Hidden Math Pattern That Makes Subtraction Easy For Everyone: S2 E19

Unknown Speaker 0:00
Steve,

Jonily Zupancic 0:03
Hello everyone, and welcome to tier one interventions podcast. You may be listening to this on any podcast platform that you choose, but we encourage you to listen and watch this podcast on YouTube at tier one interventions podcast, I'm your host today. Jay Z Joni Zupancic, my partner, host, co host, co author. Cheri Dotterer, occupational therapist is not with us today, but you've probably heard her on many other episodes. I am your math specialist, math coach, math girl, bringing you part two of a special series that we have been releasing on this podcast. This series is called Three math rules that never expire. There was an article published in 2014 and to give credit where credit is due, the article by carp bush and Dotty, Dotty, dot t i know I'm mispronouncing that, and so someone will enlighten me, as you are listening to this podcast. But the article, as to credit those three authors in 2014 was called 13 rules that expire. The premise is 13 math rules that expire as kids move to more complex number situations and into secondary school. A couple of the rules are things like, when you multiply, you get a larger number. When you subtract, you get a smaller number. Those rules only work in certain instances. And in mathematics, oftentimes, we will make a conjecture or a rule, and we have to decide whether that rule is always, sometimes or never true. So it's really good any statement we make as an educator or any statement that students make, mathematically, these statements we call conjectures, and we are trying to prove or disprove conjectures, but any conjecture or statement that we, especially as educators, make, we need to ask ourselves and the students, is this an always, sometimes or never So play the always, sometimes, never game, because with the subtraction rule that expires from the article in 2014

Unknown Speaker 2:29
the rule meaning when you subtract, you get a smaller number. That's only sometimes true, and whether you want to venture into the sometimes that it's not true with younger students, is entirely up to you, but we recommend, with our progressions of three math rules that never expire, that we introduce students to these concepts at an exposure level. We're not expecting them to master it, but exposing students to these concepts and mindsets as early as preschool and in the toddler years, the three math rules that never expire are these. Number one, division is counting. So this is very easy as kids are toddlers and preschool age, counting, wrote, counting with no meaning, 2468, who do we appreciate division is just wrote counting and skip counting. Division is also counting through sharing. And we talked about this in part one of the episode. But rule number one that never expires, division is counting. Rule number two that we're going to address today, subtraction is distance. Yeah, subtraction is takeaway. Subtraction is many other things. Subtraction could be an opposite situation, but subtraction is distance is transferable and progressive. It creates a good transformational progression from preschool through high school. It's the same no matter what types of numbers you're using. And that's what's powerful about these three math rules that never expire. The third math rule that never expires is multiplication is rectangles. We are going to address rule number two today. Subtraction is distance. If you are listening to this podcast audio only, I will be sure to let you know the visuals that others are seeing. But if you want to see the visuals and the note taking, hop on over to YouTube, tier one interventions podcast on YouTube, and you will be able to see the note page and the examples, but I'll be producing them through speaking, So you have the auditory connection as well. But subtraction is distance you are going to if you're watching this, see a notes page that's called Intro to subtraction cycles for grades two through 10, and I highly recommend that we use this in earlier grades than grade two. But definitely, absolutely.

Unknown Speaker 5:00
As is in grade two. What we mean by subtraction is distance is just these four problems. These are the only ones I'm going to address today, because this is an intro. As a matter of fact, if you are interested, if you're listening listening to this podcast on the day that it is released, tomorrow evening. When I say tomorrow, when this is going to be released? Tomorrow is going to be January 15.

Unknown Speaker 5:30
It is a Wednesday. We have a live session on this tomorrow night that you can be with us, live on impact Wednesday. We are going to extend on this topic, Intro to subtraction, and talk about the importance but the complexity of number lines. And we won't take this podcast to that level, but join us tomorrow night on impact Wednesday, we'll put the link in the podcast notes, in the show notes, but if you can't find the link, go to Eventbrite, e, v, e, n, t, b, r, i, t, e, eventbrite.com and search for minds on math, and you will find registration for impact. Wednesday, it is free, but you must register to get the Zoom link. Today, we're going to focus on these four problems, nine minus two, two minus nine, 1003 minus 998,

Unknown Speaker 6:24
and nine minus negative two and yes, this entire cycle is appropriate and necessary for grades two through 10, and we have one of our certified coaches that is a first grade teacher that also dabbles in these same Numbers, just for exposures. It's important to note that when we define subtraction as distance, it just becomes a counting exercise. All we are doing is counting, and I'm going to show you in these four examples. So if younger students don't get the exact count number, but can get close, this concept of subtraction is distance is helping them with estimation of subtraction problems, the accurate and precise answer using procedural notations and processes is not the goal here. The goal is to improve number sense and understand closely how far apart numbers are because that's the pure definition of subtraction. So let's take a look at this. We'll make this a little bigger. We're going to look first at the problem nine minus two. And as we look at nine minus two, that minus sign actually is a number line. It is the indication of where we go from and to on a number line. So if we physically make the subtraction symbol a number line, by drawing arrows at each end of the subtraction symbol, we are signifying that we are trying to find the distance between the second number which is two to the first number, which is nine. So we want to know what is the distance on a number line from two to nine. So if I draw a larger number line,

Unknown Speaker 8:11
and I place two and nine

Unknown Speaker 8:14
wherever, except we have to place them so that nine is further to the right. The subtraction problem. Every subtraction problem means that the second number is our starting number. So we're going to start here on two on our number line, and we are going to figure out what the distance is, going to nine. Now we have to make a special note, because we're starting at two and going to nine. We're going in the positive direction. Distance cannot be negative. Distance can only be positive, but direction can be positive or negative. So the distance between two and nine is absolutely the same as the distance between nine and two as in the next problem, but the direction is going to be different. So the sign positive or negative of the answer is going to be different depending on the direction and direction matters. Oftentimes, in seventh grade, teachers will have students walk a number line when they're learning integers, and we sometimes equate walking backwards with a negative, and that is creating a misconception. Walking backwards still creates a positive distance. Distance is positive. Whether we're walking forward or backward, we're still covering distance. So the distance is always a positive number, a positive concept. We can walk forwards or backwards. That's irrelevant. The way that we go left or right on the number line determines whether our answer, our result, is going to be positive or negative. Direction can be positive or negative. Distance is positive. Now this actually moves into the concept of apps.

Unknown Speaker 10:00
Absolute value. So in

Unknown Speaker 10:03
in creating this progression of subtraction is distance, starting in preschool, when we get to absolute value, which is a distance, but this time it's a distance from zero. Students have a really great foundational concept. So the distance from two to nine, we could do two to five, is three. Five to nine

Unknown Speaker 10:24
is four, three and four is seven. The distance is seven, nine minus two is seven. Two minus nine, I can actually use the same exact number line, because the distance is going to be the same. We start with the second number. So this time, two minus nine, we start at nine now. So I'm gonna star that with red, because this time, we're starting at nine and we're going to the number two. So if we start at nine and go to the number two, we're moving in the negative direction. We're moving to the left. We can also do the number line vertically, so we can go up and down instead of left and right. And I would suggest with subtraction, we change between one and the other. Toggle between those if we're starting at nine and going to the number two, the distance is the same. The distance before was seven, the distance is still seven. However, because we went to the left in the negative direction, the answer to two minus nine is going to be negative seven. We're going to use this same strategy, this same thinking, the same mindset, is really what it is. We want to ingrain this number sense mindset in students that every time they see subtraction, it's distance. When I look at this third example, we see 1003 minus 998

Unknown Speaker 11:43
I turn the subtraction symbol into a number line by putting arrows on the left and right of the subtraction symbol. And by definition, we start at not I think I said 999 but I meant 998 if I said something different, it's 1003 minus 998 so the second number is 998 that means we start at 998

Unknown Speaker 12:05
on the number line, and we're going to go to 1003

Unknown Speaker 12:10
and we are trying to figure out the distance between those two numbers. If I start at 998 and move to 1003

Unknown Speaker 12:18
my answer, my result, is going to be positive, because I'm moving to the right in the positive direction. I'm starting at 998, landing on 1003

Unknown Speaker 12:27
and if I count up 998-999-1000,

Unknown Speaker 12:32
it's two counts to get to 1000 and from 1000 to 1003 is five counts. So the distance is five. And this is a really good process for kids that are just learning about larger numbers. It's really a great process, even for our fifth, sixth, seventh and eighth graders and high school students who have never really mastered the procedure of stacking and subtracting with regrouping borrowing. Because if you stack these numbers, 1003 on top of 998

Unknown Speaker 13:04
there's a lot of room for error in crossing out, borrowing, moving through that procedure. We want kids to master that procedure, but the fact is, about five, maybe 10% of our students really struggle to master that procedure. And by secondary, middle school and high school still don't have that procedure mastered. So we want to make sure that as they're attempting to master the procedure, or as we're showing them the procedure, or as they're trying the procedure, they can also relate it to subtraction as distance. And when you do the procedure, you should get a positive five, and if you don't, there's a flaw in the procedure, so using subtraction as distance to help confirm results from the procedure. Now if we look at this final one here, and this is even important for second, third, fourth graders, even kids before negative numbers need to be mastered. They need lots of exposure. This problem is nine minus negative two. The minus sign, again, is our number line. We put an arrow on the left and right of the subtraction sign, so it turns into nine and negative two are my two numbers. My second number is negative two. And I want to know the distance from negative two to positive nine. This is where we really get the power of this mindset, belief and strategy of subtraction being distance, because this is transferable to all age levels, all grade levels, and all number types, even decimal fraction, negatives, etc. So if I put negative two to the left of nine, I could kind of estimate in here where zero would be. That's a good exercise for kids. And you can see because this is much less procedural and very much conceptual. This is.

Unknown Speaker 15:00
The exact exercise we need to improve number sense for our students. So just the exercise of plotting this and trying to figure out close to the distance, not even the exact distance, is a way for students to understand number better. We know that number sense is an innate, intuitive understanding and inborn understanding, and many of our students do not have this innate, inborn, intuitive understanding of number and they cannot be explicitly taught that the only way we can improve number sense is through experiential learning and for students to engage in the process, engage in the experience. And this process is exactly that. So you see here cycle a are these four problems and you can actually get this. There will be a link to how to get this in the show notes, if you want a copy of this notes page of examples from today's podcast. However, we do have a package bundle for purchase that not only will you get cycle a, but you'll get cycles a through almost z4, problems in each cycle. And you will have this to use with your students and with your facilitation. The number selection is important. So can you make up your own four problem cycles? Your your own cycle sets absolutely but the ones that are made here, if you end up getting the bundle, are very deliberate and intentional with the numbers that are selected and the order that the numbers are selected. I won't go into detail why there is a rhyme and reason to the selection of the number and the order. So let's go back to our final example. We start at negative two. This is a zero. It kind of looks like a degree symbol. We start at negative two because that's the second number, and we go to nine. But guess what? We're going to the right so my result, my answer, my difference is going to be positive because the direction is positive

Unknown Speaker 17:07
and the distance then that is always positive. If I go from negative two to zero, there's a distance of two, and from zero to nine is a distance of nine. So there is a distance here of nine and two a distance of 11, and the result is positive 11. Play around with this. Create your own cycles. Give it a try. Try this with students. This is the part two of a series of three math rules that never expire, appropriate for preschool through high school, and these cycle sets are completely appropriate and necessary for grades two through 10. But try these with your first graders as well. Again, this is part two in a series of three math rules that never expire. I am your host at this tier one interventions, podcast, Jay Z Joni zupanczyk, and until next time

Unknown Speaker 18:07
you.

Transcribed by https://otter.ai