Fun with e

Found a fun problem via facebook last week: find a set of whole numbers that add up to 25, and maximize the product of these numbers. It’s such a fun little puzzle that I won’t give away the solution here. But I got interested in another issue that my friend Aaron pointed out: the connection to e as the “best base”. To get some idea about what I’m getting at, let’s stay with 25 and find the set of numbers (not necessarily whole numbers) that add up to 25 and maximize the product. Actually the numbers have to all be the same — if any two numbers were different, then you could move them closer together and thereby increase their product. So basically what we’re looking at is to maximize the expression (\frac{25}{n})^n for positive integer n.

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

(\frac{A}{9})^9 = (\frac{A}{10})^{10} .

We can do some rearranging…


Et voilà! A similar calculation will show that when A=\frac{9^9}{8^8} , 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 A/n 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

\frac{9^9}{8^8} < 9e < \frac{10^{10}}{9^9}.

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…

\ldots < 7e < \frac{8^8}{7^7} < 8e < \frac{9^9}{8^8} < 9e < \frac{10^{10}}{9^9} < 10e < \frac{11^{11}}{10^{10}} < 11e < \ldots


Just go ahead and generalize that…

\ldots < (n-1)e < \frac{n^n}{(n-1)^{(n-1)}} < ne < \frac{(n+1)^{(n+1)}}{n^n} < (n+1)e < \ldots

Turn your eyes to that last piece on the right.

\frac{(n+1)^{(n+1)}}{n^n} < (n+1)e

We can simplify a little…

\frac{(n+1)^n}{n^n} < e

(\frac{n+1}{n})^n < e

(1+\frac{1}{n})^n < e

Where have I seen that before?

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Mystery of the Missing Hummus


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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meta-post / about writing

The dreaded meta-post: posting about posting. Generally I am able to resist the temptation, but now I’ll indulge myself in writing about how this is my first post in three months. While I have some blockbuster blog updates kicking around in the back of the brain — just smile and nod — here I’ll be writing about why I’m not writing.

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

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Which Spawned the Title, “Brain Open Now”

Not sure if I’ve ever explained the name of this here blog. “Brain Open Now”, you can see it right up there. So, what does it mean? I’ll tell ya. Let me take you back to a fellow you mighta heard about… fellow named Paul Erdős.

Click for source:

Open Up!

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.

Click for source, for what it's worth:

This is as good a premise as any for starting a blog!

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!”

Click for image source,

C’mon… I said open up!

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Syllabus Vainglory

Call me vainglorious, but when you read my charmingly avuncular yet somehow Hemingway-esque syllabus, I attest you will spit out that reheated breakfast okra forthwith! If not from sheer awe, then because you left it in the microwave too long. Still, it is just a draft, and I’d surely appreciate any comments.

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This post replies to the prompt, “What is one goal you have for the first week of school?” from Sam Shah et. al.’s New Blogger Intitiative.

[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”

[/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.

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.

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
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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The Garden Problem

For the beginning of the school year I wanted a short module to review and extend ideas about linear functions.  Simultaneously the module should involve students in working together and looking for patterns.  Well I found something that worked pretty well!  So let’s do the internet thing and share what works.

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.

First, the Files

The Garden Problem

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…


  1. How many border tiles are required for a garden of length 12?
  2. How many border tiles are required for a garden of length 30?
  3. How many border tiles are required for the very long garden of length 1000?  Show and explain how you get your answer.
  4. If you know the garden length, how can you find the number of border tiles?
  5. Show how to find the length of the garden if 152 tiles are used for the border.
  6. 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.

Extension Problem

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)

Group Poster Problems

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 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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