## Monday, April 30, 2018

### Teach 180: Stand & Talk, Some Success (Day 150)

Today I tried the "Stand and Talk" strategy that Sara Van Der Werf shared at the NCTM annual conference.  I told my students that we were going to try something that I learned at the conference and that I was hoping that they would amuse me by trying it.  (Shoutout to my students for letting me try new teaching ideas with them.)  We did the "Stand and Talk" strategy in 3 of my 4 classes.  In my two AP Stat classes, it was "da bomb" in terms of student engagement.  We had just gotten done reviewing some multiple choice questions. The timing for this strategy was perfect, as I noticed that a few kids were starting to nod off.  I had the students do a "Stand and Talk" related to the AP Free response 2015 question.  This was a perfect question to discuss - no calculations involved and a pair of boxplots for the students to compare.  Not only did I get more out of every student in 2 minutes related to this question, I noticed that students were more willing to volunteer what they had discussed with their partner.  Students that would have not shared their ideas before were more willing to share!

That "Stand and Talk" in two of my classes.  What about the third class?  In PreCalculus, we had reviewed the graphs of all six trig functions before I went to NCTM.  I started that class with a handout of all 6 trig functions.  The handout showed the graphs, their domains, ranges, etc.  For the PreCalculus "Stand and Talk",  I told student to pick two of the six functions and describe how they were alike and how they were different.  This went o.k., but there was definitely not as much energy
as in my AP Stat classes.  Why?  Two reasons.  First, I did not have the questions written down somewhere for the students to reference.  (Wait?  What are we supposed to do again?) Second, we did this at the beginning of class after a three days of not having classes.  I was having the students go from 0 to 60 without any sort of warm up.  Had I done our "Stand and Talk" in the middle of class, it would have probably been more successful.

The two key takeaways for me? First, it's ok to try something new.  Second, if it doesn't work, don't abandon it, but try to figure out what went awry.  Tweak it.  Try again.  Stay tuned to tomorrow's blog to see how my tweaks turn out.

## Sunday, April 29, 2018

### Teach 180: NCTM Conference Day 2 (Day 149)

This year at NCTM I saw an incredible amount of good teaching.  Here is what I mean by that.  Even in a large lecture hall with a hundred or more participants, participants had time to share with neighbors and interact with the presenter during the talk.  The presenters still had a "lesson plan" for their talk, but they were very responsive to their audience and used the interactions with their audience to further sculpt or mold the direction of their talk.  Audience participation and less presenter talking - something to think about as a teacher.

One of the best sessions I attended was led by Sara VanDerWerf.  If you have a chance to see her, she is a very engaging presenter.  One of Sarah’s goal is that “Every student talks out loud about math EVERY day.”  Another one of her goals is that "Students will see it before I show them.  Students will say it before I tell them."  If students don’t do this, students don’t own the understanding.  It belongs more to the teacher than the students.  Stand and Talks is the easiest way to get everyone talking about math every day.  We did a practice Stand and Talk and it was quick and engaging.  This is something plan on trying on Monday in PreCalculus.

In Sara's talk she also said "It is ok to not tell students the question, to give them time to notice the structure."  For example, give a parabola with two x-intercepts and a y-intercept and then ask the students what questions could be asked from the visual.  We also need to start telling me our students “convince me” rather than “can you explain that” or “tell me more”.  Sara's talk was one of those talks that I wish I could watch over and over again.  I know I could pick up more tidbits of insightful and useful information each time.

In case you are wondering more about how to do a "Stand and Talk", here is the link to the Stand and Talk blog post by Sara.

## Thursday, April 26, 2018

### Teach 180: NCTM Conference Day 1 (Day 148)

Today I attended an ok session, a good session and a great session.  The great session was in a room with about 12-14 teachers. We had various prompts we discussed around teaching, resources, technology, etc.  I wish I could remember the prompts, because they were thought provoking and everyone could contribute based on their own experiences.  What I found most fascinating in the end was that whatever type of school or level of teaching (elementary, middle, high school) we all had the common goal of doing what was best for our students.  We recognized that teaching has changed both in terms of content and pedagogy over the past 10-20 years. But it ultimately comes down to building relationships and trust with the students in front of us.  Without that foundation, it is challenging to have students learn anything, let alone math specifically.

Another topic that cropped up in our conversations was that students constantly ask "When am I going to use this?"  The truth for many students is "After high school, you will never use this."  For example, imaginary numbers or the quadratic formula are probably not useful to 90% (just my estimate) of college graduates after college.  (My husband is a Ph.D. chemist and does not use the quadratic formula on a daily basis.)  Does lack of utility for topic X mean we don't teach topic X?  If the only reason for learning something was because it was useful, we never would have fractal based antennas in our cell phones.  It was at this moment in the discussion that I said the following:

I truly believe this.  Sure there is math content that needs to be taught, but most students won't remember it unless they go into a math or science field. What will carry students through their high school years and beyond in mathematics is the willingness to discover, try, fail, try again, fail once more, make connections and finally generalize.  The generalization will not be a fill-in-the blank scaffolded note, but it will be the result of the understanding that they have discovered and constructed for themselves.  Teaching in this way takes courage.  A willingness to let go and trust your students and yourself.  It is not easy and after 25 years of teaching, it is still a conscious decision that I have to make daily.  I don't consider myself to be an "expert" or "master" teacher.  I am like my students - a learner and I am trying to improve and become better at my craft on a daily basis.

## Wednesday, April 25, 2018

### Teach 180: A Long, But Fast Day (Day 147)

Today flew by with a whiz and a blurr.  It began by being greeted by three eager students with questions on sin and cos graphs and inference for linear regression.  This was followed by giving a student a make-up test, helping another student review chi-square inference and then counting students on a bus.  We were off to the birthday lovefeast chapel downtown and returned late for lunch.  As predicted by my English colleague on the bus, Probability of a late return = 1.

After a quick salad, I met with my fifth student of the day to work on problems related to inference for a difference between two means.  It was now 1 PM and I had not officially taught anything yet.  Now it was time to help another student with making up a quiz.  She has an injured wrist and must write all her work on a vertical surface.  After a few quick email responses, at 1:45 PM I began my "teaching" day with my first class.  In AP Statistics, we reviewed the format of the AP exam and I offered some practical suggestions.  My last class was PreCalculus where we reviewed period and amplitude of various sin and cos functions through a brief Kahoot and then students examined csc, sec and cot graphs and their properties.

My day ended with a 4+ hour drive to Washington, D.C. for the NCTM annual conference. (A special thank you to my colleague M.R. for her expert driving skills.) After trying to talk to my daughter about AP Stat on the phone (the page was a bit too blurry to see in Face Time), I picked out the following two sessions for tomorrow.  One of them is seen here.  I don't insure my phone.  Looks like I'll find out if that is a good choice.

## Tuesday, April 24, 2018

### Teach 180: Comenius Presentations (Day 146)

This year we had fifteen juniors complete Comenius Independent Study projects.  The students work closely with a mentor and complete over 40 hours of study, research and synthesis.  The final product
is a ten minute presentation in the evening in front of family and friends.  On Thursday of last week, six students gave their presentations and tonight nine students presented.  What impresses me most about these presentations is that students have chosen a topic that is of interest to them to pursue.  The students will have their work mentioned in their college application next year, but there is no assigned letter grade or GPA boost with these projects.  The projects are "messy", often requiring revision based on unforseen problems.  The students quickly learn that the road to research is rarely straight or well marked.  In the end, the students take all their newly found discoveries and distill them into a short, accessible ten minute talk.  This year's presentations were, once again, outstanding!  Thank you to the mentors for guiding.  Thank you to parents for encouraging.  Thank you to the members of the Comenius Committee (Dr. Dee, Dr. Moore, Mr. Polgar, Mr. del Real and Mrs. Weems) for helping to keep the students accountable for their work. And finally, kudos to all the 2018 Comenius Scholars for a job well done!

## Monday, April 23, 2018

### Teach 180: Multiple Ways (Day 145)

One of the big takeaways I hope my students get after being in my class is there are multiple ways to solve a problem.  Depending on what information you have or what you need to figure out, some ways are more efficient than others.  This is true in life and it is true in math.  In math, we can tell that the methods are valid when they give us a similar solution.

We began Precalculus today by reviewing how to find the area of an equilateral triangle with a side of x.  We used three different methods. The formulas for these methods are shown below. Half the class was assigned to use the second method and the other half of the class was assigned to use the third method. We confirmed the solutions by finding the area with the traditional (bh)/2 formula.

Although I had students work together, they were all using the same formula in their small groups.  This meant that they didn't get to fully experience the fact that there are multiple ways to solve this problem.  Next time, I plan to have students work in trios with each person being assigned a different method.  If they don't get the same same solution, they would need to help each other to diagnose what went wrong.  In addition to seeing multiple methods and possibly determining which method they like best, they would be relying on each other to evaluate their solutions and correct their errors.  A math teaching win!

P.S. I may even introduce Pick's Rule next year.  A fourth method that gives an approximation for the area based on lattice points!

## Friday, April 20, 2018

### Teach 180: Some Kindness (Day 144)

My daughter was getting her wisdom tooth out today. (Yes, only one.  I had four and her father had none.)  That meant that I was not at school.  However, I had a sense of how the school day started, because my daughter got a text from her friend.  On the table in the senior lounge were little slips of paper with each senior's name.  Every single senior.  Typed next to each name was a kind statement about them.  My daughter's read: you're sweet and sassy at the same time. you really care about people and want the world to be a more pleasant place

It was clear that they were written by a classmate, as some referenced a specific school event or growth and maturity over the course of the student's high school years.  If people shared kindness like this with each other on a daily basis, even a weekly basis, our world would be a better place.  Whoever the anonymous student was that did this, if you are reading my blog, I say "Thank you for taking the time to recognize what makes each student special and sharing it.  Your words will have a lasting effect beyond those simple strips of paper."

## Thursday, April 19, 2018

### Teach 180: Powers of i (Day 143)

Today I was visiting another teacher's classroom for an informal classroom observation.  They were talking about complex numbers and started to work with powers of i.  When I have taught complex numbers in the past, we calculate powers of i by noticing the pattern.  We recognize that there is a cycle of 4 and that i to any power can be determined based on where you are in the cycle.  In this class, the teacher was showing the complex plane for graphing complex numbers as she was talking about powers of i.  Then it dawned on me.  Why not visually represent powers of i and consider the cycle of 4 by seeing the complex number move around the complex plane?

Was there a Desmos Activity Builder lesson that already did this?  I did a quick search and came up empty.  That's when I created this in Desmos - Powers of i. If you hit the play button in line 6, you can see the power increasing by 1 and the number being plotted in the complex plane as it cycles, 1, i, -1, -i...  Here is an short screencast to show the animation.

Two things that would make this better are for the powers to display in a little box on the complex plane, as in i12, i13, etc., and for the coordinates of the point to be displayed as i, -1, -i, 1, etc. as the point moves around the screen.  Looks like I'll need to get into the computational layer of an Activity Builder for that.  Then, I would need to think about what questions to ask students.  Perhaps it is just as easy as "What do you notice?"  "What do you wonder?" and "Can you use this to find i103?"

## Wednesday, April 18, 2018

### Teach 180: GMD Shoutout (Day 142)

If you haven't heard of the Global Math Department, then you are missing out.  It is educators sharing their ideas, free of charge on Tuesday nights at 9 PM EST.  Topics are varied and include teaching ideas, assessment, technology and more.  Plus the talks span multiple grade bands.  I coordinate the hosts for each session and often serve as host myself.  Hosts help with tech glitches and moderate the chat room.  We have had some big names in the math education world present, including Matt Larson (NCTM President), Denis Sheeran (Hacking Mathematics), Michael Fenton, James Tanton, Rose Mary Zbiek, Dan Meyer, John Stevens (Classroom Chef), David Wees and others.  Sessions are recorded and you can access any of them once you join the GMD for free.  If you are interested in presenting a session or helping with hosting, please post a comment to this blog entry.

## Tuesday, April 17, 2018

### Teach 180: The Countdown Begins (Day 141)

This time next month my AP Statistics students will be making sure their calculators have sufficient battery power and will be searching for their #2 pencils.  It is 30 days before the AP Statistics exam and it is the first day of my "Countdown to the AP Statistics Exam Review Tips".  Each day on Twitter I will be posting an exam tip for the next 30 days.  I started this a few years ago and several of my tips are based on my experience as an AP reader.  Many tips are based on common errors and common misconceptions that I have seen on thousands of free-response solutions.  If you are an AP Statistics teacher, feel free to follow my tweets with your class!  Let the Countdown Begin!

## Monday, April 16, 2018

### Teach 180: AAS Ambiguous Case (Day 140)

Today I had planned to share this video about the ambiguous case for AAS with the Law of Sines.  However, we had a guest presenter today in the other PreCalculus class and I was able to sit in on that class.  He showed the students the following from Geogebra.

I decided to use it in my class later that day, but chose to work with an angle other than 30 degrees.  It didn't work quite as well to get just one right triangle.  It snaps to one right triangle when the angle is 30 degrees, because there is an exact length of 1/2 of the hypotenuse.

There were a few students absent today and I plan on showing them the following short demo video to get them up to speed when our class meets again on Wednesday.

## Saturday, April 14, 2018

### Teach 180: Math ≠ Exercises (Day 139)

This weekend (Friday through Sunday) I am working with a group of AP Statistics teachers.  Although our work day was done at 5 PM, we naturally slid in to talking shop through much of dinner.  There are two Khan Academy representatives with us and of course, we talked about flipping the classroom a bit.

Several years ago, I flipped some of my lessons and found out that not all of my students enjoyed it.  The main reason was that I was not there beside them for them to ask clarifying questions.  Yes, they watched the video, but they wanted to know more at that moment or were not convinced they they fully understood the concepts.

Now I only flip lessons on snow days or to do some quick review of content and vocabulary at the start of a unit.  Proponents of flipping would say it is better, because the students can "go at their own pace" with the video and you can then use class time to work on problems and help individual students.  That model is fine if your teaching philosophy is "I Do" (video watching) "We Do" (working on problems with a partner) and "You Do" (individual practice) math instruction.

Consider a lesson on arithmetic sequences.  In the flipped lesson, students would watch a video where the speaker would talk about the pattern and derive the formulas for the nth term and the sum of the first n terms.  A few examples would also be shown.  In class, the student works through problems like he or she saw either alone or in groups.  The student demonstrates "understanding" by being able to mimic the examples.  After twenty-five years of teaching, I would say that this student is not doing math.

Mathematics is making conjectures and discovery.  Mathematical thinking involves reasoning and discussion.  The student watching the videos has no joy of discovery and his or her learning of the content was certainly not a social experience.  At no point did the student have to "look for and make use of structure."  The structure was explained and the video teacher made use of it.  At no point did the student get to "construct viable arguments and critique the reasoning of others."  The student never spoke to anyone.  I am not saying that these last two things couldn't happen post-video.  However, by having the content be taught by video we strip mathematics down to a mindless set of rules and procedures.

What would I propose instead?  List the following sequence on the board.

Then ask students for the 100th term in the sequence.  You can let them use their calculators.  At some point they will lose track of where they are.  If they get to the 100th term, ask for the 1000th term.  At some point, they will stop using their calculator as a hammer to solve the problem.  They will be forced to "look for and make use of structure".  Given enough time to think, work together and ask questions, they will "construct viable arguments and critique the reasoning" of their peers.  They will find the traditional formulas for themselves AND truly understand why they work.

## Friday, April 13, 2018

### Teach 180: Teach Collaboration Shout-Out (Day 138)

When I first joined Twitter several years ago, I was doubtful as to how it could be useful for math.  Tweeting live updates of faculty meetings?  Tweeting "having coffee and grading still", while true on many days, didn't have any value. But today was a day where Twitter came in handy.  It was a day for teacher collaboration.  Before school started, a colleague from my school asked me a question about graphing a function that was bounded by the x-axis.  She had read my blog from the previous day and was wondering if I could help her with her problem.  After a minute or two of trying and getting nowhere, I decided to tweet out the desmos page and get some Twitter collaboration.  Shout out to @MrCorleyMath and @mathteacher1729 and @MarchtoCharm and @Desmos for getting the job done.  My colleague and I thank you.

Here is our conversation. (I just realized that I typed "want" as "what".  Oops!)

## Wednesday, April 11, 2018

### Teach 180: Know Your Scale (Day 137)

When I am preparing my lessons and have an "Ah Ha!" moment, I often work that "Ah Ha!" moment into the next lesson.  This blog describes such an event.  Today students were working on areas bounded by curves.  One such problem had students finding the area between y = x2 and y = x2 + 3.  I was working with an individual student and was having trouble showing the area of interest with Desmos. After a minute or two, I got the correct graph by creating a compound inequality to graph the region between the curves and then using a compound inequality to restrict the domain.

After class was over, I realized it would be good to have all my students create this for themselves in desmos.  I still had the page open in desmos and decided to use desmos to find the area between the curves with a definite integral.  At first I was confused.  How can the area be 3?  There are way more than 3 squares shaded.  Is Desmos not recognizing something I typed?  Did I do something wrong with the set up?

Since I was really tired, I decided to work on something else for a while and come back to it.  I often tell my students that staring at something won't make you understand the problem.  Sometimes we need to leave a problem and come back to it later with a fresh set of eyes.  That's what I did.  Based on the title of this blog entry, you may have figured out what I was missing.  What is the scale here?  Each square is not 1 unit long, but 1/2 unit long.  This means that the area of each individual square is 1/4 of a square unit.  Each group of 4 squares is 1 square unit!  Now the answer made more sense.  When I present this to my students tomorrow, it will be interesting to see how quickly they note the scale in the graph.

### Teach 180: Math Is More Than (Day 136)

When you ask people what is mathematics, most people would probably say that it is computing or arithmetic.  However, math is so much more and it is my job to help students understand that.  While recently reviewing the National Board's AYA Math Certificate Standards for content validation, I was struck (or as the kids say "shook") by what was written on pg. 14.

It states "Thinking mathematically includes representing, modeling, proving, experimenting, conjecturing, classifying, visualizing and computing - all ways in which to approach mathematics and life."  There are definitely many times this year when this statement would describe mathematics in my classroom.  Now I wonder.  Do my students see math as more than computing?  Do they feel like they are "thinking mathematically?"  The statistician in me wants to gather some data on this and I plan to in the next few weeks.

## Tuesday, April 10, 2018

### Teach 180: College Decisions (Day 135)

Today in AP Statistics I had planned to look at an old free response problem and then allow my students time in class to work on a problem set related to chi-squared tests of association and homogeneity.  That worked well in one of my classes and not so well in the other class.  In the other class, students wanted to, actually needed to, talk about college decision making.  And so after we had finished the problem, I let them talk.

This year I have about thirty seniors and hearing them discuss their college process has really opened my eyes to the stress students feel about getting into the schools they have applied to.  The stress doesn't end once the admissions decisions are made.  Many students are deeply worried about picking the school - not only what school is right for them, but what school will make their family the happiest.  And yesterday, the discussion turned to what school will give the student the best chance at getting into a good medical school.  One of my students applied to 20 colleges and it was refreshing to hear the students in my class offering advice and insight.  They even helped her to remove two colleges from her list!  The genuine care and concern that students share for each other will be something that I will remember about teaching at Moravian Academy.

Side note: On the other side of the spectrum.  My daughter applied to two colleges and loved one more.  Her decision was made before Thanksgiving.  She already has a roommate and a college major chosen - to become an elementary school teacher.

## Monday, April 9, 2018

### Teach 180: From 0 to 60 in 1.513 Seconds (Day 134)

In Calculus, we are studying integration and on Friday we were going to look at the acceleration of a car from 0 to 50 miles per hour.  To make it a bit more interesting that just considering a generic car, we considered the acceleration and distance traveled by the world's fastest accelerating car.  The world record for the fastest accelerating car is 0 to 60 mph in 1.513 seconds.  We started by calculating the acceleration in miles per hour squared.  Then, we integrated the acceleration function to get the velocity function.  Finally, we integrated the velocity function to get the position function.  For the final step, we decided to change miles to feet. The verdict?  The car travels from 0 to 60 mph in 1.513 seconds and travels about 66 feet.

I still have the feeling that my students don't understand how fast that truly is.  Perhaps next time, we need to compare that acceleration to acceleration of roller coasters.  Or, we need to see how fast 1.513 seconds really is by having students try to write their names in 1.513 seconds.

## Thursday, April 5, 2018

### Teach 180: Riemann Sums (Day 133)

In Calculus yesterday, we began to investigate the idea of approximating area under a curve by using rectangles.  Although I can draw a diagram of this situation and students can look at static pictures showing width of the rectangles decreasing and the number of rectangles increasing, I could tell that they still couldn't fully understand the dynamic process of decreasing the width of the rectangles and increasing the number of rectangles.

Thank you to Desmos for coming to the rescue!  This page is wonderful, because you can change the function, the number of rectangles and how the rectangles are formed (left endpoint, midpoint, or right endpoint).  The next time I teach this topic, I plan on showing my students this video twice and asking them what they think is going on in terms of the area and the rectangles.  I purposefully recorded it without audio.  Feel free to use the video and put your own audio over top or play it in class and have students comment on it as it plays.

## Wednesday, April 4, 2018

### Teach 180: M&M's (Day 132)

We began our unit on the chi-square distribution in AP Statistics today by counting M&Ms.  The null hypothesis was the M&M percentages were as claimed by M&M/Mars.  The alternative hypothesis was the percentages were not as claimed, meaning that at least one of the percentages is off from what is claimed.  Initially the 8 AM class started with 2 students and by 8:15 AM, we had 5 of the 8 students that were to be in class. (There had been a bad accident that caused several students to be late.)

Here is the data we gathered:

Note: We initially had 102 candies in our sample.  I told the students that it would be easier to figure out the expected counts if we had 100 M&M's.  They suggested removing two candies from our collection "at random" with the help of random.org.

Our p-value was 0.0025 and our chi-square test statistic was 18.3951.  The image at the left is a screenshot from the TI-84 emulator.   Our conclusion was to reject the null hypothesis, because our p-value was lower than a standard alpha level, like 0.05.

So, where was the difference?  Which colors were the largest contributors to the test statistic.  When we ran the test on the calculator, it created a list called "CNTRB", which contains the individual addends that were used to create the test statistic.  The largest value was about 11.6 for yellow and the second largest contributor was 4.4 for the orange candies.  If you look at the observed and expected counts, this makes sense.

Fun Fact: M&M's are made at two different plants, Cleveland and Hackettstown.  If you look at your bag, you can see where your candies were made.  The batch we used was made in Cleveland.  Notice the CLV in the best before date box on the bag.  This is quite interesting, because we live about 30 minutes from Hackettstown.

Also, you would think that the colors would be made in the same proportions at both plants, but they aren't.  They are close, but they are not exactly the same.  The screenshot below shows the proportions of each color from each plant.  I am not sure when the percentages changed, but I do know that in 2008 the percentages were different than what is listed here.

## Tuesday, April 3, 2018

### Teach 180: A Friendly Reminder (Day 131)

Sometimes as a teacher I forget.  Today I forgot the fact that my students' brains might need a bit more review after a four day weekend. Although we had talked about angles in standard position for about 2 weeks, it was like my students couldn't remember what a radian was or the difference between positive and negative angles.  And the six trig functions in terms of x, y and r?  Yikes!   We did eventually get back into the swing of things, but it took about 30 minutes.  Note to future self.  Build in more targeted review after a long-ish break or even give students 5 minutes to look over their notes from the previous lessons.