1 00:00:00,000 --> 00:00:01,709 Jonily: distance cannot be negative. 2 00:00:01,709 --> 00:00:05,370 Distance is not negative, but direction can be negative. 3 00:00:05,699 --> 00:00:10,110 So the result of subtraction is a number. 4 00:00:10,470 --> 00:00:11,640 And a direction. 5 00:00:13,470 --> 00:00:18,420 Cheri: Welcome to Tier One Interventions podcast, where we help your, help 6 00:00:18,480 --> 00:00:24,990 you strengthen your core classroom instruction so every student has the 7 00:00:24,990 --> 00:00:33,060 chance to thrive, even those students that need additional instruction. 8 00:00:33,930 --> 00:00:39,840 I'm Cheri Dotterer, and today Jenna Lee and I are diving into the second of three. 9 00:00:40,605 --> 00:00:42,795 Math rules that never expire. 10 00:00:43,425 --> 00:00:44,205 Subtraction. 11 00:00:45,405 --> 00:00:49,245 This is one of my favorite rules, not just about the math, it's 12 00:00:49,245 --> 00:00:52,335 about how the brain thinks. 13 00:00:53,955 --> 00:00:56,835 And you'll hear Jonily use, 14 00:00:59,595 --> 00:01:03,765 you'll hear Jona Lee use subtraction across all grade levels and 15 00:01:03,765 --> 00:01:08,650 jump into tips on how to make learning stick with these brain. 16 00:01:10,650 --> 00:01:14,850 Brain-based strategies, movement, and multisensory engagement. 17 00:01:15,420 --> 00:01:20,310 Whether you teach kindergarten or high school, whether you're an occupational 18 00:01:20,310 --> 00:01:27,930 therapist, let's learn how we can change the way we instruct mathematics 19 00:01:29,455 --> 00:01:35,280 and help kids make subtraction stick. 20 00:01:36,280 --> 00:01:41,710 Jonily: Hello and welcome to Tier One Interventions podcast, where we talk 21 00:01:41,710 --> 00:01:44,200 about strengthening your core classroom. 22 00:01:44,290 --> 00:01:47,290 That is I'm your host, Jay-Z. 23 00:01:47,480 --> 00:01:51,820 Jonily Zupancic, your main Math Ander Math specialist. 24 00:01:51,820 --> 00:01:52,810 Math coach. 25 00:01:52,840 --> 00:01:53,950 Math leader. 26 00:01:54,220 --> 00:01:57,460 And at Tier One Interventions podcast, we share. 27 00:01:58,165 --> 00:02:04,225 Tips, tricks, techniques, and strategies to enhance your tier 28 00:02:04,225 --> 00:02:06,655 one core general classroom. 29 00:02:07,015 --> 00:02:12,055 This reduces the number of students that need tier two and tier 30 00:02:12,055 --> 00:02:17,995 three pullout with a strong core, tier two, tier three, no more. 31 00:02:19,555 --> 00:02:20,545 It's just a little joke. 32 00:02:20,815 --> 00:02:24,925 Not that we wanna eliminate those small group pullouts, but we want 33 00:02:24,925 --> 00:02:27,655 our tier one core regular classroom. 34 00:02:28,405 --> 00:02:33,115 Specifically for mathematics to be accessible for every student, no 35 00:02:33,115 --> 00:02:35,455 matter the ability or disability. 36 00:02:36,235 --> 00:02:39,745 We, she, we teach instructional and facilitation strategies that 37 00:02:39,745 --> 00:02:43,285 are going to increase learning memory and retention of content. 38 00:02:43,345 --> 00:02:45,265 Specifically with mathematics. 39 00:02:45,655 --> 00:02:49,645 We want to share strategies with you to improve, number, sense thinking, 40 00:02:49,705 --> 00:02:51,445 reasoning, and sense making. 41 00:02:52,870 --> 00:02:56,980 Today's focus is going to be on rule number two of three 42 00:02:56,980 --> 00:02:59,560 math rules that never expire. 43 00:02:59,590 --> 00:03:03,550 These rules are transferable from preschool through high school 44 00:03:03,730 --> 00:03:06,490 no matter what the number is. 45 00:03:07,280 --> 00:03:08,330 Hey Cheri here. 46 00:03:08,540 --> 00:03:12,470 When we teach a concept that works at every level from preschool 47 00:03:12,470 --> 00:03:17,300 through high school, we're giving students the brain pattern that 48 00:03:17,360 --> 00:03:19,760 doesn't have to be unlearned later. 49 00:03:21,890 --> 00:03:27,950 That consistency frees up working memory so we can focus on reasoning and sense 50 00:03:27,950 --> 00:03:35,420 making and remembering at any stage and building rather than tearing apart. 51 00:03:37,867 --> 00:03:42,247 Jonily: Some math rules that expire are things like when subtracting 52 00:03:42,307 --> 00:03:43,867 you can't get a smaller number. 53 00:03:44,047 --> 00:03:46,147 That's only true with certain numbers. 54 00:03:46,447 --> 00:03:51,037 We need a rule that we can teach consistently from progression. 55 00:03:51,262 --> 00:03:55,582 From preschool through high school that the rule and the strategy is not 56 00:03:55,582 --> 00:03:58,582 gonna change these specific rules. 57 00:03:58,582 --> 00:04:06,982 And there are three big ones are the rules that never expire and make high impact. 58 00:04:07,402 --> 00:04:11,872 Powerful results in improving number sense for our students. 59 00:04:12,472 --> 00:04:14,512 Rule number one is division is counting. 60 00:04:14,542 --> 00:04:16,942 Rule number two is subtraction is distance. 61 00:04:16,942 --> 00:04:19,372 Rule number three is multiplication, is rectangle. 62 00:04:19,372 --> 00:04:21,592 No matter what the numbers are. 63 00:04:22,252 --> 00:04:26,092 These rules and processes never expire. 64 00:04:26,152 --> 00:04:28,702 They are transferred from grade level to grade level. 65 00:04:29,332 --> 00:04:32,662 As we evolve and create more complexity in numbers. 66 00:04:33,727 --> 00:04:35,857 Today we're gonna focus on rule number two. 67 00:04:36,217 --> 00:04:38,317 Subtraction is distance. 68 00:04:38,677 --> 00:04:47,497 In the previous episode of Subtraction is Distance, which was aired on February 69 00:04:47,497 --> 00:04:53,647 18th, 2025, called Hidden Math Patterns that Make Subtraction easy for everyone. 70 00:04:54,517 --> 00:04:59,197 The introductory version of the three Math Rules that Never Expire 71 00:04:59,197 --> 00:05:03,787 specifically with Subtraction was released on that podcast episode. 72 00:05:04,117 --> 00:05:04,987 So look that up. 73 00:05:04,987 --> 00:05:09,487 If you wanna start at the introduction today, we are gonna do an extension 74 00:05:09,487 --> 00:05:14,047 of that introduction and we are still going to follow the subtraction. 75 00:05:14,107 --> 00:05:17,137 Is distance cycles A through Z? 76 00:05:17,562 --> 00:05:23,562 But today particularly, we're gonna focus on two of the extended cycles. 77 00:05:23,892 --> 00:05:28,302 These cycles, and these numbers are appropriate for any and all grade levels. 78 00:05:28,332 --> 00:05:34,542 And the thoughts and ideas and strategies that I share are absolutely essential for 79 00:05:34,542 --> 00:05:36,252 improving number sense in your school. 80 00:05:36,312 --> 00:05:39,642 Remember, number sense is an innate, intuitive understanding 81 00:05:39,642 --> 00:05:41,262 of the size of number. 82 00:05:41,262 --> 00:05:43,452 Another word for size is magnitude. 83 00:05:43,692 --> 00:05:45,247 Understanding the value of number. 84 00:05:45,872 --> 00:05:47,042 For subtraction. 85 00:05:47,252 --> 00:05:50,732 The question is how far apart are these two numbers? 86 00:05:50,882 --> 00:05:55,112 Subtraction is distance, subtraction is range, and yes, subtraction are 87 00:05:55,112 --> 00:06:00,062 other things, but some of those other things are not transferable, especially 88 00:06:00,242 --> 00:06:02,972 when we get to negative numbers. 89 00:06:03,212 --> 00:06:07,832 So if we want to build consistency and progressions of improving number sense 90 00:06:07,832 --> 00:06:09,122 from preschool through high school. 91 00:06:09,682 --> 00:06:13,552 This rule number two, subtraction is distance is the strategy 92 00:06:13,552 --> 00:06:15,802 and technique to do just that. 93 00:06:17,062 --> 00:06:21,802 We are gonna focus on two subtraction cycles today, f and 94 00:06:21,802 --> 00:06:25,522 p in the previous podcast titled. 95 00:06:26,047 --> 00:06:28,567 Hidden math patterns that make subtraction easy. 96 00:06:28,567 --> 00:06:35,497 From February 18th, 2025, we focused on cycle A, which is the 97 00:06:35,497 --> 00:06:38,407 introduction to subtraction cycles. 98 00:06:38,887 --> 00:06:43,207 In the show notes, you will get a copy of this notes version that 99 00:06:43,207 --> 00:06:45,157 will accompany today's podcast. 100 00:06:45,517 --> 00:06:49,507 If you are listening to this podcast, audio only, I'm gonna be 101 00:06:49,507 --> 00:06:51,787 as clear and explicit as possible. 102 00:06:52,432 --> 00:06:54,472 Telling you about what's on the screen. 103 00:06:54,712 --> 00:06:59,692 However, to catch a glimpse of what I'm writing at the same time that I'm talking, 104 00:06:59,902 --> 00:07:07,792 you can visit us on YouTube at tier one intervention podcasts, subtraction cycles, 105 00:07:07,822 --> 00:07:15,502 f and p are as follows, cycle F. These are the four problems that we are going to. 106 00:07:16,312 --> 00:07:23,962 Model today, eight minus three, three minus eight 1012 minus 995 107 00:07:24,082 --> 00:07:30,202 and negative three minus negative eight cycle P 71 minus 38. 108 00:07:30,322 --> 00:07:31,672 38 minus 71. 109 00:07:32,357 --> 00:07:36,527 Seven decimal two five minus three, decimal five and negative 110 00:07:36,527 --> 00:07:38,477 13 minus negative seven. 111 00:07:39,017 --> 00:07:45,137 As we look through some of these examples, you're going to see how this transferable 112 00:07:45,137 --> 00:07:47,777 strategy can be used with any number. 113 00:07:48,127 --> 00:07:53,467 This gives access to negative numbers earlier in the grades. 114 00:07:53,677 --> 00:07:58,177 With this one strategy, we don't have to change our strategy or technique when 115 00:07:58,177 --> 00:08:04,117 we subtract, when we use this method, again, this isn't the only method. 116 00:08:04,747 --> 00:08:08,797 There are other methods to use for different numbers that are procedural. 117 00:08:09,307 --> 00:08:11,797 Procedural focuses on answer, getting. 118 00:08:12,757 --> 00:08:16,747 And efficiency and accuracy improving. 119 00:08:16,747 --> 00:08:21,577 Number sense is negated when we focus on efficiency and accuracy. 120 00:08:21,937 --> 00:08:28,867 However, improving number sense is necessary for accuracy and fluency. 121 00:08:29,527 --> 00:08:31,567 Efficiency to be improved. 122 00:08:31,807 --> 00:08:35,857 That all sounds counterintuitive, but all of them are necessary 123 00:08:36,007 --> 00:08:37,477 for overall math achievement. 124 00:08:37,987 --> 00:08:42,787 So what does it mean to be to have subtraction as distance? 125 00:08:43,177 --> 00:08:45,547 Subtraction as distance means? 126 00:08:45,952 --> 00:08:46,912 In a problem. 127 00:08:46,912 --> 00:08:52,342 The second number is my starting point on a number line, and I'm gonna find the 128 00:08:52,342 --> 00:08:57,712 distance between that starting number and the first number in the subtraction. 129 00:08:58,162 --> 00:09:03,772 Technically this subtraction symbol, if I put arrows on either end of the 130 00:09:03,832 --> 00:09:10,192 actual subtraction symbol that shows that I'm looking at a number line. 131 00:09:11,192 --> 00:09:13,472 And the subtraction symbol 132 00:09:15,992 --> 00:09:21,902 in the depth of its meaning is how far apart are these two numbers? 133 00:09:22,262 --> 00:09:25,532 If I place these two numbers on the number line, and I go from the second number to 134 00:09:25,532 --> 00:09:27,632 the first number, what is the distance? 135 00:09:27,662 --> 00:09:28,652 What is the range? 136 00:09:28,802 --> 00:09:29,732 How far apart? 137 00:09:30,212 --> 00:09:34,022 Now, as you can see here, my second number is three, so I'm going 138 00:09:34,022 --> 00:09:35,942 to put three on the number line. 139 00:09:36,587 --> 00:09:38,357 My first number is eight. 140 00:09:38,657 --> 00:09:44,417 I'm gonna put eight on the number line, so I'm going to start at three. 141 00:09:45,302 --> 00:09:48,962 And I'm gonna go to eight and I have to figure out how 142 00:09:48,962 --> 00:09:50,642 far apart those numbers are. 143 00:09:51,002 --> 00:09:57,062 Now for these numbers and for some students, I may have to put tick marks 144 00:09:57,062 --> 00:10:01,322 to say, okay, here's four or 5, 6, 7. 145 00:10:01,572 --> 00:10:08,472 Not drawn to scale, obviously, so that then we can circle or mark the spaces. 146 00:10:08,472 --> 00:10:12,882 Kids like to count the tick marks, but we really need to count the 147 00:10:12,882 --> 00:10:18,852 number of jumps or the number of spaces from three to eight as we are 148 00:10:18,882 --> 00:10:20,892 teaching distance and measurement. 149 00:10:21,042 --> 00:10:24,252 This is a good intervention for that as well, but also 150 00:10:24,252 --> 00:10:26,652 enhancing work on a number line. 151 00:10:26,892 --> 00:10:29,077 As I've talked in previous sessions, one of our. 152 00:10:29,787 --> 00:10:35,817 Interventions for number line is to create that rectangle bar model over 153 00:10:35,817 --> 00:10:41,097 top of the number line and extend the tick marks so kids can see these spaces. 154 00:10:41,337 --> 00:10:46,077 That's a little less abstract and much more visual representational 155 00:10:46,287 --> 00:10:49,287 than the abstract number line itself. 156 00:10:50,037 --> 00:10:55,677 So the distance from three to eight is five units, and 157 00:10:55,677 --> 00:10:57,272 because we went to the right. 158 00:10:59,207 --> 00:11:04,337 We have a result of positive five if we follow the same strategy. 159 00:11:04,337 --> 00:11:08,897 For this second problem, now our starting number is eight on the number line. 160 00:11:08,897 --> 00:11:11,147 As a matter of fact, I could have used the same number line, but 161 00:11:11,147 --> 00:11:12,497 I'm gonna draw a different one. 162 00:11:12,887 --> 00:11:16,697 My starting number is eight, so I'm gonna put eight on the number line. 163 00:11:18,692 --> 00:11:21,542 And I'm going to move to three. 164 00:11:21,722 --> 00:11:23,522 I'll mark three on the number line. 165 00:11:24,062 --> 00:11:27,812 Now, keep in mind, already I know that I'm moving in the left direction. 166 00:11:27,812 --> 00:11:29,132 My result's gonna be negative. 167 00:11:29,402 --> 00:11:31,112 Distance cannot be negative. 168 00:11:31,112 --> 00:11:34,772 Distance is not negative, but direction can be negative. 169 00:11:35,102 --> 00:11:39,512 So the result of subtraction is a number. 170 00:11:39,872 --> 00:11:41,042 And a direction. 171 00:11:42,252 --> 00:11:42,772 Hey, Cheri. 172 00:11:42,792 --> 00:11:43,872 Cutting in here again. 173 00:11:44,202 --> 00:11:48,192 Breaking distances into chugs is actually how the brain 174 00:11:48,192 --> 00:11:50,202 naturally processes magnitude. 175 00:11:50,652 --> 00:11:55,872 It reduces cognitive load and letting the prefrontal cortex that says part 176 00:11:55,872 --> 00:12:00,882 right up front here hold smaller and manageable pieces, making mental 177 00:12:00,882 --> 00:12:03,762 math faster and less stressful. 178 00:12:05,730 --> 00:12:10,680 Jonily: So if I start at eight and go to three now what's super cool 179 00:12:10,680 --> 00:12:14,160 is subtraction is not commutative. 180 00:12:14,430 --> 00:12:18,750 Meaning if I change the order, I don't get the same exact answer, but I do get 181 00:12:18,750 --> 00:12:21,570 an answer that shares a relationship. 182 00:12:21,930 --> 00:12:26,220 So if the distance between three and eight is five, the distance between 183 00:12:26,220 --> 00:12:28,050 eight and three is still five. 184 00:12:29,010 --> 00:12:32,160 The distance doesn't change, but the direction. 185 00:12:32,580 --> 00:12:36,240 Changes, and that's why my overall result is negative five. 186 00:12:37,800 --> 00:12:44,310 When we explore subtraction as distance, we are actually teaching many more 187 00:12:44,310 --> 00:12:46,440 standards than just subtraction. 188 00:12:47,070 --> 00:12:50,730 That's why this improves overall math achievement because 189 00:12:50,730 --> 00:12:51,600 we're looking at number line. 190 00:12:53,025 --> 00:12:56,205 Measurement distance range. 191 00:12:56,745 --> 00:13:02,295 We are also looking at counting, which improves number, sense estimation. 192 00:13:02,595 --> 00:13:05,085 Is the distance gonna be a lot or a little like? 193 00:13:05,085 --> 00:13:06,405 Let's look at this next one. 194 00:13:06,645 --> 00:13:07,995 Grab a different color here. 195 00:13:08,505 --> 00:13:12,375 The next one is 100 1012 minus 9 95. 196 00:13:12,765 --> 00:13:15,075 9 95 is our starting number. 197 00:13:15,285 --> 00:13:20,925 So if I sketch a number line and I mark 9 95 on my number line. 198 00:13:21,925 --> 00:13:25,165 The number that I'm going to is 1012. 199 00:13:26,065 --> 00:13:32,485 Now using this strategy and technique and this thinking exercise, and really 200 00:13:32,485 --> 00:13:34,345 what this is a thinking exercise. 201 00:13:34,645 --> 00:13:38,245 It's not a procedure for answer getting and accuracy. 202 00:13:38,455 --> 00:13:41,995 That is important, but that's a different exercise. 203 00:13:42,415 --> 00:13:45,175 This exercise is a thinking exercise. 204 00:13:45,205 --> 00:13:47,005 It's a minds on exercise. 205 00:13:47,335 --> 00:13:50,875 It shapes the brain differently than working and mimicking 206 00:13:50,905 --> 00:13:52,705 a procedure step by step. 207 00:13:53,425 --> 00:13:57,955 This actually increases problem solving and critical thinking skills because 208 00:13:57,955 --> 00:14:00,415 students are constantly figuring out. 209 00:14:00,505 --> 00:14:04,495 It also improves number sense because I'm actually physically seeing 210 00:14:04,675 --> 00:14:07,015 values of number through distance. 211 00:14:08,255 --> 00:14:10,655 Hey everybody, it's Cheri again. 212 00:14:11,075 --> 00:14:15,875 If you're loving these strategies and you want to join our Mastery 213 00:14:15,875 --> 00:14:22,085 Math Method Masterclass, we hold this hour long event once a month 214 00:14:22,145 --> 00:14:24,035 and we'd love to have you join. 215 00:14:24,605 --> 00:14:28,640 What we talk about there is how to improve your math skills in 10 days. 216 00:14:29,525 --> 00:14:33,845 Can you imagine 10 days, whether you are a math teacher or an occupational 217 00:14:33,845 --> 00:14:38,825 therapist, you can see exactly how to shift teaching so that every student, 218 00:14:38,975 --> 00:14:43,955 no matter their starting point, can master core math concepts and reserve. 219 00:14:44,735 --> 00:14:50,975 So reserve your seat today and go over to disability labs.com/calendar 220 00:14:51,155 --> 00:14:53,345 to check out the next event. 221 00:14:55,620 --> 00:14:57,900 Jonily: I'm practicing my counting techniques, which 222 00:14:57,900 --> 00:14:59,610 increases rate and function. 223 00:14:59,910 --> 00:15:06,270 So all of the standards depend on this strategy and technique for subtraction. 224 00:15:07,140 --> 00:15:10,710 So if I typically look at this. 225 00:15:11,490 --> 00:15:14,820 In typical traditional procedure, I would stack and subtract 226 00:15:15,060 --> 00:15:18,180 1012, I'd put 9 95 underneath. 227 00:15:18,420 --> 00:15:20,670 I'd do a lot of borrowing, blah, blah, blah. 228 00:15:21,390 --> 00:15:26,340 In that procedure, there still is a lot of room for error and it doesn't 229 00:15:26,340 --> 00:15:29,520 improve conceptual cognitive function. 230 00:15:31,385 --> 00:15:34,605 So with this process, not only. 231 00:15:35,685 --> 00:15:41,295 Is there a little bit of room for error, but I actually can estimate better and 232 00:15:41,295 --> 00:15:44,445 improve my conceptual cognitive function. 233 00:15:44,445 --> 00:15:47,265 So I'm really working the brain and exercising the brain. 234 00:15:47,640 --> 00:15:51,690 There's brain growth that is happening through just thinking about this 235 00:15:51,930 --> 00:15:53,700 through this distance technique. 236 00:15:54,300 --> 00:15:59,550 So if I stack and subtract, there is room for error, but also in 237 00:15:59,550 --> 00:16:03,090 looking at those two numbers on top of each other, it still doesn't 238 00:16:03,090 --> 00:16:05,610 trigger to me distance how far apart. 239 00:16:05,880 --> 00:16:11,100 So I can ignore the value of those numbers, which means I'm not 240 00:16:11,100 --> 00:16:14,850 practicing a good estimate estimation. 241 00:16:15,245 --> 00:16:21,245 Prior to solving the problem here, where I'm placing the numbers, forces 242 00:16:21,245 --> 00:16:25,175 me to think about the value of the number, where it's gonna be placed. 243 00:16:25,205 --> 00:16:28,755 And then as I place this, I'm oh, you know what? 244 00:16:29,415 --> 00:16:35,040 I think 1000 might be there because I'm thinking a thousand closer to 9 245 00:16:35,040 --> 00:16:37,485 95, a little further away to 1,012. 246 00:16:37,485 --> 00:16:41,985 But really, in all honesty, neither of those numbers are too far from a thousand. 247 00:16:43,090 --> 00:16:48,220 So that thought process rather than a procedure is what is 248 00:16:48,250 --> 00:16:52,150 exercising and shaping and growing the mind to have a better 249 00:16:52,150 --> 00:16:54,220 understanding of number for students. 250 00:16:55,630 --> 00:17:00,880 So when we look at this, we can see subtraction is distance. 251 00:17:00,910 --> 00:17:07,360 I start at 9 95, I need to go to 1012, which is to the right. 252 00:17:07,360 --> 00:17:09,880 So my direction is positive. 253 00:17:10,675 --> 00:17:15,055 Distance is always positive, but I can break this distance into we're 254 00:17:15,055 --> 00:17:17,905 five away here and we're 12 away here. 255 00:17:18,025 --> 00:17:22,705 So the total difference, the total distance difference is 256 00:17:22,705 --> 00:17:25,525 a combination of five and 12. 257 00:17:25,585 --> 00:17:26,965 If I relate this to a thousand. 258 00:17:26,965 --> 00:17:28,615 Lots of ways to think about that. 259 00:17:28,855 --> 00:17:33,175 We can use manipulatives, we can use bar models, concrete. 260 00:17:34,195 --> 00:17:38,395 Paper strips, lots of instructional strategies to use, but the point 261 00:17:38,395 --> 00:17:43,555 is for students to engage in this experience of figuring out the distance. 262 00:17:43,555 --> 00:17:45,625 How far apart are these two numbers? 263 00:17:45,835 --> 00:17:47,125 So five and 12. 264 00:17:47,335 --> 00:17:49,855 17. The distance is 17. 265 00:17:50,035 --> 00:17:54,800 If I switch this around 9 95 minus 1012, I start at 1012. 266 00:17:55,120 --> 00:17:56,240 I go to 9 95. 267 00:17:56,365 --> 00:17:57,895 The distance is still 17. 268 00:17:58,105 --> 00:18:00,840 The result is negative because I moved to the left direction. 269 00:18:02,650 --> 00:18:06,940 Now you might be saying again, why don't we just teach procedures for these? 270 00:18:07,390 --> 00:18:12,820 The issue is the procedures for single digit numbers, multi-digit 271 00:18:12,820 --> 00:18:17,080 numbers, and then negative numbers are all different types of thinking. 272 00:18:17,230 --> 00:18:19,450 We teach different strategies, and that's fine. 273 00:18:19,450 --> 00:18:21,070 We don't have to stop doing that. 274 00:18:21,280 --> 00:18:26,530 But the one strategy that works for every single number is looking 275 00:18:26,530 --> 00:18:28,120 at subtraction as distance. 276 00:18:28,390 --> 00:18:29,620 Let's look at this example. 277 00:18:31,585 --> 00:18:36,565 So in this example, my second number is negative eight, and 278 00:18:36,565 --> 00:18:38,695 I'm gonna go to negative three. 279 00:18:38,695 --> 00:18:43,885 Now, typically what happens in some of these problems is if I have an 280 00:18:43,885 --> 00:18:51,115 example, like negative three, minus positive eight, kids will frequently 281 00:18:51,115 --> 00:18:54,955 ask, is that second number, is it an eight or is it a negative eight? 282 00:18:55,555 --> 00:18:58,705 And typically the teacher response is. 283 00:18:59,110 --> 00:18:59,860 What's both? 284 00:19:00,650 --> 00:19:02,360 Technically, let's look at this. 285 00:19:02,360 --> 00:19:03,140 Technically. 286 00:19:03,680 --> 00:19:04,850 Technically. 287 00:19:05,270 --> 00:19:10,160 Remember when we make that subtraction symbol? 288 00:19:11,555 --> 00:19:15,095 The representation of a number line, so I'm gonna put arrows on the left 289 00:19:15,095 --> 00:19:20,405 and right end of the subtraction symbol technically right here as this appears. 290 00:19:20,675 --> 00:19:22,685 This is positive eight. 291 00:19:23,315 --> 00:19:29,795 So I have to mark this at positive eight on the number line. 292 00:19:32,225 --> 00:19:35,315 So if this is positive eight on the number line. 293 00:19:36,920 --> 00:19:39,950 Because it's positive eight right now, but I know what you mean, that that 294 00:19:39,950 --> 00:19:42,620 subtraction can change to addition, which makes it a negative eight. 295 00:19:42,620 --> 00:19:46,150 But right now it's subtracting a positive eight. 296 00:19:46,570 --> 00:19:51,790 So that means I'm starting at eight and then I'm going to negative three. 297 00:19:51,790 --> 00:19:54,850 I'm starting at eight and I'm gonna land on negative three. 298 00:19:55,780 --> 00:19:58,150 So right away I'm moving in. 299 00:19:59,020 --> 00:20:03,250 The left direction, and I can use these number lines vertically also up and 300 00:20:03,250 --> 00:20:05,710 down, and I interchange that constantly. 301 00:20:06,400 --> 00:20:08,800 So if I'm starting an eight and going to negative three, I'm 302 00:20:08,800 --> 00:20:10,960 moving in the left direction. 303 00:20:10,960 --> 00:20:14,890 So direction is gonna be represented by negative distance is always positive. 304 00:20:15,160 --> 00:20:18,190 So now if I go from eight, let's say to zero. 305 00:20:18,470 --> 00:20:19,700 That's a distance of eight. 306 00:20:19,700 --> 00:20:22,580 And then we have from zero to negative three, a distance of three. 307 00:20:22,970 --> 00:20:27,920 There's a distance of 11, the distance is 11, the direction is negative. 308 00:20:28,310 --> 00:20:35,400 So when we look at this in that way, the same strategy, technique and thinking 309 00:20:35,400 --> 00:20:37,045 that we used with the previous examples. 310 00:20:38,145 --> 00:20:41,205 We, we end up doing the same exact thing no matter what the number is. 311 00:20:41,205 --> 00:20:45,105 We're not changing the strategy or technique because now we have 312 00:20:45,105 --> 00:20:50,055 different types of numbers and we can negate that misconception that 313 00:20:50,055 --> 00:20:54,825 this is both a negative eight and a positive eight as it stands there. 314 00:20:55,135 --> 00:20:57,085 It is a positive eight. 315 00:20:57,415 --> 00:21:02,785 So we oftentimes, with our language and how we're responding to students 316 00:21:02,995 --> 00:21:07,045 are giving them misconceptions, and that is what's causing them a 317 00:21:07,045 --> 00:21:12,235 number of errors and inefficiencies in calculation and computation. 318 00:21:12,595 --> 00:21:15,505 So now let's look at the one that was actually the example 319 00:21:15,505 --> 00:21:16,825 that was actually here. 320 00:21:17,675 --> 00:21:24,935 Which is negative three, minus negative eight, and again, that subtraction symbol. 321 00:21:24,935 --> 00:21:29,735 If I put arrows on each side, we can clearly see that my 322 00:21:29,735 --> 00:21:32,315 second number is negative eight, 323 00:21:34,505 --> 00:21:39,335 and I'm going to go to my first number, which is negative three, 324 00:21:41,825 --> 00:21:45,305 and then I can mark zero just for. 325 00:21:46,070 --> 00:21:48,710 Conceptual, just framework really. 326 00:21:49,340 --> 00:21:54,200 So now I'm going from negative eight to negative three. 327 00:21:56,075 --> 00:22:00,485 So my result, my direction is positive distance is gonna be positive. 328 00:22:00,695 --> 00:22:04,355 So now negative seven, negative six, negative five, 329 00:22:04,355 --> 00:22:06,185 negative four, negative three. 330 00:22:06,425 --> 00:22:08,915 How many hops, how many spaces? 331 00:22:09,245 --> 00:22:15,245 And again, an intervention for students to have better access to this is to create 332 00:22:15,245 --> 00:22:18,275 the bar model over top of the number line. 333 00:22:18,635 --> 00:22:19,295 And. 334 00:22:19,550 --> 00:22:25,010 Instead of tick marks, create dividing up a bar so that we can clearly 335 00:22:25,010 --> 00:22:32,000 see there's 1, 2, 3, 4, 5 spaces. 336 00:22:33,710 --> 00:22:39,890 As we look at cycle P. Same thing with all of these, but the one thing I wanna point 337 00:22:39,890 --> 00:22:43,310 out is this example in some of the cycles. 338 00:22:44,195 --> 00:22:48,245 We add in decimal numbers and fraction numbers as well. 339 00:22:48,245 --> 00:22:51,995 That's why we're giving you cycle F and cycle P today. 340 00:22:52,385 --> 00:22:57,005 There's importance in larger numbers, multi-digit numbers. 341 00:22:57,215 --> 00:23:00,665 Also importance in decimal numbers and fraction numbers. 342 00:23:01,025 --> 00:23:02,285 But let's see how this works. 343 00:23:02,285 --> 00:23:03,070 If I have a number line. 344 00:23:04,310 --> 00:23:08,780 And my second number is, oh, if I put my number line here, there's 345 00:23:08,780 --> 00:23:09,860 not gonna be any confusion. 346 00:23:09,860 --> 00:23:12,050 That's a positive three decimal five. 347 00:23:12,380 --> 00:23:15,170 So if I put my three decimal five here 348 00:23:17,480 --> 00:23:22,610 and my Landon number, my distance from three decimal, five to seven, 349 00:23:22,640 --> 00:23:27,020 decimal two five, and you can see this gets a little more complicated. 350 00:23:27,020 --> 00:23:30,530 But again, I'm not looking for efficiency and accuracy. 351 00:23:30,830 --> 00:23:33,470 I'm looking for a thinking exercise. 352 00:23:34,325 --> 00:23:37,085 That's going to contextualize and conceptualize. 353 00:23:37,745 --> 00:23:42,935 Subtraction being distance so that I can improve my estimation, my counting 354 00:23:42,935 --> 00:23:48,605 skills, which estimation and counting are fundamental for number sense. 355 00:23:48,785 --> 00:23:51,665 So if you have students that have weak number sense, yes, they're 356 00:23:51,665 --> 00:23:56,345 gonna struggle with these thinking exercises, but the more exercises we 357 00:23:56,345 --> 00:24:00,785 have them do with this, the better their sense of numbers is going to be. 358 00:24:01,340 --> 00:24:05,000 So here, this is gonna be a little tricky because it's okay, three and a half. 359 00:24:05,270 --> 00:24:06,110 Where's three? 360 00:24:06,110 --> 00:24:06,920 Where's four? 361 00:24:06,920 --> 00:24:08,330 Maybe this is three. 362 00:24:08,330 --> 00:24:09,230 Maybe this is four. 363 00:24:09,350 --> 00:24:11,300 So there's a little playing around here with. 364 00:24:11,885 --> 00:24:13,205 How much this is. 365 00:24:13,235 --> 00:24:14,045 Oh, but wait a minute. 366 00:24:14,045 --> 00:24:15,785 I know seven is here. 367 00:24:15,905 --> 00:24:18,215 Actually seven is gonna be a little closer to that line. 368 00:24:18,215 --> 00:24:19,445 I didn't draw it to scale. 369 00:24:19,445 --> 00:24:20,885 We can have those discussions. 370 00:24:21,095 --> 00:24:26,275 We can actually do draft two with three decimal five and seven decimal two, five. 371 00:24:26,545 --> 00:24:31,195 So seven is gonna be closer to seven, decimal two, five, but four 372 00:24:31,195 --> 00:24:34,615 is gonna be further like we can start to scale it a little bit better. 373 00:24:35,335 --> 00:24:40,345 And modeling this for students and then having them work independently on it is 374 00:24:40,345 --> 00:24:45,325 going to be the key to engaging them in deep thinking, reasoning and sense making. 375 00:24:45,415 --> 00:24:46,160 It is not gonna be easy. 376 00:24:47,500 --> 00:24:51,280 It's not about accuracy and efficiency, it's about playing with it so that they 377 00:24:51,280 --> 00:24:55,780 can figure it out and they can start to understand value, size, magnitude. 378 00:24:55,780 --> 00:24:59,260 Magnitude meaning how large and small numbers are and where numbers 379 00:24:59,260 --> 00:25:02,950 are in the number line, and in this case, how far apart numbers are. 380 00:25:03,370 --> 00:25:06,790 So once we have decimals and fractions in the mix. 381 00:25:07,090 --> 00:25:09,550 The thinking exercise extends. 382 00:25:09,550 --> 00:25:13,510 We wanna give kids lots of time to think about this, lots of time to play with it. 383 00:25:13,690 --> 00:25:16,030 And then what I might do is I might say, you know what? 384 00:25:16,030 --> 00:25:17,440 We're not gonna finish this today. 385 00:25:17,860 --> 00:25:19,330 We're not gonna finish this today. 386 00:25:19,600 --> 00:25:21,610 We're gonna come back to this another day. 387 00:25:21,910 --> 00:25:28,540 Because thinking exercises, if we finish thinking exercises and we confirm. 388 00:25:29,465 --> 00:25:34,505 Did you know that letting things linger actually improves thinking skills? 389 00:25:36,035 --> 00:25:40,985 By letting things linger, you're letting the brain process what you've just taught. 390 00:25:41,765 --> 00:25:44,915 Rather than give it the answer and let it stop thinking. 391 00:25:45,665 --> 00:25:48,215 Don't give them the answer, let it think. 392 00:25:48,665 --> 00:25:51,425 You'd be surprised how many kids will come back to you two or three 393 00:25:51,425 --> 00:25:53,495 days later and say, Hey, Mrs. 394 00:25:53,495 --> 00:25:56,155 Or Mr. I think I figured out the answer to that thing that 395 00:25:56,155 --> 00:25:58,345 we were talking about on Monday. 396 00:25:59,005 --> 00:26:01,315 And you'll be going, why? 397 00:26:01,345 --> 00:26:05,965 How do they, how are they thinking about this two or three days later? 398 00:26:06,185 --> 00:26:09,455 Their brain's been processing it ever since you left it linger. 399 00:26:11,067 --> 00:26:13,767 Jonily: Or deny accuracy and efficiency. 400 00:26:14,127 --> 00:26:17,547 Then the brain stops thinking about the thinking exercise. 401 00:26:17,607 --> 00:26:22,317 However, if we don't confirm or deny accuracies, we gather lots of answers. 402 00:26:22,557 --> 00:26:27,747 We let the conclusion linger and come back to it maybe three days later 403 00:26:27,747 --> 00:26:29,457 or a week later, or two weeks later. 404 00:26:29,907 --> 00:26:31,407 The subconscious mind. 405 00:26:31,762 --> 00:26:36,862 Automatically continues to think about it because there's this cognitive dissonance. 406 00:26:37,012 --> 00:26:39,082 There's not been a completion or a closure. 407 00:26:39,502 --> 00:26:43,432 What we want in mathematics, we want the brain to work over time 408 00:26:43,582 --> 00:26:44,967 too often in our math classes. 409 00:26:46,407 --> 00:26:49,887 We have this, and Dan Meyer said it beautifully a number of years ago. 410 00:26:50,067 --> 00:26:54,997 We have this sitcom approach to mathematics where, when you watch this 411 00:26:54,997 --> 00:26:59,917 show for 30 or 40 minutes and we expect everything to be wrapped up at the end 412 00:26:59,917 --> 00:27:03,667 of that time, and we're rushing to get through and we're not allowing students 413 00:27:03,667 --> 00:27:11,707 to process and have cognitive subconscious thinking that they're not even aware of. 414 00:27:12,577 --> 00:27:14,347 So we're limiting that. 415 00:27:14,377 --> 00:27:19,127 And so the brain stops working because there's no dissonance there's no conflict 416 00:27:19,127 --> 00:27:20,687 because everything has been wrapped up. 417 00:27:20,927 --> 00:27:25,307 The only conflict is some of our students that have slower processing 418 00:27:25,577 --> 00:27:27,227 or didn't get enough time to think. 419 00:27:27,492 --> 00:27:32,472 They stop thinking during math classes because they know other people 420 00:27:32,472 --> 00:27:36,252 are gonna think for them and other people are gonna finish for them. 421 00:27:36,672 --> 00:27:41,352 So there's an implication to the way that we facilitate. 422 00:27:41,652 --> 00:27:46,602 So oftentimes I will let the answer. 423 00:27:47,042 --> 00:27:50,972 The accurate answer, linger on this and come back to it two or 424 00:27:50,972 --> 00:27:54,482 three times, and that's one of the reasons for the cycle as well. 425 00:27:54,572 --> 00:27:58,352 These are thinking exercises to improve number sense. 426 00:27:58,472 --> 00:28:04,802 We use these subtraction cycles and by using those subtraction cycles, we can 427 00:28:04,802 --> 00:28:09,902 improve number sense and also increase. 428 00:28:10,457 --> 00:28:15,437 Motivation, focus, engagement, reasoning, and sensemaking. 429 00:28:15,557 --> 00:28:18,857 And those are the culture of math classrooms that we're trying to 430 00:28:18,857 --> 00:28:21,737 create to produce amazing math. 431 00:28:21,737 --> 00:28:27,527 Iners today was the extended version of a previous podcast 432 00:28:27,527 --> 00:28:29,112 that was released on February. 433 00:28:29,627 --> 00:28:35,027 18th, 2025 titled Hidden Math Patterns That Make Subtraction Easy for Everyone. 434 00:28:35,477 --> 00:28:41,507 That podcast episode was the intro version to today's extended version, 435 00:28:41,557 --> 00:28:48,997 the part two version, but this is also as part of a series of podcasts that 436 00:28:48,997 --> 00:28:52,507 are three math rules that never expire. 437 00:28:52,797 --> 00:28:56,757 Rule number one division is counting rule number two. 438 00:28:56,757 --> 00:28:58,317 Subtraction is distance. 439 00:28:58,527 --> 00:29:01,677 And rule number three, multiplication is rectangle. 440 00:29:02,127 --> 00:29:05,007 I was your host today, miss Jay-Z. 441 00:29:05,007 --> 00:29:10,047 Jonily Zupancic here with Tier One Interventions podcast. 442 00:29:12,478 --> 00:29:16,078 Hey, thanks for joining us today for this episode of Tier One Interventions. 443 00:29:16,258 --> 00:29:20,578 If you found today's strategy is helpful, imagine what your students could achieve. 444 00:29:20,758 --> 00:29:28,228 If every math lesson clicked this way, make sure you subscribe to us and 445 00:29:28,468 --> 00:29:30,778 set this a comment in the show notes. 446 00:29:30,778 --> 00:29:31,228 The. 447 00:29:33,193 --> 00:29:36,253 Send us a comment and let us know what you thought of this episode. 448 00:29:36,643 --> 00:29:40,243 Don't forget to grab your spot at Mastery Math Method masterclass, 449 00:29:40,483 --> 00:29:46,393 where you can learn how to make your math stick in 10 days. 450 00:29:47,053 --> 00:29:51,433 Go to disability labs.com forward slash calendar to register. 451 00:29:51,973 --> 00:29:54,583 Until the next time, go be awesome. 452 00:29:54,733 --> 00:29:55,573 Go be brilliant. 453 00:29:55,693 --> 00:29:58,033 You were put here for such a time as this. 454 00:29:59,353 --> 00:30:03,178 Make tier one strong so that every student can thrive.