Monthly Archives: October 2013

[Explore the MTBoS] Mission #3: SBG Makes Grading Fun & Math Mistakes

Standards Based Grading (SBG) is a different approach to how grades work at a fundamental level that will change your classroom.  To get some real background on this, read Shawn Cornally’s stuff here.

I have absolutely loved SBG so far this year. I’ve got a few posts on it, if you care to see exactly what I’m doing, but that’s not the purpose of this posts.

Something you should know about me is I get bored and distracted easily. Even before I became a teacher, I knew that motivating myself to grade would be a struggle, and early on I recognized the importance of getting grades back quickly to students. I used to be very bored while grading, checking off boxes brainlessly like a computer.  Even my open-ended problems could have been graded by a computer, and I often didn’t pay attention to how well different students did, as they all blended together when marking problems right or wrong.

Now, because of SBG, I look at their quizzes looking for understandingnot ticking off boxes saying whether their answer matches my key.  This makes me much more interested in what they’re doing, which helps motivate me to grade.

One website that is now essential for me to study and use is Math Mistakes.  It’s a great website that Michael (@mpershan) runs by getting teachers to submit student mistakes and then others comment on those mistakes and discuss what the student understands and doesn’t understand.  Many teacher contribute both student mistakes and comments on those mistakes.

It’s something that I should be visiting more often than I am, and a must for any math teacher. I hope to motivate myself to visit more often so that I can continue building my capacity to understand student mistakes and the misunderstandings behind those mistakes.


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SBG and Parent-Teacher Conferences

Although it took a while to explain my SBG scheme [1] in addition to my Participation Points setup [2], parents seemed to be really appreciative of this grading system.

Two parents of different students asked some very specific questions about SBG, which made me glad that I had reflected on these issues (through blogging) before the year started.  Here are some of their questions, which I believe I answered satisfactorily.

  1. Where did these standards come from? I created them based on the materials left behind from the previous teacher.  I also talk with the other math teacher (who teaches the subject before and after mine) to figure out which topics I should focus on more and which I can leave behind to go deeper with the important topics.  Unfortunately there’s no Common Core [3] for Precalculus content, so I’m left to figure out what’s important for students as they move on to Chemistry.
  2. Are 4’s indicators of perfection or understanding? It depends on the topic, but overall I strive for understanding rather than perfection. In the case of something like multiplication tables, demonstrating understanding is perfection, and I have my standards labeled “F” for “Foundational” and “C” for “Conceptual”.  I’m much stricter when it comes to “perfection” on the foundational standards (such as memorizing their multiplication tables) because not having those down really well can slow students down later in the class on the harder topics.
  3. Do you average all of their grades when they retake quizzes, or do you just take the last quiz grade? I actually do neither: I average the last two grades the student has received, and there are a few reasons for this.  First, I think that students should know and retain knowledge to demonstrate an understanding of a topic.  This discourages cramming and emphasizes/rewards practice in small chunks. Secondly, I also believe that one bad day shouldn’t tank a students’ grades, so I want the students to be rewarded for doing well early on in the class, but I also don’t want bad grades to hurt them at the end of the year if they are willing to put in the time to really learn the material.  Averaging the last two grades finds this balance and I believe, so far, has worked very well.
  4. What are the translated percentages for a 1, 2, 3, and 4? It depends on the class.  For your student’s Chemistry class, virtually every student in our school is required to take this class, so I believe in making it possible for every student to achieve a passable grade.  Not every student will pass with an A, but if they work hard, I think every student should pass.  Because of that, a 3 in that class is approximately an 80%, whereas in Precalculus, a 3 is 75%. In Precalculus, all the students are high-achievers that, perhaps, haven’t been too challenged in math class yet, so I’m demanding that they really understand this material in order to get an A. To do that, I’ve dropped a 4 down to a 98%, which requires them to get many more 4’s than 3’s in order to get a perfect A.  This forces the students to review topics they didn’t do really well in, which is great because they have the ability and mindset to do that and get really good at math. [4]
  5. How does this compare to the letter grades or a student having a 82% in a class, where they would know that they have to work harder? I believe that this system is better than just giving a blanket grade because it shows students not just how much they have to work on, but what they have to work on. If I was allowed, I wouldn’t even change this into an A, B, C letter grade because this information is so much more specific and says exactly how well they’re doing in a class. However, because I’m required to change this into a letter grade, I use the certain percentages to push students to their limits in this class and really help them grow.

Here are some of the comments I heard from parents about both of these systems. (All paraphrased because I can’t remember exactly what every parent said.)

I really like this system, I think it is designed to help the students out.

I wish there was something like this when I came through and took math and science.

You really have this setup to encourage the students to do well, this is great for all the students.

Needless to say, I was very happy to hear these comments about my grading system, and I hope that this is, ultimately, for the benefit of the students.  One of the things that I’ve come to realize is that I need to give more opportunities for self-practice and self-assessment in Physics.  In that class I just assume that if students do their HW, then they’ll understand the material, but some of the students are so low that they really need more individual help and more resources than I’ve given them.  (No, I haven’t handed out textbooks, and one parent thought that her son was lying to her when he told her that!) That’s something that will certainly come with experience and years teaching a subject, but hopefully I can work on providing that for this year’s class.

[1] Quick rundown: Teacher-initiated assessments (projects, quizzes, or labs), followed by student-initiated re-assessments (if necessary).

[2] Quick rundown: Students choose how to earn 30% of their grade.  Options includes homework, working on the warm-up in a timely manner, practice problems, difficult challenges, blog reflections, speaking up in class, coming to tutoring, and many others.

[3] This question & discussion occurred with a parent who happened to be a middle school teacher at our school, so she knew all about the Common Core.

[4] Part of me really wanted to tell this parent that “Percentages don’t matter!! What matters is how much your child understands, so you and they should focus on that, not on whether you have an A, B, or C!!”, but I also wanted to defend this system, so I was kinda shamefully excited to explain that.

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[Explore the MTBoS] Teaching Within a Culture

The Mission

The MTBoS Challenge for week 2 involves tweeting with some people you normally wouldn’t tweet with and then blogging about it. I saw Jason Buell (@jybuell) tweeting about Decolonization, and so I asked him about it to learn more and get his thoughts.

Decolonization: Brief Overview

The way I understand it, Decolonization is the idea that “settlers” and “colonists” who have moved in on another culture’s territory, should either remove themselves or proceed with every care to preserve the culture into which they’ve moved.  Specifically with respect to education, this emphasizes that your priorities, or the “white man’s” priorities, should not supersede the indigenous people’s priorities when it comes to educational practices, e.g. curriculum and required content. To put that in simpler terms, just because I think math is important does not mean I should “run over” another culture and teach math, or all the same parts of math, to that culture.

My Particular Situation: My School

This is a particularly interesting and important discussion to me because I teach in the midst of a Native American reservation, and so about 70% of my school’s population is Native American, primarily Navajo, but that figure also includes some Zuni Pueblo.

To complicate matters, I teach at a Christian School, which is an old school (we’re celebrating our 111th anniversary this year), and a boarding school on top of that.  The school has a very shameful and terrible past which includes things like teachers beating students for speaking Navajo instead of English, even outside of class. To give you an idea of the philosophy early in the school’s history, they used to follow a saying that went like this: “Beat the native out of the man and leave the Christian behind” or something like that.

Fortunately not only has our school turned from it’s past, but it has also repented and begun work on undoing the damage it did throughout it’s long history.  For example, we now teach Navajo, a dying language despite the fact that the Navajo population is growing, because we believe that a culture’s language is central to the preservation of that culture.  Another example is the school, just a decade ago, paid for a full-page advertisement in the local newspaper where they apologized for their past: even though all the people who perpetrated those crimes have long since passed away.

Where I Stand

So this school has come a long way, and yet has a lot of work to do yet. But I am not sure I totally agree with the philosophy behind “Decolonization” that I stated above.  Jason gave me some great links to follow to read more about it.  One of the links took me some reports and data which discussed “Redefining how success is measured in Aboriginal learning.”  The paragraph explains as follows:

Increasingly, Aboriginal communities are administering educational programs and services formerly delivered by non-Aboriginal governments. They are developing culturally relevant curricula and community-based language and culture programs, and creating their own educational institutions.

Yet as Aboriginal people work to improve community wellbeing through lifelong learning, they recognize the need to identify appropriate measurement tools that will help them assess what is working and what is not.

So I read that and I think they could mean two very different things.  What I hope they don’t mean:

Indigenous people do not excel in subjects such as math or English, so we should define what it means to be successful as an indigenous person, and make easier standards so we can assess students on those instead.

What I hope they do mean:

The current assessments do not take into account the traditions and history of indigenous cultures, and so, because we find that extremely important, would like to include that in assessments. Furthermore, the assessments have become culturally biased to the point that there is a distinct advantage of growing up in one culture over another on these assessments: they should be “normalized” or generalized so that specific cultural knowledge is not a requirement or boon for one culture over another.

And yet, even as I type that, I am worried that I am writing from too biased a perspective.  Who am I to decide that math is important for any young child?  I think that it is important, but is that too culturally specific?

At some point, I find myself putting my foot down and saying that some things transcend culture. Better healthcare, basic human rights, and a better understanding of logic are some of those things.

In the Navajo culture, there is a strong suspicious belief that “whoever tells you bad news is actually the cause of that bad news”, especially when it comes to babies.  Think about the implications of that for a second.  Mothers give birth to severely disabled babies and are totally unprepared for the consequences because doctors and nurses did not want to be the one who is seen as “responsible for the birth defect”, when a little knowledge could go a long way.

Another aspect of the Navajo culture is the stigma around death.  If someone dies in a house, you’re supposed to abandon the house and knock out a wall (I believe the North wall) to let the spirit out. I’ve talk to a doctor who has visited many very poor families, who only have one Hogan (house).  The doctor will “checks the pulse” of the person on their deathbed and waits to tell the family the person is dead until they’ve moved the body out of the house, just so the family will be allowed to keep their house.

The Navajo culture has some great and beautiful aspects, but a large part of the culture is centered around fear: from evil spirits to skin-walkers to superstitions, there is a lot to be afraid of in the culture.  And I believe it is the responsibility of people inside the culture to identify which parts of the culture are important and beautiful and which are damaging and unhealthy.

Wrap It Up, Will Ya?

So how should a belegana like me approach and interact with the culture and people of the culture? Form relationships.  Get to know people. Be respectful, of the people and of the customs. “What to teach” is actually not the first thing that comes to mind when I think of interacting with my students, some of whom have incredible needs (perhaps I should think of it more, so it’s good for me to reflect like this).

Instead, my thoughts are on the student who recently lost her only parent in a car wreck. Or the student who goes home to an alcoholic and abusive father. Or the student who is going to be a father, but couldn’t pass his classes before he had a child, so how is he going to keep up and learn anything now?

Perhaps I’m side-stepping the issue by focusing on individuals and not the culture, but I do think that anyone who decides to “redefine success” should be careful in how they do that.  Do not reduce rigor at the expense of a generation of students. (Is that the “colonist” in me speaking??)  Instead, identify what is important across cultures and keep that while changing the specifics of that rigor for various cultures.

Steam-rolling another culture is not the way to go, but neither is isolation.  Diversity is a beautiful thing when brought together, but you need differences for there to be diversity.


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Stop-Motion Parametrics: Post-lesson Analysis

I posted recently about a lesson where students created stop-motion videos to model parametric functions.  Last year I did a similar lesson, but this year it went so much better because the activity was more structured, and I had them create presentations (on Google Drive so they could embed videos) rather than just creating a video. Here are just a few of the videos (in case they don’t work in the embedded presentations).

And here are some of their presentations, with names edited out.

Thanks to the questions that students had to create as a part of their presentations, the discussions that occurred around these questions were very good: students brought up misconceptions, and other students helped them out.  I didn’t always stop them at every little thing that was wrong, but after everyone’s presentations, I think we covered all the topics where students got something wrong.

Because this (students presenting and then asking questions) was a new way of learning (at least in this class), I had to push them to ask their peers the questions, and then wait for a good discussion to evolve.  We had good discussions and some of them posed great questions, but it took too much of me asking them for it.  I hope to do something like this again, soon, so they can get accustomed to teaching each other.

It also took a full 3 days of class to get through all the presentations (only 8 presentations), but I think it was worth it with the great discussions that were taking place.

Part of the reason students payed close attention was the fact that they had to answer their peer’s questions during the presentation.  The other part is that I had them fill out peer evaluations on Google Forms, which I hope to use and share with them. Here are just some of the responses (spelling errors included).

Answers to the request for: “One thing this group did well.”

they knew how their function worked pretty well.

What they said made sense and they had a good presentation.

the video was cool

explanation of domain and range

Understood her project and graph really well.

She did very well in finding a very unique parametric equation.

I really liked their video. They did a very good job in thinking about their questions. The questions are very well thought.

They really explained what the independent and dependent variables were. They also explained domain and range.

she knew allot about how her equation worked and knew how she got it, [1]

Answers to the request for: “One thing this group could work on or do better.”

They confused me when when finding the distance horizontally. They didn’t find the distance instead they found the displacement.

Be more ready for their question maybe go over them more before they presented so they could catch their errors before their presentation.

they were a little bit scared which made them uncertain of the equation

I think that they had okay questions but i felt that there could have been a few better questions.

They did not explain there X&Y values.

Answers to the request for: “One thing you liked about the presentation.”

I liked the fact that they went BACK IN TIME!!! WITHOUT HAVING TO GO 88MPH!!!

It was smart to have the equation before the questions because it was then easier to answer them.

She thoroughly explained her equation and its domains and ranges.

I liked the way they explained their answers so that i understood.

they kept the presentation going, no awkward pauses.

i really like the graph it was creative.

cool video


This kind of reflection is great, both for students to see their own pro’s and con’s, and to think about other groups as they presented.  If students are “just supposed to watch” a presentation, then there’s much less incentive for them to pay attention to the details.

[1] That’s a new way to spell it that I haven’t seen from a student.


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Student’s Reflection on the 3D Orbitals Lesson

Students can earn “Participation Points” in my class by reflecting on their class through blogging.  Here’s what one student blogged concerning the lesson I posted the other day.

Today’s fourth hour chemistry class was very helpful for me to understand the lesson that was given about electrons and their orbital “clouds.” The work that was assigned for the class was helpful for me since we were allowed to work with the crome books. It also  challenged  my lab partner and I to think about the information given to us in order to draw a conclusion. We worked with a unique 3-demensional feature that allowed us to observe different orbitals at different energy levels and orientations. We also connected the lesson to the periodic table and how what we learned affected how it was put together. At the end of the lesson, we were given an activity that challenged us to help us locate an electrons orbital cloud, energy level and orientation. Overall, I enjoyed the lesson since it challenged me to think further and draw conclusions about the gathered information. I honestly think that more lessons should be taught throught this teaching method since its allows students to challenge themselves to think further about the information.

Needless to say, I was proud of the lesson, of the student’s desire to be challenged, and of the student’s reflection!  Here’s to hoping I can put quality time into my lesson planning to create more lessons like this in the future!

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Using Sketchfab to view 3D Orbital Clouds

Go here to view the orbital clouds that I describe below.


I didn’t really learn about Blender until about this time last year, and so I was still very new at creating 3D objects.  Since then, I’ve been able to apply my hobby of creating 3D objects to my math and science classes just a handful of times.  The two ways in which I’ve been able to share my work with students is through an Augmented Reality App on the iPad and through Sketchfab, a website that will use OpenGL to render 3D images.  Both are very cool ways to view 3D objects, with the Augmented reality app being a little more hands-on for the students with a little more “wow” factor, and SketchFab being a little easier for me to upload my blender files and for students to access (doesn’t work as well on iPads because of Apple’s restrictions, but we got Chromebooks this year from grant, wohoo!).

The highlights have been:

Using 3D & Sketchfab

I used to stand up at the board and lecture on the Quantum Numbers.  I’d draw the 3D objects which wasn’t terrible, but definitely didn’t help students picture things in 3D. [1] It was teacher-centered, lecture-based, and (no matter how excited I made it sound) boring.

So I decided to spice things up with Augmented reality.  Unfortunately our network was being stupid and blocking everything related to the app, so I had to change gears and go with Sketchfab.  I created several of the orbital shapes, orientations at multiple energy levels.

After uploading them, I embeded them on my website.  Unfortunately you can’t embed iframes into websites, so you’ll have to go to my website to check them out.  There are 5 pages of them, so don’t miss the other ones.

I gave my students a packet to work through (see below), which worked better than I had hoped!  It involved students sketching various pictures, in addition to answer questions at checkpoints and requiring that they check with me before moving on.  I was worried I’d be swamped, but the checkpoints are so easy to glance and say “yes” or “no”, that there wasn’t a big backlog.  Students developed a sense of what the atom looks like, in addition to how electrons behave within the atom.  The activity was student-centered and hands-on: much better than how I used to teach it. I overhead students struggling with and debating on the problems, asking each other what an “energy level” was and what “orientation” means.  At one point students even sketched what they thought three different shapes, when put together, should look like in 3D, and could immediately check themselves by going to the next web-page.

I could definitely improve it in little ways, such as explaining that an orbital cloud at a given orientation means both sides of the p-orbital.  Or I could better explain what an energy level is.  And I didn’t like how I phrase the question where they’re supposed to “discover” where the numbers 2, 6, 10, and 14 are on the periodic table.  But those were minor hitches that went over fine because I was constantly walking around answering questions.

The following day, we even had a great discussion from some of the students about a way to provide a “house address” for the electrons.  A few students presented how they would give students a location, and I followed up with showing students the quantum numbers.  Even when I was lecturing (for only 10 minutes or so), students were much more engaged and invested in what I was saying because they had, in the back of their mind, the system they created to locate the electrons.

[1] How important is it, in the grand scheme of Chemistry to know what the electron orbitals look like? Meh. You’d be able to understand most of Chemistry without it, but I like to do it because (1) it helps students with their spatial understanding (which is often sorely lacking), (2) it’s quite beautiful the way that these orbital clouds exist in every atom, and (3) it reinforces ideas about electrons and “where they want to be” as I put it.  This idea of “where electrons want to be” comes up more in chemical bonding.

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My Attempt at Modeling in Chemistry


For the past two years I’ve tried (and mostly failed) Modeling in Physics.  I didn’t take the class/workshop over the summer.  I was cheap and haven’t even become a member of the AMTA.  (I know, I know: I should do that.)  Finally, I came upon the awesome Kelly O’Shea and her awesome blog, in which she basically explains what she does, for some lessons word-for-word, on top of providing all of her packets of work, which is awesome.  Finally I feel like my students are learning Physics in this totally new, intuitive, cool way.  They’re solving problems by drawing graphs rather than crunching numbers. But I’ve lectured in Chemistry for the past 2 years cause I think I know what I’m talking about in that subject.  I’ve heard of modeling in Chemistry, but I haven’t had the time (to take the workshops) or money (to join the AMTA and get materials).  Now that I’m finally feeling a little good about it in Physics, I think I can picture how it should happen in Chemistry, and so one day I decided to try out what was in my head. These are Junior in Chemistry, and all we’ve done so far is background work: significant figures, scientific notation, SI units of measurement and conversions.

Part 1: Dalton’s Atomic Theory

Students broke into groups of 4, acting as scientists from previous centuries, and received a large whiteboard.  On the whiteboard, they created three columns: “What we learned”, “Our group’s 5 rules about atoms”, and “The class’s 5 rules about atoms”.  I explained the following points (putting most of these on the board):

  • At this time, nobody had heard of an “Atom”.  Your job is to come up with 5 rules that you think might be true about atoms.
  • People knew about certain elements and how they seemed to always react in certain ways.
  • For example: ~16 g of Oxygen and ~2 g of Hydrogen would react to form ~18 g of water, but if you had more Oxygen or more Hydrogen (not both) then you would have leftover of that.
  • About 8 g of Oxygen and ~1 g of Hydrogen would also react to form ~9 g of water, but again, too much of either would leave you with extra of that substance.
  • Other elements reacted in similar ways with certain amounts, where it was possible to have “too much” of one or the other.

I left students with that information and they were required to come to their own conclusions.  Students started off terribly confused by what they should do, so I shuffled around the room and asked provocative questions: “How large are these so-called atoms?  Be specific!”, “What are those numbers on the board and why might they be important?”, and “What do you think about the chemical and physical properties of atoms?”  These questions led different groups to come up with different rules, while led to a good discussion, in which the class had to decide on 5 rules.  I was impressed that both classes ended on 5 points that were comparable to Dalton’s.

Part 2: Thompson’s Cathode-Ray Tube Experiment

Here I explained Thompson’s Cathode-Ray Tube experiment, highlighting the following points:

  • Clearly something was moving through the vacuum, but it wasn’t entire atoms.
  • The thing that moved through the vacuum was attracted to a positive magnet and repelled by a negative magnet.

I asked them to make 3 columns again in groups of 4 with the following headers: “What we already knew (Dalton’s/the Class’s 5 rules)”, “What can we learn from the Cathode-Ray Tube Experiment” and “What our picture of the atom looks like now”.  After some very leading questions, groups moved in the right direction, and I made sure that at least one group in each class had an atom somewhat resembling the Plum-Pudding model before moving into the group discussion. The group discussions were a little better this time around, partly because they had done it once now and party because they had drawings to fall back on.  It was still kinda like pulling teeth, but both classes at least saw something like the Plum-Pudding model before either accepting or rejecting it. I followed up on this discussion by giving Thompson’s own model (the students really wanted to see it at this point!), which maybe I should not have done?  I want them to think that their own “Discoveries” are valuable, and those are undermined when I show them “the answer” (or what they perceive in their minds to be “the answer”).

Part 3: Rutherford’s Gold Foil Experiment

This discussion occurred on a different day, and so I was a little more prepared, but it was still foreign and difficult for them.  I introduced the experiment and asked them to make a prediction, based on our previous model of the atom, sketching where they thought the alpha particles would land. After this prediction, I highlighted the following points of the experiment:

  • The alpha particles we were bombarding them with were essentially positively charged Helium atoms.
  • Sometimes they bounced back at an odd angle, but rarely: they deflected backwards only about 1 out of every 8000 times.

Again, with some pushed upon individual groups, students came to the conclusion that there must be a small, dense, positively charged center of the atom. We decided to randomly call it the “nucleus” and I even managed to convince some of them that it was reasonable to believe that the electrons are orbiting this nucleus. [2]

Pros and Cons

Things I liked about this activity and format:

  • Very different from lecture format: students struggling with information and coming to conclusions on their own.  I know students need when I hear them say “no, we just want you to tell us how things work.”
  • More students, through the struggle, understand why we know certain things about the atom.
  • I just found a cool article about how it’s better for students to argue in science labs than for them to mundanely accept what the teacher says, or worse: finagle the data to avoid an incorrect hypothesis. More arguing = better formed foundation and understanding.

Things I didn’t like, or need to improve on:

  • Many students sitting out, both when working in groups of 4 and especially in the class discussions.
  • Very time-consuming. We took about 3-4 days in what I could have lecture on in less than an hour. But would they have gotten as much out of the 1 hour lecture?  I’m not sure, but there has to be some balance.
  • As we went along, I showed them what the conclusions of the scientists of the day was, and we discussed how it compared to our conclusions.  Should I have done this, possibly undermining their own confidence in coming to scientific conclusions?
  • Students didn’t seem to jump into the activity on their own.  I was planning on circulating the room anyway, but it seemed like it was necessary for me to be around a group to “keep them going”, so to speak, which isn’t the case in our hand-on labs or even when I give them time to do their (ungraded) homework!
  • This may be because I didn’t have good entry material, and I could have better thought out the background information to lower the entry-level.  As it was, it took a good bit of creativity and thinking to even start thinking about the experiments, and so I’m worried I loss a large percentage of the class before the activity started.

What are your thoughts on it all?  Help me out if you’ve ever done something like this or have other helpful ideas!

Here are some pictures of the students’ boards after part 2: Thompson’s Cathode-Ray Tube experiment, so you can see where the different groups took the information (with some heavy questioning and prompting on my part).


This group liked the smiley-face model of the atom.


This group jumped to the conclusion that we knew what protons are a bit early. Because of that, I continually asked them “and how do you/we know THAT?”.


This group had fascinating theory that all mass is positively charged and electrons are mass-less particles. Definitely one of the more creative approaches to solve the problem.


[1] Well, okay, so I have enough money but I’m too cheap, again.

[2] Soon I’ll explain to them that this is a lie. 🙂


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[Explore the MTBoS] Mission #1: Stop-Motion Video and Participation Points

Note: I just finished typing this blog post and re-read the prompt carefully and realized that Sam asked for one of these two prompts.  Oops.  Sorry if you’re reading this because my comment is above yours: don’t feel like you need to read both sections if you don’t want to. 🙂

Favorite Open-Ended/Rich Problems: The Stop-Motion Parametric Video

One of the topics in Precalculus that we cover is Parametric equations. One day I was watching a stop-motion video, and I had the idea to get my students to do stop-motion videos showing a parametric equation.  Here are some example videos from last year:

(The third one is my favorite…)

The students were asked to create a Parametric equation and then use stop-motion to illustrate it. I didn’t care which came first: the video or the equation, as long as they matched each other.  We used iPads and a stop-motion app, which takes pictures and puts them together to make a video for you.  However, you don’t need access to iPads because there are plenty of websites that can take your pictures and create a stop-motion video out of them, so all you’d need is one camera per group and access to a computer or computer lab for each group.

Last year that’s all I had them do: make a video and tell me what the associate equation is. This year what I want them to think about it a little more, so I’m going to have them create a Google power-point which includes various thing, such as a graph, table, equation of the situation, explanation of the variables, and a few challenging questions for their classmates.  Here’s my example Google Presentation that you can check out (yes, I used one of my student’s videos from last year as my example, but don’t worry: I’ll give them credit!).


One Thing That Makes My Classroom Distinctly Mine: Participation Points

Something I call Participation Points.  I am the only one at my school who uses this system, and I’ve yet to convince any other teachers at my school to use them, but I may have some converts in the MTBoS–we’ll see.

PPs (Participation Points) are something that students must keep track of and they must earn 100 each week. My premise:

  • Students should often be able to choose what level and type of work they complete to “earn” their grade.
  • When allowed to choose, students will typically choose work that challenges them “just enough”.
  • Homework should be option and seen as “practice if needed” (and usually it should be needed).
  • I should have the flexibility to create opportunities to learn on the spot, and have it count toward some part of their grade even if I do not have a standard for exactly what they want to do (I use SBG).

PPs are 30% of their grade, while their Standards make up the other 70%. Students hold onto a grid which documents how many points they earn each week for 9 weeks.  At the end of each class, students come up and tell me what kind of points they’ve earned.  On Friday I collect their sheets and put their grade out of 100 points into the computer, and return the sheet to them on Monday.  Here’s what their sheets look like:

Some of the things students can earn points for include:

  • Speaking up in class (making a positive contribution).
  • Reflecting about class through their blog.
  • Working on exercises on Khan Academy.
  • Working on the Warm-up on time.
  • Putting your answers to the HW on the board for others to check.
  • Answering a handful of thought-provoking questions I have on my website.
  • Coming to Tutoring (at lunch on Tuesdays and Thursdays).
  • Signing up to be a Tutor at those times.

There are many others: I have an incomplete (but more comprehensive) list on my website here.

The tutoring, especially, has helped students grow in the areas they need to grow.  At my school I am currently the only teacher to have students come regularly to tutoring, and there will be 30+ students in my classroom at lunch on those days working because they know it’s good for their grade in multiple ways.

I’ve blogged before about Participation Points, but they are something that I’ll probably do for as long as I teach because of all the little ways it helps my classes. I don’t do PPs in all my classes, but there is a big difference in the attitude of the students toward learning and work in those classes that I do use PPs. A small example of this is in the warm-up: I’ve got a facebook-style Like-Stamp that I use, which gives students 5 PPs, and I’ll stamp students’ papers who are working on the Bell-Ringer (warm-up) before the bell rings.  Students know this so they come quickly into the room and get to work right away, begging for me to come look at what they’ve done so they can earn that stamp.

Please ask if you have more questions about PPs, as I’m eager for another teacher to try them out and give feedback on them! They’re up there with SBG (Standards Based Grading) for “Most-Impact-On-My-Classroom” ideas.


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Entire Lesson Videotaped: Intro to Parametrics

On a whim, I decided to videotape an entire 45 minute lesson in one of my classes. The only thing I had on hand was an iPad (2nd Gen), so the video quality is low, but I’m also kinda banking on that so I don’t have to blur out students’ faces.

Other information possibly of importance:

  1. There are 15 students in this Precalculus class.
  2. This class is the last period of the day. [1]
  3. Yes, that is my principal who strolls in at [29:43], and yes he picks up a student’s guitar and starts playing in the last 10 minutes of class.   (Oh, and it was his birthday.) Don’t you wish you had a principal as cool as mine?

Outline of the Video

[0:00] to [3:46] Waiting for class to start. (I should have edited it out, but this is why the video starts at [3:46].

[3:46] to [5:40] Waiting for student let out of choir to get to my room (they’re late most of the time because our choir teachers let them out late).  Because I have all these students for Chemistry, we discuss a little Chemistry while waiting for everyone else to get here.

[5:40] to [8:36] I show students how they can find all the standards from the class on my website.  We also discuss Inverses of functions and they convince me to give the quiz on Tuesday instead of Monday of next week.

[8:36] to [12:30] 1st Act: I develop a reason for Parametric Equations and we do a really rough experiment of a student walking into the room.

[12:30] to [31:47] 2nd Act: Through questions I build an intuition for Parametric Equations through graphs and the measurement of our “experiment”.

[31:47] to [40:04] 3rd Act: We answer the questions I provided for them at the start of the lesson (although I didn’t vocalize them, and I didn’t have a “hook” for them like your typical good 3 Acts lesson. We look at two ways to use technology to graph this equation and I give them two of these types problems for homework (see the sheet below). I use the TI Calculator because they’ll need to know how to use that for most standardized tests. I use Desmos because it is awesome and easy to use (and way cooler than the TI Calculators).

[40:04] to [50:03] I promised them from the day that I would do the Fibonacci Magic trick.  If you haven’t seen it, here’s a video of it being done, long with an explanation of how to do it (however, I don’t like his explanation of how to multiply a 3-digit number by 11).  If you prefer to read, here’s an good explanation of how to do it.

Supplemental Materials

Since the board is hard to see in the video, here are two pictures of what I put on there.  For the most part, the black is what I had up before class started and green is what I added during class.

Picture of Classroom

Four Representations of a Parametric Equation

Here’s the worksheet I gave for homework.  Notice how I set up the board so all 4 parts match the worksheet.

Please leave feedback on my lesson and on my teaching style: both constructive and destructive comments are welcome, so please let me know what you think!


[1] The students recognize their tiredness at this point and frequently complain when I ask them to do a particularly mentally challenging problem or task.  I suppose I am fortunate that they recognize this, though I wish they had a bit more motivation and didn’t use it quite so often as an excuse.  Fortunately these are all good kids who try despite their tiredness, though they have vocalized to me that they wish this class was taught earlier in the day.

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