Ambiguous Sports Graph

“Oh shit, I think I just changed my mind.” -Student

Sports Graph

Today in class we talked about the above graph. For 45 minutes. One graph. And almost every single one of my students was engaged the entire time. Here’s what happened…

During the summer I attended an institute at Boston University on facilitating productive math conversations. We learned how to set up norms for math talk and how to use talk moves (student and teacher ones) to help us facilitate the conversation. Luckily, I attended this institute with a fantastic colleague, and we have been able to share strategies as we start to implement more math talk in our classrooms. My colleague has used the original graph for years, but now it’s more clear what we want the students to talk about and how we want them to do it.

Student goals for this lesson:
• Describe the behavior of the function using precise mathematical vocabulary
• Make a choice about the sport that can best be modeled by the given graph
• Defend their own choice and respectfully critique the choice of others if they do not agree
• Reconsider their choice if they have not taken into account the time that elapses in each section of the graph

Teacher goals for this lesson:
• Use the talk moves
• Stay neutral – don’t pick a sport to keep the kids wondering
• Say very little

I started the lesson by projecting the ambiguous graph on the front board:

Sports Graph

I asked students to “Turn and talk” with their partner about the behavior of the graph. Essentially, what is happening here? I asked volunteers to share with the class. Some pairs made up their own stories. We heard about objects, cars, runners, and animals. They were accelerating, then something happened to cause the speed to change quickly. Then “object” returned to a constant speed and stopped abruptly. There was a great debate over the vertical line at the end of the graph, with many students sharing that an object cannot have more than one speed at one moment in time. Misconceptions became clear as some students referred to the distance of the object from the starting point. Some thought the object stopped and turned around. Students jumped in to clarify. We didn’t move on until everyone understood the difference between a distance/time graph vs. speed/time graph.

Then, I threw this slide at them:

Sports Graph with Sports

Without talking, students were told to choose the sport they believe is best modeled by the graph. Convince me, I said. Students wrote their responses on this interview grid, then swapped ideas by interviewing two other students (See this post for more on the interview grid strategy).

By this point, students could barely keep their ideas inside anymore. Sharing with only two classmates was not good enough; they wanted, needed, to discuss their ideas with the whole class. I asked for one volunteer to start off. It was really hard to choose. And then they talked. I didn’t say much, just kept the conversation going by calling on students and using some of the talk moves. My above mentioned colleague observed one my lessons and wrote down some of the phrases I used to keep it going:

-Say more about ________
-Can someone rephrase what ________ said?
-Can someone add-on to what ________ said?
-Does anyone agree or disagree? Why?
-Are you changing your mind? Why?

I barely talked about the actual characteristics/behavior of the graph, because I didn’t have to. If a student said something incorrect, another student shared his thoughts to help clarify. Some students came up to the board to draw additional graphs to support their thinking. The students led the conversation and I just went with it. The goals above were always on my mind, and eventually we met them in each class.

So what did the students say? I didn’t write anything down because I wanted to give my full attention to the conversation. Here is a snapshot of their interview grids:

The student work is a great representation of the different ideas discussed during our conversation. Many students stuck with their original sport, but many were influenced by the thoughts of their peers and changed their minds. Students wanted to know what is the right sport, but, there isn’t only one right answer I told them. Many students came to this conclusion on their own as well. If we had more time, I would’ve asked students if they could think of another sport that could be modeled by this graph.

We will follow up this activity with the Desmos Function Carnival. I think students are really going to like this unit.

Update: The “Which Sport?” graph was originally published in the The Language of Functions and Graphs by Malcolm Swan.

Convince Us

convince us

Inspired by Steve Leinwand’s keynote speech at Twitter Math Camp 2014, I made a “Convince Us” poster for my classroom. One of my goals for this year is to have students practice “constructing viable arguments and critiquing the reasoning of others” (SMP #3). I plan on referencing the poster when I want students to say more about a topic and solidify their reasoning.

convince us 3

This was my first attempt at downloading and using special fonts! I might be addicted now :) Letters are approximately the same size but paper needs to be trimmed after printing for them to line up correctly.

The Interview Grid

At a Classroom Discussions institute I attended last week at Boston University (more to come on that later), one of the ideas that stuck with me most was “The Interview Grid.” The presenters learned about it in the book, Common Core Standards in diverse classrooms: Essential practices for developing academic language and disciplinary literacy, and it’s a great way to initiate classroom conversations.

Screen shot 2014-08-18 at 7.48.14 PM

The teacher poses a question to the class. The question should elicit varying responses… Compare/contrast, Explain why, Choose the best model, Convince me type questions, etc. Give students about 2-3 minutes to write an answer and then have them interview classmates. The students should listen to their classmates’ thoughts (each interview should be about 3 minutes total), and paraphrase the response onto their own paper. After speaking with a couple interviewees, students are given a chance to expand, adapt, edit their original response (2-3 more minutes). Total time = 10-12 minutes

Screen shot 2014-08-18 at 7.49.05 PM

The interview grid gives students the opportunity to externalize their own thinking, listen to others, deepen their own reasoning, and helps students work with the ideas of others. It also provides students with the chance to practice three of the domains for learning language: speaking, listening, and writing (a plus for all our students but especially our English Language Learners!). Another idea I had is to turn the document into a Google Form so that students can type their responses and submit to teacher electronically. Can’t wait to try this activity out!

#TMC14 – My Favorite

After attending Twitter Math Camp 2014 in Jenks, OK, I immediately flew to Hilton Head where I joined my family on vacation. While lounging by the pool, I had a lot of time to reflect on this year’s camp. It still ranks as the best professional development experience I have ever had, but I often struggle to find ways to articulate why TMC is so special. Prior to attending, whenever I told someone I was going to a Twitter Math Camp, the person laughed and made a joke about nerdy math teachers.

So I changed my story: “Me: I’m going to a math conference. Them: Oh cool, hope you learn a lot.” And I hated myself for those interactions. Why does everyone else think a math conference is acceptable to attend, even cool, when I (and I’m sure most TMC attendees would agree) that Twitter Math Camp is by far the coolest “conference” a math teacher will ever attend? But it’s because they don’t understand. I know that I shouldn’t take the easy way out by changing my story, and I’ve promised myself to never do that again. I want to help people (especially math teachers!) understand and appreciate what everyone in the MathTwitterBlogosphere (MTBoS) has created together. The comparison to an actual overnight summer camp might actually help the most. At TMC, you attend sessions of interest to you, listen to inspiring speakers, eat every single meal with your fellow campers, play games (some math some not) during free time, explore your surroundings, and stay up way past your bedtime. After four days, you are exhausted; you are inspired; you are passionate; you are reflecting; you are questioning; you are bonded. Luckily, the conversations that begin at TMC don’t have to end at TMC. Throughout the year, we will continue chatting on twitter, blog posts, text messages; so if you’re out there lurking, jump in and join the conversation, then join us at Harvey Mudd College next summer.

In the same spirit as last year, I will recap some of TMC as a series of “My Favorites” (in no particular order).

Favorite Airport Reading: Powerful Problem Solving by Max Ray. I wanted to read this book last summer but never got around to it. I’m now four chapters in and devouring it. Max writes about problem solving strategies directly connected with the standards for mathematical practice. He shares numerous activities you can do with your students tomorrow and shares actual samples of student work/classroom conversations.

Favorite Standard for Mathematical Practice: In Steve Leinwand’s keynote, Shifting Our Mindsets and Our Actions from Remembering HOW to Understanding WHY, he referred to SMP #3 as the “Trojan Horse” and the “Most important 9 words in the CCSS.” I would have to agree with Steve on this one. SMP #3 = Construct viable arguments and critique the reasoning of others. Students must be able to communicate their findings and respond to the conclusions of others. This is a practice that must be taught explicitly and will be one of my goals for this school year. I plan on echoing Steve, and demanding that students “CONVINCE ME” of their conclusions.

Favorite Game: Andy Pethan introduced me to the game, Q-bitz, during Wednesday’s game night. There are three different challenges where you have to recreate patterns out of cubes faster than your competitors. As someone who loves math and wants to be on Survivor one day, this game is right up my ally.

Source: Amazon

Source: Amazon

Favorite Opening Day Activity: John Mahlstedt tells his students how awesome he is on the first day of school by sharing facts/pictures about himself. It’s a way for students to learn that you’re not only a teacher, but a human too, and a fun one. I usually have students try the matching activity below, but normally I simply share the answers at the end of class. This year, I’m going to enhance this activity with John’s suggestion of making a slideshow with pictures to show my students rather than just telling them about me.

Favorite Lunchtime Adventure: During lunch one day, a group of us found Gameday Popcorn on main street and had a great time testing all the flavors.

Favorite Morning Session: Writing Real World Lessons with Mathalicious. Karim and Matt led a fantastic session on the creation process of Mathalicious lessons: “The narrative determines the standards, not vice-versa.” This was so interesting to me because I feel that most of the time, when my colleagues and I are discussing an upcoming topic/unit, we are doing the vice-versa. Ex. Tomorrow we have to teach solving systems of equations by elimination… how should we do this? Mathalicious lessons are conversations about a really interesting topic that needs math to answer the question. That’s why the lessons are so authentic and engaging. Our group spent time proposing thought-provoking questions, discussing their authenticity, and attempting to write a sample lesson out of our ideas. The experience was extremely rewarding and although it’s time consuming, I hope to bring this practice back to my own planning time.

Favorite Cupcake Locale: Smallcakes. Delicious.

Favorite Formative Assessment Explanation: John Scammell shared more than 60 formative assessment strategies that you can easily implement in your classroom. He kindly shared all of them here, but it’s how he described formative assessment that actually stuck with me the most. John said that formative assessment must be risk free. If you put a grade on it, or enter it into an online grading system, it’s no longer risk free to a student or parent. Formative assessment should be all about providing feedback. One easy suggestion is to use a highlighter to mark the last spot a student’s work was right. Another is to mark a problem with a + (better than previous work), – (worse than previous work), or = (equal to previous work) sign. And my favorite method he shared, was to actually classify a student’s error. Many students get frustrated when something is marked wrong, and they immediately think they don’t understand anything. It’s important to differentiate between whether an error is a conceptual misunderstanding or calculation mistake.

Favorite Ice Breaker: Bob Lochel discussed Meaningful Adjacencies as related to the 9/11 Memorial in NY and how a similar connection activity with tv shows can be used in your classroom. He provides a very thorough explanation on his blog post.

Favorite Phone Holder: Glenn Waddell taught us an ingenious way to use a smartphone as a video camera in your classroom. 1) Make a vertical slit in the bottom of a paper coffee cup 2) Insert smartphone 3) Record

Favorite Dan Meyer Quote: “I want to profit off what you know.” – Dan Meyer
In Dan’s keynote address, he shared tons of data on tweeting and blogging math teachers. He’s interested because he knows that great ideas are being shared, but no one knows about them. He wants to know about them. And so do I. When you post/tweet your great ideas, my students get to benefit from them. It’s okay to be selfish here, because the more students that benefit, the better it is.

Favorite Dan Meyer Slide: Dan shared a graph comparing a blogger’s velocity (posts per week) to number of subscribers. He said he’s interested in the individuals in quadrant 2, those who post infrequently, but have many readers. After examining his data, it appears that my blog falls in quadrant 2. I post infrequently because I’m afraid and therefore selective. Afraid that what I’m offering won’t be as good as what other people are offering. So I’m choosy. I like to post activities after I’ve done them so I know if they worked or not, and so I can edit them before posting. I try to include any part of the activity that is downloadable, so you can take it and use it tomorrow. I post when I want more than the 100 students on my roster to benefit from something fun. Maybe this is why some of you are following my blog, if you have other insights, please let me know. But in the meantime, thank you for reading!

Favorite Desmos Update: In his keynote, Eli Luberoff assured us that Desmos would be free forever. This is incredible news. Desmos has created API partnerships and has financial security to ensure that we will never have to pay to use this online graphing calculator. All teachers and students can benefit from this resource, so if you haven’t played with it yet, stop reading and go to Desmos now. Also, be sure to check out this new digital math lesson from Dan Meyer, Christopher Danielson, and Desmos: Central Park (and the other four lessons on Teacher Desmos).

Favorite Roommate: Rachel and I bonded over being teachers at the high schools we went to, Algebra 1, True Detective, Richard Linklater films, quiet time, beer choices, traveling and so much more. Check out her blog here and let’s convince her to post more this year.

Favorite App: Pam Wilson introduced TMC to Plickers and our classrooms will never be the same. Each student responds to a multiple choice question by holding up a unique bar code. The teacher uses a smartphone to scan the room and the responses are graphed/recorded automatically. I see myself using this as a quick formative assessment at the end of class.

Source: @approx_normal

Source: @approx_normal

Favorite BBQ: Elmer’s with this fun crew. Thank you Jason for treating us!

Favorite 3D Printing Resources: John Stevens and I talked about using 3D printers in the math classroom. We are going to have several 3D printers at my school next year and I want to do something awesome with them. If you have any advice, please share with John and me!

Favorite Book RecommendationJustin Lanier plugged the books How Children Learn and How Children Fail by John Holt. Justin’s takeaways: 1) Look Around 2) Teach Crazy 3) Trust Children.

I know there’s a lot of TMC awesome-ness that has been left out of this post, but I could never write about everything that I learned, because it would take forever. I owe a huge shout-out to all the TMC14 organizers for making this happen and providing us with this incredible experience. Thank you to everyone I met, and I hope to see you all again next year :)

Math Strategies for English Language Learners

In Massachusetts, all core subject area teachers and administrators of English Language Learners, must receive a Sheltered English Immersion (SEI) Endorsement from the state. One way for educators to receive this endorsement is by taking a RETELL (Rethinking Equity and Teaching for English Language Learners) course. One of the main goals of the course is to provide teachers with a repertoire of methods and strategies they can use to help students practice the four domains for learning language: reading, writing, listening, and speaking.

These are a few of my favorite strategies I learned from the course:

Reading Strategy: Partner Reading for Comprehension
Partner Reading

This strategy made the problems more manageable because students were able to have valuable discussions regarding the content before actually solving the problem. Since Partner #2 had to comment and respond to Partner #1’s questions, they had to pay close attention to what was being read. Most of the partners were able to choose the most important pieces from each word problem based on what Partner #1 had commented on during part two of the reading strategy. Giving my ELLs the opportunity to read aloud with a mainstream student allowed them to practice their expressions and ask for explanations. The strategy helps support both partners as they learn from each other’s observations and questions regarding the text. The students were able to determine what phrases were necessary for successful completion of the problem, and those that were not needed.

Writing Strategy #1: Cut and Grow
The Cut-n-Grow strategy provided students with an opportunity to see exemplar/non-exemplar student work samples and focus on improving their own open response questions. First, students looked at a student work sample that received a score of 2 on a standardized open response question. They cut the question apart and wrote additional explanations to turn the score into a 4. Then, students repeated the process for an open response question they had previously solved, to improve upon their own writing.

Cut and Grow

Many of my ELLs tend to leave open response questions blank on exams, so it’s important that we explicitly practice answering these questions. The strategy gave students a second chance at improving their work, and they responded very well to this strategy. The physical act of cutting and pasting pieces of the response, and then adding their revised sentences, really helped show students how to edit and model good writing.

This strategy can work with any writing sample the students produce in class, such as journal entries, AP open response answers, exit tickets, etc.

Writing Strategy #2: Write Around
Students should be divided into groups of 3 or 4. Each student starts with a blank sheet of paper and writes one sentence of a word problem. Then, the student passes the paper to the right. After reading what is written, students continue to add sentences until a word problem is created (approx 4 sentences). Each team will choose one problem to write on a large whiteboard or paper to show the rest of the class during a gallery walk. The gallery walk gives students the chance to make observations regarding other students’ work. Students can then choose one or more of the problems to solve. Teachers can scaffold this activity by providing students a list of must-haves for each word problem. For example, students might have to include the following in a quadratics word problem: a setting, the type of object being thrown/launched/dropped, height at which object starts, and the speed or distance the object travels. Each student would take turns providing one of these details.

My biggest takeaway from this course is that we all need to explicitly teach strategies for reading, writing, speaking, and listening in our disciplines. We cannot sit back and “let the English teacher handle it.” English Language Learners are trying to learn a new language at the same time we are expecting them to learn our content. It’s our responsibility to provide opportunities in our lessons to support both goals.

Solving Equations – Add It Up

I wanted a quick and fun way to assess students’ abilities to solve equations during the first week of school, so I made this “Solving Equations – Add It Up” powerpoint. Each group of four students will have one large whiteboard on their desks (purchased from Home Depot – panel board that is cut up). Each student will solve the problem in his/her quadrant, then the students will add all their solutions together to get one final number which they will write in the middle of the board. I will only look at that final number and tell a group whether they are right or wrong. If a group is wrong, they will have to look at each other’s work and figure out where the error has been made. During the activity I will walk around and monitor student’s progress, keeping notes on my clipboard for future reference.

If a group does not have 4 students, I will ask the student who finished his/her problem first, to also solve the 4th problem. If you do not have large whiteboards, you can still have students do this activity. Students can solve their problem on an individual mini white board or sheet of paper, and then combine their answers onto one sheet in the middle of the table.

A Field Trip to Italy

Last week I chaperoned the greatest field trip I can ever expect to go on… to Italy. Along with 3 colleagues, we flew 23 students to Rome for a 7 day Italian adventure. Each of the students takes Latin in school, and were far more qualified than myself to visit each of the historic sites. I’ve traveled abroad many times before, but there is something about sharing this experience with my students that makes this trip one of the best I have ever taken. It would have been impossible for our group to not become a family on this journey, as we spent numerous hours on a bus, exploring the streets of Rome, and bonding over our love/hate for various tour guides we encountered. I watched students who had never spoken to each other at school become close friends and take selfie after selfie together. I watched eight students wake up early each morning to go for a sunrise run or participate in a mini spartan workout. I watched students who have never been away from their parents for more than a night, flourish and make choices on their own. I watched our male students become protectors of the entire group as we walked the city streets.

I often found myself at the back of the group, feeling protective of our students, and wanting to make sure they were all in front of me and visible. It is during some of these times that I found myself happiest on the trip. Different students would find their way to the back of the line, some because they were tired and lagging a bit, or some because they couldn’t wait to tell me about something they just saw. I learned so much about my students and their lives during these conversations, and was able to share so much more of myself than I usually do in the classroom. There’s something special about sharing moments when you’re drenched with sweat, you’ve already walked four miles, and you’re on your third gelato of the day.

It was truly the adventure of a lifetime and I’m so grateful I was given the opportunity to chaperone. It is my hope, that some of the friendships formed will continue during the school year, and I know, that the memories of this trip is something we will all share forever. Here is a snapshot of some of the things we saw/did (minus student pictures for privacy):

The Roman Forum

The Roman Forum



Mozzarella Bar

Mozzarella Bar

St. Peter's Square and Basilica

St. Peter’s Square and Basilica



Lunch in Naples - Zeppole, Bruschetta, Crocchetta, Mozzarella Fritta, Pizza Margherita, and Foccacia with Nutella

Lunch in Naples – Zeppole, Bruschetta, Crocchetta, Mozzarella Fritta, Pizza Margherita, and Foccacia with Nutella

Early Morning Run

Early Morning Run

Blue Waters of Capri

Blue Waters of Capri



Mt. Vesuvius

Mt. Vesuvius

View of Sorrento

View of Sorrento

Tiberius' Grotto - Sperlonga

Tiberius’ Grotto – Sperlonga