Making over another typical geometry problem

It’s time to look at another typical geometry problem to make over. This time Dan Meyer (@ddmeyer) presented this problem for revision.

Dan decided to go in this direction for the revision, which, for the record I really like. I would encourage you to check it out.

I took a try at it, too. I’ll let you decide which you like better.

I like this problem’s basic core idea. Looking at the volume of a sphere (the meatball) and the volume of the cylinder (the cooking pot), in general, this is a pretty tasty set-up (pun intended). Especially considering that I am always a fan of problems that make use of food.


For this problem, food and cooking were actually more of a problem that a support.

First, the cooking pot is sitting on a hot burner and I’ll be the first to tell you, a cooking pot doesn’t have to be full to spill over. So, the question of whether or not sauce will spill over is a bit more complicated that it might seem at first.

Second, meatballs aren’t spheres. They are irregular and rarely are two of them congruent.

So, my first thought was to choose spherical objects that are all congruent: for example, baseballs. Coaches regularly carry baseballs around in 5-gallon buckets, so there is our cylindrical container.

And I figured I’d deliver the task in a video simply because videos tend to improve engagement on their own.

Now, once I made the video (and some meaningful conversation was had among those who are better at this than I am) I found that my task had one glaring drawback. When you put baseballs in a bucket, they don’t pack tightly. There is air between them. A lot of it, in fact.

So, now it seems like if we are to use this video for instruction, we would need to change the question in to multiple parts.

1. How many baseballs can we fit into the bucket? (This would likely end up being a demo or a lab where we collect data. Tricky to calculate this.)

But then we supplement the question above by…

2. How much volume is wasted by packing that many baseballs in the 5-gallon bucket.

This would get back to the original content. Likely the cylindrical volume would need to a unit conversion, and then some analysis of the collective volume of the collection of baseballs.

Now, if we could ind a way to check it. The first thought I had was to fill the bucket with water. Put the baseballs in to displace the water out of the bucket. Take the soggy baseballs back out of the bucket. Find the volume of the water that’s left.

Problems with this idea: 1. Baseballs float which is going to effect the manner in which the water is displaced. 2. Baseballs absorb water. This means that some of the none displaced water would get removed with the baseballs and not counted.

Hmm… I thought of filling the bucket with baseballs and then topping the bucket off with sand. Which would solve both of the above problems, it would also give me an opportunity to make a beach trip.

Any other ideas out there?


Developing a Calculus Course – Where I’m at so far…

As I reported back in January, I am working on developing the next generation of calculus at Pennfield High School.

To say this is overwhelming is a bit of an understatement. But the support has been strong from the math edublogosphere. To Sam Shah, Jim Fowler, Shawn Cornally, Justin Lanier, and Amber Caldwell I owe a great deal of thanks. I couldn’t be doing this without you.

Here’s what I’ve gotten done so far. At this point, I feel like I have enough material to keep my student busy for four to five weeks. Thanks most to Sam Shah, I have one unit done. Including handouts, formative assessments and summative assessments.

Also, after examining the incredible amount of resources that I have been freely given, I have decided on a couple of structures including student self-assessment sheets (a structure popular in standards-based-grading) and Friday Free-For-Alls, which give the students the opportunity to look at problems that are likely an extension of their geometry, algebra II, pre-calculus or stats work, but they may want to try to employ some of their newly-acquired calculus tools to find a better (faster, more efficient, more accurate, more realistic) solution.

The handouts and problems for unit 1 are posted under “The Calculus Course” to the right. I look forward to your constructive feedback.

Once again folks, thanks for everything and I look forward to continuing to work with all of you.


Making over a typical geometry textbook problem

So, Dan Meyer (@ddmeyer) recently introduced Makeover Mondays where we, as a math blog community, makeover a series of textbook problems that have some potential as usable contexts, but have low impact delivery for engagement or cognitive demand.

This is my first attempt to offer something to the conversation.

So, here’s the first target:

Book Problem

So, here’s where my mind went in terms of making over this problem: First, the idea of “biggest bedroom” is a fairly appealing idea, except I don’t recall competing for biggest room with my friends. So, insert a new plot twist. Rodney and Emile are brother and sister.

Second, most of the competition for biggest room comes when the rooms are both empty. That is, move-in day. Most of my students are fairly well-acquainted with the idea of moving. So, remove the detached rectangles and insert a floor plan.

Floor Plan

Third, let’s add a bit of complexity. Both of the children move in with their own furniture, but, of course, with one being a girl and one being a boy, the furniture is not the same. Emile has this dresser and this bed. Rodney has a combo piece that includes both bed and dresser.

They are going to also move in with toys, a lamp, a chair for the bedtime book, among several other things.

Finally, the new task: You are the parent in charge of making the decision over who gets which room. Who gets which room?

Reflecting on the Common Core, Part II

Creativity, flexibility, are were on the rise this year.

Creativity, flexibility, are were on the rise this year.

Around Halloween of 2011, we began to prepare to rewrite our geometry class to align with the Common Core. This enabled us to do a couple of things that I had been wanting to do for a while. First, we ditched the textbook. Then we began to move toward Standards-Based Grading (Shawn Cornally (@thinkthankthunk) has some great stuff on this). We also decided to reconsider Algebra as the backbone as had been the previous practice in favor of a more visual, experiential approach.

We made the decision to embrace the CCSS’s Standards of Mathematical Practice because they made so much sense. We imagined building a class around patient problem-solving with strategic use of a variety of materials, student-to-student geometric arguments making use of geometric modeling and repeated reasoning. We imagined a course that drew the student into an experience. We didn’t want to see student memorize facts. We wanted them to experience the relationships, explore the different figures and circumstances, and draw conclusions about the significance of their observations. The Common Core enabled us to do that.

It was a lofty goal. I thought we tried our best this year. We didn’t do all that we were hoping. Our course didn’t live up to my standards. The students still memorized. The students didn’t explore enough. I told my students too many things. They told me too few. (Dan Meyer (@ddmeyer) would say that I was “too helpful.”) Needless to say, we have some work to do and I look forward to all of you coming along side of me for year two. Your support has been unbelievable so far.

Common Core puts a premium on student-to-student discussion

Common Core puts a premium on student-to-student discussion

So that’s that. The 2012-2013 school year is over and with it, the first try at creating a geometry class filled with patient problem-solving with strategic use of a variety of materials, student-to-student geometric arguments making use of geometric modeling and repeated reasoning. Those are our targets. I get 12 weeks to catch my breath and version 2.0 gets released to another collection of eager young minds.

And to me, that’s the essence of Common Core.

I understand the political conflict that exists when a single set of educational expectations are being enforced coast-to-coast. There are a lot of different ideologies, a lot of different beliefs, a lot of different communities. There isn’t much hope of finding something that everyone is excited about.

But, Common Core or not, a class depending heavily on patient problem-solving with strategic use of a variety of materials, student-to-student geometric arguments making use of geometric modeling and repeated reasoning, is something that I suspect most people can get excited about.

The Best Classroom Management (updated)

on December 18th, 2012, I posted “The Best Classroom Management” which said this:

For the last couple of years, when I get asked about my classroom management, I’ve responded:

The best classroom management is a meaningful curriculum.

I love that line. What I want to know is if it’s a quote that I have stolen from someone that I haven’t remembered or recognized.

Can you help me out? Are you aware that the above line belongs to someone else? Let me know in the comments so that I can start giving credit where it’s due.

Well, it turns out that someone else said that and said it better than me (that is a frequent occurrence, quite frankly).

Paul Barnwell (@MindfulStew) is a teacher in Kentucky and wrote a fantastic piece on his blog Mindful Stew called “The Great Discipline Conundrum” in which he lays out some basic conflicts in looking at school discipline.

Toward the end of the piece he states:

So what is my approach to discipline?  Working at a variety of schools, I’ve found that if I focus most of my energy on building relationships with students, crafting engaging lessons, and  practicing class procedures, then I’ve usually avoided major class disruptions.  I take great pride in trying to connect with students across races and socioeconomic groups.

I don’t buy the argument that it’s the students job to sit there, be quiet, and learn.  Sure, there is always a student who is going to pop off no matter what I do.  I’ve had students who are bipolar, been rape victims, and are hungry when they enter the classroom, among other conditions I can’t control.   All I know if if I sit back and pass out work packets and expect students to comply, I’m putting myself in a tenuous position.

Like I said, he said it better than me.

Although to be fair, my little quote would fit better on a bumper sticker…

… for what that’s worth.

Reflecting on the Common Core…

photo credit: flickr user "Irargerich" - Used under Creative Commons

photo credit: flickr user “Irargerich” – Used under Creative Commons

A lot has been said about the Common Core State Standards in the last year. Some of it has been by me. Some has been by guys like Glenn Beck who is not a big fanMost (if not all) states have some sort of a “Stop Common Core” group. There is even a #stopcommoncore hashtag on Twitter that turns up quite a few results (although some use that hashtag as a means of highlighting objections in the arguments of CCSS opponents.)

The pub isn’t all negative. Some groups, The NEA among them, have come out in favor. Phil Valentine has some good things to say in support.

It is possible that both sides are probably overstating the impact that the CCSS will have. That being said, I will admit that I have some opinions on the CCSS. This year is our first year introducing a new geometry curriculum that we designed around the CCSS. I’ve written a few pieces before this one that have chronicled my journey through a CCSS-aligned geometry class. For example, I’ve documented that the CCSS places a greater emphasis on the use of specific vocabulary that I was used to in the past. I have also discussed (both here AND here) that the CCSS has present the idea of mathematical proof in a different light that I have found to be much more engaging to the students.

As I read the different articles that are being written, it seems like the beliefs about the inherent goodness or badness of CCSS has a lot to do with how you view the most beneficial actions of the teacher and the student in the process of learning. It’s about labels. Proponents call it “creativity” or “open-ended”. Opponents call it “wishy-washy” or “fuzzy”.

I suspect they are seeing and describing the same thing and disagreeing on whether or not those things are good or bad.

To illustrate this point further, a “Stop Common Core” website in Oregon posted a condemned CCSS math lesson because the students “must come to consensus on whether or not the answer is correct” and “convince others of their opinion on the matter.” The piece ends with “What do opinions and consensus have to do with math?”

The authors of this website are objecting to a teaching style. They are objecting to the value of a student’s opinion in the process of learning mathematics. Fair enough, but that was an argument long before the CCSS came around. I can remember heated discussions during my undergrad courses about the role of student opinion and discussion. (My personal favorite was the discussion as to when, if ever, 1/2 + 1/2 = 2/4 is actually a correct answer. One of my classmates rather vehemently ended his desire to be a math teacher that day.)

The CCSS have become a lightning rod for a ton of simmering arguments that haven’t been settled and aren’t new.

Consensus-building and opinions in mathematics vs. the authority of the instructor and the textbook. Classical literature vs. technical reading. The CCSS have woken up a lot of frustrations that are leading to some high-level decisions such as the Michigan State House of Representatives submitting a budget that blocks the Department of Education’s spending on the CCSS.

It is a little strange thinking that I am making a statement in a fairly-heated national debate every time I give my students some geometry to explore, but it seems like I do.

And I am prepared to make that statement more explicitly as I continue this reflection.

Reflections of a teacher who taught alongside Jim Boehmer

By the time I got hired at Pennfield, Jim Boehmer had been working there for quite a few years. He was a math teacher, like me. I was a 25-year-old novice. He wasn’t any of that. He was an experienced educator. And that was the perfect word to describe Jim… educator. I knew him almost 5 years and he embodied the label “educator” as well as anyone.

It’s easy for me to say nice things about Jim. He and I saw eye-to-eye on a lot of things. School things, home things, life things.

Jim and I were part of the District Math Team which included he and I and several of our administrators. One year, the group would meet monthly at a different school in Battle Creek with some other groups from the county. Jim and I would ride together. He would always drive because I get lost in Battle Creek. For that whole school year, once a month, Jim and I got 30 minutes to talk. Jim and I got to know each other well during that time. We talked about all sorts of stuff: school, sure, but also sports, politics, God, our similar faiths, singing, technology, theatre. It was enjoyable. We could disagree peacefully, but that didn’t happen very often.

What I did find out about him was that he was a thoughtful man who desired excellence. In the time I knew him, he never quit tinkering with his teaching style trying to find the formula that would maximize authentic student learning. He didn’t want to see his students simply pass tests. He wanted them to learn. He wanted them to enjoy real success. He knew his role in that. He was always trying to find activities that would engage the students. If you want, you can read about one of my favorite of his activities.

One more story: When the time came to reform the Algebra I classes to align with the Common Core, he and I sat down together and set out to realign the entire course. We began to review the literature and our resources and decided that we needed to create quite a lot of material… not only because we wanted them to align, but because our textbook didn’t impress us… and he wasn’t interested in a good-enough Algebra class. If it wasn’t excellent, he wasn’t done with it.

And yesterday he passed away.

And we have a giant hole to fill. But I’m sure my story isn’t unique. Jim was authentic and I’m not special. Who Jim was to me he was to so many others. And isn’t that all you can ask? Within my faith, we have a phrase that we use to honor someone special who has passed. We say, “Memory Eternal!”

If Jim was to so many what he was to me, the memories of him won’t be fading anytime soon.