## Algebra 2

The unit focused on the students' first exposure to logarithmic and exponential functions. The situation in Algebra 2 was very similar to Geometry, with one key difference. The main difference of this class compared to Geometry is that almost all of the direct instruction was outsourced to video. I decided to follow the Udacity approach of several small videos (<3 min), because that meant there was opportunity (and the expectation) that only two minutes would go by before students would be expected to do something. I like this much better because it fit my own preferences in learning material with the Udacity courses. I had 2 minutes to watch a video about hash functions in Python while brushing my teeth - my students should have that ability too. I wasn't going for the traditional flipped class model here. My motivation was less about requiring students to watch videos for homework, and more about students choosing how they wanted to go through the material. Some students wanted me to do a standard lesson, so I did a quick demonstration of problems for these students. Others were perfectly content (and successful) watching the video in class and then working on problems. Some really great consequences of doing things this way:

• Students who said they watched all my videos and 'got it' after three, two minute videos, had plenty of time in the period to prove it to me. Usually they didn't.. This led to some great conversations about active learning. Can you predict the next step in the video when you try solving the problem on your own? What? You didn't try solving it on your own? <SMIRK>  The other nice thing about this is that it's a reinvestment of two minutes suggesting that they try again with the video, rather than a ten or fifteen minute lesson from Khan Academy.
• I've never heard such spirited conversation between students about logarithms before. The process of learning each skill became a social event - they each watched the video together, rewound or paused as needed, and then got into arguments while trying to solve similar problems from the day's handout. Often this would get in the way during teacher-centered lessons, and might be classified incorrectly as 'disruption' rather than the productive refining and conveyance of ideas that should be expected as part of real learning.
• Having clear standards for what the students needed to be able to do, and making clear what tools were available to help them learn those specific standards, led to a flurry of students demanding to show me that they were proficient. That was pretty cool, and is what I was trying to do with my quiz system for years, but failed because there was just too much in the way.
• Class time became split between working on the day's standards, and then stopping at an arbitrary time to then look at other cool math concepts. We played around with some Python simulations in the beginning of the unit, looked at exponential models, and had other time to just play with some cool problems and ideas so that the students might someday see that thinking mathematically is not just followinga list of procedures, it's a way of seeing the world.

I initially did things this way because a student needed to go back to the US to take care of visa issues, and I wanted to make sure the student didn't fall behind. I also hate saying 'work on these sections of the textbook' because textbooks are heavy, and usually blow it pretty big. I'm pretty glad I took this opportunity to give it a try. I haven't finished grading their unit exams (mostly because they took it today) but I will update with how they do if it is surprising.

Warning: some philosophizing ahead. Don't say I didn't warn you. I like experimenting with the way my classroom is structured. I especially like the standards based philosophy because it is the closest I've been able to get to recreating my Montessori classroom growing up in a more traditional school. I was given guidelines for what I was supposed to learn, plenty of materials to use, and a supportive guide on the side to help me when I got stuck. I have seen a lot of this process happening with my own students - getting stuck on concepts, and then getting unstuck through conversation with classmates and with me. The best part for me has been seeing my students realize that they can do this on their own, that they don't always need me to tell them exactly what to do at all times. If they don't understand an idea, they are learning where to look, and it's not always at me. I get to push them to be better at what they already know how to do rather than being the source of what they know. It's the state I've been striving to reach as a teacher all along, and though I am not there yet, I am closer than I've ever been before. It's a cliche in the teaching world that a teacher has done his or her job when the students don't need you to help them learn anymore. This is a start, but it also is a closed-minded view of teaching as mere conveyance of knowledge. I am still just teaching students to learn different procedures and concepts. The next step is to not only show students they can learn mathematical concepts, but that they can also make the big picture connections and observe patterns for themselves. I think both sides are important. If students see my classroom as a lab in which to explore and learn interesting ideas, and my presence and experience as a guide to the tools they need to explore those ideas, then my classroom is working as designed. The first step for me was believing the students ultimately wantneed to know how to learn on their own. Getting frustrated that students won't answer a question posed to the entire class, but then will gladly help each other and have genuine conversations when that question comes naturally from the material. All the content I teach is out there on the internet, ready to be found/read/watched as needed. There's a lot of stuff out there, but students need to learn how to make sense of what they find. This comes from being forced to confront the messiness head on, to admit that there is a non-linear path to knowledge and understanding. School teaches students that there is a prescribed order to this content, and that learning needs to happen within its walls to be 'qualified' learning. The social aspect of learning is the truly unique part of the structure of school as it currently exists. It is the part that we need to really work to maintain as content becomes digital and schools get more wired and connected. We need to give students a chance to learn things on their own in an environment where they feel safe to iterate until they understand. That requires us as teachers to try new things and experiment. It won't go well the first time. I've admitted this to my students repeatedly throughout the past weeks of trying these things with my classes, and they (being teenagers) are generous with honest criticism about whether something is working or not. They get why I made these changes. By showing that iteration, reflection, and hard work are part of our own process of being successful, they just might believe us when we tell them it should be part of theirs.

## Socializing in Geometry - Similar Triangles

Another successful experiment getting my participation-challenged geometry class to interact with each other yesterday.

Each student received a cut-out triangle from the image at left. The challenge:

One (or possibly two) people in this room have triangles similar to yours. Your task is to find the person and do the following:

• Find the similarity ratio between your triangle and your match in the order big:small.
• Determine the ratio of the perimeters of each of your triangles.
• Determine the ratio of the areas of each of your triangles.

I then cut them loose. Almost immediately they started scrambling around the classroom holding up triangles and calculating as quickly as possible. (I didn't totally get why they were in a hurry, actually.) They clustered on tables and rapidly shifted partners until everyone found they were in the right place. The calculating began for perimeter - that was the easy part. Then the area question took center stage.

Some asked me how to find the heights of the triangles, and I shrugged my shoulders with the smirk of someone with ideas that isn't sharing them. (I call this my 'jerk' mode that I love taking on during class for the sole reason that it gets them finding and figuring on their own.) Some recreated the triangle in Geogebra. Some superimposed it over graph paper and counted to get an estimate. One student cleverly found Heron's formula. It was really entertaining watching them excitedly explain the formula without writing it down (something else I didn't understand) and share how it quickly and easily allows the area to be calculated. The energy in the room was apparent as they ran from person to person trying to get everyone to complete the task. Eventually they found out themselves that the similarity ratio was a square relationship. I didn't have to do a thing.

Part of my justification in doing this was to get them thinking about the important ideas necessary in solving another problem I threw their way during the previous class comparing the old iPad to the new one. The two different groups that had worked on it were generally on the right track, but there were some serious errors in their reasoning that I hinted at but didn't explicitly point out to them. I think this activity closed the gap. There should be some interesting answers to discuss in class when we next meet.

## Party games & geometry definitions

Today's geometry class started with a new random arrangement of student seats. It never fails to amaze me how the dynamics of the whole room change with a shuffle of student locations.

The lesson today was the first of our quadrilateral unit. Normally after tests, I don't tend to have homework assignments, but I decided to make an exception with a simple assignment:

Create a single Geogebra file in which you construct and label all of the quadrilaterals given in the textbook: parallelogram, rhombus, square, kite, rectangle, trapezoid, and isosceles trapezoid.

This appealed to me because I really dislike lessons in which we go through definitions slowly as a group. I also knew that giving the students some independence in reviewing or learning the definitions of these quadrilaterals was a good thing. Sometimes they are a bit to reliant on me to give them all the information they need. For this assignment, students would need to understand the definitions of quadrilaterals in order to construct them, and that was a good enough for walking into class today.

The warm-up activity involved looking at unlabeled diagrams of quadrilaterals, naming them, and writing any characteristics they noticed about them from the diagrams:

Some had trouble with the term 'characteristics', but a peek down at the chart just below on the paper helped them figure it out:

Based on what they knew from the definitions before class, I had them complete this chart while talking to their new partner. There was lots of good conversation and careful use of language for each listed characteristic.

This led to the next thing that often serves as an important (though often boring) exercise: new vocabulary. I used one of my favorite activities that gets students focused on little details - each student received one of the following four charts. The chart is originally from p. 380 of the AMSCO Geometry textbook, and was digitally ruined using GIMP.

The students had a good time filling in the missing information and conferring with each other to make sure they had it all. We then came up with some examples of consecutive vertices, angles, diagonals, and opposite sides.

From their work with the chart and using the new vocabulary whenever possible, we then did the following:

What information would you need in order to prove that a quadrilateral is... (use as much of the new vocabulary as possible!)

• a square?

• a rhombus?

• a parallelogram?

• a rectangle?

• a trapezoid? (an isosceles trapezoid?)

• a kite?

I was really pleased with how they did with this exercise - they really seemed to be interacting with the definitions and vocabulary well.

Finally, we arrived at the part that was the most fun. You know that annoying ice-breaker you sometimes are forced to do at professional development sessions where you wear something on your head and have to get the other attendees to tell you who you are?

I hate that activity. That usually means it's perfect for my students:

The students were all smiles during the ten minutes or so we spent going through it - yes, I had one too! They were using the vocabulary we had developed during the day and were pretty creative in getting each other to guess the dog names as well.

In the end, I feel pretty good about how today's set of activities went. The engagement level was pretty high and everyone did a good job of interacting with the definitions in a way that will hopefully lead to understanding as we start proving their properties in coming classes.

## Students #flipping class presentations through making videos

Those of you that know the way I usually teach probably also know that projects are not in my comfort zone. I always feel they need to be well defined in such a way to make it so that the mathematical content is the focus, and NOT necessarily about how good it looks, the "flashy factor", or whether it is appropriately stapled. As a result, I often avoid them like the plague. The activities we do in class are usually student centered and involve  a lot of student interaction, and occasionally (much to my dismay) are open ended problems to be solved.

Done well, a good project (and rubric) also involves a good amount of focused interaction between students about the mathematical content. I don't like asking students to make presentations either - what often results is a Powerpoint and students awkwardly gesturing at projected images of text that they then read to the group in front of them. In class, I openly mock adults who do this to my students - I keep the promise that I will never ask them to read to me and their peers standing at the front of the room. Presentation skills are important, don't get me wrong, but I don't see educational gold in the process, or get all tingly about 'real-world skill development' from assigning in-class presentations. They instill fear in the hearts of many students (especially those that are students of ESOL) and require  tolerance from the rest of the class and involved adults to sit through watching them, and require class time in order to 'make' students watch them.

I'm also not convinced they actually learn content by creating them. Take a bunch of information found on Wikipedia or from Google, put it on a number of slides, and read it slowly until your time is up. Where is the synthesis? Where is the real world application of an idea that the student did? What new information is the student generating? If there's very little substantive answer to those questions, it's not worth it. It's no wonder why they go the Powerpoint slide route either - it's generally what they see adults doing when they present something.

In short, I don't like asking students to do something that even adults don't typically do well, and even then without the self-esteem and image issues that teenagers have.

All of that said, I really liked seeing a presentation (a good one, mind you) from Kelly Grogan (@KellyEd121) at the Learning 2.011 conference in Shanghai this past September. She has her students combine written work, digital media, audio, and video into digital documents that can be easily shared with each other and with her as their teacher. The additional dimension of hearing the student talking about his/her work and understanding is a really powerful one. It is but one distilled aspect of what we want students to get out of the projects we assign.

The fact that it isn't live also takes away a lot of the pressure to get it all right in one take. It also takes advantage of the asynchronous capability that technology affords us - I can watch a student's product at home or on my iPad at night, as can the other students. I like how it uses the idea of the flipped classroom to change the idea of student presentations. Students present their understanding or work through video that can be watched at home,  and then the content can be discussed or used in class the next day.

It was with all of this in mind that I decided to assign the project described here:

http://wiki.hischina.org/groups/gealgerobophysiculus/wiki/57f0c/Unit_5__Living_Proof_Video_Project.html

The proofs were listed on a handout given in class, and students in groups of two chose which proof they wanted to do. Most students submitted their videos today. I'm pretty pleased with how they ran with the idea and made it their own. Some quick notes:

• The mathematical content is the focus, and the students understood that from the beginning. While the math isn't perfect in every video, the enthusiasm the students had for putting these together was pretty awesome to watch. There's no denying that enthusiasm as a tool for helping students learn - this is a major plus for project based assignments.
• Some students that rarely volunteer to speak in class have their personalities and voices all over these. I love this.

My plan to hold students accountable for watching these is to have variations of them on the unit test in a couple weeks. I don't have to force the students to watch them though - they had almost all shared them before they were due.

Yes, you heard that right. They had almost all shared their work with each other and talked about it before getting to class. I sometimes have to force this to happen during class, but this assignment encouraged them to do it on their own. Now that's cool.

I have ideas for tweaking it for next time, but I really liked what came out of this. I've been hurt(stung?)  by projects before - giving grades that meet the rubric for the project, but don't actually result in a grade that indicates student learning.

I can see how this concept could really change things though. There's no denying that the work these students produced is authentic to them, and requires engagement with the content. Isn't that what we ultimately want students to know how to do when they leave our classroom?

## Math is everywhere! - fractals on the Franz Josef glacier

One of the stops on our New Zealand adventure was at the Franz Josef glacier on the West coast. We went on the full day hike which gave us plenty of time to explore the various ice formations on the glacier under the careful eye of our guide. Along the way up the glacier, I took the following series of pictures:

All of these were taken on the way up the glacier. Can you tell in what order I took them? If you're like my students (and a few others I have shown these to), you will likely be incorrect.

I realized as I was walking that this might be because of the idea of self-similarity, a characteristic of fractals in which small parts are similar to the whole. When I showed this set of pictures to my geometry class, I then showed them a great video video zooming in on the Mandelbrot fractal to show them what this meant.

The formations in the ice and the sizes of the rocks broken off my the glacier contributed to the overall effect. Here is another shot looking down the face of the glacier in which you can see four different groups of people for a size comparison:

The cooler thing than seeing this in the first place was discovering that it's a real phenomenon! There are some papers out there discussing the fact that the grain size distribution of glacial till (the soil, sand, and rocks broken off by the glacier) is consistent throughout a striking range of magnitudes. The following chart is from Principles of Glacier Mechanics by Roger Leb. Hooke:

In case you are interested in exploring these pictures more, here are the full size ones in the same A-B-C-D order from above:

Oh, and in case you are wondering, the correct order is B-C-A-D.

## Geogebra for Triangle Congruence Postulates

It has been busy-ville in gealgerobophysicsulus-town, so I have barely had time to catch my breath over the last few days of music performances, school events, and preparations for the end of the semester.

My efforts over the  past couple days in Geometry have focused on getting in a bit of understanding of congruent triangles. We have used some Geogebra sketches I designed to have them build a triangle with specific requirements. With some feedback from some Twitter folks (thanks a_mcsquared!) and students after doing the activities, I've got these the way I want them.

Looking for an ASA postulate. Download here. (Clearly for ASA explorations.) - This one I made a quick change before class to making it so that the initial coordinates of the base of the triangle are randomized when loading the sketch. This almost guarantees that every student will have a differently oriented triangle. This makes for GREAT conversations in class. Here are three of the ones students created this afternoon:

I'm doing a lot of thinking about making these sorts of activities clearly driven by simple, short instructions. This is particularly in light of a few of the students in my class with limited English proficiency. Creating these simple activities is also a lot more fun than just asking students to draw them by hand, guess, or just listen to me tell them the postulates and theorems. Having a room full of different examples of clearly congruent triangles calls upon the social aspect of the classroom. Today they completed the activity and showed each other their triangles and had good interactions about why they knew they had to be congruent.

Last year I had them construct the triangles themselves, but the power of the end message was weakened by the written steps I included in the activity. Giving them clear instructions made the final product, a slew of congruent (or at least approximately in the case of 7-8-9) triangles a nice "coincidence" to lead to generalizing the idea.

## Testing expected values using Geogebra

I was intrigued last night looking at Dan Meyer's blog post about the power of video to clearly define a problem in a way that a static image does not. I loved the simple idea that his video provoked in me - when does one switch from betting on blue vs. purple? This gets at the idea of expected value in a really nice and elegant way. When the discussion turned to interactivity, Geogebra was the clear choice.

I created this simple sketch (downloadable here)as a demonstration that this could easily be turned into an interactive task with some cool opportunities for collecting data from classes. I found myself explaining the task in a slightly different way to the first couple students I showed this to, so I decided to just show Dan's video to everyone and take my own variable out of the experiment. After doing this with the Algebra 2 (10th grade) group, I did it again later with Geometry (9th) and a Calculus student that happened to be around before lunch.

The results were staggering.

Each colored point represents a single student's choice for when they would no longer choose blue. Why they chose these was initially beyond me. The general ability level of these groups is pretty strong. After a while of thinking and chatting with students, I realized the following:

• Since the math level of the groups were fairly strong, there had to be something about the way the question was posed that was throwing them off. I got it, but something was off for them.
• The questions the students were asking were all about winning or losing. For example, if they chose purple, but the spinner landed on blue, what would happen? The assumption they had in their heads was that they would either get \$200 or nothing. Of course they would choose to wait until there was a better than 50:50 chance before switching to purple. The part about maximizing the winnings wasn't what they understood from the task.
• When I modified the language in the sketch to say when do you 'choose' purple instead of 'bet' on the \$200  between the Algebra 2 group and the Geometry group, there wasn't a significant change in the results. They still tended to choose percentages that were close to the 50:50 range.

I made an updated sketch that allowed students to do just that, available here in my Geogebra repository. It lets the user choose the moment for switching, simulates 500 spins, and shows the amount earned if the person stuck to either color. I tried it out on an unsuspecting student that stayed after school for some help, one of the ones that had done the task earlier.

Over the course of working with the sketch, the thing he started looking for was not when the best point to switch was, but when the switch point resulted in no difference in the amount of money earned in the long run by spinning 500 times. This, after all, was why when both winning amounts were \$100, there was no difference in choosing blue or purple. This is the idea of expected value - when are the two expected values equal? When posed this way, the student was quickly able to make a fairly good guess, even when I changed the amount of the winnings for each color using the sketch.

I'm thinking of doing this again as a quick quiz with colleagues tomorrow to see what the difference is between adults and the students given the same choice. The thing is, probably because I am a math teacher, I knew exactly what Dan was getting at when I watched the video myself - this is why I was so jazzed by the problem. I saw this as an expected value problem though.

The students had no such biases - in fact, they had more realistic ones that reflect their life experiences. This is the challenge we all face designing learning activities for the classroom. We can try our best to come up with engaging, interesting activities (and engagement was not the issue - they were into the idea) but we never know exactly how they will respond. That's part of the excitement of the job, no?

The social aspect of being in a classroom is what makes it such a unique learning environment. It isn't just a place where students can practice and develop their skills, because they can do that outside of the classroom using a variety of resources. In the classroom, a student can struggle with a problem and then ask a neighbor. A student can get nudged in the right direction by a peer or an adult that cares about their progress and learning.

If students can learn everything we expect them to learn during class time by staring at a screen, then our expectations probably aren't what they should be. Our classrooms should be places in which ideas are generated, evaluated, compared, and applied. I'm not saying that this environment shouldn't be used to develop skills. I just mean that doing so all the time doesn't make the most of the fact that our students are social most of the time they are not in our classrooms. Denying the power of that tendency is missing an opportunity to engage students where they are.

I am always looking for ways to justify why my class is better than a screen. Based on a lot of discussion out there about the pros and cons of Khan academy, I tried an experiment today with my geometry class to call upon the social nature of my students for the purposes of improving the learning and conversations going on in class. As I have mentioned before, it can be a struggle sometimes to get my geometry students  to interact with each other as a group during class, so I am doing some new things with them and am evaluating what works and what doesn't.

The concept of badges as a meaningless token is often cited as a criticism of the Khan academy system. It may show progress in reaching a certain skill level, it might be meaningless. How might this concept be used in the context of a classroom filled with living, breathing students? Given that I want to place value on interactions between students that are focused on learning content, how might the concept be applied to a class?

I gave the students an assignment for homework at the end of the last class to choose five problems that tested a range of the ideas that we have explored during the unit. Most students (though not all) came to class with this assignment completed. Here was the idea:

• Share your five problems with another student. Have that student complete your five problems. If that student completes the problems correctly  and to your satisfaction, give them your personal 'badge' on their paper. This badge can be your initials, a symbol, anything that is unique to you.
• Collect as many people's badges as you can. Try to have a meaningful conversation with each person whose problems you complete that is focused on the math content.
• If someone gives a really good explanation for something you previously didn't understand, you can give them your badge this way too.

It was really interesting to see how they responded. The most obvious change was the sudden increase in conversations in the room. No, they were not all on topic, but most of them were about the math. There were a lot of audible 'aha' moments. Some of the more shy students reached out to other students more than they normally do. Some students put themselves in the position of teaching others how to solve problems.

In chatting with a couple of the students after class, they seemed in agreement that it was a good way to spend a review day. It certainly was a lot less work for me than they usually are. Some did admit that there were some instances of just having a conversation and doing problems quickly to get a badge, but again, the vast majority were not this way. At least in the context of trying to increase the social interactions between students, it was a success. For the purpose of helping students learn math from each other, it was at least better than having everyone work in parallel and hope that students would help each other when they needed it.

It is clear that if you want to use social interactions to help drive learning in the classroom, the room, the lesson, and the activities must be deliberately designed to encourage this learning. It can happen by accident, and we can force students to do it, but to truly have it happen organically, the activity must have a social component that is not contrived and makes sense being there.

The Khan academy videos may work for helping students that aren't learning content skills in the classroom. They may help dabblers that want to pick up a new skill or learn about a topic for the first time. Our students do have social time outside of class, and if learning from a screen is the way that a particular student can focus on learning content they are expected to learn, maybe that makes sense for learning that particular content. In a class of twenty to thirty other people, being social may be a more compelling choice to a student than learning to solve systems of equations is.

If we want to teach students to learn to work together, evaluate opinions and ideas, clearly communicate their thinking, then this needs to be how we spend our time in the classroom. There must be time given for students to apply and develop these skills. Using Khan Academy may raise test scores, but with social interaction not emphasized or integrated into its operation, it ultimately may result in student growth that is as valuable and fleeting as the test scores themselves. I think in the context of those that may call KA a revolution in education, we need to ask ourselves whether that resulting growth is worth the missed opportunity for real, meaningful learning.

I want to record a few things about the last couple of days of class here - cool stuff, some successes, some not as good, but all useful in terms of moving forward.

## Geometry:

I have been working incredibly hard to get this class talking about their work. I have stood on chairs. I've given pep talks, and gotten merely nods of agreement from students, but there is this amazing resistance to sharing their work or answering questions when it is a teacher-centric moment. There are a couple students that are very willing to present, but I almost think that their willingness overshadows many others who need to get feedback from peers but don't know how to go about it. What do I do?

We turn it into a workshop. If a student is done, great. I grab the notebook and throw it under the document camera, and we talk about it. (In my opinion, the number one reason to have a document camera in the classroom, aside from demonstrating lab procedures in science, is to make it easy and quick for students get feedback from many people at once. Want to make this even better and less confrontational? Throw up student work and use Today's Meet to collect comments from everyone.

The most crucial thing that seems to loosen everyone up for this conversation is that we start out with a compliment. Not "you got the right answer". Usually I tolerate a couple "the handwriting is really neat" and "I like that you can draw a straight line" comments before I say let's have some comments that focus on the mathematics here. I also give effusive and public thanks to the person whose work is up there (often not fully with their permission, but this is because I am trying to break them of the habit of only wanting to share work that is perfect.) This praise often includes how Student X (who may be not on task but is refocused by being called out) is appreciative that he/she is seeing how a peer was thinking, whether it was incorrect or not. I also noticed that after starting to do this, all students are now doing a better job of writing out their work rather than saying "I'll do it right on the test, right now I just want to get a quick answer."

## Algebra 2

We had a few students absent yesterday (which, based on our class size, knocks out a significant portion of the group) so I decided to bite the bullet and do some Python programming with them. We used the Introduction to Python activity made by Google. We are a 1:1 Mac school, and I had everyone install the Python 3 package for OS 10.6 and above. This worked well in the activities up through exercise 8. After this, students were then supposed to write programs using a new window in IDLE. I did not do my research well enough, unfortunately, as I read shortly afterward that IDLE is a bit unstable on Macs due to issues with the GUI module. At this point, however, we were at the end of the period, so it wasn't the end of the world. I will be able to do more with them now that they have at least seen it.

How would I gauge the student response? Much less resistance than I thought. They seemed to really enjoy figuring out what they were doing, especially with the % operator. That took a long time. Then one student asked if the word was 'remainder' in English, and the rest slapped their heads as they simultaneously figured it out. Everyone enjoyed the change of pace.

For homework, in addition to doing some review problems for the unit exam this week, I had them look at the programs here at the class wiki page.

## Physics

I had great success giving students immediate feedback on the physics test they took last week by giving them the solutions to look at before handing it in. I had them write feedback for themselves in colored pencils to distinguish their feedback from their original writing. In most cases, students caught their own mistakes and saw the errors in their reasoning right away. I liked many of the notes that students left for themselves.

This was after reading about Frank Noschese's experience doing this with his students after a quiz. I realize that this is something powerful that should be done during the learning cycle rather than with a summative assessment - but it also satisfied a lot of their needs to know when they left how they did. Even getting a test back a couple days later, the sense of urgency is lost. I had them walking out of the room talking about the physics rather than talking about how great it was not to be taking a test anymore.

Today we started figuring out circular motion. We played broom ball in the hallway with a simple task - get good at making the medicine ball go around in a circle using only the broom as the source of force.

We then came in and tried to figure out what was going on. I took pictures of all of their diagrams showing velocity and the applied force to the ball.

It was really interesting to see how they talked to each other about their diagrams. I think they were pretty close to reality too, particularly since the 4 kilogram medicine ball really didn't have enough momentum to make it very far (even on a smooth marble floor) without needing a bit of a tangential force to keep its speed constant. They were pretty much agreed on the fact that velocity was tangent and net force was at least pointed into the circle. To what extent it was pointed in, there wasn't a consensus. So Weinberg thinks he's all smart, and throws up the Geogebra sketch he put together for this very purpose:

All I did was put together the same diagram that is generally in textbooks for deriving the characteristics of centripetal acceleration. We weren't going to go through the steps - I just wanted them to see a quick little demo of how as point C was brought closer to B, that the change in velocity approached the radial direction. Just to see it. Suddenly the students were all messed up. Direction of change of velocity? Why is there a direction for change in velocity? We eventually settled on doing some vector diagrams to show why this is, but it certainly took me down a notch. If these students had trouble with this diagram, what were the students who I showed this diagram and did the full derivation in previous years thinking?

Patience and trust - I appreciate that they didn't jump out the windows to escape the madness.

_______________________________________________________

All in all, some good things happening in the math tower. Definitely enjoying the experimentation and movement AWAY from lecturing and using the I do, we do, you do model, but there are going to be days when you try something and it bombs. Pick up the pieces, remind the students you appreciate their patience, and be ready to try again the next day.

## Exploring Point Slope Form through Geogebra

In geometry we are studying parallel and perpendicular lines and the theorems that can be proven about them. In thinking about how to present the connection between algebra and geometry for this unit, I wanted to include an exploration of what exactly makes lines (and linear functions) so special: constant rate of change, or slope.

We did not get a chance to do this entire exploration in class, but I am expecting students to look at it at some point over the weekend. I know they have seen slope-intercept form, but point-slope form is convenient in many ways, especially in the way it applies the concept of slope between points located on a line.