## Friday, November 28, 2014

### Inequality Card Tricks

So, yesterday was Thanksgiving, and the family was all gathered. What a perfect time to engage your loved ones in a fun, educational, math game! Unfortunately, the only person who was interested was my father, and I'm sure he was just being nice. But what transpired was great material for a blog post!

The game we played was called "Greater Than," and it was created by Professor Golden. I had played the game before with my mother for my teaching class, and it was very straightforward. It's a card game for 2 players or teams and it's designed to help students practice inequalities. The dealer gives 4 cards to each team/player. Then each team/player selects a card from their 4, which is their starting value, and reveal it at the same time. Then, the non-dealer plays a turn, and the operation and value of that card gets applied to both player's cards. Then, it's the dealer's turn, and so on, until all the cards are played. The team/player who has the greatest value at the end wins! The operations were very simple: adding and multiplying by positive or negative numbers. (To see them in more detail, click the link above.)

After I explained the rules to my father, we started a game. I dealt 4 cards, and my father and I flipped our first card. We both flipped a positive 10. At this point there was no need to continue the game, since whatever cards that were played afterwards would be applied to both cards, and they would end up being equal. So, I dealt another hand, and we flipped both our cards.... We both flipped -11! At this point, we decided to break one of the rules, and thus the first variation was born! The first of three. Here is how it all went...

Variation 1: When a player plays a card, it only applies to their own card. It doesn't apply to both cards.

I dealt 4 cards to each of us. And when we looked at our hand to select our first card, it took longer than the previous two hands, since we could strategize the order of our cards to figure out the greatest value. Before, strategizing was harder, since the value of your hand was controlled by the cards that the other team/player used. Now, each player had sole control over the value of their hand. My hand consisted of +(10), x(-3), +(-4), x(10). If I played the positives first, I would be left with x(-3), which would turn my value negative. And the more negative the number, the lesser the value. So, I played the +(-4) first, followed by the x(-3), which gave me a positive 12. Then, I played the +(10), which gave me 22, and then I played the x(10), which gave me a positive 220. If I had played x(10) before the +(10), I would have gotten 12x10 = 120 + 10 = 130. And 220 > 130, and I want the greatest value possible. So, the order and combinations is very important to think about! Unfortunately, my father had the better hand with 310 (220 < 310). We played 2 more rounds, and my father won all of them. And then I decided to make the math game more challenging and changed the rules again. So, we moved on to play Variation 2.

Variation 2: Adding division to the mix.

So far, we've been adding and multiplying positive and negative numbers. But division is out of the mix. So I decided to include it, and I let the Face cards (not including the Ace, the Ace is still 11) be the dividing or multiplying-by-a-fraction cards. Kings were 1/2, Queens were 1/3, and Jacks were 1/4. With the new rules in place, I dealt 4 cards to each of us. I got x(1/3), +(-11), x(9), x(-5). The +(-11) had to be played first, so I could then use x(-5) to get +(55). Then I used x(9). This math was a little harder to do in my head. First, I added 55 +55 to get 110. Then I multiplied 110 by 4 to get 440. 2x4=8, and I needed to multiply by 9, so I had one more 55 to add, which gets me 495. Then, I divided by 3 or multiplied by 1/3, which got me the final value of 495/3. This is also equal to 165, which I did by figuring out how many 3's go into 4, drop the next number, and so on. My father this time didn't do so well, this round. He got -11/4. We played 2 more rounds, all of which I won. But then my father wanted to try his variation of the game. So, we moved on to Variation 3.

Variation 3: Diamonds now represent division.
My father wanted a whole entire suit to be division, instead of the Face cards. The rest of the rules would be the same. I pointed out that this would mean that we could only divide by a negative number. My father was fine with that. So, I dealt 4 cards to each of us again, and I got +(-10), +(-7), x(-1/2), +(3). I wanted to add as many negatives as possible, so I added (-10) and (-7) to get (-17). Then I multiplied by (-1/2) to get (17/2). Finally, I added the (9), which I converted to (18/2). 8+7 = 15, so I carried the 1, and 1+1+1=3. So, I got (35/2). My father got (-3/11), so I won this hand. We played 3 more games, which my father won all of them. At this point, Thanksgiving dinner was ready, and the game playing was over!

Reflection: All the versions of the "Greater Than" Game will be useful to teach students about inequalities. Depending on the confidence level of my students, I would have them do the original version first, followed by Variation 1. Variation 1, I believe, will teach them more about strategy and how to think a few steps ahead in order to figure out a problem. In this case, it's which card combination and order will achieve the greatest value. Then, if they are getting really comfortable, I would tell them to move on to Variation 2, and let them use a paper and pencil to help them divide/add/multiply/subtract fractions until they felt comfortable enough to do the math in their heads. I would also allow the students to use paper and pencil to figure out difficult multiplication problems with the original version, especially if I saw them struggling. All in all, the "Greater Than" Game is a great way to get students to practice math, learn about inequalities, and learn how to think steps through ahead of time to solve a problem.

## Thursday, November 13, 2014

### The Elusive Rule of Eleusis!

In my K-12 school experience, math was nothing but a chore, a series of rules, formulas and step-by-step instructions that we needed to learn in order to do the same repetitive math problems. And if we did not follow the steps or memorize the rules, we would get the answer wrong. This is a disservice to students, since this kind of teaching promotes learning by memorization, which is easily forgotten when not constantly in use. (I vaguely recall dreading those Summer Math Packets my elementary school would hand out to keep students from forgetting all the material they learned during the school year.)  It's also mindless, since it doesn't require the student to think about math or problem solving on a deeper level.

As a prospective math teacher, this is not the experience I want my future students to have. I want my students to perceive math as a process, an idea which my class has discussed very often. I need to find ways to engage my students that lets them use their math knowledge to problem solve without formulas, and one way to do that is through games. My teaching class has been exploring math games to help teach, and there is one game in particular I would like to discuss and it's called Eleusis. One player sets up a mathematical rule and plays a card that abides by that rule. Then the next player puts down a card they think fits the rule. If it does, the card is placed next to the last card, forming a straight line. If it does not fit the rule, the card gets placed underneath the last card. The rule can be very simple like "only prime numbers" or could be more complex like "all red cards need to be 5 or less away from the previous card and all black cards need to be 6 or more from the previous card." The point of the game is to eventually guess the rule by figuring out the mathematical pattern that the rule displays. To show how this game will help students in math and problem solving, I will discuss how I went about figuring out the rule, which was a homework worksheet Professor Golden provided.

Here is the setup:

The first thing I do is to see if there are any simple patterns based on color, like if all red cards are even and black cards are odd or if all red cards have to lower than the previous card and black cards have to be higher. At first glance this does not seem to be the case. The rule also doesn't stipulate that the color or suite has to alternate in any kind of pattern. Then I try to rule out more simple rules by asking myself "are these all prime numbers?" That doesn't seem to be the case either. So, the next thing I do is to look for more complex rules, like greater than or less than patterns. For example red cards can only be played if it's greater than or equal to 3 but less than 6. That also did not appear to be the case. Then I tried to focus on the suite. For example the rule could be diamonds are prime, spades are multiples of 3, clubs are multiples of 2, and hearts are multiples of 5. This would be a very hard rule since there is crossover between suites. For example, 6 is a multiple of 2 and 3. But again, the rule did not seem to involve the suites. Then, I thought I jumped to exploring complicated rules to quickly before ruling out the simpler ones, so I decided to see if there was a simple pattern between the value differences of each card. Below is my written work for this approach:

I noticed a black card, no matter what suite, could only be played if it was one value away from the previous card. So, I have part of the rule: Black cards have to be +1 or -1 from the previous card. Since, black cards had to be one value away from the previous card, I thought the rule for the red cards was that the value had to be greater than +1 or -1. This seemed to work except there is a 7 of diamonds underneath the 3 of diamonds. Both are red, 7 is +4 away from 3, and yet that card did not fit the rule. Then I thought, "Oh! 2 cards of the same suite can't be next to each other," but this was also not the case since there is a 10 of diamonds and a 3 of diamonds next to each other, and that's a difference of -7. Then, a thought hit me: maybe the rule is if a red card is played and it's the same suite then the card has to be subtraction only, not addition and the difference has to be greater than 1. That rule seems to fit, however if I was playing the next turn I would want to place a heart card next and have it's value be +2 or greater to see if my rule still holds. So, my best guess as to the rule for this pattern is if a black card is played then it has to be +1 or -1. If a red card of a different suite is played then it has to be greater than or equal to +2 or less than or equal to -2. If the red card is the same suite, then it has to be less than or equal to -2.

By explaining my thought process when playing this game, I hope I've illustrated how Eleusis builds problem solving skills and requires the use of math knowledge that isn't a formula or memorization of steps to figure out a problem. To problem solve, students will learn how to use both the correct and incorrect cards to figure out the rule. Looking at one and not the other gives you incomplete data to work with, which will lead to an incomplete rule. The correct cards gives what it could be, but the incorrect cards helps rule out what it can't be, which is more useful in narrowing down the rule. And even when I got to the end of the card play, I still wished I had more information, more cards to compare to check and verify my rule. They will also learn to continue to make guesses, and then test them like I did, a problem-solving process called trial and error, to eliminate all the wrong rules to finally reach the right one. Students will also have to think more deeply about using math to find patterns for the rule they are trying to solve. Formulas to find rules are nonexistent since each rule is different and no game will be exactly the same, even when the same rule is used again, since the cards played will be different each game. So, students will have to start from scratch to figure out the same rule again. All in all, games like Eleusis are a great tool to help students view math as less like a chore and more like a process, since it will engage them to think about math and problem solving more deeply.

***For those who are curious about whether or not my rule was right, it wasn't! The rule Professor Golden used was if the previous card was odd, then a black card must follow, and if the previous card was even, then a red card must follow. I was completely off! But this just illustrates the importance of using the incorrect cards to eliminate possibilities. If I could have played the game further, then I would have played certain cards that would have verified or eliminated my rule altogether. Trial and error and the process of elimination are very important problem solving concepts that this game illustrates!***

## Wednesday, November 12, 2014

Math can drive students mad sometimes. It helps to have a visual aid. But it's even better to have a physical aid that students can touch and manipulate. Manipulations are exactly that; they are physical representations of math concepts to help students learn. Geometric blocks can be used as manipulations, and I remember using them in my kindergarten class to learn about finding simple patterns, which laid a very important foundation for future math concepts. A big key in unlocking the mystery of math, after all, is finding and using mathematical symbols to represent patterns. So, it's very unfortunate that I never saw those blocks being used as a teaching tool in subsequent math courses again for the rest of my elementary and secondary school education.

Another great example of a manipulation tool are Algebraic Tiles. In class, we used these to explore manipulations to teach students about factoring quadratic equations. We also had a reading assignment on how to use Algebra Tiles for teaching: "Using Homemade Algebra Tiles to Develop Algebra and Prealgebra Concepts," by Annette Ricks Leitze and Nancy A. Kitt in Mathematics Teacher, Vol. 93, No. 6, September 2000 (pg 462-466 and pg 520). Students can do a quick internet search to find templates of these to print off and cut for class and their homework. Each term of the quadratic is represented: the tiny squares represent constants, the bars represent x, and the big squares represent x^2. For this blog, I will show how I would use these tiles to introduce quadratic expressions and factoring to my future students by using what I've learned from class and in the reading.

To slowly show how to use these tiles to represent linear and quadratic expressions, I would pick very simple equations for the students to represent with their Algebraic Tiles like x + 2 (Figure A) and x^2 (Figure B).
 Figure B: x^2
 Figure A: x+2

Here, I would make the connection that the bars squared equals the area of a square in Figure B, and that both x's are factors of x^2, which was a nifty trick Professor Golden showed us in class. Then, I would gradually make them harder like x^2+3x+2 (Figure C) and 2x^2+9x+4 (Figure D). Again, I would show that the sides represent the factors of the quadratic. I've circled the factors in blue (although, they aren't very visible). The factors for Figure C are (x+2)(x+3), and the factors for Figure D are (2x+1)(x+4). Here, students can clearly see the relationship between the factors and the quadratics they form. For both examples, students can count the number of big squares, bars, and tiny squares to get the quadratic expression that the factors create.

 Figure C: x^2+5x+6
 Figure D: 2x^2+9x+4

Next, I would introduce subtraction into the quadratic expressions. Tiles representing subtraction should be a different color or pattern from the tiles representing addition. To illustrate how the quadratics change when the + and - operations are flipped between factors, I chose quadratic expressions that have the same numbers in their factors. Figure E is x^2-x-6, and its factors, circled in blue, are (x+2)(x-3). Now the students might get confused and say but there are 5 bars, so it should be 5x. But the red bars mean subtraction, so for each red bar, a white bar gets paired up with it, and whatever is leftover is the value of the x-term in the quadratic expression. So, for Figure E, 2 red bars get paired with 2 white bars, and 1 red bar is leftover, so the x-term is -x. Now, some students might ask why the big square is white and not red, since the big square is flanked by a white bar and a red bar, and a positive x (+x) multiplied by a negative x (-x) should be a negative x^2 (-x^2). To clear the confusion, I would inform the students that the red bars represent an operation: subtraction. They don't represent a negative x. So, whenever there is a -x, I will say "minus x," and not "negative x." (This should also be done with plus x (+x), not positive x). Next, Figure F is x^2+x-6, and its factors are (x-2)(x+3). I would ask the students for the similarities and differences between the factors of Figure E and Figure F.  Then, I would reiterate that the numbers in the factors are the same, but the operations are flipped, which changes the quadratic expression they form. So, 2 red bars get paired with 2 white bars, and 1 white bar remains, which means the x-term is plus x (+x). Then, Figure G is x^2-5x+6, and its factors are (x-2)(x-3). Here, both factors have subtraction, so there are 5 red bars, no white bars. Then, the x-term is minus 5x (-5x).

 Figure E: x^2-x-6
 Figure F: x^2+x-6
 Figure G: x^2-5x+6

Now, if the students seem comfortable with this exercise so far, -x^2 could be introduced. This will be completely contradictory to what I have just told the students: that red tiles means subtraction, not a negative x or number. But for this portion of the exercise I am going to break that rule; that we will now be multiplying a negative x (-x), and I would tell the students so. It will be harder for the students to factor, but I think it would be a good problem they can think through on their own. I also made the quadratic expression and the factors to be very similar to make it easier. And if they can't factor it, then I would do it together as a class. First, Figure H is -x^2+5x-6 and its factors are (-x+2)(x-3). Next, Figure I is -x^2+x+6, and its factors are (-x-2)(x-3). Since we have a mix of white and red bars, we need to use subtraction. So, 2 red bars get paired up to 2 white bars, and then 1 white bar is leftover, which gives us +x for the x-term. Then, Figure J is -x^2-5x-6 and its factors are (-x-2)(x+3).

 Figure H: -x^2+5x-6
 Figure I: -x^2+x+6
 Figure J: -x^2-5x-6

## Wednesday, October 22, 2014

### Polynomial Divination!...err Division...

The Quest for the Vanishing and Reappearing Term in Polynomial Divination Division!

Last week, I had to brush up on my math magic for our unit on Polynomial Division. I had completely forgotten how to factor polynomials bigger than quadratics. It looks daunting when you first look at it. But once I found a good refresher course online, the concept came rushing back to me. Polynomial Division is just like long division except your finding factors, or dividing the bigger polynomial with a factor that's in the form of (x+4).

Starting out with a larger polynomial to factor like x^5+7x^4-3x^3-79x^2-46x+120 didn't faze me. However, it might faze my future students. So, to help my students establish a connection between finding factors for quadratic equations and polynomial division, I would use polynomial long division on a quadratic equation as my introduction to the concept. That way, they can see that polynomial division and factoring quadratic equations are about the same thing: finding factors. It's just using a different method to find them. Also, they can connect what they already know about factoring quadratics and apply it to polynomial division, which might help their understanding.

To begin the lesson, I would pick an easy quadratic, one that is easily recognizable and easy to factor, like x^2+4x+4. The factor of this equation is (x+2)(x+2) or (x+2)^2. The students might say this immediately, but the point of this is to get them familiar with polynomial division. So, we would go through it together step-by-step.

After the first one is done, I would give the students a cubic expression to factor, like x^3+6x^2+11x+6. Now, the students might ask "What do we divide it by?" and my answer would be, just like a quadratic, find and try the factors of the last number (6 in this case), and see if they work. I would also tell the students to check their work by graphing it.

Next, I would throw them a curve-ball, and give them x^3-7x-6. This expression, might give them pause. The x^2 is completely missing, vanished, disappeared! I would let the students offer suggestions or try to think it out at first. Then, I would show them how to factor it with an easier quadratic. Again, one that the students can recognize, so they can make the connection between the two and become less daunted by the process. That quadratic will be: x^2 -1. Here the x^1 part of the equation is missing; it has been canceled out. Again, the students will say almost immediately that the factors are (x+1)(x-1). As a hint, instead of writing the quadratic in the form of x^2-1, I would make the x^1 spot reappear or reveal itself by adding 0x. Then the quadratic would be: x^2+0x-1. I could also reveal more of what I did and why I did it by factoring out the factors for my students: (x-1)(x+1) = x^2-x+x-1=x^2+0x-1. Now, students can see the connection that 0x serves as a placeholder for the missing x^1, since it was canceled out in steps above. This might be the only step the students need to see to feel comfortable about dividing polynomials with missing spots or terms in the expression. If not, I would walk them through it step-by-step, and let them know that we would drop the 0x just like any other polynomial with all it's terms. Then I would give them x^3-7x-6 again to divide by themselves.

## Sunday, September 7, 2014

### Counting Circle

Welcome! Welcome to my Magical Mystifying Mathematical Metacognitive Memoir!

How is it magical and mystifying? Well, it's not. But by labeling my blog as such, I'm making a statement on how math can seem like magician's work. Just as a magician mystifies their audience by procuring rabbits out of thin air, a math teacher can mystify students by procuring solutions out of a jumble mess of numbers, symbols, and words. In both scenarios, the audience and the students wonder, "How did they do that? It was just like magic!"

But once the magician reveals their secrets and shows all the tricks behind-the-scenes, the act is no longer magical or mystifying. The same holds true for math. Once the math teacher explains the thinking, the behind-the-scenes process they took to solve the problem, then the math is no longer magical or mystifying. And that is the point of my memoir - to demystify the magic surrounding math by explaining my thinking behind my solutions to mathematical problems, and illuminate any connections and lessons-learned along the way, so I and others can learn how to explore, communicate, and teach math more effectively.

Act 1: The Counting Circle

The first day of school, the class played a game called the "Counting Circle." The class forms a circle around the room, and the first student gets a starting number and a counting number to add to it. Then the next person has to add the counting number to the last person's answer, and so it continues around the circle with each student adding the counting number to the previous person's answer. What's great about the counting circle is that it can be utilized in various ways for different math topics, not just adding natural numbers. The first one that came to my mind was for a Pre-Calculus class, which would combine the counting circle with the unit circle. Instead of counting with numbers, students would count in pi, radians, or angles. But for our first try, our class did simple addition...with big numbers. According to my recollection, the starting number was 235 and the counting number to be added was 97.

I was toward the end of the circle, so I had plenty of time to think. As the counting went around the circle, I was trying to come up with the answer for each person. I was trying to add the ones digits first, then add the tens digits, and then the hundreds digits, which is the way I was taught in elementary school. But I was so slow; I could not keep up. Others were adding the numbers in their heads so fast, (as if by magic!), and I wondered how they could do it, since I assumed they were adding the way I was.

Then it came my turn, and I struggled. First, I added the ones,  then carried the one to the tens digits, got that answer… Wait, what was my answer for the ones digit? Did the ones over again. Then the tens digit again… Thought I had the wrong answer. Needed to start over again. And so forth. Finally, after about a minute, I got an answer. Then I realized there was a better way to do it than the way I was taught in elementary school. Add 100 and then subtract 3, instead of adding 97, since 100-3=97.
At the end, we were asked how we got our answers. Many did the add 100, subtract by 3, trick. Others added in groups, such as: add 65 to 235 to get 300. Then subtract 65 from 97 to get 32. Add 32 to 300 and get 332.

Next, we were asked this question: if we continued around the circle, what would so-and-so's answer be? To solve this, I first counted how many people stood between the last person of the circle and so-and-so. I think it was 7 people. Then I modified the "add 100, subtract 3" trick, and multiplied 7 by 100 to get 700. Then added that to the last person's answer. Then I multiplied 7 by 3 to get 21. Subtracted that from the answer I got after I added 700, and got my answer. The equation would look something like this:
[last person's answer + (number of people x 100)] - (number of people x 3) = answer.

When asked how we got our answers, the majority mentioned the way I did it. A few did it some other way, but it wasn't as streamlined or elegant as the above process. In fact, some mystified me. And that's OK. In fact, that's the point. The counting circle gives students an opportunity to see different ways of looking at and solving the same math problem, so they can learn different strategies, discover which ones make sense to them, and apply those strategies to similar problems later… or for when the next counting circle comes along.