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Category Archives: Coordinate Geometry

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 …

 
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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.

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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.

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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?

What_s_My_Rule.gif

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?

 

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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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The Equation of a Circle

Expressing Geometric Properties with Equations

G-GPE.A Translate between the geometric description and the equation for a conic section

  1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

How do you provide an opportunity for your students to make sense of the equation of a circle in the coordinate plane? We recently use the Geometry Nspired activity Exploring the Equation of a Circle.

Students practiced look for and express regularity in repeated reasoning. What stays the same? What changes?

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It’s a right triangle.

The hypotenuse is always 5.

The legs change.

What else do you notice? What has to be true for these objects?

The Pythagorean Theorem works.

How?

Leg squared plus leg squared equals five squared.

What do you notice about the legs? How can we represent the legs on the graph?

One leg is always horizontal.

One leg is always vertical.

How can we represent their lengths in the coordinate plane?

x and y?

(I think they thought that the obvious was too easy.)

What do x and y have to do with point P?

Oh! They’re the x- and y-coordinates of point P.

So what can we say is always true?

Is there an equation that is always true?

x²+y²=5²

What path does P travel? (This was preceded by – I’m going to ask a question, but I don’t want you to answer out loud. Let’s give everyone time to think.)

And then we traced point P as we moved it about coordinate plane.

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So P makes a circle, and we have figured out that the equation of that circle is x²+y²=5².

I then let them explore two other pages with their teams, one where they could change the radius of the circle and one where they could change the center of the circle.

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And then they answered a few questions about what they found. I used Class Capture to watch as they practiced look for and express regularity in repeated reasoning.

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Here are the results of the questions that they worked.

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What would you do next?

What I didn’t do at this point was differentiate my instruction. It occurred to me as soon as I got the results that I should have had a plan of what to do with the students who got 1 or 2 questions correct. It turns out that it was a team of students – already sitting together – who needed extra support – but I didn’t figure that out until later. Luckily, my students know that formative assessment isn’t just for me, the teacher – it’s for them, too. They share the responsibility in making a learning adjustment before the next class when they aren’t getting it.

We pressed on together – to make more sense out of the equation of a circle. I used a few questions from the Mathematics Assessment Project formative assessment lesson, Equations of Circles 1, getting at specific points on the circle.

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And then I wondered whether we could begin making a circle. I assigned a different section of the x-y coordinate plane to each team. Send me a point (different from your team member) that lies on the circle x²+y²=64. Quadrant II is a little lacking, but overall, not too bad.

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How can we graph the circle, limited to functions?

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How can we tell which points are correct?

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I asked them to write the equation of a circle given its center and radius, practicing attend to precision.

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54% of the students were successful. The review workspace helps us attend to precision as well, since we can see how others answered.

(At the beginning of the next class, 79% of the students could write the equation, practicing attend to precision.)

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I have evidence from the lesson that students are building procedural fluency from conceptual understanding (one of the NCTM Principles to Actions Mathematics Teaching Practices).

But what I liked best is that by the end of the lesson, most students reached level 4 of look for and express regularity in repeated reasoning: I can attend to precision as I construct a viable argument to express regularity in repeated reasoning.

When I asked them the equation of a circle with center (h,k) and radius r, 79% told me the standard form (or general for or center-radius form, depending on which textbook/site you use) instead of me telling them.

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We closed the lesson by looking back at what happens when the circle is translated so that its center is no longer the origin. How does the right triangle change? How can that help us make sense of equation of the circle?

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And so the journey continues, one #AskDontTell learning episode at a time.

 
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Posted by on April 19, 2015 in Circles, Coordinate Geometry, Geometry

 

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