Vertical Value: Part 2

  • Vertical Value: Part 2

     By Cathy Yenca, posted February 15, 2016 –

    In Part 1 of this post, I shared a task that helped students understand perimeter, area, and their independence. Although in the past I used the task with seventh- grade students, concepts such as perimeter and area are typically addressed with third graders by today’s standards. However, I propose that a well-designed task has “vertical value” when it spans multiple grade levels and when it is used as a tool to cull student misconceptions.

    I recently transformed the aforementioned perimeter and area task into a digital and more open-ended format using Desmos. Desmos has several nifty tools that equip teachers to revamp lessons, placing the roles of questioner and answerer on the students. In this instance, I used Desmos Polygraph, which provides a mathematical take on a face-guessing board game that many of us played in our younger years. Playing Polygraph is ideal in the classroom with one device for every student. After students play a practice round of face-guessing against the computer, they are provided with 16 digital “math cards,” are randomly paired up, and are assigned roles in which they either ask or answer questions.

    Creating a custom Desmos Polygraph for the perimeter and area task took some planning on my part. I wanted students to initially see 16 seemingly random rectangles. However, there was nothing random about the dimensions for each rectangle I chose. The first few rectangles were inspired by the paper-and-pencil version of this task.

    2016-02-15 fig1        

    After making a plan, I created the Custom Polygraph activity for my students. Having a successful Polygraph match depends on the quality of student questioning and their use of academic vocabulary. From the teacher dashboard, student questions can be viewed in real time as well as after the game. It’s a treat to watch students succeed after a failed match when better questions and vocabulary are used!

    2016-02-15 fig2 

    I recently used this Polygraph with sixth-grade and seventh-grade students who are taking an eighth-grade math course. Why? Because I know we’re heading toward geometry units that address surface area and volume, so understanding the concepts of perimeter and area are key. Could this task be used successfully with multiple grade levels of students? In the comment section below, I’d love to hear from teachers who use this Polygraph regarding the grade-level of students, as well as a few of your favorite student questions. I’m curious about the vocabulary that students use at different grade levels when playing the same Polygraph, and hence, the “vertical value” of the activity.

    Here are a few of my favorite questions from my own students when they played Polygraph: Rectangles. Note the students didn’t mention perimeter or area explicitly at all! We had just finished a unit studying proportional and nonproportional linear functions, so it appears that timing may have shaped their questions. As we approach geometry units this spring, I plan to let them play again. I wonder how their questions might change?

    • Is it equal in all four quadrants?
    • Does it have a y-intercept of 1?
    • Is it a perfect square?
    • Is your width larger in squares than your length?
    • Is it centered around the origin?
    • Is it a rectangle? [One has to wonder, was this student trying to be funny or did this question just reveal something valuable about his or her thinking or understanding of rectangles?]

    2016-01 Yenca pic 

    Cathy Yenca,, is a math teacher in Texas. She tweets at @mathycathy and blogs about teaching mathematics in a one-to-one iPad classroom at


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    majid saadat - 2/8/2018 5:23:34 PM