Student Guest Blog Post: Sweet Treats in Deceit - Problem Solving Project

Here is second post by a different student in my Honors Problem-Solving Seminar; not every student had to do something that involved pure math. This student chose to use problem solving to figure out how to make cookies that she could eat...and the results were delicious!!

For the remaining blog posts, go to sasproblemsolving.blogspot.com

My name is Jamie and I wanted to solve the problem of substituting healthy options for unhealthy ingredients in a variety of desserts. Solving these problems are interesting to me because I love to bake, but I am gluten-free and mostly sugar-free, so I am unable to bake as much as I used to. Due to this, I am now able to bake, but with ingredients that stick to my diet. Although there are many recipes that were made to be healthy that I have tried in the past, I have never enjoyed them. Because of this project, I have created new recipes that hopefully I, as well as other people in the same situation, will always be able to enjoy in the future.

                                          

First, in order to correctly replace traditional dessert ingredients with healthy ones, I had to study the chemical reactions in baking to reproduce the same reactions, but with healthier ingredients. For example, the melting of butter in the oven causes cookies to spread out and flatten. It creates smaller air pockets, resulting in the product becoming much chewier. Coconut oil can be substituted for butter, but be careful not to use too much because then the cookies will taste like coconut.

                                    


Sugars contribute to the taste of your product, as well as caramelization and the Maillard reaction. The Maillard reaction is the combination of proteins and sugars, at 310˚, which creates rings and the brown color seen in most cookies. Also, as the sugars continue to break down, at 356˚, they turn into a brown, flavorful liquid, which is a process called caramelization. White sugar is mostly sucrose, so only dark sugars such as brown sugar are used for to create the Maillard reaction. This is why most cookie recipes call for both granulated and brown sugars because the white sugar is used for one process, while the brown is used for the other. To make up for sugar’s major part in the taste of the cookie, honey, dates, and banana can be used, or other natural sugars.

                                   


Additionally, the thickness of a cookie depends on the amount of flour used. If a thicker cookie is wanted, additional flour should be added to your product. Some substitutions for all-purpose flour consist of either almond, oat, potato, or gluten-free flour. For my project, I used almond and oat flour, but it depends on personal preference.


                              


The final ingredient I will eliminate from dessert recipes is chocolate, which for most people, is the most important part. A couple of months ago, I substituted milk chocolate for dark chocolate, precisely 86% cacao. Most people would call it disgusting because it is extremely bitter, but to me, it is my favorite type of chocolate. Because I have become accustomed to eating it, I no longer enjoy eating milk chocolate. Due to this, in my recipes, I have use 86% cacao, which is actually extremely beneficial because of all the antioxidants, as well as lowering risks of diseases. Here are my two cookie recipes. I hope you enjoy them as much as I do!


                                 
Zucchini Chocolate Chip Cookies:

3/4 cup oat flour

3/4 cups shredded zucchini

1 teaspoon baking soda

1 tablespoon almond butter

1/2 cup almond flour

1 teaspoon cinnamon

1/2 teaspoon salt

1 teaspoon coconut oil

1 tablespoon honey

1 1/2 teaspoon vanilla

2 eggs

1/4 banana


Preheat oven to 325˚. Whisk eggs and combine with almond butter, banana, zucchini, honey, vanilla, and coconut oil. Add in salt, cinnamon, baking soda, almond flour, and oat flour. Bake for 25-30 minutes.

                                 


Banana Chocolate Chip Cookies


1 banana

2 eggs

1 teaspoon baking soda

8 dates

2 1/2 teaspoon vanilla

1 ½ cup oat flour

2 tablespoon honey

1/2 teaspoon coconut oil

1/4 teaspoon salt

86% cacao chocolate

Preheat oven to 325˚. Remove the seeds from the dates and blend them until they are no longer solid. Add in the eggs, honey, coconut oil, vanilla, and a mashed banana. Mix thoroughly. Combine the remaining dry ingredients with the wet mixture. Place tablespoons of the cookie dough onto a pan with parchment paper and flatten each of them. Bake for 25-30 minutes.

Student Guest Blog Post: Drawing Mandalas with Compasses and Protractors! Geometry Art!

Here is the first guest post from my student Morgan! Morgan was in my Honors Problem-Solving Seminar class, and her final project was to learn how to draw a mandala using math! Enjoy! For the rest of the posts, visit: sasproblemsolving.blogspot.com

Hi! My name is Morgan, and because of my love for yoga, I decided to learn how to draw a mandala!

Mandalas represent an imaginary place that one's mind travels too when he or she meditates. Each object one observes in that place has significance, embodying an aspect of wisdom or reminding the meditator of a guiding principle. The mandala's purpose is to help transform ordinary minds into enlightened ones and to assist with healing just as yoga has for me with anxiety. The different movements and yoga positions leave me feeling relaxed and allow me to clear my mind. I felt a similar feeling of relaxation when I drew my mandala.

Having never drawn a mandala before, I had predicted it to be a long, complicated process that would be draining. However, I felt calm and stress-free as I allowed my mind to unravel and draw my mandala. At first, it was a little challenging to know what shapes and lines to draw after I had created the skeleton of the mandala, but by the end, I didn't even have to think much. I went with my intuition and let my mind do what it felt in the moment.

Here is a video of my first mandala drawing, showing the whole process and what goes into creating a mandala.

My first mandala completed

A close up shot of my mandala

Materials needed to draw a mandala:

• Compass
• Protractor 
• Ruler 
• Pencil
• Black pen with felt tip (thin sharpies will do)
• Blank notebook or 8.5 by 11 sheet paper

Procedure:

1. Draw “skeleton” in pencil.
• As I showed in my video, use your compass to draw a small circle in the center of the paper then continue outward drawing circles until you reach near the end of the piece of paper. 
TIP: rotate the paper while using the compass as shown in this video. 


• Once your circles are complete, use your ruler to draw straight lines to divide the circle up into several different sections. 

22.5 cm trick shown above
• In addition, although I didn't show it in my video, you can use a protractor and make markings at every 22.5 cm then use a ruler to draw lines so that your mandala is more precise.
• Also, the circles do not need to be exactly the same amount apart from each other. It actually comes out better if you organically draw some circles close together and and some further apart. 
2. Then begin using your black felt tip pen (or Sharpie) to draw the mandala. 
• Some commons designs and shapes include:
• Petals 
• Longer and thinner
• Shorter and rounder
• Triangles
• Various line lengths 
• Circles with dots 
• Squares 
• Symbols such as: 
• Flowers
• Suns
• Moons
• Trees
  
  Basic designs
• Tip: Begin with drawing simpler shapes then use dots and smaller designs to fill in shapes and the areas surrounding them!
3. After you finish drawing and designing with the black felt tip pen, erase the pencil marks underneath and admire your mandala! 
• You can also add color to your mandala to brighten it up if you want!

After learning how to draw a mandala, I shared the process with some students who are part of Saint Andrew's Mu Alpha Theta, which is a math club at my high school. Here below are some photos of them and their mandalas.

Designing the mandala

Beginning the design after creating the skeleton

Working on the first step with compasses

A colorful mandala!

A finished mandala with a unique design: a different design for each half

Another finished mandala

I hope my blog inspires you and teaches you about mandalas! I have linked some mandala designers and tutorials that I found helpful below as resources for you. :) 

Here are some accounts that I found inspiring when creating my mandala!
@courtneybetts was super helpful in getting me started with my mandala and giving me tips. She has a super cool art Instagram story on her page that features beautiful mandalas. I came across @mandalabybhagya 's Instagram page when looking for inspiration for designs for my mandala. She has lots of intricate colorful mandalas, too! 

One of @mandalabybhagya's colorful mandalas

Another one of @mandalabybhagya's beautiful creations

Lastly, here are some tutorials that I recommend to help you on your mandala journey!
https://www.youtube.com/watch?v=OiSzGBguPm0
https://www.youtube.com/watch?v=QcHDIK0E5KY&t=573s

Thanks for reading! :)

Like a Good Puzzle? Try the TED-Ed Dark Coin Riddle

Have you seen the new Dark Coin Riddle? Give it a try, but also check out the TED-Ed lesson to go even further with it.

Want to try some more? Give these a try with your classes!

First Week Lessons

Tomorrow is the first day of school! I am teaching Honors Problem-Solving Seminar (see this for the first unit of the textbook I created, problem set 1, problem set 2, and the blog post by my students from last year.), Pre-Calculus AB (our newly named accelerated Pre-Calc, for kids moving on to AP Calculus AB next year), and Algebra 2 Honors. I have four classes, and we are lucky enough to have a Math Lab staffed one period a day by a math teacher. So that's my 5th period in a 7-period block schedule.

I was going to blog AFTER I got my thoughts together about what I am doing this week (we see all students Wednesday for 30 minutes, and then we see our first set of blocks on Thursday and our second set on Friday), but I realize I need this blog to gather my thoughts together RIGHT NOW. :)

Wednesday:

I have am using Apple TV in lieu of a SmartBoard, so I am using an iPad with Notability as a teaching tool. It's a little confusing using a laptop for writing lessons and an iPad for presenting, but I am getting the hang of it. We are a 1-1 school with laptops, so students do not have iPads. 

I loved Mathy Cathy's blog about her first day, and think her use of Tackk is incredible. Since I am just getting familiar with Notability, I decided to use that but make something similar. I love the website https://www.canva.com/ and made this to display for students coming in:

Next, I am going to show students part of their homework, which is Who I Am...who can I give credit to for this? Mathy Cathy is also is doing this, so I felt good that I was doing the same. I knew my students would be confused about the self-portrait, so I like her addition in blue, but don't print that one out for the students!

Next, I am going to go over my syllabus. I know I've blogged about them before, but I can't remember where. I also am not sure who to give credit to here, so let me know if this was your idea! The "live" version has links to their text help and classroom google.

I know it's a lot of stuff to cover in 30 minutes! I am not going over the entire syllabus in one day. 

We gave summer packets in the math department this year for the first time, so then I plan on showing students the second part of their homework, which is to fill out this google form on the questions they did not understand the most. I can look at the pie chart before class and see what I really need to go over in class. I already have one response and class didn't even start yet! I love Honors!

I am going to show them my extra help times so there is no question--I also have this posted on both of my doors.

And finally, the task--for 10 minutes I am going to do to the Growth Mindset Paper Folding Activity from Tim Bowman. 

It seems really cool and I'm excited to do it. You prepare a sheet of paper cut and folded in such a way that it looks easy to create, but it actually is not. You tell students they can walk around and look at it, but can't touch it and you give them paper and scissors and tell them they must recreate it, but they only get one shot. At the same time, you walk around and write down the things they are saying, like, "I can't do this!" "This is impossible!" etc. You write what they are saying on stickies or on the board and then eventually stop them and ask them how they felt. 

Tom writes, "I then refer back to the language used during the activity and how the language we tell ourselves quickly becomes our own best friend or worst enemy. If students can start to reflect on the language they use about themselves at a (sic) young and choose a better path it is a fair bet their lives will be productive and fulfilling."

I know I will be short on time, so I will probably follow up with it in my classes the next day. 

Thursday/Friday (blocks so only meet class once):

In Problem-Solving, I first need to share how cool my workbook cover is, in the shape of our crest and designed by a student who graduated last year.

I am going to start with the "National Math Salute," which I saw at FAU Math Day last year, 

and its reveal...

I think they will love this because it seems so easy but it's almost impossible :)

Then we will do the topics on my first unit: 1-5-4-2-3 card game and Angela Duckworth's talk on grit that I posted about here. They will also get their first problem set assignment, due back in a week.

In Pre-Calculus, based on the questions that they check marked, I will go over whatever topics they need, and will have them go up to the board to practice problems. This is all we will do because we have only a 60-minute special schedule today, and they have a quiz on their summer packet Monday, and I know I may need to go over the activity and/or finish the syllabus.

In Algebra 2 Honors, I will do the same thing as in Pre-Calculus, but we have 30 more minutes. So I am going to do the 1-100 numbers activity from Sara V. I have seen this a lot, most recently from Sarah Carter, and because she said we have to do it, I'm gonna do it :)

So that's it for the week...whew! It seems like a lot, but I'm going to give it a try. Better to be overprepared than underprepared :)

Hope you all have a great week!

Rubik's Cube Mosaic...Fun, Satisfying, and Great Teambuilder

Today my Problem Solving Class unpacked the 225 Rubik's Cubes shipped from http://www.youcandothecube.com/cube-mosaics/ and began the task of creating our school crest. Last week, a student used Photoshop to turn a picture of the crest into a mosaic, and after spending time adjusting some of the squares that did not turn out quite right, we were ready! I asked Diane from the company for a bit of help, and she immediately responded and helped me to figure out what to do. My class and I divided the mosaic into 25 sets of 9 cubes. Each student (9 altogether) were assigned 2 or 3 sets of the 9 cubes, and their task was to make the top face look like the one portion of the 9 cubes...so for example, if you remember the opening of the Brady Bunch, one student had to first figure out the face of "Marcia", then "Carol", and this continued till they did the whole group of 9...then they placed them on the butcher paper I had on the floor.

 Here is how we divided up students, and we looked at the original just to make sure we fet good about the colors.

It was pretty amazing to get 225 Rubik's cubes all solved...it was quite satisfying see them all! But we had to mess them up to get them to match the picture. 

We had no idea how long it would take. Our class is 1.5 hours long, and I definitely thought we would need at least two classes to complete our task. But about an hour in, I new we could do it...it started getting really exciting as we had only a handful of cubes left. We finished with about 10 minutes to spare!

We may do one more, depending if we can find a piece of student artwork that works well with the mosaic program. Next year, we will use 400 cubes to get an even more accurate design.

I think this project really promoted teamwork, collaboration, and grit. We talked about it for months, so finally doing it was very fulfilling for students.

I am working on writing up directions to solve the cube--it will not be directions to make you the fastest, but rather to make you remember how to solve it and show other students...I am collaborating with the teacher who skyped with me and taught my class and me how to solve a Rubik's Cube, Dan VDV. He was so patient and helpful, and I hope to do the same for the #MTBoS! I have written about how solving the cube helps to develop grit here.

Teaching (and Learning) Grit by Having Students Solve the Rubik's Cube

It all started when my 17-year-old son came home from his summer teen tour and had some much needed down time. The normal teenager would probably just sleep or watch videos, but AJ decided he wanted to solve the Rubik's Cube. I remember playing with one as a child and could get one face complete pretty easily. But in "those days," the directions were all on a folded piece of paper (the horror!), and I did not have the GRIT needed back then...at least not with this puzzle. But he watched a video over and over again, and by the next day, he had it down. (As I am writing this, I hear the clicks from him solving his cube...it's a good "brain break" for him between homework assignments.)

Over same summer, I was planning for my new Problem Solving Seminar, and I thought I would make solving the Rubik's cube an assignment for the class around Thanksgiving. I thought it would be a good time for kids to practice...and perhaps get encouragement from their families over break. I decided I would have them watch the video my son learned from, and I would facilitate, but that I did not have to really solve it...after all, it was an assignment for them, and maybe I didn't really have to (gulp) solve it?! It felt really daunting to me, and yet, I knew my kids could do it. I just didn't necessarily want to--which I know does not make a whole lot of sense right now...but it somehow did to me then.

I asked the bookstore to stock Rubik's cubes...the only thing they needed to purchase for the course, and after avoiding lots of "when are we going to solve the Rubik's cube?" questions, we finally watched the above video together last week. I broke down the first few steps as such:

• The white cross on top with yellow in the middle
• The white cross flipped to the bottom with white in the middle and a partial matching T
• The entire white face with one full layer (top) complete
• The entire second layer complete

As we were watching the video and pausing A LOT, I noticed that many of the students were having trouble visualizing. Ironically, one of the top students in the class could not follow the directions at all at first. But most kids were still very interested...solving the Rubik's Cube is like a fun party trick to pull out-out of nowhere, so most were determined. Some asked me to share the video via Classroom Google, so they could watch at their own pace, which I did.

Oddly, I was able to see how to do the first three of the four steps, above, pretty easily. I say oddly because my spatial reasoning is my weak point as a math teacher. You could spin me around in my own driveway, and I will get lost. But I guess having played around with the cube a lot as a kid, I could do these steps fairly quickly. So once I realized my kids needed help, and I could help them, I started to want to solve it myself. But it was not until then that I felt the need to solve it. They needed my help in explaining it, and that I could do. But I didn't know how far I could get...that second layer blew my mind.

Then luck happened. I got an email from the Mathematical Association of America (MAA) highlighting an article about Dan Van der Vieren (known to students as Mr. VDV), a teacher who wrote his undergrad thesis on the Rubik's cube. I immediately followed him on twitter and asked if he was willing to skype with my class...and he said yes! And then the magic happened.

Via skype, and with pictures like the one to the left, he showed us how to get the entire second layer complete. This involved an algorithm. My students struggled through this (as did I), making mistakes and having to redo it all over again, but once they finally got it on their own the texts with pictures started pouring in...on a Saturday night?! And it coincided with me getting that layer complete as well. We felt so accomplished!

Mr. VDV, is skyping with us again on Wednesday (the day before Thanksgiving break) to get us to the next stage. This, more than anything else that I have ever taught, is teaching the kids tenacity and grit and stick-to-it-tiveness. I do have a student who wants to give up. I hope more than anything else, I can encourage him to stick with it and solve it. He will learn more from that, I think than anything he has learned in my class. If he learns how to do it, which will come not only from my helping him but also from his willingness to learn from his mistakes, I will feel like I have done my job.

Mr. VDV has tweeted with me regularly, sharing pictures like these to help me help my students--so incredible. I am so thankful to him...funny this is the case right around Thanksgiving.

 

Lastly, Mr. VDV has told me that his class has made mosaics with Rubik's Cubes, and now, of course, we HAVE to do this...next semester. I can't wait. He told me to register at http://www.youcandothecube.com/, which I did, and someone got back to me Sunday morning! They will ship all the cubes to you for the mosaic making; all we have to do is pay for shipping back. 

I am looking forward to my post when we actually create this. Another challenge, Mr. VDV told me, is to make our school logo as a mosaic. There is an app for that! The possibilities are endless.

I am not there yet...I haven't solved the puzzle fully. I'm 2/3 done, but I can do the entire 2/3 from memory...by Thanksgiving break, I hope to have it fully done, along with the rest of the students in my class. I finally am learning about the grit I have been talking about to my class. And it feels great.

The Four-Color Theorem and The Pumpkin Time-Bomb

"CRAYONS!? We are COLORING? Wait, can I snapchat this????"

Yes, yes you may.

Here are the directions I gave to my Problem-Solving class the other day:

Take any map and color it so that no two adjacent states are the same color. Keep in mind that you want to use as few colors as possible so that coloring the map is not too expensive. It turns out that any map can be colored in at most 4 colors! And that this was the first proof proven using a computer. 

Here is the map I gave them, without telling them that they needed 4 colors...you may want to provide extras.

Here is the quick video I showed them after they colored, that talks about the four-color theorem.

After coloring the map, we drew the graph at the right, where each vertex represents a country and an edge connecting two vertices means those countries are adjacent. Then we colored the vertices and found the "chromatic number," that is, the least number of colors we can use. We needed at least three, as you can see by the red triangle drawn between Bolivia, Paraguay, and Argentina. The chromatic number for this map was 4. (And remember, for any map, this is the most it will ever be.)

Then we went on to scheduling committees, and I showed how you could make each vertex a club and each edge would represent any club that had a member in common, so a conflict in time. The chromatic number, in this case, would be the least number of club meetings required. We also answered some interesting problems that related to the Handshake Problem.

Here are my notes from class that day, that I took from my book. 

All in all, a fun class with a theorem that they have never heard before. And I am sure that they will never forget.

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Today, we did the awesome Pumpkin-time-bomb-activity from Mr. Orr. Very fun predictions, and we were only 4 off from what we guessed! I highly recommend that you read his post, check out the data, and also watch the video from Jimmy Fallon.

The explosion took us by surprise so we didn't get a picture, but one student ended up with pumpkin all in her hair! I was amazed at what the rubberband clump looked like at the end! We didn't do anything to it. That's what it looked like after the explosion!

My Problem Solving Students Guest Blog: A Unit on Investigating Figurative Numbers

Our math teacher created a new course elective at Saint Andrew’s School called Honors Problem Solving Seminar. The class is interesting and has taught us everything from grit to fun little math tricks. So far we have discussed visual and figurative numbers, which are numbers that can be represented by a regular geometrical arrangement or sequence of evenly spaced points. They are most commonly expressed as regular polygons, for example, triangles, squares, pentagons, hexagons, etc. and also known as polygonal numbers. The class was a nice addition to the schedule providing us with a very relaxed way to enjoy math. This is what we have learned in our first big unit about figurative numbers.

(Note: this blog was collaboratively written by the entire class, originally in a Google Doc. All handouts are posted in my blog here.)

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We first started with oblong numbers.  Oblong numbers are the number of dots that can make up a rectangle with its length one more than its width.

To get to an explicit formula that can represent the development of the patterns of the number of dots on each rectangle, we first started by observing the rectangles. The first one consists of two dots; two for its length and one for width. The second has three for length and two for width. The third has four for length and three for width. We found out that as the pattern keeps going on, the new rectangle has one more dot on both its length and width than its previous term. So first, we constructed a recursive formula, using the number of dots on the previous term to define the number of dots in the current term. Since the length and width are both added by one each time, we thought that the current term is the resulting number of rows would be n and the resulting number of columns would be n+1.

Thus, the explicit formula for oblong number is n(n+1)

Then we were given triangular numbers. Triangular numbers are the dots grouped together that make up an equilateral triangle. Initially, we were trying to come up a formula based on the pattern and number of dots in each term: 1, 3, 6, 10... Obviously, the pattern does not appear directly based on the number of dots in each consecutive term. However, with Mrs. Winer’s small hint of drawing a diagonal, we found out that each term of the triangular number is exactly half of the oblong number. So the formula is basically the formula for oblong number divided by 2.

The explicit formula for triangular number is n(n+1)/2

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Now we can use what we have learnt from the oblong and triangular numbers to apply it to finding the sum of n counting numbers. If we pay attention to the triangular number, we will notice that the formation for triangular number is:

So we can find out nth triangular numbers means the sum of the nth counting numbers.

Now, for the sum of nth even numbers, (not counting zero), the first even number, 2 is two times bigger than one, the second even number, 4 is two times bigger than two, and the third 6 is two times bigger than 3.The pattern is the same for their sums. While the sum for all the counting numbers are triangular numbers, by studying the even numbers, we learned, that oblong, the combination of two triangular numbers, is the sum for the even numbers. We also can find it out by investigating the oblong numbers diagram.

After learning about the sum of n even counting numbers, we asked what the sum was for n counting odd numbers. We found out that you could find the sum of the odds just as fast as the sum of the evens. By looking at the diagram below (square numbers), we can conclude that the explicit equation for the sum of odd numbers is  n2 because in the diagram the number of dots in each picture is a perfect square of the nth term.

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To mix it up a bit, our math teacher presented to us the Pizza Problem.

Watch the video below for the Pizza Problem, plus an explanation of the Method of Finite Differences.

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Moving right along, we now faced the challenge of pentagonal numbers, which served to extend and solidify the concept of triangular numbers.

           

In a pentagon diagram, we can divide it into three parts (shown in the picture): red, blue and green. Count the dots in each part, and we can see the red part and the green part are the same; they are both the nth triangular number. In the middle, the number of dots in the blue section is also triangular number, but instead of nth term, it is the (n-2nd) term. Then we add all three parts together, but we count the first dot at the bottom twice, so we subtract 1. Simplify the equation, and we get the pattern for the pentagonal numbers.

RED:

1     1       1

2     3       1+2

3     6       1+2+3               -->    

4     10    1+2+3+4

5     15    1+2+3+4+5

…     …     …

GREEN:

1 1          1

2 3          1+2

3 6          1+2+3              -->

4 10       1+2+3+4

5 15       1+2+3+4+5

… …        …

BLUE:

1 0          0

2 0          0

3 1          1                    -->

4 3          1+2

5 6          1+2+3

…    …         …

Final Simplification

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Seeing that we were able to complete each task given so far, Mrs. Winer then challenged the class with yet another problem that once again would require an understanding of both the oblong and triangular numbers. She asked us,

“ What’s the difference between the sum of the first 2015 even counting numbers and the sum of the first 2015 odd counting numbers.”

At first the class was very confused and did not know how to solve this problem, but after much hard work, our class proudly could say that we discovered not one but TWO! very different methods to solve the problem.

Solution 1

First line up the first 5 even and first 5 odd numbers

2 + 4 + 6 + 8 + 10

1 + 3 + 5 + 7 + 9

Now you can see that the even numbers are always one greater than the odds. Since there are 2015 numbers, and each time the evens are one number more than the odds, the total difference will be the 2015.

Solution 2

This method uses direct substitution to find the first n counting even and odd numbers.

“N^2+N,” is the equation for sum of the even numbers and “N^2” for the sum of the odd numbers. The difference between the first n even and n odd counting numbers is the difference between the nth oblong number N^2+N and N^2:

N^2+N - N^2

The class then used direct substitution to solve the problem

(2015^2 + 2015) - (2015^2)= 2015

Although the two methods are very different, they both work in finding the answer. The class was split over which one they thought was easier.

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According to Pythagoras,“everything is related to mathematics. Numbers are the ultimate reality, and through mathematics, everything can be predicted and measured in rhythmic patterns or cycles.” So far, Pythagoras has been right about everything we have learned in this class and it is cool to see how everything comes together. Although there may be different ways to solve a problem, when it comes to math there is no ambiguity and everything eventually will fit together to form one solution.