Teach 180: Last Day of Classes (Day 171 & 172)

I don't normally combine two days of teaching into one blog post, but here are two posts for the price of one.  Yesterday was not a good teaching day.  About an hour into the school day, I got a call from my daughter, Cassie, saying that my dad (Pop-pop to her) was in ICU with chest pains.  I wan't teaching at the time and had the next hour to make phone calls and figure out what was happening.  He is home now and fine (tests came back negative), but the scare gave me time to think about how precious time is with family.  After school, I spent time with Cassie using her camera to take a "photo shoot" of her at the local park in her white dress for graduation.  We had fun and not once during that time did I think about teaching/school.  Plus I have some new photos for the background on my computer and phone to remind me of her when she goes to college.

Today was the last day of classes and final exams start tomorrow.  My PreCalc students requested a Kahoot to review for their last 35 minutes with me.  They chose to review logarithmic and exponential functions and based on the Kahoot, they did need more work in this area.  The fact that the students realized that they need more work in a certain area is good.  Hopefully, this will help them target their review.  I encouraged them to send me emails with questions they might have over the next few days.  As several of them left, they shook my hand and told me to "have a nice summer" and "good luck next year".  All my students have been wonderful this year and I will miss them.

Teach 180: Grow or Stagnate (Day 170)

Beginning in the fall, I will be teaching at Kent Place School in Summit, New Jersey and I spent a good portion of my day continuing to pack and organize.  The last day of school will be here soon!  My school day ended with an "exit interview" conversation with the Headmaster, reflecting on what Moravian Academy does well and where Moravian Academy could improve.  As I reflected on my twelve years of teaching at Moravian Academy, I realized that I grew as a teacher as a result of my desire to learn and continue to improve.  It was not because of anything the school, administration or Board of Trustees did.  The systems for faculty growth were missing when I started at MA and sadly, they are still absent now. 

During my tenure at MA, I renewed my National Board Certification and helped three other teachers at Moravian (and a few at other schools) obtain their National Board Certificates. (The teachers taught middle school math, PE and Spanish.)  I lead workshops on Desmos, Smartboard, plickers, Google Hangouts, flipping the classroom and twitter.  Last year I visited the classroom of about a dozen teachers, many outside my department, to provide feedback to the teachers and consider how I might incorporate their teaching strategies in my own classroom.  I tried to lead a group called "Faculty Fellows" where teachers would get together to discuss teaching strategies or articles we had read.  When that failed (teachers did not want to give up their lunch time nor did they have time meet outside the school day), I tried to start a virtual book group on the book "Grade Smarter, Not Harder".  Although twenty signed up for the group, only two have participated sporadically.  I continued to grow and lead despite being in an environment that does not invest in growth.

I am sure I will continue to lead in some capacity at my new school.  However, I want to - no, need to - be surrounded by like-minded teachers who want to risk, share, and learn from each other daily.  I hope that Kent Place School is a place where faculty professional growth is the norm and not the exception.  I hope it is a school where faculty evaluation is primarily seen as an opportunity for professional growth, instead of a way to remove ineffective teachers.  I need to be a part of an organization that values people as resources and invests in their continued journey to becoming better teachers, because I am not done learning or growing yet.

Teach 180: Exploring the Desmos CL (Day 169)

Because I only had two classes on Friday and I was tired of sorting papers and packing up my classroom, I decided to spend some time playing around with the computational layer in Desmos.  I had learned about the CL when I went to San Francisco last summer as a Desmos fellow Cohort #2.  I had fully planned to dive into the CL when I returned from that weekend, but we still had two math teachers to hire with six weeks until the start of school.  And when I go on vacation in Maine, we go to a place with limited WiFi.

Thanks to @mrchowmath I was able to begin to dabble in the CL in an exploratory way through an Activity Builder he created.  He posted his videos on his website here.  (I am now also noticing that he has some "Breakout Room" activities that he created with Desmos AB, too.  I'll definitely need to look at these this summer.)

What did I learn from Jay Chow's videos?  First, that the computational layer is not as scary as the syntax makes it look.  Second, that the built in de-bugging suggestions are very helpful.  Jay created a google sheet for people to share their "homework" and now I have about a week and a half to try out one of the following.

I am thinking of trying #1 where a student enters a parabola in standard form and the vertex is reported back in a note along with the x-intercepts.  It would be interesting to see if students notice that the x-coordinate of the vertex is the average of the x-intercepts.  Right now my students don't notice this.  It may be because I only have them focus on the ordered pairs or I only have them focus on the graph.  Perhaps having these together will help them to see the relationship better.

Teach 180: Inverse Trig Functions (Day 168)

Note: After flooding the lawnmower multiple times, doing ironing, making dinner and spending time with my daughter, I went to bed and realized that I had not written in my blog for the day.  The events I am about to describe happened yesterday.

Most of my day was spent copying final exams for PreCalculus and organizing files in my classroom.  The one class I had today had only one student. (The other student in that class was at his sister's college graduation.)  We spent about 45 minutes of class time reviewing inverse trig functions.  We looked at the graph of f(x) = sin x and g(x) = sin^-1 x and realized that we would need to restrict f(x) so that it would be one-to-one.  Desmos made this very easy to see and very easy to check the validity of our chosen restriction for the domain of f(x) = sin x.  The graph of f(x) = sin x is in purple and the graph of its inverse is in black.  The dotted line is the line y = x to show that f(x) and its inverse are reflections of each other over the line y = x.  I added the points to the "ends" of the pieces to show that x and y coordinates are swapped when graphing a function and its inverse.

What about f(x) = cos x and g(x) = cos^-1 x?  Could we use the same restriction of -π/2 < x < π/2?  We used Desmos to see what would happen if we did this.

We could easily see that the restriction of -π/2 < x < π/2 was not correct. This was visible in two ways.  First, the restricted cosine function did not pass the horizontal line test.  Second, the red graph f(x) = cos x and the blue graph g(x) = cos^-1 x were not reflections of each other over the line y = x.  We decided to change our restriction to 0 < x < π and it worked!

I love how Desmos makes it easy to construct understanding and to play with the mathematics.  If we had chosen to do this discover by hand or with a graphing calculator, we would have been bogged down by the computations or button pushing.  Thank you Desmos for making discovery more accessible to everyone!

Teach 180: Hello, Anyone Out There? (Day 167)

At the end of February, twenty faculty members signed up to be a member of the google classroom "Grading Smarter, Not Harder".  I was excited by the possibility of sharing ideas with my colleagues around a topic that really interests me - assessment.  My initial post said, "This is a virtual book group that is open to all upper school faculty. Discussion prompts will be based on the book "Grading Smarter, Not Harder" by Myron Dueck. Reading the book is not required to participate. Prompts will be posted weekly or bi-weekly."  I have created prompts on each of the chapters for the past two months.  However, only nine posts have been made since the beginning of March and six of them were made by me.  If a post happens in a discussion forum and no one responds to it, did the post happen?  

I know that my blog is essentially me talking to myself and I feel like that in the google classroom, too.  But the purpose of a google classroom forum is to have interaction with other participants.  Being a part of communities like #MTBoS, the Global Math Department and the 2018 AP Statistics Reader Facebook Group help me to grow as a math teacher.  However, I want to have discussions about more general education topics that all teachers face.  We don't have those conversations at lunch or in the faculty room.  Discussions tend to focus on trying to understand the bell schedule, when grades are due, what to watch on Netflix and rating the school coffee.  Conversations like this are a nice diversion.  However, I want to learn from my math and non-math colleagues and grow in my profession.  Setting up face-to-face meetings during lunch did not work last year (only 2 or 3 people attended).  So, I thought this might work, because it would give teachers more flexibility with participation.  But with 2/3 of the posts being written by me, it appears I am mostly talking to myself. Hello, anyone out there?

Here is the post I did today about Chapter 4 - ReTesting.  Maybe someone out there will read it and comment on it.

Teach 180: Transforming Trig Graphs (Day 166)

As students learn new concepts or skills, it is important for them to get immediate feedback.  Saying "Just check your answer against the back of the book." is NOT feedback.  Sure, a student can see if they got the right answer, but it doesn't let them know why they got it wrong.  And if they get it wrong, the student really should have a chance to try it again.

In Calculus today, we reviewed transforming graphs of trig functions.  The students were able to get immediate feedback and targeted practice through DeltaMath.  The short video below shows a little about how we used DeltaMath for this feedback and practice. (Note: It does not have any sound.)

Students worked for a full 30 minutes and completed about 25 problems each.  I was able to help students, as they needed it, but students also were also able to figure out many of their errors for themselves.

Teach 180: The Tower of Hanoi (Day 165)

There are certain math ideas or puzzles that I believe every student should experience. The Tower of Hanoi is one of those puzzles.  After we reviewed the graphs of the six basic trig functions in Calculus today, we had some time to play with this puzzle.  Our goal was to move the disks from the left peg to the right peg in as few moves as possible.  The rule is that you can't place a larger disk on top of a smaller disk.  Rather than starting with 7 disks, we started with 1 and then moved to 2 and then 3, looking for a pattern as we increased the number of disks.  It only took about 10 minutes to see that the number of moves, M, based on the number of disks D, is given by the function M(D) = 2D - 1.

If you are looking for an animation of the tower of Hanoi, there is a nice 2-D and 3-D simulation at Towers of Hanoi Animation.  I will probably show this tomorrow and we may even create a pop-up card based on the towers of Hanoi. 

Teach 180: Dance Factor Dance (Day 164)

We are continuing to look at patterns in PreCalculus.  As students walked in the room today, I had the website www.datapointed.net/visualizations/math/factorization/animated-diagrams/ open.  The short screencast here you through the first 30 seconds of what they saw. (Music is called "Sunglasses" courtesy of "Loyalty Freak Music".)

I asked them "What's going on with the dots?"  At first they noticed the pattern of colors.  Something that I didn't even notice.  But soon after that they noticed things like prime numbers forming a circle, factors of 3 forming triangles, factors of 4 forming squares, factors of 5 forming pentagons, etc. Although we weren't talking about factors explicitly today, it helped them to continue to think about patterns and was a nice way to get their brains thinking about math.

Teach 180: My Geeky Lightbulb Moment (Day 163)

In PreCalculus today, we were doing some problems involving summation.  One was the sum of n²  from n = 1 to 4.  I knew Desmos could do this one and I showed my students how this worked in Desmos.

Then, we did the following "by hand".

I asked my students, "what would Desomos do with this problem".  I wasn't really sure.  When I entered the expression, it didn't give me "an answer", it gave me a graph.  "Why is it doing that?" I wondered.

And then it dawned on me!  Desmos was graphing the polynomial that resulted from the summation.  We confirmed this as a class by entering the expression 1 + x + x²/2 + x³/6 + x⁴/24 in line 2 and it graphed over the graph that was already there.  As I was thinking aloud and having my lightbulb moment, my students asked me, "Wait, you mean you haven't done this in Desmos before?"  They thought I was acting surprised and had done this before.  I do that sometimes.  But not this time, my surprise was real.  About an hour later, I shared my discovery with the colleagues in my math department.  They were also impressed.  This is definitely something that can't happen on a TI-whatever.

Teach 180: The Law of Sines Video (Day 162)

Today I am reviewing Law of Sines with my two students in Calculus.  As I was looking through my folder, I came across this video I created using my smartboard software and Animoto.  Since I made the video back in 2010, it is not the best quality.  However, it is a quick explanation as to why the Law of Sines work.  Perhaps I should work on one for Law of Cosines to add to my YouTube collection.

Teach 180: Playing with Patterns (Day 161)

In PreCalculus today we began a short unit on sequence and series.  Today students played around with about twelve different sequences to discover various patterns.  Next we looked at a graph of the sequence 20, 17, 14, 11...in Desmos.  When I asked students what they noticed, they said it followed a line.  When I asked what the slope of the line would be, they could quickly see that it was -3 and that the slope made sense relative to the pattern of subtract 3 from the previous number. However when we graphed the line y = -3x, it didn't go through the points.  How could we get the correct y-intercept?  Natalie said that the y-intercept would be 23, because she used the pattern backwards - adding 3 to 20.  Without formally talking about arithmetic sequences and the formal notation of an arithmetic sequence, students were able to create a formula to generate the nth term.

Next students worked in groups to generate formulas for 1, 5, 25, 125... and 1, 3, 7, 15, 31,...  After graphing them, the students realized that the pattern was curved and that an exponential equation would work better than a linear equation.  With a little bit of trial and error in Desmos, students discovered the equation y = 5^x-1 would work for the first sequence and y = 2^x-1 would work for the second sequence.  We didn't formally talk about geometric sequences today, but the fact that students are already thinking about a pattern found by multiplying by the same number again and again, means that they will be ready for the formal introduction of a common ratio next week.

Tomorrow we will begin class by looking at some patterns found at Visual Patterns by Fawn Nguyen. Here are two that I plan to use.  Can you figure out how many stars and footballs are in step 43?

Teach 180: Happy Monday (Day 160)

Luckily I did not have too many copies to make this morning and could just print the 12 pages or so on the printer.  Why could I not make copies?

Thankfully I only have I have two classes scheduled today - first period and last period.  For the first period class, I only had one Calculus student.  The other student is taking the AP Bio exam.  The seniors in Calculus are done with classes and don't need to attend class.  What did I do with just one student?  We reviewed the unit circle together and then the student worked on an assignment in DeltaMath.

For my last period class, the students are not required to come to class either.  They are all seniors.  In fact, some of them may not be able to come to class, because they have a senior exam scheduled.  However, I am hoping that several students show up to ask some more last minute questions before the AP Exam on Thursday.  Time will tell.  Happy Monday, everyone!!

Teach 180: The AP Statistics Murder Mystery (Day 159)

Friday was the last official day of classes for seniors and I felt that a fun review day for AP Statistics was needed.  Although I had seen the AP Statistics Murder Mystery in the AP Statistics Reader Facebook group, I wasn't sure if I would do it.  Then, I read several posts reporting how much fun students had with the activity.  This lead me to read the questions and I concluded that it was a pretty comprehensive review in a format that the kids would find fun.  I created envelopes for each of the six worksheets.  Each worksheet led to either a weapon, location or suspect being eliminated.

Students worked in groups of 3 - 4 to solve the mystery.  My first class took about 40 minutes to complete the activity and my second class whipped through it in about 30 minutes!  Not only did they review many of the AP Statistics topics, students reviewed some of the features on their calculator, too.  As always, there will be a few things I will tweak in future years with this activity, specifically the directions on a few of the handouts required some clarification.  A special thank you to Celia Rowland for sharing this activity.

Teach 180: Marbleslide Challenges (Day 158)

Today was my last day for seniors in Calculus and I wanted them to spend their final day of math in high school just playing with mathematics.  Thanks to Sean Sweeny's Marbleslide Challenge in Desmos Activity Builder we did just that!  One pair of girls completed eleven of the challenges in about an hour.  The short video below shows the one that I think they were most proud of.

Next year I am thinking of doing this as Sean describes in his blogpost, using teacher pacing and releasing one Marbleslides challenge a week.  However, I may want to reorder the slides for this activity builder based on what functions we are studying in class.  Looks like it might be time to start my summer list of "Ideas for School This Fall".

Teach 180: The Five Pointed Star (Day 157)

The seniors last day of classes is this Friday and we essentially finished the content in Calculus last Friday.  With only 4 classes remaining, I spent the last two days showing students the documentary called "Between the Folds".  It does a great job looking at how origami blends with math and science.  It also shows that math can be learned for the sake of learning math and that applications of the math often come later.  Here is the trailer for the video:

In the video mathematics professor Eric Dermaine shows that paper can be folded and one cut made to make various shapes.  In the video, Eric folds a piece of paper and makes one cut to reveal a swan.  

After the video, I worked with students to fold paper and make one cut to form a five pointed star.  The same fold and cut trick that Betsy Ross supposedly showed George Washington when the American flag was being designed. Here are the directions we used.  My five pointed star is seen below.  We also explored mobius strips and what happens when you cut a mobius strip 1/3 from the edge of the strip and in the center of the strip. 

Teach 180: What's Going on in This Graph? (Day 156)

Today I let my students experience the new New York Times feature called What's going on in this graph?  The students worked in groups to study the graph and answered the following three questions. 1) What do you notice?  2) What do you wonder? and 3) What might be going on in this graph?  After they discussed it together, we created a class post in the comment section.  Here are two screenshots of the graph, as it changed over time.

Here is what we composed as a class response to the three questions.  Our observations received a response from the moderator and a response from someone else.  

I'll be definitely taking these comments back to my students.  Once we look at the article at the bottom of the page, it will probably help students understand why including something, or leaving something off, a graph can help the graph tell a story.

Teach 180: Which Inference Procedure? (Day 155)

Today we decided to mix it up a bit in AP Stat by using a Kahoot! to review which inference procedure should be done.  Thanks to Kahoot user david.kish for giving me something to use/modify.  After this activity, several students recognized that they may need to do a little work in this area.  So, I sent them this link in an email.  Notice the variety of concepts that students can review!!

As we continue to review for the AP exam, I think it is important for my students to figure out what they don't know well.  I give them way more practice problems and resources than they need.  The idea is that they need to start to target their areas of weakness.

Today we also started an 8 day take-home final exam.  The students are getting one question sent to them at 8 AM for the next 8 school days.  Their solution is due at 8 AM on the following school day, at which time a new question and the solution to the previous question is released.  It is about 11 PM and already about half of the students have turned in their work.  In fact, several sent me their work before leaving school for the day.  Be sure to read a future blog about this form of assessment and how I think it impacted my results on the AP Statistics Exam.

Teach 180: The Partner Quiz (Day 154)

In Calculus on Friday, students took their last quiz.  For many of them, it was the last math assessment of their high school career.  The quiz involved integration by using substitution and integration by parts.  These two topics had been taught over the past two weeks.  However, many students were out for a day or two due to field trips, illness and college visits.  Most students could be successful at solving integrals involving substitution or integration by parts, but I felt that they would be more confident, if they worked with a partner.  There are many steps in solving these problems and the students seemed to like the idea of working with someone to catch the small mistakes that often happen when doing a very involved math problem.


However, the quiz went even better than I had hoped for.  Although everyone was asking to be paired with the student they thought had the highest grade in the class, each student seemed pleased with the partner I had assigned.  The room was filled with true collaboration and discussion for almost a full forty minutes.  Students wanted to do well, because they didn't want to let their partner down.  I believe that they spent more care and time checking their work than they normally do when being assessed individually.  Individual assessment is important, but this quiz was not about to make or break anyone's grade for the course.  Students knew this, but worked to do their best.  Each pair was able to do all the problems, no problems were left blank, and students were eager to check their answers against the answer key after the quiz.  The grades for all pairs were in the A range and any errors were more related to Algebra than Calculus.  The partner quiz was a "win" in my book and something I plan to do again in the future.

Teach 180: Let the Artist in You Shine! (Day 153)

For several years now, I have been doing a project in Geometry where students create drawings in Desmos.  (Here is a drawing that was done by one of my students last year.) This year I don't teach Geometry and I am doing the project in PreCalculus with the seniors.  There are two major problems that I have had with this project in the past - one is a management issue and one is the need for a student reflection component.  Thanks to Julie Reulbach I have solved both problems.  She created a Desmos activity builder to have students do the project.  I modified what she had - included some additional directions, a recommendation to use folders and some more student reflection slides.  Now all the projects are now stored in one place! The students don't need to remember to name their file and send it to me.  I don't need to find them in my overflowing inbox!  Stay tuned for how this change turns out.  And here is the link to the activity.  Enjoy!

Teach 180: Probability is my Kryptonite (Day 152)

In AP Stat today we reviewed several ideas related to probability - discrete random variables and their probability distributions, combining random variables, mutually exclusive events, independent events, conditional probabilities, geometric and binomial probabilities, complementary events.  As we got started looking at review questions, I admitted to students that "probability is my kryptonite".  That does not mean that I can't do probability questions, but I do need to slow down and take my time with certain types of probability problems.  I recommended that students do a full practice test in the next few days to see what their kryptonite, or weakness, is.  Right now they may think all the topics are pretty easy.  However, we are only focusing on one or two chapters on a given day.  Doing a bunch of questions in a mixed order from the entire year will be more challenging and would help students to figure out what they really need to study.  Knowing what you don't know is the first step to studying for a major comprehensive assessment.

Teach 180: Stand and Talk, Again (Day 151)

I did some more "Stand and Talks" today during AP Statistics to review for the AP exam.  To begin class, I asked students to brainstorm a list of words associated with data collection (experiments and surveys).  Here is the list they created.  We didn't dwell on the words or their meanings at that point.

Next, I asked students to do a "Stand and Talk".  They were to pick a word that they weren't sure how to define.  Then, they were to stand up and find a partner across the room to discuss that word.  Like yesterday, the discussion was quite animated for about one minute.  There is something quite freeing about admitting that you don't know something to a fellow student.  (Admitting you don't know something is scarier to admit to your teacher or in front of the entire class.)  But once they shared their uncertainty with a friend, students were more willing to share with the class.  If you teach AP Stat, can you guess which words were problematic?  If you guessed strata (or stratified random assignment) and blocking, give yourself a pat on the back.