Ask, Don’t Tell (Part 2): Pythagorean Relationships

  • Ask, Don’t Tell (Part 2): Pythagorean Relationships

    By Jennifer Wilson, Posted June 8, 2015 –

    A few weeks ago, I overheard one student telling another, “Will you help me figure this out? Don’t just tell me how to do it.”

    How many of the students in our care are thinking the same thing? How often do we tell them how to do mathematics? How often do we provide them with “Ask, Don’t Tell” opportunities to learn mathematics?

    I used to tell my students how to determine whether a triangle is acute, right, or obtuse, given its three side lengths. Now I provide them an opportunity to determine the relationship between the squares of the side lengths so that they can generalize how to determine whether the triangle is acute, right, or obtuse.

    For all the triangles, we used abc, where a, b, and c are the three side lengths. Some students wrote that a triangle is acute when a2 + b2 > c2; others wrote c2 < a2 + b2. Students already knew that a triangle is right when a2 + b2 = c2 or c2 = a2 + b2. Students also determined that a triangle is obtuse when a2 + b2 < c2 or c2 > a2 + b2. Students already knew that, for three lengths to form a triangle, a + b > c, a + c > b, and b + c > a.

    I sent a Quick Poll to assess student understanding:

    2015-06-08 art1

    And I was surprised by the results. The students had determined that for a triangle to be obtuse, a2 + b2 < c2. Why did one-third of them miss the question? I had to think fast. I could have told them the correct answer. And then I could have told them how to work the problem correctly. Or I could have asked what misconception the students who marked acute had. Would everyone have paid attention?

    What I did instead was to show students the results without displaying the correct answer. I asked students to find another student in the room and construct a viable argument and critique the reasoning of others. I walked around and listened to their arguments. And I sent the poll again:

    2015-06-08 art2  

    All students answered correctly; they had all learned from the mistake of those who had chosen acute (believing that 8 + 15 > 18 was enough ensure that the triangle was acute).

    “Will you help me figure this out? Don’t just tell me how to do it.”

    “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 #AskDontTell opportunities do and can you provide?

    Resources for Pythagorean Relationships:

    Pythagorean Relationships activity for TI-Nspire

    Acute, Obtuse, Right Triangle Proof Geogebra applet

    AU Wilson JenniferJENNIFER WILSON,, 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 2:06:51 PM
    Thank you, Karen. We use TI-Nspire Navigator to send our Quick Polls to student handhelds or student computers. I like using it because of the different types of questions we can ask. This question was multiple choice, but most of the questions we send aren't. The answer can be a graph, or an equation, an expression, a point, open-ended, etc. ... I know that there are some good apps out their for polling, and Google is definitely improving its math capabilities. John Stevens recently wrote a post about using math in Google Forms:

    Karen McQueen - 6/11/2015 11:51:09 AM
    Great testimony! Thank you for sharing this. Do you prefer Quick Poll as an app for polling?