Substitution (…and Continued Fractions)

Today in Precalculus I went on a bit of a 7 minute digression, talking about continued fractions. You see, a recursive problem showed up (we’re doing sequences): Write out the first five terms of the following sequence:

$a_{n+1}=\sqrt{2+a_n}$ where $a_1=\sqrt{2}$

So obviously they go like: $a_1=\sqrt{2}$,$a_2=\sqrt{2+\sqrt{2}}$, $a_3=\sqrt{2+\sqrt{2+\sqrt{2}}}$, $a_4=\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}$, and $a_5=\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}$

So great. Awesome. NOT. Booooring. So I showed them the decimal expansions:

$\approx 1.414, \approx 1.848, \approx 1.961, \approx 1.990, \approx 1.998, \approx 1.999, \approx 1.9998, \approx 1.99996, \approx 1.999991, \approx 1.999997647$

WHOA! This is getting closer and closer to 2… Weiiiird…

And then I say I can show them this will continue, and we can find a way to show that $\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+...}}}}}$ [where the pattern continues forever] will practically become 2.

To do this, I start with something else. I don’t know why, but I really wanted to show them a continued fraction first, to get the point across easier than with the square root. This was the continued fraction.

I went through a frenetic mini-lecture, and I think I had about 40% of the kids along with me for the whole ride. I’m not sure… maybe? But later a kid came by my office, and I thought of a better way to show it. Hence, this blogpost, to show you. (I have seen teachers use this method when teaching substitution when solving systems of equations… but I have never used it myself. I’m dumb! This is awesome!) This is what I did when showing the kid how to think about this in my office.

First I took a small piece of paper and I wrote the infinite fraction on it.

Then I flipped it over and on the back wrote what it equaled… Our unknown $x$ that we were trying to solve for.

I emphasized that that card itself represented the value of that fraction. The front and back are both different ways to express the same (unknown) quantity we were looking for.

Then I took a big sheet of paper and wrote $1+\frac{1}{}$ where I left the denominator blank. And then I put the small card (fraction side up) in the denominator of the fraction…

And I said… what does this whole thing equal?

And without too much thinking, the student gave me the answer…

Yup. We’ve seen that infinite fraction before. That is $x$!

Flip.

THAT FLIP IS THE COOLEST THING EVER FOR A MATH TEACHER. That flip was the single thing that made me want to blog about this.

Now you have an equation that you can solve for $x$… and $x$ is what you’re trying to find the value of. This equation can easily be turned into a quadratic, and when you solve it you get $x=\frac{1+\sqrt{5}}{2}\approx 1.618$ (yes, the Golden Ratio). And it turns out that is close to what we might have predicted…

Because in class, we (by hand) calculated the first few terms of $a_{n+1}=1+\frac{1}{a_n}$ where $a_1=1$… and we saw: $1, 2, 1.5, 1.66666666, 1.6, 1.625, ...$

And when I drew a numberline on the board, plotted 1, then 2, then 1.5, then 1.66666666, then 1.6, then 1.625, we saw that the numbers bounced back and forth… and they seemed to be getting closer and closer to a single number… And yes, that single number is about 1.618.

COOL! [1]

BACK TO OUR REGULARLY SCHEDULED PROGRAM

So after I showed them how to calculate the crazy infinite fraction, I went back to the problem at hand… What is $\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+...}}}}}$?

Let’s say $\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+...}}}}}=x$

Then we can say $\sqrt{2+x}=x$

And even simply by inspection, we can see that $x=2$ is a solution to this!

Fin.

[1] What’s neat is that yesterday I introduced the notion of a recursive sequence that relies on the previous two terms. So soon I can show them the Fibonacci sequence (1,1,2,3,5,8,13,…). What does that have to do with any of this? Well let’s look at the exact values of $a_{n+1}=1+\frac{1}{a_n}$ where $a_1=1$.

$2, \frac{3}{2}, \frac{5}{3}, \frac{8}{5}, \frac{13}{8}, ...$. WHAAAA?!?!

Lovely. It’s all coming together!!!

TL;DR: An interactive activity having kids ask each other questions to guess the rational function graph they have on their foreheads.

***

I’m going to make a short post inspired by Twitter Math Camp 2013 (TMC13), rather than TMC14. Both @calcdave and I led morning sessions for precalculus teachers. Through that morning session, some nice end-products were created — an organization for the curricula, actual classroom activities — and you should feel free to check them out here. [1]

@calcdave and I brainstormed how we could get people in the morning session to know each other, but make sure we have math content in that activity. We came up with Rational Function Headbandz, which was inspired by this post on the agony and dx/dt.

The setup: There are a bunch of cards (they could be index cards). On the front of them is an graph of a rational function. On the back is the equation of the rational function. The cards are attached to ribbons or headbands, so that when attached to the forehead only other people can see the graph on the front of the card — not the person wearing it. Sort of like this image below. You can re-imagine how to create these cards/headbands so they work for you.

The Goal: Since this was an introductory activity, participants picked one of two goals for themselves… (a) to figure out as many features as they could of their rational function and to sketch a graph from those features, or (b) to figure out the equation of their rational function.

To Play: I put all the cards/headbands on the table, and covered up the graph with post-its so the participants couldn’t see the graphs. I wrote on the post it if the graphs were graphs I considered sort of challenging, pretty darn challenging, or wow-you’re-going-for-it challenging! Then they attached their headbands to their head, and had someone else remove the post-it note.

Before starting they were told the following things about their rational functions:

• All the graphs are of rational functions.
• Some might be plain old polynomials. (Rational functions with the a 1 in the denominator!)
• If written in the most factored form, none of the terms has degree of more than two
• If written in the most factored form, most of the coefficients are really nice

Each person carried around with them a notebook, and they were allowed to ask up to three questions about the graph to each person (and a get to know you question to each person!). The rub? All questions had to be answered with a single word or a single number.

A valid question: “How many holes does my graph have?”

A valid question: “Is my rational function a line?”

A valid question: “Does my rational function cross or kiss the x-axis at x=3?”

An invalid question: “What is the coordinate of the hole?” (Because the answer will have two numbers as an answer — an x-coordinate and a y-coordinate.) You could instead ask “What is the x-coordinate of one of the holes of my graph?” and then follow up with “For the hole with x-coordinate BLAH, what is the y-coordinate?”

After three questions, they move on to a different person. Then another. Et cetera. From these questions they were supposed to gather information about their graph, and possibly about their equation.

You stop the game whenever you want. Everyone looks at their graphs and equations, and ooohs!, dohs!, and aaahs! result.

And then if you have time, you can debrief it with students by talking about what they thought was important information to gather in order to sketch or come up with the equation for the graph (holes? x-intercepts? y-intercepts? vertical asymptotes? horizontal asymptotes? slant asymptotes? end behavior?). And then if you had time you could have individual students present their graph, their thought process, and their solution.

Our Graphs: We really varied the nature of the graphs because we were working with precalculus teachers and we didn’t know their ability level with the material. And also I know I emphasize in my class working backwards from the graph to the equation, but that isn’t a standard thing taught. So I would highly recommend creating graphs of your own based on the level of work that you’re doing in your class.

[.pdf, .docx]

Trouble Spots: One thing that was challenging for us when we played this was what someone does when they have figured out their own equation/graph. They came to us and we confirmed. But then what? We should have anticipated this because we had such varying levels of difficulty for graphs. I wonder if a good solution would be to then try to figure out the equation for the rational functions of others when they are being asked questions.

Another thing to keep in mind is that this will take a longer time than you think. We used this as a get-to-know-you activity, and so that extended everything even more. (In your class, your students probably won’t be using this as a get to know you activity.)

Alternatives: Just as I adapted this from a teacher using them for trig functions/graphs, these can easily be adapted for other topics. Some initial ideas:

Geometry vocabulary review: Students have a vocabulary word on their heads. They only can ask questions with one-word answers. (e.g. “Does it have to do with parallel lines?”)

Polynomial graphs (instead of rational function graphs), or even just parabolas [update: Mary did this!], or even just lines.

Students have derivative graphs on their heads, and they need to come up with a sketch of the original function (for this they should be allowed more than one-word answers).

[1] One thing I worked on in a group with four other people is how to get students to understand inverse trigonometric functions (a topic we collectively decided was challenging for students to wrap their heads around). I blogged about the result of our work here. I used it in class this past year, and although I didn’t use it completely as intended, it did really push home the meaning of what sine and cosine were graphically (the y- and x-coordinates on a unit circle corresponding to a given angle) and then what inverse sine and inverse cosine were graphically (the angles that are corresponding to a given y- or x-coordinate). Check it out!

“Explore Mathematics: Part II”

I felt like my first venture into “Explore Mathematics!” was so successful last quarter with my Advanced Precalculus kids that I wanted to build upon that. So this is what I’m doing for “Explore Mathematics!: Part II”

• Last quarter students scoured the web and did 5 different mini-explorations which exposed them to all the neat math that exists outside of our standard curriculum. This quarter students will be doing up to two more in-depth explorations.
• Because I don’t want this to be seen as busy work, doing “Explore Mathematics!: Part II” is going to be completely optional. I was glad to read that almost every kid who did the five mini-explorations last quarter didn’t end up finding it busy work, but I suspect doing it a second time would feel tedious.
• To have some sort of incentive for those who do it, I am going to make each of the two explorations worth 12 points. These explorations will count as a mini-assessment (normal assessments are around 50 points). This is useful for kids because our fourth quarter only has 18 days of instructional time (seriously) — so there are only two major assessments and one minor assessment scheduled. Doing these explorations can act as a way to get another mini-assessment grade in there, that will be low-stress, high-reward. [1]
• I’m not framing it around the grade boost it will likely provide, but around the fact that it’s an opportunity to do some awesome math explorations, for anyone who wishes to do so.
• It is still pretty open-ended, but I’m now looking for students to write something to get others to see what they find interesting/intriguing/awesome about something.

Here’s the document I just emailed my kids:

Here it is in .docx form in case you want to modify it.

[1] Yes, I do SBG with my calculus kids. Yes, I know how ridiculous this sounds, me playing the “point game.” I almost wanted to make it so that there was no external reward, but our kids are so busy with so many things that I know even a little incentive will go a long way. I’ve been at my school long enough, and know our kids well enough, to know this is doomed to failure without a little external reward.

Explore Math (Reprise)

At the beginning of the 3rd quarter, I did an experiment in my Advanced Precalculus classroom: Explore Math. This post is the compilation of the survey results from my kids on this experiment. So if you don’t know what the activity was, read up here, and then see what this survey is all about. I will share examples of some of student work for this experiment later. Part of the assignment for students included submitting one exploration to our school’s math-science journal, Intersections. When this year’s issue of the journal comes out, I hope to link to my kids’s explorations!

The question in the survey:

The “Explore Math” project is something I’ve never done before. I explained my reasoning behind it — which is I wanted to encourage you to see that there is so much more than our curriculum covers, and let you just have fun looking at math stuff outside of our curriculum… and get some easy credit for it (almost everyone is getting full credit for the first batch of things I’ve seen). However, as a teacher, I know something like this could easily be seen as busy work, and that was my big concern — that it would feel like a chore rather than something you actually want to do.

This is me laying my cards on the table. If I came to you in the student center and told you this and asked you for your thoughts, what would you say?

Every Student Response In Entirety:

I really liked the Explore Math project and I definitely would say it was an overall success. I loved how many options we were given for what we could do, and the fact that you gave us the options was great because otherwise it can feel like you are just trying to desperately research and find a topic to write about. My Explore Math topics I thought were extremely interesting, and it was cool to even connect some to the stuff we were learning in class. It was a lot of writing, which is something foreign for math classes, and also made it kind of difficult to grasp exactly how to format what we were writing (five page essays for each topic?). One other thing that was a little stress-inducing was the deadline and I know it was for a problem for most people that it often happens that when there are multiple assignments due on one day, students leave them all and do them in bulk. Because of this, having the deadline of the first three due in February was definitely helpful. Overall, I really loved the assignment.

I really liked this project! I found a lot of things about math that I would have never known about if we weren’t assigned this project. I learned new formulas, new (very addictive games), great youtube channels and informative popular articles. I found an entirely new community online that I did not know existed.

At first I expected it to feel like a bit of a chore but when I actually sat down and did it, it was pretty fun. I think it was great that there were multiple ways you were allowed to “explore math.” I also thought it was amazing I could play around with the project a little bit to find areas of math that are aligned with my personal interests. Being able to think about how math affects our society, in a math class, was an amazing interdisciplinary activity. I think it’s good that not every option was a math puzzle — that would have felt constrictive.

I would say as long as the students are innovative, interested and patient people the project sounds wonderful. The student, if very interest in math, should be encouraged to further their mathematical understanding, and find means in which math is even more interesting to them as it was prior. Emphasizing the point that one (the student) does not need to seek the more difficult problem or most tedious theorem is also very helpful, as the student will be encouraged to explore areas of math in which really interests them.

I would say that I absolutely love the explore math project. I have always been a person who enjoyed math that connected with the world. Being in a classroom memorizing formulas was never my interest and I was psyched when you announced the project. I think that this project can be very helpful in putting math on the global scale for students who only see it as a class in a school. This opens their eyes to new heights math can taken and how much math actually helps outside of the classroom.

I agree it felt like busy work some. I find it weird that something that’s supposed to be us having fun exploring math had a grade and time constraint attached to it. That’s one thing I didn’t like.

All I have to say is that this was not busy work; in fact it was productive and learning work. I found this to be incredibly intensive and interesting, and it broadened my horizons of the understandings of applied mathematics and sciences, and introduced me to things that I had previously trembled [at] before, like string theory, for instance. I thought this was a great project and a simple and easy way to get us thinking in a mathematical mindset, and I am definitely reaping the benefits from it, because I have come away with much more knowledge about certain aspects of math that I had previously not known. I really wouldn’t know what to change because I liked these individual explorations so much and they intrigued me so much. Thank you for giving a projected that I was thoroughly interested in, seriously!

For someone who is very interested in math in and out of the classroom, I am generally engaged with math concepts that are not a part of our curriculum. Thus, this was a good experience for me in that I was able to get credit for simply enjoying and exploring math; it also perhaps pushed me a little bit to go further than I normally would in exploring mathematical concepts online. However, for students who don’t love math outside of the classroom, I could definitely see how this might have seemed like busy-work. If you don’t genuinely enjoy math, then writing a lot about it and research about it is going to be cumbersome, but if you do, it’s enjoyable.

I really liked doing the explore math assignment. I liked that you were giving us an outlet for us to not just do the math that needs to be done in order to complete the class. This assignment allowed me, personally, to dive deeper into how math can be applied to the world and that math is actually occurring all the time. Also, I remember not really understand[ing] infinite series and then I did an explore math with infinite series that really helped me because it was a visual representation that really clicked with me.

I think that initially I thought the project might just be busy work and I didn’t really understand what we were expected to be doing. Once I read over the assignment and saw the scope of the projects we were allowed to do, I was much more interested and saw the project completely differently. I think that it is important to highlight, when giving the assignment, how broad a range of options you have when doing this, and that there are so many math projects that relate to everyday life that could be interesting if you just think about it, rather than relying on the assignment sheet completely to guide you.

Personally, I have enjoyed what I have done so far. Just recently, I voiced my concerns about the state of math in America and was able to comprehensive research about the bitcoin that I would not have done on my own. That being said, some of this has seemed like busy work and stuff “I just have to do for credit.” Since it seems like you genuinely want us to enjoy the project, it might be made better by making it extra credit. That way, we could be able to explore as much as we want without worrying about our grade.

I had a really awesome time doing my Explore Math assignments, but the one thing you could do to make it less busy work is make it 3 different assignments, rather than 5, and make them a little more in depth, and more interesting in that regard. I think that if the students only had to do 3, they could expand more on what they were interested in.

I really like the idea, but for me personally, it turned into busy work. Not because I find it boring but because I have so much other work that it gets pushed back towards the end of my load. I would like to spend more time on them, so possibly have it on top of the nightly work for math, designate a night specifically to explore math.

This is practically the farthest thing from busywork we can do! Repetitive problems often seem like busywork. Practice is always good, but once you have something down, it can be quite annoying to practice it over and over again. Sometimes i feel that way about homework, but with this project we’re choosing any math-y thing that interests us! We have a lot of freedom, and hopefully it piques an interest in math outside of the curriculum. This project is great, personally, I wish I had taken more time with it. As long as you don’t procrastinate too badly with it, I don’t see how this project could be a chore, unless you claim to hate math.

I LOVED this project, and I wish we got to do more things like this throughout the year. (I know we can do things like this whenever we want, but it’s really nice to get some recognition and the chance to formally share your math ideas with others.) As a side note, this project was also interesting to be doing while looking at colleges for the first time. I know that sounds like a really strange thing to say, but getting to enjoy math in new contexts, such as music theory, has given me new ideas of things I would like to pursue and take classes [on] while I am at college because we don’t always get to learn about things like this on a daily basis in high school.

I do admit that I wasn’t very enthusiastic at the start of the project, but as soon as I started I completely changed my mind. Most of the work that I did was stuff I had never done before and might never do again. I was genuinely interested in what I was doing, and it was great to be able to choose what I focused on instead of being told what to look at.

I understand why you assigned this project, and I think it is very important to see the relevance math has in the world. This breathes life into the abstract “why are we learning this” type that doesn’t appear to have anything to do with life outside the classroom. However the problems with this assignment are that I didn’t know what I was searching for. When I found the Sloane’s Gap video and paper I felt like I struck gold after seemingly endless mining. However the mining part is very un-exciting. Not un-exciting enough to undo the excitement of finding the cool stuff, but it’s not very encouraging either. I wouldn’t want this assignment to turn into a chose 5 of these pre-determined projects because that wouldn’t make anyone feel like anyone feel like they’re venturing outside the classroom. I’m not really sure what I would do to change this assignment, but I think it really is a good idea that with some refinement could become a really dynamic way to get into math. I think keeping it low pressure and “easy credit” is the way to go because stress + ambiguity about an assignment is a terrible combination that would only end in resentment from your students, and students not enjoying their work.

Honestly, I had quite a bit of fun with the “Explore Math” project as I saw many cool analogies of real-world applications of math. For example, one of my five “research topics” was the probability and randomly guessing on every SAT multiple choice question. I learned that the probability is horrifyingly low — I already knew this, but not to such an extent. Furthermore, I saw some very cool analogies in this SAT topic; for instance, if a computer were to take the SAT 1 million times a day, for five billion years, the chance of any of the SATs resulting in a perfect score on just the math section would be about 0.0001%. Crazy, I know!

“Explore Mathematics”

I teach an Advanced Precalculus class, and I love my kids. This is my second time teaching the course, and I get a rush seeing the kids dive into whatever we do with full intensity. Because the curriculum we teach is so chalk full of things, we don’t really get days where I can go on tangents and have students explore things that I think would be of interest to them.

Earlier this year, I was struck by this post by Fawn Nguyen. It’s rare that I read something and it just keeps rattling around in my brain, and won’t let me forget it. (Thanks Fawn, for being an annoying bee attacking my brain!) If you’re too lazy to click the link, the TL;DR version:

Fawn has her kids go to Math Munch and explore and play with mathematics it based on what interests them. She has her kids keep track of what they do with this sheet:

What I loved about this? It gave kids the freedom to explore mathematics that interested them. The assignment was fairly low-pressure.

I wanted to do something similar. I knew I wanted it to be low-pressure to do, fairly easy to grade, and really focus on what the kids want to do. Thus, Explore Mathematics! was born.

[.docx]

Students are asked to engage with mathematical things that they are interested in during the third quarter. There are two deadlines, so they are working on them continuously and not rushing at the end to finish them. (Also to make marking them easier for me.) There is a low-pressure grading structure, which reinforces the notion that this is more about just engaging and less about “doing the right thing.” In total, I’m making it worth about half a normal test.

I don’t know exactly how this is going to turn out. But I’ve already had a student present a piece of mathematical artwork he’s made, and I’ve had a couple fun conversation with kids about things they’re thinking of doing/looking at. I hope this fosters a lot of fun mathematical conversations between me and the kids about the things they’re finding (and of course, among the kids themselves).

The biggest concern is making this assignment not seem like or become busywork for the kids. I don’t want it to seem like added work just for the sake of extra work! That’s the fine line I am trying to navigate — sort of “forcing” kids to carve out some time here and there in their busy schedules to get exposed to the cool things out there. I have to figure out how I can create this feeling in the kids. Maybe that means I will give up some classtime for them to work on this every-so-often, to show them I value this sort of exploration. Wish me luck on this.

Trigonometric Pythagorean Identities

$\sin^2(\theta)+\cos^2(\theta)=1$

and derive

$\tan^2(\theta)+1=\sec^2(\theta)$ and $1+\cot^2(\theta)=\csc^2(\theta)$

simply by dividing both sides of the original equation by $\sin^2(\theta)$ or $\cos^2(\theta)$.

I did this same this year.

Except later on, a few weeks ago, I saw a post on twitter talking about introducing trigonometric identities through graphs on the unit circle — and having kids come up with their own identities. I loved this idea and planned to make a whole thing about of it. So far I’ve given students one thing I’ve made as a result of this idea (and that worked out super well).

From this, students were able to come up with the three Pythagorean Trig Identities we saw above, but also a fourth one that was totally unexpected.

I had them all pick a different angle and substitute it into the left hand side and the right hand side of the last equation. KABAM! Whoa! SAMESIES! (Note to self: Next year make a dynamic visualization of this triangle on Geogebra, like this but better/cleaner.)

Instead of doing a whole unit on Trigonometric Identities, the other teacher and I are slowly giving students a problem here and a problem there to practice with and find new strategies, over a couple weeks. I hope that works! And maybe if I have time, I’ll make a follow up activity. Maybe giving the drawing below but without anything labeled but the radius of the circle, and having kids fill in each of the lengths and find various identities? They can use the ratios of similar sides… but also if triangles are similar, they can also use the ratios of the perimeters! Or knowing that the ratio of the areas of two similar figures is simply the square of the ratio of two corresponding sides? Also, maybe just maybe kids could generate inequalities — like the area of this one triangle will always be less than the area of this other triangle?

I don’t quite know as things aren’t fully formed in my head yet. If anyone has any ideas, or existing resources, pass ’em along!

A couple years ago, Kate Nowak asked us to ask our kids:

What is 1 Radian?” Try it. Dare ya. They’ll do a little better with: “What is 1 Degree?”

I really loved the question, and I did it last year with my precalculus kids, and then again this year. In fact, today I had a mini-assessment in precalculus which had the question:

What, conceptually, is 3 radians? Don’t convert to degrees — rather, I want you to explain radians on their own terms as if you don’t know about degrees. You may (and are encouraged to) draw pictures to help your explanation.

My kids did pretty well. They still were struggling with a bit of the writing aspect, but for the most part, they had the concept down. Why? It’s because my colleague and geogebra-amaze-face math teacher friend made this applet which I used in my class. Since this blog can’t embed geogebra fiels, I entreat you to go to the geogebratube page to check it out.

Although very simple, I dare anyone to leave the applet not understanding: “a radian is the angle subtended by the bit of a circumference of the circle that has 1 radius a circle that has a length of a single radius.” What makes it so powerful is that it shows radii being pulled out of the center of the circle, like a clown pulls colorful a neverending set of handkerchiefs out of his pocket.

If you want to see the applet work but are too lazy to go to the page, I have made a short video showing it work.

PS. Again, I did not make this applet. My awesome colleague did. And although there are other radian applets out there, there is something that is just perfect about this one.