Trying to find math inside everything else

Posts tagged ‘MTBoS’

Integral Limit Game

This year when I was in my intro to integrals unit, I tried to look back at this blog for the second integral game I know I played (besides this one), and saw I hadn’t blogged about it. I had tweeted about it, but now I’m thinking, you know, I should, uh, archive things that I only tweeted about in a more permanent place, in guess Twitter doesn’t last much longer.

Anyway, this game is based on The Product Game, with the same structure of turns – players take turns moving a token on the bottom rows, that then determine which square in the top section, where the first player to get 4 in a row is the winner. (I usually have students play in teams of 2, but I’ll keep saying “player” go forward.)

The idea here is that the bottom rows represent the limits of a definite integral. One player plays as the Upper Limit, and the other as the Lower Limit. Once both limits are placed, the player who most recently went calculates the value of the definite integral on the accompanying graph, then covers the square in the top section with the area. (Remember that if the lower limit is greater than the upper limit, the sign is switched!)

Making the function that would give a variety of answers was a fun challenge. After coming up with a graph I thought looked good, I wound up making an excel sheet to calculate all the possible definite integrals to see how balanced it was, and adjusted.

I’ll include that excel sheet as well, as it’s useful for checking answers (as a teacher), although of course each team should be checking each other. After doing a bunch of different integrals on the same function, students often realize they can use their previous work to help them find new answers, reinforcing the cumulative nature of integrals.

Files

integral-limit-game-calculationsDownload
integral-limits-gameDownload

Category:

Calculus, games

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BYORF

One of the other games I made this year was during our rational functions unit: BYORF, which stands for Build Your Own Rational Function. (This was originally a placeholder name, but it kinda grew on me.)

BYORF is a drafting game, a la Sushi Go or 7 Wonders. You play over 2 rounds (because that fit best in our 45 minute period – 3 rounds might be better with more time?), drafting linear factor cards to build into rational functions that match certain criteria. Here’s an example of a round between two players.

In this example, the left player used only 4 of their linear factors (as you don’t need to use all 6). Then we can compare each of the 5 goal cards, which are randomized each round. L has 0 VA left of the y-axis, while R has 2, so that is 3 points to R. L has a hole at (-2, 1/3) while R has a hole at (-1, -2), so L gets 5 pts. They both have a HA at y=-1, so both score those 4. Then we have the two sign analysis cards, which score points if you have that formation somewhere in your sign analysis. R has the first one (around x=3) and both have the second one (L around x=1 and R around x=-3). So after one round, both players are tied with 11 points.

I hope that gets the idea across. The fact that students need to check each other’s work to make sure the points are being allocated correctly builds in a lot of good practice. After we played the game, I did a follow-up assignment to ask some conceptual questions (which is where the above example comes from). I’ve also attached that here.

I hope you have some fun with BYORF!

byorfDownload
byorf-rulesDownload
byorf-follow-upDownload

Category:

games, pre-calculus

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Letter Scramble

In our combinatorics unit in pre-Calculus, we tend to look at every problem as a letter rearrangement problem. This lets us move beyond permutations and combinations to model any problem involving duplicates. I wanted to build a game that had the students quickly calculate the number of arrangements for a given set of letters, so I came up with Letter Scramble.

The idea behind the game is that students have a set of goal cards with the number of arrangements they want to reach, and a hand of letter cards. On each player’s turn they can play a letter card to change the arrangement, and thus change the total possible number of arrangements. (They can also skip their turn to draw more goal cards, a la Ticket to Ride.)

I calculated all the possible answers you could get using 7 different letters and up to 8 slots, including if some of those slots are blank (and thus make disjoint groups), then determined which of those answers repeat at least once, and assigned them scores based on that.

Possible numbers of arrangements on the left, how many times they repeat in the second column, and the points I assigned them in the third.

One interesting thing about the gameplay is it promoted relational thinking. Instead of calculating each problem from scratch, you can based it on the previous answer you calculated. (So, for example, if the board read AABBCD, that would be 6!/(2!2!) = 180. But if you change that to AABBBD, one of the denominator’s 2! changes to 3!, which is the same as dividing by 3, so it’s equal to 60. No need to calculate 6!/(2!3!) directly.

Some examples of scored goals:

1! * 3!/2! = 3
5!/(2!3!) = 10
letter-scramble-rulesDownload
letter-scramble-cardsDownload

Category:

games, pre-calculus

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The Integral Struggle

I had an extra day to fill for one of my sections of calculus, thanks to the PSAT, so I set about thinking up a game I could play. I gotta tell you, as much as there is a paucity of good content-related math games out there, it’s extra so as you move up the years in high school. I can still find a good amount of algebra and geometry games online, but Algebra 2? Precalc? Calc? Fuhgeddaboutit. Well, I’m teaching both Pre-Calc and Calc this year, so I guess I’ll just have to make them myself. (I brainstormed two more during said PSAT, so those might happen soon enough.)

Starting layout for The Integral Struggle

We just covered function transformations and how they affected integrals, so this game hits on that topic. (Yeah, we’re doing integrals first.) Here’s how it works: there are 3 functions/graphs, each of which has a total area of 0 on the interval [–10,10]. One team is Team Positive, and the other is Team Negative. Teams take turns placing numbers from –9 to 9 that transform the function and therefore transform the area. Most importantly, they also place the numbers in the limits of integration, so they can just look at a specific part of the graph.

With three functions, it becomes a matter of best of 3 – if the final integral evaluates to a positive number, Team Positive gets a point, and vice versa. If it’s a tie at the end of the game, because one or more of the integrals evaluated to 0 or were undefined, then redo those specific integrals. (I discussed with the students whether it would be better to have it be the total value of all 3 integrals instead of best of 3, but I think that makes the vertical stretch too powerful.)

That’s it! Let me know what you think. Materials below.

the-integral-struggle-rulesDownload
the-integral-struggle-game-boardsDownload
the-integral-struggle-graphsDownload
the-integral-struggle-numbersDownload

Category:

Calculus, games, math

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Remote Teaching Math Games

I love playing math games with my students, but it’s been hard with remote teaching because so many games require physical objects. I’ve been able to play a bunch using a shared whiteboard, but there’s a limit to how many work in that method. I just discovered http://playingcards.io, though, which is a platform for playing card games online with anyone. They have built in games, but you can customize your own. I’ve taken several from the #MTBoS and my own blog and made them.

To use these files, first create a custom room. Then enter edit mode:

In the Room Options Menu, you can import a file. So download the file you want from here and import it there and the game is ready to go. You can then share the room code with students, and you can even make multiple rooms for different groups of students and jump between them.

For each game, click the image to go to the original blog post, and the title for the pcio file.

Games

Factor Draft

Math Taboo

Fraction Catch

Trig War, Log War, Inverse Trig War

Integer Deck

This last one isn’t a game so much as a resource for many other math games. It’s an integer deck, consisting of cards from -12 to 12 of each suit (and an extra 0 for each.) I colored the suits using a colorblind-friendly color palette, on top of the symbols. You can easily edit the deck (enter edit mode, then click on the deck) to remove cards from the deck or change the particular cards. It can be used for a lot of games – and helps avoid the problem of kids wondering what J, Q, and K mean. It would be a good deck to use for, say, Fighting for the Center or these Integer Games.

Category:

games

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Things I’ve Changed This Year

When I think back on my first five years of teaching, I can identify big initiatives that I took and tried each year.

  • My first year – well, it was my first year, everything was new. But I was implementing stnadards-based grading as sorta my big thing.
  • My second year, I was very focused on interdisciplinary work, creating cross-curricular lessons with my colleagues, and implementing all this new 3-Act and other stuff I had just started to find on the internet.
  • My third year, I structured my class around math labs and introduce the interactive notebook after I learned about it at TMC12.
  • My fourth year, I overhauled my grading system.
  • My fifth year, I introduced the Standards of Practice portfolios as a way to grade on those standards and, thus, have them be valued in my class. To go along with that, I had a new way to give feedback, instead of writing grades on assignments.

And this year? My big initiative? I don’t have one. It’s felt weird. Every year these big things I was trying and perfecting felt like steps I was taking towards becoming a better teacher. And if I didn’t have one this year, was I stagnating?

No. (I say it confidently now, but it took a lot of reminding myself.) First of all, my big initiative this year was teaching Calculus and Geometry for the first time. I had taught Algebra I for the whole first 5 years of my career, and the bulk of my student teaching as well. Despite the switch to the Common Core curriculum, I was still very familiar with the ins and outs of the material, and that let me focus on other things. But teaching a new course is a lot! And two, twice as much!

But, even with that…I still tried new things, tuned things, had small initiatives. And these things matter! So I’m writing a list of new things I’ve done, to remind myself. And also to keep looking forward, for new initiatives – as Black Widow says, “There is no mastery, only constant improvement.”

  • I greet my students at the door every day with a high five.
  • With the other hand, I have them pick a card so they can find their seat with their visibly random grouping.
  • I put up new boards on my walls to have even more surfaces for the students, and designed lessons around using them, facilitating group collaboration more than usual.
  • Instead of saying “Ladies and Gentlemen” to address the class, I now say “Mathematicians” (or “Computer Scientists”), to keep a gender neutral term.
  • I swapped out the Name spot on my assignments for one that says Mathematician.
  • I had up a “Good Questions” bulletin board, after going to Rachel’s session on better questioning (couldn’t find a link for this one) for a while during the year.
  • And I’ve continued the initiatives from the last two years, which were raw in idea but are now becoming fully realized structures, as I find better and more sustainable ways to do them.

I bet you’ve done a lot this year, too. More than you realize.

Category:

reflection

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The MTBoS Genome Project

Have you ever listened to Pandora and wondered what method they used to determine what songs to play for you? I did and remember writing a research paper about it back in grad school.

Pandora makes use of something called the Music Genome Project. Professional musicians will actually listen to every song in their database and tag all of the songs along different dimensions – timbre of the instruments, vocal type, volume, bpm, etc. Each song then gets a vector associated with it where each dimension is one of those categories.

Then, when you put in a seed song, Pandora will calculate the closest songs to your seed, basically using the distance formula in hundreds of dimensions. (There’s some weighting and tweaking, of course, but that’s the core premise.)

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At the Sunday My Favorites session of TMC14, Bob Lochel and Megan Schmidt show us how to find our closest buddies by filling out a survey about what movies they like. Then they calculated the correlation coefficient and the people who correlated the most were the best friends of the pair.

On Saturday, I was talking with someone (I think it was Matt Baker) about how to help people get into our community. He mentioned that while there are a lot of good ideas out there, the ideas that resonate the most with him are the ones that comes from they people he most identified with – whose teaching style was most like his. I had the idea that we could somehow make a survey that a new person could fill out and it would give them a personalized output of Twitter accounts and blogs to follow – a somewhat advanced version of the category lists we made here during TMC12.

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I want to make this, but I need help. What are the dimensions that we should ask about? What are important aspects of your teacher identity, and what are some of the things that make you feel on the same wavelength as another teacher in the MTBoS? Please let me know so I can start compiling these dimensions and building this.

(Also, if there is anyone more skilled in programming who is willing to help me, ping me.)

Category:

Explore MTBos

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TMC14

I’m on the plane on my way back from Twitter Math Camp ’14, and it was, as it was the last two years, an amazing experience.

I’m trying to process everything – of course, a lot of that is looking through all of the resources I saw, which I can’t do on the plane. More of it is writing blog posts about specific things I want to talk about – those will come later.

But i want to write about, perhaps, not #whyMTBoS but #whyTMC. Maybe a few short vignettes:

– In my algebra morning session, we had a workshop where we created assessments/tasks for certain units (you can find those here) – when I pulled up the exam I wrote for functions last year, one person told me we could just use that as a product, they liked it so much. We didn’t – we made something even better than what I made myself.

– After Steve Leinwald’s keynote on Thursday full of spit and fire, I felt really energized, even though I had been tired just before.

– Thursday night a small group of people going to get BBQ snowballed to about 30 people, and no one was bothered by that – everyone was welcomed. The restaurant was super accomodating and even made a separate check for everyone (a theme during the trip) – though that wasn’t necessary, as the wonderful Jason covered all of those bills.

– On Friday Dan expanded all of our minds about the size of our community and how much more there is out here.

– Throughout the conference different people gave us “sneak peeks” on things they were working on, and we could get to see inside the process of making these cool things.

– On Friday night I was up until 230 having deep conversations and really connecting with people. It made me realize how much I’m affected by the negativity and positivity of others – TMC is so positive, my coworkers are sometimes negative, and I need to not accept it but work to change it, if I don’t want to absorb all that negativity.

– On Saturday I saw Mary Bourassa and Alex Overwijk present their spiraled task-based curriculum. I was amazed and wanted to be there, but I was scared about it. Alex said in the session that “When you try to make small incremental changes, it is so easy for the kids to pull you back down and flip back to what you’ve always done. But if you start with the huge change, even when you slide, some of that change remains.” I thought of people like Lisa who are worried about changing and how maybe those words might help.

– The last thing I did on Saturday was to take place in a body-scale number line exploration led by Max Ray and Malke Rosenfield I got to share my insights and experiences with number lines that others may not have had, I got to see it in other people’s eyes, and I experienced new revelations and am excited to dive into them deeply.

This last things leads me to my final thought. During our work with the number line, Malke constantly pushed back – what are we actually gaining my working with the number line using our bodies, instead of just paper and pen? It pushed us to keep developing new insights and sharing them until one moment I heard Malke make an involuntary gasp – there was a moment of breakthrough, one we never would have had without using our bodies.

So you could ask the same question – what do we gain from using our bodies to meet in person at TMC, instead of just writing to each other as we do in the MTBoS? There’s this energy that infuses all of it that you can’t feel remotely, these deep experiences and quiet moments that can’t be done publicly, this sense of connection that makes all the other work we do more powerful.

There’s a reason I am always following so many more people after TMC – I need that connection and once it’s there, I want to keep it going and make it grow. And even as there are more and more old friends I want to see at TMC and so little time, I still somehow make so many new friends. And that’s why.

Category:

conference, TMC

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Growing into the Math Teacher I Want to Be

I just submitted my application to be a Math for America Master Teacher. For the personal statement, it asks me to write about how I am continuing to grow into the mathematics teacher I want to be. Of course, my main answer was #MTBoS, because, well, we else am I going to experience so much great math teaching? (Besides at MfA itself, of course, but there’s a lot of overlap, naturally!) I even had a problem when I had to submit a lesson plan, because so many lessons I’ve done this year either started with an idea from someone else’s blog or was improved by someone else. So what lesson plan can I really call mine?

Eh, I’ll just post what I wrote:

 

When I think about what type of mathematics teacher I want to be, I think about all of the wonderful mathematics teachers I communicate with regularly through Twitter. I proudly consider myself to be a member of the Math TwitterBlogosphere, or MTBoS. Every day I read the blogs and thoughts of a wide variety of teachers from across the country (and a handful of international ones). I take the ideas that I like, I give input of my own through comments and tweets.

More than just passive reading, though, is the collaboration I take part in. I have my own blog where I reflect on things that I have done in my classroom. I can get feedback on the things that I do so I can improve them. On top of that, every summer I attend Twitter Math Camp. (In fact, since last year I have been on the planning committee for the conference.) At this conference, Iʼve hosted sessions on making interdisciplinary topics and making educational math games. Iʼve co-created lessons with these teachers that are then spread to others beyond the conference.

Through this network I also often read many articles that can help improve myself as a teacher. Some of these articles are research – looking more deeply into how people learn mathematics. The blog of Christopher Danielson is usually very fruitful in this regard – digging down into number sense helps build a foundation that supports other mathematical learning. I also read articles pertaining to social justice issues – how race and gender have effects in both schools and the world at large, and steps I can take to make my classroom a safer place.

Math for America itself also provides many of those contacts that help me improve. I often sign up for more than one PLT a month because I get so many good ideas out of them.    I love having the opportunity to talk to other teachers about what they do. Teaching can often be an isolated experience, and just a window into another classroom can do a lot to improve someone.

This year Iʼve also tried new things to bring myself closer to my ideal as a teacher. I implemented a new grading system that focuses on growth instead of a static level – this way, even a student who is far behind can feel like they are making a lot of progress, and pushes those students on the top for more, beyond what they already did to get to the top. I am also using a new Problem-based Learning Curriculum. Mastery of procedures is not sufficient for learning mathematics – they need to build habits of mind and problem solving skills. Having a PrBL curriculum helps me get at those skills and find new ways.

Iʼm also often studying new math as well. Iʼm dating a PhD in math and I often have conversations about the new math he is studying or what he is teaching in his courses. We occasionally work out problems together that interest us, and that usually leads to a better understanding of topics I teach that I only thought I understood fully before.

That sums up my overall strategy on how to continue growing: look to exemplars, find what I like about what they do, question myself about why I donʼt like things they do, so as to not allow myself to get complacent, and always keep pushing boundaries.

Category:

reflection

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Estimation180 and Absolute Value Graphs

So as I was getting ready to teach absolute value graphs a little while ago, I came across this post from Kate Nowak about a lesson she did with it. I liked the idea but…I didn’t like the idea of having to “get my butt into overdrive” to collect data from staff and students about such a thing. I wanted a lesson for the next day, so I didn’t really have time for that.Screen shot 2014-03-26 at 6.52.24 PM

But then I thought, well, my students have been doing Estimation180 all year long. Maybe there’s a way I can use that? I even tweeted Kate about it, but was left to my own devices. (Though I suppose this is finally the write up I promised.)

 

I thought about what was different between what we’ve done with Estimation180 and Kate’s task, and then it hit me – Kate’s lesson is all about one guessing event, but we have loads of different ones. At that point we have done ~30 estimations. What if we could do some comparisons?

 

My premise was this – Mr. Stadel, who runs the Estimation180 site, wants to implement a ranking system where all the estimations are listed as “easy” “medium” “hard” etc. But how can he tell when one is hard or not? He knows all the answers, so he can’t used himself to judge. So I told Mr. Stadel that we have lots of data from our class and we could probably use it to come up with a system.

[Aside – this was the 3rd or 4th day of the new semester, and to complete the task I asked students to use the estimation sheets from the previous semester. They revolted, because they claimed I had told them they could throw those out! Which I vowed I didn’t…though, to be honest, it’s possible I did, since I hadn’t thought of this lesson yet. Luckily enough students had not thrown them out so that it could still work.]

So I reviewed what the estimations we did were and I told each group that they have to pick one estimation that they wanted to evaluate. Then they had to collect data from their classmates (and from the binders of other classes, through me) – the estimate each person made and what their error was. Once they have collected enough data, they have to make an Error vs Estimate graph and see what happens. Then I had them make some analysis on whether this counted as a difficult task or not. I didn’t have them compare graphs at the time, but I totally should have.

I think it worked pretty well and many of the students understood why it should be a V-shaped graph. They were at first surprised about where the vertex was, but then it made sense, especially comparing many different error graphs.

Estimation Difficulty Rating (Word format)

 

 

Category:

lessons, math

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