Category Archives: Anne-Marie Cabral

Using Desmos to have a little fun in Precalculus!

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I have found myself in a pretty lucky situation in
Pre-Calculus. For a variety of reasons I am a little ahead of where I need to
be in the curriculum so I am able to slow things down a bit while we are remote
learning. I am grateful because the particular chapter we are covering is very
challenging for most students. Polar coordinates, Complex Numbers in Polar form
and Vectors are never easy in the best of times but I want to make sure that
the students can still master these very important concepts.
Polar coordinates are
a brand new concept to the students. They are used to graphing on a rectangular

coordinate system using ordered pairs (x, y).  The polar coordinate
system looks like the figure below and the ordered pairs are now (r, θ)
where r is the length of a radius and theta is the angle.

For most students, polar coordinates are completely bizarre
and seem to have no relevance to real life. However, when I was developing the
Trigonometry curriculum for the new Math Program, I came across a Desmos
Classroom activity for introducing polar coordinates. This activity uses images and situations students have seen in the movies
for navigation on submarines and aircraft to put this coordinate system into
perspective.

For those who don’t already
know, Desmos started as an online graphing calculator but it has developed into
so much more in recent years. A teacher can set up an activity and invite their
students to join and the students can proceed through a series of screens that
have different tasks and questions. The teacher can monitor all of this from a
teacher dashboard and when the teacher wants to share out student work, the
students’ names can be anonymized by substituting in the names of famous
mathematicians! I have used this several times in class already (this will be
the subject of a different blog post) but I wasn’t sure how this would work in
a remote classroom.
The students started the activity by watching a short
movie clip showing a tense battle in a submarine with the sonar operator
shouting out the location of “bogeys” that were attacking the sub. The students
then advanced to a screen that set up the activity.

I was able to watch
their progress as they moved through the screens.
After a series of
screens where students learned how to use polar coordinates, the students were
put to the test!

The students were
racing to see who could finish first. I was able to watch the students work in
real time and was announcing over our zoom meeting who was the closest to clearing
all the bogeys.
The activity also gave me a
great opportunity to check for understanding.
  
In the end I was able
to get some final feedback. I highlighted a
couple of fun comments.


I think the class enjoyed this and I am going to
be looking for more Desmos activities for us to do in the future!

Final Exam Weekend Cram Session

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I offer a lot of math assistance for my students. I
basically make myself available before school, during lunch, during 7th
period and after school and many of my students benefit from this support. Even
with all this, as first semester finals approached, I felt the need for more. In
addition to the students who were desperately trying to improve their
performance in order to get a passing grade (C- or higher) for the semester, I
also had students who really wanted to perform well on their final exam. In the
spring, I had offered two AP Calculus “Boot Camp” sessions for my students on the
weekend before their AP exam. Because this had been successful, I decided to
offer some weekend study sessions for all of my math students on the weekend
before Final Exams. Just like last year I started out with a survey to gauge
interest and to determine what times would meet the needs of the most students.
The response was very positive and I decided to offer one 2-hour session on
Saturday and one 3-hour session on Sunday. Because there would be students attending from all 3 of the different courses that I teach, I could do no formal instruction. Instead, what came to pass was a giant study session where students were working collaboratively using the variety of resources that I had provided for them to prepare for their final exams. There was much use of the white boards and a lot of small group
discussion. While I periodically had to reteach concepts to some of the
students, I actually spent more time
Saturday end-of-day group selfie

serving as a sounding board for their
ideas and asking leading questions so they could make their own conclusions. Eleven
girls signed in on Saturday and fifteen on Sunday although more students were
actually there each day. There was really good energy in the room and the girls
seemed relaxed and productive. Many, many students thanked me before leaving
for Christmas break for giving them this extra support. I am glad that they
felt it was time well spent.

Sunday end-of-day group selfie

First (Annual?) AP Calculus AB Boot Camp

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Image result for ap calculus image

As teachers, we all do our best to reach our students and
help them to master the material in our courses. In an AP course, we have the
added challenge of an outside source judging whether our students mastered the
material or not. Every year I try to think of better ways to prepare my
students for the AP exam. A few years ago I started the “AP Review Binder”
which helps the students keep their practice materials organized while they
study for the exam. Last year I started using a subset of questions from an
actual AP exam as their final exam to help see how ready the students are. Both
of these ideas have been effective in helping to prepare my students but this
year I knew I needed something more.

I teach two all-girls AP Calculus AB courses—one in third
period and one in sixth period. The classes are not balanced and I wound up
with 12 students in 3rd period and 31 in 6th period. For
a variety of reasons my 6th period class has been a challenge. Not
only is it a large class at the end of the day, but there are several students
who either barely met the prerequisite for the class or had petitioned to have
the prerequisite waived. Overall, my 6th period class is weaker than
my classes from prior years and I am concerned with how they will do on the AP
exam.
With this in mind, I decided to offer an AP Calculus Boot
Camp to my students during this past weekend. I first surveyed the class to see
if they would even be interested in coming in to do this. The response was overwhelmingly
positive. I surveyed them again this past week to find out when they would be
willing to come. I offered 4 different 2 hour time slots to pick from—three on Saturday and
one on Sunday. Based on their responses, I chose Saturday 9-11 am and Saturday
4-6 pm as our two sessions. After checking with the facilities department to
make sure the school would be open on Saturday, I let the students know that we
were good to go!
Since several students indicated that they wanted to come to
both sessions, I needed to come up with a different plan for each session. Using
materials that I received when I attended an AP Calculus workshop a few years
ago, I made a packet of about 15 multiple choice questions and a different
packet of 6 free response questions. I had these ready to give to the students
when they arrived for the first session. I decided that we would work on one
FRQ at a time. I would give the students about 5 minutes to read the question
and develop a plan for how they would answer all the different parts. If they
finished their plan before time was up, they could start answering the
question. When time was up, I would project the problem and we would work
through it together. Using the grading rubric, I could break down the score
they would have received on the problem and could help them understand exactly
how much work they would need to show to get full credit.
While 13 students said they would come for the first
session, I really didn’t know if anyone would actually come at all. I was very
pleased that 12 students arrived at 9 am ready to work. I gave them the two
packets and we started going over the FRQs. The process of going over one question
at a time went really well. We had really great discussions and as we
progressed through the problems, I could see the girls’ confidence grow. We
actually went 15 minutes over so that we could finish the last problem! The
group was so focused on the FRQs that we didn’t have time to do the multiple
choice questions. I told them that if they were coming back in the afternoon,
they could work on them at home and I would go over them when they got back. 

Morning session

I created a second packet of FRQs for the afternoon session.
Again, I wasn’t sure how many girls would actually come and I was pleasantly
surprised when 12 girls came in for the second session. Five girls returned and
seven new girls came in. I gave the new girls all of the material that I had
handed out in the morning and then everyone got the new FRQ packet. We started
out by quickly going through the multiple choice questions and then returned to
our method of going over the FRQs one at a time. Again, we were so focused on
finishing the last problem that we went over by about 10 minutes.

Afternoon session

I am really glad that I tried this out and will probably do
this again next year. I think it did a lot to help the students as they were
doing their last minute preparation for the exam. I wish that some
of the students who I am most worried about passing the exam would have come
in, but at the same time it was wonderful to be working with students who truly wanted to be there. Even though I came in on a weekend, it barely felt like work at all!

Using Final Exams to Predict AP Exam Results

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Trying to keep AP students on track to do well on the AP exam
can be a daunting task. In the six years that I have been teaching AP Calculus,
I am constantly trying to figure out the optimum amount of guidance versus
independence to use with the students. Some students will do well no matter how
much I push them in class to prepare but others will, for whatever reason,
choose to minimize preparing for their AP Calculus exam so they can focus on
other things. One thing I chose to do differently this year was the 2nd
semester final exam.

AP Calculus students like to try to convince me to not give
them a final exam for the 2nd semester because they feel so busy
preparing for all of their AP exams. One year, I gave in and did not have a 2nd
semester final. However, when that year produced the highest percentage of
scores of “1” earned by my students on the AP exam, I recognized that there was
a correlation to the final exam and preparation for the AP exam. My first change
was to schedule the final exam to coincide with the last block period before AP
testing began. This helped somewhat. My most recent change was to the format of
the final exam itself. My 2nd semester Calculus final exam had not
been cumulative and was also completely multiple choice. This year, though, I
completely revamped my final exam to be cumulative for the entire course and I
based it on an actual unpublished AP exam. In order to get student buy-in I had
to promise to curve the final exam just like they do the actual AP test. I used
the scoring rubrics assigned by the College Board for the FRQ’s and used the
College Board’s point spreads to assign scores from 5 to 1 for each of my
students. I then crunched the data and correlated the scores to percentages to
enter into Powerschool as the final exam grade.
When the AP scores were published in July, I was anxious to
see how well I predicted the students’ scores. In over half of my students
(54%), the final exam score predicted the actual AP score. 31% scored one point
lower than predicted, 13% scored one point higher than predicted and only 1
student scored 2 points lower than predicted by the final exam. I am pleased
with my accuracy but at the same time I wonder why I was still wrong about
almost half of them. Here are my thoughts:
  1.  I only gave them the “Calculator allowed” questions in order to keep the length of test appropriate to the amount of time in a block period. Some students are better at answering questions using their calculator than they are when they don’t have access to one.
  2. Maybe I didn’t adhere to the grading rubric for the FRQ as stringently as I ought to have done.
  3. Maybe learning their “predicted” score had a psychological effect on how the students continued to prepare for the exam.

1.      I am now trying to explore the possible meanings of my 3rd
point. I think that there were a handful of students who were predicted to not
pass with a score of 2 who then took the attitude of “I’m so close. Maybe if I
work a little harder I can get a passing grade on the AP exam” and were then
able to improve their score. I also wonder though if some of the students who were
predicted a 3 on the exam then became overconfident and felt that they had
already prepared enough and then ultimately failed to pass the AP exam. This
year I plan to use the same exam format and will be extra careful on how I
grade the FRQ’s. Most importantly, though, I plan to very carefully help frame
their predicted outcome to motivate the students who need a little more work to
pass the exam and to warn other students of the dangers of complacency,
particularly if they barely earned a 3 on their final exam.

Math Department Celebrates Pi Day

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The Math department celebrated Pi Day on 3/14 (you may remember that pi, an irrational number, is commonly rounded to 3.14).  We started our day together in a Math Department meeting where Mary Beth shared this beautiful Pi-related prayer:

The beauty of pi, in part, is that it puts infinity within reach. Even young children get this. The digits of pi never end and never show a pattern. They go on forever, seemingly at random—except that they can’t possibly be random, because they embody the order inherent in a perfect circle. This tension between order and randomness is one of the most tantalizing aspects of pi.


We then departed and had many varied adventures with our students to celebrate pi.  Here are some highlights:

  • Amanda’s Geometry class was visited by Anne-Marie and four of her AP Calculus students.  These advanced students had recently used Calculus to find the volume of coca cola in a classic bottle.  The Geometry students were able to do the same task by using their knowledge of the volume of a cylinder (pi*r^2*height) to come up with a coarse estimate for the coca cola volume.  The Calculus students served as coaches throughout this project and at the end introduced the younger students to the idea of curve-fitting, area under a curve, and volumes of revolution to get a more sophisticated estimate.  It was a great activity to celebrate the many applications of pi in a collaborative way.  Both young and older students admitted they learned something new from each other.

  • Cathy’s Geometry classes derived pi empirically, by measuring the circumference and radius of various circles in the classroom and inner court and working backward using the circumference formula to derive pi.  Look how close they got!
  • Mary Beth not only treated the Faculty and Staff to many delicious pies, but she also encouraged the students to participate in a pi-tastic scavenger hunt.  See the tasks here.  
Until next year…

Using Classic Coca Cola Bottles to Understand Volume of Revolution

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The final concept covered in AP Calculus AB is finding
volumes of solids. Since solid figures and

volumes are three dimensional, it is
sometimes difficult for the students to visualize a 3-dimensional figure drawn
on a 2-dimensional whiteboard. When I attended an AP Calculus Institute, I came
home with several activities and labs that I could use with my students. Some
were helpful and some were less so, but I was intrigued by the one that had
students calculate the volume of the liquid contained in a classic Coca Cola
bottle. Calculating the volume of a can is pretty simple because cans are cylindrical and their radii remain constant. The volume would be equal to the
area of the base (a circle) multiplied by the height of the can. On the other
hand, a Coca Cola bottle is curvy and while any cross section of the bottle
would be circular, the radius changes at different heights of the bottle. In
Calculus, we learn that we can find this volume by defining the area of the
cross section (a circle) as a function of the radius (written in terms of the
variable x) integrated along the height (x) of the bottle. This is not an easy
idea to convey while drawing functions on the board and trying to simulate the
function rotating around the x-axis. I hoped that having something tangible
like a bottle to demonstrate the concept would be helpful for the students so I
decided to try the activity.

I split the activity over two days. On the first day, I
handed out Coca Cola and Diet Coke bottles (they were two different sizes),
markers, tape measures and a worksheet to record data. I explained that we
would be calculating the volume and we spent a little time discussing why
calculating the volume might be difficult. The students then worked with
partners to measure the circumference of the bottles at regular heights
measured from the bottom of the bottle to the height of the liquid inside. They
entered their data into their calculators using lists and then back calculated
the varying radii using the circumference data. This allowed them to graph
their height vs. radius of their bottles and they were able to see an approximation
of the profile of one side of their bottles. They then used a quartic
regression to find an equation that best fit the data graphed. This was
completed on day one.
On the next double block period, I had planned to teach them
the lesson on calculating volumes of revolution and then afterwards ask them to
calculate the volumes of their bottles. At the last second I decided to have
them calculate the volumes before I taught the lesson. I am so glad I did this!
We were able to spend some time hypothesizing on how we could use our data to
calculate volume. Eventually we were able to draw a picture of our bottle on
the board correctly aligned with our x and y axes. They were able to see the
relationship of the varying radii of the bottle to the volume of the bottle.
When we finally determined that we were supposed to integrate, the process made
sense to them! When I then taught them the lesson (Using the disk and washer
methods to finds volumes of solids of revolution), they were able to visualize
what was happening and I didn’t have to overcome so many objections to the
process that I normally would hear from the students each year.
Actual Volumes: Diet Coke 237mL and Coke 355mL

I will admit, the lesson did not go perfectly. Everybody
overestimated the volumes significantly. We tried to adjust by estimating the
thickness of the glass, which helped, but I still think the students were not
careful enough when measuring the circumferences. In the future, I think I
would give them lengths of string to use to wrap around the bottles and then
have them measure the string with the tape measures. I think the tape measures
were too stiff for them to work with accurately.  I also think I would let the students wrestle
with the problem longer before I guided them toward integrating the area of the
cross section. However, even without these minor tweaks, I think the lesson
achieved even more than I had hoped for!

Inverse Trigonometric Functions War

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The ability to evaluate inverse trigonometric functions correctly is a fundamental and critical skill that a student needs for Calculus. For whatever reason, many Precalculus students struggle to master this skill and many who do master it do not retain it by the time they take Calculus. I decided to step outside the box and have my students play a game that would force them to make more sense of the range of answers that can be found when evaluating inverse trig functions.
I had read on some math blogs about teachers using the card game War to help students evaluate logarithmic functions and so I thought I could make this work for my situation. War is a card game in which each player gets half of the deck of cards. For each turn, the players each flip a card. The player with the higher valued card wins both cards. The player with the most cards at the end wins the game. I adapted the game with homemade cards that had all the inverse trigonometric functions that use angles found on the unit circle. I prepared a visual aid that diagramed the allowed range of answers for the six functions. I divided the students into groups of three: two players and a referee. I explained the rules and then let them play.

It was slow at first while they got used to the rules and got comfortable with evaluating the functions. I circulated the room, checking in with students and responding to questions. It was very apparent to me just how many students really didn’t understand what they were doing. Group by group, I responded to their questions. As soon as one in the group understood, I would leave that person to explain to the others. Slowly but surely, they started to “get” it.

While I doubt that anyone would claim that this was a particularly fun game, I had many students tell me on their way out how much this helped them gain confidence with these functions. While I was circulating around the room I also witnessed many “aha” moments where students for the first time seemed to grasp the purpose of evaluating these functions. I’m not sure how I will be able to assess how well this extra day and extra activity really served to improve their understanding and retention of the subject, but subjectively I conclude that this was definitely time well spent.

Supplemental Material

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I like to have as many resources as possible available to my
students to help address their differing learning styles. For several years I
have posted pdfs of my class notes on Schoology. Originally this was intended
to be used by students who had been absent. However, my students soon realized
that if they were having trouble keeping up in class, they would either print
my notes out or pull them up on their iPads to follow along and annotate during
class. Over time, more and more students have been automatically obtaining the
notes ahead of time so that they can spend more time listening in class and
less time scribbling down the notes that are on the board.

Last year, after many of my Algebra 2 students started
subscribing to Mary Beth’s YouTube channel so they could watch her videos, I
recognized a need for my advanced math students to have access to video lessons
too. While there are many quality video lessons available on YouTube for
Algebra and Geometry, I couldn’t find good lessons that were
specifically tied to my textbooks, so I decided to make them myself.  Any student of mine can now go onto Schoology
and when they look in a lesson folder they will find a pdf of the notes, a
video lesson and the homework assignment. 
Creating the videos is a lot of work
and at first I was disappointed by how few “views” my lessons were getting.
However, as the quarter has progressed, absences have started and quizzes and
tests are becoming more challenging. Now when I check out my YouTube channel I
can see that more and more students are making use of the videos. I think that
the extra effort is paying off.