Monday, November 25, 2013

Fun with Fibonacci, Phi, and the Golden Ratio

I think the Fibonacci sequence is spectacularly amazing and mind blowing.  Even though I have taught about it for six years now, every year I am just so amazed at the intricacies of our life and how the Golden Ratio relates to so many things.

We learn about Fibonacci in my Math Analysis class when we briefly (i.e. one day) look at recursive sequences.  I could have students just do a bunch of mindless work with a bunch of recursive sequences, but instead I introduce general recursive sequences and we jump into studying Fibonacci's sequence in great detail.

See pages 7-8 of the Unit K SSS packet to see some awesome information about Fibonacci's sequence, Phi, and the Golden Ratio!

Here is a playlist of some videos and websites I have found about the "Golden Ratio" (also known as the Beauty Ratio, Phi, Golden Number, and other names) that I have students look at before class so we can jump right into the activity during class time.

  • The first step goes over the "Fibonacci Assignments" that they can do.  Some are required, some they can earn bonus points for.  
  • The second step is the required "introduction" video to the Golden Ratio.
  • The third step is the instructional video for the in class activity they do.
  • The rest of the steps are just awesome videos and websites I have found that I wanted to share.


So, what do my students think about all of this?  Here is what they said after doing their Concept 4 WSQ (BEFORE doing the in class activity)

  • It's between the body and nature because those two are radically just woah. The body is already pretty cool but then you throw in plants and well plants are already beautiful, makes sense that the Golden Ratio is in it. 
  • The information about how we find people attractive based on the proportion and symmetry of someone's face because I would just imagine ourselves thinking someone was cute by just our first sight, not by examining the facial structures.
  • That buildings can be considered beautiful, I mean the pyramids have a the number that is considered beautiful. What were the possibilities of that?!!!!
  • I found how the Golden Ratio is incorporated into music interesting because of the 13 notes in an octave (8) and so on. It's creepy... I can never sing the same again... just kidding, you can't stop me. 
  • The part I found most interesting was the part where physical attraction depends on ratio because I was thinking the other day that why do we all feel that someone is more beautiful like we all have an image of beauty.
  • The most interesting is the music. When thinking about the golden ratio, I thought it was mainly focused on appearance of something, but the golden ratio is also in music! 
  • The golden ratio and how it is used in buildings, that they saw this mathematical beauty in their early time periods.
  • I found the information about music the most interesting because I am so involved in music in my life; it is a major part of my life and it is interesting to see that the Golden Ratio exists in such a place. I am very excited about this. 
  • I find the part of PHI being an ideal type of physical attraction because of all the people I was and am attracted to, I never stopped and thought "nice ratio, girl ;)"
  • I found the part about buildings to be very interesting because I never knew that the Golden Ratio was used even in earlier times, such as in the building of the Parthenon or the Egyptian pyramids.
  • That the number is seen in nature, apparently the universe likes Italians. 
  • -snails and fingers have the beauty ratio as well
    -there are various ways to find the beauty ratio in your body such as your arms, legs, face etc. EVEN YOUR TEETH HOLY MOLY AND NOSTRILS WHAT
    -it's even in your DNA and lungs, like how cool is that?!  

Here are some pictures from our in class activity (Haiku-writing and Beauty-ratio measuring)










Here are some of the resulting blog posts turned in already

Haiku (I have many more links if you are interested in more samples)

Beauty Ratio

Reflective Essay (hoping to get more this week!)

Saturday, November 23, 2013

Week 13 Reflections - heading into Thanksgiving Break

It's wonderful to sit in my recliner with my feet up knowing that I still have 8 days left of vacation and spending time with my husband and (almost!) 6 month old son.  We strolled around the local swap meet all morning and then made 30 (yes 30!) freezer meals in less than 4 hours.  It will feed us for 2 months, which means no cooking during the week for 2 months... which means more time with Grayson!

Ok, so I've blogged a lot this week already about some "flipping" I tried in my Course 1 CCSS class (see my previous posts pre- and post-).  I also have another post coming about a really awesome activity I did on Friday in Math Analysis, but I want to wait for the student samples to come in before posting it.  I'll (hopefully remember to) link it here when it's posted.  Link is here!


This week we finished up Unit J (systems of equations with matrices) and started Unit K (sequences and series).  This is what I tell my students is finally "real" Math Analysis.  90% of Units A through I is stuff they SHOULD have seen in their previous math classes, but every year I find out that it doesn't always happen.  Regardless, Unit J is when it really starts getting tough.  I warn my students, I encourage my students to practice more, I challenge them to spend more time reviewing the material, but sometimes it doesn't quite come to a head until they take a test and don't do so hot.  That's what happened this week.  I had many students who did well, but I had much more than normal do very poorly.  It was also a time consuming test, because if you mess up on a matrix, it can take you FOREVER to figure it out.  So, I gave the students all day to work on the test (morning, class, after school) as they felt they needed.  Some students just never came back to finish and so turned in a half blank test :/...

So, what do we do when that happens?  Well, first, we had the conversation on Thursday about the difficulty of the unit and the upcoming units and had them reflect on their work ethic and focus.  Second, they are all (to the extent I can force them to) reassessing on the unit when they come back from Thanksgiving.  Third, hopefully, they learn that they can't just skate by but really need to practice the concepts to make it concrete in their brains.


I ran into one of my former students this morning at the swap meet.  She was a senior in my inaugural flipped class... that is, she was the one who had her world turned upside down in October of 2011 when I decided to flip.  I probably blogged about her.  She struggled with the change.  Still was a great A or B student, but did NOT like the flipped classroom.  She had come from 3 years of CP level math, all with the same teacher (random how that happened).  Then, she gets put into my class, an honors class, and now a FLIPPED class!  We butted heads at first but she came around by the end - not to liking the flipped class, but to putting up with it.

So, we just got to chat for about 3-4 minutes today, but she told me about her college classes and how she is in Calc 2 and how hard it is.  She told me that now she actually really misses the flipped classroom and wishes her Calc 2 class was flipped.  Whenever her friends are struggling in a math class, she points them to my class websites and videos for help as well.  All from a student who HATED to learn from video 3 years ago!


I was pulled out on Tuesday for the District Technology Stakeholders Meeting.  It was really awesome to hear about the ways different sites in our district are integrating technology (we have I think 66 kinder classes that are 1-1 with iPads, and a middle school who is in their second year of BYOD)... a lot of it I had never heard of before!  We had some great discussions with table groups (consisting of everyone from district administrators, principals, TOSAs, company partners, and a few classroom teachers like me) about technology integration, especially as we are heading towards full common core implementation in the next few years.  It was a great experience to be a part of.


I hope you all have a wonderful, restful, and blessed Thanksgiving season surrounded by those you love and are thankful for!

Monday, November 18, 2013

Flipping the Quiz Review - how did it go?

Today I tried something new in my Course 1 class.  See yesterday's post with my plan here.

And, how did it go?
 Well, here's what a few of my students said:

  • I really enjoyed it and thought it was a really good easy way to go over and understand the material. I enjoyed the videos as well, I think they were very well done and I understand pretty well what I got wrong.I don't think I found anything wrong or bad with it, overall it was helpful.
  • I think it was great we thought it was going to be complex.  No just simple.  It was good no problems.  It helped me understand my mistakes
  • I thought the iPod activity was really good, I felt it helped me.  Yes I would want to do it again! I think it helped me because it explained step by step on how you got the answers. I think its good the way it is

Students worked in pairs watching short videos and correcting their quizzes.

QR codes were posted in a few places around the room for each problem on the quiz.

I posted the QR codes in a couple places around the room.  I gave about a 5 minute intro to the technology and the iPods, and showed how to use the Qrafter app.  I put students in groups (groups of 3 in one class, groups of 2 in the other) based on how they did on the quiz.  They had to scan in the code for each question anyone in their group missed and watch the 1-2 minute explanation and make the corrections on their quiz.  As a perk, I am giving them 1/2 credit back on every question they correct.

The videos were also posted on our class blog (which we haven't used much all year, but hopefully will start using more now!),  A few students watched them on the laptops.  I also told the students to download the Qrafter app so next time they could use their phones if they had internet and we wouldn't have to share as many devices.

Overall I am happy with Day 1.  The students weren't fooling around the technology and were mostly focused.  Some changes for next time:

1. Give more time.  We only spent about 30 minutes of class on this because we had a little lesson first, and by the time I gave the info/instructions/etc, they only had about 20 minutes to work through it.

2. See exactly how many kids in each class have devices with 3G/4G internet (non-wifi) on them so I can appropriately hand out the devices so not as many students have to share them.

My plan?
Over Thanksgiving Break I would like to choose a few of the lessons for between Thanksgiving and Christmas to flip, and have students watch them in class.  This means the students would have a full assignment of example problems and practice problems and would work through it in class at their own pace.  I think it will be especially ideal for our next unit where students will be using protractors and compasses, and some students I can just see taking FOREVER and others wanting to move on... this may be perfect!

I will definitely keep updating as things go on.  I'm glad that the attitude and feedback from the first day was positive.

Sunday, November 17, 2013

My Plan For Tomorrow: Flipping the Quiz Review...

So, I haven't really done any flipping in my new "Course 1" class this year, for a variety of reasons.  
#1 being time, #2 being I don't have the curriculum for that far into the future, and #3 I needed to get to know my kids before I put hundreds of dollars worth of tech in their hands during class time.

I'm going to attempt an activity tomorrow and we'll see how it goes.

They took a quiz on Friday.  Some did good, most did okay, and several did very poorly.  They have a test on this Friday, so I really want the quiz to be a true "formative" assessment.  Let's figure out what we actually know and don't know, and learn from our mistakes so we can do better on the test.  I'm even going to give them the "perk" of earning a better grade on the quiz based on this activity.

I made short 1-2 minute videos explaining the answer to each of the 9 questions on the quiz. 

 (side note: I decided to use ShowMe for the first time and am very happy with it.  I have always used EduCreations for short quick ones, but students can't watch those videos on my iPod touches so it kind of ruined the idea behind it.  ShowMe videos work just fine!)

I made a QR code ( is what I use) for each video.  I have printed them out and will post them around the room.  I also linked to each video on our class blog under the Linear Functions Unit (; don't use too much this year but just starting to get things organized).

LINEAR FUNCTIONS QUIZ video answer keys

I have 13 iPod touches, 3 laptops, 2 desktops, and lots of student phones (assuming they have a QR reader, if not I will have them download one if we need more devices).  My plan is for students to work in partners with a device and go through the quiz questions they got incorrect (either of them) and work through the correct answer. (I have classes of 35 and 27 for Course 1 this year)

Students who don't have many corrections will finish this activity quickly and be able to move on.  Others might take the whole period.  I really want to go through the quiz errors with the students, but why waste ALL the student's time going through every problem when they don't need every problem?  

I hope this turns out to be a good activity of making the best use of the face to face time I have with my students and differentiating my support and instruction to the students who need it, when they need it.

I will let you all know how it goes!

P.S. This is the first time I'm letting these students touch my class technology all year (long backstory, don't want to go into it).  I'm a little nervous about it all and am definitely going to go over the ground rules right away... i.e. only use it for what I tell you to (in this case they will be on the QR code-reading Qrafter App ONLY), report any issues to me immediately, take care of the devices and return them to their appropriate place...

Saturday, November 16, 2013


I was so happy to be able to attend the afternoon portion of EdCampOC today.  I can't wait to go to another one and meet up with so many amazing educators.

It was great to see so many friends, some of whom I've met before in person and some who I've only interacted with online.  The power of twitter as a PLN...

So great to see again/meet you all and chat, even if it was just for a few minutes.  I can't wait to connect and chat again soon - Kate, Kevin, Chris, Dan, Randy, Jill, David, Garrett... and more but I can't remember the new names :/  So if we met and I'm not following you on twitter yet please tweet me @crystalkirch so we can stay connected!

I got to participate in an amazing hallway chat with Karl, Kate, and Kevin during session 3 (aka catch up and chat), and then went to "Things that Suck".  I've heard so much about that session but it was awesome to participate.  It was funny when David asked me to define a flipped class to those who didn't know and I guess my definition was just a little too complex.  I tried to simplify it, I really did :)

Here a few pictures from the day!
Jeanne Reed @jeanneread1
I got to EdCamp at about 12:30 and was starving since I was out all morning. I go to the food truck and order some sliders, and Jeanne comes up and tells me she is going to buy my lunch for me!  This is the first time I've met her, and she is a such a sweetheart!  And, a passionate educator!!

John Stevens @jstevens009
We missed each other at CUE this year (blame Siri), but I'm so glad I got to finally meet John and chat, even if it was for just a bit.  Looking forward to more conversations next time!

Who can resist this cutie?  Grayson attended his first EdCamp, and everyone got to meet the one who's taken over my life the last 5 1/2 months.  He's so worth it :)

KARL! @LS_karl
David tweeted this out as a "family" picture.  Talk about awkward.  But, Karl did crown Grayson as  #WSQprince, so he can be the #WSQuncle.  Sound fair?
On a random note, I won one of the raffle prizes.  When Kate announced one of them was a certificate for OCDE courses, I commented that it would be funny if I won that one since I teach a course for OCDE.  And guess what I won?  Yep... how ironic. :)

Friday, November 15, 2013

Week 12 Reflections - AND lots of student blog samples!

Another great week!

WSQ chats - 
Giving students more structure is definitely working.  This week, they were given a different handout every day to discuss and work on in their "small groups" of three students each.  What I think is best is having a handout gives students more accountability and it gives me proof of what they have talked about.  Here is what we did this week (see all the handouts here):

For more details on any of the blog posts I talk about below, go to

Tuesday - Matrices day 4 (handout 3-4d) -  The goal of this discussion was for students to understand what "infinitely many solutions" really means - that it's not just any numbers, but any ordered triples that follow the qualifications of the solution formulas.

Students also made sure their Student Video #5 was done and good.  See student samples at any of the links below:

Wednesday - Matrices day 5 (handout 3-4e) - I had students do this activity in their small groups and spent the period meeting with each table to discuss their answers.  They had been working on matrices for a week and now I wanted them to know exactly what their solutions looked like visually.  Each table is made up of two groups of three, so I looked at both group's answers and we talked through them.  I used the app QuickGraph to show them a 3-D visual of the graphs (you could move it around and look at it from all angles).  It was awesome to see students so engaged and interested in seeing the 3 dimensional graph.
It took me about 30 minutes of the class to meet with each table, so that means I spent about 5 minutes per table discussing their answers, what they thought, clarifying misunderstandings, and showing them the visual.

Thursday - Partial Fraction Decomposition Day 1 (handout 5) - The actual discussion part was purposely short (about 2 minutes) because I wanted students to focus on working through their Student Problem 4 (see video directions here).  However, it was a good quick check to see if students understood and could explain the main point from the video lesson the night before.

I thought the student problem day was awesome.  I purposely made this one pretty tough for them, because I wanted them to show me that they understand the forwards ("composing") and backwards ("decomposing") of the partial fractions.  I really wanted them to see that what we were learning was just the reverse of what they learned in Algebra 1 and Algebra 2.

It was also great because every student was working on a different problem (since they wrote it themselves), and so they all had to work individually yet support and help each other with their different problems.  They couldn't just rely on someone else to do the work and then "get help" by copying them - they actually had to do it.

Want to see some student examples?  Of course you do!  Here are the 24 that have already been turned in.  I haven't seen them yet, so I'm sure there are awesome and not-so-awesome examples.  But, take a look, and feel free to comment!

Setting up and solving this Student Problem took about 30-40 minutes of the period.  However, I have started to become more of a fan of students working longer on really hard, conceptual problems rather than spending a shorter amount of time on easier, procedural problems.  Well, I still do see the benefit of both but this week I have had them do the big conceptual one first even though it's more time consuming and then they can follow that up with as much practice as they feel they need.

Friday - Partial Fraction Decomposition Day 2 (handout 6) - Same as yesterday, the actual discussion time was short, just to make sure they knew what I meant when I say "count up the powers" for repeated linear factors.  The focus was on choosing one of the three problems to do for their Student Problem 5.  #1 on the worksheet was "hard", #2 was "medium" and #3 was "easy".  They had to choose their difficulty level.  I had about half of every class choose #1 and the other half choose #2.  It was interesting to see most of my D/F students choose #1 because 3 bonus points were offered for doing the hardest one!  However, after discussing with some of them, they also just really wanted the challenge.  I love that!  I hope they aren't in the D/F range much longer...

This isn't due until the end of the period Monday, so I only have four turned in so far... but here they are!  The first 3 are all the "medium" problem and the last one is the "hard" problem.

Ask your questions!
We did have an issue this week with students asking questions.  Every night on their WSQ, they type their HOT question into the google form.   They are supposed to label their question as one of three things:
"Confusion"  - they ask a question about something they are actually confused about
"Discussion" - they ask a good discussion question (and they are supposed to provide the answer as well)
"Example" - they make up their own example (and they are supposed to solve it as well).

Then, they jot the question in their SSS packet so they can remember it the next day.

On Tuesday, I knew there were a lot of good questions from skimming through the WSQ submissions before class.  As I went around from group to group, I asked the students what questions they had.  In my first class period, I made it all the way around to all 33 students and I had ONE student tell me they had a question.  I then projected the WSQ submission spreadsheet on the screen and counted down the question column for period 1, and counted NINETEEN actual "confusion" questions.  We had a little discussion in class about the need to ask questions and clarify confusions.  I then made my way around the classroom again, group by group, and got to answer and explain about 15 or so questions.  They were great questions, and good things to ask and discuss - but I think it really is easier for students to say, "Oh I get it, no problem".  I need to continue to work on encouraging and reminding them to ask their questions.

Grade Improvement!

This is the end of the 2nd grading period. 29 of my 105 math analysis students raised their grade by at least one letter in the last 6 weeks, which is awesome!  Students have been re-assessing, learning material they didn't understand before, and showing so much improvement.  Right now, the only students who are failing the class are ones who haven't showed up to re-assess on anything consistently.  However, I hope in the next 6 weeks even more pull through and come show me what they've learned!

I am SO PROUD of so many of my students who started the year struggling with both the content and the technology and have persevered, asked for help, and are now succeeding greatly in the class.  I hope they realize that hard work and dedication do pay off!!!  Now, I just have figure out how to help the students transition better/faster at the beginning of the year so it doesn't take some of them 3 months to come around :)
Related Posts Plugin for WordPress, Blogger...