Author: samjshah

Answer to my generalized MMM6 question

So I posed the more generalized question to  Monday Math Madness 6, which read:

Start with 500 gallons of mayonaise.
1) Mix in 3 gallons of mayonaise and 7 gallons of ketchup. Stir until completely mixed.
2) Remove 10 gallons of the mixture.
3) Repeat steps 1 and 2 until the mixture is approximately 40% mayonaise and 60% ketchup (200 gallons mayo, 300 gallons ketchup).

How many iterations will it take to do this? 

My solution below the fold.

(more…)

Monday Math Madness 6 Solutions

Here is my proposed solution to MMM6. (Questions, for the uninitiated or forgetful, are here.)

I actually found these problems to be on the easier side (solved both parts in less than 30 minutes), but I tend to be hasty when I work things out, and overconfident in my thinking, so chances are I’m not right on both counts. So here’s throwing caution to the wind and hoping that I got it right.

(Also, if you liked the problem, don’t forget to look for my generalized problem. Solution to be posted soon.)

Part I Solution:

I created a matrix X_0=[500 \hspace{12pt} 0] which represented the [mayonaise, ketchup] initially. I also created a matrix Y=[0 \hspace{12pt} 10] which represented how much [mayonaise, ketchup] I was adding at each iteration. I let k=500/510 which is our “normalization” factor. (You’ll see…)

After the first iteration, we have X_1=k(X_0+Y) which turns out to be X_1=[490.196 \hspace{12pt} 8.804]. You see now what the k does? It makes sure that we have 500 gallons total in the vat.

After the second iteration, we have X_2=k(k(X_0+Y)+Y) which simplifies to X_2=k^2X_0+(k^2+k)Y.

After the third iteration, we have X_3=k(k(k(X_0+Y)+Y)+Y) which simplifies to X_3=k^3 X_0+(k^3+k^2+k)Y.

And you can see the pattern. After the nth iteration, we will have

X_n=k^n X_0+(k^n+k^{n-1}+...+k^2+k)Y.

Now we’re almost done, believe it or not. We can use k^nX_0=k^n[500 \hspace{12pt} 0] to find out how much mayo is in the vat. (Note that since Y=[0 \hspace{12pt} 10], the (k^n+k^{n-1}+...+k^2+k)Y doesn’t contribute to the mayo.)

We want to find when the mayo is around 250. The mayo after the nth iteration is 500k^n. So we have to solve the equation:

250=500k^n

A simple manipulation (taking the log of both sides) yields n\approx 35.

Of course a simpler way to think about this is to say that you start out with 500 gallons of mayo. After n iterations, you’ll have (500/510)^n 500 gallons of mayo. That brings you to the very last step, which is quickly solvable.

But even though that’s the simpler way to do it, it isn’t the way I started thinking about it. I freely admit that I don’t always find the simplest solution… but it certainly helps when doing generalizations!

What happens if we have add 3 gallons of mayo and 7 gallons of ketchup each time… How many iterations until we get a mixture of 200 gallons of mayo and 300 gallons of ketchup?

My sort-of-involved method works for that! I dare you to try it. You get about 98 iterations. It’s only very slightly tricky, so I’ll write up the solution to that problem in a future post.

Part II Solution

Let’s take the 17 lbs of beef (“it’s what’s for dinner!”).

Let’s have (single,double) represent the number of single and double burgers that Nortrom can eat. We know that these combinations are (1,8), (3,7), (5,6), (7,5), (9,4), (11,3), (13,2), (15,1), (17,0). Let’s see how many different ways that Nortrom can eat one of these combinations of single and double burgers.

Well, let’s look at (7,5). This is 7 single burgers and 5 double burgers. How many ways are there of ordering them? Clearly there are \binom{12}{7} ways.

Let’s me be clear about this. If Nortrom eats 7 single burgers and 5 double burgers, Nortrom will be eating 12 burgers. We just have to find the number of different ways he can do this. Nortrom, before eating them, placed them down in order of eating them — in 12 slots. Nortrom puts his first burger in slot 1, Nortrom puts his second burger in slot 2, Nortrom puts his third burger in slot 3, etc. Nortrom has to put single burgers in 7 of the 12 slots. (The rest will be filled with double burgers.) The number of ways to do that is \binom{12}{7}. [1]

So then we have, using our combination of single and double burgers listed above: X=\binom{9}{1}+\binom{10}{3}+\binom{11}{5}+\binom{12}{7}+\binom{13}{9}+\binom{14}{11}+\binom{15}{13}+\binom{16}{15}+\binom{17}{17}=2584.

Let’s do exactly the same thing for our 25 lbs of beef!

The combinations are (1,12), (3,11), (5,10), (7,9), (9,8), (11,7), (13,6), (15,5), (17,4), (19,3), (21,2), (23,1), (25,0).

This leads to a similar conclusion:

Y=\binom{13}{1}+\binom{14}{3}+\binom{15}{5}+\binom{16}{7}+...+\binom{24}{23}+\binom{25}{25}=121,393.

Huzzah!

[1] As a quick aside, note that \binom{12}{7}=\binom{12}{5}. This is because if you have 12 slots and you have to put double burgers in 5 of the slots, then the rest must be filled with single burgers.

Faculty Bio

Today I was asked to fill out a little “faculty bio” of myself. In my school, students list their top 3 choices for advisers each year, and by some mysterious process that probably involves a Sorting Hat, the students get portioned out into homerooms. The students make their choices after reading the various faculty bios.

Here’s what I submitted.

  1. Department: Math
  2. Classes you will be teaching next year: Algebra II/Trigonometry, Calculus, Multivariable Calculus
  3. Two of my Favorite books: Drawing Theories Apart: The Dispersion of Feynman Diagrams in Postwar Physics (David Kaiser), The Secret History (Donna Tartt).
  4. Last book read: Dead Souls (Gogol)
  5. Favorite Movie: Igby Goes Down
  6. Favorite thing to do with your homeroom: Relax and prepare for the day. Put large ships in small bottles. Thumb my nose at those who don’t love “Veronica Mars.” Force my advisees to do really hard math problems. I mean, really, really hard.
They probably think I’m joking.

Some Places To Visit On The Interweb

I’m a blog junkie. I now read blogs daily, and I want to point out my favorite posts. Partly as an archive for myself, partly to share with others some great things out there that I’ve been struck by.

So without further ado, here I go:

  1. On Nailing/Blowing Assessment (dy/dan)
  2. End-o-Year Calculus Projects (Math Teacher Mambo)
  3. The Function Machine Game (Let’s Play Math)
  4. Teaching the Long Tail (Math Stories)
  5. Math Teacher Bingo (3 Standard Deviations To The Left)
  6. Competing the Square (Coffee and Graph Paper)
  7. Math & Art / Big Numbers (The Exponential Curve)
  8. Teflon Teacher and How Much Do They Change (Certain Uncertainty)
  9. We Know You’re Blogging (On The Tenure Track)
  10. Exceedingly Lame Final Question and Counterexamples (The Number Warrior)
  11. Quitting Teaching College (An Educator’s Blog)
  12. Classroom Management vs. Discipline (Catching Sparrows)
  13. A Motivational Experiment: Reflections on a Mohawk (I Want To Teach Forever)
  14. Help Wanted: Active Summer Learning With Technology (Dangerously Irrelevant)
Lucky 14. With that, I’m out.

Senior Letters Made of Sap

Everyone is bringing food, and we’re going to play Apples-to-Apples. Monday will be the first time my calculus class will be doing something almost-totally non-mathematical (except for the one time we watched “Numbers” before winter break). I’m convincing myself that this is okay because I have the deluxe edition of Apples-to-Apples with blank cards; we’re going to be throwing in some calculus terms.

I work them hard, and they’ve met the challenge. So we’re celebrating on Monday, and we’re going to hear presentations of everyone’s calculus projects on Tuesday and Wednesday. And then: it’s over.
I didn’t think I’d be maudlin, but I am pretty much all sap at this point. I decided to write a letter to each of my senior students thanking them. (Well, ahem, actually I wrote one letter to the whole class.)

We often expect to hear thanks for our work. But as you well know, teaching goes both ways, and I wanted to thank my students for their work. Not just for their mathematical work in class and at home, but for their positive attitude and humorous good-nature as we fought tooth-and-nail against the beautiful beast that is calculus. Being a new to a school, and being a new teacher, was made so much easier because of them.

In the envelope with that letter, I’m including two additional things.

  1. Their first day’s homework assignment — this form which they filled out (stolen from dy/dan).
  2. A juxtaposition of two quotations about Nature and Wonder. Many of my students have their grillzs all up in the humanities. I am not trying to convince them to be mathematicians and scientists. But I want them to see that the two are not mutually exclusive. So I will be giving them the poem and quotation below the cut.

I wouldn’t let them get away with having no homework. So I’m leaving them with one final homework assignment, playing on the theme of “the letter”: write a 1-page letter to yourself a year ago, giving your “old” self advice on how to succeed in this course.

After the next three days, they’re gone.

Sigh.

(more…)

When they’re wrong…, or, how the heck do I scaffold?

I started this post a long time ago (maybe two or three months ago), but scrapped it. But I’ve decided to finish it up and make a little plea for advice at the end.

What do you do when you ask a question and get a totally wrong answer? Okay, this question screams newbie, but it happens to me enough and I often get caught in an awkward situation. Let me explain.

A completely made-up but not unrealisitic example:

Me: So we now we have this: x^2-2x=-1. Where do you think we go from here? What are we trying to do again? StudentX?

[Context: We’ve learned how to graph quadratics, use the quadratic formula, complete the square, factor, and seen equations like this all year. It should be second nature to them. And for many it is, but for some it isn’t. The problem is this: we’re way beyond this. We’re working on some other concept, and these gaps force me to veer away from the current lesson and take a bunch of steps back to reteach these things to the few that don’t get it…]

StudentX:  Um… well, we could add 2x to both sides…
Me: [awkward silence while I think of what to say, because I don’t want to do that…]
Me: At every step, we want to ask ourselves: (1) why do we do that? and (2) what are we trying to find out? So why would we add 2x to both sides? What are we trying to do?

[Context: Even when they are on the right track, I will ask this question. I want them to think about every move they make.]

StudentX: I don’t know.

At this point, I’ll ask what type of equation we have, and what we know about it. StudentX will finally get it (“quadratic!”), and we’ll move on.

Sometimes I don’t make it a drawn out process. If I’m in a rush, I will ask if someone else has a different idea and call on someone who I know will have the right answer, and then move from there. And then I’ll return briefly to the original idea and explain why it won’t get us to where we want to be.

But this interaction takes 3-5 minutes, I know 80% of the students in the class are bored, some are trying to whisper the answer to the student, and we get held up.[1]

Of course, I’m all about meeting students where they’re at. And I’m happy to review. But these moments happen all too often, and using every one of them as a teachable moment takes too much time and would be bad practice. I have a curriculum to cover. Taking three steps back constantly is tough.  

That tension, between moving forward in the curriculum and making sure students are up to speed on the older stuff, is palpable. 

I often feel like I sacrifice the majority of the class when I do too many of these types of things. I don’t want to praise a wrong answer (“That’s a great idea, but I’m not sure it’ll help”), I don’t want to scare a student from speaking in class (“No”), I don’t want to spend a lot of time on a basic skill that the rest of the class knows, I don’t want to make the student feel dumb or ignored (“Anyone else have a different idea?”).

I’m afraid I’ve done all three.

To make this into a truly teachable moment would require me to add 2x to both sides, and then stick with the student and ask them what next. And just stick with them until they see that they’re stuck. But I tend to only go down really wrong paths in math when we’re learning something new and we have the time to have these dead end explorations.

Basically, when it comes down to it, I recognize that I still don’t know how to organize and manage a differentiated classroom well, how to scaffold lessons, how to keep everyone engaged and learning, while still moving forward in a fast-paced curriculum. It’s not that I don’t try. About 30% of my students have some learning difference or another, and I do think about that when I’m designing my lessons. I do. But what I’m doing isn’t working. At least not as well as I’d like.

I think that in addition to classroom management, this is one of those big topics that doesn’t often get explicitly addressed in teacher blogs. Maybe that’s because a good many teachers do it without thinking about it – it’s natural. But even though I do a lot of things naturally well, planning a scaffolded lesson for a pretty differentiated class isn’t one of my fortes. Yet.

So anyway, if you know of any blog posts or websites, or have any advice, holla out in the comments.[2]

Yeah, I know, I know. Everything about this screams “Newbie.”

[1] One of my fears is that I’m going too slow for a bunch of my kids, and I’m not sure where my focus should go. The middle of the road? Those that don’t get it? Those that do? For me, I think a complicating factor is that I was always one of those kids who did get it, and really quickly. I identify with them. I don’t want those kids to be bored. And I feel guilty because I am pretty sure they are bored.

[2] Five or so years ago, I read about differentiated classrooms in one of my teaching classes, but the readings were all academic mumbo jumbo with no connection to reality. I’m looking for something useful.

My Blogroll

I’ve been meaning to put a blogroll up for a while. But the problem is that my blogroll is constantly evolving, and I wanted something that updates as I update.

Well, my RSS reader netvibes — which I’ll tout as currently the Best. Thing. Ever. — allows you to see all the blogs I read, updated. So click on the netvibes icon on the right and check out some of the amazing blogs out there. Without further ado: my blogroll.