Tag Archives: Illustrative Mathematics Task

Is This a Rectangle?

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?

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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.

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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 …


Posted by on April 11, 2017 in Coordinate Geometry, Geometry, Polygons


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Squares on the Coordinate Grid

I’ve written before about Squares on the Coordinate Grid, an Illustrative Mathematics task using coordinate geometry.

CCSS-M G-GPE.B.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

How do you provide opportunities for your students to practice I can look for and make use of structure?

SMP7 #LL2LU Gough-Wilson

How do you draw a square with an area of 2 on the coordinate grid?

It helped some students to start by thinking about what 2 square units looks like, which was easier to see in a non-special rectangle.

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What’s true about the side length of a square with an area of 2?

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How could we arrange 2 square units into a square?

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How do you know the figure is a square? Is it enough for all four sides to be square root of 2?

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CC made his thinking visible by reflecting on his learning after class:

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“Now drawing the square root of two exactly on paper is nearly impossible unless you know how to use right triangles.”

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Posted by on September 21, 2016 in Coordinate Geometry, Geometry


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Reflected Triangles + #ShowYourWork

I’ve used the Illustrative Mathematics task Reflected Triangles for several years now. This year students practiced “Show Your Work” (from Jill Gough) along with coming up with a correct solution.

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

Which of the directions can you follow exactly to construct the line of reflection? Are there any that need clarification?

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Students wrote first and then constructed their lines of reflection. I used the Class Capture feature of TI-Nspire Navigator to watch. We’ve spent longer on this task in the past, but this year, I only had about 5 minutes for a whole class discussion. Whose work would you select for the whole class to see?

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We started with Kaelon’s construction.

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How is Holly’s construction different? Can you tell what Holly did?

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How is Phillip’s different? Can you tell what he did?

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We are well on our way towards learning how to “describe or illustrate a solution in a way that the reader understands without talking to me”, as the journey continues …


Posted by on September 8, 2015 in Angles & Triangles, Geometry, Rigid Motions


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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 …


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


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The Line of Reflection

I’ve blogged about the Illustrative Mathematics task Reflected Triangles before. I really like that it asks students to determine the line of reflection given the pre-image and image instead of determining the image given the pre-image and line of reflection.

△ABC has been reflected across a line into the blue triangle. Construct the line across which the triangle was reflected. Justify your conclusion.

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How would you construct the line of reflection?

Work on your mathematical flexibility to come up with more than one way to construct the line of reflection.

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I used Class Capture to monitor student work and selected some students to share their approach with the whole class.

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Many students approached the task like Paris:

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What can we learn about the relationship between the pre-image, image, and line of reflection from the additional line in Max’s diagram?

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I’ve always used this task towards the end of our unit on Rigid Motions, but I am thinking about using it earlier in the unit next year. Maybe the task itself could be an #AskDontTell approach for students learning what we want them to learn about the relationship between the pre-image, image, and line of reflection. We will see next year, as the journey continues …

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


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Dilating a Line

CCSS-M G-SRT.A.1. Verify experimentally the properties of dilations given by a center and a scale factor:

  1. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
  2. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

CCSS-M G-C.A.1. Prove that all circles are similar.

It probably shouldn’t surprise me that my students have a difficult time dilating lines and circles. Most of the objects that we transform in mathematics are polygons, and we often think about what happens to them in terms of their vertices, neglecting to note what’s happening to the other points on the polygons or the sides of the polygons.

We use the Illustrative Mathematics task Dilating a Line. For an animation of dilating a line, see The Mathematics Common Core Toolbox.

Suppose we apply a dilation by a factor of 2, centered at the point P, to the figure below.


Can you anticipate misconceptions that students will have?

As I was monitoring student work, I saw several approaches that I wanted the class to see. I took some pictures and thought carefully about whether to start with one that was correct or one that was incorrect.

I decided to start with a student who had used her ruler as a measuring tool and not just as a straightedge. FS drew the line that contains PA past A, measured PA, and then marked off that same measurement for AA’. She did the same for PB and BB’ and the same for PC and CC’. She hadn’t yet drawn the line through A’, B’, and C’. What will be true about that line compared to the line through A, B, and C?

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More than one student said the line would double. What does it mean for a line to double?

SC had a similar approach, except that she didn’t use her ruler to measure, she used her compass to measure. She demonstrated how she measured PA and marked AA’ the same length.

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Then I showed BB’s work. What do you think?

What do you like about her work?

What do you wonder about her work?

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BB notes that A’B’ is double AB in her answer to part (c), but her diagram isn’t convincing.

What do you think about BK’s work?

What do you like about his work?
What do you wonder about his work?

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One of the other explorations was “How would you dilate circle C by a scale factor of 3?”


Here’s the work of one student.

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Which made me realize I had not specified that A was the center of dilation.

And the last exploration was “Given any two circles, can you always find a dilation that maps one circle onto another?”


Which we didn’t get to that day.

I’ve written about Dilating a Line before, but the experience of the task changes each year, as the journey of teaching and learning continues with a different community of learners …

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Posted by on January 12, 2015 in Circles, 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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