Volume Lesson, Crowd Sourced

I was expecting to have jury duty today so I wrote sub plans that had students working on a practice state test (ours is coming up in a few weeks).  Then, jury duty was cancelled and I wanted to plan a more interesting lesson last night, but my brain was already drained.  So I turned to twitter:

I'm planning to do this activity tomorrow, but I'm not very excited about it. Help make it more interesting please? http://www.scribd.com/doc/137815960/Volume-and-Surface-Area

Then I wandered off for a bit.  When I returned, I found ideas!

Gregory Taylor(@mathtans): Your prism is growing. Dims are twice the size, now three times. When does it cover the floor? At what point can you fit inside? 

Daniel Schneider(@MathyMcMatherso): Add prediction questions about how volume & SA will increase when doubled/tripled?  

Daniel Schneider(@MathyMcMatherso): Add questions at end: Draw prism with volume _____. Are there multiple answers? 

Gregory Taylor(@mathtans): Partly comes from Alice in Wonderland. How much more fabric do you need to cover a table that's now three times as big? #DrinkMe

 John Golden(@mathhombre): geogebra similarity enlarger http://t.co/2vu5przWhk

So, I updated the document.  One thing I remembered from past years was the tendency for students to create nets with square bases since the net I provided had a square base.  Other than that I implemented the suggestions and voila, new activity!

Volume and Surface Area New

When students arrived in class they found two questions on the board: "What is area?" and "What is volume?" that I asked them to write about individually.  (First period I had them in the reverse order but kids kept talking about area first so we did them backwards and I adjusted accordingly for the other classes).  We have talked about area plenty of times, but never volume.  I had them share anything they wrote and made a list of words, formulas, definitions and ideas for each question.  In all three classes someone provided some information that allowed me to define volume of anything with a constant cross-section as base area multiplied by height.

Post discussion I distributed the assignment sheet, graph paper and scissors; the only instructions I gave were "don't make your first box too big because you'll have to double it later."  If students wanted to know if their net was correct I said "cut it out and see!"  When students didn't know how to calculate surface area or volume I directed them back to our brainstorm.

Things that made me happy:

• nearly everyone started with different dimensions, so when they helped each other there was no cop out option of "here just copy mine"
• it was easy to see the nets that didn't work
• students were surprised how much bigger their "doubled" box was (cutting out a physical box made the realization that volume doesn't double if you double all the side lengths much more real)
• after two boxes kids were tired of counting and cutting and they wanted shortcuts (great! there's no need to do that, just draw a sketch)
• if students assumed the volume doubled without calculating, they realized I probably wouldn't have asked question three if that were true and went back to calculate the volume in question two.

The best thing of all: we spent an entire 90 minute period on rectangular prisms.  I only had a couple students try to quit mid-period and I ran this lesson with three classes today.  At our department meeting yesterday someone claimed it's good to give practice state tests because it builds perseverance.  I counter that this lesson builds perseverance in ways that a multiple choice test never can.  Did you really read that sentence above?  Let me rephrase:

Students spent 90 minutes investigating surface area and volume of their rectangular prisms.

In the process of editing this document, I dropped the question about the general case.  So I asked it as the journal question today instead:  "What happens to the volume when you double all the sides?"  I know at least one student said "it multiplies by 8" since she yelled it out.  This may be my opening question next class- compare with a partner, what do you notice?

Global Math: Understanding by Design

Last night Daniel, Elizabeth and I presented Understanding by Design on Global Math.  The full recording is now available for your viewing pleasure.

People were excited about this page, so here it is in all it's glory!  We talked about the importance of knowing exactly what we want students to take with them beyond our classes (enduring understandings) before we start teaching.  I also shared how this document is a great way to think about differentiation (either within a class or between Honors and standard courses).

(written by UbD PD providers, all rights to them)

We had hoped to give you time to brainstorm essential questions by course, but got too excited talking about our experiences for that to happen.  However, google docs don't disappear after presentations!  So, I have created a folder with all the categories we created.  I would love to see these documents filled with Enduring Understandings (the big ideas we want students to leave our class with) and Essential Questions (the discussion questions or prompts we will use to get our students learning).  Feel free to reformat the documents as you see fit, and I think you even have the option to add new documents (so, for example, elementary and middle school teachers can create grade or topic based documents as you see fit!).  Check those out and start adding: Essential Question Google Docs

At the end of the evening I talked a bit about Attend to Precision/Nix the Tricks and asked for feedback on where we should go from here.  Since it's difficult to have a conversation when only one person is talking, I created a survey!  If you don't see it below, head here: Nix the Tricks Next Steps

I Will Not Quit

You've probably seen at least one, if not several of the letters teachers have written to publicly announce why they cannot abide the direction that teaching is heading in and so feel that they must leave the profession. The Washington Post shared another letter today, but this time it is a teacher resolving not to quit. That's making a statement in a way I can stand behind. In fact, I think we should do even better; how great would it be to show we are all standing next to Christine by writing our own letters? (Credit for this idea goes to Jason who commented with his intent to do just that when I shared the link on Facebook.) Mine is below. You can find Secretary Duncan on twitter and facebook, email him (arne.duncan@ed.gov), call his office (1-800-872-5327) or use good old fashioned snail mail.

Mr. Arne Duncan
Secretary
Department of Education
400 Maryland Avenue,
Washington, DC 20202

Dear Mr. Duncan,

This year has been my most challenging yet.  I have been teaching for six years and never have I felt so micro-managed as I have in the past 7 months.  You see, our district was categorized as Level 4 last year, which means that we are on warning and risk state takeover if there aren't drastic changes in our test scores.  While there would be several plans available to the district if we don't improve, all of them include firing the administrators.  They are feeling pressured to do something, which can result in the decision to do anything and everything to change, and fast.  The culture of the school has changed and the students know it; anxiety abounds and at an assembly before the state test a student asked what will happen if they don't do well enough.  It's bad enough to have their own diplomas depend on the results of three tests, but if students are feeling that the success of the district is sitting squarely on their shoulders?  That's a burden that no one should have to bear.  This comic shows exactly the situation we are facing now, which isn't helping anyone to improve:

SOL testing: No pressure, by Chris OBrion June 18, 2012

Despite this label of Level 4, there are amazing things happening in our schools.  Just after the designation, the high school was awarded full accreditation by the New England Association of Schools and Colleges (NEASC).  What's the difference between the state designation and NEASC's?  NEASC asked the school what is going well and what we need to improve on.  We compiled evidence and interpreted data pulled from a variety of sources (not just tests!). Then, after our self-assessment, teachers and administrators (currently working in schools throughout the region) visited our school.  They walked around the building, spoke to students, interviewed teachers and attended a weekend reception so that parents could participate.  Finally, they sat down with the evidence we provided and looked at it all, asking questions and requesting more examples or information as they needed it.  You see, NEASC looks at all aspects of a school, including students' "collaboration, systems thinking, empathy, communication (including reading, writing, speaking, listening skills), technological skills, civic engagement, intrapersonal intelligence, and problem-solving."  That list shows concern with the well being of students and the skills that will carry into their future regardless of the path they choose to take.  Based on these criteria, including ones that a test cannot assess, our school is succeeding.  Which criteria do you think are more important?

The life of a teacher is a challenging one on a good day.  Many of use work with over a hundred students and each of them needs our attention.  There is never enough time during a school day to complete the most basic of teaching tasks.  When you add on extra requirements you'll see why most of us work more than the 35 hours some claim we "get away with," more than the 40 hours of a 9-5 job, until you see that we work 6 or 7 days a week.  Some teachers wanted to share exactly what it is that teachers are asked to do and they came together to create "A Day in the Life of an Educator" where you can read about a typical day from the perspective of teachers around the country.  These narratives give you a glimpse of what it is that teachers are asked to do.  Our jobs are about so much more than test scores; basing our pay or evaluations on test scores alone devalues the entire profession.  

We are teachers.  We are professionals.  Give us the respect we deserve and ask us what we need.  I can promise you the answer won't be "more tests."

Tina Cardone

Exponentials In Context

Between all the talk about noticing and wondering, taking a course on differentiation and doing fun projects with Geometry, I decided it was past time to do something exciting with my PreCalculus classes.  When students arrived in class they saw this on the board:

Interest based differentiation!

In front of the board I pulled up a small table and spread out slips of paper (I cut up 3 copies of the document below so that there could be a maximum of three people in each group).

Exponential Topics

From there, I instructed students to start asking questions, as many as they could think of.  While most groups got started, I called up the students with a link instead of information on their paper to show them the corresponding Act 1 video.  Once everyone was started making a list of questions, I distributed the instructions and checklist (I hate rubrics- so many words! Everything was worth 1 point except the questions/answers were worth 4).  Some students had trouble coming up with interesting questions so I gave examples of "How long would it take to double?" and "What would happen after 2 years?" but recommended they pick something more significant than doubling/two years.  Mostly, though, they came up with interesting questions without assistance.  The sunglasses groups weren't confident in their answer, so we watched the end of the video to be extra sure, but everyone else was able to come up with an equation or interpret their equation without much difficulty. We had done a few examples of compound interest, but otherwise were working off of memories from last year.  I was impressed with how many people pulled out logs with little to no prompting when they needed to solve for an exponent (way to go Algebra 2 teachers!).

Grading these projects was surprisingly fun.  First, they were all different!

Title of Report

Second, they came up with cute stories. They named characters, identified the source of bacteria (yogurt dropped in an agar tray by an unreliable intern, naturally) and one student even wrote a multi-page story. The premise of that one is a student who wants to work at a car dealership, but his parents are scientists so he needs to provide a proof this is a wise career choice. He gives his parents graphs of the exponential function and cites data. The concluding sentence is: “Now I know why I took PreCalculus, he thought.”

Another favorite sentence: “In the alleyway behind the snack-packing factory is a row of dumpsters where all the mispackaged or expired snacks go to die.” This obviously leads to a growth in the rat population. Another reason the rat population might grow is toxic waste:

At the local dump there are many rats. The rats love all the trash and debris that is so readily available to them. However, there is one thing that makes these rats grow and repopulate so quickly; it is the radioactive trash someone has been throwing away! At the current rate of P(t)=648e^.016t the rat population will double in about 18 years! Not to mention the rats have radioactive superpowers.

Third, the questions they asked were truly interesting.  Groups thought to go forward and backwards in time, compare points, add complications (a fire meant all the library books were lost!) and change the rate.  Most kids asked different types of questions (after t years vs. when will it reach y value) but it may be worth specifying that the questions shouldn't all be the same type.

The group who looked at the photocopied dollar came up with this question, debated if it even made any sense, and then decided to go with it anyway because they found it interesting!

When shrunk two times how much will the dollar be worth in size?
15.5(.75)²=8.7 cm long
8.7/15.5=$.56

Finally, the reflections were amusing.

One student wrote “our group is like a well oiled machine, and I was the on button.” When he shared this sentence with his group after school, the discussion became animated as they tried to figure out the type of machine they were:

If we’re a simple machine I can’t be the on button; there is no on button.
If we’re a car you’d be the key.
Would I be the driver?
No, we’re the machine. You can be the engine.
Ms. Cardone can be the driver.
So that makes (student 3)… everything else!
So, I’m the body of the car, and the air conditioning – in winter.
Everyone laughs.
End scene.

Rational functions and limits

Way back when, over February vacation, I gave my Honors PreCalculus students a rational functions packet (thanks largely to CME). One of the tricky things about teaching this class for the first time is I'm never sure how much of any topic students studied or internalized last year. Turned out vertical asymptotes and holes came back to them easily, success! One of the things I love about teaching PreCalculus is that the end goal is "do cool math, be ready for calc." The other teacher and I were chatting and said limits were just sitting there just beyond horizontal asymptotes, how could we not study them? So we did! (She's also the AP Calc teacher so we're not messing up someone else's curriculum.)

I started with the chart below which is pretty much exactly what JackieB shared. The difference was in the follow up; I didn't want to scaffold their conclusions as much as she did. Which resulted in most students not understanding what I wanted for generalizations. Next time I need to reword that section, should I ask students what the possible values are for the limits first and then ask them to generalize how to know what the limit will be?

Horizontal Asymptotes, Limits

One of my favorite responses to question 3- it's color coded!
Albeit not accurately, but we got there eventually.

After studying limits for a bit I wanted to make sure that everyone had internalized the phrasing I was using. I asked a student to translate from symbols "the limit as x approaches 2 from the right" and in both classes they used that exact phrasing. In one class I made them all say it in chorus, just because they're all so quiet and I needed to liven things up. Some students were mixing up whether the answer was the x or y-value, so I frequently found myself saying "as x gets close to blah, what does y get close to?" And of course every time there was a hole "do we care that there's a hole? Do we care about that value at all? No! Just what it's approaching"

I did limits at infinity, one sided limits at vertical asymptotes, direct substitution, and algebraically solving removable discontinuities, in that order. It made sense to me since we were approaching limits from the context of rational functions, but I'd be interested in hearing rationales for a different sequence. (Math puns!)

The blogosphere has some cool limit activities that I set up as stations on the last day of the unit.  Bowman covered up the value in question and asked students to find the limit, lift the cover and then reassess.  Quite a few students called me over to this station to tell me that the covered up part of the chart was wrong.  Even after all those time I said holes don't matter, that we did graphical piecewise functions, that I emphasized the word approach, students were still confounded by the hidden values not matching their expectation.  However, once I asked them what a limit was, or showed them a graph from the homework that matched this situation they returned to their original answers with confidence and said "this station is easy!"  Sam is right, this really is the best question ever.

I also found a cool geometry application on Irrational Cube which had some students really digging deep into their recollection of approaches to finding area and using a lot of problem solving skills.