Tuesday, November 30, 2010

(How to) Get a Job

Dr. Thomas Reeder, Associate Superintendent of Wyoming Public Schools, was kind enough to come talk to our secondary mathematics student teachers about life after student-hood.  He has had an amazing career already, starting as a K-12 certified teacher and becoming a literally award-winning administrator.  He gave me permission to take notes to share.

You might here someone say, "The best thing about teaching is June, July and August", but that is nowhere near true.  The best thing is the students.  Decent pay and benefits, but it's the students that are why you teach.

This is a good time to be a math teacher because of retiring teachers (state buyouts) and Michigan's new 4 year math requirement.  There are some jobs.  Science minors are excellent, too.  As you're getting ready to apply, you want to think about what sets you apart.

There are different certifications: state certification and highly qualified.  Major or minor, plus certification test equals highly qualified.  Sciences and social sciences are content specific.  In your materials, be clear about what you are qualified to teach.  Once you're hired, the district will put you where you are needed.  Know the Michigan Merit Curriculum, which has changed even since you were in high school.  Four years of math, English and science, a world language, health, etc. 

Figure out which students you want to teach.  Figure out where you want to teach.  The state, the type of area, urban or near-urban... but evaluate where you want to be.  Rural teachers whose schools have not been able to hire new teachers for consecutive years.

Michigan teachers are paid well (top 3) with much lower cost of living than other high-paying states.  You will never be affluent, except for what you get from your students.

L Hollis @ Flickr
Succinct, no errors.

What to do to get ready: be involved in educationally relevant experiences.  What do you do outside of your teaching?  Conferences, tutoring, coaching, special events like a math night or tutoring night, etc.  The more you have those experiences the better.  I'm looking at between 100 and 1000 resumes.  If you're a rookie, what else have you done that makes you not a rookie. 

The two crucial areas of relationships are with parents of students, and with your peers.  100% of parents believe that there kids are 100% innocent... what will you do?  Get experiences with conferences.  We have a principal who can remain calm in the face of full out screaming and cursing.  You don't need to take it: "I'm not going to talk with you now if you can't speak civilly..." give a warning.  But then you can hang up.

Other work experiences can be valuable if they build up the image of who you are.  Never lie on your resume.  Never - that's a firing offense.  If you were fired from a job, you can still put it on.  When you're asked for employment history, you need to be complete.

I've only put my educationally relevant jobs - should I put the other ones?  Sure.  Maybe going back to senior year of high school.  Does your experience show the ability to balance?  Loyalty?

Should I list volunteering in high school, or is that too far back?  I would put it in.

Should I list a full education history?  Yes.  High school plus all your colleges.

You can list or not list you GPA on your resume, but I always look at transcripts.  I don't want all A's, but I want to know about retakes, how long, what kind of student you were.  Some of the teachers who have struggled for us, also struggled in college.

Personalities are more important now than before; maybe you've heard about speed interviews - we might use those as kind of a screener before more in depth interviews.  Now while you're a student, belong to the professional organizations.  It's dirt cheap and they are valuable.  Then make sure they're listed on the resume.

Get your resume down to two pages.  Include relevant personal interests.  One person got a job because they listed skiing and they needed someone for ski club.  Schools will want you to have multiple roles, coaching, clubs, drama, etc.  Plus, as a teacher, it is going to help you see your students in a different way.  And help them see you in a different way.

The number of mistakes on resumes is unbelievable.  You need to proof them and clean them up totally.  If you're not good at it, get help.  Even a professional.  The resume isn't tailored to a specific job... that's the...

Cover Letter
With internet applications we went from dozens to hundreds of applications.  The cover letter is how you tailor your application.  Be specific to the school or district.  Don't be generic.  Share what you know about the school and district.  A question we ask is:  tell us about yourself, (2) tell us what you know about Wyoming Park.  Personalize your cover letter.  If you were dating someone, you would write a personal letter.  Will that mean you write a lot of letters?  Yes.  Go there or at least do your homework.  You need to know if you want to be there anyway.  What will set you apart?

Three paragraph essay: a little about yourself, what you know about the district and your fit; an overview of yourself and what qualifies you; a wrap-up.

Is the cover letter and resume it?   Each district have specific information they request.  Follow their instructions exactly.  The old adage was go in person.  Don't do that anymore.  It's annoying and might even count against you.  In our district, resumes go to our HR person.  They put the piles together and vet the resumes.  Then they go to the principals.  The only thing to check with the central office is to make sure they received all your materials.

How many references?  Sometimes it's specific, some will be general.  If they don't specify, 3 or 4 is appropriate.  Make sure your references know what you're doing.

Start putting your materials together now. You can still tweak.  Two periods for hiring are a little bit in January, and then June-August or even Septemeber.  We hired 6 people at the start of school.

Use those weeks after you graduate to sub.  Get your name out there.  Subbing is the best and worst job.  Maybe 80% of our hires come out of our sub pool.  Why?  There's safety in hiring people that you've tested out in some way.  Maybe you have another job, but is that going to lead to the teaching job you want.  You don't have to sub every day.  It will give you experiences to help you decide what you want and don't want.

What are red flags?  It's not supposed to be about your personal characteristics.  No picture, for example.  No gimmick.  Got one with a plate of chocolate chip cookies.  Some might like it, I don't.  I don't want any preconceived ideas about you as a candidate.

Some teachers don't want to teach the high level classes.  I got my first job because none of the current teachers wanted to teach AP calc.  I was an A student, but that calc class was a shock.  The kids caught on quickly, and then after a year it finally, totally clicked.  "I thought this was hard, but it is so easy."

If you want to go out of state:  check out the particular standards.  Michigan standards are pretty high, so shouldn't be too much of an issue.  Be careful about pay and benefits.  If you're young, go anywhere you want, but look into it.  There are good websites for out of state jobs.  MASB is a good source for Michigan jobs.

Resume, cover letter, portfolio (paper or DVD), put a lesson together that you could use.  You might hear on a short timeline, so you want to have things done ahead of time.

If you get called, be very courteous.  They may give you some options as to time.  What you want to do is change anything you have to do to get to that interview.  If there is a reason, work around it, share the reason and make sure it's important.  I had a conflict with a game I was coaching and they were willing to interview me on Saturday.

Bring whatever they ask.  It is okay to ask a question, "do you want me to bring..."  Don't ask anything else.  You need to show initiative to get the information you need.  Now it's time to do your homework.  You have got through the biggest hoop, you have your foot in the door.  Find out anything you can.  School Improvement Plan, classroom management system, special district programs, anything they put online.  Go visit the town and the school.  My first job I went to the town for two days, talked to people in the coffee shop, found out a ton.  Plus I found out that it was a job I wanted.  But it would have been just as good to find out if I didn't want it.

When they ask you if you have questions, have some.  Be specific with what you've found out about the school.  What about your poor grades in math, your SIP, your ...

Don't ask about salary and benefits.  With the one exception that if you're choosing between offers and need to know that information.

Dress appropriately.  Comfortable.  Suit okay? Yes.  But dress up at least equivalently of a dressed up teacher.

I will never tell you not to be yourself, but nose rings, tattoos, piercings ... be aware of the effect.  There are places where a man with an earring can not get hired.  Impressions are natural and always happening.  The interviewers will be forming impressions immediately.

Should I introduce myself to each person, shaking hands?  Absolutely.  Plus it will help you relax.  Good firm, practiced handshake.  I've interviewed people so nervous they broke out in hives.  Know yourself.  If you speak quickly when nervous, pay attention and slow down.  Don't be afraid to do what you need.  If you need to doodle, take notes, then bring a pad of paper.

Sometimes we'll ask a three part question to see if the candidate can track all three parts.  Good time to have paper.  If you bring in something extra, don't be afraid to leave it anyway.  (If it's something to be left.)  Or ask if they want to look at it before you go.

Every district interviews differently.  Short, long, test class to adults or a summer program.  The hard part of the interview is that you're looking for the best candidate, but it's insufficient to really know what kind of teacher you're going to be.

Typically, there are a dozen questions.
  • Starting with tell us about yourself.  Don't stick to the resume, complete it.  Extra training, share your personality.  
  • Tell us about the district.
  • Scenario questions:  how would you handle...
  • Philosophy: what do you believe about...
  • Management: what do you do to make a learning environment?  Not about having your thumb on the students.
  • Past experiences
  • Expertise areas
  • Maybe: content questions. 
  • Try googling teacher interview questions.

  • be succinct in your answers but completely answer the question.  Okay to pause and think.  If you're going on and on, catch yourself.  Not so short that you leave blanks.  
  • Be punctual.  (If something happens, call on the cell and inform.)  Be in the area early.  Don't have to go in the building, but be close.  
  • Don't be so nervous that you can't communicate who you are.  
  • Be aware of yourself.  Are you a loud talker?  Too quiet? People have probably told you already things you can work on.
  • Writing samples might happen on 2nd round.  Sometimes tech skills, although that's not an issue for novice teachers.  I had an interview where they left me with 2 questions and 2 sheets of paper and an hour to answer them.
  • Pay attention to their time if they've given you their limits.  Too fast, too slow - monitor.  Be done with questions a few minutes early to leave room for dialogue.

The most impressive things are: being confident and comfortable.  (Not too confident.  Eg the interviewee who asked when they would be signing the offer since they were sure to get it.)  Lean in towards the interviewers, make eye contact, look at everyone.  At the end, ask a good solid question.  OK to have written it ahead of time.  (Maybe have 2 in case they ask you one.)  Ask about resources that are available, ask about tech.  This would impress me: " I notice you use Prentice Hall.  Do you use..."  "I know you are on trimesters..."  "For your after-school tutoring, do teachers..."

You have worked hard to get here - don't skimp on the work for this last thing to get the job you want.
shareski @ Flickr

Can I take chances the first year?  Not sure what you mean.  But there are two grounds for immediate dismissal: (1) don't touch a kid (2) you're young; know about what is appropriate and inappropriate with a student.  I was 21 at my first job, with students who were 18 and 19.  Now we have parents that have hit on our teachers.  The third thing would be finances; be careful if you're responsible for any school funds. 

Michigan has curricula for academic content.  But some schools have policies like abstinence-only education or creationism... you should know that.  If you don't know, check with someone else.  You can always tell a student, "I will check."

You will make mistakes, how are you going to grow from them.  You might be miffed at how you see colleagues teaching or treating kids.  Classroom management is hard to develop as a student teacher.  When you start from the beginning, you'll need a way to have control. 

The four years towards tenure are for the school to evaluate your performance and your growth.  By law, every teacher needs to be evaluated every year.  You want feedback, though.  I'd rather know my struggles in November than in June.  Have people come in and watch you teach.  We have a lot of team teaching now with ELL and special education students.  You want to be the best you can be.

Looking back at my career, I would have gotten a Spanish degree in addition.  When I started, there was small need, but now, working in an urban district, it's a great need.  If you're interested in administration ever, you have to be involved in all areas of the school.  Get on a School Improvement Team (now required in each school by state law), be active in all areas of school life.  Teacher, union, coaching... when I went into administration there were 10 positions they had to fill!  I took that as a good sign that I was involved.

Monday, November 29, 2010

Math and Multimedia 5

Welcome to the Mathematics and Multimedia Blog Carnival, 5th edition.  (Hope you enjoyed the Fantastic Fourth Edition.)  Five seems like a fortunate number, since we have five senses.  Our five fingered hands are a good start to mathematics.  We all love a high five, are happy to take five or are glad when it's five o'clock (somewhere).  But my favorite five fact features 5 for the fifth Fibonacci number.  Far out!

This blog carnival seeks to promote seven principles:
1. Connection between and among different mathematical concepts

Sol Lederman at Wild About Math shares a video of an Incredible Magic Square.

Antonio Gutierrez at Go Geometry has a fascinating golden rectangle puzzle that connects with the Droste effect.

2. Connections between math and real life; use of real-life contexts to explain mathematical concepts

John D Cook at the Endeavor shares that there isn't a googol of anything.

 Grrrl Scientist suggested this article from her guardian.co.uk blog about "How the leopard got its spots" that has some literally beautiful mathematics.

Consider this beautiful film by Cristóbal Vila - Nature by Numbers.  Or this collection of Hands On Math Movies.

David Cox has just been sharing a ridiculous amount of great stuff lately.  For example, projectile motion.

It's been widely shared, but you have to check out Kate Nowak's money take on special right triangles at f(t).

 3. Clear and intuitive explanation of topics not discussed in textbooks, hard to understand, or difficult to teach

James Tanton has two videos explaining the principles for math genius thinking.  Hat tip: Denise at Let's Play Math.  You might also like Sue Van Hattum's interview of Dr. Tanton for the Math 2.0 interest group.

For that matter, Sue's post at Math Mama Writes about E is for Eigenvectors and Eigenvalues belongs in this category.  Has the great first sentence: "This post is about fear."

I spent some time recently looking at trigonometric function visualizations and making a couple Geogebra sketches for them and their inverses.  Seems silly to link, since it's right down there.

4. Proofs of mathematical theorems in which the difficulty of the explanation is accessible to high school students

No one nominated entries in this category, but it makes me think of work like James Tanton's explanation of Euler's proof that every even perfect number is triangular, or Alexander Bogomolny's proofs at Cut-the-Knot of the addition and subtraction formulas for sine and cosine.  (Both of these sites I've had occasion to look up recently!)

 5. Intuitive explanation of higher math topics, in which the difficulty is accessible to high school students

Derek Bruff has put together a fascinating interactive Cryptography Timeline.  I'd love to see some of these for some important math concepts.

6. Software introduction, review or tutorials

Guillermo Bautista, the founder of this here carnival, at the Math and Multimedia blog, has a roundup of essential tools for every math blogger.  Also be sure to check out his terrific Geogebra tutorials while at his site.

Maria Anderson has video of her presentation from MAA-Michigan up, Math Technology to Engage, Delight and Excite.  Also watch as her new blog, Edge of Learning, gets up and running.

Chris Betcher has some terrific Scratch (the programming language) resources and videos.

You might try one of these 15 mind-mapping tools.  (Many are free.)

 7. Integration of technology (Web 2.0, Teaching 2.0, Classroom 2.0), in teaching mathematics 

David Wees has an interesting meditation on the importance of interactivity in math teaching.

Cybraryman has a long list of math/tech integration resources and lessons.

If you're looking for Five tunes, Take 5, or try Gimme 5 from Sesame Street, High Five from They Might Be Giants, Dino 5 (who also have a great counting rap called What About 10)?, or maybe the best (in terms of math) ...

If you have ever posted a blog carnival, you know that you receive a lot of obvious spam.  But some can seem relevant, so I like to have a Best of the Spam category.  For example, the Top 40 sources for open courseware video.

And if you are mistakenly put here, or your post did not appear, please let me know and I will correct it posthaste.

Images were obtained from Creative Commons search.  Attributions are in the picture title - click on the image and you will see the source from Flickr.

If you enjoyed the carnival, please consider nominating a blogpost, your own or someone else's, for the sixth edition, to be held at Great Maths Teaching Ideas by William Emery.  It's a very active blog with many K-12 activities, so don't wait a month to check it out!

Congratulations if you recognized the five connection for each of the images above.  I tried to slip in some tricky ones.

If the carnival is done, must be time to head over to the Five Bells.  Cheers!

Monday, November 22, 2010

Trig Visualizing

Rebecca Walker and I modeled a lesson for our secondary student teachers on trigonometric equations, based on the first chapter of the Precalculus book from the very interesting CME Project curriculum.  While it has some interesting applications, this curriculum really does a good job of letting the mathematics be the context and addressing mathematical habits of mind.  The lead developer is Al Cuoco, who has a great history of interesting math and math ed work.

The lesson is a bit of a stretch, because we're just touching on one section, using a bit of information from three or four.  We did unit planning one week, lesson planning the next week, and finally the lesson.  The TAs read The Teaching Gap, so then we connected it to the idea of lesson study, and a discussion both about how to revise this lesson, and why lesson study might work as professional development.

We have two Geogebra sketches to help with visualization.

 As a sketch or a webpage.  This sketch supports visualizing sine and cosine with unit circle connections.
As a sketch or a webpage.  This sketch lets you invert trig functions using the Unit Circle representation.

This is my first attempt at a WCYDWT.  When I was making these sketches (don't worry, I disinfected them before posting) I had a bad cold, so was constantly reheating my tea.  Watching it go round and round.  Thinking, "so when do we know a position and want to know the angle, with possible multiplicities...hey, wait a second."  If I was using this, I think I would start with the video, and use that to motivate the idea of solving for information based on the circle position, as well as how periodicity relates to multiple solutions.

This has to be the world's most boring video.  Enjoy!

Here's a slightly more polished version of the handout we used with the sketches.  There was some discusssion with the student teachers as to whether the inverse trig or the algebraic solutions part should come first.  I think they could be switched, depending on what you wanted to emphasize with the students and how strong their trig background is.  Also, the handout is written as if the teacher is demonstrating with the computer, which is what we wanted to model for them, (no lab is no reason to no have technology) but the ideal would be to have the students have access to the sketches.

Solving Trig Equations

Sunday, November 21, 2010

Images of Teaching

There was a nice (if long-windedly titled) article in the Decemeber '03 Teaching Children Mathematics called "Metaphors as a Vehicle for Exploring Preservice Teachers' Perceptions of Mathematics," by Brenda Wolodoko, Katherine Wilson and Richard Johnson.  In the article the preservice teachers made images to display themselves as teachers or learners of mathematics.  The majority of their images revealed anxieties about the content as learners, but hope for themselves as teachers.  The researchers liked the way that the images created an opportunity for dialogue and created a potential for change.  One interesting sidenote is that students used the idea of puzzles both positively and negatively, modeling both frustration and engagement.

My preservice elementary teachers recently made images of what does it mean to teach mathematics in small groups, and our secondary student teachers envisioned their future classrooms.  There were many neat ideas to share.

 The mullet came later, but I think they thought the "Business in the front, party in the back," slogan did relate to teaching.

 This and the next chart were imagined classrooms, which you'll see a lot of with the secondary teachers as well.  They value group work, technology, manipulatives and whole class time.  I often wish college classrooms had room for a carpet section for students to sit down.

 These next two charts are more like concept maps.  This group focused on the most important aspects to them.

While this group saw their concept map evolving into a hierarchy.  There's some pretty interesting connections here to look into.

Somehow missed my favorite poster here.  Clever use of Facebook, and really made me think about that page as a representation of who someone is.  I think there are lots of idealized people I'd be interested in seeing Facebook pages for.

The secondary teacher assistants made quick individual sketches at the end of a seminar.  So don't expect the artistic commitment we got from the elementary teachers.  One thing that came across in classroom images is the presence of the kind of technology to which they've been exposed.  It's becoming clear to me that we need to do a better job of teaching technological pedagogical content knowledge, primarily by explicit modeling.

Let's start off with a few of the text descriptions.  This teacher is worried about the content they will be forced to cover.

This teacher is thinking about classroom management as the start of learning.
I'm not sure if this teacher was describing life as it is or as they envision.  Somebody definitely considering the different models with which they've been presented.

Next come several visions of cooperative learning.  There seems to be a clear value on student discussion, and varying images of what the teacher's role is in relation.

And this sketch merges a vision of the classroom with a concept map of what is important to them.

I'd be very interested in knowing what you think about the images here and what you notice, if you'd care to leave a comment or drop an email.  Thanks!

Wednesday, November 17, 2010

To Understand, Book Club 3

Ellin Oliver Keene
Today my preservice K-8 teachers are discussing Chapters 5 and 6 in To Understand.  I gave them four focus questions to try and help guide discussion:
  1. What is the author's main point in the chapter?
  2. How does she support that point?
  3. How does this idea apply in mathematics?
  4. What's your reaction or connection with this.
Chapter 5 - To Savor the Struggle
Mentor: Reynolds Price, author and poet who has written and spoken eloquently about his response to an inoperable spinal tumor, as well as just been more artistically productive than he was before.  (Sample of his poetry available at Google Books: The Collected Poems.)  What enables some people to flourish in adversity?

  • Greatest gift to a student is the experience of struggle leading to understanding.  We don't do it in younger grades; students may not experience struggle till college.  
  • Intentional struggle for students requires time and modeling.
  • Teaching kids how to learn to think more critically requires struggle, versusthe students just doing without reflection.  In math, struggle leads you to know why you're solving, and where the answer comes from.
  • Part of conceptual understanding is problem solving.  If it's too easy there's no problem solving.  Struggle requires problem solving.
  • How do you motivate students to struggle?  They'll sit there until someone else does it.  One student, or the teacher, or.. (several stories to back this up).
  • Once you get them willing to struggle, then they can gain understanding.  They will know why they got that answer.
  • A lot will blow off the work b/c of the struggle.  They didn't have a need to struggle until college.
  • When you conquer through struggle you're more likely to remember how, remember your process, and be able to use it again.
  • Does more struggling now mean less struggling later?  It means you will be less frustrated with struggle later.  Got through it, rather than going through it.  
  • Frustration is a problem.  You have to continue to struggle to add new understanding.  My elementary struggles make me able to struggle for a long-time now.  
  • With struggling comes a work ethic.  It was a shock in college to have to learn to do the work.
  • Struggle motivates me to want to figure it out.  
  • How much is too much struggle?  Different for every student.  Know your students and differentiating.
  • Mixed emotions.  Why don't we want to find a way to prevent struggle, for them to get it without struggle.  What happens in real life?  Remember waiting for others to solve and just get this lab done; later on, on my own, I had to do it for myself.  Want our students to get content, but more so to be able to solve their real problems in life.  They will have struggles, I want first struggles to be in the safety of my classroom.  Help them now or help them later?
  • Isn't a struggle enough now? Some of the kids struggling will ask, but some just shut down.  Do nothing.  How motivated is each student? 
  • Why don't we have leveled math problems like leveled readers?
  • How does it happen - when a student moves from resisting to engaging struggle? 
  • If you create struggle, what about that 10% that cannot even begin.  Everyone's different. 

She discusses these systems in literacy instruction, and I think it would be worth investigating in math.
  • Semantic System - word meaning and use
  • Schematic System - process focus on what it means to read
  • Pragmatic System - what do you do with the content

Leonardo's horse at Meijer Gardens, Grand Rapids

Chapter 6 - A Renaissance of Understanding
Mentor: thinkers and artists of the renaissance.  Considering the idea of intellectual liberation.  What conditions are necessary for a renaissance?
  • Allow the students to go beyond your expectations.  Challenge my students.  A teacher who designed tests so no one could get an A - that's not challenge.
  • Be interesting to see reading levels transferred to math.  A math corner... and then science.  Time and effort from teachers too much?  Maybe they don't have the background to do it.
  • Choice is a part of the answer to this.
  • The title renaissance: something new.  The testing is a problem because it's based on a norm.  Students should be awarded for original thinking.  Writing you can imagine this.  What is it like in math - original thinking?
  • Grading is fear inducing, restricts creativity.  Especially when it rewards conformity.
  • Are you going to take the classes where you struggle, or where you can get an A.  It looks good on your GPA.  Could grade technical difficulty and performance.

In this chapter she considers the importance of text and genre in literacy instruction.  What are these things in mathematics?  Is it our content areas?  Problem types?  Different processes?  Different uses like investigation, homework, tests?

The Chapter 5 discussion was energetic and purposeful.  Definitely one of the reasons to use this book.

Photo by cliff1066 @ Flickr

Saturday, November 13, 2010

Scratch and 8th Grade Geometry

From twitter I found out about the K12 Online Conference. In particular about a Scratch session by Chris Betcher (@betchaboy on Twitter).  The session has a nice 22 min video about using the Scratch programming language with Year 5 students.  Very worthwhile info in a sitcom sized bite.  Chris also has a Scratch wiki devoted to getting students going with Scratch, and all the resources you need to get going.  Scratch is perpetually one of those things I'm going to investigate when I have obtained some sparemomentium.

I got a chance to work with a local 8th grade teacher who's looking into Geogebra.  He sent me some of the state standards he was interested in exploring, and I made up a couple sketches to play around.

The first is just a demonstration of the area formula for a parallelogram.  It seems so unreasonable that all parallelograms with the same base and height have the same area.  That connects with one of Geogebra's strengths to me - providing essentially infinite examples so that students can notice.

The next is probably pointless.  I was considering how to make a dynamic area measuring sketch.  I thought the advantage would be being able to change the figure, and easily check your answer regardless of how you've changed the shape.  I used sliders to build the shapes so that the distances would be nice.
Webpage and geogebra file

The third sketch is for similarity - I'll post that later this week with the world's easiest activity.

Monday, November 8, 2010

M&M is coming!

The Mathematics and Multimedia Blog Carnival is coming to Mathhombre.

Just not yet!

Today was the originally scheduled date, but the guiding director and founder of the Mathematics and Multimedia Blog Carnival has reset it to be in the fourth week of each month. It will be here in 3 weeks, November 29th.

That means 3 more weeks to submit your own ideas or report a blogpost that you think would be good.  Here's the link for submissions.

In the meantime, we can watch this video:

Thank you, Mr. Lehrer!

Sunday, November 7, 2010

Tessellating Kites

Our study of motions led to one of my favorite topics in all of mathematics:  tessellations.  I've posted some previous work on this blog, have an old webpage with some good tessellation resources, and found a new source of beautiful Alhambra images to share with students.

As a math topic I just love them.  The visual aspect, the geometry, the connections with algebra, the historical context, the art connections with Escher... it's darn near perfect.  Working with 2nd - 12th graders they are amazing for rigid motions because you use the motions to make the tiles, then to repeat the tiles in a pattern, then can see the motions in the finished tessellation.  It's visual and kinesthetic.  There ought to be a song.

Instead, how about some geogebra.  For some reason, this time around, I got interested in kites.  Which tessellate by side to side rotation and what would a mixed glide reflection/rotation tessellation look like.

The question of which kites tessellate by double rotation boils down to what happens at the joint vertices between the two (possibly) different edges.  We can pick the angles so that the kites blossom (tessellate around a point) at the vertices between congruent sides.  In this sketch, you set those numbers, then observe the effect on the other angles.  What condition is necessary for the angles to work out?  Is it sufficient?

Webpage or geogebra file.

This sketch lets you make alterations to that classic 60-120-90-90 kite tiling.  The sketch will adjust and give you a chance to both design and watch the effects.

Webpage or geogebra file.

 This sketch does a tiling that can be done with any kite.  (Pretty good question as to why it works for any kite!)  Two of the congruent sides rotate to themselves around a midpoint, and the other two fit together with a glide reflection.  Escher was fond of this pattern, as it allowed him to create creatures going in opposite directions for his contrasting tilings.

Webpage or geogebra file.

I would love to hear from readers if they prefer these geogebra sketches as links or embedded applets.  Could you take a second to comment?  Also, I love making geogebra tessellations, so if you have any ideas for ones you'd like to see, let me know.