I hope you’ll play around with this REALLY COOL GEOGEBRA APPLET (screenshot below) before continuing.

The thing that really interested me in this problem was how changing the value of A (originally 25) would change the optimal value of n. Sure, for 25 we see that the best value is n=9, but apparently for some larger value of A, then we need n=10. What is the cutoff? When do we switch from 9 to 10?

In fact, you can try it: increase A gradually until the line for n=9 is as high as the line for n=10. You should find yourself at A=25.81 or so. To find the exact value we can do a little algebra. Suppose we have A such that

.

We can do some rearranging…

Et voilà! A similar calculation will show that when , both 8 and 9 groups give the same product.

Another issue implied by the applet is the link to e. Apparently, we would like the ratio to be close to e to get the maximal product. It’s no coincidence that our original value of 25 is close to 9e (about 24.46). This leads to the interesting inequality

.

And based on the context, this is the range where we would choose to divide A by 9, and 9e should be more or less in the middle of this range.

Of course, there’s no reason not to extend this a little on both sides…

Just go ahead and generalize that…

Turn your eyes to that last piece on the right.

We can simplify a little…

Where have I seen that before?

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Mmmmm…. edamame hummus, two delicious flavors in one perfect cylindrical container. 8 ounces of goodness.

Well this is alarming. Couldn’t they fill it all the way? How much hummus am I missing?

Wait, that’s cheating!

Time to put on your geometric modeling caps! What questions would you ask? What do you need to know?

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Thesis: I lack internal motivation.

Some nuance: Even though I can think of myriad moments of internal motivation, at a basic level I am waiting for external forces to spur me into action.

- Evidence 1: My last three blog posts are due to participation in the new blogger initiative.
- Evidence 2: Last year when I took that class about strategies for ELLs, my teaching became stronger as a result of being forced to use these strategies in my classroom. Now, I’ve mostly let this habit fade away.
- Evidence 3: On a large scale, my best professional work in (let’s say) the past five years has been a result of the pressure of developing curriculum and plans to use in my class.

Of course (is obvious nuance possible?) most folks who aren’t freelancers and/or vagrants might write something like item 3. This is the nature of a professional career, and as humans we use each other to accomplish more than we would individually.

But to bring a merciful end to an indulgent post, I’ll relate a story my friend told to a group of aspiring writers (I went incognito). It was some question about, what is the most important lesson you learned in the process of getting published? Response (and excuse my poor sense for reconstructing remembered dialogue):

I think the big thing is, no excuses. I had this idea about writing a story about LA, and I thought that I should move to LA to write the book. A friend said, ‘why do you have to move? Just write the book here.’ I was like, … oh yeah! I don’t need to move to LA to write the book. I don’t need to wait for

xto be able to do the work — just do it now. If you want to write — well, start writing.

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Interesting guy, and I can point you to the obvious places to to read about him. Seems like he would come into a classroom to hear or give a lecture and declare, “My brain is open.” A rather pedestrian virtue of being open-minded is filtered through the Hungarian (and whatever else) to become this somehow medical/mechanical statement about his inner workings. Say this, and you say that you are willing to entertain and give energy to any line of thought. And to me, this also has connotations about being transparent with thought, like in the phrase “open source”, as if one could observe how the thought is formed, developed, and executed.

This sounds suspiciously like blogging.

So I thought to title my blog “My Brain is Open” and get on with that. But then I found that someone on WordPress had already done so, so I struck out for (relatively) more original ground.

For some reason I chose ~~“Open is My Brain”~~ “Brain Open Now”, which could be part of the phrase, “My brain is open now”. But I prefer to read it as a command: “Brain! Hey brain! You there? Ok …. Open Now!”

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[Apologetic Opener]

It’s been a while since I wrote a good blog post, and probably this is not one of them. As usual, what I thought was a tidy idea ended up with more loose ends than, um, unloose ends. You’ve been warned…

[/Apologetic Opener]

[Short Version]

The note I made to myself during pre-writing probably suffices. So if nothing else, check out the link, and then ask, “Really?”

introduce a HOT TOPIC statistical issue — something that seems to enrage with its ridiculosity. could be related to immigration or other hot-button issues.

Example: “The Deporter-In-Chief”

https://www.youtube.com/watch?v=6kopRTfWeNo&feature=player_embedded

[/Short Version]

Going into my sixth year of teaching, you might think that I’d have these first week goals on lock down. For sure, I’ve felt fine with my usual combination of community building and interesting introductory math problems. But this prompt lead me to reassess what I’m trying to do for the first week. Simply enough, my goal is to have a completely KICK ASS First Day Of Class.

What are the components of a KICK ASS class, aside from holding down Shift as you type those sweet letters? Like a good book that immerses you immediately in the story, or an album that lets you know right away that “this one’s a keeper”, in this KICK ASS class students will dive right into the issues that define the class as a complete entity.

[Context]

I teach students who have immigrated to the US within the past 1-10 years. The class I’m thinking about in this discussion is targeted toward Seniors who have completed Algebra 2, but would probably not do well in Pre Calculus for a variety of reasons. My only directive in this class is that it fit into the Dept. of Education’s code of “Advanced Topics in Math” and attached subcode “Research and/or Projects”. Since most students have seen only bits and pieces of statistics, I thought that a course that focuses on numeracy using statistical techniques would be valuable to the students.

[/Context]

Pushing back against my desire to dive in is my sense that it’s the first day of class, and there are certain “first day of class” (FDOC) management duties that I need to take care of. Chris Luzniak’s post was helpful in reminding me what an effective FDOC might look like:

1. Assign seats

2. Collecting information on index cards

3. Introductions

4. Go over the Syllabus

5. Do a math problem

Item 5 seems to be the best area to up the KICK ASS-NESS of my FDOC. One end goal of this class is that students will learn how to question the quantitative claims that are too often used to manipulate their opinions. We’ve all heard the lessons of “How to Lie with Statistics”, and while Statistics might have some other uses, surely the act of lying with math will pique a few of my students’ interest.

Enter the Deporter-In-Chief, President Barack Obama.

To the class: What do you think about when you see this video? What questions come to you mind? Brainstorm with your group.

I will write down the statements and questions that the students come up with, and then I’ll ask them to think about any information they would want to know. Hopefully this includes information related to the number of deportations each year, which I will have preprinted for their perusal.

The infamous “Table 36”, from the Dept. of Homeland Security, via Politifact.org:

http://www.dhs.gov/xlibrary/assets/statistics/yearbook/2010/table36.xls

Department of Homeland Security, “Aliens Removed or Returned: Fiscal Years 1892 to 2010,” accessed Aug. 9, 2012

Of course this only raises more questions! Muahahaha…

1. Who was the president in each of these years?

2. Why did the number of deportations decrease from 2009 to 2010?

3. What’s the story with the massive number of “Returns”?

4. Do we need any more information?

Does this video tell the truth? Or is it an example of lying with statistics?

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It might be helpful to know that my class is quite heterogeneous in terms of math background, and all of them are ELLs. It’s an Algebra 2 class so they’ve “seen” linear functions before.

I found the Garden Problem while searching for components of algebraic thinking. In her aptly named webpage, “Just WHAT IS ALGEBRAIC THINKING?“, Dr. Shelley Krieger offers a fine introduction and a beautiful problem that I swiped for my own use:

*I introduced the pattern like this…*

*And then I changed the questions slightly…*

Questions:

- How many border tiles are required for a garden of length 12?
- How many border tiles are required for a garden of length 30?
- How many border tiles are required for the very long garden of length 1000? Show and explain how you get your answer.
**If you know the garden length, how can you find the number of border tiles?**- Show how to find the length of the garden if 152 tiles are used for the border.
- Can there be a garden that uses 2011 tiles? Explain your reasoning. (When you’re done with this question let me know!)

Question 3 is something that I’m starting to see as the bridge between “Length 12” and “Length x” from the original problem statement. If you can do it with 1000, then you can do it with x. Question 6 was also nice because some students used their formula to show that the garden length would not be a whole number if it used 2011 tiles, while other students made the reasonable observation that every garden tile number is even.

Knowing that I have some students who would breeze through these questions and appreciate something a little trickier, I devised an extension problem for this scenario, as follows… (I’ll leave the solution to the reader)

To build on their work with the Garden Problem, I designed a group task for students to solve and present solutions for a new patterns. The patterns were copied over from an activity written by the nice people at the Math Learning Center, who appear to be having some website problems. The activity that was formally available at www.mathlearningcenter.org/media/ATVP_Gr6-12_Lesson1.pdf can be downloaded from my dropbox here.

*Here’s a sample of what a group might encounter…*

You might notice that this pattern is not linear! So we start the discussion for comparing linear and nonlinear functions.

Finally, if you paid attention to the files at the beginning, you might have noticed some sort of “Audience Participation Activity.” I wanted something that helped the students get to know each other (in this case writing down names), and to help them get engaged in the presentations. I made the “write an interesting question” part optional because it can be difficult to come up with such a question, and I didn’t want to stress out the students about this non-critical part.

If I get my act together I’ll take some pictures of the posters — there were some nice ones. This whole introduction to linear problem-solving occupied the greater part of four days at the beginning of this school year. As I start looking at exponential functions, I’m wondering how I can engage in the same sort of patterns-based problem-solving and algebraic thinking. Time will tell, and I hope to share it if it works.

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So I sat down the other day (as I often do), and thought, “What about exponential and logarithmic functions might a high-school student find interesting?” Yes yes, exponential functions are essential in a number of financial math applications, but figuring out mortgage calculations is not the most engaging activity for the average teenager. Surely I could find something more worthy of student attention.

Soon enough, I found my rabbit hole: “Malthusian catastrophe.” Since stumbling on this half-familiar phrase nearly 24 hours ago, I have devoted an unusual proportion of my time to flesh it out through internet research. This post is an attempt to organize this information and maybe even get some suggestions for new resources. Well anyway, behold! :

“The power of population is so superior to the power of the earth to produce subsistence for man, that premature death must in some shape or other visit the human race. The vices of mankind are active and able ministers of depopulation. They are the precursors in the great army of destruction, and often finish the dreadful work themselves. But should they fail in this war of extermination, sickly seasons, epidemics, pestilence, and plague advance in terrific array, and sweep off their thousands and tens of thousands. Should success be still incomplete, gigantic inevitable famine stalks in the rear, and with one mighty blow levels the population with the food of the world”.

—Malthus T.R. 1798.

An essay on the principle of population. Chapter VII, p61 [via Wikipedia]

Famine, pestilence, and plague … I’m in!

[side note: Check out the Math Pickle and their values for student learning: “Mathematics and Beauty are Inseparable” ; “War, Death, and Nastiness Engage Students” ; “Hard Fun! That’s How We Learn Best” ]

Right away we have debatable questions: are humans doomed to catastrophe? are there too many of us? what is the maximum possible human population, and what will happen when we get close to that level?

**Resources**

**1. Introduction.** The best resource I’ve found so far is this site: http://www.math.montana.edu/frankw/ccp/modeling/discrete/snooping/learn.htm

It’s a sequence of modules that have taught me a good deal about different models for population growth. This particular section focuses on discrete methods (most appropriate for my class), but there are sections invoking continuous data and more Calculus-based approaches as well. Anyway, the discrete model module naturally leads to the study of recursive sequences, arithmetic and geometric scenarios, and other goodies. The algebra requirement is robust but not insurmountable. And there’s the sense that the advancing models show different levels for differentiation: the earliest models definitely hit at my core curriculum expectations for the semester, while later models become as complex as you want them to be. For example, I would expect all student to be able to use the exponential model, while more advanced students can use logistic models and beyond.

Model 1: . Simple exponential growth.

Model 1b: . Allows for “immigration/emigration”

Model 2: . C represents “carrying capacity,” so as P approaches C, the possibility for growth is diminished, with interesting behavior based on the value of R.

[I’ve had some difficulty with disparities in quoted discrete logistic models. This site was immensely helpful in straightening things out.]

Now that I feel like the math is appropriate for my class, I can look for my next requirement: there should be the possibility to play — play with calculations to look for patterns, play with a simulation to make conjectures, and play with an equation to see how a model works out.

**2. M&M’s in their natural environment. ** First, place 20 M&M’s (representing fish) on their fishbowl (plate) [sidenote: I’ve also seen this done with coins, but M&M’s are much more inherently interesting (and edible).]. Each time, put the fish in the cup, shake ’em up, and pour them on the plate. The ones showing “M” get eaten. Second scenario, start with two, and for each one showing an “M”, add one to the bowl. The PBS activity sheet clearly tells you when the fish should be eaten. The follow-up activity leads the students to develop NOW-NEXT models to match the fishbowl data.

Supplementing this activity, there is a simulation of the fish population for different growth coefficients. The page also gives a model including logistic growth, so this could be explored later.

**3. Dice simulation.** As an extension to the M&M’s simulation, I want to use dice to let students invent different scenarios that would vary the growth rate. For example, start with 2 dice. Each turn, roll the dice, and for each 6 you see, add one dice to the population. What is the long-term behavior? Can you write a model?

I’ve spent an inordinate amount of time trying to make up a variation that would lead to logistic-like behavior. Here’s my best attempt at such a game:

Start with any number of dice. Each time, roll the dice onto a sheet of paper (or other region). If the dice land on the paper, they survive, then they may reproduce according to whatever rules you like. If they do not land on the paper, they die. What is the maximum population you can achieve? What is the long-term (and short-term) behavior? Can you write a model? (the model is not nearly as clear in this case)

This game is not perfect in terms of reproducing a lot of the quirks of the logistic model, but I’m awaiting my bulk shipment of dice to try it out.

**4. The model should be intuitive.** The Wikipedia article on “Matrix Population Models” explains that “all populations can be modeled by one simple equation:

Which is often called a BIDE model (Birth, Immigration, Death, Emigration). This is a rather intuitive equation that can let you build some good models. Also of note is that when considering the world population, there is (currently) no immigration and emigration from our planet! This is rather convenient.

**5. Vast selection of computer simulations.** By no means a complete or optimized list.

- Rabbits and Wolves
- Same idea, sustainability game
- Fertility dynamics – nice applet simulation
- Demographics/population simulation
- Animation / interactive showing births deaths CO2 use worldwide

**6. For further reading.**

- Readable explanation of exponential and logistic models
- Periodicity and Chaos — includes TI – 83 fun (part of a nice set of “vignettes”)
- Policy issues, math and teaching notes too
- Incidentally, the Montana website was produced as part of the “Connected Curriculum Project,” and I found the Duke site here.

**7. For further thought / processing.** Thanks for your hard work, intrepid reader!

Have students mark their data in a Google Doc spreadsheet, which can allow for easier aggregation of data and class comparisons. This could also aid discussions about random variation and deviation from the NOW-NEXT model.

BINARY FAMILY TREE: Create tree with you at the trunk, and biological parents above. How many great-great-great-great-…-great-grandparents do you have? Realization: This number is greater than the human population. You must be related!

RABBIT PROBLEM: Note that Fibonacci is proposing his problem at the dawn of population growth! Fibonacci – c. 1200. World pop. in 1200: about 400 million.

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At the same time, I never developed a homework system that I liked. For the last school year, I essentially didn’t assign homework at all, which is certainly one solution to the issue. However, my students were expected to do DAILY homework in their previous class, and I’m not sure what message they’re getting by getting no homework in a class that’s supposed to be harder. So… homework is on the agenda for the next year (my fifth year of teaching!) I hope to use this blog post to share my thinking and propose a system that I could imagine working for me next year.

**Guiding principle**: Homework can let students use as much time as they need to try out some math questions. Therefore it can be an effective forum for practicing and receiving feedback. Feedback is the focus, not grades and the pressure they bring.

I highly recommend looking at Sam’s blog post HOMEWORK SURVEY RESULTS. It’s a thought provoking synthesis of what different teachers out there are doing. In the survey data I came across a great counterpoint to homework review in class, as follows:

**Counterpoint**: “Homework review, next to disciplinary conflicts, has been the most efficient way I know to waste classroom time and sink classroom morale.”

A handful of homework problems are assigned each night. Students complete them in their journal. At the beginning of the next class, randomly selected students put problems on the board. (Not optional.) They sit down. We look, discuss, correct. Teach a respect for what goes on the board. I love mistakes. Teach how to respond to the work and not the student.

At the beginning of class, the rest of the students have out their homework to compare with the work on the board. The teacher sees who has done their work in a rough sense. I make a note of who has an incomplete collection of work and/or missing notebook.

At the end of class, collect a selection of notebooks to look at. (Ideal world, I would get to see this during class, but not happening I think.) At least collect a few. Emphasize that I just want to look closely at their work and see how they’re doing. The students who get their work collected should work on their homework at a later date.

Also at the end of class, assign, say, four homework questions. Students make a four square in their notebooks and put one in each. In their notebooks, write: “HOMEWORK DUE ON WEDNESDAY, OCT. 25th” or whatever. This gives me a clear heading to see that their work is organized, and it makes it easier to scan the class.

(Discarded idea: write this on the board at the BEGINNING of class. Students copy down the problems in their notebooks; this lets me put it on the board in an organized space. Discarded because this doesn’t let me be that flexible with the types of problems I assign, and some students will just automatically start solving the homework problems instead of reviewing their previous problems.)

The grade will be given on a completion basis. Perhaps I give a weekly homework grade: 10 points per week, and they start at 10/10. I subtract points for incomplete work or missing notebooks. No late work accepted in general. When I collect the notebooks of a few students, I see if there is an agreement between the what I have observed and what’s actually in their notebook. (Of course this is also the best time for providing written feedback to the student.)

What I like in the system described above is that it’s something I can actually imagine doing on a daily basis. For the most part, it provides quick feedback in terms of seeing student solutions, which in turn can spur some revealing conversations. I’ll need to read the class to make sure we don’t kill the homework review. The major pitfall of the homework review is that it can become an endless process if a few students have a number of questions, while the majority of the class has no interest in hearing how this or that problem is solved. With this in mind, I might start out with a lower number of problems (like 1 or 2), and then build up to more when I get the pacing down.

Any thoughts on this, intrepid reader(s)?

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But for now I thought I’d share a little problem that I saw today. This summer I’m an adviser to a group of high school students who are taking college classes. A student in a college algebra type course showed me this “bonus” question on the bottom of his quiz:

Find the sum, .

I won’t spoil your fun by posting a solution, but I encourage you to do so. I’m definitely interested in different solution methods on this one.

Enjoy!

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