As I was watching 163 Ramanujan Constant - Numberphile besides wondering why they were filming so intensely close to his face I also wondered how mathematicians from the 19th Century were so advanced in their way of thinking. And even non-mathematicians were contributing to explorations in math. With this being said let's talk about the number 163.

So, apparently Heegner found that sqrt(-163) was the last number that could be written as a product of primes. The numbers jumped fairly quickly as you can see in the following list: sqrt(-1), sqrt(-2), sqrt(-3), sqrt(-7), sqrt(-11), sqrt(-19), sqrt(-43), sqrt(-67), sqrt(-163). He conjectured that 163 was the last number of this nature despite the fact that the numbers seemed to follow a pattern of jumping by larger and larger quantities. He came up with a proof that was not accepted to be true until years later. The really fascinating part about this is that when e is raised the the power of sqrt(163)pi it is oddly close to a whole number. These numbers are referred to as the Heegner numbers and have cool connections with amazing results in prime number theory. In particular, when e is raised to the power of sqrt(Heegner#)pi there are stunning connections between e, pi, and the algebraic integers.

This video had me so confused but in a curious way. I had so many questions for the man presenting the topic. I wish he would have went more into why these numbers worked out the way they did. How does one even think to try raising e to the product of the sqrt of these new found numbers and pi? It is so interesting to learn about all the ways people came up with new discoveries in math.

]]>So, apparently Heegner found that sqrt(-163) was the last number that could be written as a product of primes. The numbers jumped fairly quickly as you can see in the following list: sqrt(-1), sqrt(-2), sqrt(-3), sqrt(-7), sqrt(-11), sqrt(-19), sqrt(-43), sqrt(-67), sqrt(-163). He conjectured that 163 was the last number of this nature despite the fact that the numbers seemed to follow a pattern of jumping by larger and larger quantities. He came up with a proof that was not accepted to be true until years later. The really fascinating part about this is that when e is raised the the power of sqrt(163)pi it is oddly close to a whole number. These numbers are referred to as the Heegner numbers and have cool connections with amazing results in prime number theory. In particular, when e is raised to the power of sqrt(Heegner#)pi there are stunning connections between e, pi, and the algebraic integers.

This video had me so confused but in a curious way. I had so many questions for the man presenting the topic. I wish he would have went more into why these numbers worked out the way they did. How does one even think to try raising e to the product of the sqrt of these new found numbers and pi? It is so interesting to learn about all the ways people came up with new discoveries in math.

Learning about Ramanujan’s Magic Square was interesting to say the least. The square looks like another normal magic square but this is square has not just every column and row adding up to 139 but also every diagonal, four corners, four middle squares adding to 139 as well. He got the number 139 by adding his birthdate numbers together which makes it all the more impressive.

We were challenged to try to create our own Ramanujan’s Magic Square but with our own birthdates. I attempted to do so and will take you through my thought process along the way. Seeing as I was born on October 1st, 1994 I started out with the top row with a sum of 124. From that point on I knew that 124 was my special number that I was now working toward for other sums. I chose to first work on creating just row and column sums of 124. I did so by taking 124 and subtracting 94 to get 30. I now knew that I had 3 squares left to fill to get a sum of 30. Knowing that 15 and 15 together give 30 I wrote down 15 and filled the other 2 spots with a sum of 2 numbers that gave 30; 12 and 3. I then saw I had 15 in the bottom right corner and subtracted 15 rom 124 to get 109. I now knew that I had 3 squares left to fill to get a sum of 109. I divided 109 by 3 to give me an idea of 3 numbers that add together to get 109. Seeing as we can’t use the same number twice I could not use what the calculator gave me. So I then decided I would use a number that had a 9 in the ones place so then I could add a multiple of ten to it to get 109. Exactly why on the bottom row we have 13 +17=30+79=109+15=124. I then went up the left column and immediately saw that 10 and 13 gave 23. I knew that in order to get 109 I needed 101. I had not used 51 or 50 yet so decided to use them. Then I attempted filling in the 4 middle squares. I did so by taking the 2 numbers already within the column, for instance 1 and 17 give 18 and 18 subtracted from 124 is 106. I knew that within the 2 squares I needed to have them add to that 106. It worked out to be 49 and 57. The next column included both 19 and 79 already to give 98. By subtracting 98 from 124 I knew that I needed to get two numbers that when added together gave 26. This worked with 22 and 4.

When attempting to alter my magic square in order to get diagonals, four corners, and the four middle squares to add to 124 I had much difficulty. I then realized that I probably should have started with the four corners after I wrote out my birthday in the first row. I also should have done the diagonals first before filling in the rest of the 4x4 square.

There are many different methods of going about creating a magic square but when it comes to creating a super magic square I almost feel that there are fewer ways to go about that seeing as there are many more requirements that are involved.

]]>We were challenged to try to create our own Ramanujan’s Magic Square but with our own birthdates. I attempted to do so and will take you through my thought process along the way. Seeing as I was born on October 1st, 1994 I started out with the top row with a sum of 124. From that point on I knew that 124 was my special number that I was now working toward for other sums. I chose to first work on creating just row and column sums of 124. I did so by taking 124 and subtracting 94 to get 30. I now knew that I had 3 squares left to fill to get a sum of 30. Knowing that 15 and 15 together give 30 I wrote down 15 and filled the other 2 spots with a sum of 2 numbers that gave 30; 12 and 3. I then saw I had 15 in the bottom right corner and subtracted 15 rom 124 to get 109. I now knew that I had 3 squares left to fill to get a sum of 109. I divided 109 by 3 to give me an idea of 3 numbers that add together to get 109. Seeing as we can’t use the same number twice I could not use what the calculator gave me. So I then decided I would use a number that had a 9 in the ones place so then I could add a multiple of ten to it to get 109. Exactly why on the bottom row we have 13 +17=30+79=109+15=124. I then went up the left column and immediately saw that 10 and 13 gave 23. I knew that in order to get 109 I needed 101. I had not used 51 or 50 yet so decided to use them. Then I attempted filling in the 4 middle squares. I did so by taking the 2 numbers already within the column, for instance 1 and 17 give 18 and 18 subtracted from 124 is 106. I knew that within the 2 squares I needed to have them add to that 106. It worked out to be 49 and 57. The next column included both 19 and 79 already to give 98. By subtracting 98 from 124 I knew that I needed to get two numbers that when added together gave 26. This worked with 22 and 4.

When attempting to alter my magic square in order to get diagonals, four corners, and the four middle squares to add to 124 I had much difficulty. I then realized that I probably should have started with the four corners after I wrote out my birthday in the first row. I also should have done the diagonals first before filling in the rest of the 4x4 square.

There are many different methods of going about creating a magic square but when it comes to creating a super magic square I almost feel that there are fewer ways to go about that seeing as there are many more requirements that are involved.

I read the *The Number Mysteries: A Mathematical Odyssey Through Everyday Life *by Marcus Du Sautoy. I enjoyed reading this author’s take on the relevance of math in our everyday lives. Although I am aware that most all math is very relevant in our daily lives, it is comforting to see the connections he makes within this book.

One topic that I found very interesting, personally, was the idea of thinking of a clock to be in modulo 12. Since there are 12 numbers that round the clock and there are 24 hours within our day we can think of the clock in such a way. We repeat 12 two times. I thought of an example that is super relevant in our daily lives as when you go on a driving trip. Say you leave the house at 4pm and it takes you 9 hours to get to your destination. How do we know what time it will be when we arrive to our destination? Well because of the way modulo works, we would say we’d arrive at 1am. 4+9=1 (mod 12) Anything that would equal zero would be either noon or midnight depending on when you started your trip! Thinking of modulo in this relevant light made it easier to understand.

Another topic that I found to be interesting is the idea of nature being lazy. The example found in the book refers to bubbles! No matter the shape of the bubble wand the bubble blown will always be a sphere. It’s odd that I have never wondered why, but this book shed light on just that. A sphere is the easiest shape to create due to the fact that a sphere takes the least amount of energy to create. This is because the sphere is the shape that has the smallest surface area that can contain that fixed amount of air. We have been trying for years to mimic nature’s effective sphere making abilities and finally were successful in 1783 when an English plumber, William Watts, realized that he could crack the code for nature’s tendency for spheres.

I would say overall the book was very informative and challenged my critical thinking abilities by pointing out obvious math in every day life that was not so obvious to me, such as the clock and bubble examples. I did enjoy reading this book due to my own curiosity of the significance of math in our day-to-day lives. I don’t know if I would say that all future or current teachers should read this book but I don’t think it would hurt at all. Seeing the relevance of random math is cool and could potentially help us when students ask the beloved question of, “Why does this matter?” but I am reluctant to say it is necessary for all teachers to read.

When we had our classroom share about the books we read I feel there were other books that portrayed relevance in different ways that oddly enough seemed more relevant than this unsystematic compilation of relevant math.

]]>One topic that I found very interesting, personally, was the idea of thinking of a clock to be in modulo 12. Since there are 12 numbers that round the clock and there are 24 hours within our day we can think of the clock in such a way. We repeat 12 two times. I thought of an example that is super relevant in our daily lives as when you go on a driving trip. Say you leave the house at 4pm and it takes you 9 hours to get to your destination. How do we know what time it will be when we arrive to our destination? Well because of the way modulo works, we would say we’d arrive at 1am. 4+9=1 (mod 12) Anything that would equal zero would be either noon or midnight depending on when you started your trip! Thinking of modulo in this relevant light made it easier to understand.

Another topic that I found to be interesting is the idea of nature being lazy. The example found in the book refers to bubbles! No matter the shape of the bubble wand the bubble blown will always be a sphere. It’s odd that I have never wondered why, but this book shed light on just that. A sphere is the easiest shape to create due to the fact that a sphere takes the least amount of energy to create. This is because the sphere is the shape that has the smallest surface area that can contain that fixed amount of air. We have been trying for years to mimic nature’s effective sphere making abilities and finally were successful in 1783 when an English plumber, William Watts, realized that he could crack the code for nature’s tendency for spheres.

I would say overall the book was very informative and challenged my critical thinking abilities by pointing out obvious math in every day life that was not so obvious to me, such as the clock and bubble examples. I did enjoy reading this book due to my own curiosity of the significance of math in our day-to-day lives. I don’t know if I would say that all future or current teachers should read this book but I don’t think it would hurt at all. Seeing the relevance of random math is cool and could potentially help us when students ask the beloved question of, “Why does this matter?” but I am reluctant to say it is necessary for all teachers to read.

When we had our classroom share about the books we read I feel there were other books that portrayed relevance in different ways that oddly enough seemed more relevant than this unsystematic compilation of relevant math.

The image above represents part of the reason why I really have always liked math. There is, if proven, a correct answer that can be concluded from most every problem at hand. It is also convenient that math does not have to be memorized, but rather, solved. I recently took a 4-8 grade teacher certification exam and was having a difficult time understanding how this exam was really assessing my abilities as a teacher. I know it’s important to be able to do the tasks you ask of your students but instead, I felt as though the exam assessed how well I could memorize facts about history and science. When it came to English and math it asked me to apply my knowledge. We have done well differentiating learning for young students but where is that same differentiation when it comes to higher education and entrance exams? I understand the importance of summative assessments. They serve as a tool to get an idea of where the person is at academically within that topic. However, I do think we need a new approach that does not ask the surface questions of memorizing facts but rather our philosophy of education and how we plan on implementing these philosophies in our classrooms. There was no writing portion at all, only the 4 core subjects. While there were situational questions thrown into the mix of factual questions they were scarce. I do not know the best way to go about changing this assessment process but I do know it needs to be altered.

This has got me thinking further about the connection between English and math. I've always heard the phrase, "math is its own language". I have always agreed with this statement because of the fact that math uses symbols to represent meaning to others. Just as you are reading and comprehending this blog post we also read equations and (sometimes) comprehend their meaning. Our alphabet consists of 26 letters and the number of numbers is infinite. Think about all the complexities of the English language, such as spelling and reading. Now think about all the complexities of the Mathematical language, such as exponential and quadratic functions. By making note of the fact that math is its own language to students you allow them to understand how difficult math can be to learn so they do not get discouraged when trying to persevere in problem solving!

]]>This has got me thinking further about the connection between English and math. I've always heard the phrase, "math is its own language". I have always agreed with this statement because of the fact that math uses symbols to represent meaning to others. Just as you are reading and comprehending this blog post we also read equations and (sometimes) comprehend their meaning. Our alphabet consists of 26 letters and the number of numbers is infinite. Think about all the complexities of the English language, such as spelling and reading. Now think about all the complexities of the Mathematical language, such as exponential and quadratic functions. By making note of the fact that math is its own language to students you allow them to understand how difficult math can be to learn so they do not get discouraged when trying to persevere in problem solving!

My tessellation that I had worked on in class was simple but that is most likely what everyone thinks because a tessellation is just one shape transformed throughout a plane. The shape can be reflected over lines (reflection), rotated about a point (rotation) or moved without losing its position (translation). Who would have thought that by applying different transformations to a shape could result in such awesome designs?

I personally chose to translate my shape, which consisted of the two short bases of the trapezoid coming together with the two rhombi fitting into each end. By repeating this pattern of translating my shape we can see other shapes besides just the rhombus and trapezoid such as a regular hexagon and a parallelogram, along with different looking hexagons.

My next tessellation that I created included two hexagons and two rhombi as the transformation piece. The familiar shapes that I see are stretched down hexagons that are skinnier in relation to a regular hexagon if you look at it as in the picture from top to bottom. These two tessellations are similar because they both use rhombi and since two trapezoids make up a hexagon we can see them as such. But the tessellation itself is different because of the pattern we see. In the red and blue tessellation we see two rows of rhombi lined up but in the yellow and blue tessellation we only see one two of rhombi lined up. That has to do with what we are calling our base piece or the piece that you are transforming.

Keeping my future students in mind I can see myself using this activity to integrate math and art as we did in class. I like the idea of letting the students be creative with the types of patterns they make while also allowing them to make mathematical connections unknowingly. When math is being done without it being too obvious students seem to benefit. They benefit because they aren’t as intimidated because it’s not “really” math. But when they can draw connections on their own about their tessellations that is when learning is being done.

In my MTH 322 geometry class we engaged in an activity that used pattern blocks to explore fractions. I thought that this activity was awesome to really show what fractions are all about, a part of a whole. By asking questions like, “how many green triangles can fit into this hexagon?” we allow students to explore the relationship between the triangle and the hexagon. The triangle is now some portion of the hexagon. I would love to explore an activity using a tessellation out of pattern blocks to teach fractions to students. I could have them come up with the different proportions that are seen in the tessellation and question them about the ratios they see.

]]>In my MTH 322 geometry class we engaged in an activity that used pattern blocks to explore fractions. I thought that this activity was awesome to really show what fractions are all about, a part of a whole. By asking questions like, “how many green triangles can fit into this hexagon?” we allow students to explore the relationship between the triangle and the hexagon. The triangle is now some portion of the hexagon. I would love to explore an activity using a tessellation out of pattern blocks to teach fractions to students. I could have them come up with the different proportions that are seen in the tessellation and question them about the ratios they see.

One major way that I personally see important that we acquired from Greek mathematicians is the use of the Pythagorean theorem. When we learn this formula we often learn it out of context and just as another formula to memorize pertaining to right triangles. But this theorem has done a lot for our very industrialized world, such as aiding in construction of buildings and findings of unknown distances. Formulas mean nothing unless they are given a context for which they should or are most likely used. In math classes today we see that push toward bridging the gap between a given formula and the context for which it belongs. I think that it is so beneficial to students to see the why and how because then instead of memorizing they are more likely to recall from the context why it works. By providing them with that we are really giving them the tools they need to be successful when working with the theorem.

Democritus, a Greek mathematician was most famous for his prescient ideas about all matter being composed of tiny atoms, was also a pioneer of mathematics and geometry and he produced works with titles like "On Numbers", "On Geometrics", "On Tangencies", "On Mapping" and "On Irrationals”. These works are important but the fact that Democritus was recording his work in writing is more interesting to me. Not many mathematicians of this time period were recording their work and were being published. Without publication of new findings these new discoveries were relayed via word of mouth and we all know what can happen to information that way. Having the original document that was written by whomever discovered it is valuable and at this point in time, that was recognized. Think of all the resources we would be missing out on without documentation. Many of us going into teaching would have to have photographic memories to try to recall the information we are expected to teach. This documenting development did a lot for our future society.

Even though these were only two instances in Greek mathematics it is very apparent of the imperative role they played in how we use and do mathematics in every day life. Math is a tool and in order to use that tool we have to know what kinds of problems the tool fixes, just as the Greeks knew when coming up with the Pythagorean theorem. Once we know that important information the rest is more bearable. ]]>

- What is Math?
- Math cannot be summed up in one sentence. One definition of mathematics is not sufficient and some books don't even capture the true meaning fully. Today we were posed with this daunting question and we as a class of mathematicians had a difficult time describing specifically what mathematics defined looks like. We came up with many different words to describe math and phrases to help describe what math is but when deciding what the top/best answer was there was no real winner. I personally think that math can be described as a language of its own that aids in problem solving both quantitatively and qualitatively. I also believe that the previous sentence is just a snapshot of what I actually mean and to articulate what math is in its entirety. For that exact reason, when asked to place a check mark next to the "best" answer to define math I checked all the statements.
- Biggest Moments/Discoveries in History of Math
- I am not familiar at all with math history, I have got to be honest. With that being said, I intend on working on learning math history throughout this course.

- Math cannot be summed up in one sentence. One definition of mathematics is not sufficient and some books don't even capture the true meaning fully. Today we were posed with this daunting question and we as a class of mathematicians had a difficult time describing specifically what mathematics defined looks like. We came up with many different words to describe math and phrases to help describe what math is but when deciding what the top/best answer was there was no real winner. I personally think that math can be described as a language of its own that aids in problem solving both quantitatively and qualitatively. I also believe that the previous sentence is just a snapshot of what I actually mean and to articulate what math is in its entirety. For that exact reason, when asked to place a check mark next to the "best" answer to define math I checked all the statements.

As the final days of the semester drew near I was able to take what I learned from my geometry class and put it to the test in a sculpture classroom at the West Michigan Academy of Arts and Academics. I was excited to be part of an environment different to what I was used to, or at least I expected it to be different just by hearing the name of the school. Integrating art and academics in general seemed interesting to me, coming from someone who doesn't necessarily consider herself to be very art inclined. I still was eager to see what this school was all about while having a chance to teach a mini lesson on nets to a class of 7th and 8th grade students.

Arriving to the school, a bunch of emotions ran through me. I was anxious , nervous, excited and most of all ready. I felt prepared to do what it was I had come there to do, which was quite comforting. Our lesson really stemmed from the fact that we were in a sculpture class, having students build their own polyhedrons with provided "polytiles". This gave them a chance to work with shapes, maybe unfamiliar to themselves, and attempt to create a net or two dimensional representation of their own shape.

The students strolled in and immediately saw the tiles on their desks. A slew of excitement was apparent in the classroom, hearing students say, "Do we get to play with these today?". It was nice to see how eager the students were to get started before we had even told them what their task for the next hour would be. The announcements were over and it was time to begin with our lesson. We first had students explore with the polytiles to create their own polyhedrons. While this was going on I walked around the classroom, mostly answering questions about how to go about connecting the polytiles together. Once they got the hang of how to connect them, the building really got started. Observing the students work was a cool experience because I had a chance to see the advantage of being a part of an academy of arts.

There was one student, in particular, that thought outside of the box to create a polyhedron that looked different from everyone else's which was valuable to recognize because he ended up making the task at hand more challenging for himself without even knowing. This ended up being the case for a quite a few students. When students were asked to create a net for their polyhedrons one student immediately said, "So, would we just unfold what we made to get a net?". At that point I was both excited and bummed; excited because she understood exactly what was being asked of her and visually saw in her head what to do, but bummed because that idea was now planted into every students' mind instead of them coming up with different ways to create their nets. This reflected what we saw happening in the classroom. All of the students ended up unfolding their polyhedrons to lay flat on their tables and used that as a reference to draw their nets.

There were, however, differences when it came to the how. How they decided to draw their nets from that representation differed tremendously. Some students traced it exactly, with ridges and all, while other students used a straight edge or ruler to get straight lines. One student had an approach that really used geometric terminology well. She was using a ruler to measure the sides of the square base she had used to make her polyhedron and said that because she was working with squares and equilateral triangles that in order to all fit together evenly each side would have be the exact same length. So instead of tracing, she used her ruler and measured out 4 inches on each side of her square base and 4 triangular faces, creating a net for her pyramid. She also mentioned that the perpendicular bisector helped her to better draw a more precise net. She even included tabs on her net so that she could glue it together instead of tape it.

]]>Arriving to the school, a bunch of emotions ran through me. I was anxious , nervous, excited and most of all ready. I felt prepared to do what it was I had come there to do, which was quite comforting. Our lesson really stemmed from the fact that we were in a sculpture class, having students build their own polyhedrons with provided "polytiles". This gave them a chance to work with shapes, maybe unfamiliar to themselves, and attempt to create a net or two dimensional representation of their own shape.

The students strolled in and immediately saw the tiles on their desks. A slew of excitement was apparent in the classroom, hearing students say, "Do we get to play with these today?". It was nice to see how eager the students were to get started before we had even told them what their task for the next hour would be. The announcements were over and it was time to begin with our lesson. We first had students explore with the polytiles to create their own polyhedrons. While this was going on I walked around the classroom, mostly answering questions about how to go about connecting the polytiles together. Once they got the hang of how to connect them, the building really got started. Observing the students work was a cool experience because I had a chance to see the advantage of being a part of an academy of arts.

There was one student, in particular, that thought outside of the box to create a polyhedron that looked different from everyone else's which was valuable to recognize because he ended up making the task at hand more challenging for himself without even knowing. This ended up being the case for a quite a few students. When students were asked to create a net for their polyhedrons one student immediately said, "So, would we just unfold what we made to get a net?". At that point I was both excited and bummed; excited because she understood exactly what was being asked of her and visually saw in her head what to do, but bummed because that idea was now planted into every students' mind instead of them coming up with different ways to create their nets. This reflected what we saw happening in the classroom. All of the students ended up unfolding their polyhedrons to lay flat on their tables and used that as a reference to draw their nets.

There were, however, differences when it came to the how. How they decided to draw their nets from that representation differed tremendously. Some students traced it exactly, with ridges and all, while other students used a straight edge or ruler to get straight lines. One student had an approach that really used geometric terminology well. She was using a ruler to measure the sides of the square base she had used to make her polyhedron and said that because she was working with squares and equilateral triangles that in order to all fit together evenly each side would have be the exact same length. So instead of tracing, she used her ruler and measured out 4 inches on each side of her square base and 4 triangular faces, creating a net for her pyramid. She also mentioned that the perpendicular bisector helped her to better draw a more precise net. She even included tabs on her net so that she could glue it together instead of tape it.

Rotation and reflection are two R's that are very simple to mix up for elementary students. The main difference between a rotation and reflection is the fact that rotation has the students essentially turning the shape while the reflection has the students flipping it over. As a daily assignment we were asked to correct/grade an assessment that involved reflections, rotations and translations. A group of elementary students were asked complete Aaron's picture by following the directions asked of them, for example, "To finish drawing Aaron's second design, rotate the gray shape 1/4 of a turn in a clockwise direction about the origin. Then draw the second shape. Rotate the second shape 1/4 of a turn in a clockwise direction about the origin. The draw the third shape. Rotate the third shape 1/4 of a turn in a clockwise direction about the origin. Then draw the fourth shape."

I had noticed after a few different students did, just as the above picture shows, and reflected about the vertical and horizontal lines when asked to rotate 1/4 of a turn clockwise. This is a great example of the confusion that takes over students when trying to decipher between a rotation and a reflection.

Another observation I made when grading these students' work was that all of the students, when asked to reflect the shape, got all 3 reflections correct. This shows me that the students understand what it means to reflect over a vertical and horizontal line. This could be due to the fact that we are used to seeing reflections daily, whether that be in a mirror or in the store of a window, which make the topic obviously relevant. The fact that a good majority of students had trouble when asked to rotate the shape 1/4 of a turn shows that the understanding is not all there. Using this observation I would suggest that we, as teachers, look into this misunderstanding and dig deep to better serve our students.

One way to represent what a rotation is includes using manipulatives to help students visualize what it is they are being asked to do. Using manipulatives on top of an enlarged handout of a grid would be beneficial to students because they are feeling what is happening instead of having to visualize it on their own. This, in-person, visualization allows them to see the relationship between what they have in front of them physically and what they are drawing on the page.

Another way to further help them with both of these confusing R's includes making the topic into an interactive game as we did in class with the game, "Square Match". This game took two non-symmetric objects, which allows for noticeable reflections, and make 6 game pieces (3 of each object). You then get into teams of two people and try to get all of your objects in the exact same position by either rotating to the right or flipping forward, while trying to get all of your opponents objects in different positions. After playing this game myself, I can say that it was fun and challenging at the same time. I could see elementary students enjoying this game, not even noticing how much thinking they are actually partaking in.

Students have the means to learn these topics it just takes a little extra work for us as teachers. By challenging students we challenge ourselves as teachers to work a little harder and think a little deeper. Being timid and working with topics familiar to us is easy and less work but to see real results in our classrooms it is important to not only challenge the students but to challenge ourselves to be the best that we can be!

]]>I had noticed after a few different students did, just as the above picture shows, and reflected about the vertical and horizontal lines when asked to rotate 1/4 of a turn clockwise. This is a great example of the confusion that takes over students when trying to decipher between a rotation and a reflection.

Another observation I made when grading these students' work was that all of the students, when asked to reflect the shape, got all 3 reflections correct. This shows me that the students understand what it means to reflect over a vertical and horizontal line. This could be due to the fact that we are used to seeing reflections daily, whether that be in a mirror or in the store of a window, which make the topic obviously relevant. The fact that a good majority of students had trouble when asked to rotate the shape 1/4 of a turn shows that the understanding is not all there. Using this observation I would suggest that we, as teachers, look into this misunderstanding and dig deep to better serve our students.

One way to represent what a rotation is includes using manipulatives to help students visualize what it is they are being asked to do. Using manipulatives on top of an enlarged handout of a grid would be beneficial to students because they are feeling what is happening instead of having to visualize it on their own. This, in-person, visualization allows them to see the relationship between what they have in front of them physically and what they are drawing on the page.

Another way to further help them with both of these confusing R's includes making the topic into an interactive game as we did in class with the game, "Square Match". This game took two non-symmetric objects, which allows for noticeable reflections, and make 6 game pieces (3 of each object). You then get into teams of two people and try to get all of your objects in the exact same position by either rotating to the right or flipping forward, while trying to get all of your opponents objects in different positions. After playing this game myself, I can say that it was fun and challenging at the same time. I could see elementary students enjoying this game, not even noticing how much thinking they are actually partaking in.

Students have the means to learn these topics it just takes a little extra work for us as teachers. By challenging students we challenge ourselves as teachers to work a little harder and think a little deeper. Being timid and working with topics familiar to us is easy and less work but to see real results in our classrooms it is important to not only challenge the students but to challenge ourselves to be the best that we can be!

First, I want to address a few questions about measurement that are key to our investigation on why measurement is so important. Why do we need these measurements? What do these measurements tell us? What important information are we obtaining? The reason behind the measurement is something that we need to think about before we start measuring so that students understand that what they are doing has a specific relevant purpose and is used as a way of communication. The specifics of what we are measuring differ from problem to problem but all in all the question is always, "how much?". We are obtaining information about how much and able to communicate that measurement all because of units.

We, as college students, have been exploring these questions during our class sessions. One activity that I thought guided me along in my own thinking about measurement was one where we had to measure our own stride and use that stride as a unit of measure to measure the hallway. My group thought the best way to go about this was to take 10 consecutive steps as normally as possible and measure from toe to toe. After measuring, in centimeters, the length of my combined steps we would divide that number by 10 to give us our individual strides. We measured the hallway in number of strides, individually, and then multiplied that by our stride in centimeters to get a standard measure of unit measurement.

We then gathered the classes’ data all together and saw a lot of variability in our data. We then struggled with the question of whether one unit of measure was more precise or accurate than the other, between the steps and centimeters. We, having been taught this, assumed centimeters were obviously the more accurate and precise choice at first. After collaboration and communication, we came to the conclusion that it mattered what we were measuring in order to see what unit to use. One unit is not necessarily better than the other; it depends on the task at hand. Although everyone had a different stride that was not what was accountable for our variability. We did not all agree upon a starting and stopping point for the hallway, as well as other students disrupting our strides in order to obey "hallway traffic laws". Measurement is a tricky topic because of the amount of error that can appear. It depends on who is measuring it, how they measure it, where they measure it from, and what units of measure they are using. By exploring without a standard unit of measure it allowed us to think outside of the box and truly appreciate standard units for communication as well as simplicity reasons of being able to use a ruler. By doing this type of inquiry with students you allow them to come up with that realization on their own and appreciate when they are given a standard measurement tool to better communicate with their peers as well as the rest of the world. In my classroom I intend on letting my students search for their own reasoning to better understand the importance of measurement and what it does for us in our every day lives. And with that key idea, my students will feel comfortable communicating with mathematics and measurement.

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