Friday, March 30, 2018

Teach 180: 14% (Day 130)

On Thursday, seven of twenty-two students didn't take the AP Statistics test that was scheduled for that day.  Absenteeism and apathy among my seniors is rampant.  If I recall correctly, I gave 13 of my seniors incompletes for the current reporting period due to work not being turned in or an assessment that had not been taken yet.

Some of the students have missed multiple days of class and the fact that they had to learn half of the current chapter via screencasts on snow days did not help.  As a stat teacher, it made sense for me to gather some data.  The table below shows the overall average percent absence was 14% over the past two weeks.
A lack of "touch points" has been a problem for many teachers this year.  One Algebra 2 teacher in my department greeted her students on March 26th with "Who are you again?"  She hadn't seen the students since March 9th due to days when she wouldn't normally meet with them, spring break, snow days and the standardized testing day.

Our math departments (middle school and upper school) will be meeting on April 19th to discuss what we have chosen to leave out of our curricula this year and the impact it will have on students as they move through the sequence of math courses.  In future years, we may offer AP classes in math, but we may no longer have any time to review before the AP exams.  This will be especially true if we need to take content from lower level courses and move them into upper level courses.  Our current bell schedule will be the death of our AP math program without a coordinated, major overhaul of the curriculum in the next two years.

Wednesday, March 28, 2018

Teach 180: Bounce Off! (Day 129)

One of the things I love about our new schedule this year is that I have more time to spend with my advisees.  They are seniors this year.  As college decisions come in, I can sense them starting to relax.  Today in advisory we spent about five minutes discussing the assembly period from the previous day. Then, we played a game "Bounce Off!" (Thank you AP Stat Secret Santa for the gift!)


Students read the rules and then decided to make their own rules.  The rules became modified as we played the game for a nearly 30 minutes.  After the first game, they were laughing and having so much fun that I decided to join them.  There wasn't much math involved in this game, but it did get me to thinking math-related questions, like "What is the best bounce height?", "What is the best bounce angle?" and "What is the optimal distance from the board?"   These questions will need to be saved for another advisory meeting.

Tuesday, March 27, 2018

Teach 180: Discovering the Antiderivative (Day 128)

Today in Calculus students were introduced to the idea of the antiderivative.  I wanted them to discover how to take the antiderivative of xn where n ≠ -1.  In addition, I hoped they would discover the reason why +C is necessary when finding an indefinite integral.  I began by having students fold a piece of paper into thirds.  On the top third, they wrote a polynomial function of their choice.  On the middle third, they wrote the derivative.  Then, I had them fold the paper in such a way that the original function was inside.

Next, students traded folded papers with their neighbors and were instructed to find the original function - the one that was now folded inside the paper.  After they had written down what they thought f(x) was, they were allowed to open the paper and see if their function matched the original function.  Here, you can see a sample paper.

This led to the discussion of three main ideas.  First, did you get an exact match to the original function?  If not, why not.  If the original polynomial function had a non-zero constant term, then it was very likely that there was not an exact match.  This led us to adding +C to show that there could be constant added to the end of the original function.  Tomorrow we will consider this as a family of functions.

Second, how did you actually figure out the exponent and coefficient in front of the exponent when you were trying to undo the derivative?  What procedure did you use?  Most groups could articulate adding 1 to the exponent.  But only two groups were able to describe dividing by the new power.

Finally, we looked at why n ≠ -1 when we wrote our general rule for taking the antiderivative  of xn. What would happen if we added 1 to the power and divided by the new power?  Raising a number to the zero power isn't a problem, but dividing by zero IS a problem.  It was at that point that a student recalled that the derivative of ln(x) is 1/x.  So, the antiderivative of 1/x would be ln(x).

Monday, March 26, 2018

Teach 180: These Lights (Day 127)

At the beginning of the year, I had two seniors in my PreCalculus class who did not think they could do math.  Today, one of them told a math joke and the other explored a feature in Desmos.  They were engaged and they were learning.

For part of today's lesson, we were learning to use calculators to find values like sin(43o) and cot(π/12).  About 1/3 of the class had graphing calculators and 2/3 of the class had their phones set to landscape mode.  Although I had only planned to show how to use the graphing calculator, I found myself giving instructions on using the calculator in the phone.  It took me back to the days of using my scientific calculator in 1986.  "Hit 43 first and then sin".  Or "do π divided by 12 and hit equals. Then, hit tan and then 1/x."  Very clunky.

I had one student using Desmos for his calculations and he ran into a slight problem.  He was not getting the same answers.  Perhaps he was in Degree mode.  Yes, that was it!  Is it easy to switch to Radian mode?  What this short video and you decide.   

My other student told a math joke.  He said to me, "Mrs. Nataro, I think there is something wrong with these lights."  I looked up at the ceiling and then asked what he was talking about.  He pointed to something he had written on his arm.  He had written sin/cos.  "The lights are making my arm tan," he said and then grinned.

While some of my seniors have developed a serious case of "senioritis", these students are still willing to participate in class and they may not realize it, but they are actually learning something, too.

Sunday, March 25, 2018

Teach 180: But What Question Do I Ask? (Day 126)

Part of Friday's lesson involved reviewing special right triangle lengths.  I had given the students a 30-60-90 triangle with a hypotenuse of 1.  The students remembered numbers like the square root of 2 and the square root of 3 were involved, but didn't know how to figure out what the sides of the right triangles were.  I told the class to get out their laptops and look it up.  "Google it."  One student said "But I don't even know what question to ask."  Some students quickly found an answer, but it wasn't quite the answer we needed.  The right triangle they found had a hypotenuse of 2.  With a little bit of discussion, we were able to figure out that the triangle with the hypotenuse of 2 was twice as large as the triangle we were considering.

One argument against teaching basic concepts or facts is that students can always "Google it".  And while it is true that students have a vast resource of information quickly at their fingertips, if they "don't even know what question to ask", they can't get the answer.  It had never crossed my mind that students don't know how to ask questions or evaluate sources.  This is a skill that is easy for me and a skill that I assumed my students had.  Asking the right questions, evaluating resources and modifying searches.  Those are life skills that all students need.  I know we don't have a course where this is taught, but perhaps I can model this more.  It is definitely a problem that needs to be addressed across the curriculum.

Thursday, March 22, 2018

Teach 180: Yet Another Snow Day - Ugh! (Day 125)

Technically, this is snow day #7 for us.  (Snow day #6 was on a day of conferences and teachers had to reschedule the conferences on their own time.)  All of our snow days have happened since January 1.  If you also include the regular days off, like President's Day and MLK Day, we have had a very inconsistent schedule in 2018.  The graphic below shows this inconsistency.  An Blue box with an X means NO SCHOOL; this was a planned day off from school.  Red means "Snow Day" and Yellow means 2 hour delay or early dismissal.  When we have a 2 hour delay or early dismissal, the classes that were scheduled to meet at the beginning of the day or the end of the day don't happen. 
The class I am most concerned about is AP Statistics.  The AP exam is scheduled for Thursday, May 17.  However, the seven school days leading up to that include senior exams or other AP exams.  This means that I might not see my students in class in the few days just before they take their AP exam.  Therefore, the bulk of their review needs to happen prior to May 1. 

Since January 1, I have seen my period A class for a total of 28 one-hour class periods.  My period F class I have seen a total of 30    one-hour class periods.  I have lost 6 hours of instructional time for period A and 4 hours of instructional time for period F. 

That translates into one less week of time for review.  For the past eight or nine years, my students have reviewed one chapter a day for 12 consecutive school days.  Now they will need to review 2 or 3 chapters each time we meet.  The fact that on Monday I had one of my "A" students ask me, "What is the median, again?" has me very, very concerned.

I know that there are other AP teachers that get zapped by snow days in the Northeast and rush to complete the curriculum and review.  If you have any thoughts on minimalist reviews, I would love to hear from you.

Wednesday, March 21, 2018

Teach 180: Snow Day = Online Teaching Day (Day 124)

Outside my front door at 10:30 AM today
We have a snow day today and potentially a delay or snow day tomorrow.  If I didn't teach my AP Statistics class today, we would have fewer days for review.  With only five class periods to review without today's snow event, I can't afford to NOT teach.  Unfortunately, my two sections are on different days in the current chapter due to our new bell schedule this year.  This means I had to create two different screencasts for my two classes. 

Screencast 1 is on "Confidence Intervals and Hypothesis testing for Two Proportions".  Screencast 2 is on "Confidence Intervals for Two Means".

Tuesday, March 20, 2018

Teach 180: Cleaning Out and Moving On (Day 123)

Today I spent time cleaning out and organizing my classroom.  After 12 years of teaching at Moravian Academy, I will be moving on teach at Kent Place School in Summit, New Jersey.  I accepted their offer about 2 months ago.  Over spring break, I realized how much I have accumulated in 12 years of teaching and I decided to spend 10-15 minutes each day over the next few weeks pitching, recycling and organizing.  Today's 10 minutes turned into 45 minutes.  In the process, I found two items that were particularly interesting. 

Item #1 was an original TI-Nspire with the interchangeable TI-84 faceplate.  I got this when I went to a T-Cubed Summer Institute about 10 years ago, I think.  Since it belongs to the school, I'll leave it in my closet and let the person who inherits my room decide what to do with it.  Right now it doesn't work and should probably be e-cycled along with the TI-92s that were in my closet when I arrived 12 years ago.

Item #2 is shown at the right.  It is my collection of Mathematics Teacher journals.  I am thinking that I'll reach out to my college alma mater and see if they would like them.  I could see having each math education student in a math seminar class pick one journal "at random" and then report back on an article the student read.  

I'll miss many of my colleagues and friend from Moravian Academy.  I'll also miss many of the students.  However, this change was necessary for me to continue to grow as a teacher and to do what is believe is best for student learning.

Teach 180: Becoming Mathematicians (Day 122)

On Day 117, I posted that I wanted to take a class period to have my students look at the following problem.  You can read more about the blog post here.

My students have never formally worked with geometric or arithmetic sequences before and I thought that I might use this problem to introduce them to those ideas.  But as my ideas for the lesson took form, I decided that I wanted them to play with the mathematics and that too much formality might squash that.  Here are the 7 steps that were part of that discovery.

Here are two main items that my students noticed.  For some reason I was so focused on the relationship between the log of a geometric sequence and the resulting arithmetic sequence that I didn't notice the very first thing that my students noticed! (Note: The bases of 2 and 4 were chosen to make it easier for students to make observations.)

Here is what my students wondered:

1) Is the ratio of f(x):g(x) always 2?  

2) What if a and r weren't even, positive integers?  

3) What happens if 0 < r < 1?

Here are screenshots from Desmos to show some of the student's investigations.

This screenshot shows one instance where a and r (initial term and constant ratio) are positive odd integers.  Students noted that the ratio of log2 x and log4 x is still 2.  Then, the question becomes "why?"

We proved this was true by rewriting log2 x and log4 x with change of base formula.  We had log2 x = (log x)/(log 2) and log4 x = (log x)/(log 4).  After writing the quotient of f(x)/g(x) and simplifying, we had (log 4)/(log 2).  Since 4 can be thought of as 2 squared, log 4 can be rewritten as log 22 or 2(log 2).
So, 2(log 2)/(log 2) = 2.
Next, we investigated how to use a slider to look at many different values for the base.  At one point, my students were even able to predict what values of b would produce a ratio that is an integer.  They recognized that since f(x) was log base 2, that b would need to be a power of 2.

We also looked at how the table of values changed if r was a value between 0 and 1.  This led to an arithmetic sequence that was decreasing by a constant difference, instead of increasing by a constant difference.

Although we did not get around to creating a more formal proof, I was pleased that my students felt comfortable with making observations, not knowing for sure if there was a right answer and checking their conjectures both informally (with desmos) and more formally with properties of logarithms.  We finished class with me telling my students that they really were "thinking like mathematicians" that making observations, forming conjectures and deciding if they are true or not was what it really means to "do math".  One of my students noted that Desmos made it much easier to easily test our ideas.

Sunday, March 11, 2018

Teach 180: I Want Proof (Day 121)

A few days ago I was talking to a colleague about the fact that her Honors Geometry students weren't as curious this year.  When she asked the students if they wanted to know why something was true or why it made sense, most of them said "No".  These same students want their math teachers to tell them what strategy they should use or what the assignment is so they can do it and get it over with.  These students say, "But how are we supposed to know how to do that problem on the test. We didn't have any like it on the homework."

When I was in high school, I always wanted to know why something was true and my teachers were very good out outlining steps to a proof or leading us in reasoning why something was true.  Just because some of my students don't share the same curiosity I did doesn't mean that I shouldn't expose them to mathematical arguments.  In fact, one of the Common Core State Standard for Mathematical Practices states that "Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments." In addition, "Students at all grades can listen to or read arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments." 

Most of my Calculus students are seniors and simply want to be told formulas for the derivative.  Few of them care to understand why the derivatives of f(x) = ax and f(x) = logax are what they are.  The fact that they don't care, doesn't mean that I should not share the argument with them.  In class today, I presented the following argument to show that the derivative of f(x) = ax is (ln a)(ax).  Enjoy!

Thursday, March 8, 2018

Teach 180: Project Based Learning (Day 120)

At my current school, we meet about five times a year (approximately an hour each time) in cross-divisional groups to discuss something we ranked as being of interest to us.  Unfortunately, I was one of several people who did not have one of his or her top 3 choices in the list of options.  This meant that I had to choose among the following options: project-based learning, social emotional learning, intrinsic motivation, well-being or integrating play in learning.  I was really interested in focusing on assessment as a tool for learning and the closest group that was related to assessment was Project-Based Learning. 

My group consisted of a middle school math teacher, a middle school history teacher, a middle school English teacher, an upper school Spanish teacher and myself.  We read the book "Setting the Standard for Project Based Learning".  The practical advice offered was mainly to not rush, make sure all stakeholders buy-in and to allow for plenty of time for collaboration.  Little was offered in the way of assessing projects or assessing group skills.  The one project that was specifically given as an example for math would have required about two weeks to complete and it covered a topic that we typically spend two days on in class.  Although the book said that PBL is better for students, especially for learning collaboration skills and for at-risk students, at no point did it offer any empirical evidence.  All the evidence that was presented was anecdotal.  At no point did it offer suggestions for what content should be removed from the high school mathematics curriculum to make time for PBL.

Today was the last day our group met and we put our "book report" in google slides to share with our colleagues.  Here was the slide I created.

Overall, our group was not impressed with this book and felt like it was a sales pitch for the Buck Institute for Education, one of the two main publishers of the book.  Would PBL work in a large scale at my current school?  With so many different initiatives happening and faculty being drained of time and energy, I would say it would be unwise to attempt a PBL initiative at this time.

Wednesday, March 7, 2018

Teach 180: SolveMe Mobiles (Day 119)

It's not a teaching day today, but I will still spend about 4 or 5 hours at home today working on school related things.  Next week is spring break and I would love to spend a minimal amount of time on school related items over spring break. (I'd rather spend my spring break spoiling my niece and nephew.)

One of the things I'll be doing today is playing at the site SolveMe Mobiles.  Your progress can be saved and you can create your own puzzles.  Here is puzzle #15 from the explorer collection and puzzle #71 from the puzzler collection.

We know that to keep the mobile balanced what is on the left must equal what is on the right.  As a matter of fact, if you click on the horizontal bars and drag them, you get systems of equations. And then if you "pull down" with your mouse on the right heart, it subtracts one heart from both sides of the equation.


Finally, when you enter the values for each shape, you get instant feedback about your solution.  By the way, the answer to this one is not trapezoid = 3, heart = 1 and square = 5.

Thanks to Kevin Smith for reminding me about this resource at his Global Math Department session last night.  His session was called "Gamify the Math Classroom" and you can view the video of the webinar here.  Note that if you are not a member of the Global Math Department, you will be prompted to set up a free account.  However, when you do that you'll be able to see many of the other wonderful free webinars that have been recorded over the past several years.  I serve as host for many GMD sessions and we are always looking for quality presenters.  If you are interested in presenting, send me an email at

Tuesday, March 6, 2018

Teach 180: I Call Time Out!!! (Day 118)

Thanks to winter storm Riley the Friday evening performance of the musical had to be moved to Sunday evening.  Students involved in the musical were allowed to miss school on Monday. (They were at school from about 9 AM to midnight, or later, on Sunday.)  This meant that I missed 43% of my AP Statistics class on Monday.  Initially we were going to review for a test on Chapter 9.  It is an AP course and students who are absent are expected to learn things on their own when they are absent.  But the students who missed class would have no clue what I was talking about.  The class they missed was on one-sample t-tests for means and paired t-tests.

Time out!!!  Change of plans.  I dismissed the 57% who had been in class on Monday and sent them to lunch.  Then, I taught the lesson on one sample t-tests to the 43% that had missed class.  When we reconvened from lunch all students had time to work on the problem set on one sample inference for means.  I am fortunate that I work in a school that allows me the flexibility to work with the students in this way.  However, with more snow in the forecast, I am nervous about students learning all the content needed before AP exams start.  Right now, we have 5 days on the schedule to review for the exam.  This is about half or one-third of the time I have had in previous years.

Teach 180: A Sequence of Logs (Day 117)

Rarely do I have time to do something extra in my classes and that is especially the case this year.  Even though we are about 2 weeks behind where we were last year, my class of PreCalculus students is pretty hardworking and inquisitive.  They are ahead of the other PreCalculus classes and it looks like I'll end up doing something fun after spring break to allow the other PreCalculus classes time catch up.

Today I was starting to purge my files at school and I ran across this gem of a question.  I have some thoughts about how I will teach it.  My students aren't familiar with the concepts of geometric sequences and arithmetic sequences, but they do understand exponential and linear functions.  I'll have plenty of time between now and March 19th (the day back from spring break) to design a lesson around this question.  One of my goals will be for the language of sequences (terms, initial term, geometric, arithmetic, common ratio and common difference) to develop naturally as a consequence of playing with this problem.  Any comments or thoughts on this welcomed!

Friday, March 2, 2018

Teach 180: You Fool. That's Wrong (Day 116)

It's currently 1 PM and I am writing this blog entry from the comfort of my dining room table thanks to Winter Storm Riley.  Assuming we don't lose power, I'll also be able to get my laundry done, clean the bathrooms and vacuum before 4 PM.  I only taught the first two periods today and those classes met.  This is very good news.  It means I won't be having to revise my schedule and shuffle things around for classes that did not meet.  This winter has been very harsh and it has made it challenging to have continuity. "Whenever it was we last met" has been a common phrase coming from my mouth this year.

Today I showed the "You Fool. That's Wrong." video in AP Statistics.  If you aren't familiar with it and you teach AP Statistics, you should be.  Although we have only been doing hypothesis testing for about a week, I showed the video to my students today and it was met with some chuckles.  One student even said, "Now I finally understand p-value!"  So, thanks to Steve Willott for bringing levity to my classroom and helping one of my students today.  (Also, here is a link to a handout that is related to the scenario in the video.)

Thursday, March 1, 2018

Teach 180: Is the Coin Fair? (Day 115)

Although students can spout off what Type I and Type II errors are for hypothesis testing, they often have trouble understanding what would lead to making this type of error.  Today I gave them information on two coins.  Students had to determine if the coin was fair or not.  The null hypothesis corresponded to the fair coin and the alternative hypothesis corresponded to the unfair coin.  What would you conclude about Coin 1 - fair or unfair?  What would you conclude about Coin 2 - fair or unfair?



I had students discuss this in groups and one group of students couldn't decide.  They thought both coins were fair.  So, I asked them to look at the 95% confidence intervals.  If they could only choose one coin to use for the NFL Superbowl coin toss, which one would it be?  They said Coin 2.  The confidence interval contains .50 and the proportion of heads is closer to 50% than the other coin.

The big reveal came when they flipped the papers over.  Coin 1 was actually the fair coin and Coin 2 was actually the unfair coin.  With Coin 1 they had made a Type I error and with Coin 2 they had made a Type 2 error. 

We'll see if this helps student understanding when they have their test next Friday - the day before Spring Break!  (NOTE: I created this simulation in Fathom and kept re-randomizing until I made errors for both the fair and the unfair coin.  If you would like a copy of this Fathom file, let me know in the comments below or send me an email to