# Tutor profile: Matt M.

## Questions

### Subject: Geometry

We're starting proofs, and I don't get it. The teacher showed us some examples of "proving" equations using two-column proofs, and she said if we don't get this then we're going to have even more trouble when we start doing real geometry proofs. Usually math is kind of fun, but proofs are weird. I really want to get this, but I'm stuck. So, the first thing I'm trying to get is this list of rules. My teacher said that these are the basic rules to get first: Reflexive Symmetric Transitive Substitution The first two basically make sense, I think. They're kind of obvious, really. The last two are harder. They kind of seem the same as each other, but I don't know. I guess that's where I'm really stuck.

It's good that math is usually kind of fun for you. It's a good feeling when you solve a problem and you really understand what you're doing. I think with a little work you'll start feeling that way about proofs, too. I agree with your teacher that those four rules are the first things to get. Geometry proofs will add a lot more rules to that list as you go along, but those four will always be special. I think of those four as tools for putting things together. A proof is like a building you put together. The long list of geometry rules that come later will be like all the different materials you'll build with, but these first four rules are the basic tools you'll use over and over to work with those different building materials. That's one way to think about it. Learn how to use your tools well, and you can build a lot of different stuff. So, yeah, the first two rules are simple. They're kind of like, "Duh, no kidding." I mean: Reflexive: $$x=x$$ Okay, I got that. Then: Symmetric: If $$x=y$$ then $$y=x$$ That's obvious, too. Just flip the equation around. You can't be that simple with an inequality. You can't say: If $$x<y$$ then $$y<x$$ But with the equals sign, sure, no problem. Now, the last two are more involved. And, yeah, I also think they're close to each other. Let's do some simple proofs to see first how Substitution and Transitive can look the same, and then see how they're different, okay? 1. $$a=b$$, Given 2. $$a=c$$, Given 3. $$b=c$$, Substitution, using lines 1 and 2 Substitution here says that since $$a$$ is $$b$$ (line 1), then anywhere I have an $$a$$ I can replace that with a $$b$$. That's all I did to get line 3. I just looked at line 2 and replaced an $$a$$ with a $$b$$. That's Substitution. Substitution says: "If two things are equal, then one can be substituted for the other." Now, here's Transitive: 1. $$a=b$$, Given 2. $$a=c$$, Given 3. $$b=c$$, Transitive, using lines 1 and 2 This is the exact same proof as the first one, but I just wrote "Transitive" in line 3 instead of writing "Substitution." What's up with that? Transitive says: "If two things are both equal to the same thing, then they're also equal to each other." In line 1 $$b$$ is equal to $$a$$, and in line 2 $$c$$ is equal to $$a$$. So, since $$b$$ and $$c$$ are both equal to the same thing (they're both equal to $$a$$), then $$b$$ and $$c$$ are equal to each other. That's Transitive. So, I see why you were wondering if Substitution and Transitive were the same thing. They can look the same. But they are different in how we state each rule. Look again at the statement of each rule: Substitution: "If two things are equal, then one can be substituted for the other." Transitive: "If two things are both equal to the same thing, then they're also equal to each other." What do you think the main difference is between these two rules? How did we think differently in each proof? The way I think about it is that with Substitution you just look at one line, and you see from that one line the substitutions you can make. Then you go around anywhere in your proof making those substitutions. In the first proof we looked at line 1 by itself, and we saw that anywhere we have $$a$$ in our proof we can replace it with $$b$$. Then we did that replacement to line 2, changing line 2 to line 3. With Transitive, though, you look at two lines at the same time, and from those two lines together you see what the third line is. In the second proof we looked at lines 1 and 2 at the same time, and then we saw that $$b$$ and $$c$$ were both equal to the same thing, and so we wrote down that $$b$$ and $$c$$ were equal to each other. Take a minute to look back at the proofs and think about that difference. Do you see the difference? As you go through more complicated proofs you'll see when it's best to use Substitution and when you can use Transitive. They're not always interchangeable like they were in our simple example, but you'll see that as you go along. Right now, as you're first getting started with proofs, just thinking through the difference between Substitution and Transitive will help prepare you for what's to come. Have fun with it! Well, that's a lot of talk about two little rules. But, like we said at the beginning, these rules are your tools, and more you know your tools the less you'll be a fool, and the cooler you'll be in school. (Okay, you can forget that last part. It won't be on the test.)

### Subject: Databases

For my database design class I came up with a three-table schema to store orders that come in to a retailer and are then fulfilled from multiple warehouses. The order header table stores the basic order info like the customer name and the ship-to address. This header table has a one-to-many relationship with an order product table where each order can have many products. Finally, the order product table has a one to many relationship with an order product fulfillment table where any individual product ordered can potentially be fulfilled from multiple warehouses. I thought it was good, but my teacher said the keys weren't normalized. The key to the header is order_no, an automatically assigned unique number. To allow any product to be ordered more than once on a single order, say, if the customer wants different quantities of a single product to ship on different ship dates, I made the key to the order product table (order_no, order_line_no), where order_id is a foreign key to the header table and line_no starts at 1 for a given order and keeps increasing for each line added to that order. Finally, the key to the fulfillment table was (order_no, order_line_no, warehouse_id), to allow any given order line to be fulfilled from multiple warehouses. So, what's not "normalized" about that? What's my teacher talking about? I've read through the five normal forms, and I don't really see how I'm violating any of them.

It looks like you've thought this through carefully, and built in a lot flexibility. I think your schema could work as intended. However, let's think how you might get the same level of flexibility in a more "normal" way. Perhaps this will lead to a design more pleasing to your teacher. Since you've already read the definitions of the normal forms I won't use our session time rehashing that. I encourage you to make sure you're understanding them, especially the first three forms, but, instead of going over that here, let me give you a simple rule that's related to normalizing and that I've come to live by: "Composite primary keys are reserved for intersection tables." In other words, composite primary keys should be composed entirely of foreign keys. Looking at the order product table we see a multi-part primary key, but order_line_no is not a foreign key, and so it violates this rule. How can we fix that? We can take order_id out of the primary key and demote it to be a not null foreign key. Then we change order_line_no to be an auto-incrementing number that's unique across the entire order product table, not just unique to its individual order. This accomplishes the same level of flexibility, and doesn't enable violation of any other constraints you might have in the schema. This change means the primary key to the fulfillment table will now be (order_line_no, warehouse_id). The primary key there is still composite. Does it violate our composite primary key rule? No, because there are nothing but foreign keys there. In this case the fulfillment table is a true "intersection" table, defining what warehouses are used for each order line. So, that's my suggestion. Take that schema change back to your teacher and see what he says. Now, if you want to spend a few minutes making sure that what we did here makes sense, can you tell me what schema change you could make if you wanted to allow a single order line to be fulfilled multiple times from the same warehouse? I think you see that your current design doesn't allow that, and think that's your intent. But, what if you did want to allow that? How would you follow the pattern of what we did with the order line table? In the fulfillment table you would demote both order_line_no and warehouse_id from primary key to not null foreign key, then add a new auto-incremented column, say, order_line_fulfillment_no, and make that the new primary key. Keep having fun in the world of RDBMS schema design.

### Subject: Algebra

When I complete the square, why do I add "half the b term squared" to both sides? I mean, it works, I get that, but, it seems so random. Why does it work?

It's great that you want to know "why." Knowing why helps you remember how to do it, and makes the whole thing more fun and interesting. I like pictures. They say, "A picture is worth a thousand words," but we can't draw good pictures with the tools we have in this session, so, let's see if we can build a mental picture, hopefully using less than a thousand words! Let's build up our equation term by term, and, as we go, we'll build up in our mind a picture for it. If you want to follow along by making your own drawing on your own paper, that might help, too. So, here's the first term of our equation: $$x^2$$ What does that mean in a picture? I see a square with sides of length $$x$$. $$x^2$$ is the area of an "x by x" square, right? So, when you see that $$x^2$$ term, then picture an "x by x" square. Let's keep going: $$x^2+bx$$ Next to our square we now have a "b by x" rectangle. See it there? Turn this rectangle so its "x" side is next to the square. If fact, put the "x" side of the rectangle right up next to the right side of the square, touching the square. Got it? Now the whole thing is a new rectangle that's $$x$$ tall and $$x+b$$ wide. But we don't like rectangles so much. We like squares. Squares are simpler. All the sides are the same, you know? Boring? Maybe. Easier? Definitely. Let's try to get back to a square. We can get closer to a square by slicing that $$b*x$$ smaller rectangle in half, down the middle from the $$b$$-length top to the $$b$$-length bottom, and putting half on the right of the $$x*x$$ square and half on the bottom of the square. It's like we changed the equation to: $$x^2+\frac{b}{2}x+\frac{b}{2}x$$ That's really the same equation, but now the $$bx$$ term is broken up into two skinnier rectangles that add up to the same area. Leave one skinny rectangle on the right of the $$x*x$$ square, but put the other skinny rectangle on the bottom, turning it so the $$x$$ sides touch, and the $$\frac{b}{2}$$ side is lined up with the $$x*x$$ square on the left. Now we almost have a new big square. The only piece we need to "complete" our big square is a tiny square missing from the bottom right corner. Can you tell the area of that tiny missing square? What's the length of each side of that tiny missing square? The area of that tiny missing square is $$\frac{b}{2}*\frac{b}{2}$$, or, $$(\frac{b}{2})^2$$. There's that "random" term you need to add to "complete the square." Now you not only have a mental picture to help you remember, you also have the reason this is called "completing the square." Great question.

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