Category Archives: Coordinate Geometry

Blending Technology with Paper and Pencil

My geometry class is 1:1 this year; each student has her own MacBook Air. Students share responses to questions digitally in class using TI-Nspire Navigator for Networked Computers. Students explore mathematics using TI-Nspire dynamic graphs and geometry software. Students explore mathematics and share responses digitally using Demos Activity Builder. We use Canvas, an online learning management system, for assignments. We use Google Drive for sharing electronic documents with each other, and we use MathXL, online homework with built-in learning help, to practice mathematics. What place does pencil and paper have in my students’ learning and understanding of mathematics?

Even though many of the tasks that my students do for geometry take place digitally, I am convinced that pencil and paper plays an important role in how much mathematics my students not only learn but also remember. In a Wall Street Journal article, “Can Handwriting Make You Smarter?“, Robert Lee Hotz reports that students who take notes by hand usually outperform students who type notes when assessed more than one day after the class period. Students who type notes quickly type everything the professor says, but students who handwrite notes have to process the information while they are hearing it to select what is important to remember (Hotz 2016).

Hotz cites the work of Mueller and Oppenheimer published in Pyschological Science. Their research studies showed that students who took notes by hand performed better on conceptual questions than those who took notes on a laptop. Students performed about the same on factual questions. Their hypothesis for why is that students who take notes by hand choose which information is important to include in their notes, and so they are able to study “more efficiently” than those who are reviewing an entire typed lecture (Mueller and Oppenheimer 2014). Note: These studies are on college students; I have found little research on grade school students.

For several years now, my students and I have been learning how to learn mathematics using the Standards for Mathematical Practice. MP8, “look for and express regularity in repeated reasoning”, has pushed me to think about having students record what they see instead of just noticing and discussing it.

SMP8 #LL2LU Gough-Wilson

One of the ways that I’ve learned to talk about “look for and express regularity in repeated reasoning” is to ask students to notice what changes and what stays the same as we take a dynamic action on a geometric figure. Consider a recent learning episode from my classroom.

Students were told that our learning intention was “I can look for and express regularity in repeating reasoning”. The content was conceptual development of the equation of a circle in a coordinate plane using the Pythagorean Theorem. I did not share that specific content with students up front, however, because I wanted it to be revealed as the lesson progressed. I showed them a dynamic right triangle in the coordinate plane.

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What changes? What stays the same?

I could have let them simply discuss what they noticed. But instead I asked them to “Notice & Note”, using words, pictures, and numbers to write and sketch what they saw.

Then I asked them to share what they noticed with a partner and add to their own notes as desired.


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Our classroom discussion revealed that the equation of the circle formed by tracing point P was x2 + y2 = 52.

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Students continued to “Notice & Note” as they moved a circle around in the coordinate plane. What changes? What stays the same?


Notice & Note 3Notice & Note 4

As we moved P around in the coordinate plane, and then as they later moved the circle around in the coordinate plane students noted what they saw. Eventually, students generalized the center-radius form of an equation.

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Notice & Note, by Kylene Beers and Robert E. Probst, is a guide of signposts (strategies) for close reading of text. Students are taught signposts to notice while they are reading, and they are asked to stop reading and note what the signpost might imply. “Again & Again” is one signpost. Do you notice an event in the text that keeps happening again and again? Do you notice a phrase in the text that is repeated again and again? Stop reading, note it, and think about what that might mean (Beers and Probst 2013). How might we take advantage of the ways that students are learning to read text in their English Language Arts (ELA) classes to guide students in inquiry based exploration of mathematics?

In her online course, Sunni Brown, author of The Doodle Revolution, states that “Tracking content using imagery, color, word pictures and typography can change the way you understand information and also dramatically increase your level of knowledge and retention” (Brown 2016). How do we make tracking content using words, pictures, and numbers a reality in the 1:1 classroom? My experience is that it doesn’t happen without deliberate emphasis on its importance.

In Reading Nonfiction, Beers and Probst write “When students recognize that nonfiction ought to challenge us, ought to slow us down and make us think, then they’re more likely to become close readers” (Beers and Probst 2016). Our ELA counterparts are on to something. Effective classroom instruction is not just about creating learning episodes for our students to experience the mathematics using the Math Practices. Effective classroom instruction incorporates practices that will help students remember what they are learning longer than for the next test.


As I think about our district’s continued implementation of 1:1 technology, I am convinced that we need to pay attention to when we are asking, encouraging, and requiring students to use pencil and paper to create a record of what they are learning. I am interested in thinking more about how we might blend the use of dynamic graphs and geometry software with Notice & Note – using words, pictures, and numbers, along with color, so that students not only have a record of what they are learning but also have a better chance of remembering it later. And so, the journey continues …


Beers, G. Kylene, and Robert E. Probst. Notice & note: Strategies for close reading. Portsmouth: Heinemann, 2013. Print.

Beers, G. K., & Probst, R. E. (2016). Reading nonfiction: Notice & note stances, signposts, and strategies. Portsmouth: Heinemann.

Brown, S. (n.d.). Visual Note-Taking 101 / Personal Infodoodling™. Retrieved April 25, 2016, from

Hotz, Robert Lee. “Can handwriting make you smarter?” The Wall Street Journal. 04 Apr. 2016. Web. 25 Apr. 2016.

Mueller, P. A., and D. M. Oppenheimer. “The pen is mightier than the keyboard: Advantages of longhand over laptop note taking.” Psychological Science 25.6 (2014): 1159-168. Web.


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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 …

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


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Hinge Questions: Dilations

Students noticed and noted.

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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 18th National Conference (2010) “Innovation that works: research-based strategies that raise achievement”.

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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.

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Next we looked at this question.

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Students worked on paper first.

Then some explored with technology.

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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.

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


Posted by on December 20, 2016 in Coordinate Geometry, Dilations, Geometry


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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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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”?

SMP8 #LL2LU Gough-Wilson

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.

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What is significant about the coordinates of the centroid?

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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.

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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 …


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What’s My Rule?

We practice “I can look for and make use of structure” and “I can look for and express regularity in repeated reasoning” almost every day in geometry.

This What’s My Rule? relationship provided that opportunity, along with “I can attend to precision”.

What rule can you write or describe or draw that maps Z onto W?


As students first started looking, I heard some of the following:

  • positive x axis
  • x is positive, y equals 0
  • they come together on (2,0)
  • (?,y*0)
  • when z is on top of w, z is on the positive side on the x axis


Students have been accustomed to drawing auxiliary objects to make use of the structure of the given objects.

As students continued looking, I saw some of the following:

Some students constructed circles with W as center, containing Z. And with Z as center, containing W.

Others constructed circles with W as center, containing the origin. And with Z as center, containing the origin.

Others constructed a circle with the midpoint of segment ZW as the center.

Another student recognized that the distance from the origin to Z was the same as the x-coordinate of W.

And then made sense of that by measuring the distance from W to the origin as well.

Does the redefining Z to be stuck on the grid help make sense of the relationship between W and Z?



As students looked for longer, I heard some of the following:

  • The length of the line segment from the origin to Z is the x coordinate of W.
  • w=((distance of z from origin),0)
  • The Pythagorean Theorem

Eventually, I saw a circle with the origin as center that contained Z and W.

I saw lots of good conversation starters for our whole class discussion when I collected the student responses.

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And so, as the journey continues,

Where would you start?

What questions would you ask?

How would you close the discussion?


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Team Sorting – Coordinate Geometry

Last year we sorted students into teams using cards at the beginning of each unit.

All cards can be found at this link.

The following are some of the comments that we overheard when students sorted for their Coordinate Geometry unit.

Equations! I can do equations.

If this is algebra, I’m going to do great in this unit.

Are we a team because we both have 18x?

If your slope equals -7, you are here.

Oh! We are doing slopes!

Is your slope -5/3?

Do any of you know how to do this?

Ours all look like 4x-y=2.

2015-02-23 08.27.38  2015-02-23 08.34.43 2015-02-23 08.36.092015-02-23 08.33.25When we asked students at the end of the year what to stop, start, keep, and change, many said that we should keep the Team Sorting Cards. They enjoyed changing teams for each unit and getting to know and work with most of the students in the class.

You can read about previous team sorting here and here.

This year’s class has their first test on Wednesday, and so we look forward to their first team sort on Friday (even though our Tools of Geometry/Construction Unit team sort is lame and needs to be changed before then), as the journey continues …

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Posted by on August 30, 2015 in Coordinate Geometry


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