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.

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’.

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.

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’.

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.

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 …

 

Hinge Questions: Dilations

Students noticed and noted.

I wanted to be sure that they could answer a dilations question based on their observations. I had two questions premade in my set of Quick Polls. Which question would you ask?

In the past, I would have asked both questions without thinking.

I am learning, though, to think more about which questions I ask. If we only have time to ask a few questions, which questions are worth asking?

From slide 34 in Dylan Wiliam’s presentation at the SSAT 18^th National Conference (2010) “Innovation that works: research-based strategies that raise achievement”.

I decided to send the second poll. I decided that if they get that one right, they can both dilate a point about the origin and pay attention to whether they are given the image or pre-image. If I had sent the second poll, I wouldn’t know whether they could both do and undo a dilation.

Next we looked at this question.

Students worked on paper first.

Then some explored with technology.

What do you want your students to know about the relationships in the diagram?

What question would you ask to see whether they did?

I asked this question to see what my students were thinking.

And so the journey to write and ask and share and revise hinge questions continues …

 

MP6 – Mapping a Parallelogram Onto Itself

How do you provide your students the opportunity to practice I can attend to precision?

Jill and I have worked on a leveled learning progression for MP6:

Level 4:

I can distinguish between necessary and sufficient language for definitions, conjectures, and conclusions.

Level 3:
I can attend to precision.

Level 2:
I can communicate my reasoning using proper mathematical vocabulary and symbols, and I can express my solution with units.

Level 1:
I can write in complete mathematical sentences using equality and inequality signs appropriately and consistently.

CCSS G-CO 3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

We continued working on our learning intention: I can map a figure onto itself using transformations.

Perform and describe a [sequence of] transformation[s] that will map parallelogram ABCD onto itself.

This task also requires students to practice I can look for and make use of structure. What auxiliary objects will be helpful in mapping the parallelogram onto itself?

The student who shared her work drew the diagonals of the parallelogram so that she could use the intersection of the diagonals as the center of rotation.

Then she rotated the parallelogram 180˚ about that point.

Could you use only reflections to carry a parallelogram onto itself?

You can. How can you describe the sequence of reflections to carry the parallelogram onto itself?

How else could you carry a parallelogram onto itself?

 

MP6 – Mapping a Figure Onto Itself

How do you provide your students the opportunity to practice I can attend to precision?

Jill and I have worked on a leveled learning progression for MP6:

Level 4:

I can distinguish between necessary and sufficient language for definitions, conjectures, and conclusions.

Level 3:
I can attend to precision.

Level 2:
I can communicate my reasoning using proper mathematical vocabulary and symbols, and I can express my solution with units.

Level 1:
I can write in complete mathematical sentences using equality and inequality signs appropriately and consistently.

CCSS G-CO 3:

Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

Our learning intention for the day was I can map a figure onto itself using transformations.

Performing a [sequence of] transformation[s] that will map rectangle ABCD onto itself is not the same thing as describing a [sequence of] transformation[s].

We practiced both, but we focused on describing.

I asked the student who listed several steps to share his work.

1. rotate rectangle 180˚ about point A
2. translate rectangle A’B’C’D’ right so that points A’ and B line up as points B’ and A. [What vector are you using?]
3. Reflect rectangle A”B”C”D” onto rectangle ABCD to get it to reflect onto itself. [About what line are you reflecting?]

What if we want to carry rectangle ABCD onto rectangle CDAB? How is this task different from just carrying rectangle ABCD onto itself?

What about mapping a regular pentagon onto itself?

Many students suggested using a single rotation, but they didn’t note the center of rotation. How could you find the center of rotation for a single rotation to map the pentagon onto itself?

This student used the intersection of the perpendicular bisectors to find the center of rotation, but didn’t know what angle to use for the rotation. How would you find an angle of rotation that would work?

What can you do other than a single rotation?

This student reflected the pentagon about the perpendicular bisectors of one of the side of the pentagon.

The descriptions students gave made it obvious that we needed more work on describing. The next day, we took some of the descriptions and critiqued them. Which students have attended to precision?

It’s good work to distinguish precision from knowing what someone means as we learn to attend to precision. And so the journey continues …

 

Join Me for #ObserveMe?

Robert Kaplinsky recently issued a challenge to welcome other educators into your classroom by posting a sign and asking for feedback on at least one of your classroom goals.

Mississippi has recently released an update to the Professional Growth System, formerly known as the Mississippi Statewide Teacher Appraisal Rubric (M-STAR). The name is a huge upgrade, and thankfully, so are the indicators.

In the Teacher Growth Rubric, the difference between Level 4 and Level 3 for indicator 8, Engages in Professional Learning, is serves as a critical friend for colleagues, both providing and seeking meaningful feedback on instruction.

 

How are you already serving as a critical friend for colleagues?

How might you serve as a critical friend for colleagues?

 

How are you already seeking meaningful feedback on instruction from your colleagues?

How might you seek meaningful feedback on instruction from your colleagues?

 

I wrote Changing Our Practice, Slowly earlier this year reflecting on Dylan Wiliam’s chapter in Embedding Formative Assessment called Your Professional Learning.

Wiliam says, “A far more likely reason for the slowness of teacher change is that it is genuinely difficult.” (Wiliam & Leahy, p. 17) He suggests teachers need to take small steps as we change our practice. We need accountability, and we need support.

The Professional Growth System recognizes that all of us can and should improve our practice in the classroom. Robert provides us a way to make public what small steps we are going to work on and seek the accountability and support we need to truly change our practice.

 

What small step(s) are you working on in your teaching practice this year?

Who will hold you accountable?

What support will you need?

Will you join me by participating in Robert’s #ObserveMe call to action?

Here’s my sign. If I see yours, I’ll be sure to stop by.

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Wiliam, D., & Leahy, S. (2015). Embedding formative assessment: Practical techniques for k-12 classrooms. West Palm Beach, FL: Learning Sciences.

 

MP5: The Traveling Point

How do you give students the opportunity to practice “I can use appropriate tools strategically”?

When we have a new type of problem to think about, I am learning to have students give their best guess of the solution first. I’ve written about The Traveling Point before.

Students sketched the path of point A. How far does A travel?

Students used paper and polydrons, their hands and string.

I sent a poll to find out what they were thinking about the distance traveled.

Students then interacted with dynamic geometry software. Does seeing the figure dynamically move help you better see the path?

Does seeing the path help you calculate how far A travels?

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

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And the journey for my own learning continues. Thanks to Howard for correcting me. The second two moves do not travel a distance of 6, but the length of the circumference of the quarter circle.

One student figured that out by the time the bell rang.

I look forward to redeeming this lesson this year, as the journey continues …

 

When Your Team Is Working Well Together

Have you seen Jill Gough’s blog post Strategic Teaming: leadership, voice, our hopes and dreams? Jill reminds us that strong teams both set norms for their work together and then self assess to ensure that they are functioning within their norms.

How do you provide your students the opportunity to set norms for the work that we have to do together?

I asked my students what it looks like when your team is working well together.

Here’s a wordle of their responses.

I see communicating, cooperating, talking, participating, strategies, but what strikes me most from their suggestions is everyone.

Some lengthier responses from the students:

We are all talking about our strategies. Everyone considers all possibilities presented by the team. Everyone is contributing and listening to what each other has to say, respecting each other. We communicate reasons the answers may be correct or wrong. We will work together to figure out multiple solutions, or the one correct solution, or if there is no solution.

We’ve agreed to these norms.

Everyone …

Respects

Contributes

Listens

Questions

Collaborates

Communicates

Since I want to be transparent about formative assessment being for students as well as teachers, I showed them Popham’s levels of formative assessment.

We are working well together when the whole class is using formative assessment (and not just the teacher). We want all students in our class to meet the learning goals. Not just the “smartest”; not just the fastest. This isn’t survival of the fittest where some can adapt and others will grow extinct. Everyone can learn. Everyone will learn.

The start of another school year has come and gone as the journey continues …

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Popham, W. James. Transformative Assessment. Alexandria, VA: Association for Supervision and Curriculum Development, 2008. Print.

 

MP8: The Centroid of a Triangle

We had been working on a unit on Coordinate Geometry.

How do you give students the opportunity to practice “I can look for and express regularity in repeated reasoning”?

When we have a new type of problem to think about, I am learning to have students estimate the answer first.

I asked them to “drop a point” at the centroid of the triangle. We looked at the responses on the graph first and then as a list of ordered pairs.

What is significant about the coordinates of the centroid?

Students then interacted with dynamic geometry software.

What changes? What stays the same?

Do you see a pattern?
What conjecture can you make about the relationship between the coordinates of the vertices of a triangle and the coordinates of its centroid?

Some students needed to interact on a different grid setup to see a relationship.

After a few minutes, I sent another poll to find out what they figured out.

And then we confirmed student conjectures as a whole class.

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

 

MP8: The Medians of a Triangle

How do you give students the opportunity to practice “I can look for and express regularity in repeated reasoning”?

When we have a new type of problem to think about, I am learning to have students estimate the answer first.

I asked for their estimate in two slightly different problems because I wanted them to pay attention to what was given and what was asked for.

Students then interacted with dynamic geometry software.

What changes? What stays the same?

Do you see a pattern?
What conjecture can you make about the relationship between a median of a triangle and its segments partitioned by the centroid?

As students moved the vertices of the triangle, the automatic data capture feature of TI-Nspire collected the measurements in a spreadsheet.

I sent another poll.

And then we confirmed student conjectures on the spreadsheet.

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

 

Notice and Note: The Equation of a Circle

I wrote in detail last year about how our students practice I can look for and express regularity in repeated reasoning to make sense of the equation of a circle in the coordinate plane.

This year we took the time not only to notice what changes and what stays the same but also to note what changes and what stays the same.

Our ELA colleagues have been using Notice and Note as a strategy for close reading for a while now. How might we encourage our learners to Notice and Note across disciplines?

Students noticed and noted what stays the same and what changes as we moved point P.

They made a conjecture about the path P follows, and then we traced point P.

We connected their noticings about the Pythagorean Theorem to come up with the equation of the circle.

Students moved a circle around in the coordinate plane to notice and note what happens with the location of the circle, size of the circle, and equation of the circle.

And then most of them told me the equation of a circle with center (h,k) and radius r, along with giving us the opportunity to think about whether square of (x-h) is equivalent to the square of (h-x).

And so the journey continues … with an emphasis on noticing and noting.