SMP7 – The Triangle Sum Theorem

G-CO.C.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

How do you provide opportunities for your students to look for and make use of structure? I’m finding that deliberate practice in looking for and making use of structure is making the practice a habit for my students.

SMP7 #LL2LU Gough-Wilson

We ask: “what you can you make visible that isn’t yet pictured?”

We practice: “contemplate before you calculate”.

We practice: “look before you leap”.

We make mistakes; the first auxiliary line we draw isn’t always helpful.

We persevere.

We learn from each other.


Months ago, our goal was to prove the Triangle Sum Theorem.

We thought first about what we already knew … what we had already added to our deductive system.

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Then we practiced “I can look for and make use of structure”.

And so the journey to make the Math Practices our habitual practice in learning mathematics continues …

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Posted by on July 1, 2016 in Angles & Triangles, Geometry


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Rereading 5 Practices

I’ve been rereading Smith & Stein’s 5 Practices for our #T3Learns slow chat book study.


We used daily prompts for one chapter each week.


I have failed to keep up with the last few chapters because of exams, graduation, a litter of 8 abandoned puppies, and a family vacation to the mountains of western North Carolina.

I like to finish what I started, so I reread the last few chapters and wrote the tweets I would have posted. Rereading this book a few years after the first time has been a valuable experience – reminding me of not only the importance of the 5 practices but also some of the challenges in implementing them – so I am including my reflections as a blog post to reference every once in a while, as the journey continues …

Chapter 1: Introduction and Introducing the Five Practices

Sentence/Phrase We learn through a process of knowledge construction that requires us to actively manipulate and refine information and then integrate it with our prior understandings. #T3Learns
Connect Love the idea of our community of learners participating in the “joint construction of knowledge”.
Extend When going over a task, how do we turn “show-&-tell” into an opportunity for Ss to learn how solutions are connected to the math we want them to know? #T3Learns
Challenge As noted in intro, challenge is “aligning students’ developing ideas & methods with the disciplinary ideas that they ultimately are accountable for knowing.” #T3Learns
I wonder I wonder how many Ts plan lessons thinking about “launch” phase, “explore” phase, & “discuss & summarize” phase. #T3Learns #5Practices


Chapter 2: Laying the Groundwork: Setting Goals and Selecting Tasks

Sentence/Phrase “productive discussions that highlight key mathematical ideas are unlikely to occur if the task on which Ss are working requires limited thinking & reasoning.” #T3Learns p20
Connect Ts often think of lessons in terms of what Ss will do instead of what they will come to know & understand about the math. #T3Learns
Extend Extend: If the mathematical idea is explicitly written in the learning goal, will that influence the way I plan/teach the lesson? #T3Learns
Challenge How might we get Ss to think about lessons in terms of what they know & understand about the math instead of just what they are doing? #T3Learns
I wonder I wonder what % of tasks I give my Ss would be considered higher-level. I wonder when a lower-level task is helpful. #T3Learns


Chapter 3: Investigating the Five Practices in Action

Sentence/Phrase T avoided show&tell in which solutions are presented in succession w/o rhyme or reason, often obscuring point of the lesson. #T3Learns p.29
Connect Rereading this reminds me of where I’ve picked up new habits while teaching: “By referring to notes that she had made during the monitoring process” #T3Learns p.27
Extend I’m reminded how helpful reading through a vignette is to see #5Practices in action. I need to be sure my team of Ts has this opportunity. #T3Learns
Challenge Challenge: Deliberate selecting, even w/teams w/same solution. Have clipboard but don’t always keep up. #5Practices #T3Learns
I wonder @elsdunbar has me wondering how #5Practices can be connected to other disciplines: #T3Learns


Chapter 4: Getting Started: Anticipating Students’ Responses and Monitoring Their Work

Sentence/Phrase “His preparatory work would help him make sense of what he did see and free him up to consider more deeply the things that emerged that he had not anticipated.” #T3Learns #5Practices p. 35
Connect So simple & yet so important for Ts planning lessons: “Once he had determined what he was going to do and why …” #T3Learns #5Practices p. 35
Extend “Developing Qs only “in the moment” is challenging for a teacher who is juggling the needs of a classroom full of learners who need different types and levels of assistance.” #T3Learns #5Practices p. 36
Challenge Challenge: Solving problems using nonprocedural methods to anticipate what Ss might do. I find this best done in the company of my coworkers! #T3Learns #5Practices
I wonder I wonder whether I can find Ts willing to share a classroom experience through the lens of #5Practices. The vignettes are so helpful! #T3Learns


Chapter 5: Determining Direction of Discussion: Selecting, Sequencing & Connecting Students’ Responses

Sentence/Phrase Selecting is “purposefully determining what math Ss will have access to beyond their own initial thoughts”. #T3Learns p.43 #5Practices
Connect Selecting gives T control over what class discusses–not left up to chance of who raises hand or whom T randomly calls. #T3Learns p.44
Extend I love “orchestrating” in title. Select*Sequence*Connect really does make me feel like the conductor of an orchestra of math Ss. #T3Learns
Challenge Unfiltered S contribution hard to follow or causes unproductive direction #T3Learns p.44 Reminds me of @PamWHarris:
I wonder HMW teach every lesson so that “the goals for the lesson serve as a beacon toward which all activity is directed”? #T3Learns p.59


Chapter 6: Ensuring Active Thinking & Participation: Asking Good Questions & Holding Students Accountable

Sentence/Phrase Giving Ss time to compose responses signals value of deliberative thinking, recognizes that deep thinking takes time #SlowMath #T3Learns p72
Connect “What Ss learn is intertwined with how they learn it.” Which is why we start with #AskDontTell learning episodes. #T3Learns #5Practices p.61
Extend Classroom discussions “do not materialize out of thin air. Rather, they are planned …” #T3Learns #5Practices p.69
Challenge Moving from S sharing solution to revealing connections is a challenge. Takes practice & do-overs. Blogging helps me process. #T3Learns
I wonder I wonder whether my team would be willing to listen/record each other’s Qs so that we can improve # that push/probe S thinking. #T3Learns


Chapter 7: Math – Putting the Five Practices in a Broader Context of Lesson Planning

Sentence/Phrase Planning “is a skill that can be learned and greatly enhanced through collaborations with colleagues.” #T3Learns #5Practices #MTBoS
Connect How do you create a permanent record of the decisions you make in your lesson? How many of us blog for this reason? #T3Learns
Extend Beginning Ts need even more support planning – so thankful for our team teaching opportunity. #T3Learns
Challenge Challenge: Our Ts dutifully submit lesson plans but few find value in completing. HMW use them to improve teaching & learning? #T3Learns
I wonder HMW change lesson planning at our school to focus on Qs we’ll ask to drive instruction instead of what we’ll cover. #T3Learns


Chapter 8: Working in the School Environment to Improve Classroom Discussions

Sentence/Phrase “All teachers have the capacity to be stars—they just need access to opportunities to learn, reflect, and grow.” #T3Learns #5Practices p94
Connect I am thankful to work in a school that values common planning time for Ts, even when scheduling is a challenge. #5Practices #T3Learns
Extend Thoughtfully & thoroughly planning instruction for tasks culminating in discussion so math learned is salient to Ss isn’t easy #T3Learns p94
Challenge I am thankful to work in a school that values common planning time for Ts, even when scheduling is a challenge. #5Practices #T3Learns
I wonder I look forward to continued work with #5Practices. I’ve seen our work w/tasks improve over several yrs of sequencing & connecting. #T3Learns



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Preparing for an Ignite: Slow Math

What makes you passionate about mathematics education? Suzanne Alejandre asked me this question a few months ago as she was preparing for the 2016 NCTM Ignite Talks.

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This is Jill’s sketch from the Ignite session, highlighting the message from each of the ten speakers.

In an Ignite Talk, you get 5 minutes to share your passion using 20 slides that advance automatically every 15 seconds.

I’ve been thinking about Slow Math for a while now. I started a Slow Math blog sometime last year. My family and I live on a farm and share lots of Slow Food meals with our family and friends. My husband is a minister, and he was awarded a Lilly Endowment clergy renewal grant several years ago to think about what Slow Church might look like. Why not Slow Math, too? I’ve said many times that using technology in the classroom slows me down – because I find out what students really know when I send them a poll during the lesson – because the dynamic action technology that we use to interact with data and graphs and geometry causes my students to ask questions that weren’t asking otherwise. Slow Math seemed to fit.

Now I had a topic, but even so, the thought of preparing for an Ignite Talk was daunting. Figuring out how to talk about Slow Math in 5 minutes was only part of it.

I have been learning to share stories from my classroom with others for several years now, but I almost always have my talks written out word for word, and I often use notes when I give them. I saw a Twitter exchange between Robert Kaplinsky and Andrew Stadel a few months ago about Weekend Language, and so I ordered my copy and read it. Which made the thought of preparing for an Ignite Talk even more daunting. I did it anyway.

An exchange with Tracy Zager over Twitter made me realize that it’s kind of like wordsmithing 20 Tweets – one per slide – as it apparently takes me about 15 seconds to read 140 characters out loud. For a week or so, I timed myself with the slides to be sure I had the timing right. I added a phrase. I subtracted two phrases. I added a word. I subtracted three words.

Once I had the timing right, I started memorizing, one slide at a time. I said it as I fell asleep at night. I was still saying it when I woke up in the morning. I said it in the shower (which made me look up the average shower length in the US … 5 minutes is below average). I said it once per mile every mile I ran. (I certainly don’t run 5 minute miles … just gave myself a break between practices.) I said it to my family before I left. I said it to Jill when I got to San Francisco. When we got to the session, I found out I was 9th. Against my will, I continued to say it while the first 8 speakers spoke. (Luckily I was able to watch them on video as they were released.)

And then it was my turn. After practicing so much, the experience was surreal. The audience reactions weren’t completely as expected. I heard a few gasps as I shared some of the troublesome #SlowMath tweets. There was a laugh or two, one expected, one not. Mostly I heard affirmation that maybe Slow Math could be a movement.

How have you already incorporated Slow Math into your teaching and learning? What can you do to further the Slow Math Movement? How can you make sure teachers and students know they have time to enjoy a slow math lesson, asking questions and engaging in productive struggle? Let’s continue that conversation at the SlowMath hashtag.

As the journey continues, I am thankful for the good work of our friends at The Math Forum, providing many opportunities for us to learn and think about math together.


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For my students, on their graduation

You might have heard me say before that I believe that my students and I enter into a community of learning together at the beginning of each school year. While it is our tangible goal to study the measure of the earth (geometry) and change in motion (calculus), our intangible goal is to enter into the practice of learning.

We are not here to celebrate because you’re smart. We are here to celebrate what you’ve accomplished because you are committed to the practice of learning. We are here to celebrate the perseverance that you’ve shown, all of the hours of studying and practice that you have put in. We are even here to celebrate the synapses that have fired in your brain every time you’ve made a mistake – every time you’ve learned something new. We are here to celebrate your kindness to us and to each other. And we are here to celebrate the questions you have asked.

In The Falconer – What We Wish We Had Learned in School, Grant Lichtman suggests that “Questions are the waypoints on the path of wisdom”. We are here to celebrate your journey towards wisdom. I know your parents agree that you are well on your way to learning the art of questioning. You have been asking questions since you could talk: Why is broccoli green? Why is 2+2 equal to 4? Why do dogs bark and cats meow? What if my internet goes out on the night of the deadline? Who came up with the number e? What if it snows and we don’t have class tomorrow? What if my alarm doesn’t go off? Why does the unit circle go counter-clockwise?

As your journey continues, we urge you to keep asking questions – to keep learning –to seek peace and defend justice – to live responsibly – but we also want to warn you away from only doing enough to get by.

In his book about ethics, Sam Wells insists over and over that you cannot know what to do and how to act without preparation. Don’t expect to be able to lead later unless you’ve done the hard work of becoming a leader. You can’t sleep now and expect to do the right thing later. So, he tells the story of a surgery that took a tragic turn in an Edinburgh hospital in the 50s resulting in the death of a young child. Later that week two friends were discussing the tragedy, and one of them expressed sympathy for the surgeon who had run into a completely unexpected complication. The other friend disagreed.

I think the man is to blame. If somebody had handed me ether instead of chloroform, I would have known from the weight it was the wrong thing. You see, I know the surgeon. We were students together at Aberdeen, and he could have become one of the finest surgeons in Europe if only he had given his mind to it. But he didn’t. He was more interested in golf. So he did just enough work to pass his exams and no more, and that is how he has lived his life – just enough to get through but no more; so he has never picked up those seemingly peripheral bits of knowledge that can one day be crucial. The other day [at that table] a bit of ‘peripheral’ knowledge was crucial and he didn’t have it. But it wasn’t the other day that he failed – it was thirty-nine years ago, when he only gave himself half-heartedly to medicine. (74)

Our hope for you is for you to be who you are called to be – to find something to which to give yourself whole-heartedly – something about which you are passionate – and for which you learn all of the peripheral knowledge crucial to doing the right thing. (We aren’t suggesting you should never play golf.)

We’ve spent the past several years convincing you that the part plus the part equals the whole. You remember, right? In geometry – the Segment Addition Postulate – If I have a piece of wire that is 4 m long and another that is 3 m long, then together, I have 7 m of wire. It works for angles, and it works for area. When it comes to learners, though, you give us evidence that maybe Aristotle knew more than Euclid: The whole is greater than the sum of its parts. You are better together than you are alone. And we are better teachers and learners because of you.

Take good care of yourselves and keep in touch with us and each other. Don’t ever wonder whether there’s someone who’s cheering for you. We are, and we look forward to hearing about the next part of your journey.


[I shared this with some of my students at a luncheon celebration last month and last year.]

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Posted by on May 18, 2016 in Student Reflection


Collaboration & Perseverance: What Do They Look Like?

I recently wrote about this year’s circumference of a cylinder lesson.

As I was looking through some pictures, I ran across these two from last year’s lesson.

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What do you see in these pictures?

I was struck by what I saw: collaboration and perseverance.

What do collaboration and perseverance look like in classrooms you’ve observed? What about in your own classroom?

How do you create a culture of collaboration in your classroom?

How do you make sure your students know that we want them to learn mathematics by making sense of problems and persevering in solving them?

Thank you to all who share your classroom stories of collaboration and perseverance, so that we might add parts of those to our own classroom stories, as the journey continues.

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Posted by on May 8, 2016 in SMP1, Student Reflection


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Making a Better Question Worse

I recently read a post on betterQs from @srcav with an area question from Brilliant that I added to today’s opener.


I knew something was up when I saw my students’ results.

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My mistyping was a good reminder of the importance of nouns.

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Without showing any results, I sent the corrected question as a Quick Poll.

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(I wonder whether the way the first question was asked prompted the misconception in the wrong answer, but I won’t find out until I have another group of students.)

We are learning to look for and make use of structure.

We are learning to contemplate, then calculate.

And we are learning how to ask better questions, as the journey continues …


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The Surface Area of a Sphere

The Surface Area of a Sphere

I’ve written before about making sense of the surface area of a sphere. The lesson this year unfolded (unpeeled?) a bit differently.

I’m not sure how students might guess that the surface area of a sphere has something to do with the area of a great circle of the sphere. We talked about what a great circle must be, we used fishing wire to measure the circumference of a great circle of the sphere (orange), and I asked them to estimate how many great circles would cover the orange. You can see the huge variety of responses.

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We cut the orange in half. I showed them the surface of the great circle and the act of “stamping” it onto the orange peeling. Do you want to keep your estimate? Or change it?


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We went from 20 responses below π, 9 infinites, and 3 correct to 13 responses below π, 5 infinites and 7 correct.

I hesitated about what to ask next. We were ready to peel the orange to see how many great circles we could cover and figure out what the surface area formula would be, but I was curious about whether students could make sense of the formula before we did that if I told them what the formula was. So we waited on peeling and I sent another poll.

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68% understood that the sphere’s surface area formula, 4πr^2, meant that 4 great circles could cover the sphere. We peeled it to be sure.


And so the journey to figure out what questions to ask when continues …


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