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5 Practices: Dilations

5 Practices for Orchestrating Productive Mathematics Discussions might be the book that has made me most think about and change my practice for the better in the past 10 years.

At the beginning of our second day on dilations, I asked students to work on this.

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Because of the 5 Practices, I pay attention differently when I walk around and monitor students working. I know that I looked for different student approaches before I read the book, but I didn’t consciously think about selecting and sequencing them for a whole class discussion. I often asked for volunteers. And then hoped that another student would volunteer when I asked who worked it differently [who had actually worked it differently and correctly].

I asked a few questions of students while I was monitoring them to clarify what they were doing and selected and sequenced a few to share. The student work above looks similar at first glance, but there are subtle differences in their thinking that make important connections about dilations.

TM shared first. She used slope to find the vertices of the image. She went down 1 and to the right 3 from C to X, and then because of the scale factor of 2 went down 1 and to the right 3 from X to get to X’. She went down 3 and to the right 2 to get from C to Z, and then went down 3 and to the right 2 from Z to get to Z’.

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JA shared next. He focused on the line that contains the center of dilation, image, and pre-image. He knew that X’ would lie on line CX and that Z’ would lie on line CZ.

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MB shared next. He also used slope, but a bit differently from TM. He noticed “down 1 and to the right 3” to get from C to X and so because of the scale factor of 2 then did “down 2 and to the right 6” from C to get to X. He noticed “down 3 and to the right 2” to get from C to Z and so then did “down 6 and to the right 4” to get from C to Z’.

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I had not seen additional methods while monitoring. This exercise didn’t take too long, and so I didn’t get around to everyone. [This is where Smith & Stein’s advice about keeping a clipboard to pay closer attention to whom you check in with and whom you call on helps so that you aren’t checking in with and calling on the same few every time you have a whole class discussion.] I hesitated before I asked, but I did then ask, “did anyone find X’Y’Z’ a different way?” [This is also where I am learning to trust my students to recognize when their method is different.] TC raised his hand. I treated C as the origin and used coordinates. He shared his work and showed that the coordinates of X (3, -1) transformed to X’ (6,-2) with a dilation about the origin for a scale factor of 2.

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And so the journey continues, thankful for friends like Gail Burrill [one of my voices] who recommend authors like Smith and Stein to help me think about and change my practice for the better, making me feel like a conductor rehearsing for a beautiful, exciting mathematics masterpiece …

 
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Posted by on December 21, 2016 in Dilations, Geometry

 

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Rereading 5 Practices

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

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We used daily prompts for one chapter each week.

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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 https://www.google.com/search?q=smith/stein+task+analysis+guide&safe=strict&client=safari&rls=en&tbm=isch&tbo=u&source=univ&sa=X&ved=0ahUKEwiNup2AkJnNAhVJIlIKHVsFDS0QsAQINg&biw=1325&bih=949

 

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: https://elsdunbar.wordpress.com/2016/05/01/teach-math-as-a-story/ #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: http://www.mathisfigureoutable.com/down-the-rabbit-hole-2/
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. https://easingthehurrysyndrome.wordpress.com/2015/08/22/the-best-professional-development-ever/ #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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#T3Learns Slow-Chat Book Study: 5 Practices for Orchestrating

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After the success of the slow-chat book study on Embedding Formative Assessment we plan to engage in another slow chat book study.

A few years ago, as we embraced focusing our classrooms on the Standards for Mathematical Practice, a number of our community began reading and using the book by Margaret S. Smith and Mary Kay Stein, 5 Practices for Orchestrating Productive Mathematics Discussions.

This book has been transformational to many educators, and there is also a companion book focused on the science classroom, 5 Practices for Orchestrating Task-Based Discussions in Science, by Jennifer Cartier and Margaret S. Smith.

Both books are also available in pdf format and NCTM offers them together as a bundle.

Simultaneous Study
: As our community works with both math and science educators, we are going to try something unique in reading the books simultaneously and sharing ideas using the same hashtag.

We know that reading these books, with the emphasis on classroom practices, will be worth our time. In addition to encouraging those who have not read them, we expect that those who have read them previously will find it beneficial to re-read and share with educators around the world.

Slow Chat Book Study
: For those new to this idea of a “slow chat book study”, we will use Twitter to share our thoughts with each other, using the hashtag #T3Learns.

With a slow chat book study you are not required to be online at any set time. Instead, share and respond to others’ thoughts as you can. Great conversations will unfold – just at a slower pace.

When you have more to say than 140 characters, we encourage you to link to blog posts, pictures, or other documents. There is no need to sign up for the study – just use your Twitter account and the hashtag #T3Learns when you post your comments.

Don’t forget to search for others’ comments using the hashtag #T3Learns.

Need to set up a Twitter account? Start here.

If you need help once we start, contact us (see below).

Book Study Schedule
: We have established the following schedule and daily prompts to help with sharing and discussion. This will allow us to wrap up in early June.

The content of the Math and Science versions line up fairly well, with the exception of the chapters being off by one.

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Daily Prompts

Book Study 2

Contact Information

Moderators will be Jill Gough, Kim Thomas, and Jennifer Wilson.

Please contact kspry@ti.com if you have any questions.

-Kevin Spry

@kspry

 

[Cross-posted on T^3 Learns]

 
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Posted by on April 17, 2016 in Professional Learning & Pedagogy

 

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Placing a Fire Hydrant

Placing a Fire Hydrant

We’ve used the Illustrative Mathematics task Placing a Fire Hydrant for several years now. Each year, the task plays out a bit differently because of the questions that the students ask and the mathematics that students notice. Which is, honestly, why I continue to teach.

I set up our work for the day as practicing I can make sense of problems and persevere in solving them and also I can attend to precision. If you don’t know how to start at Level 3, use Levels 1 and 2 to help you get there.

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SMP5 #LL2LU

In an effort not to articulate all of the requirements ahead of time, I simply asked: where would you place a fire hydrant to serve buildings A, B, and C. Students dropped a point at the location they thought best.

It was then obvious from the students’ choices that they thought equidistant was important.

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This year I didn’t put out tools that students might choose to use. Instead, I set the timer for them to work alone on paper for a few minutes and told them to ask for what they needed. Before I could get from the front of the room to the back, almost every hand was raised to request either a ruler or a protractor. (No one asked for a compass this year. Last year, when I had them out on the tables, lots of students used them.)

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I gave students a few more minutes to work individually with the option, this time, of working with the TI-Nspire software to show their thinking. And at the end of that, I added a few more minutes, asking students to focus on how they could justify that their solution always works. Then I gave them a few minutes to discuss their thinking with a partner.

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I watched (or monitored, according to Smith & Stein’s 5 Practices) while they worked using the Class Capture feature of TI-Nspire Navigator. During that time I also selected and sequenced for our whole class discussion. I wanted some of the vocabulary associated with special segments in triangles to come out of our discussion, so I didn’t immediately start with the correct solution.

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We started with Autumn, who had constructed the midpoints of the sides and then created both a midsegment of the triangle and some medians of the triangle. She could tell that the intersection of the midsegment and medians was “too high”.

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C chimed in that she had constructed lots of midsegments. In fact, she had created several midsegment triangles, one inside the other.

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Next we went to Addison, who not only had created all three medians of the triangle but had also measured to show that the medians weren’t the answer.

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That led to S, who had been trying to figure out when the intersection of the medians would be a good location for the fire hydrant.

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Arienne told us about her approach next. She had placed a point inside of the buildings, measured from the point to each building, and she was moving the point around to a location that would be equidistant from the buildings.

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Reagan talked with us about her solution next. She had constructed the perpendicular bisectors and measured from their intersection to each vertex to show that it always worked.

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I wonder what that point has to do with the vertices. What do you see in the diagram? (I was expecting students to “see” a circle. But they didn’t. They saw a triangular prism.) I wasn’t ready to show them the circle, though. How could I help make the circle visible without telling them? A new question came to me: What if we had a 4th building? Where could we place the building so that the fire hydrant served it, too?

I quickly collected Reagan’s file and sent it out to all of the students so that they could create a 4th building that was the same distance from the fire hydrant as A, B, and C.

While they were working, Janie said, “I have a 4th building the same distance, but how do I place it so that it always works?” (On the inside, I was thrilled that Janie asked this question. It is exciting for students to realize this early on in the course that we are about generalizing and proving so that something always works and not just for one case.)

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How do you place the 4th building so that it always works? What is significant about the location of the 3 buildings and the fire hydrant?

Sofia volunteered that her 4th building always works. (I have to admit that I was skeptical, but I made her the Live Presenter and asked how she made it.) Sofia had rotated building C about the fire hydrant to get d. (How many degrees? Does the number of degrees matter? Would rotating always work? Why would it work?) She rotated C again to get a 5th building between A and B. What is significant about the location of the 5 buildings and the fire hydrant?

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And then they saw it. It wasn’t yet pictured, but it had become visible. All of the buildings would form a circle around the fire hydrant! The fire hydrant is the circumcenter of ∆ABC. The circle is circumscribed about the triangle.

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And so the journey continues … every once in a while finding a more beautiful question.

 

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The Center of Rotation

This is the first year we have tried Identifying Rotations from Illustrative Mathematics.

△ABC has been rotated about a point into the blue triangle. Construct the point about which the triangle was rotated. Justify your conclusion.

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This reminds me of the Reflected Triangles task, which we have used now for several years.

I got a glimpse of students working on the task using Class Capture. I watched them make sense of problems and persevere in solving them.

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We looked at all of the auxiliary lines that LJ made, trying to make sense of the relationship between the center of rotation, pre-image, and image.

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We looked at Jarret’s work, who used technology to perform a rotation, going backwards to make sense of the relationship between the center of rotation, pre-image, and image.

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We looked at Justin’s work, who rotated the given triangle about A to make sense of the relationship between the center of rotation, pre-image, and image.

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We looked at Quinn’s work, who knew that if R is the center of rotation, then the measures of angles ARA’, BRB’, and CRC’ must be the same.

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Students took those conversations and continued their own work.

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The next day, Jared shared his diagram. What can you figure out about the relationship between the center of rotation, pre-image, and image looking at his diagram?

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In my last two posts, I’ve wondered what geometry looks like if we start our unit on Rigid Motions with tasks like these instead of ending the unit with tasks like these. Maybe we will see next year, as the #AskDontTell journey continues …

 
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Posted by on March 10, 2015 in Geometry, Rigid Motions

 

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Transforming a Segment

The task:

Given segment AB, construct a regular hexagon ABCDEF with segment AB as one of its sides.

-You may not use any Shapes tools.

-You many not use any Measurement tools.

When you are finished, we will use Measurement tools to justify your construction.

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This task is a good one for working on math flexibility. You can construct the hexagon one way? Great! Now find another way to construct the hexagon.

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I used Class Capture to monitor students working.

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Whose would you select for a whole class discussion?

 

This year, we started with someone who did rotations only.

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Then moved to someone who did rotations and reflections.

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Then moved to someone who had a number on their page that was different from everyone else.

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I’ve written about this task before, here and here. As I think about the students I will have next year, I wonder whether we could start our unit on Rigid Motions with this task, instead of ending the unit with it. We’ve been using our new standards for 3 years now in geometry, but it is really only next year that we will have students who had most of the new standards in middle school. I’m beginning to think about how that changes what we’ve been doing.

And so the journey continues, constantly making adjustments to meet the needs of the students we do have and not those we did have …

 
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Posted by on March 10, 2015 in Geometry, Rigid Motions

 

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The Side-Splitter Theorem

CCSS-M G-SRT.B.4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.

One of the theorems that we prove in our unit on Dilations is the Side-Splitter Theorem.

At the end of class one day, I checked to see how intuitive the Side-Splitter Theorem is.

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Not so intuitive, apparently.

(I didn’t show them the correct answer.)

But we started class the next day with our diagram and our learning goal, look for and make use of structure.

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What do you see that isn’t pictured?

We set the learning mode to individual. Students worked alone for a minute or two to see what they could see.

I monitored. And selected. And sequenced.

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And then we talked.

Which auxiliary lines help us determine a relationship between the segments?

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Our work with dilations and similarity has been mostly with triangles. Do you see any similar triangles? Why are they similar?

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We generalized our results. And then I sent back the original question.

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And so the journey to make sense out of what isn’t always intuitive continues …

 
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Posted by on January 21, 2015 in Dilations, Geometry

 

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Angle Bisection and Midpoints of Line Segments

As we finished Unit 2 on Tools of Geometry this year, I looked back at Illustrative Mathematics to see if a new task had been posted that we might use on our “put it all together” day before the summative assessment.

I found Angle Bisection and Midpoints of Line Segments.

I had recently read Jessica Murk’s blog post on an introduction to peer feedback, and so I decided to incorporate the feedback template that she used with the task.

The task:

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What misconceptions do you anticipate that students will have while working on this task?

 

What can you find right about the arguments below? What do you question about the arguments below?

Student A:

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Student B:

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Student C:

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Student D:

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Student E:

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Student F:

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Student G:

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Student H:

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Student I:

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Student J:

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The misconception that stuck out to me the most is that students didn’t recognize the difference between parts (a) and (b). I’ve wondered before whether we should still give students the opportunity to recognize differences and similarities between a conditional statement, its converse, inverse, contrapositive, and biconditional. We decided as a geometry team to continue including some work on building our deductive system using logic, even though our standards don’t explicitly include this work. We know that our standards are the “floor, not the ceiling”. We did this task before our work on conditional statements in Unit 3, and so students didn’t realize that, essentially, one statement was the converse of the other. Which means that what we start with (our given information) in part (a) is what we are trying to prove in part(b). And vice versa.

The feedback that students gave was tainted by this misconception.

Another misconception I noticed more than once is that while every point on an angle bisector is equidistant from the sides of the angle, students carelessly talked about the distance from a point to a line, not requiring the length of the segment perpendicular from the point to the line and instead just noting that that the lengths of two segments from two lines to a point are equal.

It occurred to me mid-lesson that maybe we should look at some student work together to give feedback. (This happened after I saw the “What he said” feedback given by one of the students.)

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I have the Reflector App on my iPad, but between the wireless infrastructure in my room for large files like images and my fumbling around on the iPad, it takes too long to get student work displayed on the board. A document camera would be helpful. But I don’t have one. And I’m not sure how I’d get the work we do through the document camera into the student notes for the day. So I actually did take a picture or too, use Dropbox to get the pictures from my iPad to my computer, and then displayed them on the board using my Promethean ActivInspire flipchart so that we could write on them. And then a few of those were so light because of the pencil (and/or maybe lack of confidence that students had while writing) that the time spent wasn’t helpful for student learning.

Looking back at Jessica’s post, I see that her students partnered to give feedback, since they were just learning to give feedback. That might have helped some, but I’m not sure that would have “fixed” this lesson.

So while I can’t say with confidence that this was a great lesson, I can say with confidence that next year will be better. Next year, I’ll give students time to write their own arguments, and then I’ll show them some of the arguments shown here and ask them to provide feedback together to improve them. Maybe next year, too, I’ll add a question to the opener that gives a true conditional statement and a converse and ask whether the true conditional statement implies that the converse must be true, just so they have some experience with recognizing the difference between conditional statements and converses before we try this task.

And so the journey continues, this time with gratefulness for “do-overs”.

 

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More Reflections

We spent a second day on reflections in our Rigid Motions unit.

We had started by really thinking about what a reflection buys us mathematically, and so we were ready to generalize what happens when we reflect an image over the lines y=x and y=-x and over the lines y=a and x=b.

I used a Numerical Input Quick Poll to let students explore the reflection of a point over the line y=x. The graph side of the page was dynamic. So students could grab and move point R, observe what happens to R’, and then make a conjecture about what would happen if we reflected the point (13,-8) about the line y=x (note that the point was purposefully not within the given window). Students moved to pencil and paper to reflect a triangle about the line y=x and were ready to generalize what happens when we reflect the point (x,y) about the line y=x: (x,y)→(y,x).

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When we started thinking about a reflection about the line y=–x, I asked students to predict their generalization before interacting with the TNS document.

One student said that (x,y)→(–y,–x). Did anyone have a different conjecture? Another student wanted to use words instead of symbols for her conjecture: x and y are going to flip and be opposite in sign. We went through the Quick Poll, reflected a triangle about the line using pencil and paper, and generalized the results.

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Next up: reflecting a triangle about the line y=1. The goal was to eventually generalize our results for reflecting the point (x,y) about the line y=a. We started on paper. Students could pretty easily get the coordinates for reflecting the triangle about the line, and then they began to look for regularity in repeated reasoning. Can you find a pattern for what happens?

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I’m glad that I read Smith & Stein’s “5 Practices for Orchestrating Productive Mathematics Discussions” this summer: anticipate, monitor, select, sequence, connect. I have talked in detail about them in a previous post.

Sequencing was important here. I had some students who generalized the reflection of the point (x,y) about the line y=a. But I had some students who generalized the reflection of the point (x,y) about the line y=1. And I had some students who were making sense of the relationship between the coordinates of the image, the pre-image, and the equation of the line who were thinking more geometrically and quantitatively about distance and were not quite ready to reason abstractly. I didn’t start by calling on the students who had generalized completely. And I might have had I not been deliberately engaged in the 5 Practices. We started with the students who began to articulate what was happening geometrically.  The x-coordinates of the pre-image and image are equal. The distance from the pre-image point to the line of reflection has to equal the distance from the line of reflection to the image point.

F.J. shared his work. He began to reason abstractly and quantitatively, using Y for the y-coordinate of the pre-image, L for the y-coordinate of a point on the line of reflection, D for the distance between the pre-image and the line, and then y’ for the y-coordinate of the image. He worked out how to get y’ in two steps.

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Then B.E. shared what his group had worked out. They focused on the pattern when reflecting a point about the line y=1. What was happening every time? What is the same going from 5 to –3, from 4 to –2, and from 2 to 0? They tried it going from their pre-image to the image. They tried it going from the image to the pre-image. They generalized that result to be y’=–y+2.

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So what happens if we are reflecting a point about a horizontal line that is different from y=1? M.A. wanted to share what her table discussed. They had generalized the result to be (x,y)→(–y,–y+2) for y=1, but upon further reflection decided that (x,y)→(–y,–y+2a) would work for any line y=a.

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Did anyone else get that result? R.E. said that their group used the expression 2a–y instead, so we talked about whether those were equivalent. He actually knew they were equivalent, but I think he really wanted me to take a picture of his work to show on the board 🙂Image

So what did we learn?

I sent two Quick Polls to formatively assess student learning.

The first was a “Drop Points” poll where students placed a point at the image of the point (–4,3) about the line y=–2.

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ImageThe second was a “Numerical Input” poll where students entered the coordinates of the image of the point (3,–4) about the line y=7. About half of the students used their general rule to get the coordinates of the image, and the others sketched a graph of the point and the line to get the coordinates of the image.

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So I’ll be honest. After all of this work, I’m not actually sure it is important for students to know a rule for reflecting a point about the line y=a or x=b. What is important is the work the students did to come up with a rule. Within a minute of being asked to explore a rule, M.A. said that she couldn’t do it. M.A. is not used to having to make sense of problems and persevere in solving them. She usually figures out how to solve problems quickly. She isn’t used to being challenged to think in her mathematics class. The bell rang during our work the first day, and so we left class without a rule (easing the hurry syndrome), taking time to think more about it before reconvening for the next class. M.A. came back to class the next time and announced that she had come up with a rule.

We could take a look back through the Standards for Mathematical Practice and find evidence of most of them in this exploration. Ultimately, that is what is important to me…and not that students remember that they can use (x,y)→(–y,–y+2a) to reflect a point about the line y=a.

And so the journey continues…

 
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Posted by on August 30, 2013 in Coordinate Geometry, Geometry, Rigid Motions

 

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Hopewell Geometry – Right Triangle

A while back we gave our students the Mathematics Assessment Project assessment task called Hopewell Geometry.

I have just finished reading Smith and Stein’s book 5 Practices for Orchestrating Productive Mathematics Discussions, and so I have been thinking a lot about sequencing.

Students are given a set of Hopewell Triangles (along with a historical explanation, which you can see at the link above).

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And they are given a diagram with the layout of some Hopewell earthworks.

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The second question is for students to explain whether or not the shaded triangle is a right triangle.

With which student explanation would you start in a class discussion?

How would you sequence the student explanations? Are there any you would be sure to include? Some you would leave out?

Student A

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Student B

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Student C

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Student D

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Student E

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Student F

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Student G

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Student H

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Student I

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Student J

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Student K

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Student L

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And so the journey continues …

 
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Posted by on June 18, 2013 in Geometry, Right Triangles

 

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