The Seven Bridges of **Königsberg** is such an important problem in the history of mathematics because the negative resolution essentially laid the foundations of graph theory and prefigured the idea of topology. The solution to this problem was considered to be the first theorem of graph theory. And, when Euler recognized that the important information was the number of bridges and the list of their endpoints rather than the exact location and layout of each bridge this was the beginning development of topology. The idea of topology is not concerned with rigid shape of objects.

This problem was based in the city of **Königsberg** in Prussia along the Pregel River. The city had two large islands within which were connected by seven bridges (Hence the name of the problem.) The problem Euler proposed was to find a walk through the city that would cross each bridge exactly one time. The islands couldn’t be reached by any route other than the bridges, and every bridge must have been crossed completely every time, in other words, you couldn’t walk half way across a bridge and turn around then cross the other half at a later time. A final criterion was that the walked didn’t have to start and end at the same spot. Euler had eventually proved that the problem didn’t have a solution, but the question was why?

Euler noticed that whenever someone enters a island, or vertex, by a bridge they will then have to leave by a bridge (an edge) except for the end points of the walk. Therefore, the number of times someone enters an island equals the number of times they leave it. From this observation Euler was able to conclude that the number of bridges touching a specific land mass must be even (even degree) in order for this walk to be possible or have a zero or two odd degrees. Each of the islands had an odd degree thus providing a contradiction to the proposition of crossing each bridge only once.

Such a walk as this is now referred to an Eulerian path which was later proven by mathematician Carl Hierholzer. The lack of an Eulerian circuit, where you begin and start at the same point is another reason this problem had no solution.

To put more of a stamp on the start of graph theory, Euler also discovered the formula V – E + F = 2 which relates the number of vertices, edges, and faces of a convex polyhedron and now known as the Euler characteristic for the graph.

The Seven Bridges of **Königsberg** and its impact on graph theory and topology is just one of the many impacts Euler made on the history of mathematics and is why I find him so amazing. Without his works, who knows where our math would be today. Euler is definitely a Math Avenger in my book.

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We immediately started to plan once we were informed what we would be teaching. It was a struggle at first to come up with ideas that had students working with technology throughout the entire lesson. None of us, (except for Dr. Yu) had much experience with using technology in the classroom. After some exploration on GeoGebratube.org we discovered a student worksheet applet that was a Pantograph. A pantograph allows you to drag one point to draw a figure, and then is enlarged by the pantograph. We made our first draft of the lesson plan, and then had many meetings revising it to make it as good as possible trying to anticipate different scenarios that could occur. The students would have access to Chromebooks, so we decided the easiest way to give the students all the information they would need for the lesson would be to create a Weebly that had tabs for each part of lesson. The Weebly for this particular lesson can be found at gvsujenison.weebly.com where there are multiple tabs to gain access to the launch, exploration, and reflection pieces of the lesson.

The format of the teaching was that I would teach first hour, take an hour to reflect, and then Becky teach third hour with the revisions made during our reflection time. I believe that the lesson I taught went well overall. The group of students I had would not engage in much conversation throughout. I think this had to do with being first hour, and a complete stranger was conducting the lesson. The cooperating teacher Jill says that is a normal day with those students, just sit and stare. Even though the students were quiet, I could tell by what I did get out of them that they knew there stuff. It wasn’t till the last portion of the lesson that they got really engaged. I decided to have every two set of rows change the sliders on the geogebra worksheet and then convince their partners whether or not the two shapes were still similar. It was great to see the reactions of the students after they measured the corresponding parts to find they were not similar in some cases in which by looking visually, the two shapes looked similar. After the lesson was concluded, I had students fill out an exit slip. They were asked to answer 5 questions.

- What are two things you learned today?
- What did you like about the lesson?
- What did you dislike about the lesson?
- What could have been done differently?
- Do you prefer lessons with or without use of technology?

An overall consensus, this group was very much so in to the idea of having technology based lessons. Through out the lesson I would ask checkpoint questions that told me whether or not they understood a certain topic. At the end of the lesson when we summarized what we learned, students really responded well by demonstrating they now understood the concept of proportions/similarity. I was happy with the results.

During the reflection time, we discussed some of the modifications that should be made from my lesson to Becky’s. Some of these included:

- Use the terms magnitude instead of scale factor because this is what the students are familiar with
- Utilize the board to write definitions and show things on the figures. I did not do this during my lesson. Remember, the board should be a log of what happens throughout a lesson. (Teaching Gap)
- Instead of saying “Convince your partner your figure is similar” say “Use your CALCULATIONS/MEASUREMENTS to convince your partner…” this way students don’t just say looks similar and the move on to a different topic of conversation.
- Have students move their desks together for the duration of class versus having students move during moments of collaboration.

Becky utilized all of these modifications and it worked wonderfully. The only parts of her lesson I would change would be time management. There were times where too much time was spent on a specific part of lessons and students got bored of the task at hand. I would also close/summarize throughout lesson verse just stopping and moving on.

It was interesting to see that there were more students in this class that did not like the technology in the lesson. They recommended not using technology the entire time, and to have more things to do throughout lesson. Which gave mixed feelings for this particular class verse the generally unanimous love for technology during first hour.

The next step in the project is to collaborate and write an article displaying our lesson and results to hopefully be published.

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Most of the instructional shifts I have learned about at Grand Valley State University, so it was really beneficial to reiterate important ideas about the classroom. Leinwand did a great job of providing examples of dialogue within a classroom with and without the suggested shift. This provides a good idea of what our students are not getting out of math class and why so many students are bored with math.

Chapter 2 discussed the importance of constant review. This is one topic we did not discuss in my education classes and I found to be interesting and something I want to bring with me to my future classroom. It discusses the waste of the first few minutes of class when the teacher takes attendance, checks for homework completion, and then attempts to get class going. A simple solution and easy way to take full advantage of the short class periods is to have mini-math “quizzes” to get students minds warmed up for the rest of class. There could be 1 to 7 problems that cover topics from the first week of school or even foreshadow next weeks lesson. This provides students with a chance to master the variety of math skills. Let’s face it, its rare for a student to master something after half of a week of instruction. Plus, the teacher is able to check homework and do attendance while the students are doing this.

I specifically see my self using the idea of constant review in my future class room but adjusting it to fit the needs of the class I see my self having and to most importantly fit my students needs. I see my future class as being a very much activity based class which asks for as much time as possible for students to work on these activities. Therefore, I think I would stay away from any number of questions that exceeds 3 or 4. I believe it would be very beneficial to use the five minutes that it takes for me to take attendance as a time to have 3 questions for my students to work on. One of these questions I would pull from previously learned concepts earlier in the year. This as described above will allow for mastery of the information rather than memorizing it for the test and immediately forgetting it. The second question I would think would come from the current unit being taught to keep it fresh in my students minds. Possibly directly from the lesson previous to that days lesson just to reiterate what we learned the day prior. The third question I foresee coming from the future unit. Students I would expect to struggle and possibly have no clue how to work on this problem and I think this would be great for the class. I would anticipate the discussion of the first problem to go fairly quick because hopefully they would have it mastered. The second problem I see spending more time on simply because we would of just discussed it the day prior. The third we may not even come to a final conclusion. I see it being more meant to get the students thinking ahead so that when it comes time to actually teach that concept, students are already aware of it which I think will make the learning experience much better.

An important instructional shift Leinwand brings to our attention is having real world connections and situations for students to apply their math skills in. This will engage the students in their learning instead of falling asleep while counting down the minutes to go to phys. ed. class. With having these real world situations students can discover math and have in depth conversations amongst their peers and really learn the material. I see my self using a lot of videos to bring a real world situation into the classroom. This will get my students hooked onto that video, and then I can supply activities that feed off the video shown.

I see my classroom being a very activity and discussion based. After class activities, I would like to have class discussions about the activity where I can ask students questions such as “why?” or “how do you know?” In my experience as an educator thus far, I have found when I do this type of thing students learn a lot more. One, because students have to develop enough knowledge to be able to explain their thought process. And, secondly the students who are having trouble have multiple opportunities to hear explanations of the topic we are covering. This type of environment is tightly related to Leinwands 10th instruction shift of making “Why?” “How do you know?” and “Can you explain?” apart of your classroom mantras.

Another shift Leinwand talks about is having multiple representations. This is something I have near the top of my list as far as how I would like to teach my students. There are so many different learning styles out there and a big reason so many students don’t like math is because there teacher only stands in front of the class spewing out information that the students have no clue what it means. I am going to make it a priority to constantly give my students opportunities to draw and describe their math. The board will be a log of the class period full of pictures, tables, and number lines to give my students a visual to go off of to hopefully help them make sense of the information.

This book did a fantastic job of pointing out important shifts teacher need to make to better the learning of their students. would recommend this book to any one entering the field of teaching or even current teachers. These are 10 minor adjustments that can be made in a classroom that can make major impacts for the betterment of our students. Leinwand does a phenomenal job of portraying the importance of each.

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I am not an artsy person to say the least. Thus, it was really cool to see some art being made using mathematics such as Fibonacci numbers. So I decided to make a pine cone exactly like ViHarts. With 5 spirals going one way, and 8 spirals going the other. Yes, I agree this isn’t very creative, at least I should have used two different Fibonacci numbers but I am trying to learn how to be a decent artist, so I was using the visual aide to help me. I struggled to create a decent looking piece.

Some of my issues included:

- Making my spirals with the right amount of space between them
- Knowing where to make the “petal” of pine cone and how big to make it

The steps of how to create a pine cone include:

- Pick two Fibonacci numbers (I chose 5 and 8)
- Make 5 spirals going one way, and 8 spirals going the other way
- Create your pine cone by going 5 one way, and 8 the other using your spirals as a guide (This was very tricky for me)

Below is a picture of the best outcome I was able to come up with.

As you can see, even with the help of Fibonacci, I am not the best artist. In the middle of the pine cone, you can count and see that I am following the Fibonacci numbers 5 and 8. Although, as I get wider, I start to lose where I should be drawing the petals and count nine of them on the outside instead of 8. I would say this is a start into the right direction of creating a decent looking pine cone. I am going to continue to draw these using a variety of Fibonacci numbers. I will be posting the results on here. Maybe I will have to resort to Geogebra to get something that looks good. Even if I did enjoy drawing free hand using mathematics that I have learned.

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When I first started to create these tessellations my first thought was to start out with a single shape then build off of that shape. For the first one, I decided it would be easiest if I can use different figures off of my initial shape, the octagon, so that the end result would be a square with multiple shapes within. I then started to play with the reflection over a line tool in Geogebra and started to make my tessellation bigger along with having symmetry. As I started to make the tessellation bigger into a bigger rectangle I thought it looked boring. Therefore, I decided to bring the octagon from the start all the way to the finish by making the shape of the final tessellation exactly that. An octagon. Well almost an octagon, the 4 corner sides are not equal to the other sides. But, either way I think it brought some more spunk to my tessellation.

For my second tessellation, I wanted it to be much more creative. I wouldn’t consider my self the most creative person when it comes to artsy type tasks but I was happy with the outcome. I started with the same mentality as the first by making a single shape my starting point. I used the triangle, which eventually turned into a hexagon. I went to the reflection over a line tool and just started to play with the different directions of reflection. I admit, I accidentally came up with the shape of the upper half just my reflecting and thought it looked cool, especially have I reflected it over a line to create the bottom half. I then referred to some tessellations we looked at in class and noticed how colorful they were. I wanted the symmetry to still be there even within the color selections, so I made sure to put color in the same spot on the other side of the symmetrical line as if I was painting the wings of a butterfly.

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Books one thru six discuss plane geometry

Books seven thru nine have to do with number theory

Book ten deals with Eudoxus’s theory of irrational numbers

Books eleven thru thirteen discusses solid geometry

Euclid was said to compile these Elements from a variety of men before him such as Hippocrates of Chios and Theudius.

I am most familiar with the 5 assumptions from Book I, that Euclid called postulates or axioms.

1. Given two points there is one straight line that joins them

2. A straight line segment can be prolonged indefinitely

3. A circle can be constructed when a point for its centre and a distance for its radius are given

4. All right angles are equal

If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if

5. produced indefinitely, meet on that side on which the angles are less than the two right angles.

These to me are some of the foundations for understanding geometry, specifically Euclidean geometry.

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These are just a few of the ideas that come to mind when I hear the question, “What is Math?” But, what about where math came from? The history? This is an idea that I don’t know much about, and I look forward to learning through out the MTH 495 Capstone course. I did read a graphic novel during the semester of Fall 2013 named, “Logicomix” which was about a mathematician/logician by the name of Bertrand Russell who struggled to make sense of logic. Throughout the story it discussed names such as Euclid.

Some of the top 5 discoveries/moments in the world of mathematics in my eyes include the following:

1. The Fundamental Theorem of Calculus

2. The Pythagorean Theorem

3. Fibonacci Sequence

4. Math Specific Technology (online applets, calculators, computer programs, educational tools)

5. Pi

The five discoveries above are not in a specific order, these are just the 5 that I feel are important to mathematics and how we teach the subject. I don’t know exactly where these were first discovered or by who, but hopefully these are some of the topics we discuss throughout the course.

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