Friday, August 1, 2014

One Resource a (Week)Day #19: Using Desmos in Calculus

I am starting to focus in to think about what I want to do with my classes during the first week of school. For Calculus, I think the choice is obvious. I should start them off by playing with Function Carnival, over at Desmos.com!! (For those of you who, like I, did not attend Twitter Math Camp and have missed the demo / all the blogosphere buzz, you should sign up for an account at teacher.desmos.com to play with their interactive applets immediately. The interface is thoughtful and it allows you to collect data regarding what your students are creating as graphs and how they analyze other people's errors, and you can use it at the start of a unit to build understanding from recognizing and addressing misconceptions.) I will be starting them off easy, on Day 1, by playing with the regular algebra version to warm up. And then, on Day 2, plunge them right into predicting and testing velocity function graphs in the Function Carnival, in order to jump-start their interest in thinking about rates of change.

To reiterate things that others have said, I really like the ability to both toggle through individual students' responses and to look at the whole class's responses at once. To start with this activity at the very beginning of the year can help me to set the tone that it is okay to experiment and to make mistakes, because that is how we learn math.

Yay Desmos! This is really a fantastic tool that incorporates real mathematical instruction, not by replacing the whole-group experience but to reinforce it by giving each student a structured think-time before the whole-group discussion. I really hope that this is the future of digital instruction!

Thursday, July 31, 2014

Thinking About Calculus

If you teach Calculus in high school, you probably have a similar observation as mine that the student population is highly heterogeneous (a fact that is perhaps surprising to outsiders). Many students feel pressure from their parents and peers (ie. social pressure) to stay on the Calculus track, so within the same group they can span truly a wide range, from those who really could have benefited from another year of Precalculus-type skills-building, to those who are somewhat okay with procedures but who really need more experience/exposure with integrated problem-solving (something like a "Math 4" course that involves college-level math language), to those who are truly ready -- and eager -- to tackle new topics, both conceptually and algebraically.

I stumbled across this research summary from years ago (dated 1988) that noted a shift in college Calculus curricula away from conceptual understanding and toward procedural manipulation. Even though the article is dated, I find myself torn in the middle of a similar tug-of-war in my own thinking of Calculus. Fortunately, I teach a non-AP class, but my juniors (within a class mixed with seniors) do largely feed into another year of Multivariable Calculus after my class (combined with students from another class), so I have to take that into consideration in organizing my course. But, the statement from the article that we're trying to make the course "be all things to all people" really resonates with me. In the same class, I am supposed (and trying!) to differentiate so much as to help all the kids who need an extra year of algebra practice; to help all the kids who need an extra year of problem-solving; to cover all the major Calculus concepts; and to develop all of their skills as learners and to nurture their mathematical practices (all of which, mind you, take time). It was really half a miracle that many of the kids ended up enjoying and appreciating the experience of my first year of trial-by-fire to accomplish ALL of this at a new school; the task set for us can be daunting, frustrating, or exciting, depending on how you choose to look at it. 

One of the things that I did this year which actually helped a lot with teaching a heterogeneous group was to make sure that I made time for projects in this class. The projects gave the kids time to self-differentiate. Many of the kids kept working on their projects until the last minute, throwing in extra features or sometimes even working on it after it was due and presented. The next year, I will keep most of the projects but re-arrange the pacing so that they are better timed and hopefully a little less stressful for the kids. 

Projects I did:
* Economics mini-project (but this one I will switch out for something better, more individualized)
* Graphical organizer showing connections between all the learned skills
* Rollercoaster design using piecewise functions
* Function pictures including shaded definite integrals (calculated via GeoGebra but also by hand)
* 3D-modeling using vases that they created and volumes of revolution / scaling up to find and test against real volumes

I also did a group quiz this year on the minimization of coordinate distances (to help review the distance formula) and on related rates, which was assigned to be completed mostly outside of class. It was such a great experience for me and most of the kids, that I'll definitely have to find some opportunity to repeat it. I decided that next year, I'll have to bump up the Function Pictures project to the first grading period, to help the kids review functional forms and inverses. That way, when we revisit their projects to fill in the integrals later on in the year, their focus will be more on the integrals and less on the outlines.

Since I need to replace the economics mini-project, I am going to try doing a sustainability project this year, that involves some regression and rate-analysis at the start of the year, and then have an individual component where the kids look for something else (an interesting data set) that does not have a time domain, to extend their understanding of rates beyond the time domain.

Another choice that really helped me with approaching the heterogeneity was teaching Calculus in reverse. First, we did a lot of graph sketching and graphical analysis by calculator. This evened out the playing field because the kids who were weak with prerequisite algebra skills could still access and feel successful immediately about the new concepts. Then, we learned differential Calculus skills via exploration, which helped the kids build some of those valuable mathematical practices and to build their confidence in playing around with math. I took our time on this part of the course, to go through old algebra skills and to practice things that were challenging as they came up. The really intuitive students started during this unit to peek ahead on their own at how to procedurally un-do differentiation. After the derivative skills, I took a significant amount of time to work on related rates with the kids, to give everyone some quality time with problem-solving. (It wasn't nearly enough time though. Obviously, you can never spend enough time on teaching and practicing problem-solving.) Later, through explorations, the students were able to gather most of the core concepts about integrals. We then used projects to reinforce their algebra skills. Limits came last in my course (because it is the most abstract, and I think it made the most sense to introduce it last as a way to prove the things we thought were true), and at that point, it was a really nice tie-up of the whole year, reinforcing the definitions we had learned throughout the year regarding derivatives and anti-derivatives by proving them via limits.

Even though I have been thoughtful with my course, I wish I could feel more certain that I am making the right choices for my class. For everything that I decided to spend more time on, it was a choice to leave something out. I tried to keep the class fluid, so that if an interesting question came up during a project with a bunch of kids, I expanded that during the next class to go into the relevant material, even if it's not part of the classic Calculus 1 curriculum. (For example, my students really wanted to know how to integrate circular areas on their projects, so we did an example together and about a third of the class then followed suit to use trig-substitution to help them on their projects.) Never in my class did I feel like I had wasted time, but I couldn't shake the feeling that Calculus 1, the way that I was teaching the class, could easily have spanned 1.5 years.

What do you think? What choices have you had to make for your own Calculus classes? Do you think they are worth it?

Wednesday, July 30, 2014

My Work-in-Progress Algebra 2 Sequence (for Next Year)

Last year, I felt pretty good about my Algebra 2 sequence. Some of the students struggled with the formal assessments in the class, but I don't think the fact that they struggled was tied to the way the material was sequenced. (Re-quizzes definitely helped them, but I also saw them learning things like managing their responsibility as the year went on and the material ramped up in complexity.) This year, as usual, I want to do better.

I think that my ideal Algebra 2 sequence, assuming that the students come in only knowing how to solve basic one-variable equations, would look like this:

Unit 1. Review solving equations, clearing fractions, and manipulating formulas. The reason why I would start with this topic is that it allows me to see immediately who is struggling with topics from the years prior, and who isn't. Introducing fractions at this point of the year also gives me an opportunity to keep spiraling back to it in every other topic. A nice problem to use at the very start of the year is the classic pool border problem or any visual pattern that extends linearly (to review the idea of inductive thinking and what a variable means for generalizing patterns).

Unit 2. Linear Functions. After reviewing the meaning of a solution in Unit 1, I feel that it's super important for kids to see that when there are multiple variables, you can now intuit an infinite number of solutions to an equation! Using exploration to plot some of those solutions allows us to see a linear pattern emerge. Unit 2 is all about understanding the connection between predictability of elements and algebraic forms. In Algebra 2, I cover both slope-intercept and point-slope form, the latter I start with letting the kids figure out that collinearity has everything to do with slope, and then from there they can simplify the slope formula m = (y2-y1)/(x2-x1) into the point-slope form. This year, I think I am also going to throw some unit analysis in there to help explain why slope m has to be change of y over change of x, and not vice versa (in order for mx + b to work out to have the same units as y.) Along with linear functions last year, we did a bungee-jumping regression project including a significant lab write-up, and I plan to repeat that next year.

Unit 3. Systems of Equations Setup and Solving via Graphing Calculator. Following basic review of things from Algebra 1, the most important baseline skill for a student's algebra success is the ability to go from language to symbols. I always take some time (even in Algebra 2) to go through how to write basic equations of the form PART + PART = TOTAL  or  PART*PART = TOTAL, and then I give them a chance to put that to use by writing various systems equations and solving by graphing. A trick I learned from a former rock-star colleague is that you have to always teach and thoroughly practice the graphing skills first, if you want to have a fighting chance of the kids using the graphing calculator later on. If you make the choice of teaching algebraic approaches first, most kids who are afraid of thinking flexibly will always resort to the algebra, even in the cases when the calculator is clearly more efficient and less error-prone. Similarly, kids will be reluctant to check their answers using technology, unless the mechanics of doing so is already second-nature. In this unit, I teach them how to graph, zoom, trace, find intersection, and look at the table to help them with figuring out the appropriate zoom. (I don't like the Zoom Fit feature of TIs, since they're a bit buggy.) For you GeoGebra-lovers, don't worry, the kids will use graphing software later on. 

Unit 4. Systems of Equations Algebra. Now that the kids already know how to set up word problems and to solve by graph, we are ready to delve into the various methods of manually solving a system. This year, I will start with the puzzle explorations for systems to help the kids really get what it means to substitute. After they learn both elimination and substitution methods, they will then practice setting up and solving systems involving fractions (spiraling back to fractions is always a good idea) and word problems, and to use their graphical solution from the calculator to check. I wrote about this before, but I always require on tests that kids solve each complicated problem twice, using two different methods, to reinforce their understanding of graphical and algebraic connections.

Unit 5. Inequalities in the Coordinate Plane. If time allows, I want to spend a short amount of time on inequalities this year. (I did so last year as well, but it was sort of scattered.) Following systems is a good time, because I can then use linear programming problems to drive home the usefulness of the constraints and the graphing.

Unit 6. Quadratic Functions. The way I teach quadratics is by building it up from linear patterns, and I drive home the connection between dimensionality and degrees via this type of side-by-side comparison. The recurrent problem sometimes is that kids don't really understand dimensions from Geometry. (If you're a Geometry teacher, please give some TLC to this very important idea!) We do go into various forms of quadratics and I teach them both how to factor and how to sing and apply the quadratic formula. We do some completing the square, but not enough to master it in Algebra 2, only to see that you can get things from standard form into vertex form. It's important for them to recognize that the quadratic formula can be broken down into various useful parts (discriminant and axis of symmetry) before we move on, so that they could sketch graphs based on any given function equation. Last year, I really drilled the kids to be able to sketch linear-quadratic systems, which, although they probably will not remember the specifics of the procedure, definitely helped to reinforce the idea of connections between graphs and algebraic forms. I didn't do a quadratic-specific project last year, but this year I plan to do a bridge modeling project using all three forms of the quadratic function, as I have done previously in other classes. Dan Meyer's pennies and circles task is also nice to use during this unit to review the idea of regression in the context of quadratics. Sometime early in the quadratics unit, I feel that it is very important to explore the idea of constant second differences between the sequence elements. This sets the stage for other types of patterns to come and helps to reinforce the difference between linear and quadratic patterns.

Unit 7. Transformations. Following quadratics is a good time to talk about general function transformations. The same rock-star colleague had advised me that kids think this topic is too abstract. They will not retain it if you start by teaching g(x) = a*g(x - c) + d, but they will retain it if they can think of a concrete (ie. quadratic) pattern that they already are familiar with. I do these with explorations on the computer, and I have a pretty scaffolded plan if you want to grab it to take a look.

Unit 8. Exponential Functions. I think after quadratics as a big unit, the most natural next major topic is exponential patterns, if your students are following the trajectory of discussing various sequences. Kids can see geometric sequences everywhere, and it is so useful in their lives to understand compounded growth, that I think exponential sequences should be introduced as early as possible to contrast with linear patterns. Here, sustainability issues should really be discussed, both in terms of inflation of costs (of living and education and debts) and our unsustainable human growth / depletion of resources. I teach exponent rules inductively, and they go along with this unit but are assessed separately. I had the idea last year of asking the students to do a sustainability PSA (public service announcement) project, and I will really try that this year with more careful planning / pacing. I have not decided if I will teach logs this year yet, only because not all the teachers in our department can agree where in our curricula (Algebra 2 vs. Precalc) that should be taught.

Unit 8. Inverses of Functions. I didn't do a full unit on this last year, but I think a full unit on inverses, domain, and range should logically follow the introduction of basic function types, because it helps to introduce all the interesting forms that the kids may wish to use in their function pictures project. The wrap-up of this unit should be a functions picture project (via GeoGebra or Desmos), in which they show at least two types of "sideways" functions as part of their included functions. For me, the functions pictures project in Algebra 2 needs to be accompanied by explanations of the transformations, to reinforce the connections between algebraic form and graph.

Unit 9. Polynomials. I taught polynomials as the very last topic this past year, and absolutely loved it! I loved the particular placing of this unit at the end of the year because it allowed us to spiral back to factorization and quadratic formulas, while covering deeper ideas like u-substitution and complex roots. (I didn't do complex roots during the quadratics unit, since there were already so many skills there.) The kids also learned to apply the Rational Root Theorem, of course, and reinforced their understanding of the root. We did a quick maximization problem and some backwards problems working from remainders and factors to finding missing coefficients, and I was so happy to see how well the kids did! If we have time, we'll do a stocks project along with this unit. If so, I'll have to dust that one off from the archives...

Ok, that is a lot! I've just pretty much laid out my entire Algebra 2 curriculum for next year. Whew! Let me know what you think and where you think the missing corners are! xoxo.

Tuesday, July 29, 2014

One Resource a (Week)Day #18: Geometry Activities

When I taught Geometry last, I found that it was very feasible to structure most of the geometry class like this:

* First, some exploratory activity meant to introduce a new topic and important vocabulary terms
* Some project- or lab- based learning that lasts about 2 or 3 weeks, interleaved with skills taught as needed
* Concentrated skills practice / "review" after the project
* Quiz or test on the skills

Just off the top of my head, the units where this learning structure was very applicable included: 
* Tessellations (we made triangular, quadrilateral, and custom tessellations using rulers and protractors, which motivated some triangular congruent properties) 
* Measurement and conversions (we learned to measure everything from lengths to volume to mass, and practiced some unconventional or indirect methods as well)
*Right-triangle trig (lots of outdoors measurements involving angles of elevation and depression and inclinometers)
* Quadrilateral trig (using KFouss's problems and some paper folding to see why quadrilaterals are built from non-right triangles, which are built from right-triangles)
* Scaling (we did logo projects and calculated how that impacted the perimeters and areas) 
* Perimeter and area (using blueprint of houses on coordinate planes, with circular and concave portions)
* Surface area and volume (kids designed and built their own 3-D composite solids)
* Construction of reflections (mini-golf course designs)
* spatial projections (going from 3-D views to drawing 2-D views, and vice versa, using the computer to verify their hypotheses)

Some of the other traditional topics (integration of algebra with geometry; some coordinate-plane concepts; proofs and counterexamples; and basic geometry visualization based on language) we didn't do through projects, but I tried to still make those parts of the class as interactive as possible. Most of the topics you can illustrate through patty paper, move-around demos, and just plain fun things. Geometry is definitely my favorite class to teach, but I am always looking for new ways to spice it up! I find that after doing a project, the kids are solid with the basics and are ready for me to push them a bit farther on the paper assessment.

Here are some more ideas of projects, from a school in Columbus, NJ. (Sorry but I couldn't find the teacher's name!) I like this list. It has a variety of ideas, so that if our Geometry team decides to do different sets of core topics this year, I can still incorporate projects into my class. I think that it will make a nice complement to the visual / artistic activities from the beginning of Discovering Geometry: An Inductive Approach. 

That's it for today. Till tomorrow! 

Monday, July 28, 2014

Math and History: A Look at Slavery

Geoff and I decided to go on a plantation tour on Saturday. Before we had committed to this, I was feeling uneasy about how morally gray this experience might be, so I did some research and found out online that there is a plantation that runs its tours with historical accuracy and talks about slavery with candor. So, that was the one we decided to go on. The tour was combined with going to another plantation also in the area, and the experience was one that I will not forget.

This is what I gathered from both plantations tours:

* During the Antebellum period, the sugar cane industry down in New Orleans had boomed. The farmers invested in slaves to help them expand their business. Both plantations we visited had roughly 100 slaves in the 1830s. Most of them lived in small slave quarters a short distance away from the main house, but the house slaves lived and worked closer to their masters.

* Among the slaves on the plantation, there existed a hierarchy both depending on their skills and where they came from. The Creole slaves spoke French, and therefore were able to work inside the house and/or communicate with their masters, and therefore were valued more highly (and bought/sold for more money, especially if they were also highly skilled in things like metalsmith). The "American" slaves that were brought in after the Louisiana Purchase were generally valued less, because if they did not happen to speak the same African language as the other Creole slaves, the owners often had trouble communicating with them and they would struggle on the job.

* The slaves worked in grueling conditions, sometimes for up to 16 hours a day. I cannot imagine working in the fields when it was 90 degrees and super humid. I was sweating up a storm just from the short walk in between the buildings during the tour. At one of the plantations, we saw a huge shackle that the slaves would wear around their necks to prevent them from running away.

* The slaves who were considered of least value slept on the floor of their slave quarters, without a bed. Those who were valued more, had more complete furnishing.

* The slaves grew their own foodstuff on the farms, in order to feed themselves and to live off the land. 

* During the Postbellum period, the slaves were essentially kept in slavery because each plantation only paid the "freedmen" in tokens that only worked on the plantation, at the plantation store. This way, the freedmen could never really leave because they could not save up money to do so. This continued on one of the plantations until 1900s and on the other all the way until 1940s. The latter plantation, for this reason, still has the original slave quarters that you can visit today. When the current owner bought the plantation in 1940s, the tenement farmers who still lived in those quarters were still largely the descendants of the former slaves. The big difference before- and after- the war was that the freedmen could send their kids to school.

* Both plantations have a list commemorating the slaves who once lived there, first names only (because they didn't have last names as slaves). The slaves were listed with values, some as little as $25 (in the 1830s) and others listed as $1500 if they had specialized skills.

* The tour guide at one of the plantations told us that after the war, the public schools in New Orleans were actually initially desegregated until the Jim Crow laws came into effect in the 1880s. Afterwards, the schools remained segregated until the Civil Rights era. Although I found this article about the re-integration in schools in 1960, the tour guide had explained that the schools weren't integrated here until the 70s. (Anyway, now everyone goes to charter schools in NOLA, and people who could afford it send their kids here to Catholic schools.)

Although I was ambivalent about these trips, and I was bothered by the way one of the plantations seemed to brush the slavery issue under the rug, I still think that going to see a plantation firsthand was a very educational experience. If my students end up learning about slavery this year (and they must, in one of the grades I teach), I could tell them about this experience and when I tell them that people were bought and sold for as little as $25, we could do the math to figure out how little that money is in today's terms.

So, anyway, that was my rumination on math's role in history.

PS. I found this interesting article about Creole slave-owners who were themselves black in Louisiana. It definitely helps to explain some of the things people have said about Creole blacks feeling superior to African-Americans in NOLA. There are lots of numbers in this article to use for calculations of current-day value.

PPS. I've been doing some recreational reading about WWII for my book club, and one of the factoids I learned was that the initial funding for the Manhattan Project was $6000. This was in 1940. This is also a real exponential growth application, to figure out how much that funding would be worth today.

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?