Tellagami  – Think of the Possibilities….

tellagami

While on Twitter I recently learned about the Tellagami App  and immediately I started thinking about how much fun it would be for students while allowing them to create ways to share their learning.  Formative and summative assessments might just be more engaging using Tellagami!

Tellagami reminds me a little of creating a Voki Avatar (http://Voki.com), a web-based tool.  Tellagami seems to be easier and quicker to learn and share, but has less free customization options that a Voki avatar offers.

Tellagami is a mobile app that lets you create and share a quick animated video called a Gami.  This app is designed for Android devices, iPhones, and iPads.  Get it on iTuneshttps://itunes.apple.com/us/app/tellagami/id572737805?mt=8 or Google Play – https://play.google.com/store/apps/details?id=com.tellagami.Tellagami

This app is EASY TO USE
Create a Gami in 3 easy steps:
1 – Customize a character and choose your background
2 – Record your voice or type a message for your character to say
3 – Share your Gami on Facebook, Twitter or send via text or email

Some of its FEATURES and a video tutorial makes it easy to find and use them.
• Mix and match your character & background
• Record your voice or type a message
• Resize character and place it in the scene
• Personalize with a photo background
• Share via Facebook, Twitter, email or SMS
• View Gami as a web URL on all devices

Possibilities for using Tellagami with students.

Here are some ideas for students in different classes:

  • Math – create mini lessons on shapes, theorems, or proofs; create an animated dictionary of mathematical terms.
  • Language Arts – create mini book reports or character studies for readings; prepare grammar lessons; describe characters from a book and let other students guess the character being described; create an animated dictionary of vocabulary.
  • Social Studies – create newscasts reporting on historical events;  give directions from a given spot on a map and have others follow the directions to find the destinations;  pretend to be a historical figure and tell their famous historical events from their points of view; create a debate between characters on two gamis
  • Science – prepare a documentary on elements in the periodic table;  describe biological attributes of things in nature/body parts; prepare reports on constellations and the stories that go with them; create an animated dictionary of scientific terms.
  • Art – pretend to be a famous artist and create an abbreviated autobiography; be a museum curator and describe a piece of art that could be displayed next to it in the museum.
  • Music – Create a musical tribute to a composer including a brief musical recording of his/her work.

What other possibilities can you think of for your students?

Resources for even more ideas and information.  This article was inspired by:

Posted in Algebra, Calculus, Common Core, Geometry, Technology | Tagged , , , , , , , , , | 1 Comment

Real Math + Students = Engagement

RealMathNeonStudents always ask, “When are we ever going to use math?”  They often say, “I don’t need math for the job I want.”  As teachers, we know that almost all jobs use math.  This assignment helped my students realize that math was everywhere and also introduced them to careers they may not have known existed or what people do at these jobs. This assignment REVOLUTIONALIZED my math classes. Students began to realize that they were in no position to decide what math they needed and what math they didn’t.  They just better learn the math and be prepared for anything!

When I started teaching, it became more important to have examples to share with students. As the Internet became more available,  it became easier to find examples.

Videos, the Web, the Library, and the Assignment

I wanted to get students thinking about this and came up with an assignment to help students start investigating the answer to these questions. Students in ANY math class can do this assignment and the same video can be viewed by different classes and you’ll get different observations. Little did I know how powerful the assignment was going to be, how much I would learn, and how fast it was to grade. Students needed to watch two-four videos and papers per semester.  For the first two, I took students time in the computer lab.  After that, students did this as an assignment outside class time with little, if any, resistance. Papers were saved to their class Moodle site to make accessing their papers easier than via emails.  A shared drive, Google Docs, or a DropBox would be great alternatives to Moodle for this assignment. For students without computer access at home, there were computers available before and after school and during the day in the media center. Below is the actual assignment and the grading rubric.  Please customize #2 before giving it to students. I changed it to Algebra concepts for Algebra, Geometry concepts for Geometry, Calculus concepts for Calculus classes, etc.

What’s Math Got to Do with It?

Each week when there is a movie to watch (on the Internet) and each movie averages about 5 minutes long.   Submit your final papers electronically. Take a few minutes (these will be short movies) and watch the movie, do a little research (use a book, a magazine, another Internet site, etc.), take some notes, and then write a paper at least one page long if word processed or 2 pages if hand-written. Please do not hand in a list of questions and answers. 1. What’s MATH got to do with the subject of the movie? (2 points) 2. Be more specific and find course concepts (i.e. replace this with “Algebra” concepts if you’re in Algebra, Geometry if you’re in geometry, Calculus if you’re in Calc, etc.) appear to be related to the subject of the movie? For ideas, look in your math book after watching the movie. Be specific and use correct mathematical vocabulary. (3 points) 3. Would you like to have this job? Why or why not? (1 point) 4. What other occupations (2 or more) might be related to this job or the math used in this movie?  (2 points) 5. What did you find most interesting, what was new to you, and/or did it make you think differently about math in the “real world”? (2 points) There must be at least one other source in addition to the movie website – another related website or magazine/newspaper article or other print material.  Please remember to cite your sources. (-1 if they are not given!) If your paper does not meet the following requirements -1 point . *Word-processed written paper must be at least one page long using a 12-point or smaller font (such as Times New Roman), not more than 1″ margins, and double-spaced. (submit these on Moodle) *Hand-written paper must be two pages hand-written. Reading papers can be time-consuming but these went really quickly and were easy to grade.  The time invested reaped great rewards! MovieEval Download the assignment as a Word Document: MovieAssignment Download the evaluations sheet as a Word Document: MovieEval

What I learned from student papers:

(examples of student work)

  1. Students stopped asking the question about when were they going to use the math. They started realizing that until they knew what they were going to do, they needed to be prepared to meet any math challenges and learn the math. Some of the people in the movies said that they didn’t know that they were ever going to use the math that they used in their jobs until they needed it.
  2. Some students didn’t even know some of the careers they saw in the movies existed and got excited that they were REAL jobs. I used more career related problems in my examples/homework.
  3. I learned more about my students, their experiences, their hopes and dreams, their interests, and more. I used this information in interactions with students, later assignments and when developing future projects.
  4. I realized that English teachers probably learned a lot more than I had about my students by reading their papers.  This might be why they were probably perceived as more “interested in them [students] as a person” than math teachers.  I got to know my students even better and it made a positive impact in the classroom.

Sites for good videos and resources for student research Here are some recommendations.

  • My all-time favorite: The Futures Channel http://thefutureschannel.com/- Do your students ever ask “What’s math got to do with it?” Here is a site that has videos that you can use to answer some of their questions. The “stars” are real people – some famous, some not, some young, some not, but all interesting and current topics. Movies are about 2 – 6 minutes in length – just perfect to begin or end a lesson. The videos change from week to week so sign up for their free newsletter that keep you up to date on the movies that are new for the week. (Just so you know, they are generally accessible for two weeks and then they are replaced so use them fast!) The Futures Channel uses media technologies to link scientists, explorers, and visionaries with today’s learners and educators. Videos on this site are very high quality and there are often lessons and activities to go along with the videos, making them easy to use in the classroom.
  • Math at Work Mondays http://www.mathforgrownups.com/category/math-for-grownups/math-at-work-monday/ – Look on this site on Mondays (and other days) for Math at Work Mondays for interviews (some videos) with people at work and how they use math in their jobs.
  • Head Rush Cool Jobs http://www.sciencechannel.com/tv-shows/head-rush/videos/cool-jobs-in-science.htm  – While this site is technically about cool jobs in science, there are several videos that relate to mathematics. For example, there is one video where a skate park designer describes how he uses shapes, angles, and trigonometry to create his skate parks. It is a great video for geometry classes.
  • Mathematics for Economics http://www.metalproject.co.uk/ -ME:TAL or Mathematics for Economics: Enhancing Teaching and Learning is an organization that creates videos and lessons illustrating to students the use of mathematical topics in the business world. For example, there is a video that shows two industries that use linear programming. The website is great to show students real life applications.
  • Careers Information http://www.mathscareers.org.uk/16-19/career_profiles.cfm.html – There are many career profiles/interviews at this site
  • Maths in Work videos https://www.ncetm.org.uk/resources/11329 – This includes many links to jobs including Designing Aircraft, Listening to Music, Experimenting with the Heart, Revolutionizing Computing, Beating Traffic, Scanning the Unseen Cat scan, Packing It In, Unearthing Power Lines, What to do with a Maths Degree, and much more  Learn a lot about 40+ careers and answer when math is used in each, including possible salary predictions
  • WeUseMath http://weusemath.org/?page_id=800 – Learn a lot about 40+ careers and answer when math is used in each, including possible salary predictions
  • 10 Amazing Jobs You Could Land with the right STEM Education http://mashable.com/2013/02/05/10-awesome-stem-jobs/

=================================================================== Additional resources you might be interested in

  1. NAPmathhttp://NAPmath.wordpress.com – this is my blog. I write when I have something to say/share.
  2. Green Appleshttp://GreenApples.wikispaces.com – a website full of resources for new MATH teachers (and veterans). Join the wiki and contribute to the pages. I use it with new teachers in my department and pre-service teachers. Teachers that helped developed the site include Nationally Board Certified in Math, Presidential Awardees in Math Teaching, state and local teaching awards, and most important, they are successful in the classroom.
  3. GeometryGemshttp://geometrygems.wikispaces.com/ – this site is dedicated to all things Geometry including activities, projects, SMART board files, paper folding, common core, and more.
  4. SmartBoardSmarty http://SmartBoardSmarty.wikispaces.com – If you have a SMART board in your classroom, go here and download free Math Notebook files, get step-by-step directions, tutorials, updates on SMART, resources, and lots more.
  5. ActivInspireAdventures http://ActivInspireAdventures.wikispaces.com – If you have a Promethean board in your classroom, check this site for resources, tutorials, free downloads, and more.
  6. Pinterest http://www.pinterest.com/napmath/ – find links to Math sites, math blogs, math journal and foldables, technology, iPad (and other Apple problems) resources, activities for QR codes, SMART/Promethean/Interactive white board resources, assessment and much more.
  7. Resource Garden - http://resourcegarden.wikispaces.com/Planted+Gardens – need to find resources for graphics, sign and fancy Word Generators, photos, sounds, music, videos, flash/java, and free ways to edit sound, video, and graphics? This site has great links, copyright information, and lots more.
Posted in Algebra, Calculus, Common Core, Geometry, Math Education, Technology | Tagged , , , | Leave a comment

My Message to Parents about The MYTH of “I’m Bad at Math”

I just read, The Myth of “I’m Bad at Math”: Basic ability in the subject isn’t the product of good genes, but third work. By Miles Kimball and Noah Smith (Oct. 28, 2013) and it brought me back to a newsletter I wrote to parents in the 1990’s.  I’ve updated that message to parents and you’ll find it below.

My Message to Parents about The MYTH of “I’m Bad at Math”My Message to Parents about the Myth of "I'm Bad at Math"

Math is a subject that all students CAN learn and it will enhance their opportunities and their self-esteem later in life. Think about it.  When people find out that a person can do math, there is some sense of instant respect.

The Power of Mathematics opens doors to new knowledge and understanding. Math is not hereditary – there is no math gene (at least I don’t know of any!). One of the worst things any parent can tell their child is that they were “bad in math”, inferring or saying that it is why they are having trouble. Do not make excuses for students. If you do, their expectations (and yours) cost some children their math education.   I don’t remember ever hearing anything negative from either of my parents about mathematics.  We were expected to learn whatever we were supposed to. Later in life, I remember my mom say she didn’t know where I got my ability to do math but she was always doing some hands-on activity with her kindergarten students and my dad was always busy with some sort of problem, diagram, calculation, or busy building something.

As parents, we need to instill confidence in students. That confidence gives students the power to be persistent, work through challenging problems, and know with hard work, any progress on a challenging problem is success.  Over 37 years of teaching math, I watched how confidence transformed learning.  I have letters from students telling me that because I believed in them, that they now believe they can do math and are willing to put more time into doing math.  The best sense of achievement and boost of confidence comes from working hard on a “problem”, being frustrating with it, and persevering unit you figure it out by yourself!  No one can give anyone that feeling.  They have to earn it themselves.  (Good thing to remember as a teacher or a parent!)

A quick thought about math “problems”…. Is a real problem ever solved quickly or without thinking deeply? Keep this in mind – Quick “problems” are usually practice and a time to explore the different twists that might happen, not what I would consider to be “problems”.  Just because a person isn’t able to solve a problem as fast as someone else, does not mean that they are not good at math. Some people need a little more time to understand it, make sense of it, and mentally store it for access later. (By the way, that’s me.  Once “I get it”, I can build on it, retrieve it, and recreate that knowledge. Memorizing is not my forte but when I get it I can explain it to others!!)

“If you can’t explain it to a six year old, you don’t understand it yourself.” – Albert Einstein

Everyone needs math to understand the world around them and make sound decisions on the future using probability and statistics.

Encourage your child.

Expect nothing but the best! 

Believe that they can do math and tell them!

The tough part of math is the practice required to learn it well. But, let’s put it in perspective… Would a basketball coach put a player on the floor that didn’t show up to practice? Did Michael Jordan get good at basketball by watching other players play? Why should math be any different? Math takes practice – For most, 30-45 minutes per night is not expecting too much (a basketball practice is often a couple of hours per afternoon!). In the preparation for life, math should have a high priority, right? Every basketball player won’t be a “Michael Jordan” and every math student won’t become a professional mathematician, but everyone can develop a deeper understanding and a love or at least respect for both with practice.

Technology is changing the job market. No longer will there be jobs for students without mathematical reasoning skills. Students will not only have to do math but they will also have to communicate math and work together to solve problems. Jobs are being redefined. Employers are looking for a new kind of employee. Once students reach the job market, more and more of them will have to create their own job and market their own skills. Downsizing is beginning to make this a reality today. Many years ago I read a good book – The Monster Under the Bed by Stan Davis and Jim Botkin and it helped me to start to see the future with new eyes. Now there are more resources to read and think about.

Below are some additional sources (in no particular order) that I’ve found that might give you some good information as parents. I will continue to update this list to provide you with the best tools to be the best parent.  

What tip or thought do you have for parents? Please share your ideas with students and parents.  I welcome your comments and tips!!

Posted in Math Education, Technology | Tagged , , | 1 Comment

Twitterverse – You’ve changed my world!

It’s been 1 year since I’ve been actively using Twitter and made it a part of my PLN (Personal Learning Network).  Here’s what I’ve done to extend my PLN, keep learning, and start giving back to teachers and the teaching profession in my retirement. I love being a connected educator and it is great to write this article during Connected Educator Month #CE13.

STEP 1:  I started by seeking out people to follow in order to learn new things and find new resources.

STEP 2: As I went across the country doing workshops for teachers, I shared my Twitter contact information with my participants and all of the wonderful things I was learning from Twitter and my PLN.

Visit my Wikis!

Visit my Wikis!

STEP 3:  I started sharing some of my resources on my blog and wikis.  Teachers that used my school websites to access some of my projects started sending me emails when they were no longer available (now that I’m retired).  So, I tweeted where to find where to find them.

hastagSTEP 4:  I started looking into hashtags (#) and found that they were an easy way to search for conversations and collaborations on Twitter.  I started seeing conversations between people on Twitter and discussions centered around certain topics.

STEP 5: Hashtags (#) also led me into twitter chats and found these are amazing ways to connect with excited educators doing innovative things in their classrooms to enhance the learning of their students.  I have to admit I am a lurker on new chats to get the feel for the conversation and while I determine if  the chat is for me.  So, if you’re not sure, just do a search on the hastag for the chat and read the conversations – P.S. It’s a  great way to find others that you want to learn from too!  When looking for new chats/groups, I look at what is trending and that’s how I’ve found some of these amazing chats (and before I knew organized lists existed!)

TwitterChatshttps://docs.google.com/spreadsheet/ccc?key=0AiftIdjCeWSXdDRLRzNsVktUUGJpRWJhdUlWLS1Genc#gid=0

Some of my favorite chats:

  • Sundays:
    • #21stedchat – 21st Century Education Chat – 7 pm CST
    • #alg1chat – Algebra 1 Chat 8 pm CST
  • Mondays:
    • #edtechchat  – Educational Technology Chat – 7 pm CST
    • #tlap – Teach Like a Pirate – 8 pm CST
    • #iledchat – Illinois educators – 9 pm CST
  • Tuesdays:
    • #makered –  Maker Ed (k-12 classrooms built around Maker/DIY principles) – 5 pm CST
    • #PATUE – Palo Alto Twitter Using Educators (pedagogy and technology) 7 pm CST
    • #SMARTee  – SMART  Exemplary Educators  – 8 pm CST
    • #hsmath – high school math chat – 8:30 pm CST
  • Wednesdays:
    • #ntchat – New Teacher Chat – 7 pm CST
    • #geomchat – Geometry Chat (transitioning to Common Core) – 8 pm CST every other week (next on Oct 30)
  • Thursdays:
    • #mathchat – Math teachers chat – 6 pm CST
  • Fridays:
    • #calcchat – Calculus chat – 12:30 – 1:30 AM EST (yes right after midnight EST)
  • Saturdays:
    • #satchat – Saturday Chat (general topics)  – 6:30 – 7:30 am CST and 9:30 – 10:30 am CST

Step 6: I gave my first webinar yesterday!  It was by no means perfect but I found out what it’s like not to see your audience or hear from them – very strange.  I built a nice #SMART notebook file for the participants, put it on the web for them to download, and have been hearing from them since the workshop.  Thank you for your comments and support if you were at the webinar.  I don’t have to wonder if you learned something new and found that it was not a waste of your precious time.  At last check over 90 people have downloaded the file (http://smartboardsmarty.wikispaces.com/ed2ed) and I today I woke up and saw this:

twitternumber

YAY!!!! Thank you to all of my followers on Twitter for having faith and interest in what I have to say.  Thanks to all of the people I’m following – I’m learning SO much.  So, if you’re on Twitter, consider following me.  Chances are that I’ll follow you back.  If you’re not on Twitter, consider joining and expanding your PLN.

For more information on Connected Educator Month (CEM), visit http://connectededucators.org/about-connected-educators-mission-goals/ and I love the resources and activities in the CEM Starter Kit – a free download at http://connectededucators.org/cem/cem-getting-started/ .  I’m participating and earning badges.  It’s sort of fun but I really don’t know what to do with them other than collect them.  I guess there must be more to learn.  If you know what I should do with these badges, let me know in a comment on this blog or tweet me @NAPmath.

ConnectedEd

Posted in Math Education, Technology | Tagged , , , , , , | 2 Comments

Academic Honesty in Question

Cheating? Really? Are you positive?

Cheating is sometimes a hard call and other times it is not even questionable.

Here’s my story:

One day I decided once to give my students extra time on an exam the next day because I had made it too long .  I was trying to be nice but maybe not one of my wisest decisions.  Looking back, it changed the way I wrote tests and administered them in the future – but I’m getting ahead of myself.

Someone took a copy of the test when he/she left class and shared it with friends. It was very evident that this had happened and a couple of students came to see me when they found out this out. It was also obvious that not all students saw it but most of them knew about it.  I couldn’t eat or sleep while I was trying to figure out what to do.  (Now students don’t even have to take the physical test if they can get pictures with their mobile devices.)  I gathered my facts and took them to the principal.  I laid out my plan, got his input, and made sure I had his support for what I was going to do. (Don’t forget to do this if you get in this situation! )  Next, I sent a letter home to parents, wrote another exam, and required all students to take it over since I couldn’t tell for sure who all were involved. It wasn’t a popular decision with all students or parents, but in my opinion, it was the most fair thing to do. The funny thing is, that some of the students I know were involved  had the parents who were most outraged that their child had to take the test over.  My whole body was a mess for a long time after and the relationship that I had with that class was back to the “prove to me you’re worthy of my trust.”   BTW, did I mention it was an AP Calculus class with potential valedictorians in the class, and that it was when I had a student teacher?

To make things worse

Now there are all sorts of other complications.

from http://kythana.deviantart.com/art/Cheat-sheet-nails-267741507 (check out the conversation on this page if you need more to think about!)

from http://kythana.deviantart.com/art/Cheat-sheet-nails-267741507 (check out the conversation on this page if you need more to think about!)

Here are some articles to check out and maybe help you think this through.  Warning: Don’t expect that all of your questions will be answered.

  1. Digital Tools Raise Questions about what is and what is not cheating by Katie Ash (August 21, 2013) -http://blogs.edweek.org/edweek/DigitalEducation/2013/08/students_need_more_guidance_on.html
  2. Read the stats about Harvard’s freshmen in the class of 2017 By  (Posted Thursday, Sept. 5, 2013 ) http://www.slate.com/blogs/moneybox/2013/09/05/harvard_cheaters_crimson_survey_finds_lots_of_cheating.html
  3. Moving from Cheating to Academic Honesty By Eugene Bratek – http://www.edweek.org/ew/articles/2012/10/17/08bratek.h32.html
  4. Cheating Runs Rampant:  No Child Left Behind has unleashed a nationwide epidemic of cheating. Will education reformers wake up?  by Daniel Denvir ( May 25, 2012) – http://www.salon.com/2012/05/25/cheating_runs_rampant/

What would you do if you saw or suspect students of cheating? 

Posted in Calculus, Common Core, Math Education, Technology | Tagged , , , | 1 Comment

Read a Test to Students without Saying a Word!

qrScavengerHuntonGeomGems

I’ve always thought there would be some very cool educational applications for QR codes.  I’ve make QR scavenger hunts that are fun for students while being great learning activities.  There is an example of a Quadrilateral Properties Scavenger Hunt Activity at:  http://geometrygems.wikispaces.com/MMC+Geometry+files+2013

NAPMATHqrCodeBut today… I think that I have very helpful application for ELL / ESL students,  students with special needs, and for teachers that need to be in many places at once.  My students didn’t like to leave the classroom to have tests read to them.  They didn’t like being singled out.  My mission was to figure out how to get them the help they needed while keeping them in the classroom with their peers.

QR Codes, Mobile Devices, Assessments, and Empowering Students

With more and more students and schools having access to mobile devices, technology can help out when students need a test read for them.  They can listen to the questions as many times as they need to and move at their own pace.

Prepare the Test

What is needed to MAKE the quiz or test:

  1. Create your test as usual, leaving a place to enter a QR code.  See/download a sample assessment below.
  2. Access  http://recordmp3.org or another site that allows you to record the question and save it to a website as RecordMP3 does.  RecordMP3 is very user friendly and does everything you need it to do with ease!  You can use it for other things, but that’s another post.  Record your voice, reading a question as clearly as possible. (Okay you got me! You do have to talk but not during the test and your voice can be “on demand” by multiple students at once!!!!)
  3. Use a microphone to make your recordings as clear as possible.  Make a document (and/or add the website to the test) and save each questions’ websites – make sure to label them to make them easy to find again in order to create your QR codes.
  4. Access  a QR Code Generator like http://www.qrstuff.com/ that will allows you to enter the web address for your each of your recordings and generate the QR code for each of your recordings/questions.  Copy the web site for each questions’ recording and paste it into the QR generator and generate a QR code for the web site for the recording of each question, one at a time.
  5. Paste the QR code into the document and continue until all questions have a QR code.  Save the document each time as you enter codes.

 SampleQuizDownload this sample quiz as a

Give the Assessment

What students need to TAKE the quiz or test:

1. A copy of the quiz or test and writing device (if taking this as a paper/pencil assessment).

2. A device (such as an iPad, iPhone, Android, etc.) with a QR Reader such as the

3.  Earphones

Practice makes perfect

Now it’s time for students to take the assessment.  Before students take an assessment for the first time, give them some practice examples with QR codes until they are comfortable with the process.  You could use the sample quiz or make some review questions that take them to websites, to voice recordings, or other activities on the web.

Individualize the experience

To allow the students to remain in class during the assessment, have them use headphones so they won’t bother students near by.  Now you have EMPOWERED your students to complete their assessments on their own!!!!

One final idea for all your assessments!

Oh… one final thought – wouldn’t QR codes be great for making different forms of a test for all of your students, with or without voice recordings? Try http://www.classtools.net/QR/ for making QR codes for just text entries.  You might still need to include the figures for some of your questions, but Classtools makes it easy to get the QR codes!

Websites used:

Additional resources for QR codes

Posted in Algebra, Common Core, Math Education, Technology | Tagged , , , , , , | 1 Comment

Origami + Math + Mobiles

image001So exactly how do origami and math relate to each other? The connection with geometry is   clear and yet multifaceted; a folded model is both a piece of art and a geometric figure. Just unfold it and take a look! You will see a complex geometric pattern, even if the model you folded was a simple one. As a beginning geometry student, you might want to figure out the types of triangles on the paper. What angles can be seen? What shapes? How did those angles and shapes get there? Did you know that you were folding those angles or shapes during the folding itself?  Can you come up with any relationships between a fold and something you know in geometry?

purple3Any basic fold has an associated geometric pattern. Take a squash fold – when you do this fold and look at the crease pattern, you will see that you have bisected an angle, twice! Can you come up with similar relationships between a fold and something you know in geometry?

On the other hand, if you are a person who likes puzzles, there are a number of great origami CubeOrigamichallenges that you might enjoy trying to solve. These puzzles involve folding a piece of paper so that certain color patterns arise, or so that a shape of a certain area results. But let’s continue on with crease patterns… For instance, the traditional crane unfolded provides a crease pattern from which we can learn a lot. Pick a point (vertex) on the crease pattern. How many creases originate at this vertex? Is it possible for a flat origami model to have an odd number of creases coming out of a vertex on it’s crease pattern? How about the relationship between mountain and valley folds? Can you have a vertex with only valley folds or only mountain folds?  

image06How about the angles around this point? You can really impress your teacher (or your students) with this…of course, you will need to understand it first! There is a theorem called Kawasaki’s Theorem, which says that if the angles surrounding a single vertex in a flat origami crease pattern are a1, a2, a3, …, a2n, then:

a1 + a3 + a5 + … + a2n-1 = 180      and        a2 + a4 + a6 + … + a2n = 180

In other words, if you add up the angle measurements of every other angle around a point, the sum will be 180. Try it and see!   Can you see that this is true, or, even better, can you prove it?

The study of origami and mathematics can be classified as topology, although some feel that it is more closely aligned with combinatorics, or, more specifically, graph theory.  You can investigate these connections further on your own.

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Origami, a Japanese word, combines the word oru (to fold) and the noun maki (paper).

A Bit Of HistoryOrigami origin is unclear. Some historians claim the origin of origami began as a Japanese tradition of folding important documents/certificates. “Origami Tsuki” means “certified or “guaranteed”. The phrase stems from their ancient custom of folding certain special documents – such as diplomas for Tea Ceremony masters, or masters of swordsmanship – in such a way as to prevent unauthorized copies from being made.  Others claim origami came about after the invention of paper and made by Cai Lun in AD 105. This was the first usage of the word “origami” traced in Japan. The word “origami” came to be used occasionally for another kind of ceremonial folding, namely for “tsutsumi”, or formal wrappers, by the beginning of the 18th century. However, its use for recreational origami of the kind with which we are familiar did not come until the end of the nineteenth century or the beginning of the twentieth. Before that, paperfolding for play was known by a variety of names, including “orikata”, “orisue”, “orimono”, “tatamgami” and others. Exactly why the switch came to “origami” is not clear, but it has been suggested that the word was adopted in the kindergartens because the written characters were easier for young children to write. Whatever the origin, Japan has fully practiced the art. It is so valued in Japan that it has become part of religious ceremonies.

What is origami? Origami is the art of folding paper and creating three-dimensional figures of: people, animals, object, and abstract shapes. The material for creating origami is a piece of thin paper; although any paper may be used. The paper is normally cut into a 15 cm. square that is plain white on one side, and decorated on the other (color or decoration). Some creative origami artists try to experiment with cardboard, cloth, wire mesh, sheet metal, and even pasta. I bet there a million other possible ways to be creative in origami.

The four most common bases of origami are the kite base, fish base, bird base, and the frog base. Bases are the starting shapes for different figures. Adding additional folds you can create figures of virtually any shape. Some of the folds specialize in modular origami, or making multiple copies of a simple single shape and forming the pieces to make an elaborate structure.  See the cube to the right.

In 1999, Joseph Wu provided this simple yet encompassing definition.

Origami is a form of visual / sculptural representation that is defined primarily by the folding of the medium (usually paper). 

Here’s another interesting theorem:  
Theorem:
 Every flat-foldable crease pattern is 2-colorable.

In other words, suppose you have folded an origami model which lies flat. If you completely unfold the model, the crease pattern that you will see has a special property. If you want to color in the regions of your crease pattern with various colors so that no two bordering regions have the same color, you only need two colors. This may remind you of the famous map-maker’s problem: what is the fewest number of colors you need to color countries on a map (again, so that two neighboring countries aren’t the same color)? This is known as the Four Color Theorem, since the answer is four colors. As an interesting aside, this theorem was proven in 1976 by American mathematicians Appel and Haken using a computer to check the thousands of different cases involved. You can learn more about this proof, if you like.

But back to our theorem since I know that it is buggin’ you (look to the right!!).  Can you see that you need only two colors to color a crease pattern? Try it yourself! You will see that anything you fold (as long as it lies flat) will need only two colors to color in the regions on its crease pattern.   Here’s an easy way to see it: fold something that lies flat. Now color all of the regions facing towards you red and the ones facing the table blue (remember to only color one side of the paper). When you unfold, you will see that you have a proper 2-coloring!

MovingOnToMobiles

image021   Mobiles are not just to hang on a baby’s crib. They’re modern art.  Mobiles were first made by Alexander Calder, an American artist who started as a boy making wire and wooden toys.  Calder created a whole circus of animals of and performers with wire and cloth.     image020    Mobiles are sculptures with parts that move. Mobiles mean movement. You’re mobile when you move and jump and play. Mobiles move when they are suspended freely in space.

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Mobiles are shapes – circles, triangles, rectangles and squares – floating in space, suspended by a string or wire. If the shapes are balanced, they will spin and float, turn and twirl.  Touch them or blow on them, and watch them move.

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Generally mobiles hang from a ceiling, but some are mounted on pedestals.  All parts of a mobile should swing freely so that the movement of the mobile is maximized!

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Links – More about Origami

purple3Links – More about Mobiles

Download the Make your own custom made origami mobile.

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