Squares, Square Roots & Student Sense-Making – A Desmos Activity

In this post …

Part 1:  The Activity’s Inception

Saturday, November 7, 2015, Palm Springs, CA.  I was at CMCS15,  listening to Eli Luberoff  (creator of DESMOS) discuss Technology and Intellectual Need.  His message was motivating.


Me and Eli

My colleagues are well aware of my competitive nature. I’m constantly setting goals and pushing myself to achieve them. After Eli’s session, as J.J, Tim and I walked to lunch, I announced my next goal:  Create an activity builder where the pieces of individual student work combine to reveal a larger discovery.  Eli had just demonstrated this structure in his CMCS15 session and I wanted to bring it to the classroom.

Screen Shot 2015-11-16 at 5.50.08 PM



Being a math coach, I don’t have a classroom of my own.  I do have wonderful teachers who are willing to let me test out ideas.  Allan, one of those teachers, attended CMCS15 as well and he bravely agreed to let me create a lesson using ideas from the conference. My deadline – Tuesday November 10, 2015.


Sunday morning, November 8, 2015.  Not one to wait, I started tinkering with an activity builder lesson involving square roots.  Out of excitement, I tweeted it prematurely. The activity was rough – not well planned.  A short while later, I received a notification:

Screen Shot 2015-11-16 at 5.41.36 PM

I made some edits.  Heard back from Leeanne.Screen Shot 2015-11-16 at 5.44.06 PM

I had a diamond in the rough – A solid concept that required refinement. That night, I emailed my colleagues the link, so we could work on it the next day.

My Colleagues and Collaborators

Screen Shot 2015-03-15 at 4.34.29 PM       caP2      Capture     J.J.                                       Tim                                              Mari


Monday morning, November 9, 2015

  • J.J.  Just got here and Jenn’s already sent us 2 emails in 5 minutes.
  • Me:  Yup, need feedback…  (I wanted to use this the very next day)
  • J.J.  Class code?
  • Me:  fnuf

J.J, Tim and Mari all hop on to check it out.  Feedback was given. Clarifying questions asked. Explanations expressed. Views shared. Views debated. The activity morphed.

The day progressed and my collaborators tended to their own responsibilities. Periodically, J.J would ask, “Do you have a class code for me? or I’d ask for feedback. In both cases discussions, debates or tangents would ensue.

At one point, I had a slide that showed x=y².Screen Shot 2015-11-10 at 9.09.46 PM

  • Leanne tweeted: “Why show the negative half of the parabola? ” Soon after I hear…
  • J.J.:  Why show the full parabola?
  • Me:  That’s 2 people with the same question, guess I need to think about this
  • Tim: What’s the goal of your lesson?

Tim and J.J proceeded to discuss the graph.  I tuned them out (sorry guys).  My focus was on Tim’s question.

My Goal:  Create an activity where the students realize the need for square roots.

Screen Shot 2015-11-15 at 6.43.49 PM

Slide 6 – After the edits

As the guys debated, I went to work. Although I loved the connection to x=y², the end goal of the lesson to arrive at the equation y =√x.  Therefore to avoid confusion, I dropped the negative half of the parabola.

During the lesson, the equation x = y² was an integral discussion point when determining whether the graph represented linear or non-linear information.

Monday afternoon, November 9, 2015

Allan previewed the slides and we discuss the lesson’s progression.  He asked to include questions on domain and range (slides 11 and 12).  Activity completed.


 Part 2 –   An Activity Guide:  

Squares and Square Roots

  • Slide 1 – 2:  Pick a number from 1 – 10 and square it.
  • Slide 3:  Organize the information from slides 1 and 2 into an ordered pair and graph. Be careful: Students may reverse the information.
  • Slide 4:  Enter a Prediction.  We gave students 4-5 minutes to make a prediction.  Many of them wanted to go back to slide 3 to plot more points prior to predicting.
  • Slide 5:  Stop!  Time for whole class discussion.
    • Our discussion topics
      • Does the graph represent linear or non-linear information? Explain your choice.
      • What happens if we pick a negative number to square?

Allan and I pursued the question of linear vs non-linear:  Groups easily determined that the graph represented non-linear information.  The intriguing aspect of the lesson stemmed from student reasoning and understanding. Here’s a mashup of our interactions…

  • Student: The line isn’t straight. It’s curved
  • Teacher: Why?
  • Student: It doesn’t have a constant rate of change.
  • TeacherProve it.
  • StudentThe line is curved
  • Teacher: Now prove it using the x/y chart
  • Student:  The numbers are different.Screen Shot 2015-11-15 at 8.37.11 PM
  • Teacher:  Which ones?
  • Student:  The ones in the x column.
  • TeacherWhy do they seem different?
  • Student:  They don’t increase the same?
  • Teacher:  How do they increase?
  • Student:  I don’t know
  • Teacher:  Figure out how they increase.  I’m going to check in with another group and come back to you.
  • Teacher:  (going back to the group) How do the numbers increase?
  • Student:  The numbers increase by 3, then 5, then 7,  9 …

The x column did not increase at a constant rate of change.  To highlight this point, we asked the groups to calculate the slope between various points on the line. Depending on the points used, the slopes were 1/3, 1/5 1/7 etc… One student made a great observation.

Screen Shot 2015-11-15 at 6.39.36 PMLogan:  As the numbers in the x column increase by larger numbers, the distance between the plotted points increases too.

More student reasoning:

  • Student:  Non-linear, b/c of the exponent.
  • Teacher: Tell me more
  • Student:  The exponent is 2.  Linear equations have an exponent of 1
  • Student:  Non-linear, b/c we are squaring numbers.
  • Student:  It’s quadratic.
  • Teacher:  How do you know?
  • Student:  Since we are squaring the y-column, there’s an exponent of 2.


Slide 6:  Students are asked to find points on the graph in between the given ones.  We anticipated the following strategiesScreen Shot 2015-11-15 at 6.43.49 PM

  1. Trace the line
  2. Guess and check
  3. Pick a decimal, square it.


Below are various interactions I had with students.  My questions were designed to push student thinking.

  • Me: How did you  determine out the points in your x/y chart?
  • Student:  I traced the line.
  • Me: Ok. Now how would you determine a point, if you didn’t have the line?
  • Me: How did you  determine the points in your x/y chart?
  • Student:  I picked a number and kept changing it until the point was on the line.
  • Me: Ok.  Now how would you determine a point, if you didn’t have the line?
  • Me: How did you determine points in your x/y chart?
  • Student:  I picked a number and squared it.
  • Me: Why did you square it?
  • Student:  That’s what we did on earlier slides.

Allan and I called the students whole group to discuss the three strategies used.


Slide 7: Introducing the concept of square roots  Capture

  • Me: Why is this slide different from the others?  Talk with your group.
  • Student:  We are starting with the x values
  • Me: Why is that different?
  • Student:  In the other slides, we started with the y values.

And with that, the students went to work…  We anticipated the following strategies

  1. Dividing by 2 (a common misconception)Capture
  2. Tracing the line
  3. Guessing and checking
  4. Finding the square roots

More student interactions

  • Me: How did you determine the y-value when x is 3?
  • Student:  I divided by 2.
  • Me:  Is your point on the line?
  • Student:  No
  • Me: Is it close?
  • Student:  Yes
  • Me: What could you try next?
  • Me: How did you determine the y-value ?
  • Student:  I divided by 2 and got 1.5. Then kept changing the value ’til the point landed on the line.
  • Me:Capture How did you get 1.75?
  • Student:  I thought, what can I multiply by itself to get 3?  I used the calculator to help.  But watch. (student proceeded to show me his  calculator) When I multiply 1.75 by 1.75, I get 3.0625.  I’m having a hard time getting 3.
  • Me: (Thinking – YES!) Interesting process.
  • Me: How did you determine the y-value?
  • Student:  I used the square root button
  • Me:  Why?
  • Student:  I have to “unsquare” the 3.
  • Me: (smiling) Explain what you mean by “unsquare”
  • Student:  I have to do the opposite of squaring which would be unsquaring.
  • Me: Why did you use this symbol?  (I point to the square root symbol √)
  • Student:  The inverse of squaring is finding the square root.

The small group discussions set up the whole class discussion of square roots.


Slide 10:  Writing the equation, that passes through the given points.  The equation is:  y = √x

Slides 11 – 12:  Domain and range (Due to rich conversations and student sense making, not all classes reached these slides)



Desmos is phenomenal resource.  A purposefully designed Desmos task in conjunction with time for student discussion and sense-making is POWERFUL!

The students in Allan’s classes weren’t simply told what a square root is. They

  • calculated, organized and plotted information
  • predicted the graph’s shape
  • determined that the information collected was non-linear.
  • proved why the information was non-linear.
  • discovered patterns.
  • were presented with a need for the square root symbol
  • learned the inverse function of squaring a number is finding its square root.
  • pushed their thinking, tested ideas, analyzed, retested, shared, discussed, connected and discovered
  • enjoyed themselves in the process 🙂


Here’s a link to my Desmos Page which contains other Activity Builder tasks, Polygraphs and blog posts written about lessons incorporating Desmos


About jgvadnais

Instructional Technology Coach. Desmos Fellow. Google Level 1 Certified. SoCal transplant. New Englander at heart. Lover of yoga, dogs, green smoothies and coffee creamer
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3 Responses to Squares, Square Roots & Student Sense-Making – A Desmos Activity

  1. Leeanne Branham says:

    Great write up! So enjoyed reading the rest of the process and how it went with kids.

    note to self: Put sign on wall above computer ” What is your goal for this lesson?”
    (Isn’t it funny how often we lose sight of this while in the middle of creating mode?)

    Thanks for letting me in on a tiny bit of the experience. Better together.

    Liked by 1 person

  2. Pingback: Desmos: Introduction to Irrational Numbers | Communicating Mathematically

  3. Pingback: dy/dan » Blog Archive » 2015 Remainders

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