from it. 167 00:08:40,460 --> 00:08:49,460 So I go 1, 2, 3, 4, 5, 6, 7, 8, 9. 168 00:08:49,460 --> 00:08:52,370 And once again, we are left at 8. 169 00:08:52,370 --> 00:08:55,659 Now this was, at least in my head, a little bit cleaner 170 00:08:55,659 --> 00:08:57,110 and faster than this one. 171 00:08:57,110 --> 00:08:59,190 But in either case, you don't want to do this every time you 172 00:08:59,190 --> 00:09:02,160 have to subtract 9 from 17 or want to find the difference 173 00:09:02,159 --> 00:09:03,990 between 17 and 9. 174 00:09:03,990 --> 00:09:05,000 And then to realize that's 8. 175 00:09:05,000 --> 00:09:06,909 So this is something that eventually you want 176 00:09:06,909 --> 00:09:08,079 to internalize. 177 00:09:08,080 --> 00:09:10,720 You'll want to know by heart that, oh, 17 minus 9? 178 00:09:10,720 --> 00:09:12,200 I know that is 8. 179 00:09:12,200 --> 00:09:15,320 And by the way, 17 minus 8? 180 00:09:15,320 --> 00:09:17,270 What's 17 minus 8? 181 00:09:17,269 --> 00:09:19,710 Well, that is 9. 182 00:09:19,710 --> 00:09:21,820 And now why does all of this make sense? 183 00:09:21,820 --> 00:09:27,270 Because 8 plus 9 is equal to 17. 184 00:09:27,269 --> 00:09:31,860 So 17 minus 9 is 8. 185 00:09:31,860 --> 00:09:35,330 Or 17 minus 8 is 9. 186 00:09:35,330 --> 00:09:39,110 When I say 17 minus 8, I'm essentially saying that is 187 00:09:39,110 --> 00:09:42,550 equal to some number that if I were to add to 8 will equal 17. 188 00:09:42,549 --> 00:09:43,579 Well, that's 9. 189 00:09:43,580 --> 00:09:46,720 When I say 17 minus 9 that's saying, there's some number, 190 00:09:46,720 --> 00:09:49,250 that if I were to add it to 9, I'll get 17. 191 00:09:49,250 --> 00:09:50,169 And that's 8. 192 00:09:50,169 --> 00:09:53,620 So all of these, all of these statements, are kind of 193 00:09:53,620 --> 00:09:54,759 saying the same thing. 194 00:09:54,759 --> 00:09:56,500 That 8 plus 9 are 17. 195 00:09:56,500 --> 00:09:58,659 Or the difference between 17 and 9 is 8. 196 00:09:58,659 --> 00:10:03,120 Or the difference between 17 and 8 is 9. 197 00:10:03,120 --> 00:10:04,560 Hopefully I'm not confusing you. 198 00:10:04,559 --> 00:10:08,659 So for most of these subtraction problems where the 199 00:10:08,659 --> 00:10:12,620 answer is a one-digit answer, you should eventually have them 200 00:10:12,620 --> 00:10:15,980 memorized, but in your head it's good to be imagining 201 00:10:15,980 --> 00:10:16,639 this number line. 202 00:10:16,639 --> 00:10:18,889 Let's do a couple more of these. 203 00:10:18,889 --> 00:10:22,470 And then, once we have these memorized or at least be able 204 00:10:22,470 --> 00:10:25,879 to do a number line if we forget, I'll show you have to 205 00:10:25,879 --> 00:10:29,460 do any subtraction problem, arbitrarily for super 206 00:10:29,460 --> 00:10:30,860 large numbers. 207 00:10:30,860 --> 00:10:37,050 So let's say we're going to do 13 minus 5. 208 00:10:37,049 --> 00:10:39,659 So once again, I'm not going to do the whole circles 209 00:10:39,659 --> 00:10:40,539 or the berries this time. 210 00:10:40,539 --> 00:10:43,459 I'm just going to draw the number line. 211 00:10:43,460 --> 00:10:45,720 Just draw the number line like that. 212 00:10:45,720 --> 00:10:58,450 Let's start at 14, 13, 12, 11, 10, 9, 8, 7, 6, 5-- and you 213 00:10:58,450 --> 00:11:00,460 just can keep going lower and lower. 214 00:11:00,460 --> 00:11:02,509 You can go to 0 or you can even go past 0. 215 00:11:02,509 --> 00:11:03,830 We'll talk about that in the future. 216 00:11:03,830 --> 00:11:05,810 But we start at 13. 217 00:11:05,809 --> 00:11:08,959 We're starting at 13. 218 00:11:08,960 --> 00:11:10,620 And we're going to take 5 away from it. 219 00:11:10,620 --> 00:11:13,919 So this is the subtraction view of subtraction; 220 00:11:13,919 --> 00:11:15,199 we're taking away. 221 00:11:15,200 --> 00:11:21,340 1, 2, 3, 4, 5 and we land at 8. 222 00:11:21,340 --> 00:11:26,009 So 13 minus 5-- let me do this in a new color. 223 00:11:26,009 --> 00:11:29,809 13 minus 5 is equal to 8. 224 00:11:29,809 --> 00:11:31,579 Now another way we could have thought about that, 225 00:11:31,580 --> 00:11:33,980 I plotted where 13 is. 226 00:11:33,980 --> 00:11:36,060 I can plot where 5 is. 227 00:11:36,059 --> 00:11:37,849 I could say look, this is 5. 228 00:11:37,850 --> 00:11:39,759 5 is right here on my number line. 229 00:11:39,759 --> 00:11:42,735 What do I have to add to 5 to get to 13? 230 00:11:42,735 --> 00:11:43,279 So let's see. 231 00:11:43,279 --> 00:11:49,250 I would have to go 1, 2, 3, 4, 5, 6, 7, 8. 232 00:11:49,250 --> 00:11:52,179 I have to add 8 to 5 to get to 13. 233 00:11:52,179 --> 00:11:55,789 5 plus 8 is equal to 13. 234 00:11:55,789 --> 00:11:59,939 So that tells me that 13 minus 5 is equal to 8. 235 00:11:59,940 --> 00:12:06,210 This also tells me that 13 minus 8 is equal to 5. 236 00:12:06,210 --> 00:12:08,690 All of these, are on some level, telling me the 237 00:12:08,690 --> 00:12:09,650 exact same thing. 238 00:12:09,649 --> 00:12:11,634 But the difference between 13 and 5 is 8. 239 00:12:11,634 --> 00:12:14,409 The difference between 13 and 8 is 5. 240 00:12:14,409 --> 00:12:17,079 5 plus 8 is 13. 241 00:12:17,080 --> 00:12:18,780 So hopefully you have the hang of that and if you haven't done 242 00:12:18,779 --> 00:12:22,939 so already, it'll be good to practice all of these. 243 00:12:22,940 --> 00:12:26,240 Taking a teen number and then subtracting any of the 244 00:12:26,240 --> 00:12:28,470 one-digit numbers from those teen numbers. 245 00:12:28,470 --> 00:12:32,110 That's in general, very, very good practice for you.