Category Archives: calculus

Using Delta Math to See 3D Volumes

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Delta Math has incredible sets for seeing volume of cross-sections and solids of revolutions. If you instruct your students to click on “show solution“, they can pull a slider to see the region become a solid! For the cross-section set, it shows the shape (square, rectangle, or triangle) growing along the base. For the disks, it includes the radius with the 3D shape. The washer set shows the washer inside the 3D shape. There is also a matching set, where students can practice matching up the 2D region with a resulting 3D shape, and vice versa.

The five sets I use are in the image below. I have also included videos of the volumes being created when you drag the slider so you can see how cool it is! Delta Math is free for teachers!

Delta Math – Volumes of Known Cross-sections
Delta Math – Volume, Solids of Revolution, Disks
Delta Math – Volume, Solids of Revolution, Washers

Discovering Volumes of Known Cross Sections Using Play-Doh

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My AP Calculus students had a tough time with volume last year. So this year I wanted an activity where they could actually see the cross sections. I have seen many awesome activities where students actually measure the base between curves, and then create cross sections using paper, like the ones from Bowman and Rebacka. I would love to do these with my students, but unfortunately we are very short on time this year because of our school schedule.

So, I decided to try it out using Play-Doh. I had students mold each type of cross section, and then let them slice it with either dental floss or a plastic knife. Once they cut their solid, I had them pull out cross sections so that we could develop the area and then volume formulas together.

First of all, kids loved playing with Play-Doh! It worked beautifully, and only took about 40 minutes to do the entire activity. Most of my students didn’t need to do all of the cross sections after the doing the first one. Doing the first one really helped them see and understand what was going on. So, if you are short on time, you could just do one cross section. However, my students really enjoyed trying to make the shapes for all four cross sections, and cutting them into the sections. And, again, they really enjoyed playing with the Play-Doh! They were so sad when I asked them to put away the Play-Doh for the quiz that I let them keep it out and play with it during the quiz. Sometimes I forget that high school students are still kids.

I pulled together a Desmos activity that I planned on using the same day. It is full of amazing graphs created by the amazing Suzanne von Oy. However we did not get to it as we ran out of time, and were just too excited about the Play-Doh! So I will share the Desmos with them when we do examples, and to remind them of what we did in class.

Procedure:

  • Class time: 45 minutes
  • Materials
    • Graph sheet and worksheet – I put the graph sheet into slip in sleeves, so I could reuse them for my classes. You could also use plastic page protectors or laminate the graph sheet.
    • Play-Doh – I used one 4oz. can per every two students.
    • Dental Floss (or plastic knives, but dental floss works better as it doesn’t squish the top of the shape).
    • Rulers (they don’t need them for this activity, but it helps them see why they shouldn’t measure with a ruler).
    • Desmos activity
  • Directions:
    • I let the kids just play with the Play-Doh first. Don’t skip this important first step, or you will never get their attention. lol!
    • I had two students work together on one mat.
    • While they are playing we talked about volume formulas. I found asking them how do you get the volume of a cylinder was especially helpful as they all knew it was the area of a circle time the height. I pushed them on this, and then they told me that you have lots of circles to get the volume of a cylinder.
    • I told them to use all of their Play-Doh to make a solid out of the area between f(x) and g(x). Then, I had them mold their shape into the different cross sections.
    • After molding, they sliced their solid, and took out two different sized cross sections, one from the middle and one towards the end. This was incredibly helpful as when I said, what is the base of the square? A couple of tables started measuring it with a ruler. (Rulers had been left on the table from the class before). This was accidental, but a great opportunity to talk about why measuring this is not the best method.
    • This led to a discussion about how to get the base measurement we needed. And moved to the worksheet to write down the area and the volume.
    • I did not give them the area of an equilateral triangle. We used our 30-60-90 knowledge to develop this.
    • Once we finished everything we compared the volume formulas on their worksheet. We talked about how they were different and how they were the same.
    • I planned on using the Desmos activity, but we ran out of time. So, I will use this the next day so they can see and manipulate the cross-sections of the examples we use in class.

Calculus Class Playlists for +C and U-Substitution

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I am one of those annoying people who sing everything. I can’t help it, and I can’t stop it. As such, I also love to play music in class. Kids really like it as well, and I love it when they are working hard and singing along. It also seems to help my more talkative classes focus. “Music soothes the savage beast.” 

+C Playlist:

Some calculus concepts can be easy to forget, especially adding C. So, I created a +C Spotify playlist full of songs with anything that sounded like the letter C in them, see, sea, me,… As we first learn about why we need to add C, I play, and SING, these songs, loudly emphasizing the C. I also change lyrics when necessary, as in, “Don’t you, forget about C!”  Because I am crazy. 

U-Substitution Playlist:

I also have a U-substitution Spotify Playlist, that I save until we get to Integration. But, you could use this as early as Algebra 2 if you wanted to!

 

Songs to Sing in the Shower is a great Spotify playlist for class for any day. I did not make it, but I really like it. It’s fun, and kids sing along to almost every song. 

I am always looking for great songs to add to these playlists, so if you have one, please add it to the comments! 

 

AP Calculus Pacing Spreadsheet (Follows the CED)

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Here is the spreadsheet that I used to figure out and keep track of my pacing for AP Calculus throughout the year. The spreadsheet includes AB and BC. I follow the CED (page 27, Course at a Glance) so that I can use the AP Classroom Progress checks throughout the year. I really like using the Progress Checks throughout the year, instead of saving them for review time. It shows how my students are progressing and what they still need to work on. It gives them consistent practice with AP level timed multiplication problems. And, it shows students (and me) what they still need work on, and which students need the most help.

I picked the lower number of suggested days from the CED, because I never have enough class time! The spreadsheet is for AB and BC. We have about 80 class periods before AP Exam week begins. I need to review more this year, so I blocked off 12 days of review, leaving 68 days for content.

** Please note that the CED recommends 22-23 days for Unit 1, Limits. However, our CPM Precalculus book does a fantastic job with limits, so I cut that in half for my schedule.

Example AB Schedule

The Can I? section is how many days I plan on using for each unit. However, things always change, which is what the Used section is for.

Used is the days I actually used, so I can adjust as I go along. Last year I stole from review days, I am hoping to do that less this year!

I try to stick to this schedule, even if it feels like we are going way too fast. I tell my students that we have to, in order to get it all finished. I have math help almost everyday for students who need more time or help.

What I Learned as a Newbie AP Calculus Teacher

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This year, I taught AP Calculus AB for the first time, ever.

I was definitely more terrified than excited when I was offered the class. I had wanted to teach it for a couple of years, but remote learning had been incredibly difficult and time consuming for me, and the thought of teaching a very difficult new course during the pandemic was even more terrifying. Oh, and I hadn’t taught Calculus since 1998.

Most of all, I was worried because I wanted to do a really great job for my students. Yes, of course I always feel this way, but I was moving up, again, with the same group of students. I had taught all of them in Pre-Calculus the year before, and most of them Algebra 2 as well. For my juniors, I was the only high school math teacher they had ever had. I did not want to let them down. But, my students all told me our school has had a low passing rate on the AB test in past years. So I was pretty doubtful myself. Additionally, in order to lower screen time and stress for our students during the pandemic, we reduced class time. I would only see my students for one or two 80-minute class periods per week, and about half of that needed to be asynchronous. I was very worried.

Fast forward a year later. We just got our exam results back. I was elated, shocked, and most of all so relieved that my students did well on the exam. (I actually FaceTimed my teacher friend at school crying.) It was like a giant weight had been lifted off of my chest. Most of all, I was so happy for them, and grateful that they trusted me and worked so hard to pass.

I have seen many posts from teachers who will be teaching AP calc for the first time next year. They are as excited and anxious as I was, and like me, they have many questions about how to do this! So, I am writing this to tell other AP Calc newbies out there what I did, as a newbie. Please keep in mind that I am a newbie, and you should also talk to a seasoned AP teacher for much better advice! I was fortunate to have a close friend who is a reader, and got so much great advice from her. I am going to do many things differently this year, but I listed the things below that will stay the same.

  1. I joined the AB Calc Teachers AB/BC Facebook group. This was the single more important thing I did. Even though most of my math teacher friends communicate via Twitter, FB is where you find the AP Calculus teachers. Many teachers in this group are AP readers (which means they grade the exams), so they know what they are talking about. I rarely posted, but many people post great material and/or ask questions, which I greatly benefitted from!
    * You have to be an AP Calculus teacher to join this private FB group.
  2. I mapped out the entire year so that I could be sure to cover every unit, and have 2 weeks for review. The CED lists how many days every topic should take, so I used that as a guide. I stuck with my schedule even when it was going too fast, because I knew I had to. Some students really struggled with the pace, and I would meet with them before or after school often, but it was still tough. I mapped out my pacing with a Google Spreadsheet.
  3. I used AP Classroom Online extensively. They need to work with the AP material to get used to it, especially the FRQ’s. I gave almost every Progress Check in AP Classroom, as we moved along. The students did the MCQ’s at home, timed, and assigned through AP Classroom. The next day, I would go over the most frequently missed questions in class. I gave the FRQ’s in class, timed, and then we went over them together.
  4. I relied on Mark Sparks free calculus curriculum. It has a good amount of discovery in it, and is conceptual. Most of the AP questions are very conceptual, so I think this helped my students. I modified it heavily, mostly by shortening it as we did not have enough class time. The textbook we had was outdated, and many students didn’t have it anyway, so I didn’t use our textbook at first, then I never used it all year. I also used material shared in the FB group by Bryan Passwater, and Circuits by Virge Cornelius. I loved the Calc Medic material, but I didn’t have enough class days to incorporate their discovery lessons.
    *Be careful about using Mark Sparks assessments, as kids will find them on the internet.
  5. I timed all of their assessments, all year long. I started out giving them 4-5 mintues per question, and gradually moved the time down. By January, I was using AP Exam timing. They hated this at the time, but they all said it really helped them on the exam.
  6. I gave a full practice exam to kick off our review. I graded the MCQ, and then they graded their own FRQ’s. I put up the solutions and told them, “When in doubt, don’t.” which meant to not give themselves credit for FRQ answers that didn’t match the key pretty closly. I would rather the student underestimate their score than overestimate their score. This showed the kids where they “were” and especially what they needed to work on in the last weeks before the exam. I had a BC student who wasn’t even in my class (who took the practice exam with us) text me after he got his exam score back to thank me for administering the practice exam. He said it really helped.
  7. I used Patrick Cox’s kahoot reviews. They are amazing. My kids LOVED these and they really helped keep review fun. Thank you Patrick!
  8. I taught them how to use their TI’s, not extensively, as we did not have the time. But they knew the basics. I’ve only used Desmos with them in the past, so this one was a tough one for me, and I definitely need to do better next year.
  9. I didn’t grade homework, and I provided fully worked out solutions to all homework I gave them. There is no point in practicing if you have no idea what you are doing, or are doing it wrong. I instruct my students to do a problem, then check the answer. They would come in before or after school if they couldn’t figure out questions (and we didn’t have time in class that day to go over all of their questions).

    The following things were my personal preference, and I doubt they made a difference. But I was so short on class time and these things really helped me.
  10. I followed the CED. Since I was new, I was starting at ground zero. I wanted to use AP Classroom, so this seemed like the best route for me.
  11. Assessments: I didn’t give any assessment longer than 30 minutes all year because I just didn’t have the class time. I made my assessments from AP Classroom material and curved when needed. I usually use a square root curve or a kennedy curve.
  12. I didn’t do precalculus review at the beginning of the year. I jumped right into Unit 1, Limits, and reviewed as I went. Students are usually very motived at the beginning of the school year, so I wanted to take advantage of that with new material. Also, since they had me for one or even two years prior, I wanted my students to view this class as a new adventure, and not the “same ol, same ol” stuff.
  13. I used Delta Math for extra skills practice only. I always made this optional.

What I am doing differently and/or need to do better next year:

  • Chain rule. They had a much harder time with that than I thought they would. And it haunted us all year long.
  • More work e and ln earlier in the year, and all year long.
  • Integration using substitution, especially helping kids know when they need to use it. I gave them a ton of Delta Math for extra practice, but they told me that didn’t help as the sets are literally named “integrals with substitution”, so they always knew when they were supposed to use it. This is something that I didn’t realize until almost the end of the year.
  • Giving more calculator active problems overall, that they actually HAVE to use their calculators on. This was tough this year with remote testing and photo math. So my students did not get enough practice.
  • More review time at the end.
  • The AP Classroom videos are available now, so I plan on assigning some of them as required at the beginning of the year. Then, I will make them optional if they need them as the year goes on. I want to require them at first so they will see how helpful the videos can be!
  • Senioritis: I had a handful of seniors “quit” on me second sememster, after they got into college. We do trimesters, so they weren’t concerned about the grade drop. It espcially broke my heart for kids that worked so hard first trimester, as they could have passed and gotten their college credit. I don’t know how to fix that, but have some ideas in mind for next year. Suggestions are always welcome!

My Challenges

  • The last time I taugh Calculus was in 1999. And I only had 1 – 2 80 minute class periods per week, and half of that was supposed to be asynchronous to give zoom kids a break.
  • Apparently our school has a low pass rate. I haven’t seen the scores, as those aren’t shared, but students talk. At the beginning of the year, many kids told me that “They knew they probably wouldn’t pass the test, so why was I making the class so hard? Why should they even try?” Since I had taught them before, I asked them to trust me, and most of them did. Unfortunately Some seniors gave up on after Christmas, after they had been accepted to college. Even the ones that didn’t quit really didn’t believe they had a chance to pass until they did well on the full practice exam I gave in April.
  • Fifty percent of our students were remote for most of the year, which was an advantage and a disadvantage. I at least got to see most of my kids in person every other week, where many other schools were fully remote.
  • Our text book was outdated and does not follow the CED, oh, and only a few kids had a textbook to start the year since they were all backordered.

My Advantages

  • I teach in a private school. Even though we have a very wide variety of students academically, the majority of my kids are privileged. This is huge, becuase even though kids weren’t very successful on the exam in the past, they had strong potential to be very successful.
  • I have taught most of these students for two or three years. For my nine juniors, I was the only math teacher they have had in high school. This was great for me, as I knew what they had been taught, and we could move right along. But, they were also really comfortable with me, which also was a disadvatange at times. I’m a softy and they know it.
  • We use CPM at my school. Even though the kids don’t love it, it is very conceptual and challenging, espeically the precalculus series. There are two chapters on Limits in the Precalculus book, so I was able to skip most of that chapter to save precious time.

The school where I teach is moving away from AP. So, that is a culture barrier to get through. Although I discovered this year that many more kids really do care about the AP scores (and getting college credit) than I had previously thought. At the beginning of the year, many of my students (and their parents) told me that they really wanted to pass the exam. Afterwards, every single student who passed contacted me to thank me for helping them pass the exam. So, it really did matter to my students. My students worked incredibly hard this year to pass. I am so proud of their consistent hard work this year. And I am so grateful that I did not let them down.

I don’t believe that I have never worked harder in my life. It had been 20 years since I last taught Calculus, so I did every problem I assigned, and many more problems that I did not assign, so I could see which problems I wanted to give them. I also did all of the MCQ and FRQ questions, in advance. I haven’t had time to write a blog post in almost a year. I even ended up in physical therapy for three weeks in April because I had apparently been grinding my teeth so much at night my jaw completely locked up. But, I have also never had more fun in my life. Calculus was when I first fell in love with math in high school, and it was great to revisit it. I loved teaching Calculus. It was a completely rewarding experience and I can’t wait to do it all again next year!