Ask, Don’t Tell (Part 4): The Equation of a Circle

  • Ask, Don’t Tell (Part 4): The Equation of a Circle

    By Jennifer Wilson, posted July 6, 2015 –

    I used to tell my students how to write the equation of a circle, given its center and radius. Then I would give them the center and radius of a circle and ask for an equation. Now I provide my students an opportunity to figure it out by practicing The Common Core’s Standard for Mathematical Practice 8: Look for and express regularity in repeated reasoning.

    Jill Gough and I have worked this year on leveled learning progressions for giving students a path to using the Standards for Mathematical Practice when they don’t know where to start. This is the progression that we created for SMP 8.

    We used a Geometry Nspired activity as a guide for our exploration. A good GeoGebra exploration is Equations of Circles.

    2015-07-06 art1b 

    We started by noting what changes and what stays the same as we move point P around in the coordinate plane:           

    • The triangle is always right.

    • The hypotenuse is always 5.

    • The legs change.

    • One vertex is always at the origin.

    Then I ask some questions: What path does P follow? What can we say that’s always true about the coordinates of P? We trace point P as it moves around the coordinate plane and students begin to generalize the results, connecting the equation of the circle to the Pythagorean theorem: x2 + y2 = 52.

    Eventually, students move the circle around in the coordinate plane, changing the radius and center, noticing and noting how the equation of the circle changes. How does the equation of the circle change when the center is translated from the origin?

    Finally, I ask my students how to write the equation of a circle, given its center and radius.

    2015-07-06 art2

     

    “Ask, Don’t Tell” learning opportunities allow the mathematics that we study to unfold through questions, conjectures, and exploration. “Ask, Don’t Tell” learning opportunities begin to activate students as owners of their learning.

    What are some of your favorite places to find “Ask Don’t Tell” learning opportunities for your students? Here are some of ours:

    • We use many of the formative assessment lessons from the Shell Center.

    • Mary Bourassa started a site for sharing Which One Doesn’t Belong? items.

    • Nanette Johnson, Robert Kaplinsky, and Bryan Anderson have created a site to share Open Middle problems.

    • Jo Boaler and the Youcubed team have shared a Week of Inspirational Math for beginning the school year by inspiring students with inquiry-based activities.

    Toward the end of the school year, one student reflected, “I think that interacting in class helped me further my learning experience. I was nervous to speak up because I was afraid of being wrong but now I understand that instead of getting in trouble for being wrong, the others in my class will help me find the right answer without actually telling me the answer.”

    What #AskDontTell opportunities do and can you provide?

    AU Wilson Jennifer JENNIFER WILSON, http://www.easingthehurrysyndrome.wordpress.com, a National Board Certified Teacher, teaches and learns mathematics at Northwest Rankin High School and is a curriculum specialist at the Rankin County School District in Brandon, Mississippi. She is an instructor for TI’s Teachers Teaching with Technology (T3) program. She enjoys learning alongside the Illustrative Mathematics community, and she is a recipient of the Presidential Award for Excellence in Mathematics and Science Teaching.

     

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    Jennifer Wilson - 7/17/2015 1:53:03 PM
    Hi, Peggy. I am glad that it is helpful!

    Peggy Welch - 7/9/2015 5:53:12 PM
    Jennifer, this is a GREAT resource for me. I am developing an AgSTEM unit for Precision Farming. One of the activities is a kinesthetic modeling of how GPS works. After students model the procedure, then they draw a diagram of the three intersecting circles to scale and calculate the equations of the 3 circles. I will definitely use the progression that you and Jill developed when presenting my activity.