So let's extend our arithmetic to dealing with integer multiplication. So everything you need to know about the multiplication of integers can be found in the following. First, we have to know the definition of the additive inverse of A. Again, so that A plus its additive inverse gives you 0. The multiplication of integers is associative and commutative. This is actually a theorem, but the proof is somewhat complicated, so we're not going to go into it. And our definition of multiplication, if A is a whole number and B is an integer, then A times B is, well, this is essentially the same definition we had before. It's the sum of A copies of B and conversely. So, for example, let's say the problem is 5 times negative 3 without using any property of integer multiplication to defend your steps. So you might have learned at some point that the product of a positive and a negative number is a negative number, but that's a property of integer multiplication, and the problem very specifically says don't use those properties. So what can we do? Well, a definition is not a property. We can use the definitions, and again, since we know the definitions, we can do mathematics. If you don't know the definition, you cannot do mathematics. So let's see, 5 times negative 3, well, my definition of times says this is 5 copies of negative 3 added together. So 5 times negative 3 looks like this. Now, again, my addition is part of my definition of multiplication, so I can find the sum without comment. So at this point, I can say, well, the right-hand side here is what I get when I add a whole bunch of negative 3s together. That's going to be negative 15. And important to note, this problem we can only solve by writing these two things down. That 5 times negative 3 is this sum, which is the same as negative 15, and this is from our definition of multiplication. Well, it turns out that there are some useful results that will make multiplication of the integers a little bit easier. Now, here's the primary one. If a is a whole number, then the additive inverse of a is the same as a times the additive inverse of 1.  Now, so what can we do with that? Well, let's see if we can prove it. So again, if you know the definitions, then we can do this pretty easily. If you don't know the definitions, you can't do mathematics. So what do we know? Well, we know the definition of additive inverse. a plus its additive inverse is going to be equal to 0. However, a as a whole number is the sum of a whole bunch of ones, a times, which says that the additive inverse of a has to be the sum of the same number of negative ones. Because when I add these two things together, I can cancel out each negative one. It takes away one of the positive ones. So I know something about a and also the additive inverse of a. And so that says I can rewrite a plus the additive inverse. Well, additive inverse is this sum. They are equal. So if I see the one, I can replace it with the other. So I'm going to take this additive inverse of a and I'm going to replace it with this sum of negative ones. a plus a whole bunch of negative ones is equal to 0. But if I compare those two statements, what I see is that they're the same. a plus a plus equals 0 equals 0. The only difference is the one statement has this additive inverse of a. And the other statement has this sum of negative ones. And so that's the only difference of two things that are otherwise identical. And so it must be that additive inverse of a is the same as a whole bunch of negative ones added together. Well, because I know the definitions, I can do mathematics. This thing, when I add a whole bunch of things together, is a multiplication. And specifically, this is a copies of negative one. This thing on the right is a times negative one, and there is my result. The additive inverse of a is the same as a times the additive inverse of one, which is what I was looking to prove. Well, let's use only this property, definitions, associativity, and so on, and find negative five times four, defend your steps. And again, a quick wrong answer. Negative five times four equals negative twenty because the product of a negative and a positive is a negative. And well, we defended our steps, but the important thing is that our defense here was not this property, was not a definition, was not associativity and communitivity, and was not whole number arithmetic. This defense uses something that we're not allowed to use. And so this is not a correct answer. So let's take a correct answer. Well, we do have that property, negative a equals a times negative one. So this negative five, I can rewrite as five times negative one, and associativity and communitivity holds, so I'll rearrange things. I can do whole number arithmetic, five times four is twenty, and I have my property, anything times negative one is the same as negative times the number. And so there's our final answer.


is a much better approximation of what a mathematician actually does. I think unfortunately the way math is generally taught, people see math as just a sequence of procedures or rules that have to be followed and they don't really see the creativity that goes into it. I mean really mathematics is sort of a process of discovery. You're kind of figuring out what's going on, how things are connected, what the patterns are, and you're recognizing these things and then trying to make general statements that represent what's happening. And then once you understand why it's happening, the proof is just sort of the finishing touch where you write a logical argument that explains why something is true. And so I think when they're working on the worksheets and they're able to sort of figure things out for themselves they get much better kind of concrete understanding of why things are the way they are. In terms of getting them prepared, a lot of onus is on them to really keep a record of their work because I don't collect a lot of this stuff. The investigations they make on the worksheet, I don't collect any of that, that's for them to understand and discover for themselves. And then even with the theorems they get graded on their proof presentations but they don't do that many of those in a quarter. So they really are responsible for keeping a good record of their work, keeping a sort of a polished version of each theorem even if they're the ones that they don't present, they really should be keeping their written proofs for all of the theorems. Another thing that I do to help them have sort of kind of a class set of finished proofs is we use our learning management system for the course and so each person that presents a proof is responsible for after they've had the feedback from the class writing their final version as a page on the learning management system so that everybody can go and view that and actually they can even edit on there and give responses on there if they want. But that way we have sort of one version of kind of a polished proof for each of the theorems that everybody can access and so they can use that to study from as well. The vibe can it's it's really dependent on the group of students I mean I've had I've had quarters where unfortunately whatever the whatever the chemistry of the class of the group of students is it just somehow they don't gel and it's kind of harder and they're more quiet and more reticent and those are tough quarters. This quarter is actually great from the get-go they were just ready to go and really excited about it. When it's going well you know I pass on a worksheet and they just dive right in and they're excited and they're talking about it right away and they're trying things out and then they're kind of checking in with each other to see you know if they're in agreement on certain questions and then sometimes someone will go up to one of the boards because they're trying to work something out. So yeah it is messy and it's kind of it's not really loud because they're sort of working in groups but it's you know people are sort of all over the place and sometimes working on the board and sometimes doing what they're doing. That's on the worksheet days the presentation days look a bit more normal because we're all at least sitting facing the front and someone's at the board.


This study aimed to create an SLM-level map of Batafa region in northern Iraq using random forest and extreme gradient boosting models with remote sensing data, supplementary variables, and machine learning algorithms. The XGBoost model exhibited higher accuracy than the RF model, and Band 10, DEM, SAVI, and NDVI were identified as the most important predictors for both models. The methodology has the potential to map a SLM in similar settings and offer significant insights for soil management stakeholders. This article was authored by Helmat S. Kalef, Yasim T. Mustafa and Mohamed A. Fayat.