You can use the link to the right under Pages, or click here.

If you'd like to just take a look at the collection of my GeoGebra worksheets for the project, click here.

At this point, I am finished with my project - a translation of Book III of the elements with links to GeoGebra manipulatives (no proofs at this point, however). Please take a look, and feel free to leave feedback.

You can use the link to the right under Pages, or click here.

If you'd like to just take a look at the collection of my GeoGebra worksheets for the project, click here.

As the semester wraps up, here are some of the big things I will take away from Math 641, Modern Geometry:

*Get students asking questions.*

As a result of our work in class and our readings for Math 641, I am trying to give my students more opportunities to ask questions, develop curiosity, and explore ideas. On a day to day basis, I am asking my students "why" more often, and asking them to defend their answers. I have increased my wait time, too. On a broader scale, I am mindful of the value of letting students explore and discover new ideas, rather than having everything new presented to them directly. I want to eventually work more exploratory days into the classes that I teach at opportune points in the curriculum. This can be a step toward helping students develop arguments to defend their thinking. Having them reach their own conjectures and then supporting their reasoning could be a more natural way to introduce them to proof.

*GeoGebra is an excellent tool and resource*.

I knew what GeoGebra was before Math 641, but I had not taken the time to try it out. Especially once we took a look at GeoGebra Tube, I was hugely impressed by how broadly applicable GeoGebra is to all math classes. I have spent days in the computer lab with my geometry classes using GeoGebra this year, but I plan to do the same with my calculus class as well. It can be used to illustrate and explore such a wide variety concepts. It is also simple enough to be accessible to students - the learning curve is not nearly as steep as that of a program like Maple.

*There are ***far **more ways to present and explore mathematical ideas than I am currently aware of.

That is an obvious statement. What Math 641 opened my eyes to is some of the better ways to learn about new ways to teach concepts and new resources. I had never used Twitter before this class, and I had never done any blogging. I wasn't thrilled to have to get a Twitter account at the start of the semester, or start a blog. But I've grown to appreciate what a resource Twitter is as a forum for discussions about math ed, among other things of course. Twitter and blogging are great means for sharing ideas, and there are lots of people with far better ideas than me maintaining blogs. Rather than sticking only to publications like*Mathematics Teacher*, resources like the mathtwitterblogosphere are another great place to find new ideas.

*Geometry and mathematics in general are far more broad than most students get a chance to see*.

Working with new topics week to week in Math 641, and exploring those topics from a variety of angles reminded me that there's always a big and interesting math horizon to explore. As a high school teacher, it's easy to remain relegated to the standards and the narrow math sphere they encompass. Modern Geometry was a refreshing contrast. There were many times during the semester when I thought, if I could spend three days doing*this *with my students, they would be way more engaged, and a few more of them might even decide they like math. I know I can try to work some of those things in to my regular classes, but in general I want to give my students more glimpses of how many different things are mathematical. Maybe I'll have to keep up a good bulletin board for once.

As a result of our work in class and our readings for Math 641, I am trying to give my students more opportunities to ask questions, develop curiosity, and explore ideas. On a day to day basis, I am asking my students "why" more often, and asking them to defend their answers. I have increased my wait time, too. On a broader scale, I am mindful of the value of letting students explore and discover new ideas, rather than having everything new presented to them directly. I want to eventually work more exploratory days into the classes that I teach at opportune points in the curriculum. This can be a step toward helping students develop arguments to defend their thinking. Having them reach their own conjectures and then supporting their reasoning could be a more natural way to introduce them to proof.

I knew what GeoGebra was before Math 641, but I had not taken the time to try it out. Especially once we took a look at GeoGebra Tube, I was hugely impressed by how broadly applicable GeoGebra is to all math classes. I have spent days in the computer lab with my geometry classes using GeoGebra this year, but I plan to do the same with my calculus class as well. It can be used to illustrate and explore such a wide variety concepts. It is also simple enough to be accessible to students - the learning curve is not nearly as steep as that of a program like Maple.

That is an obvious statement. What Math 641 opened my eyes to is some of the better ways to learn about new ways to teach concepts and new resources. I had never used Twitter before this class, and I had never done any blogging. I wasn't thrilled to have to get a Twitter account at the start of the semester, or start a blog. But I've grown to appreciate what a resource Twitter is as a forum for discussions about math ed, among other things of course. Twitter and blogging are great means for sharing ideas, and there are lots of people with far better ideas than me maintaining blogs. Rather than sticking only to publications like

Working with new topics week to week in Math 641, and exploring those topics from a variety of angles reminded me that there's always a big and interesting math horizon to explore. As a high school teacher, it's easy to remain relegated to the standards and the narrow math sphere they encompass. Modern Geometry was a refreshing contrast. There were many times during the semester when I thought, if I could spend three days doing

Working on my project for Math 641 has forced me to learn a lot more about GeoGebra. Some of the manipulatives for the propositions in Book III were quite simple, but others had a lot going on. Here are some of the new things that I learned:

__How to define a figure made up of two or more other objects__

I first ran into this issue when I was trying to make a picture for a circular segment. I couldn't find a tool (and maybe there is one?) to make a segment that I could fill and color. After a few Google searches, I found instructions for how to create a*list*. In the input line at the bottom of the GeoGebra window I used set braces to enter the following, for example

{Segment[A,B], Arc[c,B,A]}

This would define the figure bounded by line segment AB and the arc on circle c between B and A. It showed up as a*list* in the algebra pane, which I could select and give a color to for emphasis. Of course, different types of figures could be included in a list, but I used these mainly to create segments, since my work was with circles.

__More varied ways of assigning conditions to show objects__

In an earlier post, I explained how I had learned to use check-boxes and sliders to show and hide figures. For many the manipulatives that I was making for the propositions, I wanted things to change color or have tick marks added to them when lengths or angles were the same. I learned how to put measures other than a slider value in the conditions to show object line in object properties. I used the distance between points,

Distance[A,B]

frequently. I also learned how to include measures of objects that were in a worksheet by using their letter names, which required using the "alpha" button on the right side of the dialogue box for angles. If I wanted measures to be within a certain range of one another, I used the absolute value of their difference, for example

abs(a-b)<0.1

As I mentioned, I often used these to make segments change color when they were congruent, or to have tick marks added. Most of the time, I had constructed completely separate segments with the different colors or marks. It wasn't until I was almost done with the project that I realized if color was the only thing I wanted to change, I could put conditions in the "Dynamic Colors" settings right below "Condition to Show Object" (so obvious!). However, it was still nice to be able to change line styles and thicknesses in addition to color.

I understand that these are very simple things for more seasoned GeoGebra users, and they do seem simple to me now. They were new ground for me as I worked on my project, though, and each one was a hurdle that I had to work through to make each of my manipulatives work the way they were supposed to.

If you know of easier ways to do the things described above, I'd love to know! Please leave a comment.

I first ran into this issue when I was trying to make a picture for a circular segment. I couldn't find a tool (and maybe there is one?) to make a segment that I could fill and color. After a few Google searches, I found instructions for how to create a

{Segment[A,B], Arc[c,B,A]}

This would define the figure bounded by line segment AB and the arc on circle c between B and A. It showed up as a

In an earlier post, I explained how I had learned to use check-boxes and sliders to show and hide figures. For many the manipulatives that I was making for the propositions, I wanted things to change color or have tick marks added to them when lengths or angles were the same. I learned how to put measures other than a slider value in the conditions to show object line in object properties. I used the distance between points,

Distance[A,B]

frequently. I also learned how to include measures of objects that were in a worksheet by using their letter names, which required using the "alpha" button on the right side of the dialogue box for angles. If I wanted measures to be within a certain range of one another, I used the absolute value of their difference, for example

abs(a-b)<0.1

As I mentioned, I often used these to make segments change color when they were congruent, or to have tick marks added. Most of the time, I had constructed completely separate segments with the different colors or marks. It wasn't until I was almost done with the project that I realized if color was the only thing I wanted to change, I could put conditions in the "Dynamic Colors" settings right below "Condition to Show Object" (so obvious!). However, it was still nice to be able to change line styles and thicknesses in addition to color.

I understand that these are very simple things for more seasoned GeoGebra users, and they do seem simple to me now. They were new ground for me as I worked on my project, though, and each one was a hurdle that I had to work through to make each of my manipulatives work the way they were supposed to.

If you know of easier ways to do the things described above, I'd love to know! Please leave a comment.

The basic idea is this: How to motivate students to help one another improve their understanding.

In our most recent department meeting, one of my colleagues described a group work system that he has been trying in his classes. Here are some basics about the structure of our homework that I need to mention before I describe the system further:

- We have two-day homework. That is, there is a day in between the lesson and the due date to allow for questions to be asked in class.

- Actual homework (what is done out of the book) is not part of students' calculated grade. On the due date for an assignment, they take a short half-sheet quiz that we call a daily assessment, and their grade is based on the score they earn. They may use their homework as a reference on daily assessments.

- If students have completed a short list (10-12 problems, usually) of "required" homework, they have the opportunity to retake their daily assessment to earn back lost points. They have to complete some additional practice problems first. We call the retakes "reworks".

At the start of a unit, my colleague creates groups of 3 to 4 students. He puts one top student and one low-achieving student in each group, and tries to pick the other two students based on who might work well together. On the day between the lesson and the daily assessment, he has these groups work on a few practice problems based on the previous day's lesson. While students work on these, they often get their homework out to reference and compare, and many of the homework questions get answered within the groups. So, while more time is spent in groups, there is a reduction in how much time is spent on homework questions as a class.

The especially interesting part of this group system is the incentive my colleague attaches to it. He calculates each group's overall average grade at the start of the unit, and again at the end of the unit. If a group's average rises between the start and close of the unit, he awards all of the members of that group extra credit points on their unit tests.

The great thing about this incentive is that it gives excelling students a good reason to help out the ones who are having difficulty. After all, the students with the lowest grades are the ones that have the most room for improvement. Alternatively, it's hard to bring a 98% up a whole lot. Helping a struggling group member is the most promising way of earning those extra credit points. Teaching is the great way to learn, so there's a benefit to the excelling students also. My colleague says he has seen students taking an interest in each others' progress, and that group members begin holding one another accountable for doing their homework and doing reworks as needed.

I'm not a big fan of extra credit points in general, and I might be even less comfortable with adding them to test scores. In some form or another, though, this sounds like a great idea. Does anyone else out there have a good method for getting students to invest in each others' learning?

In our most recent department meeting, one of my colleagues described a group work system that he has been trying in his classes. Here are some basics about the structure of our homework that I need to mention before I describe the system further:

- We have two-day homework. That is, there is a day in between the lesson and the due date to allow for questions to be asked in class.

- Actual homework (what is done out of the book) is not part of students' calculated grade. On the due date for an assignment, they take a short half-sheet quiz that we call a daily assessment, and their grade is based on the score they earn. They may use their homework as a reference on daily assessments.

- If students have completed a short list (10-12 problems, usually) of "required" homework, they have the opportunity to retake their daily assessment to earn back lost points. They have to complete some additional practice problems first. We call the retakes "reworks".

At the start of a unit, my colleague creates groups of 3 to 4 students. He puts one top student and one low-achieving student in each group, and tries to pick the other two students based on who might work well together. On the day between the lesson and the daily assessment, he has these groups work on a few practice problems based on the previous day's lesson. While students work on these, they often get their homework out to reference and compare, and many of the homework questions get answered within the groups. So, while more time is spent in groups, there is a reduction in how much time is spent on homework questions as a class.

The especially interesting part of this group system is the incentive my colleague attaches to it. He calculates each group's overall average grade at the start of the unit, and again at the end of the unit. If a group's average rises between the start and close of the unit, he awards all of the members of that group extra credit points on their unit tests.

The great thing about this incentive is that it gives excelling students a good reason to help out the ones who are having difficulty. After all, the students with the lowest grades are the ones that have the most room for improvement. Alternatively, it's hard to bring a 98% up a whole lot. Helping a struggling group member is the most promising way of earning those extra credit points. Teaching is the great way to learn, so there's a benefit to the excelling students also. My colleague says he has seen students taking an interest in each others' progress, and that group members begin holding one another accountable for doing their homework and doing reworks as needed.

I'm not a big fan of extra credit points in general, and I might be even less comfortable with adding them to test scores. In some form or another, though, this sounds like a great idea. Does anyone else out there have a good method for getting students to invest in each others' learning?

For my Math 641 project, I am working on a "translation" of Book III of Euclid's Elements. As my source for the Elements, I am using Joyce's Elements webpage. It is obviously in English already, so by translate I mean that I am trying to use more modern terminology for the things described in Book III. It has been a challenge though, particularly for some of the longer, multi-faceted propositions. It is difficult to know how to state some of those more clearly, without getting much too wordy (as I'm prone to do).

For visuals to accompany the definitions, and for manipulatives to accompany each proposition, I am using GeoGebra. I am still a rookie when it comes to GeoGebra, so it is probably taking me longer to develop these things than most, but I have been surprised at how time consuming making these manipulatives has been. I've been learning a lot about how to set conditions to show objects, and having to think carefully about what conditions I can set in a diagram to emphasize each proposition. I had originally imagined that I would make an illustrated step-by-step GeoGebra file for the proofs of each proposition as well, but that might be an ongoing task for some time in the future.

To organize and display my work, I am adding an additional page to this blog per Dr. Golden's suggestion. I had considered using Google Sites, but after looking that over it seemed just as well to lay out the definitions and propositions in the blog, with links to the GeoGebra pages for each manipulative.

For visuals to accompany the definitions, and for manipulatives to accompany each proposition, I am using GeoGebra. I am still a rookie when it comes to GeoGebra, so it is probably taking me longer to develop these things than most, but I have been surprised at how time consuming making these manipulatives has been. I've been learning a lot about how to set conditions to show objects, and having to think carefully about what conditions I can set in a diagram to emphasize each proposition. I had originally imagined that I would make an illustrated step-by-step GeoGebra file for the proofs of each proposition as well, but that might be an ongoing task for some time in the future.

To organize and display my work, I am adding an additional page to this blog per Dr. Golden's suggestion. I had considered using Google Sites, but after looking that over it seemed just as well to lay out the definitions and propositions in the blog, with links to the GeoGebra pages for each manipulative.

Compass and straight-edge constructions are a part of my school's curriculum for Geometry A. The constructions that we teach and assess include:

- copying a segment

- copying an angle

- perpendicular bisector of a segment

- bisecting an angle

- constructing a line perpendicular to a given line through a point not on the line

- constructing a line parallel to a given line through a given point

I personally enjoy doing and teaching these constructions. It's hands on, and I enjoy seeing how they work out. I'm intrigued by all that can be done with just a compass and straight-edge, especially when it comes to things like quadrature and more complicated constructions. I like the history of it, too, and it seems amazing what early geometers and mathematicians were able to figure out with these tools.

I wonder sometimes what value these constructions have for my students, though. Some of the basic constructions can be used to draw other things that we study, like a triangle with certain lengths or angles, or the incenter of a triangle, although we usually have them explore concepts with difficult constructions like that on GeoGebra. I'm not sure whether these types of constructions really enhance students' visual-spatial reasoning relative to the things they are constructing either. I know I could do a better job emphasizing the fixed distance feature of arcs, but even if I did that, I'm afraid most of my students would still learn these as mechanical processes.

Eric Pandisico, in his article *Alternative Geometric Constructions: Promoting Mathematical Reasoning* suggests that students can take the most away from constructions when they have the chance to try them with a variety of tools. Rather than stick primarily to compass and straight-edge constructions, Pandisico promotes the use of several tools to do the same constructions, including Miras, a simple right angle source, like a notecard, and even a two-edged straight-edge. He suggests that because each tool has the potential to emphasize a different feature or property of each construction, doing one construction three different times with different tools can give students a deeper and more full understanding of the geometry underlying each construction.

We only teach these constructions with a compass and straight-edge. We don't use any other methods for creating these things. After trying these constructions and others with patty paper in Math 641, I could see the value in using more than one method. The patty paper made the symmetry of the constructions much easier to see, and even though the results were the same, the thinking I was doing about each construction was much different, and for some of them more intuitive.

Classmate Trevor Kuzee did a nice job of comparing and contrasting patty paper and compass/straight-edge constructions in this blog post. I'm not necessarily drawn toward one method over another, but I will try to give my students a chance and using more than one method the next time I teach constructions.

To help my students review the compass and straight-edge constructions that they had learned, I made the GeoGebra worksheet pictured below. This didn't represent much new GeoGebra learning for me. It was a lot of assigning values to show objects again, and there were lots of hidden circles underneath those arcs. I did set points to which I fixed the text boxes for the instructions, which allowed me to keep those well aligned even between constructions.

Reference:

Padisico, Eric A. (2002) Alternative geometric constructions: Promoting mathematical reasoning.*The Mathematics Teacher, 95*(1), 32-36.

Padisico, Eric A. (2002) Alternative geometric constructions: Promoting mathematical reasoning.

One of the things I find most interesting about conic sections is their reflective properties. Here are those properties, as I understand them (for the sake of a common description, I'll suppose a "beam" is bouncing off each conic section):

1) Any beam parallel to the axis of a parabola will be reflected to its focus. Or, any beam leaving the focus of a parabola will be reflected to a path that is parallel to the axis.

2) Any beam that passes through or leaves one focus of an ellipse will be reflected to pass through the other focus.

3) Any beam directed toward one focus of a hyperbola will be reflected toward the other focus.

These are a nice talking point when teaching students about conic sections, since they provide some interesting reasons for locating the foci or finding equations for sections with specific foci or focal distances. Flashlights and satellite dishes are familiar objects for kids, and it's always nice to have an answer to the question "Who uses this stuff?"

One of these days I'd like to team up with the engineering teacher at my school and make a model that demonstrates these properties. I imagine setting up pieces of conics with shared foci, as sketched out below. Our technical education department has a computerized lathe, so I think we could get the appropriate grooves cut into a piece of particle board and then put a thin strip of reflective material in each one. There would be a track for a laser pointer that runs perpendicular to the axis of the parabola, so the beam could move back and forth and always be parallel to the axis. If it were set up correctly, the beam would always hit the last focus (the second focus of the hyperbola) even as the laser pointer is moved back and forth. I'm not sure how tight the precision would have to be, but I guess that last focus could just be as big as necessary.

Maybe students wouldn't get as excited about this as I would, but I think it would be a pretty cool demonstration for class. Here's my attempt at creating the same in GeoGebra, but I'm not sure how to plot only part of a conic section yet.

1) Any beam parallel to the axis of a parabola will be reflected to its focus. Or, any beam leaving the focus of a parabola will be reflected to a path that is parallel to the axis.

2) Any beam that passes through or leaves one focus of an ellipse will be reflected to pass through the other focus.

3) Any beam directed toward one focus of a hyperbola will be reflected toward the other focus.

These are a nice talking point when teaching students about conic sections, since they provide some interesting reasons for locating the foci or finding equations for sections with specific foci or focal distances. Flashlights and satellite dishes are familiar objects for kids, and it's always nice to have an answer to the question "Who uses this stuff?"

One of these days I'd like to team up with the engineering teacher at my school and make a model that demonstrates these properties. I imagine setting up pieces of conics with shared foci, as sketched out below. Our technical education department has a computerized lathe, so I think we could get the appropriate grooves cut into a piece of particle board and then put a thin strip of reflective material in each one. There would be a track for a laser pointer that runs perpendicular to the axis of the parabola, so the beam could move back and forth and always be parallel to the axis. If it were set up correctly, the beam would always hit the last focus (the second focus of the hyperbola) even as the laser pointer is moved back and forth. I'm not sure how tight the precision would have to be, but I guess that last focus could just be as big as necessary.

Maybe students wouldn't get as excited about this as I would, but I think it would be a pretty cool demonstration for class. Here's my attempt at creating the same in GeoGebra, but I'm not sure how to plot only part of a conic section yet.

As I mentioned in my first blog post, on Van Hiele levels, I have some misgivings about the way proofs are presented in the curriculum from which my geometry classes are taught. The work that we have been doing in Modern Geometry this semester has opened my eyes to the value of conjecturing. Ideally, proof in my students' geometry classes would develop naturally out of discussion and defense of their own conjectures. Instruction on how to construct proofs more formally could follow.

As it is, my students face formal geometric proof head on. They start by writing "algebraic proofs," which basically entails setting up and solving equations with justifications (maybe a theorem or two, mostly properties of equality) for each step. Then they practice proofs that are more based in geometry concepts. Early on most of their work is filling in blanks in two-column proofs (which is the format we stick to, for the most part), providing missing statements and reasons. The next chapter in the course includes parallel and perpendicular lines, and proofs are included again. The big finish for proofs in Geometry A (although there are more in Geometry B) is triangle congruence proofs.

Proofs are always a struggle, for the majority of my students anyway. To help them along, I try to point out common patterns in the structure of the proofs we write in class, such as "*the final statement is whatever you were asked to prove*" or "*to change a word into an equation or an equation into a word, use its definition.*" I try to remind them to keep the end result in mind, and sometimes we even start at the end of a proof and work backwards a few steps. A lot of these things that I do to help students "get through" proofs don't really get at the meaning involved, and to a large extent I often feel I'm just teaching them the rules of a game.

It was hard not to be somewhat encouraged though, when by the end of our chapter on triangle congruence, many of my students were able to put together a pretty solid triangle congruence proof on their own. It made me wonder if teaching formal proof head on might have some value. The big question, of course, was whether or not my students could transfer any of the skills they were demonstrating to more varied proofs. I got a partial answer to that question in my class' response to a bonus question on one of their unit tests.

The bonus problem asked them to write a proof of the Triangle Sum Theorem (the sum of the angle measures is 180 degrees). Not every student attempted the proof. Among those who did, only two gave fairly conceptually solid proofs, and theirs weren't entirely rock solid as far as our formal standards from class went. Below is a copy of the problem on the test (I think more could have been said of what is "Given:"), and the two best proofs from my class.

As it is, my students face formal geometric proof head on. They start by writing "algebraic proofs," which basically entails setting up and solving equations with justifications (maybe a theorem or two, mostly properties of equality) for each step. Then they practice proofs that are more based in geometry concepts. Early on most of their work is filling in blanks in two-column proofs (which is the format we stick to, for the most part), providing missing statements and reasons. The next chapter in the course includes parallel and perpendicular lines, and proofs are included again. The big finish for proofs in Geometry A (although there are more in Geometry B) is triangle congruence proofs.

Proofs are always a struggle, for the majority of my students anyway. To help them along, I try to point out common patterns in the structure of the proofs we write in class, such as "

It was hard not to be somewhat encouraged though, when by the end of our chapter on triangle congruence, many of my students were able to put together a pretty solid triangle congruence proof on their own. It made me wonder if teaching formal proof head on might have some value. The big question, of course, was whether or not my students could transfer any of the skills they were demonstrating to more varied proofs. I got a partial answer to that question in my class' response to a bonus question on one of their unit tests.

The bonus problem asked them to write a proof of the Triangle Sum Theorem (the sum of the angle measures is 180 degrees). Not every student attempted the proof. Among those who did, only two gave fairly conceptually solid proofs, and theirs weren't entirely rock solid as far as our formal standards from class went. Below is a copy of the problem on the test (I think more could have been said of what is "Given:"), and the two best proofs from my class.

For only two students to come this close took some wind out of my sails again. But at the same time, it is still the slightest bit encouraging that a couple of them were able to go "off pattern" and prove something fairly different from the triangle congruence cases we had been working with (maybe my hopes for students' proof fluency are way too low).

So I still think that a heavily conjecture-based lead in to proof would serve more of my students better. But having gone through the usual proof sequence with my students once again, I don't believe what they've done will have *no *value for them as they continue in mathematics. If they aren't able to transfer their proof writing to more novel proofs now, might some ground work still have been laid for future proof writing? It seems like writing proofs formally will always be somewhat awkward the first time, even if it is preceded by lots of practice with informal arguments.

I will have many of the same students again next trimester in Geometry B, and the first unit is on quadrilaterals and their properties. We are planning to give students a few days to hopefully conjecture some of these properties on their own and then try to prove them with what they already know. This will be another opportunity to see how much they were able to take away from their education in proofs this trimester.

Schwan Smith, M., and Stein, M.K. (1998). Selecting and creating mathematical tasks: From research to practice*.* *Mathematics Teaching in the Middle School, 3*(5), 344-350.

This article stresses the importance of thoughtfully selecting/assigning/creating mathematical tasks for students. The tasks presented to students should facilitate the intended learning, and to be sure that happens, teachers need to pay close attention to the level of cognitive demand tasks will potentially place on students. Low demand tasks may help students learn math facts and develop procedural fluency, but tasks with greater complexity and even procedural ambiguity have the potential to engage students in high level reasoning and connection-making.

It would be very interesting to have the "Task Sort" discussing with my own math department.

- Here's an example of a low-level task from my precalculus class:

Rewrite | pi - sqrt(5) | without the absolute value symbol.

- Here's an example of a high-level task from precalculus from one of our problem solving days:

A line *tangent* to a circle is a line that intersects a circle at one point and is perpendicular to the radius at the point of intersection.

The line 2x-3y=-11 is tangent to a circle at (2, 5).

The line 3x+2y=29 is tangent to the same circle at (7, 4).

Find the standard form equation of the circle.

This article stresses the importance of thoughtfully selecting/assigning/creating mathematical tasks for students. The tasks presented to students should facilitate the intended learning, and to be sure that happens, teachers need to pay close attention to the level of cognitive demand tasks will potentially place on students. Low demand tasks may help students learn math facts and develop procedural fluency, but tasks with greater complexity and even procedural ambiguity have the potential to engage students in high level reasoning and connection-making.

It would be very interesting to have the "Task Sort" discussing with my own math department.

- Here's an example of a low-level task from my precalculus class:

Rewrite | pi - sqrt(5) | without the absolute value symbol.

- Here's an example of a high-level task from precalculus from one of our problem solving days:

The line 2x-3y=-11 is tangent to a circle at (2, 5).

The line 3x+2y=29 is tangent to the same circle at (7, 4).

Find the standard form equation of the circle.

One of our assignments for class this week was to do some math art, and one of the suggestions was to try making an ambigram. Ambigrams are words or phrases that can be read in more than one direction, or from different points of view. In class, we had looked at several examples by Scott Kim. Having never made an ambigram before, I decided to try what seems to be the simplest type - one that reads forward and backward (upside down).

After thinking about the geometry involved for this particular type of ambigram, I understood that I was aiming to make a word that would look the same under a 180 degree rotation. With this in mind, I tried to think of words whose first and last letters might bear some similarities under rotation. Obviously, every letter would have to have its rotated partner, but I had to start (and end) somewhere.*Geometry* seemed like it could work well. It wasn't too much of a stretch to imagine a capital *G* that looked something like a *y*.

Having settled on a word to try, I listed out the pairs of letters that would have to be rotations of one another. In retrospect, I suppose the best ambigrams don't necessarily match one letter with one other letter, but carefully weave shapes and features even between "letters" to achieve the desired effect. To keep it simple for myself, though, I tried to work with one letter pair at a time:*G* and *y*, *e* and *r*, *o* and *t*, *m* and *e*.

I filled up the margins of my notepad with sketches of my attempts at making symbols for these pairs. I basically tried to write each letter with the other in mind, and then I would rotate my notebook around 180 degrees and add more marks to the letter to make it look like the other one. There were lots of attempts that I didn't use. The challenge felt like walking a fine line between ambiguity and specificity in what letter(s) each symbol represented.

I had the hardest time with*e* and *r*. I was tempted to just borrow Scott Kim's solution to this one in his "SuperTeacher" inversion, but I kept trying and came up with something else that worked. My solution to *m* and *e*, right in the middle of the ambigram, was to go with a basic lowercase *m* and turn it 45 degrees. Not fancy, but again, it worked. Once I had each symbol figured out (four of them altogether, since it was two letters for each), I set about drawing a large copy of the ambigram.

To draw it accurately, I decided to mark the center of a piece of paper and then to draw out half of the word (the first, and the last half :) ) on one side of this center. Then I used an architect's scale to make a connect-the-dots copy of this half as a reflection through the center point. The resulting point symmetry is the same thing as 180 degree rotation symmetry. This was tedious, keeping the ruler lined up through the center and mirroring distances for lots of points. While I worked I kept thinking of more efficient ways to do this, like scanning the image and rotating it on the computer, or photocopying it, or even just tracing it on another piece of paper and attaching the two. But for the sake of having my finished product completely hand drawn, and on one, intact piece of paper, I persisted.

The finished product turned out fairly well. Here are two photographs of the completed ambigram. I lined up the corners of the paper with the ruler and the pencil the same way in each, but the paper does get rotated 180 degrees. Fairly similar?

After thinking about the geometry involved for this particular type of ambigram, I understood that I was aiming to make a word that would look the same under a 180 degree rotation. With this in mind, I tried to think of words whose first and last letters might bear some similarities under rotation. Obviously, every letter would have to have its rotated partner, but I had to start (and end) somewhere.

Having settled on a word to try, I listed out the pairs of letters that would have to be rotations of one another. In retrospect, I suppose the best ambigrams don't necessarily match one letter with one other letter, but carefully weave shapes and features even between "letters" to achieve the desired effect. To keep it simple for myself, though, I tried to work with one letter pair at a time:

I filled up the margins of my notepad with sketches of my attempts at making symbols for these pairs. I basically tried to write each letter with the other in mind, and then I would rotate my notebook around 180 degrees and add more marks to the letter to make it look like the other one. There were lots of attempts that I didn't use. The challenge felt like walking a fine line between ambiguity and specificity in what letter(s) each symbol represented.

I had the hardest time with

To draw it accurately, I decided to mark the center of a piece of paper and then to draw out half of the word (the first, and the last half :) ) on one side of this center. Then I used an architect's scale to make a connect-the-dots copy of this half as a reflection through the center point. The resulting point symmetry is the same thing as 180 degree rotation symmetry. This was tedious, keeping the ruler lined up through the center and mirroring distances for lots of points. While I worked I kept thinking of more efficient ways to do this, like scanning the image and rotating it on the computer, or photocopying it, or even just tracing it on another piece of paper and attaching the two. But for the sake of having my finished product completely hand drawn, and on one, intact piece of paper, I persisted.

The finished product turned out fairly well. Here are two photographs of the completed ambigram. I lined up the corners of the paper with the ruler and the pencil the same way in each, but the paper does get rotated 180 degrees. Fairly similar?

My wife's family is very serious about Artprize. Her parents and sister volunteer several times during the event. My wife and daughter and I usually make two to three, if not four, trips downtown during Artprize. The first day we went, I took our camera along, in hopes of finding some geometrically interesting art. I had hoped to notice some geometry that was very subtle yet deeply tied to the art. Most of what I photographed, though, was pretty obvious in its geometry. Nothing too profound here. And this is only from one visit, so I'm sure there were many more "geometric" entries that I never ran across. But here are a few things that I saw.

In the Gerald R. Ford Museum, one of the entries was *Reflection*, by Josemiguel Perera. Aptly named, I thought, since it has reflection symmetry across a horizontal. I added these circles to the picture to emphasize the concentric circles in each fan. The little circles are tangent to one another. Of course, then the larger circles with the same centers intersect. I thought it was interesting that connecting the points of intersection on the larger circles created a segment that passed through the point of tangency of the smaller ones. I suppose this would be the case any time two congruent larger circles are concentric with two smaller, congruent, externally tangent circles. That's a fairly specific situation, though.

I also found some of the Artprize signs interesting. They suggest geometry at a pretty basic level, so this was no great discovery on my part - all circles, triangles, and squares. From the right side of the sign, this pair of shapes stood out to me (right end, middle row of figures). It's a quarter circle with an isosceles right triangle attached to the side, with legs equal to the radius of the circle. It's divided up differently than that, though, with a long diagonal across the middle. My drawing from GeoGebra shows three triangles, and all of them have the same area. Each of them has the radius of the sector as its height, and half of the radius as the base (the vertical radius is bisected by the diagonal - the two line segments are diagonals of a parallelogram). Each of the three triangles has an area of (pi*r^2)/4. The light blue portion of the this figure (as seen in the Artprize sign) has the same area as that of a full quarter sector of the circle with this radius. These are somewhat random musings, but it seems there is probably more in this picture that could be uncovered and might be interesting.

At another venue, someone had put together a bunch of square Artprize signs into an interesting design. This got me thinking about squares inscribed inside one another, with vertices tangent to the midpoints of the outer squares' sides. It's related to the special, isosceles right triangles that show up when these midpoints are connected, and square roots of 2 canceling one another out, but there was an interesting relationship in these, too. As more inscribed squares were added, every second square would have its side length halved. In other words, the second inscribed square would have side lengths half as large as the outermost square. The side lengths between were, according to the special right triangle relationship, the outer square's side length times square root 2. I thought it was interesting that every second square had half the side length (and one fourth the area).

There were some other geometric aspects of entries that caught my eye. *Silkwaves in the Grand*, by Al and Laurie Roberts consisted of flags in the Grand River. The flag poles were connected near their bases, which made it apparent that they were arranged in star formations - pentagons with triangles attached to each side. I wonder if there was a particular aesthetic reason this arrangement was chosen.

At the B.O.B., *Through the Iris*, by Armin Mersmann was my favorite entry. It was impressive to me because I thought the pictures were photographs, but they were pencil drawings! I usually take for granted the perfect (or near perfect, considering my astigmatism) circular geometry of human eyes. It's pretty amazing, and it was nice to be reminded of it.

Most of my geometric noticing was fairly mundane, but it was fun to spend the day being attentive to the subject in my surroundings. Here are a few other Artprize entries that stood out as being more geometric than others:

Sacrifice, by Tom Panei |

Facing Al Aquaba, by Maurice Jacobsen |

Hilo - Sacred Geometry, by Kimberly Toogood |

I failed to write down the exhibit name and artist for this one. Nice to see pi in my search for geometry, but unfortunately it was backward in every place it appeared in this exhibit. |

Our in class activities in Modern Geometry as well as several of the assigned readings have heightened my awareness of the absence of conjecturing in the geometry curriculum that I present to my students. In general, my students read or are presented with the postulates and theorems that are important to a topic, and then we work with them. Very rarely are they presented with a relationship that they are asked to uncover on their own.

In their article Geometry and Proof, Battista and Clements (1995) suggest that students own conjectures are an ideal segue to the process of developing proofs. Providing students with an opportunity to discover a geometric relationship, and then leading them to defend their discoveries through class discussion and debate can demonstrate the importance of proof to students before they are asked to engage in it. The process of developing and testing a conjecture before proving it is also more true to real mathematical development. The article left me wanting to completely restructure the way proofs are presented in my geometry course, with ample time for exploration leading into informal proofs, and eventually some practice with formal deductive proofs. A progression like this would be more in line with Van Hiele‘s levels of geometric reasoning as well. Letting conjecture be the impetus for proof would make the learning of proof more intrinsic for students, and it would also make their learning a more genuine mathematical experience.

The value of conjecture in geometry courses stood out to me again in Diemente’s (2000) article about the Euler line and its relationship to the sides of a triangle. The mathematics presented in the article are certainly interesting, but I was most struck by the fact that the content of the article was all prompted by students’ questions and conjectures. Taking the time to explore a students’ question about whether the Euler line can be parallel to a side of a triangle, Diemente and his class covered a wide swath of mathematical ground. Their exploration and proof included midpoints, slopes, systems of linear equations, and even conic sections. What a great example of how valuable students’ curiosity can be! Not only were these and other concepts used, reviewed, and learned, but they were shown in connection with one another, and moreover, with a real purpose! While an exploration like that of Diemente’s honors geometry class might be over the ability levels of my geometry students, it might be a great study for students in my precalculus class.

So now I have a heightened sense of the value of providing students opportunities to make and investigate conjectures, and at the same time a greater awareness of the absence of such opportunities for my students. So now I am looking for places to create these opportunities. There are a couple on the horizon.

Tomorrow we will begin our unit on triangle congruence. Our department has an activity designed to help students discover the “shortcut” congruence theorems, like Side-Angle-Side, etc. It’s called “Triangle in a Bag”. Students can ask for one side length or angle measure of a certain triangle at a time, and the goal is to construct a triangle congruent to the one “in the bag” with as few given pieces as possible. The hope is that after doing this for several triangles, they might start to think critically about what key measurements are necessary to guarantee the same triangle, and develop the shortcut theorems. It’s a nice activity, but at the same time, I’m not sure if there’s much learning in it. A good class discussion might need to follow it, with some time for students to defend their conjectures as to why each set of three measures is enough. Any suggestions out there to make this activity better?

The chapter following this one includes the centers of triangles (incenter, circumcenter, centroid, etc.). For the first time, our department is planning to get students working with this with the aid of GeoGebra. I am looking forward to seeing how students respond, and I hope some of them will get excited about what they’re learning. Depending on how much time is available, it would be great to have students or groups of students share what they find about the relationships between these centers, or just the centers themselves. This might also be a good opportunity for some informal/formal proof writing.

I took my class to the computer lab for the first time this past week just to do some basic GeoGebra constructions and find their way around the program. They liked it, and more than once I ran across students experimenting with lines and figures on the program - being curious. I hope I can find more opportunities to let them be curious about geometry and learn through their curiosity. For that matter, I hope I can do that for all of my math students, and not just my geometry students.

Battista, M.T., & Clements, D.H. (1995). Geometry and proof.

Diamente, D. (2000). Algebra in the service of geometry: Can Euler's line be parallel to a side of a triangle?

I am finally reaching the point at which I can put GeoGebra to good use. I am getting to the steeper portion of the learning curve. I am sure I have much more to learn, still. But I have learned enough to make GeoGebra a useful tool for exploring geometry, and also for creating learning activities for my students.

My first week of trying to use GeoGebra was frustrating. I was having a hard time making shapes that were adjustable and movable, or dynamic. At a basic level, I did not understand the*dependency *of points in Geogebra. Points can be free (completely movable), semi-dependent (only able to move along the object they are placed upon), and fully dependent (not movable, unless the object(s) they lie on move). I didn't realize that if I placed a point at the intersection of two lines, I wouldn't then be able to move those lines by moving that point. The point needed to come first. In my early attempts at figuring this out, I tried to make a rectangle with adjustable dimensions. I went back and forth between creating rectangles that were entirely rigid, to trying to use four free points and creating a quadrilateral that didn't maintain the properties of a rectangle.

A Google search for how to make a dynamic rectangle in GeoGebra led me to a YouTube video presenting a challenge to do just that. The rectangle in the video was dynamic in three points, that is, three points could be moved and the shape would remain a rectangle. I had imagined that four points could be adjustable, so seeing that only three could move gave some more direction to my attempts, of which there were many. However, I still wasn't able to figure it out until I watched the longer challenge video, which provided a key hint: one diagonal of the rectangle must lie on the diameter of a circle. Then I was able to figure out that the vertices on either end of that diagonal are the only free points on the rectangle. Another vertex needs to lie on the circle, and a right angle is guaranteed at that vertex, according to Thales' Theorem. Here's a link to the rectangle on GeoGebra Tube.

My first week of trying to use GeoGebra was frustrating. I was having a hard time making shapes that were adjustable and movable, or dynamic. At a basic level, I did not understand the

A Google search for how to make a dynamic rectangle in GeoGebra led me to a YouTube video presenting a challenge to do just that. The rectangle in the video was dynamic in three points, that is, three points could be moved and the shape would remain a rectangle. I had imagined that four points could be adjustable, so seeing that only three could move gave some more direction to my attempts, of which there were many. However, I still wasn't able to figure it out until I watched the longer challenge video, which provided a key hint: one diagonal of the rectangle must lie on the diameter of a circle. Then I was able to figure out that the vertices on either end of that diagonal are the only free points on the rectangle. Another vertex needs to lie on the circle, and a right angle is guaranteed at that vertex, according to Thales' Theorem. Here's a link to the rectangle on GeoGebra Tube.

My dynamic rectangle with semicircle shown.A and C are freepoints, B is semi-dependent,and D is fully dependent. |

Working through the rectangle challenge problem helped me to understand how objects and figures behave and relate to each other in GeoGebra, and how to carefully put a sketch together to allow adjustment of certain dimensions or positions. Another thing that I particularly wanted to understand, especially for the sake of creating activities for my students, was how to show different steps in a GeoGebra worksheet.

More Google searches on this topic helped me fumble my way through using *check boxes*. Several objects on a worksheet can be selected to be shown or hidden when a box is checked or unchecked. Here's a GeoGebra worksheet that I created using check boxes. With the help of Professor John Golden, I learned how to use a *slider *to achieve the same results, but with more control of what is being shown or hidden. With a slider set to integer values, *conditions to show objects* could be assigned in the properties for each part of a drawing, according to the values on the slider. This also made it easier to have objects, like text, appear for only one step, or for several steps, without the need for unchecking a box again.

A worksheet with check boxes. |

A worksheet with a slider to show steps. Plus, the rectangle's length and width can be adjusted! |

Being able to use a slider will be key to making worksheets for students to use that contain several different examples. In the near future I am hoping to make a GeoGebra worksheet to help my geometry students discover the four triangle congruence theorems (*SSS*, *SAS*, *ASA*, and *AAS*). In a similar vein, I think a GeoGebra worksheet could be very helpful for my precalculus students learning about the Law of Sines and the *SSA *case, and when two triangles are possible.

“School geometry that is presented in a [Euclidean] axiomatic fashion assumes that students think on a formal deductive level. However, that is usually not the case…” (Van Hiele, 1999)

This excerpt from Van Hiele’s article, Developing Geometric Thinking through Activities that Begin with Play, immediately made me think of one topic in particular, the bane of (most) high school geometry students’ existence: proofs. The type of formal reasoning demanded by deductive proofs is seldom easy for students to attain, even in high school. If students have one complaint about geometry, it is usually the proofs.

As such, I am always looking for ways to make proofs easier for students to understand. Before we get into geometric proofs, we always play a “proof game” with simple rules for manipulating strings of letters, to get them used to justifying every statement they make. I also try to emphasize the sequence and structure that is common among many proofs, like using definitions to change words into equations, after which properties of equality can be used to show a range of things. We also try to scaffold heavily as students learn how to write proofs, slowly bringing them along from filling in blanks to eventually writing complete proofs on their own.

But Van Hiele’s article led me to think about students’ struggles with proof in a different light, and question whether my students have been given or are given enough opportunities to progress through Van Hiele’s levels of geometric reasoning. Perhaps some of their difficulty with proof stems from having too little time to engage in informal reasoning.

Van Hiele’s Levels of Geometric Reasoning

1 - Visual

2 - Descriptive

3 - Informal Deductive

4 - Formal Deductive

5 - Axiomatic

I know that my students do not have many opportunities to do this, to take their time and consider what they know about geometric figures, and to discover out of their own knowledge further relationships and geometric truths. On the first day of class, they are introduced to four postulates, and at least one or two postulates or theorems follows in each subsequent lesson. They do come in a logical sequence, but still they are presented for students to take at face value most of the time, unless we take the time to prove them. It would be better if my students could be presented with new ideas along the way, and allowed to explore and struggle to find new relationships on their own, before having the formalized theorems presented to them. And they could practice justifying their discoveries, without immediately being tied to structure of a proof. I think they would much more thoroughly internalize important concepts than they do now.

Formal proof writing will always be an adjustment, though. It is a new style of writing for students. But if they have had more time to explore their way through new concepts, they might be better at making the connections that tie given information to a desired conclusion.

Van Hiele’s mosaic and the activities he describes to go with it are impressive. Its careful design provides a basis for a wide range of discussions and student discoveries. The example of the mosaic makes me want to find more exploratory activities that can apply to the concepts my high school geometry students will work with this year. I will certainly look for these, or try to come up with some, and look for opportune places to tie them into our curriculum.

Time might be the biggest challenge to integrating more exploratory activities. It is at such a premium already, with all that is supposed to be covered in a term. Giving students a day here or there to explore, rather than presenting material directly and moving on is not the most efficient way to go, time-wise. Our math department already has moved to include days dedicated to problem solving group work near the end of each unit. I wonder now if in geometry it might be better to use some of those days for exploratory activities at the start of a unit. That is a suggesting I will make to our geometry team. At the very least, I will be more apt to provide more time within lessons to have students discuss key ideas with each other.

Teaching Children Mathematics, 5(6), 310-316.

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