Category Archives: group work

Second semester

I’ve been enjoying second semester so far.

In geometry, we just finished up the unit on triangle congruence. I felt like it was kind of rushed, but progress was definitely made. We revisited proofs after a brief introduction back in October, and I enjoyed seeing kids reasoning again.

One practice that I used several times was having small groups write out proofs on the whiteboards and then sharing them with the rest of the class. I asked each person to contribute one thing to the board (the diagram, marking the diagram, writing the congruence statements, using cpctc, etc) which was a good way to make sure no one was dead weight. One thing I’d want to improve is what the kids in the. audience are doing while the other groups share. Too many were tuned out and I felt like I was the only one listening and asking questions. However I think everyone was listening when one student ended up saying FU is congruent to FU.

I’m now teaching stats in FST. The curriculum is… lacking to say the least. I feel like stats could be so cool, but these crappy worksheets with blurry images are pretty uninspiring. My mentor said she’d help me come up with some cool stats projects, so I have to remember to follow through on that.

Speaking of curriculum, during the last inservice, the math department met and had a pretty kickass discussion on equity in our curriculum. Nothing concrete came out of it, but it was refreshing to hear everyone’s ideas, and I was glad to learn that I’m not the only one who thinks much of our curriculum is really out-dated, non-rigorous, and inequitable. I really do feel like I’m teaching something that was written in the late 80s or early 90s. I can’t relate to it all, and I sympathsize with my students when they can’t either.

So there’s much work to be done, but so far second semester has been shaping up nicely. It’s frickin cold out and we won’t have any vacation until spring break, but I’ve been enjoying every day so what more could I ask for? I’ve been feeling pretty lucky lately.

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Filed under collaboration, Geometry, group work, productive struggle

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.


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

A Flop

Had a pretty big lesson flop today. I was frustrated at the end of the particular class, but I’ve thought about some things I can do to improve. Fortunately, I’m on an A-B schedule, so I get to re-do the lesson on Monday with another class. (My A-day kids always get the flops.)

I’m almost too embarrassed to write about the lesson because so much was wrong with it. I am tweaking every part of it for Monday. I thought about scraping the main task altogether, but it’s a good task that was ruined by poor implementation.

First, I am not going to assume the students remember how to do something even though I know they studied it last year. Flying through one example and saying, “This is familiar, right?” isn’t going to cut it. It turns out what they learned last year was a “trick” anyway, so I definitely need to do a better job explaining explicitly what is happening conceptually.

Second, I am not going to throw a handout at them and expect them to get to work. I am going to do a better job explaining the task and modeling how they should get started. I just read the phrase “model curiosity” while surfing some blogs, and I think it perfectly describes what I need to do at the beginning of a task.

Third, I’ve got to follow through on my behavior expectations. I had too many non-participating, off-task students. Worst of all, I let them behave that way. I let them get out of their assigned seats. This is my problem. I don’t like telling people what to do. I just want them to do the right thing. But I have to remember that high schoolers are still kids, and they still need guidance. Basically, I’ve got to toughen up. I’ve got to enforce my expectations.

Fourth, I want to do a better job structuring group work. I think this will also help me with my classroom management issues. I think I need to bring the groups back for a whole-class check-in more often. If there are four parts to the task, then I think I should bring everyone back together to go over each one before we move on to the next. In contrast, today I just said “do it” and consequently lost a lot of people, who never came back when I tried to go over everything at the end. So, on Monday, as students make progress on part 1, I’m going to bring us back together and have groups share. Then I’m going to explain part 2 and let them go. Then I’m going to bring them back again for a whole-class discussion on part 2. Then I’m going to explain part 3, and so on.

The tricky bit will be bringing everyone back. They’ll want to keep talking to their friends, but I need them to pay attention to me or whoever is sharing. I really need something to get everyone’s attention back. Maybe a timer, but students might work more slowly or more quickly than I anticipate. Another new teacher, who is in the English department, shared with me her method for bring everyone back. She simply says, “I need everyone back up here in 3.. 2… 1.” That sounds magical to me.

I can probably pull it off. I can do anything, right? I think what will work for me and for my students is to explain to them at the beginning what it’s going to look like. I will explain that I will let them work on part 1 for a bit, but that when I say “I need everyone back up here in 3, 2, 1” they need to stop where they are, turn to the front, and listen because we are going to share ideas at that point.

Overall, I think I need to be a better communicator. Specifically, I need to be more explicit with my directions and my expectations. More explicit with some of my explanations of content would also be good. Again, these are kids, not adults. They are learners, not experienced mathematicians. They are relying on me to communicate well.

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Filed under classroom management, group work, planning

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

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

A Favorite Math Lesson Resource

I want to share one of my favorite resources for rich math lessons: the Mathematics Assessment Project, which is a collaboration between the University of Nottingham and UC Berkeley. I love the structure of their Formative Assessment Lessons. According to the website, these lessons are designed “to reveal and develop students’ conceptions, and misconceptions, of significant mathematical ideas and how these connect to their other knowledge”, as well as “to assess and develop students’ capacity to apply their mathematics flexibly to non-routine unstructured problems, both from the real world and within pure mathematics”.

When I taught middle school summer school, I used a couple of these lessons with great success. I love how they include matching and sorting activities that require the students to come up with their own mathematical justifications. When I tried these out in summer school, I had small classes and had the students do it individually, but this year I am definitely going to follow their suggested structure of using small groups because I want to see more mathematical discourse among the kids. I also want to do a better job of creating the final product and displaying the work on posters. In my summer school class, once the kids finished sorting they would just toss the cards in the trash (or worse, leave them scattered on tables or the floor for me to clean up). So this year I will definitely use the small group format and require the posters, which will hopefully get more students to explain their thoughts and reach a deeper understanding.

Conveniently, I was able to find a MAP lesson for each of the first four units in my Functions, Statistics, and Trigonometry (FST) class. (Students at my school have the option of the Geom–>Advanced Algebra–>Pre-Calc route, or the Geom–>Algebra 2–>FST route. So the curriculum for this class is comparable to the second half of an Algebra 2 class, I think. First year teacher here, so I can’t comment too much on “typical” math curriculums.)

Unit 1 – Our curriculum calls this first unit “Symbolic Manipulation” which is a decent description for it. It’s basically a review of Algebra topics they’ve previously encountered, such as the distributive property, combining like terms, and factoring. The MAP lesson I plan to use is Interpreting Algebraic Expressions. I like how this lesson helps students distinguish between expressions like (5n)^2 and 5n^2, and it also uses area diagrams that will come up again when I’m teaching factoring. I plan to try this lesson right away the first week of school, and it might be a fairly simple for this class content-wise, but that’s okay because it will also serve the purpose of introducing the students to this type of group work and what it’s like to share their thinking, which will probably be a very new experience for them.

Unit 2 – This one is all-things quadratics. The MAP lesson I want to use is Forming Quadratics. This is a great matching activity, and I like how it incorporates all of the different forms for quadratic functions and helps the students identify key features of the graphs.

Unit 3 – Polynomials. Not my favorite unit (I find long division, rational root theorem, etc to be somewhat tedious), but I can’t let that show. I think I will definitely emphasize SMP #6 – Make sense of problems and persevere in solving them during this unit. The MAP lesson here is Representing Polynomials, which is another great matching activity which connects what students already know about finding zeros to 3rd-degree polynomials and their graphs. The extension activity is also intriguing.

Unit 4 – Functions and Function Transformations. There are two awesome MAP lessons that could be incorporated here. The first one, Interpreting Distance-Time Graphs, was the one I used with my summer school class last summer (8th graders who would be taking Algebra 2 in the fall… so a very different group than the one I’m talking about now). Perhaps this one would be too simple for my FST kids, so the second option, Functions and Everyday Situations, might be a better fit. I like the open-endedness of Distance-Time, but I like how Everyday Situations has students translating between the algebra and the graph.

These lessons will be great formative assessments, and  I’m glad I can incorporate one into each unit. Hopefully you got a chance to check out the MAP website and lessons if you aren’t already using this great resource. If you’ve used these in your classroom before, what do you like about them? What about them is difficult to implement? Thanks in advance!

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Filed under formative assessment, FST / Algebra 2, group work