Is This a Rectangle?

One of our learning intentions in our Coordinate Geometry unit is for students to be able to say I can use slope, distance, and midpoint along with properties of geometric objects to verify claims about the objects.

G-GPE. Expressing Geometric Properties with Equations

B. Use coordinates to prove simple geometric theorems algebraically

1. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

We recently used the Illustrative Mathematics Task Is This a Rectangle to provide students the opportunity to practice.

We also used Jill Gough’s and Kato Nims’ visual #ShowYourWork learning progression to frame how to write a solution to the task.

How often do we tell our students Show Your Work only to get papers on which work isn’t shown? How often do we write Show Your Work next to a student answer for which the student thought she had shown her work? How often do our students wonder what we mean when we say Show Your Work?

The Show Your Work learning progression begins to help students understand what we mean when we say Show Your Work. I have seen it empower students to ask each other for feedback on their work: Can you read this and understand it without asking me any questions? It has been transformative for my AP Calculus students as they write Free Response questions that will be scored by readers who can’t ask them questions and don’t know what math they can do in their heads.

We set the timer for 5 minutes of quiet think time. Most students began by sketching the graph on paper or creating it using their dynamic graphs software. [Some students painfully and slowly drew every tick mark on a grid, making me realize I should have graph paper more readily available for them.]

They began to look for and make use of structure. Some sketched in right triangles to see the slope or length of the sides. Some used slope and distance formulas to calculate the slope or length of the sides.

I saw several who were showing necessary but not sufficient information to verify that the figure is a rectangle. I wondered how I could steer them towards a solution without telling them they weren’t there yet.

I decided to summarize a few of the solutions I was seeing and send them in a Quick Poll, asking students to decide which reasoning was sufficient for verifying that the figure is a rectangle.

Students discussed and used what they learned to improve their work.

It occurred to me that it might be helpful for them to determine the Show Your Work level for some sample student work. And so I showed a sample and asked the level.

But I didn’t plan ahead for that, and so I hurriedly selected two pieces of student work from last year to display. I was pleased with the response to the first piece of work. Most students recognized that the solution is correct and that the work could be improved so that the reader knows what the student means.

I wish that I hadn’t chosen the second piece of work. Did students say that this work was at level 3 because there are lots of words in the explanation and plenty of numbers on the diagram? Unfortunately, the logic is lacking: adjacent sides perpendicular is not a result of parallel opposite sides. Learning to pay close enough attention to whether an argument is valid is good, hard work.

Tasks like this often take longer than I expect. I’m not sure whether that is because I am now well practiced at easing the hurry syndrome or whether that is because learning to Show Your Work just takes longer than copying the teacher’s work. And so the journey continues …

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Angle Bisection and Midpoints of Line Segments, Take Two

Last year’s lesson using the Illustrative Mathematics task Angle Bisection and Midpoints of Line Segments had plenty of room for improvement. This year, students left with a better understanding of proof and giving feedback on proof.

Our goal? SMP3: I can construct a viable argument and critique the reasoning of others.

Students started by reading through both parts of the proof, noticing and wondering.

I’ll admit, I really wanted someone to notice that parts a and b were converses. (I didn’t expect them to use that language … I was just looking for anything about the parts being “opposite”.) I wasn’t ready to tell them, so I specifically asked, “what is the difference between parts a and b”.

In triangle a thhey already give you the midpoint of line QR and asking you to draw the angle bisector, but in triangle b they are giving you the angle bisector and are asking you to find the midpoint of line QR.        1

In part a, you’re trying to find the angle bisector from the midpoint, but in part b, you’re trying to find the midpoint using the angle bisector. So they’re basically the opposite of each other, but you have the same point and the same line. They were just found in different ways.  1

Part a starts of with finding the midpoint to segment QR and then creates a line from P to go through the midpoint while part b starts with an angle bisector PS then goes to see if it intersects the midpoint to of segment QR.       1

in part a your contructing a midpoint, in part b you are constructing a bisector         1

In part a you are justifying that PM is a bisector of QPR, but in part b you are justifying that PS meets QR at its midpoint.         1

The difference is that part a to show that the bisector will go through the midpoint, while part b is asking to show that the bisector does go through the midpoint rather than just some random point.       1

In part A the midpoint is labeled M and in part B the midpoint is labeled S, but it is the same point. Also part A and part B make the same image, but the just switch the order they made the image. like finding the midpoint first then the bisector, vice versa    1

Students spent a few minutes creating an argument for part a. Then we looked at some of the student work from last year to critique the arguments.

In Embedding Formative Assessment, Dylan Wiliam suggests that students learning how to give feedback should start with anonymous student work … and eventually move towards student work from peers in the same class. This seemed to work well for this task. Additionally, I had the opportunity to purposefully select and sequence the work for giving feedback ahead of time, which gave us more time for learning during class.

My geometry students are 1:1 this year with MacBook Airs, and so I sent a PDF of the student work samples through TI-Nspire Navigator for Networked Computers, which gave them an up-close look at the student work instead of my having to stand at the copy machine for a while or students trying to decipher from it only being displayed on the board at the front of the room.

We looked at one student work sample at a time using Think-Pair-Share to make student thinking visible. What feedback would you give this student?

M is the same distance from Q and R, but points on the angle bisector are the same distance from the sides of the angle. How do you know M is the same distance from ray PQ as it is from ray PR? We represent distance from a point to a line as the length of the segment perpendicular from the point to the line.

What is a perpendicular bisector of an angle?

What is the difference in saying segment QR is a perpendicular bisector of ray PM and saying ray PM is a perpendicular bisector of segment PM?

Before we looked at the next student work sample, I asked students to practice look for and make use of structure, asking what they saw when segment QR was drawn.

An angle bisector.

A midpoint.

Triangles.

How many triangles?

3 triangles.

What kind of triangles?

The big one is isosceles.

What do you know about isosceles triangles?

They have two congruent angles.

Eventually we showed that the two triangles were congruent using SAS.

Then we looked at another student work sample.

This student noted that the triangle is isosceles, but jumped from one pair of corresponding congruent sides to the angle bisector.

And one other student work sample, where the student noted that the triangles were congruent, but didn’t give a reason why.

Students looked at part b for a few minutes. Then we looked at one last student work sample. What do you wonder about this argument?

Does S have to be the midpoint?

After working for a few more minutes, students gave each other feedback and then revised their argument based on the feedback.

Are we going to look at a correct argument for this?

Will you check mine to be sure that it is right?

Last year, students didn’t care so much whether their argument was correct, nor did they care about seeing a “viable argument”. Somehow, figuring out how to improve some of the arguments for part a got them more interested in their argument for part b.

We plan to look at the following five arguments tomorrow.

With what do you agree?
With what do you disagree?

And so the journey continues … thankful for do-overs from one year to the next.

 

Popcorn Picker + #ShowYourWork

Towards the end of our geometry course last year, we focused on students being able to say:

I can show my work.

How often do our students understand what we mean when we say, “show your work”?

Jill Gough’s Show Your Work learning progression has been an important addition to our classroom.

Level 4: I can show more than one way to find a solution to the problem.

Level 3: I can describe or illustrate how I arrived at a solution in a way that the reader understands without talking to me.

Level 2: I can find a correct solution to the problem.

Level 1: I can ask questions to help me work toward a solution to the problem.

A correct solution isn’t enough … we want the reader (and sometimes grader) to understand our solution without having to ask any questions.

We continue to use Dan Meyer’s Popcorn Picker 3-Act, even though I keep thinking we shouldn’t need to do this in high school. The Quick Poll results, however, provide evidence that we aren’t wasting our time.

You can read more about how the 3-Act lesson played out in last year’s post.

My purpose for posting about the lesson again is to consider the value added when we ask students to practice “show your work” and provide students the opportunity to give other students feedback on what they’ve shown … when we provide students an opportunity to practice SMP3, “I can construct a viable argument and critique the reasoning of others”.

We used the “I like …, I wish …, What if (or I wonder) …” protocol for providing feedback.

Kato Nims recently posted Illuminating Success and Growth, where she shares her students’ first attempts at giving feedback to each other this year. She writes, “It is clear to me that as our work together continues that it will be important for me to model how to give effective feedback so that we can all benefit from the unique perspectives that are represented in our classroom.”

Some of the feedback that my students gave each other is helpful, but more of it is not. How do we teach students to give productive feedback to each other? Would my giving feedback on the feedback have been helpful? At what point have we spent too much time on this activity and need to call it “done”? But then how often am I so focused on teaching content that I neglect to provide students the opportunity to grow as learners? Isn’t learning how to give feedback important for all of us? The tasks we choose are so important … which ones will further both our content and our practice learning goals?

And so the journey continues … with many more questions than answers.

 

Visual: SMP-8 Look for and Express Regularity in Repeated Reasoning #LL2LU

Many students would struggle much less in school if, before we presented new material for them to learn, we took the time to help them acquire background knowledge and skills that will help them learn. (Jackson, 18 pag.)

We want every learner in our care to be able to say

I can look for and express regularity in repeated reasoning.
(CCSS.MATH.PRACTICE.MP8)

But…what if I can’t? What if I have no idea what to look for, notice, take note of, or attempt to generalize?

Investing time in teaching students how to learn is never wasted; in doing so, you deepen their understanding of the upcoming content and better equip them for future success. (Jackson, 19 pag.)

Are we teaching for a solution, or are we teaching strategy to express patterns? What if we facilitate experiences where both are considered essential to learn?

We want more students to experience the burst of energy that comes from asking questions that lead to making new connections, feel a greater sense of urgency to seek answers to questions on their own, and reap the satisfaction of actually understanding more deeply the subject matter as a result of the questions they asked.  (Rothstein and Santana, 151 pag.)

What if we collaboratively plan questions that guide learners to think, notice, and question for themselves?

What do you notice? What changes? What stays the same?

Indeed, sharing high-quality questions may be the most significant thing we can do to improve the quality of student learning. (Wiliam, 104 pag.)

How might we design for, expect, and offer feedback on procedural fluency and conceptual understanding?

Level 4
I can attend to precision as I construct a viable argument to express regularity in repeated reasoning.

Level 3
I can look for and express regularity in repeated reasoning.

Level 2
I can identify and describe patterns and regularities, and I can begin to develop generalizations.

Level 1
I can notice and note what changes and what stays the same when performing calculations or interacting with geometric figures.

If we are to harness the power of feedback to increase student learning, then we need to ensure that feedback causes a cognitive rather than an emotional reaction—in other words, feedback should cause thinking. It should be focused; it should relate to the learning goals that have been shared with the students; and it should be more work for the recipient than the donor. (Wiliam, 130 pag.)

[Cross posted on Experiments in Learning by Doing]

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Jackson, Robyn R. (2010-07-27). How to Support Struggling Students (Mastering the Principles of Great Teaching series) (Pages 18-19). Association for Supervision & Curriculum Development. Kindle Edition.

Rothstein, Dan, and Luz Santana. Make Just One Change: Teach Students to Ask Their Own Questions. Cambridge, MA: Harvard Education, 2011. Print.

Wiliam, Dylan (2011-05-01). Embedded Formative Assessment (Kindle Locations 2679-2681). Ingram Distribution. Kindle Edition.

 

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:

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:

Student B:

Student C:

Student D:

Student E:

Student F:

Student G:

Student H:

Student I:

Student J:

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.)

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”.

 

Circles – Student Reflections

Unit 8 Student Reflections

I can statements:

1. I can use relationships between angles and arcs in circles to solve for missing measures. (100% strongly agree or agree)
2. I can use relationships between secants, chords, and tangents in circles to solve for missing measures. (100% strongly agree or agree)
3. I can use similarity to calculate arc length and area of a sector. (92% strongly agree or agree)
4. I can prove relationships between secants, chords, and tangent in circles. (96% strongly agree or agree)

Standards:

Circles: G-C

Understand and apply theorems about circles

2. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

4. (+) Construct a tangent line from a point outside a given circle to the circle.

Find arc lengths and areas of sectors of circles

5. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

 

Which Standard of Mathematical Practice did you use most often in this unit?
In which other Standard of Mathematical Practice did you engage often during the unit?

Students answered that they used make sense of problems and persevere in solving them the most and look for and make use of structure second most.

 

Think back through the lessons. Did you feel that any were repeats of material that you already knew? If so, which parts?
8A Angles and Arcs
8B Tangents and Chords
8C Angle Measures
8D Segment Lengths
8E Pi
8F Arc Length and Sectors
8G Performance Assessment
8H Mastering

• The only part that was a little familiar was 8E pi.
• I’ve never gone in depth with circles, so no, none of the material was repeats, except maybe a few spots when we learned what pi was.
• 8E was a similar of a repeat of what we learned in Algebra because we had to find pi in multiple situations. It made it much easier for me to understand. Before then, I did have a small misunderstanding of what I needed, but after remembering what I had learned in eighth grade and then I was able to understand more. This was a lesson that showed me that algebra and geometry were very identical.
• The Pi lesson also had some repeated information, but it was nice to learn about the Pi ratio in detail.
• I have had little experience up until this year with circles, other than memorizing the formulas.

 

Think back through the lessons. Was there a lesson or activity that was particularly helpful for you to meet the learning targets for this unit? If so, how?
8A Angles and Arcs
8B Tangents and Chords
8C Angle Measures
8D Segment Lengths
8E Pi
8F Arc Length and Sectors
8G Performance Assessment
8H Mastering

• It helped me understand circles more when we used real life situations.
• Pi was an integral part of the unit and it helped me kind of piece things together. Finding degree measurements in a circle was like a puzzle, when you got one piece you could move on to the next.
• I believe the day where we thought through ways to find the center of the circle gave me insight into the relationship between the circle and the accuracy necessary to correctly talk about it. It also gave me a chance to apply what I had learned previously to find an accurate solution.
• All of the lessons were very helpful for the understanding of the chapter, especially 8A, the beginning lesson, which became the “backbone” for the later lessons.
• I think lesson 8B helped a lot because it introduced tangents and chords, which was a totally new concept for me.

Some students feel like the practice assignments really helped them make sense of the unit as a whole:

• All of the homework activities helped me understand the unit.
• I find it wasn’t the lesson itself, but the homework. The homework challenged me to really think and on a few of them I had to search how to do them. It really helped me make sense of problems on the test and on the lessons after some homework.
• The 8H homework was actually probably the most helpful of any of the things I did. I thought I knew this unit pretty well, until I went and made a 5/10 on my first try. It really helped me go back, review, and relearn the things we’d been doing because I had some very skewed ideas about circles before that lesson.

Some students feel like the Performance Assessment tasks really helped them make sense of the unit as a whole:

• The performance assessment was very helpful to me because in a sense it was a combination of all the previous lessons. This also was a way to make myself quicker at solving the problems and figuring out short-cuts. This was a great review to prepare for my test, which I believe that really was an asset for me. The test was much better with the performance assessment.
• The Performance Assessment was very helpful because it let me see which parts of the unit I didn’t understand as well as the others and showed me what I needed to work on before the test.
• Every lesson this unit was helpful, but the Performance Assessment and the Mastering lessons helped the most. In the lessons before these, we learned a couple things, but in these lessons, we learned how to combine everything that we learned to find the correct answers to challenging problems.

What have you learned during this unit?

• I learned how to calculate the arc length of a circle using the relationship of it to the whole circle.
• In this unit I have deepened my knowledge of Pi, which is one of my favorite numbers and how to recognize tangents. I have also learned how to find arc measures and how to find an angle measure in a circle.
• I have learned the relationships between radii, chords, tangents, and secants in circles. I’ve also learned so many different ways to solve for missing links within circles. I can also find missing lengths outside of the circle. I find that this unit was the toughest to this point.
• … I’ve also learned that’s circles are more complicated than they seem at first.
• During this unit, I learned about the relationships between arc lengths and the angles that intercept them. I also learned how tangents and chords relate to circles, and how to figure out their lengths. Another thing I learned was what Pi was and where it comes from. A fourth thing I learned was how to determine the arc length and area of sectors in circles.

 

And so the journey continues … trying to determine what is best for students. Do I take out the pi lesson next year, since students had some knowledge of pi coming in to high school geometry? Or do I leave it, since several students noted that they enjoyed learning about pi in detail? Eventually, I’ll have students who should have an informal derivation of the relationship between the circumference and area of a circle in grade 7 (7.G.B.4), but until then, I’ll probably keep going over at least part of the lesson.

 

Reflections on Unit 1 – Rigid Motions

I created a survey in our Canvas course for students to reflect on what they had learned in this unit.

CCSS-M Standards:

G-CO 3

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

G-CO 2

Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

G-CO 4

Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

G-CO 5

Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

G-CO 6

Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

G-CO 7

Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

I Can statements:

Level 1: I can identify and define transformations and composite transformations.

Level 2: I can perform transformations and composite transformations.

Level 3: I can determine the congruence of two figures using rigid motions.

Level 4: I can apply transformations and composite transformations to figures in the coordinate plane.

Level 4: I can map a figure onto itself using transformations.

Every student answered “strongly agree” or “agree” to each of the I can statements.

We focused on learning math using the Standards of Mathematical Practice. I asked students which practice they used most often during this unit.

55% of students said that they most used make sense of problems and persevere in solving them.

The students have some experience with transformations in middle school. Could they tell that we were deepening their understanding of rigid motions? Did any of the lessons feel like repeats of material they had previously learned?

• None of them were complete repeats because I learned about them more detailed than before.
• Before now, I had a very general knowledge of transformations, but actually putting this knowledge to practice was new.
• I had already learned most of the basics of transformations. I knew reflections were flips, translations were slides, and so on and so on. However, I’d never gone in depth with the mathematical reasoning behind these transformations and their causes and effects, which brought a whole new light on the subject.
• I believe that the basic ideas of 1A-E I knew but I still had room to grow on the knowledge these subjects.
• I felt that I already knew how to do translations and reflections, but in this unit, I noticed new things about them such as that a reflection changes the orientation of the pre-image and that a translation keeps the same orientation as the pre-image. I also knew about rotations, but I wasn’t great at actually preforming a rotation.

Most students answered that What’s My Rule was most helpful. Several others noted that the lesson on Rotations was helpful.

• I learned how to really think about math differently. I learned to double check my work- A LOT. I learned how to tell counterclockwise and clockwise apart. I learned translations and how they move, reflections and how they work, and rotations and how they twist. I learned about matrices and how complex they can be, although I’m still a little confused (I think I have it, but I’m not so sure). I learned the first parts of geometry.
• I have learned how to reflect a figure across the y=x line and the y=-x line.
• I learned how to map figures onto itself and reflect on lines other the the x and y.
• I learned some formulas to help with reflections. I also learned what directions a figure should be rotated if I’m only give the degrees to rotate.
• I learned how to see if shapes on a coordinate plane are congruent, and how to map something onto itself using rigid motions.
• A lot of things in this unit were repeats of what I have learned in the past, but there was so much more that I didn’t know about all the transformations. For one, I didn’t even think about the distance from each side to the line of reflection in the past. I just knew the simple things like the points were congruent. I’ve learned a lot and couldn’t type it all. People who don’t learn anything within a whole unit are crazy because you can always learn something even if it’s just a small little thing. I would say that I’ve decided not to give up on hard problems like I would in the past. I didn’t really learn that, but I think it’s important to geometry and, really, everyday life.

I wonder (maybe too often) whether what we are doing in class is important. The student reflections give me evidence that my students are recognizing new content and also connecting it to what they knew before our high school geometry class.

And so the journey continues…