So, as a follow-up to Erastothenes using geometry to calculate the circumference of the Earth, this week I plan to go over how we can use the ratio to Earth to calculate the circumference of the moon!

The lesson idea came from my colleague John. I fleshed it out to scaffold it for my kids. It looks like this, and it ties in nicely both with our school's Grade 9 science curriculum (which teaches astronomy for all of next term), and our current circles unit. I plan to use this lesson the day after our basic circles quiz, when a few of the students will have family members visiting our class. (At our school, this is called Grandparents' Day, even though it is quite possibly not the grandparents that are coming.)

I'm excited!! I've never done this lesson before, but I like how it revisits perpendicular bisectors and makes them seem useful in application.

Addendum 3/26/15: I prepped for this lesson today and it REALLY BOTHERED ME that I got the estimate that the Earth's circumference was about 2.5 times bigger than that of the moon, when in reality it should be about 3.6 times bigger. I did some more digging and worked out a ratio to find out how big the Earth's shadow would be by the time it reaches the moon, and I think it's 9200 km in diameter at that point! That makes my ratio make a lot more sense, because this is only 2.6 times bigger than the moon!!! Go Geometry!

# I Hope This Old Train Breaks Down...

## Wednesday, March 25, 2015

## Saturday, March 21, 2015

### Geometry to Algebra Transition

Our school is trying out a new thing this year (one that I think is fabulous). We're re-shuffling 9th-grade kids' Math classes for the last 8 weeks of school this year, depending on whether they intend on taking Algebra 2 or Precalculus after finishing Geometry this year. The classes in our last term of the school year will prep the kids for transition into their choice of algebra classes for next year, and we'll assess them at the start of the term, end of the term, and again at the end of summer to determine whether their achievement and commitment-to-hard-work together seem to predict success in their choice of classes (particularly those intending on skipping Algebra 2), in order for us to advise them and their parents about whether they should be working over the summer, and what seems to make sense for their course placement. In prepping for this transition, we are including lots of algebra into our current circle unit to help kids "warm up" in thinking about algebra skills.

Below is what I have so far. The kids are definitely hitting their edge, but I am able to motivate them by explaining that quadratics is the next logical thing for us to practice, since we have done already a lot of work with lines and systems this year. Even those who have taken Algebra 2 in Grade 8 and who are intending to take Precalc next year did not have an easy time solving for points on a circle, so this is great stuff for all of them!

Here is my intro to circular equations, which most of the class is about finished with. Following it, I plan to spend a few days doing this, which is a modified version of a worksheet that two of my colleagues had created. We want the kids to get familiar with circular vocabulary (as preparation for Calculus) and to do some algebra practice involving circles, but besides it, we're not too attached to teaching all of the circle theorems, since we only have 7 more school days left of this term. I am excited to see the kids' transition to algebra after all the work we've done with them this year in terms of problem-solving. I hope it'll pay off when they get to Algebra 2 or Precalc next year!

Below is what I have so far. The kids are definitely hitting their edge, but I am able to motivate them by explaining that quadratics is the next logical thing for us to practice, since we have done already a lot of work with lines and systems this year. Even those who have taken Algebra 2 in Grade 8 and who are intending to take Precalc next year did not have an easy time solving for points on a circle, so this is great stuff for all of them!

Here is my intro to circular equations, which most of the class is about finished with. Following it, I plan to spend a few days doing this, which is a modified version of a worksheet that two of my colleagues had created. We want the kids to get familiar with circular vocabulary (as preparation for Calculus) and to do some algebra practice involving circles, but besides it, we're not too attached to teaching all of the circle theorems, since we only have 7 more school days left of this term. I am excited to see the kids' transition to algebra after all the work we've done with them this year in terms of problem-solving. I hope it'll pay off when they get to Algebra 2 or Precalc next year!

## Sunday, March 15, 2015

### Some Calculus Worksheets

I took some time during the previous school term to observe some of my colleagues, in order to educate myself about the ways in which they encourage inquiry in their classrooms. Among other observations, I enjoyed seeing how the other Calculus teacher (who is about 20 years more experienced than me) structures his worksheets to always circle back to applications and interpretation of answers. Since my visit to his classroom, I have been working on modifying my handouts from the previous year in order to put in more application into every concept.

Here are two of them: I used this to help kids wrap their minds around basic integral Calculus applications, and this one reviews some algebra skills from earlier this year, plus introduces the necessity of going in between algebra and the graphing calculator sometimes. The problems are not ground-breaking, but I think they've definitely helped to break up the skills practice, so I'm happy to share them if they might be useful to someone else.

From earlier months of this year, one thing that I did that totally helped with teaching Related Rates is that I first taught implicit differentiation with respect to time, and formally assessed students on this skill, prior to starting Related Rates word problems. (Sorry if this sounds obvious; it wasn't that obvious to me last year, teaching Related Rates for the first time!) Here is how I introduced implicit differentiation, using the analysis of non-functional relationships as a premise. After this, I had the students do some pure skills practice in converting geometric formulas to differential equations with time as the domain, before introducing my scaffolding for related rates problems and the many related problems I took from Bowman last year. I felt really good about this sequence of skills this year, because I noticed that it really made the problems more accessible to ALL students (as in, by the time they got to the word problems, they were really only focused on parsing the word problems process, rather than simultaneously struggling with the algebraic skills of differentiating implicitly). I recommend trying this, if your students get baffled by Related Rates problems.

I am also trying to place a general focus on vocabulary and communication this year. I've been doing this in all classes by giving the kids a list of essential questions at the start of a unit, and then having them journal their responses to those essential questions throughout the term. For example, for our current term in Calculus (which is short, only about 5+ weeks), I gave the kids the following questions. The questions are a mix between related rates (which we did at the start of the term) and intro to integral Calculus.

Students have shown a varying degree of enthusiasm about the journal assignment, even though I have been doing it since the start of the school year and explaining periodically its purpose. Part of the purpose of this journal is to get them to record, in their own words, examples and explanations to important concepts, so that they can have a succinct set of notes for future years. Another purpose is for me to see what they write periodically, so that I can informally gather information about common misconceptions for the topics that we have finished learning, and clarify them with the class. As it turns out, however, the naturally reflective students are thoroughly utilizing the journal to dialogue with me about their understanding, and the rest of the kids see it as a drag to have to keep revising their explanations until the end of the term, so the work that I receive is kind of a mixed bag in terms of quality. It has been a somewhat tough sell, but one that I think is important, because from time to time, students would comment on how they notice that by answering questions in their journal while learning the concepts (instead of putting it off until the end of the term), their understanding improves in real time. Do you do something like this in your classes? How do you drum up enthusiasm for such a revision-based assignment?

That's it for now! My Geometry students are wrapping up their 3-D project, which is very interesting as per usual. They have some really neat designs this year, which I might share at some point. Algebra 2 kids are knee-deep in thinking about the domain and range of different function types, and thinking about transformations on the various functions. It is nice to hear them go, "Ooh, ahh..." as they realize that they can connect information from different types of functions. Not much to write home about, but a productive time of the year nonetheless!

Here are two of them: I used this to help kids wrap their minds around basic integral Calculus applications, and this one reviews some algebra skills from earlier this year, plus introduces the necessity of going in between algebra and the graphing calculator sometimes. The problems are not ground-breaking, but I think they've definitely helped to break up the skills practice, so I'm happy to share them if they might be useful to someone else.

From earlier months of this year, one thing that I did that totally helped with teaching Related Rates is that I first taught implicit differentiation with respect to time, and formally assessed students on this skill, prior to starting Related Rates word problems. (Sorry if this sounds obvious; it wasn't that obvious to me last year, teaching Related Rates for the first time!) Here is how I introduced implicit differentiation, using the analysis of non-functional relationships as a premise. After this, I had the students do some pure skills practice in converting geometric formulas to differential equations with time as the domain, before introducing my scaffolding for related rates problems and the many related problems I took from Bowman last year. I felt really good about this sequence of skills this year, because I noticed that it really made the problems more accessible to ALL students (as in, by the time they got to the word problems, they were really only focused on parsing the word problems process, rather than simultaneously struggling with the algebraic skills of differentiating implicitly). I recommend trying this, if your students get baffled by Related Rates problems.

I am also trying to place a general focus on vocabulary and communication this year. I've been doing this in all classes by giving the kids a list of essential questions at the start of a unit, and then having them journal their responses to those essential questions throughout the term. For example, for our current term in Calculus (which is short, only about 5+ weeks), I gave the kids the following questions. The questions are a mix between related rates (which we did at the start of the term) and intro to integral Calculus.

*How are "implicit differentiation", "chain rule", and "related rates" all related? Illustrating this with a simple algebra example may help to clarify your thinking.*

*Take one of our Level 2 or Level 3 Related Rate problems from class and explain/describe, step-by-step, how you are able to find the missing information.*

*What is integral Calculus? Describe a couple of situations where this concept is useful.*

*Choose an exponential function of the form f(x) = a*e^(x - k), by assigning values for a and k. Estimate the area underneath the curve of f, from x = 0 to x = 5, using a total of 10 rectangles. Show both left-hand sum and right-hand sum, and draw labeled diagrams to show what your numbers mean.**Show, step-by-step, how you would calculate the enclosed area that lies between two functions f and g, where f is a quadratic function of the form f(x) = ax^2 + bx + c and g is a trigonometric function of the form g(x) = m*sin(n(x - k)) + p, where the value of n is not 1. You get to choose the parameters a, b, c, m, n, k, p to start, but make sure n is not 1.*

Students have shown a varying degree of enthusiasm about the journal assignment, even though I have been doing it since the start of the school year and explaining periodically its purpose. Part of the purpose of this journal is to get them to record, in their own words, examples and explanations to important concepts, so that they can have a succinct set of notes for future years. Another purpose is for me to see what they write periodically, so that I can informally gather information about common misconceptions for the topics that we have finished learning, and clarify them with the class. As it turns out, however, the naturally reflective students are thoroughly utilizing the journal to dialogue with me about their understanding, and the rest of the kids see it as a drag to have to keep revising their explanations until the end of the term, so the work that I receive is kind of a mixed bag in terms of quality. It has been a somewhat tough sell, but one that I think is important, because from time to time, students would comment on how they notice that by answering questions in their journal while learning the concepts (instead of putting it off until the end of the term), their understanding improves in real time. Do you do something like this in your classes? How do you drum up enthusiasm for such a revision-based assignment?

That's it for now! My Geometry students are wrapping up their 3-D project, which is very interesting as per usual. They have some really neat designs this year, which I might share at some point. Algebra 2 kids are knee-deep in thinking about the domain and range of different function types, and thinking about transformations on the various functions. It is nice to hear them go, "Ooh, ahh..." as they realize that they can connect information from different types of functions. Not much to write home about, but a productive time of the year nonetheless!

## Saturday, March 14, 2015

### Hello, World!

Sorry, web, I have been away! It has been a busy few months.

Geoff and I went to Hawaii in December, followed by some busy weeks at work for both of us while shopping in our off-time for a house. In February, I went home to visit my parents, and almost immediately afterwards, my in-laws came to stay with us for 10 days in our 700-square-foot apartment. They have just left, and Geoff and I have finally begun planning for our big summer trip. This summer, Geoff plans to take off 2 months from work and travel with me. Tentatively, we will start in New Zealand, then go through Philippines, Indonesia, South Korea, Japan, Estonia, Russia, Greece, Italy, Germany, England, Ireland, Iceland before coming home to Seattle. Our house's closing date is still set for the end of March, so in the mean time, there are just a lot of things keeping us busy. Hence, the radio silence...

But, I have been reminded recently that I have a blog! Through the grapevine, two friends of friends have mentioned this blog to me. Funny, small world. So, let me think about what I can put on the blog that is worth sharing. Stay tuned.

Geoff and I went to Hawaii in December, followed by some busy weeks at work for both of us while shopping in our off-time for a house. In February, I went home to visit my parents, and almost immediately afterwards, my in-laws came to stay with us for 10 days in our 700-square-foot apartment. They have just left, and Geoff and I have finally begun planning for our big summer trip. This summer, Geoff plans to take off 2 months from work and travel with me. Tentatively, we will start in New Zealand, then go through Philippines, Indonesia, South Korea, Japan, Estonia, Russia, Greece, Italy, Germany, England, Ireland, Iceland before coming home to Seattle. Our house's closing date is still set for the end of March, so in the mean time, there are just a lot of things keeping us busy. Hence, the radio silence...

But, I have been reminded recently that I have a blog! Through the grapevine, two friends of friends have mentioned this blog to me. Funny, small world. So, let me think about what I can put on the blog that is worth sharing. Stay tuned.

## Friday, December 12, 2014

### 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!

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!

## Friday, October 31, 2014

### 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.

## Thursday, October 2, 2014

### 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.

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