Monthly Archives: April 2013

Polar Coordinate Introduction: Battleships!

I was a bit distracted during the sermon in church today and started thinking about polar coordinates and how I was going to teach them this year.  I mostly lectured on this topic last year (as I did for most topics), but I had a flash of insight to play battleship with my students to help build their intuition.  We played battleships to introduce Cartesian coordinates when I taught 6th grade, and it was far and away my most successful lesson that year.

The Setup & Plan

You are the Commander of a small battleship (I suppose they’re called destroyers?) which has only a single gun that rotates on a pivot.  You receive information from your Radarman (do they have a special name?) who operates the radar and have to let the Gunner know where to shoot.  The Radarman sees his screen and reports the location of enemy ships in Cartesian coordinates (yes, I know that radar actually works essentially using polar coordinates, but hopefully my students will breeze by this detail and we can discuss that later).  However, the Gunner just needs to know the angle (looking from above) to rotate the gun, and how far to shoot.  Yay, we’ve found a situation, a game at that, where we need to move from (x,y) to (r,theta).

In addition to this, I did 2-3 situations with them that were really simple, and then we launched into a game of half the class vs the other half.

Game Setup for Students

I had the students sketch 10 x 10 boards (using Cartesian coordinates) onto a 3′ x 2′ whiteboard, so the x and y values went from -5 to 5.  (See below for one class’s placement of their ships).

The students then placed the following 4 ships onto the board.  Unlike traditional battleship, the center of the ship had to land on the Cartesian coordinates.  So I sketched the following diagram for them.

Photo 2013-04-22 09.33.40 PM

The students then sketched their ships onto the whiteboards, after which I promptly took a picture with my iPad so I could ensure that there was no cheating.

Photo 2013-04-22 09.33.37 PM

Photo 2013-04-22 09.33.42 PM

Results and Student Interaction

The students really took to the game.  One team even built a “fort” in the classroom so the other team couldn’t see their board!

Photo 2013-04-22 09.33.45 PM

So students had their own boards in Cartesian coordinates, but would “fire” on each other only using polar coordinates.  They were then responsible for saying whether their opponents had “hit” or “missed” their ship.  If they were wrong, then their opponents got another shot, so accuracy was critical.

The teams took turns, and we actually had about 10 volleys on each side with no hits!  I was wondering about the probability of that happening, with both teams [1] when I realized that could be our warm-up tomorrow.  So I had them lay the boards to the side and we’ll continue the game tomorrow.


The thing I liked best about this game was how much it had students converting from Cartesian to polar and back, and them trying to do it quickly, yet accurately!  And to think they’ve done all this before I gave them the equations (actually, I gave them some of the equations, but they all forgot them in the face of a game, and resorted to problem-solving–yay!).

[1] Let’s see, 14 “hit-able ship spots” out of 100 possible points.  The probability of a miss is 86/100, so to do that 10 times (about) would be (86/100)^10 or about 22.1%.  However, neither team hit, so squaring that would be about 4.9%. Hmm, much less likely and interesting!

Edit: Ooh, as I was showing my students this probability, I realized this is wrong because the smart player wouldn’t shoot in the same place twice!  Also, from -5 to 5 includes 11 points, not 10 (fence post problem/gotta include 0!).  So the right math would start like this:

\frac{107}{121} \times \frac{106}{120} \times ...

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[SBG] Looking out for Creativity

My father sent me an article [1] from Scientific American Minds about innovation, and came across an interesting report.  Because I don’t know about copyright laws when it comes to unpublished articles, I won’t attach it to my blog, but you can look it up using the info below.  The article argued that while people in leadership positions (like teachers) encourage creativity verbally, they are actually subconsciously adverse to the impracticality that often  is associated with that creativity.

How should SBG interact with (look out for) creativity?  Should it be its own standard?  Are math teachers responsible for instilling and encouraging creativity?  I would say “heck yeah”, but to what extent should we grade our students on that?

I’m toying with the idea of just having a standard “creativity”, just to help me keep looking out for that and rewarding that.  But is that too broad a category?  Could I feel confident that I distinguished between mathematical creativity and lunacy/randomness?


[1] I’ve forgotten how to cite and don’t feel like looking it back up (one of the reasons I’m a math teacher, not an English teacher), but I’ll provide all the necessary info for this article if you desire to seek it out:

Mueller, J.; Melwani, S.; Goncalo, J. The Bias Against Creativity: Why People Desire But Reject Creative Ideas Psychological Science. Sage Publications. (Final version forthcoming–in press)

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[SBG] Student Grades & Karate Belts

I was brainstorming about my transition to SBG next year, and I happened to also be thinking about how I reeeeally want to use the ninja board, and so (naturally) I started thinking about karate belts.

What if grading worked like these karate belts? The article I just linked had some interesting points–I especially liked the Pros and Cons, as they are something for teaching to think about, regardless of the way grades are done. Here’s a quote from that article:

The awarding of Karate Belts should be a fantastic time for both the examiner and the student. When the result of long diligent practice of Karate technique is displayed by a student at a serious Karate Grading examination, and when the performance is assessed by a competent, responsible examiner – everyone wins. Any other way and Karate loses.


I also found it interesting how apparently there are only so many belts outside of Japan–the Japanese actually limit themselves to just three colors.  It’s almost as if they’ve tried out different things and know what works. Hmm.

Lastly, I found it somewhat comedic to search online and see how different dojos advanced their students, and I found some mothers complaining about how “they were paying good money but such-and-such dojo only advanced their child slowly” while other dojos advanced students on a quarterly basis–everyone together so they’re always with the same students who entered with them.  Funny to see parents complaining about that kind of thing.

Glad to know we’re not alone.


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Socrative to the Rescue?

I’m terrible at printing tests and quizzes before they’re due, and one day I’ll learn my lesson.  I thought that would be today because the printer jammed on me 4 times in a row.  I only had 45 minutes until the start of the day, and I was giving a test AND a quiz in two different classes.

All week, students had been using Socrative for the iPad, and so I thought “hey, I could probably copy and paste all the questions from Word to Socrative in time.”  So I did.  About 40 questions later, I was ready to go, and my thought was “whew, Socrative saved me!”, but I hadn’t quite thought through all the difficulties taking a quiz or a test like this might pose.


Saves paper (trees?).  Students still use scrap paper to show work, but I didn’t run 80 copies to be handed out.

All my math assessments are open-ended responses (not multiple choice).  Socrative has an open-ended (short answer) response option so it was actually pretty fast to transfer the quiz questions.

As students are working, they can’t cheat off each other because they’re only working on one question at a time.  Some students tried, but quickly gave up when their peers outpaced them because they hadn’t studied.

The Excel sheet it outputs gives me a nice overview of where the class stands.  Even the open-ended response, which I have to grade myself, I can use background colors on excel to mark wrong answers and see where everyone is at a glance.


I have to grade looking at paper (where they showed their work) and the computer (Excel).  I believe work is essential to understanding a student’s misconceptions and the simple input box format just isn’t good for showing math calculations.

Students may retake my quizzes, and I encourage them to use the quizzes to study for the retake (I try to make them very different).  However, unless I do something fancy with Google Spreadsheets, and/or print off a bunch of quiz results, students will not be walking home with a quiz that they can study.

Students also do “test corrections”, where they write down (on a separate sheet of paper) what they got wrong and why they got it wrong (reflection).  Again, figuring this out with a single Excel sheet for the class is difficult.

Student can only work on one question at a time.  They can’t “skip and come back” (good test-taking strategy) nor can they go back if they remembered a previous question.  Also, with my Tiered Assessments, students like to see “Okay, what’s worth an A?  What’ worth a B? etc.”  Here, they’re forced to take it one at a time.  I suppose one could argue that this is a good thing, but I don’t think so on these assessments.

If students hit the wrong button (“Submit” too early, or click the wrong answer), because there is no way to go back, then students could get a wrong answer even if they knew how to do it correctly and did it correctly.  I got around this by telling them to write on their “work” paper if they did this, and to tell me what the correct answer is, so I can override their accidental entry.  However, this is one more thing I have to check when comparing the Excel sheet and the paper.


You can decide for yourself whether this would be a good situation or not, but unless I can figure out some better way for students to record more information per question, in a nice mathematical format, I think I’ll be sticking to pen/pencil and paper.

I’ve also asked students to blog about their experience, so we’ll see what they say about that.  I’ve already gotten the obligatory “I like this” and “I hate this” responses, so we’ll see how well they can articulate those reactions.

Of course, right as I finished typing the final question, I heard the copier start up across the hall and roll of several dozen sheets without jamming.

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Student Blogs: Update

I don’t have much time tonight, but I wanted to give a quick update as for how the student blogs are going.

Edit: Oh yeah, mad props to Jim Pai for suggesting Kid Blog.  Already it’s come in handy because 2 students have forgotten passwords and it’s so easy to reset their passwords for them in that program.

Some of the students are really taking these and running with them!  Others (as expected) find the activity dull and pointless.  I’m requiring that they blog for 1 week to try it out.  After that, they may continue to do so for Participation Points, and I think many will continue to do that.

Also, I wanted to mention at how fast I’m able to “grade” these!  I just type the points awarded in the comments, so they can see, and then I also type it into a Google Spreadsheet that I’ve shared with the students so they can see their running totals.  I can skim a blog pretty quickly now and decide if it’s worth 5, 10, or 20 points (the 20 points are saved for students who really blow me away with their insight and reflection).

Some of the students are really doing a great job reflecting, but I thought I’d let the blogs speak for themselves, so here are just a few of the better daily posts from my students.  Enjoy!

It today’s class we started of with a race that I did horrible in. The questions were about probability and for some reason I messed up on a lot of them. One question that I did get right(thanks to Benita and Chad) was – In poker, you dealt a 5 cards out of 52. How many different ways can 5 cards be dealt out of the 52.( I hope I worded that right.) We solved This question by realizing it is a combination because order didn’t matter. So in our calculator we typed in 52C5. The answer was 2598960. After the race we took a class quiz, which really helped because Mr. Newman explained how to do each problem. I wish Newman would do this more often!!

After the class quiz we took 9.6 Properties of Probability notes. We learned intersection of events- If A and B are events, then the intersection of A and B, is the set of all outcomes in event A AND B. Newman also explained union events- if A and B are two events then th union of A and B is the set of all outcomes in event A OR event B… At first I had no idea what Newman was talking about, but then he used a ven-diagram to help us. For intersection of events everything in the middle of the diagram would be an example. Union of events is everything inside and outside of the diagram.

Think of it like this…

Intersection of Events- Suppose you draw 2 cards for a 52 card deck without replacing. What is the probability that both cards will be black?(looking for the intersection of the 1st card black AND the 2nd card black.)

P(black AND black)= 26(black)/52(total cards) * 25(26-1)/51(52-1)= 25/102

Union of Evnts- Suppose a bag contains 7 chocolate chip cookies, 11 macadamia nut cookies, 12 oatmeal cookies, 4 gingersnap cookies, and 9 oatmeal-chocolate cookies. if you select 1 cookie at random, what is the probability it will contain oatmeal OR chocolate?



Oatmeal and Chocolate-9   <= 28

28/43(total cookies)= 65%

What about ginger or macadamia?


Macadamia-11  <= 15

15/43(total)= 34.8%

These were the examples that helped me the most through this class!

Yes, that was all one post.  Some students decided to go a more humorous route:

Today in Pre-Calculus we first set up these oh-so-beautiful blogs and learned how touse them, though I had to figure out how to insert pictures on my own. (How? I’ll never tell!) After that we ventured to a wild and educational acivity in which we created propability related problems for each other and answer them, therefore cleverly shortening the workload of a certain professor that shall be called Mr. Jim to protect the lives of the innocent. I found heiroglyphics as the best method of conveying my questions, but it seems that my colleagues, especially Mr. Haha, have a difficult time appreciating the sentiment. In any curcumstance, good examples for questions that we made are, “Is this specific orientation of cars a permutation or a combination?” And “What is the propability of choosing any specific 4-card hand out of a normal deck of 52 cards?” The former is a permutation and the latter is 52 nCr 4. I greatly hope that I will be able to grasp the concepts that have grown fuzzy from distance as clearly as I had weeks ago, but for now I must say farewell. Untill next time.
The Cryptographer

Some students decided it was best to simply take a photo of their notes (love the calculator in the image!).  I’m cool with that!

Screenshot from 2013-04-11 22:38:38

Others decided to be a little more creative and less practical:

APRIL 8, 2013

Location: Disclosed

Time: Forgot

Today we reflected back on the the different properties of mass and types of equations we can use to figure out a problem. An example of a equation was PV=nRT. How we reflected, We first got iPad’s and went to a app labeled “Student”. In here we typed the class code and waited. Newman would then put a question on the board for us to respond with the iPad  our answers would then at the same time go into an infinite matrix were data is stored. In this matrix we could see what all others responded with and can “Vote” which one is better, funny, or even CORRECT! And because I forgot my glasses at home, limited me from see and knowing which one was actually correct. The questions Mr. Newman posted was something like “Name the properties of mass”, Something like that. And it would be Volume, Pressure, Number and something else that I can not remeber right now. also a word problem that made us use a equation (PV=nRT) to  find out. Yup! That was mainly it for today, Not much happened except Shannon wasn’t here and has my headphones… :( But today was a good Day! (Minus the wind….and the suffering that came with it. Also track practice…..and Advance PE. But the wind… oh the wind. *Slowly Dramatic Music* Lunch was Fun! :D

Or other creativity:

If there was a theme to today, it would be reviewing. Almost like preparing for a battle. But although not everyone studied for the test, I did. Some of this stuff was a little bit of a blur, and some of it was a breeze. If there was one thing that I could not fully grasp, it would be Intermolecular Forces. That may have been the only thing I did not fully understand. But after a lot of reading (and re-reading) I sorta got the basics of it. Like Dipole Forces is when a partial positive and partial negative charge attract between molecules. And a London Dispersion when a electron group closes together and causes a partial charge. It took awhile to understand this but I got it now. Along with reviewing for the test, I was asked by my fellow peers for assistance. Ms. [Edited out] and Ms. [Edited out] both asked me to assist them with correcting their old Gas Laws Quiz. It was more of a review and correction session. Now that I look at it, I regret not signing up for tutor. But in the end, they learned a thing or two and so did I. I think I am ready for the test tomorrow, but who knows.

Until Next Time, Martza Out!

So I think that, for at least some students, it’s a really good experience.  I let the creative ones know that I enjoy their creativity, but that they still need more substance.  Also, I’m still giving them plenty of time in class to do this, so I hope that as I decrease that time in class, many of them continue to do this at home.

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Popsicle Sticks: A Tip

Many teachers, myself included, use Popsicle sticks (in a coffee mug) to select students randomly.  I still think it’s a great idea, but you have to be careful of one thing.

My Tip

Add positive incentives to the Popsicle sticks.  My first year I only used them to call on students to answer questions, often difficult ones when nobody was talking in class.  This created a negative vibe around the sticks and I could almost feel the negative energy emanating from the mug when I reached for it.  Students would come to dread the clink-clink of me shuffling the sticks in the mug.

This year one student even added several more of her name to the mug! I caught her (I *think* she was only joking), so she didn’t benefit from it, but it made me realize that I had done something, almost by accident, that turned out to be a very good thing for my class.  Why was she adding her name more times?  Well, I still use it to call on students, but I suppose the good outweighed the bad to the point that she wanted to add her name & increase the chances of my drawing her name.  You see, I also use the sticks to call on “homework writers” and select “best notes”–positions that students can earn participation points if they’ve been doing their HW and taking notes in class.

Even if you don’t use that system in class, I don’t care if you have to draw a Popsicle stick out and give candy away to make them positive–you should definitely help students to associate that sound more positively so they don’t dread being called on so much.  It has certainly helped my classroom become a more positive place this year.

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[Augment] Probability Review

I did this lesson with Precalculus students, but it would work easily in any subject that studies probability.

We just had spring break, and the week prior to that was an “Alternative Curriculum” week, where teachers get to teach things other than their typical classes and students sign up for what they’re most interested in. Great idea, and was really good this year. Maybe I’ll write a blog post about it sometime. Prior to that, we let out school after Wednesday due to parent/teacher conferences.

Anyways, all that to say, my students have been 2 and a half weeks away from school, and I needed something to kick-start their memory concerning the probability unit we left off half-way through. So here’s my plan for review.

The Plan

  1. The students will use the augmented reality to view one of many objects I have created for them. The objects include: a glass with various colored marbles, a group of colorful stick-figures, or a deck of cards, among other things.
  2. They must write a question concerning that item. Example: “What are the chances of drawing an Ace?”
  3. Everyone will rotate to the next station and view the object, as well as answering the question the previous group left behind. The new group must now create a different question, and turn in their answer (with the question) into a box I will have in the middle of the room.
  4. Later, I’ll sort through the questions and choose good ones to re-use when assessing, as well as check for understanding and comprehension.

I will explain to them that the purpose of this activity is review, so they shouldn’t make the questions too hard or too easy. The perfect question is a challenging question that makes the next group think, but they can still answer it.

EDIT: Here are the QR Codes you can scan to view the Augmented Objects.  Below that is the question/answer template I used.  Feel free to take them and use them yourself.

Note: when moving the students, I wanted to shift them around in a unique pattern so that they were not always following the same students, and thus always answering the same classmates’ questions.

The Result

Students enjoyed this activity, in part because of Augment, in part because they got to walk around the room. Here are some photos and video of their interaction.

Augmented Reality in Use

Augmented Reality in Use

Augmented Reality in Use

Pros and Cons

One pro of this activity are that students are actively engaged in the creation process as well as the answering process. They have to think critically to make the question at just the right level, and yet they have room to think creatively. I had a question or two in mind for each augmented object as I created them, but I was hopeful that some students would come up with creative and interesting questions for each other.

The activity got the students asking the right questions and had them discussing important points, but it didn’t quite push them as much as I had hoped.

This was also a very quick activity (about 15-20 minutes), which was good because we had a lot more administrative-classroom stuff to do (like starting our blogs).

One con is that students couldn’t always read (legibility) what other students wrote.  You got to see that in one of the videos above.

Another con is that students didn’t always ask the best questions.  Even when the questions weren’t 100% clear, students just tried to answer “as best as they could” instead of asking for clarity and helping each other perfect their questions.

Advantages of Using Augment

You could argue that this could be done just as easily with real objects. However, I think that the surprise of “oh, this is what we’re looking at now” adds a level of focus and intimacy with the creation of the questions. Otherwise, each student would be able to see the other stations from across the room and know “what was coming”, reducing interest and letting them perhaps feel like “well, the next group knows what’s coming, so what’s the use?”

You could also argue that I could just as easily have used QR codes, link to Dropbox, and shown an image instead. This would be fun, but somewhat removes the “wow” factor. I really think that viewing the objects in 3-D amps up the interest just enough to make the task more than worthwhile. Furthermore, with pictures, I would be limited in certain situations. For example with the glass holding the marbles, students would simply be counting a 2-D image. Instead, students had to consider “do I really need to know how many of each there are here?” and then attempt to count in 3-D, which improves their spatial reasoning, however slightly. It also generates discussion and the need to check each other on the results.

Summary & Improvements

I was glad the students got to experience this, but it perhaps wasn’t the best activity to do after such a long break because the students weren’t able to challenge each other enough to recall the questions.  My warm-up asked them “what’s the difference between a combination and a permutation”, and I was hoping to get more questions oriented in that direction, but there wasn’t a whole lot of that happening.  Next year I might do it as a review activity just before a test, so students will be more prepared to ask (and answer) tougher questions.

I had a TA type up the questions and answers into a Google spreadsheet, so you can see the level of engagement and the depth and  creativity of their questions (or lack thereof) by clicking on that link.  We might do something with those in Precal soon.

Ideally I would use something like Google Forms so students are entirely using the iPads, and we can review the questions afterwards (without requiring that the TA type up the questions).  However, I couldn’t figure out a way for the students to only see the latest question & answer exactly that one.  If you find a program or website for that OR can get Google Forms to behave in that way, let me know.


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