Tuesday, May 3, 2016

Modeling in 3-D with Play-doh: Part II

For part I of this activity click here

Today we wrapped up the model-making activity and ended with up a couple more models to help visualize this 3D construction of a solid object with cross-sections of known shapes. Some people didn't finish all but I made the executive decision to stop and start talking about how to find the volume of the solids in the last 15 minutes of class. (We are also in the midst of AP testing so some students had missed class the previous day)

To have them "save" their models (since we deconstructed each one to save materials) I had them take a picture of each one and dump it into a Google Slide Deck and I will go through and pick the best ones to display on our Haiku Class Webpage. Since they did this in partners, they shared the document with their partner so they could both access it.

 It is especially helpful for those slower students who made less of the models, or for students who were absent.

This is what the assignment looked like. I pushed it out through Google Classroom .
(So every student had a copy of this already ready to go, I made the template)

Equilateral Cross-Sections!

In Summary:

I loved trying out this activity for the first time this year! (Especially something this hands-on)

I definitely wouldn't do this 100% the same for next year, however. Specifically I was not entirely satisfied with the sheer amount of class time it took to make these, and how I felt at certain times there was a large gap between the students who were on task, or just worked faster, and how much they had completed as compared with the others.

Things that went great! (and I want to keep):

  • Students getting hands-on practice constructing models
    • Thinking in 3D is hard, and this allows an entire day to "play" and see what different types of scenarios would look like on a physical model that they can touch. I want to still have this be a conserved day with no official "formula" to speak of.
  • Students seeing how even objects with the same base can look different with different cross sections
    • Having them all use the same base function and interval region allows them to see how just changing the type of cross section can effect the solid. It also allows them to see and predict whether or not they think they will all have the same volume or not... (they don't)
  • Students seeing the repeated shape generates the entire solid
    • This means the discussion of where the volume formula comes from is more fluid and seamless, as they cannot look at a single model without seeing the same cross section repeatedly shown.
  • Reviewing the concept of integration before the activity 
    • I could feel that they needed a reminder of what an integral really represented, even though we had come out of a unit of calculating them. The important thing we discussed as a sort of warm-up and again after we finished the models, was that an integral allowed us to take the infinite sum of repeated things (the "things" before being rectangles for Riemann sums)
  • Organizing the materials into buckets 
    • This allowed for easy distribution and cleanup. I go from Applied Calculus to Algebra 2H and then back to Applied Calculus so I had to collect and distribute materials often.

Things I would change for next time:

  • The number of models they build and what they do with them 
    • It would be more efficient if each table group had a different cross-section model to build, and then we had a "gallery walk" at the end to see them all to take pictures and answer questions about them. I could then save the best models from the last period of the day to have them go back through and calculate the volume of each one the next day, in a station activity.
  • The distribution and subsequent collection of the Play-doh
    • This might not be an issue for you, but for my classes they are 5 weeks away from graduating and in the midst of AP testing so given any opportunity to "play" in class they will run with it. (I love them but at the same time I don't at this time of year...) I had several groups slack off and have one person build all the cross sections, which set them far behind every other group, all because they wanted to play with the play-doh. Next year I want to collect it as soon as they have molded their base, or have them put it in the bucket and put that on the floor. Your results may vary, and classroom management is definitely a growth area for me.
  • The "Storytelling" aspect of building this concept
    • I didn't do as good of a job as I could have with creating a sense of "need" for this sort of application. I kind of just said "Look at this neat solid we created! How can we find the volume?" That isn't the worst thing I could have done, I suppose, but definitely I wanted students to have more interest in what this could be used for than they did in this unit.

Let me know if you have any feedback when you try it out! (And thanks for reading if you got this far!)

Monday, May 2, 2016

Modeling in 3-D with Play-doh: Part I (Because Calculus Can Be Fun Too!)

Today I did a pretty neat lesson.

As neat lessons go, there were moments that I could see spending the time to break out play-doh to build a physical model was worth it. Students were excited to be in class (a rare treasure for Seniors about to graduate and in the midst of taking AP tests) and were able to describe what complex solid objects would look like after seeing it firsthand, instead of "imagining it" or seeing 2D computer models of 3D objects.

There were also moments that included students building snowmen or pizzas out of the building materials. But hey, did I mention how they were excited to be in class 5 weeks away from graduation? Disclaimer: I did allow 2 minutes of play time to "get playing with it out of their system" but I'm not sure that had any influence. Next time I should have a control class to test it.

This whole lesson is adapted from an activity purchased from Teachers Pay Teachers but I think next year I will adapt it to do my own thing, as well as include a student instruction sheet as the activity only had the materials. It was a great starting point to go off, however, and encourage you to go to the teacher's page to check it out. (link above) I can't link the resources here, unfortunately.

I started the lesson with a Pear Deck slide deck:

Cue: Discussion about area of a single card and off-topic but still cool discussions of  flexibility, color on the electromagnetic spectrum, flammability (?!), density, malleability

Cue: Discussion about volume of the deck, and how volume of a deck is probably more relevant than volume of a single card, since cards are fairly thin

Cue: Discussion about how as the height/thickness of the card decreases, we can fit more cards into the stack. The thinner the cards, the better; we are then adding up just the area of the slices to get the volume!

At this point we stop. I don't bombard them with formulas. I had decided beforehand that my purpose of the lesson today was purely for them to gain spacial awareness of what these 3-D cross section models look like, otherwise the procedure they will eventually learn means nothing. I had thought we would have time at the end to discuss how to build a formula, but we didn't get there, so I didn't push it in an inauthentic manner just to be "able to do the homework." Next year I want to create an assignment that just has some debrief and thinking questions from the day to frontload the idea for the formula for the next.

I present them a task creating a solid using repeated shapes that we know how to take the area of, and we like the idea of adding up repeated things infinitely in Calculus. (mmm... Integrals)

They start with a set of materials:

Room layout, all materials in buckets so that I can easily grab them to prep for other class

Play-Doh, Rulers, Scissors, Markers, Task Cards, Blank Laminated Graph Paper

Definitely recommend Sharpies for next time, or having them draw the graph and then put it in a sheet protector

 The students work on building a solid where the base is the area under the graph of y = x^(2/3) from x = 0 to x = 8, by first graphing the function:

Featured: New scientific calculators I got from the 99 cent store (for $2) that have 2 lines of text!

They start to add play-doh to the base, showing both the area under the graph, and allowing a surface in which to stick the cross-sections. 

Note: in future years I would have them graph the function and the put a sheet protector over it for when the play-doh comes, instead of laminating them. I had them use wet-erase markers over the laminated ones and the playdoh definitely picked up some of the icky black color. Oh well! (Laminating or using a sheet protector is a must, to protect and reuse materials from class to class).

They used plastic knives to cut the play-doh to fit, and it worked really well!

Students worked together in partners to measure the graph and construct semicircles that were the length of the function at various points on the graph. I had originally said "Make 10-15 cross sections somewhat evenly spaced out to see the 3-D shape better" thinking they could handle planning that all out. (They are 18 years old) That changed 5th period when I decided to offer a suggestion to split up making them, having one person do the whole number markings on the x-axis (1,2,3,...8) and one person to do the half-number markings (1.5, 2.5, 3.5,... 7.5) and that worked out better.

At this point, it all starts to take shape!

A mostly-finished product!

It took much longer to finish all that then I had thought, almost 50 minutes just to set up the graph and make 1 solid with semi-circle cross sections. (I think AP students would be faster at constructing the shapes) I would set timers for each type of cross section that you do. BUT I do believe it was worth the time for them to be able to see and visualize what these models would look like, and to see the repeated shapes within them.

Ideally I wanted them to be able to "cover" their models when they were done, so they could see the smooth 3D object, but I couldn't figure out a way to do that. Another teacher had suggested colored cling wrap (which I know they sell red and green colors around the holidays) but I couldn't find any and other alternatives seemed messy and difficult.

This will now be a 2-day lesson, where Day 2 will have them continue constructing using the same base region, but different cross sections instead. (Equilateral triangles, squares, etc.) At the end of tomorrow we will develop the procedure for how to calculate the volume using Integration.

Stay tuned for Part II!

A Beginning (?)

Hello World!

This is me finally getting around to making a blog, which I only said I was going to do...about a year ago (so, you know, a pretty good introduction to what kind of person I can be).

I've been stealing (shamelessly) from amazing people like Sam J Shah (Continuous Everywhere but Differentiable Nowhere) and Amy Gruen (Square Root of Negative One Teach Math and Dan Meyer (dy/dan). BUT now I will start giving back! (*disclaimer: the quality of my submissions is not necessarily equivalent to theirs*)

A little about my student context: I am a 3rd year teacher of math (Applied Calculus and Algebra 2 Trig Honors) at an affluent public school in Orange County, highly ranked and with an active parent presence. I have a digital learning coach this year, which means I was assigned to a cool math tech wizard who helps me make awesome things! (Desmos Activity Builder is quickly becoming my favorite...)

At my school, Applied Calculus is 85-90% seniors who mostly earned B's, C's (and some D's) in PreCalculus and can't move on to AP Calc AB but still want a fourth year of math. (there are a few that earned A's but want an "easy time")

Applied Calculus is my favorite class to work with, but it wasn't always that way. As a first year teacher I was handed this class to teach pretty much by myself with no standards and no curriculum besides a student copy of a textbook. That much freedom was daunting! Unfortunately (due the the workload I had in addition to that course) I stuck with "what I knew" and approached the class from a direct instruction approach, following the order of the textbook making sure we "covered everything" even though we didn't even have the pressure of a standardized test to worry about! The following year I rewrote everything from scratch, not following the textbook but making my own "Units" of study based on topics I thought were more beneficial for my students, allowing time for application projects.

It's not perfect, but I'm much happier with my student outcomes and behavior when we don't just follow the textbook for the sake of doing so because that's we are (and I am) used to doing. I am still working on making it meaningful for them and engaging them more as self-directed learners.

Some things I am interested in improving are:
  • Planning student centered lessons
  • Creating a positive, supportive classroom environment
  • Increasing student talk time
  • Improving student writing and explanations of core concepts
  • Classroom management of "situations" and problem students with positive reinforcement

I'm not sure what this blog will turn into, but I hope you are along for the ride!