Author: samjshah

Will the fish bite?

Today at the end of Precalculus today I asked if any kids had any questions/topics they wanted a quick review on for an assessment we’re having tomorrow. (We lost a week due to Hurricane Sandy, so it’s been a while since they’ve worked on some of the topics.) One of the topics was inverse functions, so I gave a quick 3 minute lecture on them, and then we solved a simple “Here is a function. Find the inverse function.” question. They then wanted an example of a more challenging one, so I made up a function:

And then we went through and solved for y. And we found that …

… the function is it’s own inverse. Yup, when you go through and solve it, you’ll find that is true. (Do it!)

I said: “WAIT! Don’t think this always happens! This is just random! Really! This is random!”

But I just had the thought that this might be good to capitalize on. So I sent my kids the following email:

I don’t know if anyone will bite, but I hope that someone decides to take me up on it. I already have a few ideas on how to have them explore this! (Namely, first exploring \frac{x-b}{x-d} and then exploring \frac{ax-b}{cx-d}.)

We’ll see… I’m trying to capitalize on something random from class. I hope it pans out.

A biology question that is actually a probability question

Now that the hurricane issues are slowly dissipating, I made it back to Brooklyn today, back to the place I spend most of my time… my school… I suppose you could call it home.

I’m doing some work here before turning to my apartment, and I ran into a science teacher who asked me a question:

Let’s say you have a sequence of 3 billion nucleotides. What is the probability that there is a sequence of 20 nucleotides that repeats somewhere in the sequence? You may assume that there are 4 nucleotides (A, C, T, G) and when coming up with the 3 billion nucleotide sequence, they are all equally likely to appear.

I liked the question, but I haaaave to work on my own work and not this problem at this moment. So I thought I’d throw it to you.

A. What’s the answer to this question?

B. How would you explain it to this biology teacher (who knows basic math stuffs)?

and for the bonus…

C. How would you design a lesson that would make a student understand the process and your answer. You can assume that the student understands combinations and permutations.

If I get some work done today, I may think through this problem as a treat. If none of you beat me to the punch. But I’d rather you beat me to the punch.

PS. I might as well throw in the additional question of: “how long does the length of the sequence have to be before you are guaranteed a repetition of a sequence of 20 nucleotides?”

UPDATE: My friend Jason Lang sent me his solution, which is amazingly written and cogent.

What does it mean to be going 58 mph at 2:03pm?

That’s the question I asked myself when I was trying to prepare a particular lesson in calculus. What does it mean to be going 58 mph at 2:03pm? More specifically, what does that 58 mean?

You see, here’s the issue I was having… You could talk about saying “well, if you went at that speed for an hour, you’d go 58 miles.” But that’s an if. It answers the question, but it feels like a lame answer, because I only have that information for a moment. That “if” really bothered me. Fundamentally, here’s the question: how can you even talk about a rate of change at a moment, when rate of change implies something is changing. But you have a moment. A snapshot. A photograph. Not enough to talk about rates of change.

And that, I realized, is precisely what I needed to make my lesson about. Because calculus is all about describing a rate of change at a moment. This gets to the heart of calculus.

I realized I needed to problematize something that students find familiar and understandable and obvious. I wanted to problematize that sentence “What does it mean to be going 58 mph at 2:03pm?”

And so that’s what I did. I posed the question in class, and we talked. To be clear, this is before we talked about average or instantaneous rates of change. This turned out to be just the question to prime them into thinking about these concepts.

Then after this discussion, where we didn’t really get a good answer, I gave them this sheet and had them work in their groups on it:

I have to say that this sheet generated some awesome discussions. The first question had some kids calculate the average rate of change for the trip while others were saying “you can’t know how fast the car is moving at noon! you just can’t!” I loved it, because most groups identified their own issue: they were assuming that the car was traveling at a constant speed which was not a given. (They also without much guidance from me discovered the mean value theorem which I threw in randomly for part (b) and (c)… which rocked my socks off!)

As they went along and did the back side of the sheet, they started recognizing that the average rate of change (something that wasn’t named, but that they were calculating) felt like it would be a more accurate prediction of what’s truly going on in the car when you have a shorter time period.

In case this isn’t clear to you because you aren’t working on the sheet: think about if you knew the start time and stop time for a 360 mile trip that started at 2pm and ended a 8pm. Would you have confidence that at 4pm you were traveling around 60 mph? I’d say probably not. You could be stopping for gas or an early dinner, you might not be on a highway, whatever. But you don’t really have a good sense of what’s going on at any given moment between 2pm and 8pm. But if I said that if you had a 1 mile trip that started at 2pm and ended at 2:01pm, you might start to have more confidence that at around 2pm you were going about 60 mph. You wouldn’t be certain, but your gut would tell you that you might feel more confident in that estimate than in the first scenario. And finally if I said that you had a 0.2 mile trip that started at 2pm and ended at 2:01:02pm, you would feel more confident that you were going around 72mph at 2pm.

And here’s the key… Why does your confidence in the prediction you made (using the average rate of change) increase as your time interval decreases? What is the logic behind that intuition?

And almost all groups were hitting on the key point… that as your time interval goes down, the car has less time to fluctuate its speed dramatically. In six hours, a car can change up it’s speed a lot. But in a second, it is less likely to change up it’s speed a lot. Is it certain that it won’t? Absolutely not. You never have total certainty. But you are more confident in your predictions.

Conclusion: You gain more certainty about how fast the car is moving at a particular moment in time as you reduce the time interval you use to estimate it.

The more general mathematical conclusion: If you are estimating a rate of change of a function (for the general nice functions we deal with in calculus), if you decrease a time interval enough, the function will look less like a squiggly mess changing around a lot, and more and more like a line. Or another way to think about it: if you zoom into a function at a particular point enough, it will stop looking like a squiggly mess and more and more like a line. Thus your estimation is more accurate, because you are estimating how fast something is going when it’s graph is almost exactly a line (indicating a constant rate of change) rather than a squiggly mess.

I liked the first day of this. The discussions were great, kids seemed to get into it. After that, I explicitly introduced the idea of average rate of change, and had them do some more formulaic work (this sheet, book problems). And then  finally, I tried exploiting the reverse of the initial sheet. I gave students an instantaneous rate of change, and then had them make predictions in the future.

It went well, but you could tell that the kids were tired of thinking about this. The discussions lagged, even though the kids actually did see the relationships I wanted them to see.

My Concluding Thoughts: I came up with this idea of the first sheet the night before I was going to teach it. It wasn’t super well thought out — I was throwing it out there. It was a success. It got kids to think about some major ideas but I didn’t have to teach them these ideas. Heck, it totally reoriented the way I think about average and instantaneous rate of change. I usually have thought of it visually, like

But now I have a way better sense of the conceptual undergirding to this visual, and more depth/nuance. Anyway, my kids were able to start grappling with these big ideas on their own. However, I dragged out things too long. We spent too long talking about why we have to use a lot of average rates of changes of smaller and smaller time intervals to approximate the instantaneous rate of changes, instead of just one average rate of change over a super duper small time interval. The reverse sheet (given the instantaneous rate of change) felt tedious for kids, and the discussion felt very similar. It would have been way better to use it (after some tweaking) to introduce linear approximations a little bit later, after a break. There were too much concept work all at once, for too long a period of time.

The good news is that after some more work, we finally took the time to tie these ideas all together, which kids said they found super helpful.

A Day In The Life, Math Teacher 2012 Edition

For anyone out there — I’m fine here in New York City. I spent the hurricane  [2] at a friend’s place in the city, and we have power and all good things. When I was trying to pass the time, I decided to do one productive thing.

I would like to present to you the start of a one day blogging initiative.

We are busy. We do a lot. We are professionals. And you know what happens when we talk about what we do… most people who don’t teach just don’t get it. That’s why we go to each other for support — either in real life by unwinding over a glass of wine (or a mocktail) at a local watering hole, or by talking with each other virtually using blogs, twitter, email, or something else.

But I think that needs to change. Three years ago, my school got me a sub for a day and I shadowed a tenth grade student. I met up with her before homeroom, and I went from class to class until 3:15. I sat through various lessons (I even answered some questions in French!), I took a chemistry exam (D+, but it would have been a C if I had remembered the conversion from Celsius to Kelvin!), and I ran from room to room seeing what it was like to go from teacher to teacher with differing expectations, to see if the passing period was enough time to get from point A to point B, what a student was to do if they had no lunch period (like the student I shadowed… answer: eat yogurt and string cheese during class), etc. The real shame was that I didn’t shadow the student after school. Balancing extracurriculars and the work assigned to see what a true full day was like. The experience was golden. I got to be a student again. (Plus the student I shadowed and I got to make up a rad handshake… the shadower-shadowee handshake.)

I believe that others out there can know what it is like to be a math educator, at least for one day, from start to finish. I think we can explain to them about what we don’t and not leave the conversation saying “yeah, they don’t get it.” What are the big things we do, and more importantly, all those little things we think about and deal with? Not only am I beyond curious what a day in the life is like for all y’all, but I would like to take up the challenge of trying to get across what it’s like to be a teacher to someone who isn’t a teacher. Verbal explanations — even to my parents who are interested and care — hasn’t quite done the trick.

Thus, Tina — the author of Drawing On Math — and I have decided that we’re going to post about one of our days — from start to finish — during the week starting on November 12, 2012. Personally, I don’t know what I’m going to do. Maybe I”ll have some video, some photos, to accompany words. Maybe I’ll just write. Maybe I”ll have a timeline. Maybe it’ll be accounting of things, maybe it’ll be an accounting of thoughts. Who knows — but I am going to try to get across the big and the small of my day. [1]

Here’s the thing: we’re professionals. Let me say that again: we. are. professionals. There are many of us, which maybe makes what we do undervalued. There is this disturbing cultural war on teachers which is disheartening and just sucks to bear witness to. And there is this hidden side to teaching that everyone who has had teachers, but never been a teacher, doesn’t know about. And so I’m hoping that this might help people understand.

We’d love for you to join. Do a day in the life from any day in the week of November 12. Post about it on your blog. And maybe by the end, we’ll have a collection of some good things to share with someone who just doesn’t get it. If you write about it on twitter, use the hashtag #DITLife … Throw your blogpost link down in the comments on this post or in the comments of Tina’s post, Submit your form on this handy dandy form that Tina created, and Tina will compile them in one of her mathemes

I suspect there will be various banners for this floating about. You can take the one from the top of this post, or if you don’t like the pastel flowers, you can take this one below. Or any of the others that are oot and aboot (Tina’s are here).

And if you’re not a math educator, and want to do this, have I got two banners for you. Yup, they look almost identical. Yup, I’m lazy. Deal with it.

[1] Of course, we have to be careful to make the post about us. There are issues of kids privacy, and sensitive things we deal with, that we can’t explicitly write about. That’s probably why I haven’t done this before… because I don’t yet have a good way to get around this and still give an honest accounting of what happened. But that’s a lame excuse, and I will come up with a good way to give an honest picture of a day in the life, while still respecting my school, colleagues, and students.

[2] Want to see a vector valued function related to the hurricane? (I’m teaching about vector valued functions in multivariable calculus… so this resonated.) This is a live wind map. And here are some screenshots from around 1am last night:

Approximating the Instantaneous Rate of Change in Calculus

I’ve been trying something new this year in calculus… really having students grapple with the concepts of what they can definitively know, what they can definitively not know, and what they can know with some certainty (but not total certainty) when they are given some information about a car trip. I’m hammering home the conceptual underpinnings of average and instantaneous rates of change. And I’ll blog about that soon I hope. But today’s post comes from where we went with this…

This week, we got to the point where we were estimating the instantaneous rate of change of a function at a point by using the average rate of change for a small interval near the point. And we’re used to seeing things like this in a textbook:

We’re getting our interval smaller and smaller and seeing the average rate of change get closer to some value. This value it is getting closer and closer to is the instantaneous rate of change.

That’s a deep and important thing. And we all know that.

But when we were generating a table like this, one of my students asked “Why do we have to do this? Why can’t we just pick two points really really close together instead of doing this horrible calculation like 4 times? Like super close together. Then we only have to do it once if we’re just trying to estimate the instantaneous rate of change.”

Brilliant!

Because who wants to do that horrible calculation like 4 times? It’s tedious, even with a calculator. I wasn’t ready to talk about the derivative but I did want to answer his question. Why do we have to do so many calculations instead of just one?

Unfortunately, I fumbled through it.

And as always is the case, a genius idea strikes me right after class ends. So I decided to use it for my other section.

In that section, I have them think about what the use is of doing this calculation for smaller and smaller intervals, instead of just one interval… one student came up with the idea that “it gives us more certainty… more data to work with…” but that was ambiguously stated. More certainty about what?

So here’s where the idea came in. I had each student individually use only one small interval of their choice (instead of four) to estimate the instantaneous rate of change of y=sin(921,364x) at x=0.

What was great is that some students picked intervals like [0,0.0001] and others [0,0.00001]. Were they similar? Different? WHOA they were very different. Students got VERY VERY different estimates even though everyone used really small intervals. So what’s going on?

When we looked at the average rate of changes for various intervals, we saw this:

So yeah, if you happened to choose two numbers really close to each other, they might not be close enough! You just don’t know. Even if they’re really close. So doing a series of smaller and smaller intervals indeed gives us more certainty that we have a good estimation.

This was just sort of thrown into my lesson, so I don’t know exactly how much they got out of it. But I hope that next year either I use it as a do now, a new conceptual skill that I add to my calculus Standards Based Grading skill list, and make it a little more formalized [1]. Maybe after doing this next year, have a sheet with a few different functions, some which are wildly erratic and fluctuate a lot and some which are nice — and have students pick out merely from the graph and the point I want to estimate the average rate of change, if they can make do with two points “pretty close together” to estimate the instantaneous rate of change, or if they truly do need two points “very very close together.” That would be a good check to see if they understood the conceptual underpinnings of what’s going on.

[1] Idea. Have a sheet with two columns. On the left column, the function y=x^2. On the right column y=\sin(921,364x). Have them use the interval [0,0.001] to estimate the instantaneous rate of change at x=0. Then say: “You have $5 to bet on which one is closest to the true instantaneous rate of change. What are you going to bet on, and why?” Have groups whiteboard their ideas/thoughts for 5 minutes and present. Then show the graphs of the functions. Have then talk for 2 minutes to see if the graphs change their thoughts. Finish up student discussion.

I’m alive, I’m alive, but I’m sinking in! No Drama / Drama!

The worst thing ever is when I look back at when I last posted, and it’s over a month ago. Shame on me. After spending all that time trying to inspire others to post semi-regularly, I have fallen amiss. I only feign busyness. Lots and lots of busyness. The reason? I’m teaching a new course. And I’m collaborating with teachers this year. And I’m exhausted all the time.

But I do have some things to write about! Today I’m going to write about something the other calculus teacher and I spent a while emphasizing. It’s the idea of holes and vertical asymptotes in rational functions. There is something about how formulaic students learn about holes and vertical asymptotes in precalculus. And they memorize rules — and not why.

No Drama, no more no more drama! (Holes)

In class I throw up the following slide and have half the kids fill in the top table, and half the kids fill up the bottom table.

It’s a race (those that fill in the top function win!)! And of course, they start filling it in and see that the top function has all y-values are 1 (except for the one y-value that does not exist)… And the bottom one, they start seeing that the y-coordinate is always one less than the x-coordinate. (I sometimes have to do a little prodding.) I ask them why those things were happening, and we have a big discussion.

I forbid the use of the word “cancel.” It is like “Voldemort.” Verboten.

For things like \frac{(3)(2)}{(3)}, I have all kids say “the 3s divide out to equal 1.” Why? Because kids don’t really know what cancelling is, and by “crossing” terms out in the numerator and denominator, they don’t think about what it is they are doing.

Also, they think that \frac{x-1}{x-1} is the same as 1. Which is not true. So they learn to say \frac{x-1}{x-1} is the same as 1 except for when x=1.

What’s nice is once they get that distinction down, we look at the graphs and see that we have a missing point. Which is simply the one point where the function is undefined. So when I ask them why the graph of y=\frac{(x-2)(x-1)}{(x-2)} looks like the line y=x-1, they can say that they can rewrite y=\frac{x-2}{x-2} \cdot \frac{x-1}{1}, and they can say that for all x values except for x=2, the function is essentially y=1\cdot \frac{x-1}{1}. And at x=2, the function simply is undefined (why? because \frac{x-2}{x-2} is undefined at that single value).

So since for almost all x values, the expression \frac{x-2}{x-2} is just 1, and 1 times anything is itself. So the presence of this term doesn’t cause any drama. Just that one undefined point.

It’s no drama.

Drama! (Vertical Asymptotes)

Very early on in the algebra bootcamp for limits, I threw this on as a do now:

And they saw (and we talked about) why a big number (compared to the bottom number) divided by a small number is a huuuuge number.

Later in the class, I asked them what they remembered about vertical asymptotes. Someone inevitably said “they occur when the denominator of a rational function is zero!” I go “ooooooh, right, okay!” and I throw this up and ask them where the vertical asymptotes are…

… and they respond with what I wrote about. Never have I had a class say “oh there might be a hole!” (even though we had previously talked about them… sometimes during the same class!). But I really lead them on, and I give positive affirmation.

Then I throw up the pictures of the graphs, and we see if we were correct.

And so we saw that that wasn’t right. And they go “oh, right, the holes!”… so we modify our page…

… and talk about how it is not sufficient to say that vertical asymptotes occur when the denominator is zero. 

From this, I really emphasize the importance of truly understanding what a vertical asymptote is, what a hole is, and gaining deep conceptual understanding as to what is going on. Memorization equals death.

Finally, to talk about vertical asymptotes, I show y=\frac{1}{x-2} and talk about why a vertical asymptote occurs at x=2. And we relate this to the original do now.

The simple argument goes that for values near x=2, the denominator gets close to zero. And we see that 1 divided by a number very close to zero is going to be very very very large (the function is blowing up!) or very very very negative (the function is blowing down!). Thus, we can clearly see as we get x values closer and closer to 2, the function is getting closer and closer to positive or negative infinity.

Thus, we have drama. Tons and tons of drama — especially near x=2. Thus, vertical asymptotes are the consequence of a term that provides high drama.

(I also throw up y=\frac{1}{x^2+1} and we talk about whether this has any vertical asymptotes and why.)

So we get that. Finally, I encourage students to rewrite rational equations to see all the drama and no drama parts so when they analyze a rational equation, they can understand why certain things are happening…

For example m(x)=\frac{(x-1)(x+2)}{(x-1)(x+5)(x^2+3)} can be rewritten m(x)=\frac{x-1}{x-1}\cdot\frac{1}{x+5}\cdot\frac{1}{x^2+3}\cdot\frac{x+2}{1}. And then they can analyze each part separately to get a sense of how it contributes to the whole function.

That’s all. Nothing special. But I really like the idea of no drama / drama. The other calc teacher and I came up with that idea. The rewriting came out of that approach, and I really like it! I think I’m going to make it more formalized / have specific practice with rational functions based around this approach next year. This year it sort of came into being, so it as clear as I really wanted it to be to my students. But they, for the most part, got it.

A High School Math-Science Journal

In my first year of teaching, fresh from my haze from history grad school, I remember approaching the history and English department chairs about creating a high school level journal for those subjects. I mean, our school has a literary magazine, and also even a publication for works in foreign languages (seriously!). But nothing for amazing critical analyses and interpretations in English and history. I figured having something like this might encourage students to revise already excellent work for publication, and also make the audience of their paper be an audience of more than one. I even contacted the literary magazine student editors to see if they would feel like the journal would encroach on their domain (they said no). For reasons that are still quite beyond my understanding (because I still think it’s an amazing idea), both department heads rebuffed my idea. (Also, if they said yes, they would have gotten an enthusiastic first year teacher who would have taken on all this work!)

And so, I let this idea pass. One of many that I have, think are awesome, and then languish and die, either due to my own laziness or due to external circumstances beyond my control.

Until last year. When I was thinking: I’m a math teacher. Why not start a math and science journal? It’s so obvious that I don’t know why the idea didn’t hit me over the head years ago. So I found a science teacher compatriot who I knew would be interested, and we came up with an initial plan. And at the end of last year, we presented it to some students who we thought might have been interested (as this was something that is something that has to be for them, by them… if they don’t want it, there’s not point in doing it… it’s not about us…). They were, and we were officially off to the races.

We shared with the students the following document we made, with a brief outline of one vision for the journal. But with the understanding that this was their thing so their ideas reign supreme. This was, in some sense, a mock-up that the science teacher and I made to show them one possibility. The one thing that the science teacher and I were really aiming for in our mock-up was that the journal shouldn’t just be for superstar students. We wanted to come up with an journal that has a low barrier of entry for students submitting to the journal, and that if a student has interest or a passion for math or science, that’s really all they need to get started. To do this, original and deep research wasn’t really the primary focus of the journal. So here’s our brief proposal:

The additional benefit of having this journal is hopefully it will cause curricular changes. Teachers will hopefully feel moved to create assignments that go “outside of the box” — and that could result in things being submitted. Students who express an interest in some math-y or science-y idea (like why is 0/0 undefined… something that came up in calculus this week) could have a teacher say “hey, that’s great… why don’t you look it up and do a 3 minute presentation on what you find tomorrow?” … and if they do a good job, encourage them to write it up for the journal. Or a teacher might assign a group project on nuclear disasters, and encourages the students who do extraordinary work to submit their project to the journal. (Which can be showcased by teachers the following year!) Or a student who notices a neat pattern, or comes up with an innovative explanation for something, or who wants to try to create their own sudoku puzzle, or decides to research fractions that satisfy \frac{1}{a}+\frac{1}{b}=\frac{2}{a+b}. Or whatever. Knowing there is a publication you can direct the student to, as a way to say “hey, you’re doing something awesome… seriously… so awesome I think you kind of have to share it with others!” is going to be so cool for teachers. (As a random aside, I was thinking I could enlist the help of the art and photography teachers, because of the overlap between math and art… They might make an assignment based around something mathematical/geometrical, which students can submit…)

I honestly have no idea how this is going to turn out. What’s going to happen. How the word is going to get out. If anything will be submitted. If kids get excited about it. Lots of questions. But I have a deep feeling that the answers will come and good things are going to happen with this.

I’m soliciting in the comments any thoughts you might have about this. If your school does a math journal, a science journal, or a math-science journal, what does it look like? What works and what doesn’t? Do you have a website/sample we could look at? If you don’t have one, and you are inspired and think of awesome things kids could put in there (e.g. kids submitting their own puzzles! kids writing book reviews of popular math/science books, or biographies of mathematicians/scientists! getting kids to create photographs or computer images of science or data visualization or just making geometrical graphing designs! trust me — brainstorming this is super fun!) I’d love to share any and all ideas with the kids involved with this project at my school.