All of the Lights

One of my family's favorite Christmas movie is National Lampoon's Christmas Vacation. So since it's getting close and it's that time of the year, I thought I would look into something Christmas related. This weekend, since the weather was nice (no more snow and not too cold!), my family dug out the Christmas lights to put them up. And it got me thinking of that movie and all those lights. How many and how much would it cost to decorate a house as done in the movie - the entire roof and all around the entire sides.

First, I went to find out what sort of lights were used. Based on a picture in a movie, they're larger than the sort I'm used to. So then I searched Christmas lights. I know that there are all sorts of places to buy lights but I choose Home-Depot. The lights cost $8.48/each for a strand of 25 lights at a length of 25 feet. You can however only connect at most 2 sets. But I'm guessing by the looks of it with any lights you wouldn't be able to create that long of a strand. So we're just going to assume that we can attach them all together no matter.

So now how to figure out how many strands are needed. After a little searching, hoping to find the house dimensions (and then go round trying to figure out how many strands), I was able to find that house was decorated with 25,000 lights! That's a ton and no wonder. So divide that by the nice easy 25 lights per strand and you would need 1,000 strands of lights. And based on the website, you'd have to travel to several stores to pick them up - the one near me only has 20 on stock. These lights being bigger than usual, already cost a lot. Multiplying the cost of a strand at $8.48 by the number of strands, it ends up costing $8480.00 dollars. No thank you. I could pay to study abroad instead!

And the length of all those strands, at 25 feet per strand ends up being the same as the number of lights - 25,000 feet.

So then I thought: how tall of a tree would you need to use that many lights? I'm going to use a cone to represent a tree to find this out. Since lights go around the tree, I'm going to look at the surface area of a cone minus the base: pi*r*l. The variable "l" of the cone represents the slant height, which in terms of height equals sqrt(h² + r² ). So our formula becomes: pi*r*sqrt(h² + r² ). Just looking at my own tree, the height is about 3 times the radius of the tree. Using this I can narrow the formula down to having only one variable, height: pi*h/3*sqrt(h² + (h/3)² ). Now to simply the equation a bit:

= pi*h/3*sqrt((4h/3)² )

= pi*h/3*4h/3

= pi*4h²/9

To make things simple, just like the house we'll cover the entire tree with lights! Which I think would hurt to look at up close haha. The width of the light strands is about 2 inches or 1/6 of a foot, so the area of all the light strands is  feet squared. Setting this equal to the surface area and solving:

4166.67 = pi*4h²/9

37500 = pi*4h²

2984.15 = h²

54.63 = h

So the height of the tree is about 55 feet. A little too small to be the tree at Rockefeller Center (69 to 100ft). If the spacing between the lights was increased from nothing to something, the height of the tree would just continue to grow. So again, like I thought, I'm going to pass. I'd rather keep with tradition and continue to put up the fake tree and string a few lights.

A Jumble of Geometry

Recently in class, we began with some talking points - questions related to geometry to spark discussion. One of the questions asked, "From the diagram given below, you can find the measure of angle D":

After a brief look at the diagram, the answer was clear: no. The measure of angle D can not be found because the diagram does not tell have parallel lines - if this was given the answer would be yes! As a follow up, a new diagram was given. Below I will outline my steps in the process of solving for each of the 4 angles presented.

The Process

The first thing I noticed when looking over the above diagram was the 44 deg. angle at the bottom. Since this time the lines are parallel and recalling that opposite side exterior angles are congruent, I knew the value of A is 44 deg. as well. At this point, recalling the vertical angle theorem, I was able to fill in 2 other 44 deg. angles(in blue).

After writing in the values, I noticed the triangles at the top and bottom of the diagram. They now had 2 of there 3 angles filled in. Knowing that the angles in a triangle always add to 180 deg., I was able to find the third angle in each:

180 - (12 + 44) = 124

180 - (44 + 30) = 106

Once those angles were filled, my focus became on B. By the vertical angle theorem would also we 124 deg. And by the same theorem I was able to find another angle of 106.

If you can't tell, the tiny orange dot in the small triangle represents the other 106.

Next, I looked to solving C, because I felt like if I'm going in order thus far I should keep with it. But as I looked at the diagram, D felt like the clearer next step. All I would have to do is find all the angles of the little triangle and I'd be a step away from finding the measure of D. I recalled that angles on the same line add up to 180 deg. This allowed me to find the angle beneath 112 deg, because they are on the same line - so the angle must be 68. By the same reasoning I was able to find another angle of 68 deg.

Now there's a tiny pink dot in the tiny triangle. 

Again, I know all the angles of a triangle add up to 180 degrees. So 180 - (106 + 68) equals the measure of the third angle, which is 6 deg. Then I noticed that the measure of angle D with the angle I just found makes a circle, which has an angle of 360 deg. So using this fact and the found angle, D is 354 deg.

So now I finally had to face C. This one took me a minute. As I looked at the diagram, I kept thinking "how will I find C?" and wishing "if only I could find the measure of the angle that with C results in 360 deg". So how could I find this angle? The shape was irregular and I didn't know anything extra about the lines around C (no parallel lines to be had). I kept thinking, if only I could somehow break the shape in triangles and find other angles that could lead to C. I wished the shape was regular, then maybe I could find the sum of the angles and then find C. As I thought about this I realized that the regularity of the shape was irrelevant. This shape was five sides, so it's a pentagon, and pentagons (all and any) have 540 deg as the sum of their interior angles! 

Since I now have 4 of the 5, finding the fifth became extremely simple: 540 - (124 + 22 + 68 + 52) = 274. To find C, all I had to do was subtract this angle from 360. So C ends up being 86 deg.

Thoughts:

For not having done high school geometry in a while, I felt good about working through this problem. I especially liked the way in which it seems a little daunting (all the lines and angles and shapes) but as you dive in, you realize that each piece of the puzzle isn't too bad. It was also helpful in using several different theorems that I was taught in high school geometry to figure everything out. While the pentagon through me off a little in my search for C, I enjoyed that aspect. It was nice to struggle and face a minor challenge along the way - once reaching the solution, how obvious it now seemed! I think it would be fun to present the problem to high school students, since a lot of the knowledge and understanding required for the puzzle they are in the process of learning. And because it's a process, that one thing leads to the next and has more purpose than simply filling in the blank for one question, students could get a lot out of it - applying the skills they are gaining and diving into critical thinking and problem solving!

Visual Mathematical Laws

Inspired by the following blogpost, I decided to make my own visual representation of mathematical laws. Art being my tied for favorite subject with math, the idea of visual representations sounded not only fun to create but potentially really helpful for visual learners. The visual representations take away the daunting idea or confusion variables bring to many students. If students are confused about variables or just have a general dislike, with the visual representation they are still receiving the same information just in a more understandable way. Along with this, students should be brought to discover or shown how the laws have come to be, since understanding the laws is more important and helpful than just remembering them.

In addition, to me, the visual representation is more interesting. Students see numbers and variables all the time in their math classes - of course! Color, on the other hand, not so much. So maybe students will have a tendency to recall the laws better because of the uniqueness with which they were presented. Plus it's always fun to see math and art come together! - even if in such a small way.

Through the process of making the laws, I became more aware of how some of the laws work. Before I just took it for what I was told, memorizing but not really seeing the connections. The colors helped me to make the connections and clearly see where each piece is coming from. It's difficult to say if I would personally ask my students to create their own - it might take some students a lot of time and they may not see the purpose in it. I would however in going over the laws with a class encourage them to use colors in place of the variables and have a poster of the laws in this way displayed in the classroom.

So for mine, I decided to link it to those seen in the blogpost(which focused on laws of exponents) and visually represent the laws of logarithms :)
They are as follows:

• Logarithm to Exponential
• Canceling Exponentials (2nd and 3rd)
• Product
• Quotient
• Power
• Changing Base
• BONUS!

Working with Algebra Tiles

After using them in class and reading an article advocating for the use of algebra tiles in classrooms, I was left with two main questions.

 The first, deals with completing the square. The article gave a long list of things the tiles could be useful to help teach. Having been tutoring for awhile, I've realized that many students do not know what it is to complete the square or get confused about how. So I wanted to put it to the test and see how the tiles would be able to show the concept of completing the square.

I did a quick Google search to find some problems that require completing the square:

1. x^2 - 4x + 6 = 0
2. -x^2 - 2x - 5 = 0
3. 4x^2 + 4x - 3 =0

Then I completed the square to find the solution, so I could compare when working with the algebra tiles.

1. (x - 2)^2 + 2 = 0
2. -(x + 1)^2 - 4 = 0
3. (2x + 1)^2 -4 = 0

So I began, and at first, I was confused. I collected all the tiles I would need, but somehow could not form a rectangle/square. It simply was not possible.

After a moment, I realized, to complete the square I will have to create a square with the tiles. So first, after placing the x^2 tile, you have to divide the x tiles evenly on either side.

Then I filled in the units to complete the square, with 2 of the six leftover. So when writing the solution you get (-x + 2)^2 + 2 = 0. Which is equivalent to the solution above. As I solved the others, I realized it got easier as I went along. You just have to be careful with the units (that the total tiles is the total units you had in the original equation). Both ways, visual and symbols seem like fine ways to teach it. It seems like it would be helpful for visual/struggling learners and interesting to others. But potentially a waste of time for students who already grasp the idea symbolically, if they must do a lot of work using the tiles.

Here are the visual representations of the other 2 equations:

 

My second question, was if algebra tiles could make realizing if a given binomial can not be factored easier? So again, I Google searched and found two equations that could not be factored and got out my tiles.

1. x^2 + 2x + 5 = 0
2. x^2 + 4x + 1 = 0

After setting up both, the answer is yes! If the equation can not be factored, you will have more or less units than what is necessary to create a rectangle with the tiles. Since students are sometimes asked to factor an equation, or say if it can not be factored, this method seems really helpful. Students can see the solution quickly, whether it can be factored or not. It also allows students to pay attention to the units of the equation when deciding if something can be factored. When they face larger equations, which the tiles would not work well for, students know where to look.

 

So both questions resolved, I recommended the use of Algebra tiles. The algebra tiles help to introduce ideas of factoring (which can feel and seem abstract) in a more concrete way. Students can see, visually, why something works and how to fix the problem. They become more comfortable with the ideas of factoring and gain insight into recognizing when things can and cannot be factored. So hopefully, as a result, factoring will not feel like a daunting task, but an achievable one.

Teaching and Technology

      I have never been the biggest fan of technology. When I was younger my siblings would play video games or computer games, while I only turned the computer on to write stories or play solitaire. As I grew up this stayed about the same. I got my first phone sophomore year of high school and then never really used it until the end of senior year (my friends would call the house because that was the more reliable way to reach me). It's not that I don't appreciate technology or am bad with technology; I can learn things pretty quickly. I've just always had a slight disinterest towards it.


    Thinking about the use of technology in my future classroom, online activities and web-based programs, I leaned toward the side of no. I feel like there is a lot to said for doing things by hand. But the world is changing. While teachers of the past did not have access to such things, I will. Using sites like Desmos and programs such as Geogebra have shown me technology has a lot to offer a math classroom. Students can experiment with graphing equations, how to reflect/translate/transform them. Data collection can be increasingly simplified. There is an abundance of resources for activities and projects. Technology has the ability to help introduce students to new, difficult to grasp ideas, providing a transition into the topic. Students are able to make references to real situations and see the actual motion a graph is depicting.

   So my mind has changed. However, along with this desire to incorporate technology, which can do amazing things, I still have some hesitance. Yes students are seeing ideas in new ways, and yes they can work at their own pace, and yes they get the chance to explore and discover on their own. Yet, if not careful, the message and ideas you are hoping students to see will become lost. Some of the activities - and I know I have not seen them all - were fun and provide a good introduction to a topic; however, they felt semi-easy. I did not feel I was presented with a challenge or something asking me to think deeply and critically. And this is what I think is truly important. I see technology is helpful, but without in-depth discussions to follow up or asking students to create something of their own (to work through the process of solving a complex problem), it is not going to be used at it's full potential. We need to use technology to challenge our students, not just simplify how things can be done.

Eleusis: Problem Solving

The Problem

 
 Given the following Eleusis card set-up, find a rule to describe it. Then, list three more not yet played cards that can follow the set-up.

The Process

  
   My first initial thought when viewing the Eleusis card set-up was that color was not the sole factor of the rule. There were two blacks, two reds, black, red, four black. This did not appear to be any sort of pattern. Next I noticed how color affected cards of the same number being right and wrong. I found these four instances:

• A black 10 could not follow a black 8 but a red 10 could.
• A black 2 could follow a red 3 but a red 2 could not.
• A black 6 could follow a black king (or 13) but a red 6 could not.
• A black 8 could follow a black 7 but a red 8 could not.

   After discovering this, I was able to eliminate adding/subtracting of the cards to be part of the rule. Color would not affect the resulting sum or difference of the two numbers. Getting on the thought of addition and subtracting, I decided to look at the differences between the cards. At the time, I was thinking anything I discover will be helpful - whether I find a pattern or not. As a result I ended up with the following:

   In hindsight, I discovered what I already knew: that the difference between the numbers is irrelevant. Otherwise, if this was untrue any 10 should have been able to follow the 8 because the difference would still be the same. However, I stumbled upon something important. What is the value of the ace? In this rule, was the ace being viewed as 1 or 14, odd or even? To figure this out, I started to propose guesses as to what the rule may be.

• If you go from even to even, you must change suite.

This instance only occurred once. So I ruled it out and reformulated the hypothesis.

• If you go from even to even or odd to odd, you must change suite.

There were however cases not covered by this rule, so this was not it either. For example it could not explain why the 2 of hearts could not follow the 3 of diamonds. While neither of these were the rule, thinking and testing my guesses was helpful. I saw what did not work and began to notice my focus on odd/even and the suite/color of the card. So I made the following chart (assuming the ace is 1):

   This simplification of the information given, brought me to a pattern and a rule to describe the Eleusis set-up

The Solution

  
   The rule for the pictured Eleusis is: An even card must be followed by a red card, and an odd card must be followed by a black card. So based upon my rule I can play the following 3 cards:

Evaluation

 
    Describing my thought process was the easy part of solving this problem. Although, it took an effort to remind myself to record what all I had been thinking. Having stepped back and re-evaluated the problem several times over a couple days and still no answer, I thought to myself: perhaps it is unsolvable. Perhaps the diagram  is too limited to reveal the rule. Unwilling to give up and a strong desire to overcome the problem - it couldn't be that difficult could it? - I continued to search for the solution from where I left off each time. A lot of thinking, as well as various methods, went into solving this one problem. Even when I was not looking at the problem, I tried to think about it. It wasn't so much as arriving at a correct answer. After all, my rule only holds if the ace is 1 and there was no example in the diagram to confirm an even red card can follow an even red card. I kept trying because I enjoyed the complexity of the problem and simply did not want to give up. I wanted to make sense of the problem at hand and I did.

Eleusis: A Card Game

Eleusis is a fun and challenging card game that at it's heart encourages critical thinking and problem solving. Always having been a fan of card games and puzzles, I was excited to learn how to play.

The Rules:

Each Player is dealt 10 cards, expect the dealer (so the game is best played with smaller groups).

The dealer is then left in charge of coming up with a rule that the cards played must follow. The remaining cards are sat in a pile. The dealer will flip the first card over, then the players will take turns placing a card from their hand to test their hypothesis of what the dealer's card rule may be. If the players card does not follow the rule, it is placed beneath the card it could not follow. The player must then draw a card. 

The Object of The Game:

To guess the dealer's rule! After each correctly played card, a player can take a guess as to what the dealer's rule is. Once they get it, they have successfully won the game. From here you can pick a new dealer and the game starts over.

Example:

The first card played in the example is the 5, followed by the correctly played 7. The 10 was played after the 7, however, because it was incorrect it is placed under the 7. Then the 2, followed by the 9 are played after the 7 correctly. In this example the rule is very simple: Red, Black, Red,.....

Helpful Hints:

• Don't make the rule too complex. If your rule is too complex it will restrict a lot of cards from being played. The players will likely be unable to solve your rule and there's no fun in that.
• Don't make the rule too simple. The players will guess the rule quickly and the challenge of the game is lost.
• Play wrong cards. When you think you have the rule, play a card that goes against it. If the dealer says it works then your initial guess isn't quite right yet.
• What's the Ace? The ace can be viewed as high or low, 1 or 14, odd or even.
• Remember the Characteristics. You're searching for a pattern, and in finding this you should remember to consider all the characteristics cards have. Cards are even/odd, black/red, different suites and numbers. Any one of these or a combination can go into the rule.

    While playing Eleusis you get caught up in the game, you forget that you're thinking deeply and critically about each card that is played. Whether a card is right or wrong, you are taking it into consideration. You take multiple steps to try to solve the problem, testing your hypothesis, reformulating it and testing it again. In your head you are adding numbers, subtracting them, analyzing the relationships. Yet, Eleusis is just a card game and you are having fun. After playing the game in MTH 229 (Mathematical Activities for Secondary Teachers) and discussing our thought processes, it does seem like a great game to play in the classroom. Since it is a card game, negative perceptions and connotations of math disappear for the time being. At the same time, students gain and engage in mathematical thinking.

Check out my follow-up blogpost (Eleusis: Problem Solving) where I discuss my thought process in solving the rule for the below Eleusis card game. Try it for yourself and compare results.