Friday, July 25, 2014

When Linear Sequences Coincide

I was playing around with this idea yesterday: How do we figure out when two linear sequences will eventually have the same value? How do we know when they will not? Randomly, I came across Amy Gruen's question on Twitter from a while back (but it also relates to the last NRich task I had posted yesterday, in trying to figure out how to find numbers that would light up multiple colors in the applet):

I am a number less than 3000, Divide by 32, the remainder is 30. Divide by 58, the remainder is 44. Who am I?

(If you haven't had a chance to play with this problem, I encourage you to do so and to get back to me if you have a different method than the one I have described below!)

I started by listing some multiples of 32 (32, 64, 96, 128, 160, 192, 224, ...) in Excel and then adding 30 to find numbers that satisfy the first condition ("Divide by 32, the remainder is 30"). These are the numbers in the first linear sequence, we'll call it sequence A: 62, 94, 126, 158, 190, 222, 254, ... A 5th grader can do this as well (as Amy had stipulated), but probably not in Excel but by hand.

And then, similarly, I listed out some of the sequence of numbers that would satisfy the second condition ("Divide by 58, the remainder is 44"): 102, 160, 218, 276, 334, 392, 450, ... We'll call this Sequence B.

My first instinct was to write two expressions 32n + 30 and 58n + 44 and to set them equal, but of course that doesn't work because the sequence values are likely not going to coincide at the same position n. (This is probably a common misconception, so I thought I would point out that it's a natural one to make.) Also, algebra isn't part of the Grade 5 curriculum.

Then, I thought if I started iterating through elements of Sequence B, I would probably reach the coinciding element faster, only because sequence B takes "bigger steps" and skips more of the in-between, irrelevant values. And, instead of listing every element from each sequence, I thought that maybe keeping track of how far "off" sequence B is from the closest element of sequence A might help me.

I made a table that looked like this. I decided to use shorthands in column 3 to help me focus on seeing a pattern. Originally I didn't have the a and b, but the numbers by themselves didn't seem helpful. Once I added the a and b (for above and below), the pattern was much more recognizable, because I was essentially assigning positive and negative signs to the distances.

Sequence B ElementDistance from Nearest Sequence A ElementsShorthand representation of distance
1028 above 94; 24 below 1268a (24b)
1602 above 158; 30 below 1902a (30b)
2184 below 222; 28 above 1904b (28a)

I observed what I think was a linear pattern by this point, and decided that I could predict what the next few sequence B elements' distance would be from the nearest element of sequence A. I also noticed that the above and below nearest distances added up to 32, as you would expect. (Since sequence A elements are separated by steps of size 32, if you're 8 values above the nearest value in that sequence, you must be 24 values below the next one, like being suspended in between two rungs of a ladder of fixed space between the rungs.)

So, based on this I hypothesized and tested that the next elements of sequence B will continue to follow this pattern and be located at a predictable distance from the nearest sequence A elements. 

Sequence B ElementShorthand representation of distance from nearest Sequence A elements
27610b (22a)
33416b (16a)
39222b (10a)
45028b (4a)

A pause here. I haven't reached any repeats yet. If I had reached any repeats in my table in terms of distance from the nearest elements (for example if I saw 8a and 24b appear twice in the table before reaching 0a or 0b), I would conclude that the two sequences will never meet. In the case of this problem, I should continue the table since we haven't reached any cycles yet.

Sequence B ElementShorthand representation of distance from nearest Sequence A elements
50830a (2b)
56624a (8b)
...... skipping some rows here, since I can see that 24a is a multiple of 6 and it will eventually decrease to 0a perfectly...
7980a (32b)
So, if this pattern holds, 798 should be the first time that sequences A and B converge, which means this number should leave me a remainder of 30 when divided by 32 and leave me a remainder of 44 when divided by 58. And it does! 

To find the subsequent elements is much easier (more of a standard math problem), because we know that the two sequences move by paces of 32 and 58, respectively. All we need to do is to find the least common multiple of their steps, which will be the distance that separates pairs of coinciding elements. The factors of 58 are 29 and 2, the (partial) factors of 32 are 2 and 16. So, 29x2x16 = 928 should be the least common multiple. That means that after 798, the next time the sequences converge to satisfy both conditions is at 798 + 928 = 1726, and the next time is 1726 + 928 = 2654.  Since 2654 + 928 > 3000 and 798 - 928 < 0, our complete set of solutions is {798, 1726, 2654}.

Just in case, I tested all three values against the two given conditions (since I don't trust myself with arithmetic and book-keeping). Also just in case I didn't mess up the LCM calculation, I tested the value halfway in between 798 and 1726 to make sure that it doesn't satisfy both given conditions. 

Now, here are the follow-up questions: Do you think this problem is doable by a student? What type of scaffolding would they need in order to accomplish this type of task? Are there other ways of doing this problem?

Thursday, July 24, 2014

One Resource a (Week)Day #17: Interactive Tasks from NRich

I have been doing some more playing of the secondary-school tasks from NRich, and I noticed in that process that they actually have some really nice interactive applets. I think that making an effective teaching applet is tricky, because:

1. If you make an applet that has too many features, even if you have the best of intentions, it can end up distracting from the actual mathematics.

2. If you make an applet that has too few features, on the other hand, it does not necessarily support the student's need to generate more data points and to test their conjectures.

Anyhow, here are a few tasks that have quite nice connections to high-school topics, each with a useful interactive applet.

http://nrich.maths.org/2293 has to do with finding (and predicting) areas of tilted squares, with a specified tilt k. The problem is accessible with just basic geometry, but it is extendable to a function of two input variables. You can generalize the pattern A(t, h) to describe the area of a square with a tilt t and whose two leftmost vertices differ by h units in height. The applet at the bottom of that page is very user-friendly. It only has two togglable points for you to construct squares of a certain tilt and height, and it is only there to help students construct newer instances and to observe their resulting shape and area concretely.

http://nrich.maths.org/2281 is a super easy-entry puzzle on building a pyramid of numbers. The guiding questions are gentle but they effectively get the kids to start thinking about how the position of a bottom number affects the final value at the top of the pyramid. They can make conjectures and test them repeatedly using the applet, thereby deepening their observations along the way. And then, the plot thickens when the pyramid gets to be bigger -- with 4 or more elements at the bottom level. Eventually, it could be generalized to show connections to Pascal's Triangle, a topic often touched upon in Algebra 2. Tres cool!

http://nrich.maths.org/7016 is a quite high-level task suitable for thinking about sequences. The applet is there for the students to try and gather data about which numbers will light up each color, and the really nice thing is that each group can be working on different patterns, without extra work on your part to generate different data. The entry to this task is a fairly straight-forward practice of linear equations / sequences, but when you start asking questions about how to light up multiple colors, the question gets rich really fast. When we dig even deeper into how to generalize relationships between sequences, I at least found myself in a quick sand. Besides some trial-and-error, I couldn't find a systematic way of predicting the first sequence element where two lights (of known pattern) will both light up. (After the first coinciding lighting, the rest is easy to obtain.) Can you help?

By the way, I am loving the various Twitter quotes from the Twitter Math Camp y'all are at. Keep them coming! You guys are so inspiring!

Addendum 7/24/14: I did a bit of playing and figured out how to find the first coinciding element of two linear sequences! For example, this problem from Amy Gruen goes nicely with the problem #7016 from above (hits the same type of idea). I leave it for you as an exercise to find all the numbers that satisfy this within the range 1 to 3000, but I'll answer it in a few days if you haven't already figured out how to do it...

I am a number less than 3000, Divide by 32, the remainder is 30. Divide by 58, the remainder is 44. Who am I?


Wednesday, July 23, 2014

Flow

I have been doing a fair amount of yoga down here in NOLA. It was part of my commitment to myself that I would find a way to remain active during my stay here. (Especially with all the excellent fried cuisine and cocktails, it seemed a real priority to exercise.) I signed up for an unlimited introductory deal, and I have been doing an hour of fairly rigorous yoga every weekday without missing a beat. At first, because I had been away from the mat for so long (months, really), I had a hard time keeping up with the classes. As much as I hated to admit it, my core was weak and it was hard for me to hold the poses for long, let alone thinking about what my fingers were doing. But, gradually it got easier and more enjoyable, and now I can genuinely look forward to going to class each day, even though I know that parts of each class will continue to challenge me.

It got me thinking about the idea of "flow", which is a psychological term for being fully immersed in an activity. If you have ever done yoga for a sustained period of time, you know that feeling of being fully present and focused on the mat. To me, yoga is relaxing and rejuvenating because it is one of the few times when I can completely be in the moment and to be silent / looking inward at the same time. It is a time when I have the luxury of not rushing off to do something, and I am simultaneously pushing and surrendering to my body's needs and limits. It is an entirely intimate experience, a tug-of-war between what is comfortable and what is challenging. For me, the feeling of flow keeps me returning to the mat, even though I can go through months of hiatus when I let other priorities get in the way.

Psychological studies have shown that being in a state of flow on a regular basis is necessary for us, in order to maintain our sanity and our state of joy. In his book Drive, Daniel Pink talks about the state of flow inside the classroom or at work. People experience the same state of therapeutic immersion, or flow, when they are challenged with a task that is of an appropriate level (challenging, yet still accessible) and when they have increased autonomy (ie. during a creative process). In other words, if we can achieve that fine balace in our classroom, our students would look forward to each day the same way I look forward to my yoga class.

What do you do to achieve flow in your life? What do you do in your classroom to facilitate flow?

The other thing about yoga class is that most yoga classes I go to are very heterogeneous. Within the same basic framework, the teacher can address the needs of people like me who have been away for a while, newbies who are coming on to the mat for the first time, and advanced yogis who practice daily and maybe teach their own classes. Each person is advised to look inward at what they need to get from the mat and how the pose feels, rather than looking outwards and comparing themselves to others and thinking in terms of what they "should" look like. This, to me, is a core element of yoga teaching and why it brings flow no matter what level you are practicing at.

Now, how do we replicate that in our classrooms?

Tuesday, July 22, 2014

One Resource a (Week)Day #16: Math in Wind Turbines

I got off on a tangent today looking at ways to build a simple electric generator and an electric motor (both operate on the same concepts, but one converts mechanical energy to electrical energy, and the other does the opposite). I think it would be awesome to do some RPM math associated with motors after building them in class, but the electric motor math turns out to be pretty physics-heavy. (I could see us building a simple DC motor with a variable resistor attached to it to allow adjustment of the rotational speed of the resulting electromagnet, but the math from there is just basic formula manipulations.) So, I dug around a bit and found, instead, some accessible mathematics related to wind turbines.

If you think about it, wind turbines tie nicely to simple geometry. This website from the Minnesota Municipal Power Agency has a nice set of fairly basic math activities related to wind turbine analysis. In particular, I was intrigued to learn that it's not always the more turbine rotational speed, the merrier! In fact, the ratio between the "tip blade speed" (how fast the tip travels) and the actual linear wind speed needs to maintain a healthy ratio (which differs based on the number of blades in the turbine design), in order for the drag to be minimized and the maximal amount of wind to be propelling the turbine. So, modern turbines have the ability to control their orientation and pitch in order to tweak the tip speed within a range of wind conditions. (Pictures below taken from the modern turbines link above, entitled "Wind Turbine Control Methods.")


Neat, eh? (A far more thorough explanation of the wind turbine power calculation is available here, but that's firmly in the realm of mechanical engineering, I think, and not appropriate for secondary students.)

Monday, July 21, 2014

One Resource a (Week)Day #15: Math in Appraising Properties (and Companies)

It occurred to me today that property appraisal is a problem of mathematical interest. Before you read further, I challenge you to think about how you would assign monetary value to a piece of property, if you were an appraiser. Considered the different types of buildings: private houses, rental property, commercial property, and public spaces.

The heuristic brainstorm is the richest part of this math problem, I think. I was so intrigued to learn that there are actually 3 different processes for evaluating property value, each suitable for a different type of property.

This page talks about the three ways of appraising property, which I will summarize below:

1. The way you're probably most familiar with is sales comparison. You basically look at similar houses in the vicinity of the house being appraised, and use recent sales histories, with some up-or-down adjustment factors (for granite floors, kitchen islands, or below-ground oil tanks) to estimate the market value of the current property. Most private home sales involve this type of appraisal.

2. The income approach involves something called the "cap rate", and can be used to appraise buildings that are primarily intended as rental property. This method involves calculating the expected net income from the rent per year, and then using the cap rate of similar rental buildings in the neighborhood to calculate the total value of the property. Cap rate is a fancy way of saying ROI (return on investment percentage), which means if you know the cap rate and you know the net annual income, you can do basic proportional reasoning to find the value of the property. (The only thing that is a bit tricky is that the definition of cap rate is not always consistent from listing to listing, in terms of which costs are subtracted when calculating net income.)

3. The cost approach involves calculating the value of the empty lot, then estimating the cost of a new construction of the same house, then factoring in depreciation to find the current value of the house. They use this method for public buildings such as churches or libraries or school buildings, which have no easy sales comparisons to be made in the vicinity.

I read online that the people who are studying to become property appraisers are often intimidated by the math involved. If you have the opportunity to introduce some of these terms in your class, it can really help your students to learn some long-term investment terms that can help them be a bit more savvy with their own investments*, or ease their fear of entry into the field of appraisal. (*For example, the cap rate is extremely important when you glance at a bunch of listings in search of rental property investment, if you are looking to make a monthly profit on your personal investment.)

Here and here are some real-estate percent problems that appraisers have to be able to answer on their licensing exams. You can rephrase them and use most of them for your secondary students! 

And, as an extension, how do companies determine how much another company is worth? (How did Facebook determine that What's App is worth $19 billion?) Have your students consider the problem and brainstorm some possibilities before researching the process of appraising a company. 

--As another extension, have students look at how neighborhoods affect things like the cost of land and cap rate. What social justice questions can they ask about this?

Calculus of Steam Engines and Steam Boats

I had the good fortune of taking a ride on a classic steam boat over the weekend on the Mississippi River while listening to some great Dixieland music. Being curious about these classic industrial-revolution-era designs, Geoff and I went down to the steam room to see how the steam engine works. The most interesting part to me was that the engine has only one incoming steam pipe (connected to the boiler room), which means that in order for the piston to move both forwards and backwards, there is a sliding valve that determines which chamber adjacent to the piston is being filled with steam, and therefore which direction the piston will be pushed.

It looks something like this (although I couldn't find a diagram exactly similar to the design on the particular boat. I am pretty sure the boat we were on had an engine whose slide valve moves more symmetrically than this site indicates).



Anyhow, it made me think about the Calculus that must be involved in steam engines, since the shape is changing dynamically. With a little research, here is an authentic physics problem (adapted from answers.yahoo.com since I needed a little physics refresher) that I can see giving to my Calculus students next year:

You know that within a steam engine, both the pressure and volume of the chamber are changing constantly. But, the pressure and volume are related at any given moment by PV^1.4 = k, where k is a constant. In order to calculate the amount of physical work put out by the steam engine, you need to know that Work = constant pressure times change in volume, or Work = constant volume * change in pressure. Since in this case both are changing, we will need to use Calculus to determine the total work done as volume changes, via integrating the equation dW = p*dv, where p is the steam pressure as a function of the instantaneous volume v inside the steam engine.

For any given steam engine, we can measure a starting volume and pressure to give us something to work with. For simplicity, we'll say that the original steam engine conditions are volume V= 100in^3 and pressure P = 160 lb/in^2. It follows then that any new combination of (v, p) is pv^1.4 = 160(100)^1.4. 

a.) Assuming that the volume of 100 in^3 is the smallest engine chamber that exists in this engine (ie. when the piston is fully compressed towards the starting side), and that the chamber can expand to 800 in^3, can you find some pairings of volume and pressure that the engine will necessarily experience during its movement? 

b.) Sketch the curve from Part A. Is the curve continuous and differentiable? Explain why or why not.

c.) In order to find the physical work done as the initial chamber expands (and pushes on the piston), we will need to find the formula that describes p as a function of v, and then integrate dW = p*dv from v = 100 in^3 to v = 800 in^3. Do that, and carefully write down the resulting units for your answer.

d.) In physics, it is easiest to relate parts of simple machines using the SI unit "Joules", which is equivalent to lb*ft. 1 Joule is approximately the same as the energy required to lift a small apple by 1 meter. Can you figure out how many Joules this steam engine will complete in one complete cycle (through expansion and then through compression, or expansion of the opposite chamber)?

d.) An early steam boat may be powered by an engine of several hundred horsepower. (See http://lakegeorgesteamboat.com/about/boats/previousboathistory/ for some sample points.) One horsepower is the same as 745.7 Joule/second. If your engine is the size described in this problem, then how long (in seconds) does it have to complete 1 full cycle, in order to achieve 200 horsepower?

e.) Is this a reasonably sized steam engine for a fair-sized steam boat? If so, explain. If not, find more reasonable specs and justify your choice through calculations!

What do you think? Is this authentic? Is it rigorous? Is it interesting? Help me improve it!

Friday, July 18, 2014

One Resource a (Week)Day #14: Low-Floor, High-Ceiling NRich Tasks

I started looking at the NRich website today and just loved their tasks for secondary students! Just on their front page alone, I found these excellent tasks.

http://nrich.maths.org/1019 looks very different from, but is actually very similar to the beehive traversal problem I had written about here. You can actually couple the two problems on the same day(s), to give one problem to one half of the class and the other problem to the other half of the class, and they can come together later to discuss/make connections.

http://nrich.maths.org/1785 is a really nice and sneaky lead-in to quadratics. Low-floor, high-ceiling indeed!

http://nrich.maths.org/7405 is also super fun to help us think about modular arithmetic. A nice extension task to the problem would be for the students to come up with values that could go into each bag, that would allow you to draw a set of 4 numbers and then have the result be divisible by 4, or to allow you to draw a set of 5 numbers and be divisible by 5, etc. Then, have the students go around and experiment this with other students or with their parents as a "Gee, Whiz!" homework assignment. As part of the homework, they should also explain how it works to their parent.

NRich also has a fun Problem of the Week, which also promotes inquiry and flexible thinking but is still very accessible to kids. This week's problem is reproduced as follows:

A snail is at one corner of the top face of a cube with side length 1m. The snail can crawl at a speed of 1m per hour. What proportion of the cube's surface is made up of points which the snail could reach within one hour? (Originally from UKMT Mathematical Challenges)

This snail problem is a simple introductory Geometry problem that all students can access in Grade 9 Geometry, that also reviews some basic proportional reasoning concepts. It is rich because it helps to raise questions such as, "What happens if the snail stops and turns direction? How do we find the maximum covered distance?"

If you are looking for a more challenging version of the surface traversal problem for use later in the year, I recommend this one from Exeter's Math 3 (thanks to Thomas Seidenberg for the generous sharing of their de-spiraled Exeter conical material). This one is not a low-floor problem, as you've probably figured out. But, I still like it because I don't think it's that intuitive without some hands-on manipulation of 2-D nets, that the shortest distance from the spider to the fly would not be along the conical rim.

A spider is on the rim of a conical cup when it spies a fly one third of the way around
the rim. The cone is 36 cm in diameter and 24 cm deep. In a hurry for lunch, the
spider chooses the shortest path to the fly. How long is this path?

I love NRich! (And Brits in general, but perhaps that is out of the scope of this post.)