A Fun Lesson on Similarity

This term, I have been mixing in some old Geometry lesson material into my current Geometry class, now that we're no longer doing purely Exeter problems. It has been so much fun!!! My kids are delighted every time I throw in something that I had used and liked in El Salvador. It makes me very happy to observe.

The last couple of days, we have been doing a fun little similarity activity on the Geoboard, that I had made back then and then revised for this year. See it here. I am using it to introduce similarity, as a lead in to special right triangles and right-triangle trigonometry. The kids are having so much fun with rubber bands that they don't realize I am sneaking in significant learning.

I still like the Exeter problems for their incredible richness, but balancing them out with other modes of learning is the way to go,  I think! 

PS. This has been a good teaching week. Two of my low-confidence students who have been working their BUTTS OFF for weeks each got an 100% on their requiz. HOT DAMN! I'm so, so proud!  

How it Must Feel

Imagine yourself as a student who has always had a hard time with math. Honestly, you are not even sure why there are sometimes multiple variables in the same formula. When the teacher starts a new topic, you may need to see it 4 different ways and have it be repeated a lot of times in order to become somewhat proficient at the skills involved, even though the general idea is sometimes (not always) accessible to you. When other people discuss mathematics in groups, the speed at which they are discussing the ideas often flies right over your head, but you feel embarrassed to ask them to explain every part to you, because other people all seem to understand it fine. So, you do ask some questions during class, enough to make some progress on the work, and then you wait until you can find the teacher to ask for more help. You can only find the teacher to meet outside of class about once a week, and by then you already have more questions accumulated than can be answered in one session, so that although you think the concept is probably important, you just want to know the bottom line of how to get through the various problems you are going to see on the quizzes. So, the quiz comes and goes and you feel defeated by your score, even though you are working hard everyday in class and your teacher can also see that, too. You go home with the quiz and you don't want to even look at it for now, even though you know that you probably should start thinking about the re-quiz and that the teacher will probably start something new either tomorrow or the next day, and that the cycle will likely repeat itself.

What do you do?

In every class I teach, there are some kids whom I meet outside of class on a weekly basis. For many of them, that is enough. For some, that isn't. I love the idea of heterogeneous grouping, because I believe in all the things that other math educators believe, which is that it promotes safer learning environments, teaches diverse students to work together, promotes growth mindset, etc. But, at the end of the day, I don't know what to do in a practical sense to help these kids who experience those cycles of disappointment in every unit. I try to vary up what I do to help them build the conceptual understanding of underlying topics (ie. Math Talk, problem-based learning, visual representations), but realistically I have to balance too much going-back-to-basics against the majority of the class's need to develop other, more sophisticated, skills and concepts. Open-ended problems sound awesome in theory, but in reality we still need coherent conceptual and skills development the majority of the time. 

So, what do you do besides trying to be empathetic?

I don't have an answer right now. In the past, I have had great success in homogeneous grouping at helping the slower-paced classes build confidence and feel successful with a smaller set of topics, but I find that goal very challenging/elusive when those lower-confidence students are situated in an environment that is perhaps just faster-paced than they can handle. I am fortunate that most of my students give me 100%. I have no doubt about that. But, how can I grade them all on an absolute scale based on what they know, if they are all starting off at very different places along the spectrum of prior algebra experience? And, more so than grading, how can I serve them all? 

Those are open questions in my mind. Would love if you could chime in to enlighten me in your thoughts about this. 

What to Do When You Are Busy

Today, one of the things I read over the summer came back to me. The thing I read was written by a therapist, regarding his patients who are stressed and unhappy because they feel like they're spread too thin among work, family, friends, etc. The therapist's advice is that you cannot make more minutes in a day, but you can increase the quality of your minutes in the day. If you're thinking of work when you are at home, and thinking of home when you're at work, then you're not making anyone feel valued and therefore the quality of your minutes spent with them is low. This way, everyone around you will feel dissatisfied, and you'll feel unhappy as a result. The wisdom he shares is to focus on making the people you are presently with feel like you are 110% present. Be proactive about it, instead of trying to coast by with little engagement. (In terms of active engagement, he distinguishes between your son telling you about his upcoming soccer game, and you remembering it yourself and mentioning, without being prompted, how much you already look forward to it. In both cases, your time commitment is the same -- you're still going to watch his game -- but just by being a little bit more thoughtful, the reception of your time spent will be very different.)

I thought of this today because I noticed that I have been working a lot of extra hours. But, when I meet with my students outside of class, it's not helpful for me to think even fleetingly about the many things that I still need to do that day. Instead, being fully present and taking an extra 10 minutes to let the kid slowly organize their binder or to tell me about their week before we start to look at where they need help in math, is so valuable. It slows me down and it slows them down, and it allows me to make them feel important before, during, and after our meeting. Maybe I cannot increase the amount of time that I have for them on a regular basis (and maybe I cannot solve all of their math troubles in one meeting), but I can make each encounter outside of class more meaningful by just taking a little extra time.

So, here is a reminder to myself to keep building the quality of my minutes in a day, especially as it gets really busy. 

The Start to a Great Year!

It has been 2.5 weeks since we got back to school! During this time, because of various back-to-school grade-level bonding activities, we only had 2 good weeks' worth of classes in each grade. Still, I am feeling somewhat settled into my classes, having learned all the kids' names, and am starting to figure out who needs help and to meet regularly with some students outside of class. So, things are good!

In addition to revising curricula, I am trying a lot of new classroom procedures this year. At first I thought these changes wouldn't be manageable, but so far, although I am working a fair amount, my work time has been very productive! I can clearly see the impact of various small changes on my classes, and that leaves me to spend my energy where it counts the most (ie. trying to tackle the big challenge of having many more high-need students this year).

Here are some things that I am trying this year:

* Assigning group roles for each group member: The one benefit that has stood out the most to me is the role of the recorder. The group's facilitator and time-keeper help me ensure that every student in the group tries the problems independently and jots down their individual work, but at the end of the activity/discussion, I only collect 1 copy of the worksheet and notes (from the recorder) to make sure the answers they went over are in fact correct. I also ask the recorder to write down additional items as per their discussion, such as a description for the algebra process used and definitions for key terms. I then correct just that one copy (from a group of 3 or 4), and then make copies for the other group members to keep. This way, everyone has the conceptual explanations, to help them remember what was discussed in the group besides just the answers they have on their paper. It also cuts down on the amount of grading I have to do, while being able to give written feedback to the entire group. Playing the role of recorder has also been very motivating for the kids who are the weakest performers, stay actively engaged. They know that their role is highly relied upon, so there is some unsaid peer pressure for them to make sure that they understand the process enough to write down important pieces of it on paper. Besides the recorder, I also have a facilitator, a time-keeper, and a questioner in each group. (The basic construct comes from Complex Instruction, but I am not sure if I am following their protocol exactly.)

* Green, yellow, red stickers: In lieu of grades, this year I have been, thus far, giving feedback in terms of colored stickers. Eventually we will have quizzes that will count for points, but in the mean time, "mock quizzes" are just about 3 questions I write on the board, and I collect them to see how the kids are doing (and to take notes on who needs reinforcement on what), and we discuss the answers as a class. The next day, I return the mock quizzes with a colored sticker to show whether each student needs to keep working on that concept. It's de-coupled from grades, but my secret hope is that kids will just want to get green stickers. And, so far, that is pretty true!! I have been using this colored-sticker system on the collected discussion notes (from recorders), mock quizzes, and conceptual "check-in" homework assigned individually. The kids who got red stickers from me on a quiz or an important assignment have approached me to automatically resubmit the assignment after revision, to ask for help, or have been very happy when I proactively approached them to set up a time to meet outside of class. They treat the red stickers very seriously, which is great. It's such a visual way to alert them that there is a gap. The fact that the assignments are de-coupled from grades makes it possible for me to focus on meeting with them for their learning, rather than the conversation being about grades. It also makes it possible for me to have multiple "mock quizzes" in the same week without it being stressful for the kids. So far, I love it! I got a pack of round stickers from a local drug store's stationary/school-supplies section, and it already has all the colors I needed. One of the science teachers started using the same system with his students after hearing me talk about it, and he thinks it's much more clear than giving kids check plus, check, and check minus. Eventually, my goal is to have the kids self-assess via a sticker when they turn in certain assignments. And then I'd give a second sticker upon returning it, to confirm or correct their self-assessment. 

* Green, yellow, red cups: I know this is not news for other teachers, but I wanted to try the green, yellow, red cups in groups as a passive indicator to me whether the students need help. Sometimes, I think as a teacher, it's hard for me to tell when students are positively frustrated or negatively frustrated by a task. The cups are a clear way to indicate to me whether I should intervene. My secret hope in introducing this was that if they looked around the room and the other students all have green cups, maybe the group that was already about to give up would push themselves to persevere just a little more. Also, because they could only indicate green, yellow, or red to me as a group, they would be forced to communicate amongst themselves before they reached out to me for help on a problem. I have thus far only used this in my lower-grade classes (Algebra 2 and Geometry), because I am not sure whether it's a bit too cheesy for my 11th- and 12th-graders. But, I may roll it out to them eventually.

* Self-reporting math efforts outside of class: I feel strongly that as juniors and seniors in my Calculus class, the students need to be doing self-directed learning outside of class. Practically, that means that I don't assign problems everyday, because I want them to use their at-home time to make flashcards, concept outlines, re-do tricky problems, do new problems on their own, see me for help, etc. It does not mean that on those days when I don't assign specific problems, they shouldn't be working on math at home!!!! But, I am also a realist; I know that most teens will do the minimum unless there is some visibility into their action. So, I made a Google Form that collects data about what the students are doing. (You can click on it. This is a copy of the link I sent out to the kids, so even if you fill it out, it won't mess up their data.) I created a bit.ly link to the form I made for my students, and I asked them to fill it out every time that they do math outside of class. I stated that, as "homework", I expect that they're doing/logging 20 minutes of math outside of class, 5 times a week. Of course, they can clump the times if that is not possible, but if they are doing 100 minutes of math once a week, that is probably something I should talk to them about. Anyhow, I think the information collected this way will be a great resource to allow me to have productive learning dialogue with my students, while encouraging them to be self-directed learners, asking me questions like, "What else can I be doing with my studying time?" I've only rolled this out on Thursday, and already I can see some data being logged by some students, with comments on what they might need from me as next steps. I'm very excited about this!!!!! If this is successful, I will extend it to my Algebra 2 class. Throughout the term, it'll be a very valuable resource for me in terms of giving them specific feedback on their learning strategies. It'll also make writing narrative report cards a breeze at the end of the term.

So far, these are the "systematic" changes that I am trying to make. I have already noticed a tremendous difference in my classroom culture this year, in terms of how equitably and actively students participate, and how positive they are. I will observe a bit more and write a big post about that, maybe next week! 

Experimenting with Structured Group Work

I am trying to change my classes this year drastically in the way that I handle group work! I want to be intentional, intentional, intentional.

So far, it has been one day and that one day has been excellent. I decided that for Day 1, I would talk to the kids about inquiry-based learning vs. direct instruction, and then pass out a task that is accessible for everyone to start. All they had time for on Day 1 was the individual thinking time, which I am going to build into every group activity this year.

In Geometry and Algebra 2, I did Mark Driscoll's folding task on Day 1, but emphasized that the content of the task is not nearly as important as the students practicing the expectations for group work. In Calculus, I made a custom sequence from Desmo's Function Carnival that consisted only of the parachute height vs. time, followed by misconception analysis, and parachute vertical velocity vs. time, with subsequent misconception analysis. The students in all classes then were asked to go home and finish the rest of their individual thinking.

Tomorrow, when we come back, we will do verrrry structured group work, and I will ask the students to do first one round of just making observations with no comments from their peers (Round 1). Then, they will go around and ask clarifying questions or challenge each other to justify their thinking (Round 2). Then, the group will engage in an open discussion while the recorder continues to take careful notes, to turn in later and to distribute to the group. Afterwards, they will summarize the findings and record what questions they still have as a group. To help the Calculus students focus in on the misconceptions I saw today, I will give them this handout as generated from their parachute graphs, and ask them to brainstorm as many observations as possible for each graph before we discuss as a whole class. The recorder of each group will record all the accuracies and inaccuracies that they notice about each graph.

Here is some language I will offer the groups tomorrow to help them with their structured discussions:

Beginning of discussion (Round 1):

"What have you tried so far?"

"I noticed that ______________"

"I tried __________ and found that to be (un)helpful, because _____________"

Middle of discussion (Round 2):
"I was confused about how to _______________"

"Are you saying that _______________?"

"Can you explain why you think ___________ ?"

"I don’t get that. Can you explain it in another way?"

"If we changed __________, then what would the result look like?"

End of discussion:

"In conclusion, we agreed that ______________"

"We found it hard to agree on _____________"

"As a group, we still have trouble understanding ____________"

I am really excited about this! Having structured group work is allowing me to slow down the pace at the start of the year and to emphasize quality over quantity of work done, in order to set the right tone for the rest of the year. I plan to have the kids stay in these groups for about a week, and then we'll discuss what have been the most helpful parts of the structure, and then switch into new groups. Wish me luck!!! 

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!

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?

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.

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! 

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.

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 Element	Distance from Nearest Sequence A Elements	Shorthand representation of distance
102	8 above 94; 24 below 126	8a (24b)
160	2 above 158; 30 below 190	2a (30b)
218	4 below 222; 28 above 190	4b (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 Element	Shorthand representation of distance from nearest Sequence A elements
276	10b (22a)
334	16b (16a)
392	22b (10a)
450	28b (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 Element	Shorthand representation of distance from nearest Sequence A elements
508	30a (2b)
566	24a (8b)
...	... skipping some rows here, since I can see that 24a is a multiple of 6 and it will eventually decrease to 0a perfectly...
798	0a (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?

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?

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?

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.)

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!

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.)

One Resource a (Week)Day #13: Rich Tasks from YouCubed.org

I have loved these rich tasks from YouCubed.org. I found myself puzzling over this particular puzzle about flipping coins this afternoon. It ties nicely to modular arithmetic. I believe in the end that for any number of coins n, its minimum number of flips is a function of (n mod 3) and floor(n/3), but I'll let you try it and see for yourself! As usual, I find that starting with smaller problems really helped me to generalize and see a pattern.

I also enjoyed the much simpler but accessible paper-folding task from Mark Driscoll. What I find to be really enjoyable to read is also Jo Boaler's commentary on each task, what questions she asks, and when she allows the kids to work independently versus talking to each other.

If you haven't already signed up to be on the YouCubed.org mailing list, I highly recommend it! I find myself jumping with excitement to open their emails. Even though some of the site's content targets middle-school teachers, it is so enjoyable to hear and read of the way Jo talks about mathematics.

On the upper end of open-middle problems, I had a lot of fun today thinking about the size of the tiny sphere that can fit snugly inside the space of a tetrahedron built from 4 larger spheres. As in, if you packed 4 equally sized spheres together like shown here (this is the top-down view) and then in the middle space fit in another tiny sphere, then how big is that tiny sphere's volume relative to the other bigger spheres? PCMI had problems like this in their Geometry sets from this year, and I enjoyed playing around with this particular problem. It's probably too challenging for most of our students to do, but a good one to keep in my arsenal nonetheless.

I hope you have enjoyed the tasks from today. Ciao!

First-Pass Brainstorm for SY2014-2015

For some reason, my sleep pattern has been erratic here in NOLA. Some nights I sleep beautifully and still can go back to sleep after I wake up in the morning. Other nights I get only around 5 or 6 hours of sleep, and any noise like the AC kicking in or a bug buzzing around or the train passing by a few blocks away is enough to wake me up. 

Two nights ago, I made the mistake of thinking about what I would do on Day 1 of the next school year just as I was lying down, and that was enough to keep me wide awake most of the night. The next day, in an attempt to calm my nerves, I pulled open a spreadsheet and started jotting down some of those floating ideas and to attempt to organize them, while staying away from the areas that could get me into trouble. (As in, I really don't have full control over what I will teach in Algebra 2, since there will be 4 of us teaching Algebra 2 at our school next year. Similarly, there will actually be 5 Geometry teachers next year, and we'll all have to agree on what to teach, which seems daunting to me.) Thus, I started my brainstorm with the bigger-picture things that I can fully control -- like how I plan to build a collaborative and reflective classroom culture, and ended with the more curricular things like which enduring concepts I would want my students to walk away with. Instead of focusing on each class individually and going into all the curricular details of that class, I tried to figure out which ideas or themes seem to flow through more than one class and are therefore the most important and transferable. I looked afterwards at both Kate's and Anna's lists of essential Algebra 2 understanding questions, just to make sure that I didn't leave out anything majorly important. (Thanks, gals, for the fabulous lists!) I decided not to separate the process goals from the content goals in the end, because realistically, each type of goal will require from me the same level of concrete steps and careful planning in order to implement it with gusto.

This is a work in progress, but now that I have a basic framework, I think the next step is to find and refine some rich tasks that I can use. I would love any resources you may have for me! 

One Resource a (Week)Day #12: NASA's Sustainability Math Curriculum

NASA offers a great collection of resources for teaching real math topics embedded in real science. I was skimming through their archive of materials today when I came across these interesting lessons:

Did you know that the length of an Earth day changes over time, meaning that the ratio between Earth's revolution and its rotation around the sun is not constant? That is so cool!

Also, based on the idea of cell (waste) equilibrium, we can estimate cell sizes.

Here is another nice tie-in to geometric proportionality: Estimating rate of glacial retreat via photos.

They, in fact, offer an entire book of well-organized "Earth math" for free to help educators with teaching sustainability math. A lot of the concepts can be tied easily to middle-school math.

Yay, NASA!

Just For Fun: Cake Problems

I was thinking about the variety of cake problems that are open-ended, accessible, and that encourage mathematical thinking and reasoning.

Easy: How to cut a cylindrical cake, Salvadoran-style, into n equal pieces.

Medium: How to cut a circular cake using 2 parallel cuts, such that each of the 3 resulting pieces has the same volume. 

Medium: How to cut a square cake into n pieces, so that each piece has the same amount of cake and same amount of frosting.  

Hard: How to cut a cake that already has a hole in it into 2 equal parts (from PCMI's pizza session, originally from Car Talk Puzzler). The PCMI version of the problem contains a diagram, so I give it to you here. Note that you don't know the sizes of the rectangles nor the relative angle of rotation of the inner rectangle.

Expert: How to cut a cake while maintaining maximum freshness.

Random Puzzler: How can you cut a cylindrical cake 3 times to make 8 equally large slices? 

Yum, cake! Lots of math fun in sharing things!

One Resource a (Week)Day #11: Math and Sustainability

I started looking into resources on teaching math and sustainability together, and it is like going down a rabbit hole. I think this is going to take me the rest of the week, possibly, to cull through all the fabulous resources out there. I'll review them piecewise.

Since I moved back to Seattle, I have really been thinking more about sustainability in the curriculum. Part of it is because I live in Seattle and most of the residents of the city recycle and compost on a very regular basis. (Since we moved to Seattle, Geoff and I have learned a neat trick from our friends for composting. We put our partially filled compost bags in the freezer, and that helps to make composting do-able for us. If you have been on the verge of composting, this could really make a difference for you! We have been amazed by how much our trash is cut down, once we started composting last year.) At my school, we have an extensive environmental stewardship program wherein every single student and full-time faculty member (including most of the non-teaching support faculty) gets out and cleans the school 3 times a week, which includes scrubbing bathrooms, cleaning classrooms and hallways and offices, emptying recycling and compost bins, etc. We do it because we want to instill in the kids a sense of caring for their environment, and this sentiment extends to the greater environment around us. The school's cafeteria institutes Meatless Mondays and provides compostible to-go utensils for the occasions when the staffers need to eat during a meeting. At some point this year, we used an all-school assembly to teach the students how to properly sort trash, and another meeting was devoted to looking at what happens to our sewage waste and where it goes. We encourage the students to minimize packaging on any food that they bring to school (ie. during "parties"). Geoff and I try to walk, bike, and bus everywhere in Seattle, and on the occasions when we do need to drive, we typically use a smart car-sharing program called Car2Go that both saves us money from having to maintain a car ourselves and minimizes our carbon footprint over time. Although we're not vegetarian, we subscribe to a bi-weekly CSA basket for fresh produce to encourage us to eat sustainably (it's local, organic, and cheap... win-win-win!). As a math department, we talked at length about what happens to those textbooks we order and don't use. This coming year, most of the teachers opted away from ordering textbooks as a result. So, at least I don't feel like a hypocrite when I talk to kids about sustainability these days.

Just like teaching math together with social justice, teaching math in the context of sustainability can encourage our students to think critically about issues around them, using the lens of mathematical reasoning. That said, teaching sustainability with math is certainly a challenge, for me at least. The summer is a great time for me to delve into these resources, and I would love to hear what you already do in your classroom and what has been successful.

I found a couple of terrific resources today. This website has some model lesson plans, and I looked specifically at #8, #9, #10 which pertained to the higher-level math classes. I really liked them. I think they are thoughtful and relevant, and with little or no modification can be used in our high-school classes to align to existing content.

On a separate note, Professor Pete Kaslik has written an excellent book on math and sustainability intended for the college level. (If you scroll down on that web page, there is a link to download the book. I linked to the webpage because it includes some copyright disclaimers from the author.) I think that with some modification, you can adapt most parts of Kaslik's great content to be teachable at the high school level. (The statistics part is the only part that I think is difficult to adapt without leaving behind most of the juicy bits.) More generally, his book is a great, in-depth view of math in the real world. Each topic feeds into the next one, covering an array of math content that is solidly rooted in real-world application.

For example, Kaslik starts by investigating individual sustainability in terms of basic financial education and savings -- classic Precalculus stuff. After that, he extends the idea of exponential growth to population and limited resources, in that process investigating a new type of pattern (logistics curve). He talks about the geometry of maximizing living area while minimizing loss of energy (minimizing wall space) in designing architecture, and then takes you through the math of looking at the carrying capacity of towns, based on other living requirements. You can then compare this carrying capacity with population patterns. To investigate sustainability issues further, he introduces the idea of surveying the population, goes through the mathematics of sampling, and then ties it altogether with complex analysis of dynamic systems and how you can model the many input variables using technology (Excel programming). All in all, all of the math is authentic and motivated with real analysis of real issues that can be scaled to critically consider the national or global implications. Brilliant!

I really enjoyed today's foray into sustainability and math, and I look forward to more digging around tomorrow!

One Resource a (Week)Day #10: Sustaining Student Motivation

After the last post, I went and spent some time reading up on motivation. One resource I decided to read is Daniel Pink's Drive, because even though that sounds really familiar to me (I am a bookstore browser), I couldn't easily find any reviews of the book from educators via Google. I read (most of) it between last Friday and today. The book follows a sort of 20-80 rule; the first 20% of the book contains 80% of the central ideas, and the rest of the book kind of just goes over the concepts in finer granularity, and I found my interest steadily tapering as I got farther into the book.

I am kind of iffy about recommending this book. Like I said, because it's so front-packed, I think you can more or less borrow it from a friend and glean the applicable ideas in a day. One thing I did like about reading this book is that it breaks down the best motivation for specific tasks in quite some detail, beyond the one-liner that is most commonly relayed ("extrinsic motivation is bad for kids").

Pink actually provides a quite detailed break-down of types of tasks. Say you're a teacher and you have a complex problem that you want your students to solve creatively. In order to be successful, they need to be inquisitive, resourceful, and to consider a wide range of approaches. (Yes, yes... Ideally, all of our teaching looks like this.) In that case, you definitely don't want to introduce any extrinsic motivation. The task, if given at the correct level of intellectual challenge ("just beyond the comfort zone of the student by 1 or 2 levels"), will be its own reward. The effort and process required to solve the problem are its own reward, because that level of concentration is necessary to our well-being as people, and the kids will enjoy purely being in the "flow" of the moment by engaging in the task. Giving them an "if-then" extrinsic motivation (like mentioning the impact this will have on their grades before they start the task) will negatively impact their performance, by limiting their ability to be flexible and open to all cognitive options. In a more long-term impact, it'll also diminish their intrinsic motivation to do work without grades attached. If, for some reason, after class you decide to collect the task and to grade them (a sort of extrinsic motivation after the fact), since it hasn't impacted their experience during the active learning, it will have minimal impact on their intrinsic motivation to learn. (They will still associate that learning task/experience with being intrinsically motivated.)

On a more rote task (such as skills practice that cannot be avoided before an exam), you should provide flexibility in timing and method as much as possible, to retain the level of autonomy of the student and therefore to increase their motivation. Pink asserts that autonomy is a central human need, which in turn encourages motivation. An example of this might be having multiple review sheets, and letting kids choose which ones to work on, and for how long. When they start to get bored, they should be encouraged to switch to another task to help to break up the task. Explaining directly why this more rote task is necessary/purposeful instead of relying on a reward/punishment system ("I will count it as extra points on your upcoming quiz!") is also more beneficial to encouraging quality work, as well as not impacting the students' sense of intrinsic motivation.

As for verbal feedback, Pink asserts that students' intrinsic motivation is encouraged by direct praise of their efforts and specific (positive but well-earned) feedback on their work. If you have to give rote homework assignments as additional practice, for example, besides explaining why this is necessary, be sure to also praise their efforts (individually, as you go around the class) and to collect the homework for individual feedback.

If all else fails, Pink recommends that an "if-[you-do-this]-then-[this-will-happen]" extrinsic motivation should only be used as a last resort to encourage rote tasks that have no meaning (ie. stuffing envelopes and putting stamps on... I had trouble picturing what you should use this for in education, that could be truly that meaningless).

Very importantly, Pink believes that all people (employees and students and even some higher-order lab animals) are intrinsically motivated. We have the natural need to grow and improve. What we need to do as organizations and teachers is to find ways not to suppress that natural need, but to encourage its natural expression over time.

In a separate reading of a short handbook on motivation (available free to me through Amazon Prime), the authors Albert and Robbins had a practical idea for helping to work through fears of failure. I think this is a common problem that affects a fair number of students. Talking them in advance through the worst-case scenario can help to alleviate the panic that often comes with doing poorly on a test. For example, take the day before your first quiz or test to go over what they should do, if they do fail or struggle on the first exam. Should they come to talk to you? Should they do corrections on their own to prepare for a re-quiz? Remind them of the growth mindset then. Reviewing the assessment policy in your class before a test can help to minimize the panic / fear of failure and to establish trust early on in the class.

Setting small, achievable goals is also important for sustaining motivation over the long run (after the initial energy invested in the "newness" of the task has run out). Achieving smaller goals builds up the confidence and stamina required to tackle bigger goals. I shared a strategy from Albert and Robbins with my husband, about keeping a log of current incremental goals and also a log of the goals that you have already achieved. He says that he does this on his computer, and sometimes when he feels frustrated about work, he would still pull up the "Already Finished" goals list and look at it, to help sustain his energy and to feel good about how far along he has come. (He is really crazy about goals. His current work goals list runs about 44 pages, and his personal goals list runs another 22 pages. So, I don't know how big his archived goals file must be.)

That's it! I hope this has been as enlightening and practical to you as it has been for me. I'll be back here tomorrow for another Resource of the Day.

One Resource a (Week)Day #9: Open-Middle Problems

Hi, I live under a rock so I am sure you all already know about this, but I just discovered this fabulous collection of open-middle problems, which is my offering to you as today's resource of the day. I was disappointed, however, that there were few problems under the High School category, so I am going to look and try to locate some to add to their collection. I already wrote them about one set of polynomial problems that I had used this year, but I'm not sure whether they'll post it up since the problems (as I noted to them) are not original problems from me, and I had found them on a website that is not operating under Creative Commons.

Anyhow, I should at least write about those problems here, and I'll just link you to them!

During the last weeks of this year, I had taught two methods of polynomial division (long division and the box method) in Algebra 2 and then decided to hand out these "backwards" problems to my students in randomly assigned groups without any hints. I gave each group a big piece of butcher paper, and told them to try to work out the problems, in any order, on that sheet. I'd go around and check them off for the problems that were completely correct, with all work shown on the butcher paper. I also encouraged them to talk to their team mates if they successfully completed a problem and got it checked off. Because they were working off of the same piece of paper, the students were looking at each other's work (especially the problems that I had checked off as being correct) in order to share approaches and to help look for procedural errors. If I do this activity again next year, I'll definitely escalate the level of challenge by asking each group to try to find two different ways of solving each problem.

Since we had just learned and practiced polynomial division, I was surprised and delighted that they came up with really a variety of approaches to these "backwards" problems. Some of them used the idea of a root to set up systems of equations, which provided for a rich discussion afterwards when we compared methods across groups. They also understood that if you know the remainder already, then you can figure out what the exact (perfect) product was during the division process, even though it was something that we had really not talked about explicitly. I loved the creativity they had!

It helped me appreciate that every unit, I should be providing some substantial "backwards" assignment (similar to this one, given with no hints) in order to help them think more flexibly and to help transfer the learning. If you can go forwards and backwards, then that's how you know that you really understand it, right?

Hasta el lunes (si Dios quiere??)!