Volume is can be a tricky concept for my fifth graders to understand. They always want to revert back to their fourth grade knowledge of area and perimeter. I’ve finally settled on a hands-on activity that allows the kids to explore volume and discover the volume formula on their own.
I blame @bowmanimal for all of this. A few months ago he blogged about conceptualizing volume in calculus before formalizing. At the time, I had just started looking over past AP Calc exams, wondering how I was going introduce volume (solids with known cross sections and solids of revolution). Volume is a calculus topic I've not taught before, and I want my students to do more than memorize the formulas. Because, as our own textbook puts it so beautifully: "Some students try to learn calculus as if it were simply a collection of new formulas. This is unfortunate. If you reduce calculus to the memorization of differentiation and integration formulas, you will miss a great deal of understanding, self-confidence, and satisfaction."[1] Anyway, I tucked Bowman's post in my "Summer Projects" folder, and, well, it's summer now, so I thought I'd best get on with it. Solids with Known Cross Sections Here's what came out of solids with known cross sections: So pretty! The idea is a type of think-pair-share activity where students conceptualize what's going on before throwing the actual mathematical definition at them.[2] I love this because these visuals get glued to your brain. Now when kids see: Find the volume of the solid whose base is the region bounded by y=x^2, y=0, and x=1 and whose cross sections are semicircles perpendicular to the x-axis. They're less likely to throw in the towel because of all the scary words and more likely to remember that green tornado-looking thing. I hope the conversation that goes on in their darling little heads is, "Need to add an infinite number of infinitesimally thin semicircles...no prob...I've got the tool for infinite sums, an integral, baby! Thanks, Uncle Leibniz!" Bowman shares some great tips for constructing these solids (be sure to read his responses in the comments section, too), but I thought I'd add some hints that helped me, if you want to make these as well: I could not, for the life of me, get my cross sections to stand using tabs, so I resorted to a hot glue gun, which worked marvelously. The cross sections seem pretty sturdy (my cat even tried to snuggle with one of them, and it endured her voracious nuzzling, so I think they might just last a few years...cross your fingers). I splurged and got Ghostline foam board because I'm both anal and a terrible free-hand artist. I did not trust myself to draw nice parabolas without it. They come in packs of two at Hobby Lobby for about $3.50. SQUARES ARE EVIL. Bowman mentioned they were floppy. Indeed, they are. I ended up only taking the squares from 0 to 0.7, instead of 0 to 1 like the other cross sections. This anti-symmetry was deeply depressing, but I couldn't get those darn squares to stay up once they reached a certain size. Le sigh. After students converse about what they've seen on the poster boards, I plan to explore this applet with them as well, so they can see a 2D visualization of the 3D object we've created. Solids of Revolution Since I was already on this volume kick, I started to wonder how I could create a visual that would help students understand the formulas for solids of revolution (taking a graph and rotating it about a given line). The answer? Foam sheets and a wooden skewer: The graph I chose was y=sin(x)+2.5 (from 0 to 2pi) because I wanted the finished product to look kind of a like a vase and I also wanted to maximize the amount of area available to me on my foam sheets (bought in a pack of 65...of which I used all but 2). And also because I wanted to show something other than a polynomial function. Again, I was pretty Type-A about this little project. The foam sheets were 2 mm thick, so I let 2 mm=1 unit and used a compass and my handy dandy graphing calculator to create circles of approximately the correct radii (overboard? Yeah...probably so...). The skewer was the best idea of this project because not only does it serve as a visual for the axis of rotation, but it was super easy to center all the little foam circles on it because of the imprint the compass had already made: 11 down, 52 to go... Again, instead of being scared of the weird words and sometimes weird, unhelpful figures that go along with rotation problems, I hope my students will think, "Just adding up a bunch of super thin (dx!) circles...gonna need pi*r^2 and an integral for that. Thanks again, Uncle."[3] And here is a fantastic Geogebra interactive worksheet we can explore as well. I hope my students gain a great deal from these two summer projects. I know I was really thrilled by the mathematics that was being exposed while constructing them. For example, with known cross sections, decreasing the base by the same amount each time did not create even gaps between cross sections since the rate of change is smaller as we get closer to (differentiable) mins and maxs. I had a similar struggle with the vase: the closer I got to a min or max, the harder it was to get the right radius because the radii were changing so slowly. Update The next year I had my kids make their own solids. See post here. [1] Larson and Edwards, Calculus of a Single Variable 9th ed., p. 42. [2] Anytime I can use the phrase, "There are no right or wrong answers here," I know it's a good activity. [3] In the words of Steven Strogatz, "Infinity to the rescue!"
Here you will find our new set of printable fun Math Worksheets for Kids. There are a range of math addition sheets, printable math comparing game, and other fun math activities.
The Volume Zoo project is all about tapping into our students' creativity and mathematical skills. With simple materials like cardboard boxes and a lot of imagination, they'll craft 3D replicas of an animal of their choice, all while exploring the exciting world of math. I put together step-by-step directions for the project build as well as planning sheets for each part of their build, so every student can follow along seamlessly and you can check in on their project progress easily.
Are you looking for visual, hands-on and interactive ideas for teaching nets, surface area and volume? In this post are some cool ideas, a jazzy video, math word wall references (one of them is a free math word wall for volume and surface area!) and a couple math pennant activities
Looking for a way to help your kiddos better understand cubic volume? Head over to my blog to read about a neat activity that really works and to grab your free sheets with volume models for practice. Math Coach’s Corner You Might Also Like:K/1 Measurement Lesson IdeaSpring Measurement Interactive Bulletin BoardMeasurement Activity: Are You A ... Read More about Volume Made Easy!
Kids have a lot of misconceptions about capacity, and they can really only be cleared up through hands-on exploration. Sadly, I don't think they get opportunities for this like we did when we were kids. Instead of playing in a sand box or with cups and bowls in the bathtub, kids are playing with iPads and Wii's. And so many districts (mine included) are taking away sand and water tables from kindergarten, in favor of more "rigor." Because of this, kids are not building the schema necessary to understand more abstract concepts about capacity. So please, please give your kids lots of time to explore by filling up cups with water, sand, rice, beans, cubes...whatever you have! If you are lucky enough to still have a sand or water table, it is the perfect excuse to use it. Don't let anyone tell you it's not "rigorous"! Day 1 I start my unit by showing a quick little powerpoint, to get a discussion started. You can download the PPT by clicking the link below: Download Will an Elephant fit in a Bathtub The kids are able to quickly give the correct answers. But I push them--I want to know why! (Because...
Volume and capacity worksheets and displays. Work with litres and millilitres, estimate capacity and volume, read scales and more.
Set up a math discovery centre using water play to learn about volume with these top tips for hands-on math lessons.
I teach high school inclusive ed, and it is the reason I started designing my own resources in the first place. Most of my kids are working at an early elementary level but are high school aged. So…
Kids have a lot of misconceptions about capacity, and they can really only be cleared up through hands-on exploration. Sadly, I don't think they get opportunities for this like we did when we were kids. Instead of playing in a sand box or with cups and bowls in the bathtub, kids are playing with iPads and Wii's. And so many districts (mine included) are taking away sand and water tables from kindergarten, in favor of more "rigor." Because of this, kids are not building the schema necessary to understand more abstract concepts about capacity. So please, please give your kids lots of time to explore by filling up cups with water, sand, rice, beans, cubes...whatever you have! If you are lucky enough to still have a sand or water table, it is the perfect excuse to use it. Don't let anyone tell you it's not "rigorous"! Day 1 I start my unit by showing a quick little powerpoint, to get a discussion started. You can download the PPT by clicking the link below: Download Will an Elephant fit in a Bathtub The kids are able to quickly give the correct answers. But I push them--I want to know why! (Because...
Use popcorn to explore math activities and science concepts - counting, volume and physical change experiments for elementary & middle school!
Amazing poster for teaching volume! *Now available in color AND black and white!!* Terms of use: You may use this in your classroom or digitally with your students. You may NOT resell these products as your own or use these products for commercial use. Backgrounds from: https://www.teacherspayteachers.com/Store/Alina-V-Design-And-Resources and https://www.teacherspayteachers.com/Store/Lovin-Lit Fonts from: https://www.teacherspayteachers.com/Store/Amy-Groesbeck
If I were in Harry Potter World, I would expand time between when I teach Volumes of Revolution and the AP Exam. This way, I could do a hands on project to actually embed the knowledge in the student brains. But alas, I'm just stuck with regular old days and a time-crunched teacher and students who are teenagers. This year, I had my students do this project for the days they were in class. It's the first time I've done it, so I made notes in my document for when the inevitable things went wrong this year that I want to improve upon for next attempt of this project: You'll be shocked to learn that students can't convert between ruler tick marks and decimal numbers. SHOCKED, I tell you. You will also be floored by the fact that directions are for "other people", when you are doing a project, you should just keep asking about the next step. Anyway, I liked how they turned out: It was a good mix of freedom for their creativity, an in-depth practice of regression and degrees of polynomials and piecewise functions and graphing. It was a sad awareness of just how shallow some of the students' knowledge was of how to find a volume of revolution. I don't have a grading rubric (everyone is a winner!), but I think I may add one next time.
Try this weight guessing game to help your second grader get to know liters.
