Tag Archives: education

Motivating 1/x

We’re deep in a functions unit in FST (year two of a decelerated Algebra 2 course), and I love it. I love the concept of a relationship that takes inputs and produces outputs. I love visualizing functions with graphs. I love that functions feel natural and intuitive. I’m trying hard to share this enthusiasm with my students. They’re doing well with it so far, and it’s interesting to see how they think about functions.

Last class, I wanted to introduce f(x) = 1/x. I love this function. I love the discussions about division by zero and division by really large numbers and how the graph represents those ideas visually. My colleague shared a fun investigation with me, and I am so glad that I tried it out. At first I was hesitant because I know that I don’t explain directions well, but I focused on being very explicit and modelling each step. The kids investigated the breaking point of spaghetti. I wish I had some photos, but the students placed a dry spaghetti noodle over the edge of the table and hung a paper cup on the end of the noodle and added pennies one by one until the noodle snapped. The fun factor was definitely there- the kids enjoyed predicting when it would snap and liked watching the pennies crash to the floor.

Besides being fun, the activity modelled the function effectively. The kids recorded their data (length of spaghetti vs number of pennies), and I used Desmos to display some class data.

hyperbolaPenniesSpaghetti

Voilà, a hyperbola. The investigation gave the kids a good understanding of how the function behaves and why the graph looks the way it does. In retrospect, I should have done more of a “Noticing and Wondering” activity with the graph, but instead I just asked some questions like “What happened as the length of the spaghetti got longer?” and “What happened if the length of the spaghetti was really small?” which probably did too much of the thinking for them, but oh well.

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Filed under FST / Algebra 2, fun, graphing, group work

Keep Yo Head Up

I came home from school on Tuesday and told my boyfriend that I wasn’t going back. Of course, I was overreacting to one bad day, and after a good night’s rest, I went to school as normal the next morning. The rest of the week went great, so there you go. It’s so easy to focus on the negative and forget the positive that I have to continually remind myself of what’s going well in my classroom, or I’d never make it.

Last week, my administration granted me the privilege to watch an experienced colleague teach a class. I watched a Biology class that was so well taught that I left feeling inspired. I used to think classroom management was my biggest issue, but after watching that lesson, I realized that time management might actually be my biggest problem. I wasn’t managing time at all. (That sounds crazy, but it really shouldn’t surprise me. I won “Most Likely to be Late to their Own Funeral” in high school. The clock in my car is always set to a random time. I never rush anywhere or feel a sense of urgency.) For example, I used to set the kids off on a task with absolutely no time-frame. I guess I thought that every kid had to finish before moving on.

To avoid wasting so much time, I need to keep things moving fairly quickly. I need to set time limits on activities. I can’t have kids messing around because they think they have the rest of the class period to accomplish something that really only needs 10 minutes of concentrated effort. I need to have tasks for kids who finish early, and I need to recognize that there’s not enough time to wait for 25 kids to each discover something on their own. One thing that has been effective for me is to have a list of two things that students need to accomplish within a given time-frame, with the second item being one that doesn’t require all students to get to before moving on.

Lessons learned there for sure, and that’s a good thing. Other good things: Had a good 1-1 conversation with a student who acts out in my class and likes to challenge me. Two students I’ve been trying to get a hold of for awhile finally came in after school to take missing tests and even waited 10 minutes extra for me because I was in a meeting that went long. Did a a fun investigation in FST (blog post coming soon, hopefully). Taught some algebra-heavy topics in Geometry and no one threw a fit, including me.

Did a long problem in Geometry, and at the end a student pointed out “that answer doesn’t make sense”. She was right, it didn’t, but I couldn’t see a mistake in my work. I stepped back and told everyone to try and find the mistake. Finally, one of my favorite delinquents (he’s not really) piped up, and said “The point is labelled wrong in the diagram. It should be (1,0) not (0,1).” Dang it, a typo on the handout. But all was well because I got to model mistake-making in front of my students. Then I had to go through and basically re-do the entire problem which was also good because the students got to see it again and hopefully understand better.

Oh yeah, and I’m the advisor of the newly formed snowboard and ski club. Held a brief meeting in my classroom after school just to see how many people were interested, and within a few minutes, my room was packed. It was awesome. One of my Geom kids has stepped up as the student leader of the club, which is exactly what I needed.

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Filed under classroom management, productive struggle

Always, Sometimes, Never

I debated some Always, Sometimes, Never statements with my Geometry kids today. In groups, they had to choose the word that they thought went in the blank, as well as draw a picture to explain their choice.

Some example statements (taken straight out of our textbook):
Two planes ________ intersect in a line.
Lines ________ have endpoints.
Lines that are not parallel ________ intersect.
Two points _________ determine a line.

That last one created some interesting discussions, particularly in my last period. Many students wanted to put Sometimes in the blank. I didn’t look at the textbook’s answers, but I assume the authors wanted Always in the blank.

Why did so many students think Sometimes? Well, I think the statement was kind of confusing to them. What does it mean to “determine” a line? Does “a” line mean one line or does it many any line? I tried to resolve the matter by putting two random dots on the board and drawing a line through them. “Look, I can draw a line connecting any two points.” Not particularly convincing.

The students then told me to draw a line going through each of the points (parallel lines, for example). “See,” they told me, “there’s two lines, not a line.” I didn’t really know how to respond to that. I told them yes, I can draw different lines through each point, but only one line will connect them.

Well, I think I convinced them that any two points could be connected with a line, but we just left the Always, Sometimes, Never question unanswered. Which is okay. Of course, some kids insisted, “But what’s the answer?” and I replied, “Well, I think it’s Always, but I don’t think it’s totally clear.”

Perhaps the answer would have been less ambiguous if the original statement was Two points can _________ be connected with a line. But that statement seems way less powerful. So now I am intrigued by the word “determine”. I definitely think it’s important. It’s hard to explain to the kids what is meant by “determine” though.

One instructional difference I would have made during the activity was to require new people to be the writer and the speaker for each statement. In a couple groups, it was very obvious that two or three students were doing all the work while the others checked out, so some sort of rotation would have been smart.

I want to start the next class by playing Sarah Rubin’s Draw It game because some of the drawings I saw today were definitely off the mark, but that’s okay. Visualizing lines and planes and space can be tricky. I love seeing their eyes widen when they begin to “see” it.

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Filed under conversations, Geometry, group work

Factor Craze

I didn’t have the greatest teaching day today, so I thought I’d try to remind myself that I CAN do this teaching thing by describing an activity from a few days ago that was successful.

Factor Craze, which I think I found via Fawn Nguyen, is one of NCTM’s monthly “Problems to Ponder”, and it asks:
Which numbers have exactly three factors?
Which numbers have exactly four factors?
Which numbers have exactly five factors?

This problem was a great introduction to factoring with my FST (2nd half of an extended Algebra 2) kids. They saw factoring last year, but this year I wanted them to really understand how they were coming up with the equivalent expression instead of following a list of steps from the teacher. So I used Factor Craze to spark some conversations about factors.

I have my students seated in groups, but I had them think on their own for a minute before working with their group. I actually started with the question Which numbers have exactly two factors?, which may seem rather elementary for high school juniors and seniors, but as I suspected, many had very little knowledge or experience with the concept of prime numbers.

Most groups started by writing down examples of numbers that had the required number of factors, but I prompted them with, “What’s a way to describe ALL numbers that have exactly ___ factors?”. All groups eventually came up with prime for exactly two.

When they got to exactly three, most groups found out that 4 and 9 worked. I asked them if there was anything special about numbers 4 and 9. “Oh, oh! They’re perfect squares! Perfect squares have exactly three factors!”

So I respond with, “Do all perfect squares? What about 16 and 25?”

“16 doesn’t work. Oh. But 25 does!”

So I say, “Nice. So some perfect squares but not all perfect squares. What type of perfect squares work?”

And so on. Most groups figured out that squares of prime numbers have exactly three factors. Only one group in each class was able to delve into exactly four factors before we ran out of time.

I really liked how this problem posed a challenge for every student. For some, just remembering what prime numbers are like was a challenge. For others, it was recognizing a theme for exactly four factors. Either way, all students were developing an understanding of factors.

We later moved on to greatest common factors and factoring expressions, and I think laying the ground work with Factor Craze made a difference.

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Filed under FST / Algebra 2, group work, productive struggle

My Automathography

I’m taking Justin Lanier’s smOOC called Math is Personal, and one of our first assignments is to write our “automathography”. So here’s mine. Enjoy!

Mary’s Automathography

I love math, but I didn’t fall in love with it until college. I was good at math in high school, but I was good at all my classes, so nothing stood out about math in particular. I definitely had a fear of getting the wrong answer in math class, and I was happy to just follow the procedures given to me by my teachers. At this point in my life, I don’t think I understood what mathematics actually was. I won the conference quiz bowl in math my senior year, and it was great to get that recognition, but I graduated high school thinking I would study chemistry in college.

I soon discovered that I did not enjoy working in the lab, but that I did enjoy my math courses, so I ended up majoring in math. I went to a huge university (40,000+ undergraduates), so my first two years of math classes consisted of lectures with 300 students. Despite this, I found myself completely inspired by the professors. I was enamored with how passionate and genuine they seemed. In other subjects, I felt like the professors and TAs were egotistical or arrogant. In contrast, everyone in the math department seemed friendly and easy going. I’ll always remember when one of my calculus professors introduced Euler’s identity. His voice wavered, and I thought he might even cry when he described how this one equation related the most important numbers in mathematics.

Even those first few years of college, I was still focused on answer-getting. This quickly changed when I started taking courses like Real Analysis and Modern Algebra. In these classes, I was finally challenged to think for myself. There were no recipes to follow, and it was completely up to me to decide how to prove or demonstrate something. It was both terrifying and liberating. Math became a creative endeavor for me, and I loved it. I truly came to understand and appreciate Georg Cantor’s quote: “The essence of mathematics is its freedom.”

Besides the creative aspect of math, I also thrived on its collaborative aspect. Getting to know the other students in my classes was so much fun, and struggling with them on math problems late into the night will always be one of my favorite college memories. I also always appreciated how there wasn’t a competitive atmosphere in math, compared with most of the science classes I took. Simply put, I learned so much from doing and talking math with my peers. I became more confident and began to embody the mathematical habits of mind.

In particular, I will never forget the group I worked with in Real Analysis. The professor assigned problems every class which were due the following class (this course required more of my time than any other), so the five of us would get together almost every day, sometimes for several hours, to struggle through them. We would meet in the student union in the evenings, staying later than everyone else and having conversations about math or maybe not about math. Before class, we would meet in the math library to share any last minute insights, often getting looks from others for being too loud. Naturally, a strong bond formed between the five of us. On weekends (or Thursdays, or whenever we could no longer stand to stare at our papers) we would go out and get drinks together.

The experiences I had in classes like Real Analysis really transformed my idea of math. I learned the value of productive struggle and collaboration. I learned how to be creative in math and make it my own. I really felt mathematically strong at the end of it all.

Fast forward to the present- five years after that Real Analysis class. I am now about to start my first-year teaching high school math. I hope I don’t suck.

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Filed under culture, fun, productive struggle

Exit Tickets and Group Work Norms

Ever experience kids packing up 5 minutes before the bell? I definitely did during student teaching, but half-way through, with support from my cooperating teacher and help from another math teacher, I decided to implement exit tickets. It definitely made a difference. Kids worked until the end of class when I handed out their exit ticket sheets and put the prompt up on the Smartboard. It was great. The other math teacher who did exit tickets used a weekly sheet with a space for each day’s answer, so dutifully followed her lead, but I didn’t really enjoy keeping track of the sheets for a whole week and it was a pain to pass them out at the end of each class. The kids complained that it took too long to get their sheets back so they didn’t have enough time to answer the question.

So anyway, this year I plan to just have a bunch of half-sheets of paper printed out for exit tickets. I am wasting more paper this way, which is a concern of mine, but it’ll have to do for now. On the back of the half-sheet there is a participation reflection. I’m  focusing on group work and creating healthy math culture in my classroom this year, so I want to remind the kids of our group work norms every day, and I want them to do some reflection on the day, hence the three questions on the back of the exit ticket.

I really like all of the norms I’ve decided to use, but unfortunately there are twelve of them, which is probably too many. I should try to shorten the list, but I don’t know which ones to give up. They’re all important to me!

Well, here’s the file with the exit ticket on the front and the reflection on the back. Nothing fancy, but check it out, and I’m interested in hearing your thoughts. Do you use exit tickets? How do you implement them, and what do you like about them? What classroom norms do you use? How do you get your students to think about your norms?

(I don’t have any word processing software on my computer, so I just use google docs for everything, but in the process of uploading my documents to scribd, the spacing gets a little weird, but you should still get the idea.)

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Filed under culture, formative assessment, group work

NYC trip and some thoughts on INBs

(Great acronyms in the title.)

I just got home from a fantastic four day visit with my sister in New York City. I live in a much smaller city myself, so I wasn’t sure what I would think of NYC, but I loved it! I loved all of the beautiful people, the amazing and delicious varieties of food, and the daily hustle and bustle. It was so much fun hanging out my sister and seeing where she lives and works. It was also my first time in the Big Apple, so of course I did all of the fun tourist stuff.

Central Park

Central Park

Brooklyn Bridge

Brooklyn Bridge

Statue of Lib

Statue of Lib

My sister and I also went to the Museum of Math, which I highly recommend. All of their displays are very interactive and let you experience and discover the math. They’re also super fun! We spent almost four hours there! Going with my sister was perfect because she has a great natural curiosity for math even though she went the finance and accounting route. She can solve all those fit-these-shapes-into-this-box and disconnect-this-metal-loop-from-this-other-metal-thing puzzles that I never have the mind for.

Of course the Math Museum has pi door handles.

Of course the Math Museum has pi door handles.

Overall, it was an awesome trip, but it’s so nice to be back home in spacious, natural, beautiful Wisconsin. #tmc14 took place over the same weekend, so while I was spending 24 hours each way on the train to and from NYC, I got to catch up via Twitter on all the excitement and mathematics happening in Oklahoma. Hoping to attend #tmc15 next summer!

So now that I’m home, I’m thinking about what sort of procedures I want in place this year in my classroom. I definitely want to do something with interactive notebooks (INBs), a popular topic in the MTBoS these days.

There are people with far more experience than me (which is zero) who write about INBs, so definitely check out Math=Love, Kalamity Kat, and Infinite Sums for great ideas and advice.

My reasoning for INBs is to help my students process information, organize their work, and have a resource that they can refer back to. Even when I student taught an Advanced Math 2 (like a Pre-Calculus Honors, maybe?) class where the kids furiously took notes on their own, I still think they needed help with organization and actually getting something useful out of their notes.

I’m planning on implementing a very low-maintenance version of INBs. Foldables are definitely NOT my thing and there’s no way I could keep track of a table of contents, let alone make my students do it. So really these are just going to be regular old notebooks. NBs, if you will.

Basically, my plan is to have my students put all the math they do into their notebooks. That’s… it. Maybe this is too unstructured (I’ll find out), but I really don’t care what the format is or how pretty it looks, I just want them to record the math they do in an organized manner and all in one place.

I’m going to require that they write the date and the topic on the top of each page. Below that they do the warm up. Below that they show the work for whatever activity or investigation we do. Below that we sum up the investigation or do notes or additional examples. Below that goes homework or some other sort of output. That’s the plan. I’ve requested some glue sticks so that the students can paste in any handouts. Yay, glue sticks.

It sounds simple. Hopefully it is. I plan to do regular notebook checks so that they know I’m serious about them picking up their pencils and doing the work. Accountability for them, accountability for me. Can you tell I’m a first year teacher? I’m constantly worried that they’re not going to take me seriously. Well, I guess I just better be serious when I need to be serious, right?

Along with the notebook, I’m requiring a folder for graded work or extraneous handouts. Hopefully I won’t be handing out much that won’t go in the notebook, but they need a place to store quizlets (my department’s name for our formative assessment), instruction sheets for projects, practice tests, etc.

Eh. I’m at the point where I have all these PLANS but have no idea how they’re actually going to work until the school year starts. The anticipation is killing me!

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Filed under notebooks, planning, travel