Taking a geometry course for the the first time in a decade or so, I was nervously optimistic. I always know the math department does an excellent job of making the work in the classroom relevant to our own classrooms as well as enhance us as mathematicians. This semester did not disappoint.
I do not teach a whole ton of geometry to my 7th graders but the unit I do teach in now ten times better than it was before. I have a "toolbox" full of resources I am able to use to enhance and engage my students. The skill I feel will support my teaching the best is Geogebbra. I had little to no experience with the program four months ago, but today I am using in my classroom on a regular bases. I am confident in the foundation skills it uses and am comfortable enough to use it with the students. The readings in the class where relevant to both the classroom and mathematician side. As many of the choices in the class where between a "mathy" article and a classroom article.
The mathematician side of me was strengthened in many ways as well. I was in awe of the many higher level math problems completed in the course. They where presented in a way that was enjoyable and engaging. My eyes were opened to many real world math questions I would not have had an opportunity to have think about without this course. I can not give enough praise for the enjoyment I had coming to this class. I was truly good to escape the rush of daily live and do something I enjoy.
It saddens me the course will no longer be taught at GVSU. It was more productive than any seminar I have every attended. I hope and pray the cyclical motion of education will one day find value in promoting higher learning for its teachers.
Saturday, December 7, 2013
The Great Result: Pt. 2
this blog is a second attempt to an earlier post "The Great Result:"
As I spoke about it in the Celtic Knots blog post, our school has a "Flex" block period where students are placed in classrooms based on data collected from screener assessments. I am able to work with the "high flyers" and we so some enrichment to get them thinking about math beyond the walls of our classroom. I am able to work with students on topics they might not see.
I was bound and determined to have them have the "shock 'n awe" of how amazing the Euler Line was when I first saw it in my MAT 641 class at GVSU. We got the computers out and the students have the Geogebra program on it already so I was going to be able to "kill two birds with one stone." The student could get an introduction to both geoebra and would be able to see math in new way with a new appreciation.
I did a great job of hyping up the concept and gave it my best sales pitch telling them they where going to be "mind blown" by the end of class. So we began, they where having an enjoyable time working with geogebra and creating the different centers for their given triangle. Students were very exited about finding the orthocenter, curcumcenter, and centroid. They had their struggles with geogebra but were very optimisc about what was going on and truly enjoyed the learning the process on finding different centers to a simple triangle. They loved the hands on experience with manipulating the different vertices of the triangle and they would comment on how all the lines would move. They like to see how each center points always worked, how all the lines would change, but the 3 points would but still hold true to their intended purpose.
Then it was time to deliver the big moment, the culmination of the big sales pitch. Students had started to see it after the three points where found and they began to move the triangle into different forms. we then created a line through the three points and continued to manipulate the triangle and we did notice it always works.
Now I didn't receive the overwhelming joy I was hoping like they had just won a state championship but I did spark a little awe in their eyes. A few students thought it was pretty cool that three seemingly independent points of a triangle would form a straight line. So I was relatively excited I had made a few believers. We would continuing the next time we meant I was determined to make more believers.
The next Flex Block class the student continued with questions about the Euler Line so we took second look at it. We improved our geogebra skill by adding icons that hide/showed lines, plus we also introduced the nine point circle. When we where done we noticed the center of the nine point circle also fell on the Euler Line. I had made a few more converts! From where I had began, I made a good majority of my student believe that math was something greater that what was taught in a classroom.
As I continue to become a better mathematician I am in awe about how math is intertwine in everything around us. One thing that I believe I need to do as an educator is pass that love onto my students. Let them know that everything around them is related in some why to what we do in class. Give them the passion they deserve.
As I spoke about it in the Celtic Knots blog post, our school has a "Flex" block period where students are placed in classrooms based on data collected from screener assessments. I am able to work with the "high flyers" and we so some enrichment to get them thinking about math beyond the walls of our classroom. I am able to work with students on topics they might not see.
I was bound and determined to have them have the "shock 'n awe" of how amazing the Euler Line was when I first saw it in my MAT 641 class at GVSU. We got the computers out and the students have the Geogebra program on it already so I was going to be able to "kill two birds with one stone." The student could get an introduction to both geoebra and would be able to see math in new way with a new appreciation.
I did a great job of hyping up the concept and gave it my best sales pitch telling them they where going to be "mind blown" by the end of class. So we began, they where having an enjoyable time working with geogebra and creating the different centers for their given triangle. Students were very exited about finding the orthocenter, curcumcenter, and centroid. They had their struggles with geogebra but were very optimisc about what was going on and truly enjoyed the learning the process on finding different centers to a simple triangle. They loved the hands on experience with manipulating the different vertices of the triangle and they would comment on how all the lines would move. They like to see how each center points always worked, how all the lines would change, but the 3 points would but still hold true to their intended purpose.
Then it was time to deliver the big moment, the culmination of the big sales pitch. Students had started to see it after the three points where found and they began to move the triangle into different forms. we then created a line through the three points and continued to manipulate the triangle and we did notice it always works.
Now I didn't receive the overwhelming joy I was hoping like they had just won a state championship but I did spark a little awe in their eyes. A few students thought it was pretty cool that three seemingly independent points of a triangle would form a straight line. So I was relatively excited I had made a few believers. We would continuing the next time we meant I was determined to make more believers.
The next Flex Block class the student continued with questions about the Euler Line so we took second look at it. We improved our geogebra skill by adding icons that hide/showed lines, plus we also introduced the nine point circle. When we where done we noticed the center of the nine point circle also fell on the Euler Line. I had made a few more converts! From where I had began, I made a good majority of my student believe that math was something greater that what was taught in a classroom.
As I continue to become a better mathematician I am in awe about how math is intertwine in everything around us. One thing that I believe I need to do as an educator is pass that love onto my students. Let them know that everything around them is related in some why to what we do in class. Give them the passion they deserve.
Celtic Knots
At the last MAT 641 class at GVSU, I was introduced to the geometric art of Celtic Knots. I have always had an interest in art and especially math related art. The systematic approach to creating Celtic Knots is very in line with my personality. I am a very linear thinking and process oriented art is right up my ally. I spend some time the next few days after class sharpening my knot art skills and was ready to bring it into my classroom.
In my school we are blessed to have a "Flex" block period that meets twice a week for a half hour. The time is so students can receive the enrichment they need in different subjects. They are grouped up based on achievement level. I am fortunate enough to have the students who are highly successful in math. We are able to do some activities that are not in the curriculum and show them math outside of just structured curriculum. So this past Thursday I introduced them to Celtic Knots. The initial response was "I don't get this," or "this isn't fun," but after continued demonstration and I completed a few on the board they deemed "cool" their interest level increased. After a few minutes, a few students came with smiles on their face and showed me their completed knots. Their passion for math art increased just as mine had when I was introduced to Celtic Knots. Before I knew it the class was silent and everyone had their heads down and where working away.
The greatest moment happened after we talked about adding different walls to manipulate the knots and see what happens. Students tried simple 4x4 knots with one wall. Then the great moment happened. I overheard a student speak to his neighbor, "I don't know what is going to happen but I am going to try it." We stopped the whole class right here and we had a conversation that this is what math should be like. School has become a chore for adolescences and not an inquisitive exploration of gaining knowledge. I was excited and passionate about their level of interest by the end of the hour. It was great to see them take the Celtic Knot and run with it and try it on their own.
In the future I want to wrap my head around some of the math that is involved with Celtic Knots. I have done a little research and spent some time pouring over the articles. I would also like to bring it down to the level where a middle school student can comprehend what is going on and it's application to other real world items.
Tuesday, December 3, 2013
Making Geometry Simpler: "Volume of a Pyramid"
Reflecting on the reading for this week there are some good ideas Masha Albrecht talks about that I can apply to my own class. I am always looking for ideas on Low-Tech "hands on" activities where students can dive into their work.
I always have a hard time with students making the transition with v = lwh in a rectangular prism to using v = Bh. I like using v = Bh because it makes the understanding and application of cylinders and triangular prisms that much easier. We can use the same formula v = Bh for all three 3-D figures instead of switching between independent formulas. Students always have a hard time making that transition. The worksheet that is used and the technique the writer uses makes this transition very fluidly, and in a way middle school students can handle.
I also like how she uses the blocks to understand where the formula for pyramids come from. I believe I can use this for my students as well. We can create the "cube pyramid" and work our way to getting to the 1/3 of the prism formula. Using the technology piece will come in handy but I am having my doubts on how 7th graders of all learning levels will be able to comprehend the formula by Misha uses. I would like them to discover the patter for finding the volume on their own but squaring the side length and adding it to the last pyramid's volume seems to be a little stretch for my students. I will continue looking into finding a more attainable method for 7th graders.
I am excited to use this article in my class. It is the fourth or fifth different resource I will be able to use in the upcoming geometry unit.
I always have a hard time with students making the transition with v = lwh in a rectangular prism to using v = Bh. I like using v = Bh because it makes the understanding and application of cylinders and triangular prisms that much easier. We can use the same formula v = Bh for all three 3-D figures instead of switching between independent formulas. Students always have a hard time making that transition. The worksheet that is used and the technique the writer uses makes this transition very fluidly, and in a way middle school students can handle.
I also like how she uses the blocks to understand where the formula for pyramids come from. I believe I can use this for my students as well. We can create the "cube pyramid" and work our way to getting to the 1/3 of the prism formula. Using the technology piece will come in handy but I am having my doubts on how 7th graders of all learning levels will be able to comprehend the formula by Misha uses. I would like them to discover the patter for finding the volume on their own but squaring the side length and adding it to the last pyramid's volume seems to be a little stretch for my students. I will continue looking into finding a more attainable method for 7th graders.
I am excited to use this article in my class. It is the fourth or fifth different resource I will be able to use in the upcoming geometry unit.
Monday, November 18, 2013
Smarter Balance Practice Test:
The topic to begin every staff meeting is how are we preparing the students for the Smart Balance Assessment that correlates to the Common Core State Standards. Even to day in the department meeting our department chair was talking with us about the use of Depth of Knowledge (DoK) questions so students are accustomed to high level level questions. This evening I spent a good majority of time dissecting the practice test found at the Smarter Balance website.. This is was a good experience for me in preparation for my project in m641. In order for me to create a outstanding lesson plan I need to create an outstanding summative assessment. My goal to create a similar assessment to the one provided by the smarter balance assessment for the students at the end of their Geometry Unit that will have the Common Core State Standards as their foundation.
As I continued to take the assessment at Smarter Balance I was amazed by the number of different learning targets I was using to solve a particular problem. I was often thinking the question was written to trick the students. After continued reflection I came to the realization the assessment wanted to assess the students on a wide variety of learning objects for a single question. This is going to be a major hurtle for the way I have been teaching and for the way I was taught. So often growing up for me and seeing the released items from older MEAP assessment the questions where basic problems (i.e. add these fractions). Now they ask the students to perform a number of operations with fractions then identify if it is larger or smaller than a given value or it doesn't' ask the student to find the mean from a set of values it asks the students, "how much did the mean number of push ups increase in these two line plots from gym class."
Creating questions of this nature are going to take some time, I believe creating one unit a year would be a good goal. For this year and this class, I am going to use a majority of my last few weeks of m641 to prepare a summative assessment based on the smarter balance questions.
As I continued to take the assessment at Smarter Balance I was amazed by the number of different learning targets I was using to solve a particular problem. I was often thinking the question was written to trick the students. After continued reflection I came to the realization the assessment wanted to assess the students on a wide variety of learning objects for a single question. This is going to be a major hurtle for the way I have been teaching and for the way I was taught. So often growing up for me and seeing the released items from older MEAP assessment the questions where basic problems (i.e. add these fractions). Now they ask the students to perform a number of operations with fractions then identify if it is larger or smaller than a given value or it doesn't' ask the student to find the mean from a set of values it asks the students, "how much did the mean number of push ups increase in these two line plots from gym class."
Creating questions of this nature are going to take some time, I believe creating one unit a year would be a good goal. For this year and this class, I am going to use a majority of my last few weeks of m641 to prepare a summative assessment based on the smarter balance questions.
Tuesday, November 12, 2013
3 Different Methods to Conic Sections
As the article "Paper Folding and Conic Sections" by Scott G. Smith starts out, "One sign of a good problem can be approached in more than one way." I spent my time this week trying to achieve this goal. The first way was provided by the article by using Patti Paper, the second is hinted at by Smith but doesn't go into detail and that's by using a sketchpad, and lastly I discover on my own by having a conversation with our art teacher by creating String Art.
Paper Folding Conics:
The farthest left conic is by folding a dot off a line onto a line multiple times and its result was a parabola if we were to be done an infinite amount of times. The middle figure is a dot inside of a circle and the folding the dot onto the circle. The resulting figure when done it an infinite amount of times an ellipse. The third image is a dot outside of the circle folded onto the circle and its resulting image is a hyperbola.
As I was working on these folding sections I was thinking about what would happen if I moved the points around the circle or farther/closer to the line. I did cheat a little bit and some students after school saw me folding the paper and wanted to join, but this still only gave me a limited number of results.My next option was to work with Geogebra and see what happens.
In comparison to the Patti Paper the final image is the same. The parabola, is still created when we use the graphing software program. To create this parabola, I placed a number of points on the vertical line. Next I created bisecting lines from each point to the point off the line. I repeated the process through all the points on the vertical line. Once all the bisecting lines are gathered together you can see the shape of the parabola starting to come together.
Because I was using the Geogebra program I was able to answer some of the questions I couldn't with Patti paper. Like how the closer you get to the line the sharper the curve is of the parabola and the farther you get away the wider the parabola becomes.
To create the ellipse I did a similar process to the parabola above. I created a circle and a point within the circle. Next I placed a number of point on the circle. Just like the project above I found the bisecting line between each point and the point outside the circle.
I was able to move the point inside the circle around and observe the different shapes the bisecting lines created. The closer you get to the center the closer the shape of the lines get to becoming a circle, only when the point we are moving around is the center do we get the perfect circle.
The hyperbola was created using the same program as the ellipse. The only difference is when the point is outside the circle does it turn into a hyperbola.
Using Geogebra allowed me to observe what happens when when we change the circumstances of the problem. It was very quick for me to see what happens when I place the dot I'm moving around on the line. I noticed all the lines crossed in the center of the circle. There was no true shape created by all the trend of the lines. It was one of the benefits of Geogebra. The ability to instantly move items around allow for quick feedback.
The last work with conics came to mind after a conversation at the lunch table with the art teacher. I came up with the idea after coming across the a Geoebra program were the student can design their own string project before they begin. Working with the string art was good may have been the most time consuming but it was definitely the most rewarding.
When comparing the three versions of I worked through this week, all have their highlights. If I were to judge them against another, the Geogerba program would take the cake. It is works best when it comes to manipulating the figure and seeing overall concepts. Although all would be great to work in class. I can see a future cross curricular project with the art teacher to incorporate all learners.
Paper Folding Conics:
The farthest left conic is by folding a dot off a line onto a line multiple times and its result was a parabola if we were to be done an infinite amount of times. The middle figure is a dot inside of a circle and the folding the dot onto the circle. The resulting figure when done it an infinite amount of times an ellipse. The third image is a dot outside of the circle folded onto the circle and its resulting image is a hyperbola.
As I was working on these folding sections I was thinking about what would happen if I moved the points around the circle or farther/closer to the line. I did cheat a little bit and some students after school saw me folding the paper and wanted to join, but this still only gave me a limited number of results.My next option was to work with Geogebra and see what happens.
Conics with Geogebra:
In comparison to the Patti Paper the final image is the same. The parabola, is still created when we use the graphing software program. To create this parabola, I placed a number of points on the vertical line. Next I created bisecting lines from each point to the point off the line. I repeated the process through all the points on the vertical line. Once all the bisecting lines are gathered together you can see the shape of the parabola starting to come together.
Because I was using the Geogebra program I was able to answer some of the questions I couldn't with Patti paper. Like how the closer you get to the line the sharper the curve is of the parabola and the farther you get away the wider the parabola becomes.
To create the ellipse I did a similar process to the parabola above. I created a circle and a point within the circle. Next I placed a number of point on the circle. Just like the project above I found the bisecting line between each point and the point outside the circle.I was able to move the point inside the circle around and observe the different shapes the bisecting lines created. The closer you get to the center the closer the shape of the lines get to becoming a circle, only when the point we are moving around is the center do we get the perfect circle.
The hyperbola was created using the same program as the ellipse. The only difference is when the point is outside the circle does it turn into a hyperbola.
Using Geogebra allowed me to observe what happens when when we change the circumstances of the problem. It was very quick for me to see what happens when I place the dot I'm moving around on the line. I noticed all the lines crossed in the center of the circle. There was no true shape created by all the trend of the lines. It was one of the benefits of Geogebra. The ability to instantly move items around allow for quick feedback.
Conics with String Art:
The last work with conics came to mind after a conversation at the lunch table with the art teacher. I came up with the idea after coming across the a Geoebra program were the student can design their own string project before they begin. Working with the string art was good may have been the most time consuming but it was definitely the most rewarding.
When comparing the three versions of I worked through this week, all have their highlights. If I were to judge them against another, the Geogerba program would take the cake. It is works best when it comes to manipulating the figure and seeing overall concepts. Although all would be great to work in class. I can see a future cross curricular project with the art teacher to incorporate all learners.
Tuesday, November 5, 2013
Exploring Escher
I have been very fascinated with the art of M.C. Escher the last few years. His designs will captivate your mind. In our October 29th meeting of class, the professor gave us some background on the geometry behind his work. For my outside of class this week, I would do some exploration on my own. The following is my "Metacognative Memoir" or thought process, on a few of Escher's pieces of art..
My First Piece.
I started by using a "duel" technique.
Steps:
If you find the center of each of the small lizards and place a dot in the same location on each of them. For me I used the top of the back right before the the neck started to turn to the head. Once you have all the locations found, I connected each dot to its neighbor. This formed a series of triangles over the whole paper. I could tell there was a pattern starting.
For it to be "duel" you repeat the process only using the shapes your found from the first step. I took the center of each of the triangles and connected those dots to one another. When I did this last step, I used patty paper over top or the original pictire. Below is my result.
The new insight gained from these pictures gave me an appreciation for his work. I will continue to explore some additional art of M.C. Escher.
My First Piece.
I started by using a "duel" technique.
Steps:
If you find the center of each of the small lizards and place a dot in the same location on each of them. For me I used the top of the back right before the the neck started to turn to the head. Once you have all the locations found, I connected each dot to its neighbor. This formed a series of triangles over the whole paper. I could tell there was a pattern starting.
For it to be "duel" you repeat the process only using the shapes your found from the first step. I took the center of each of the triangles and connected those dots to one another. When I did this last step, I used patty paper over top or the original pictire. Below is my result.
It turns out the figure behind the lizards is an octagon and a rhombus. He used this general out line for the design of his picture.
I was very excited to see this work. With my excitement I decided to show my advanced class to students the Escher piece. There enthusiasm was greater than the experience with the Euler Line (see blog post "The Great Result"). When I showed the student the work I showed them another piece I was working on with Geogebra the same way. Geogebra of Echer's Work
This piece took me quite some time to look over. I had about three of four different trials. I don't know if this is even the right path, but it seems to work well.
Analyzing the Angels and Demons picture on Geogebra. It woks perfectly until you get to the rhombus' and then there are more dots on the paper than are used to make the figures. Also the figures are not as regular polygon as I would think they should be.
The new insight gained from these pictures gave me an appreciation for his work. I will continue to explore some additional art of M.C. Escher.
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