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!
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. Iwould 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.
Class time: 45 minutes
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).
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.
“The negative screams at you, but the positive only whispers.”
Most of my Facebook and all of my Twitter friends are teachers. And I have seen so many posts that tell me that my teacher friends are not ok. Teachers are overwhelmed, overworked, stressed and sad. Well, in none of the posts do teachers actually say that they are sad. Instead, they say they exhausted, lethargic, and talk about the lack of motivation. And I have been feeling many of these same things, especially the lack of motivation in this very cold winter. I am sad, as I am every January and February. But I really worry about my teacher friends, most of whom have it so much harder than me right now. Dearest colleagues, I am worried about you.
Today we had a professional development day and in our morning meeting, our school counselors gave an amazing presentation about depression, so that we can watch out for our students and help them. But then they brought it back to us. They said to us, “Put on that oxygen mask first!” I think we forget that. I think that we are working so hard to make things better for our students right now that we are NOT taking enough care of ourselves. And we are burning out, hard. Our counselors shared the Duke University “Three Good Things” Program on Addressing Physician & Staff Burnout Presented by HRA. The researchers at Duke developed an app and had physicians and staff briefly record “three good things” about their day in the app, for just 15 days. It only took about 1 minute each evening to do. Upon completion of the program, they found that burnout was reduced, and that the positive effects of their program lasted for up to 12 months. You can read the article above, to watch this 3 minute video overview, to learn more about the program and their findings.
Listening to our counselors today, I was reminded of the One Good Thing blog started by Rachel Kernoodle in 2013, where various mathematics educators blogged about one good thing that had happened to them that day. This branched out to Twitter, where math educators blogged about “One Good Thing” using the two hashtags, #onegoodthing #iteachmath.
I loved reading these posts and tweets. And right now, I think we could all use a boost of positivity! I am hoping that this works, and will bring us all some joy and positivity. So please join me, and Tweet about ONE GOOD THING today, and for the next two weeks! BONUS: at the end, it wiill be two weeks closer to SPRING!
Also, if you want to go all in, you could try the Duke One Good Thing Smartphone App. I decided to try it. You will answer a few questions on their website, and then receive a text from them. After you type in your three good things, you are presented with a page of other peoples “three good things”. I loved reading through this!
I often get asked how students can easily type symbols into Desmos. I don’t teach Geometry, so I have not worked to find the best way to do this. I am including what I currently do below, but would love to know if you have a better way. Please comment below if you have a great way that easily enables students to type Geometry symbols.
I use Emojis & Symbols to get things into Desmos that are not in their math keyboard (yet!). I use this often for the arrow key with limits statements.
To include symbols, I go to my top bar (on the browser, not Desmos) and select, Edit, Emoji & Symbols, and then I can input them there. You can copy and paste with Desmos. So to make it easier for the students, you could put the symbols into the text box question, and the students can copy the symbols from your question and paste them into their answer box. Alternatively, you could also code the text input box so students could have the symbols they need already in the box. I created a Desmos to illustrate what I am talking about. It still isn’t great, as you can’t put the symbols over the numbers.
If you have a great way you have students type symbols, please comment below! Thank you!!
I say this every year after a summer of sleeping more, exercising more, and eating better. “This year, I will keep this up. This year, I will take better care of myself once school begins.”
Last year, I even made it one of my official professional GOALS, that I typed into a FORM. I still did not make enough time for myself.
But this year has to be different. There is the stress of hybrid, distance learning, or face to face, and the worry about getting sick. I MUST try harder to take care of myself. We all must. I have seen your tweets and your Facebook posts. I know how you are feeling. I feel it too. We need to be healthier, more well rested, and in good shape, so that we will feel better physically and mentally. Then we can better take care of ourselves, and also our dear students and colleagues that will need our support this year.
I know this, we all know this. We just need to try harder to do this. It won’t be easy, as I have worked more this summer than I ever have in my career. I do not know what this year will be like. I feel like a first year teacher. And that is stressful. You know what helps relieve stress?Sleep! And exercise!!
So, I am pledging to myself to do better for myself. Because only then can I be better for everyone else.
I am going to get enough sleep.
I am going do my 30 minute workout at least 5-6 days a week.
I am going to take 30 minute walks with my dogs a few times a week.
I am going to meal plan on the weekends.
I first typed “I am going to try to”. But then I changed it to I am.
I don’t know who else needs to hear this right now, BUT I SURE DID. So I am going to return the favor and tell you. Teachers, you are STILL enough. Whether or not you are making videos, bitmoji classrooms, or researching every available technology for this fall, you are enough. For some people, doing these things is what they need to do right now. For others, it is not. Both of those choices are ok. It is great if you are doing those things, and it is great if you are not. I am not doing those things. And I am enough. And you are enough, especially if you are doing what you need to do to take care of yourself, before another semester of teaching in an actual worldwide pandemic begins.
Last week I found out that we would be going back face to face, with a hybrid model, 50% of the students at a time. Our students will be in school for a week, and then home for a week, with at home students logging on synchronously. I thought it would invigorate me to know, but instead it has me stressing.
I am feeling overwhelmed and anxious about the coming school year. I’ve been having back to school nightmares. I have been short with my kids, and especially my poor husband. I cry easily. But this just isn’t me. I think that many, if not most, teachers are feeling this way right now.
Teachers, it is OK if you are feeling overwhelmed and have anxiety. Every single time I tweet about feeling this way, I have many replies of teachers feeling the same way. So I believe that many more of us feel this way, probably ALL of us. My school is working hard and doing a great job getting us back. We will be 50%, have masks, swivls, and so much support. And yet I still feel this way. And that is ok. It does not mean that I don’t love my job or my students. It does not mean that I am not a good teacher. In fact, I think it is the opposite. It is because I love my job and students that I am worried. I want to do the best job for them, but I feel like I don’t have a complete handle on what I need to do in the fall. And honestly, I may not know until I start actually doing it. I have not done hybrid teaching before, so I don’t know what to expect. I don’t know best practices. And I have a new prep this year. I feel like a first year teacher again. And that is uncomfortable, and scary, especially for an experienced teacher.
I love my work. I have spent most of my summer working through the material for my new prep. And usually, I am especially invigorated and industrious this time of year, right before school starts. But I hadn’t planned for how I am going to teach remotely or hybrid, as I didn’t know what we were doing until last week. As soon as I found out that we were going hybrid, I went to teacher Twitter to ask for suggestions. I got so many wonderful suggestions that I was almost immediately overwhelmed, as I had not done most of the things that so many people were suggesting. I felt paralyzed.
I am reading about teachers who have spent their summer making video lessons for the entire year. I am in awe of the creative bitmoji classrooms I keep seeing everywhere. Other teachers have already used Classkick, Jamboard, Whiteboard, SeeSaw, Flipgrid, One Note … I have done some of these things, but not all of these things, and it makes me feel that I have not done enough. I makes me feel like I am not near enough.
But as Sara VanDerWerf and Elizabeth Statmore recently reminded me, I AM enough. And you ARE enough too. They reminded me that in these unprecedented times we cannot expend our energy comparing ourselves to other teachers. Just like in pre-pandemic times, we do not have to do ALL of the things to be enough. Self-care, kindness, and support to each other is what we ALL need to get ourselves mentally ready for this unusual school year. So then we can figure out how to do the things that work best for our classrooms.
I am also scared about getting sick, or one of my students or colleagues getting sick. I haven’t seen my parents since February, as my dad is an 80 year old diabetic, so I’m being cautious. And once I go back to school, I definitely won’t be able to see them. I’m sad. I am so, so sad. I miss my parents. I miss my old life. I don’t like wearing a mask, but I always wear one. I’m so mad at people who don’t wear masks. I’m upset about many things right now.
So please teachers, give yourself a break right now. Hug your family. Take long walks, or do what ever it is that brings you peace. Breathe. We have a busy fall in front of us, and we need to be rested in order to be ready. When the time comes, we will do an incredible job for our students, like we always do, even if we have to work many extra hours each day. We are professionals. Trust yourself. We have got this.
I was using a great locker shelf hack that I saw somewhere on the internets. I highly recommend this set up! But when my internet was running slow it was lagging. And with five people online all day, it was lagging quite frequently, especially during class times. So, I NEEDED to get the iPad thing figured out.
My handwriting is atrocious when I try to write on my iPad because I can’t put my palm down to write. I thought the Apple Pencil with palm rejection could save me, but my iPad is too old, and thus not compatible with either Apple Pencil.
Next stop, a great, inexpensive stylus. I’ve had trouble writing with my iPad since I got it, so I’ve tired every stylus out there. The Apple knockoffs w Bluetooth work and write well, but they make a clicking sound that not only drives me crazy, but is picked up when I’m recording instructional videos. I don’t like the rubber tips, as they are large and don’t slide well across the surface of the iPad. I finally found this amazing fiber stylus for only $14 that I love! It also has that fine tip end with the disk it you need it.
** Note: If you have an iPad Air 2, screen share is found by swiping from the top right corner. That took me forever to find. It’s been a long month….
My house isn’t set up for ergonomic teaching, and my back has been KILLING me after a couple of weeks sitting in a chair that was too low, at a “desk” that was too high. Thanks to Jenn White I reinflated an underused exercise ball to sit on. Then,I bought this small inexpensive desk with an adjustable height top, and wheels! I can sit or stand at it! I love that it is small, but has a place for storage where I am keeping my binders as I roll it around the house.
Desmos JUST released a new feature that give teachers the ability to give feedback to students INSIDE Desmos activity builder! This will be such a helpful feature now that we cannot interact with our students in person, and especially for classes that are only able to do asynchronous distance learning.
** Your students MUST be signed in to the activity, with either with a Google or Desmos account, to be able to go BACK into an activity and read the feedback. **
Yes! It works on previous activities! Just today I went into our last activity and added some feedback for students to look at on our first day back. I figured it would be a great way to review, and ease into distance learning.
The feedback is very easy for students to access, as it puts all of the feedback you give them in one place, and they can click to go directly to the screen.
1. You have to TURN on the feedback option in Desmos Labs. Click on your name in the top right corner > Desmos Labs. Then check all the Labs you would like to have.
2. Once in an activity, click the text box at the top of the screen to leave feedback.
3. ALL of the feedback is in one convenient place for your students. And they can just click on the “Go to screen 3” to look at the screen or edit their work.
I am planning on putting a Feedback screen at the end of each activity where students can ask questions once they finish an activity. I am so excited to be able to answer my students questions, right in the activity!
I’m still on “Spring Break”, and we do not start online learning at my school until March 26th. I am glad that my school community has this time to process, as I am not ready to think about online learning yet. I don’t know what online learning looks like for my school yet. And I think that some students may crave the structure, but that others will need compassion, and space.
I do not want to assess my students. I am sure that they are feeling the stress of uncertainty just as I am, and I do not want to add to their stress. I want kids to feel connected to each other, especially if they need that. I would also like to mix up my students online learning, so they they are not just watching online videos and doing practice problems. I would love for their online learning to be creative, and not too monotonous.
I plan on using Desmos Activity Builders that connect students to each other, like Point Collector, and especially activities that contain Gallery Slides.A Desmos Gallery is a screen at the end of an activity where students get to create their own challenge for their classmates. And then their classmates pick each others challenges to solve. I feel that doing these galleries will not only allow students to be creative, but also feel connected to other students. I plan on encouraging them to reach out to the “creator” while working on a challenge, or providing feedback after.
I’ve created a Collection of Desmos Activity Builders that contain gallery slides here. I am sure there are more I am not aware of! Please let me know and I will add to this collection. I would love to know how it works out for your students, and the ways that you are incorporating galley slides, and all Desmos in general, in your online learning. Enjoy!