How can algebra tiles show the properties of equality?

What are the equality axioms?

Say there's a relation ~ between the two objects a and b such that a ~ b. We call ~ an equivalence relation if:i) a ~ a.ii) If a ~ b. then b ~ a.iii) If a ~ b and b ~ c, then a ~ c.Where c is another object.The three properties above are called the reflexive, symmetric, and transitive properties. Were those three properties all that was needed to define the equalityrelation, we could safely call them axioms. However, one more property is needed first. To show you why, I'll give an example.Consider the relation, "is parallel to," represented by . We'll check the properties above to see if is an equivalence relation.i) a a.Believe it or not, whether this statement is true is an ongoing debate. Many people feel that the parallel relation isn't defined for just one line, because it's a comparison. Well, if that were true, then you would have to say the same thing for everybinary equivalence relation; e.g., a triangle couldn't be similar to itself, or, even more preposterously, the statement a = a would have to be tossed out the window too. But, just to be formal, we'll use the following definition for parallel lines:Two lines are not parallel if they have exactlyone point in common; otherwise they are parallel.So, with that definition in hand, i) holds for .ii) If a b, then b a. True.iii) If a b and b c, then a c. True.Thus, the relation is an equivalence relation, but two parallel lines certainly don't have to be equal! So, we need an additional property to describe an equality relation:iv) If a ~ b and b ~ a, then a = b.Let's check iv) and see if this works for our relation :If a b and b a, then a = b. False. But, does it hold for the equality relation?If a = b and b = a then a = b. True. This is what's known as the antisymmetricproperty, and is what distinguishes equality from equivalence.But wait, we have a problem. We used the relation = in one of our "axioms" of equality. That doesn't work, because equality wasn't part of the signature of the formal languagewe're using here. By the way, the signature of the formal language that we are using is ~. So, any other non-logical symbol we use has to either be defined, or derived from axioms.Well, we have three possible ways out of this. We can either:1) Figure out a way to axiomize the = relation through the use of the ~ relation.2) Define the = relation.3) Add = to our language's signature.Well, 1) is not possible without the use of sets, and since the existence of sets isn't part of our signature either, we'd have to define a set, or add it to our language. This isn't very hard to do, but I'm not going to bother, because the result is what we're going to obtain from 2).Anyways, speaking of 2), let's define =.For all predicates (also called properties) P, and for all a and b, P(a) if and only if P(b) implies that a = b.In other words, for a to be equal to b, anyproperty that either of them have must also be a property of the other. In this case, the term propertymeans exactly what you think it means; e.g. red, even, tall, Hungarian, etc.So, the million dollar question is, by defining =, are our properties now officially axioms? For three of the properties, the answer is no. In fact, because we just defined =, we've turned properties ii), iii), and iv) from above into theorems, not axioms. Why? Because, property iv)still has that = relation in it, which we had to define. So, iv) is a true statement, but we had to use another statement to prove it. That's the definition of a theorem! And, since the qualifier for iv)'s truth was that a ~ b and b ~ a, we can now freely replace b with a in ii), giving us "If a ~ a, then a ~ a." Well, now ii)'s proven as well, but we had to use iv) to do it. Thus, both ii) and iv) are now theorems. Finally, iii) can be proven in a similar was as ii) was, so it, too, is a theorem.However, our definition of = only related a to b, it never related a to itself. Thus, we need to include i), from above, as an axiom.Just for kicks, let's try plan 3) too.The idea here is to make = a part of our signature, which means now we don't need to define it. In fact, we can't define it if we put it in our signature; because by placing it there, we're assuming that it's understood without definition. Therefore, iv) must now be assumed to be true, because we have no means to prove it; that sounds like an axiom to me! However, just like before, we can prove both ii) and iii)through the use of iv), so they get relegated back to the land of theorems and properties. Interestingly though, iv)makes no mention of reflexivity, and since our formal definition of = is gone, we have no way to prove i). Once again, we have to assume that it's true. Thus i) is an axiom as well.So, to paraphrase our two separate situations:In order for the relation ~ to be considered an equality relation between the objects a and b, oneaxiom must be satisfied if we define =:1) For all a, a ~ a,as well as three theorems:1) If a ~ b, then b ~ a2) If a ~ b and b ~ c, then a~ c, where c is another object3) For all a and b, if a ~ b and b ~ a, then a = b.Additionally,In order for the relation ~ to be considered an equality relation between the objects a and b, twoaxioms must be satisfied if we put = into our signature:1) For all a, a ~ a2) For all a and b, if a ~ b and b ~ a, then a = b,as well as two properties:1) If a ~ b, then b ~ a2) If a ~ b and b ~ c, then a~ c,where c is another object.What the one right above did is include "=" into our formal language, but "=" is equality, so he actually came up with a fairly well axiom before he finishes with the circular looking one.His axiom: We say ~ is an equality relation means whenever x ~ y, for any condition P, P(x) iff P(y)The axiom is the bolded part.After discussion with my Math prof. this morning, that axiom becomes a properties follows from this more formal definition. It does not need to include any more things then what we already have for the formal language.We say ~ is an equality relation on a set A if (a set is something that satisfies the set axioms)For any element in A, a ~ a.If follows that P(a) is true and y ~ a, then P(y) is also true, vice versa. Because in this case, y has to be a for it to work.You might argue well the definition for an equivalence relation have this statement in it too, does that mean equivalence IS equality?No! It's the other way around, equality is equivalence. Equality is the most special case for any relation, say *, where a * a.Take an equivalence relation, say isomorphisms for instance (don't know what that word mean? Google or as it on this website), we know any linear transformation T is isomorphic to T, in particular this isomorphism IS equality. Of course it would be boring if isomorphism is JUST equality, so it's MORE.The other axioms in a definition of a relation are to differ THEM from equality, because equality is the most basic. Equality must always be assumed, it always exist, any other relation is built upon it. It is the most powerful relation, because ALL relations have it. (I mean all relations, say *, such that a * a for all a must at least be equality)

What is algebraic tiles?

==UsesAdding integers=== '''Algebra tiles''' can be used for adding [[integers]].Kitts, N: "Using Homemade Algebra Tiles to Develop Algebra and Prealgebra Concepts", page 463. MATHEMATICS TEACHER, 2000. To demonstrate this ability you can consider the problem 2+3=?. In order to solve this problem using '''algebra tiles''' a person would group two of the positive unit tiles together and then group three of the positive unit tiles together to represent separately 2 and 3. In order to represent 2+3 the person would then combine their two groups together. Once this step is complete the person can then count that together there are 5 unit tiles, so 2+3=5. Since adding a number with the negative of that number gives you zero, for instance -2+2=0, adding a negative unit tile and a positive unit tile will also give you [[zero]]. When you add a positive tile and a negative tile it is known as the zero pair. In order to show that any [[integer]] plus its negative is [[zero]] a person can physically represent this concept through '''algebra tiles'''. Let us take the example used earlier where -2+2=0. A person would fist lay out two negative unit tiles and then two positive unit tiles, which would then be combined into two sets of zero pairs. These two sets of zero pairs would then be equal to [[zero]].[http://www.phschool.com/professional_development/teaching_tools/pdf/using_algebra_tiles.pdf] Understanding zero pairs allows you to also add positive and negative integers that are not equal. An example of this would be -7+4=?, where you would group seven negative unit tiles together and then four positive unit tiles together and then combine them. Before you count the number of tiles that you now have you would have to create zero pairs and then remove them from you final answer. In this example you would have four zero pairs which would remove all of the positive unit tiles and you would be left with three negative unit tiles, so -7+4=-3. ===Subtracting integers=== '''Algebra tiles''' can also be used for subtracting [[integers]]. A person can take a problem such as 6-3=? and begin with a group of six unit tiles and then take three away to leave you with three left over, so then 6-3=3. '''Algebra tiles''' can also be used to solve problems like -4-(-2)=?. First you would start off with four negative unit tiles and then take away two negative unit tiles to leave you with two negative unit tiles. Therefore -4-(-2)=-2, which is also the same answer you would get if you had the problem -4+2. Being able to relate these two problems and why they get the same answer is important because it shows that -(-2)=2. Another way in which '''algebra tiles''' can be used for [[integer]] [[subtraction]] can be seen through looking at problems where you subtract a positive [[integer]] from a smaller positive [[integer]], like 5-8. Here you would begin with five positive unit tiles and then you would add zero pairs to the five positive unit tiles until there were eight positive unit tiles in front of you. Adding the zero pairs will not change the value of the original five positive unit tiles you originally had. You would then remove the eight positive unit tiles and count the number of negative unit tiles left. This number of negative unit tiles would then be your answer, which would be -3.[http://www.phschool.com/professional_development/teaching_tools/pdf/using_algebra_tiles.pdf] ===Multiplication of integers=== [[Multiplication]] of [[integers]] with '''algebra tiles''' is performed through forming a rectangle with the tiles. The [[length]] and [[width]] of your rectangle would be your two [[factors]] and then the total number of tiles in the rectangle would be the answer to your [[multiplication]] problem. For instance in order to determine 3×4 you would take three positive unit tiles to represent three rows in the rectangle and then there would be four positive unit tiles to represent the columns in the rectangle. This would lead to having a rectangle with four columns of three positive unit tiles, which represents 3×4. Now you can count the number of unit tiles in the rectangle, which will equal 12. ===Modeling and simplifying algebraic expressions=== Modeling algebraic expressions with '''algebra tiles''' is very similar to modeling [[addition]] and [[subtraction]] of integers using '''algebra tiles'''. In an expression such as 5x-3 you would group five positive x tiles together and then three negative unit tiles together to represent this algebraic expression. Along with modeling these expressions, '''algebra tiles''' can also be used to simplify algebraic expressions. For instance, if you have 4x+5-2x-3 you can combine the positive and negative x tiles and unit tiles to form zero pairs to leave you with the expression 2x+2. Since the tiles are laid out right in front of you it is easy to combine the like terms, or the terms that represent the same type of tile. [http://www.phschool.com/professional_development/teaching_tools/pdf/using_algebra_tiles.pdf] ===Using the distributive property=== The [[distributive property]] is modeled through the '''algebra tiles''' by demonstrating that a(b+c)=(a×b)+(a×c). You would want to model what is being represented on both sides of the equation separately and determine that they are both equal to each other. If we want to show that 3(x+1)=3x+3 then we would make three sets of one unit tile and one x tile and then combine them together to see if would have 3x+3, which we would.[http://www.regentsprep.org/rEGENTS/math/realnum/Tdistrib.htm] ===Solving linear equations=== Manipulating '''algebra tiles''' can help students solve [[linear equations]]. In order to solve a problem like x-6=2 you would first place one x tile and six negative unit tiles in one group and then two positive unit tiles in another. You would then want to isolate the x tile by adding six positive unit tiles to each group, since whatever you do to one side has to be done to the other or they would not be equal anymore. This would create six zero pairs in the group with the x tile and then there would be eight positive unit tiles in the other group. this would mean that x=8.[http://www.phschool.com/professional_development/teaching_tools/pdf/using_algebra_tiles.pdf] You can also use the [[subtraction]] property of equality to solve your [[linear equation]] with '''algebra tiles'''. If you have the equation x+7=10, then you can add seven negative unit tiles to both sides and create zero pairs, which is the same as subtracting seven. Once the seven unit tiles are subtracted from both sides you find that your answer is x=3.Kitts, N: "Using Homemade Algebra Tiles to Develop Algebra and Prealgebra Concepts", page 464. MATHEMATICS TEACHER, 2000. There are programs online that allow students to create their own [[linear equations]] and manipulate the '''algebra tiles''' to solve the problem. [http://my.hrw.com/math06_07/nsmedia/tools/Algebra_Tiles/Algebra_Tiles.html Solving Linear Equations Program] This video from Teacher Tube also demonstrates how '''algebra tiles''' can be used to solve linear equations. [http://www.teachertube.com/view_video.php?viewkey=7b93931b2e628c6e6244&page=&viewtype=&category= Teacher Tube Solving Equations] ===Multiplying polynomials=== When using '''algebra tiles''' to multiply a [[monomial]] by a [[monomial]] you first set up a rectangle where the [[length]] of the rectangle is the one [[monomial]] and then the [[width]] of the rectangle is the other [[monomial]], similar to when you multiply [[integers]] using '''algebra tiles'''. Once the sides of the rectangle are represented by the '''algebra tiles''' you would then try to figure out which '''algebra tiles''' would fill in the rectangle. For instance, if you had x×x the only '''algebra tile''' that would complete the rectangle would be x2, which is the answer. [[Multiplication]] of [[binomials]] is similar to [[multiplication]] of [[monomials]] when using the '''algebra tiles''' . Multiplication of [[binomials]] can also be thought of as creating a rectangle where the [[factors]] are the [[length]] and [[width]].Stein, M: Implementing Standards-Based Mathematics Instruction", page 98. Teachers College Press, 2000. Like with the [[monomials]], you set up the sides of the rectangle to be the [[factors]] and then you fill in the rectangle with the '''algebra tiles'''. Stein, M: Implementing Standards-Based Mathematics Instruction", page 106. Teachers College Press, 2000. This method of using '''algebra tiles''' to multiply [[polynomials]] is known as the area modelLarson R: "Algebra 1", page 516. McDougal Littell, 1998. and it can also be applied to multiplying [[monomials]] and [[binomials]] with each other. An example of multiplying [[binomials]] is (2x+1)×(x+2) and the first step you would take is set up two positive x tiles and one positive unit tile to represent the [[length]] of a rectangle and then you would take one positive x tile and two positive unit tiles to represent the [[width]]. These two lines of tiles would create a space that looks like a rectangle which can be filled in with certain tiles. In the case of this example the rectangle would be composed of two positive x2 tiles, five positive x tiles, and two positive unit tiles. So the solution is 2x2+5x+2. ===Factoring=== In order to factor using '''algebra tiles''' you start out with a set of tiles that you combine into a rectangle, this may require the use of adding zero pairs in order to make the rectangular shape. An example would be where you are given one positive x2 tile, three positive x tiles, and two positive unit tiles. You form the rectangle by having the x2 tile in the upper right corner, then you have two x tiles on the right side of the x2 tile, one x tile underneath the x2 tile, and two unit tiles are in the bottom right corner. By placing the '''algebra tiles''' to the sides of this rectangle we can determine that we need one positive x tile and one positive unit tile for the [[length]]and then one positive x tile and two positive unit tiles for the [[width]]. This means that the two [[factors]] are x+1 and x+2. Kitts, N: "Using Homemade Algebra Tiles to Develop Algebra and Prealgebra Concepts", page 464. MATHEMATICS TEACHER, 2000. In a sense this is the reverse of the procedure for multiplying [[polynomials]]. ===Completing the square=== The process of [[completing the square]] can be accomplished using '''algebra tiles''' by placing your x2 tiles and x tiles into a square. You will not be able to completely create the square because there will be a smaller square missing from your larger square that you made from the tiles you were given, which will be filled in by the unit tiles. In order to [[complete the square]] you would determine how many unit tiles would be needed to fill in the missing square. In order to [[complete the square]] of x2+6x you start off with one positive x2 tile and six positive x tiles. You place the x2 tile in the upper left corner and then you place three positive x tiles to the right of the x2 tile and three positive unit x tiles under the x2 tile. In order to fill in the square we need nine positive unit tiles. we have now created x2+6x+9, which can be factored into (x+3)(x+3).[http://www.regentsprep.org/Regents/math/algtrig/ATE12/completesq.htm]

How can algebra tiles show the properties of equality?

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