Category Archives: Quadratics

Math + Art + Desmos… Connections.

“I love math and art, and I’m glad that I was introduced to Desmos, a way to use both subjects at the same time.” – Marianna, Grade 10

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Callisto, Grade 10

Drawing with graphs has been a powerful way to motivate students’ interest in understanding how equations relate to functions, and how manipulations of equations lead to transformations from a parent function.  I jumped on to Fawn Nguyen’s Des-Man project as soon as I saw the idea, and have done some incarnation of this work each year.  Each time I’ve guided students through this process, it’s gotten better and deeper, both through the development of my own approach, and from improved tools like the Desmos Des-Man interface (…which I’ve heard is currently “in the shop” undergoing some improvements) and more recently tweaks to this idea like the “Winking Boy” challenge, created by Chris Shore (@MathProjects), and posted on the Desmos Activity Builder by Andrew Stadel.

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Amit, Grade 10

This year’s work was definitely the strongest yet, and I owe the major improvements to my reading of  Nat Banting’s post, which extended this project to another level for my students. In the past, I have asked students to create a graph, which had features of a face or a building or a plant.  This year, I asked my grade 10 students to choose a graphic, photo, or work of art, which they had to replicate using only equations.  I asked that they choose an image that was meaningful to them for some reason, and then helped to guide them to something that was challenging, but that they could accomplish – a natural moment for differentiation, built in to the process.  In the earlier versions of this project, students had been motivated by trying to make their face look angry or happy or sad, but they didn’t have a specific place where their equations had to end up. Asking kids to commit to re-creating something forced them to be purposeful and deliberate in every choice.

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Ilyas, Grade 10

They took the responsibility of recreating their chosen image seriously, and honestly, their work exceeded my expectations. There were regular exclamations of satisfaction echoing around the room as we worked on this. They persevered. They definitely attended to precision. They argued with each other about the best equations to use. They reflected about how to make the best use of Desmos. They practiced the habits of mind of successful mathematicians.

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Anastasia, Grade 10

When we shared the in-progress work for some peer feedback, kids were actually applauding each other when their work came up on the screen.  Not because I reminded them to be a supportive audience, but spontaneously.  Seriously.  And when they saw the staff creative picks at Desmos, they asked me whether they might be able to submit their work.  The whole class was taking pride in creative math work.

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Karim, Grade 10

I asked that students reflect in writing on their learning during and at the end of the project.  I haven’t asked for students to do enough writing in math so far this year, so when they seemed to be really struggling with this, I made a fill-in-the-blank “reflection assistant” to scaffold their thinking and writing, and to give them some ideas about what to include in their written analysis.

A few highlights from their reflections:

  • “I was quite surprised that I could replicate a drawing by using graphing.  If somebody asked me to do it last year, I would say that it is a “mission impossible.”   However I was able to do it.”
  • “As my piece of art, I chose the logo of the football club Barcelona because I am a big football fan and FC Barcelona is a club worthy to be recreated through the use of quadratic equations in vertex form. In addition, the logo was an appropriate challenge for me, containing easy and smooth curves but also difficult shapes, like letters or circles. When the project was assigned, I was skeptical that it was possible to recreate an artwork, just by using equations. But now that I am done and a proud owner of a recreated art piece, I strongly believe that it is possible (obviously).”
  • “I found out that desmos is a really good tool to practice and sharpen your understanding on any equation and in my case it was the vertex form of a quadratic. Desmos allows you to experiment and find new ways to fix the problems or even work more efficient in order to surpass the problems in the first place. I am proud of the detail and sharpness of my work in general. I tried really hard to make the whole piece smooth and detailed. In order to do so, I zoomed in a lot and by doing so, I identified minor mistakes and was able to fix them.”
  • I chose “Pumpkin Pepe” as the subject of my project because it provided the right level of challenge for me and it was really fun to do. Overall, I really liked this project because it solidified my knowledge of graphing equations and has made me more comfortable using parabolas. I found that my understanding of quadratic equations really improved while I worked on this project because before, I wasn’t sure which variable shifted the parabola which way, but now I understand.
  • “I found that my understanding of parabolas and linear equations really helped me improve, and made me more confident during my work on this project. At first parabolas seemed to not make any sense to me, but now I feel like I really understand the way they work. Now I have the capability make connections with all these equations in the real world.”
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Marianna, Grade 10

Here are the project guidelines, the rubric, some peer editing forms, the “reflection assistant,” and a .pdf, which has a range of student work.  My rubric borrows from the I.B. Math Internal Assessment Guidelines, as one of my tasks as a grade 10 teacher at my school is to do some specific preparation for the I.B. program in grade 11.  Thanks in advance for any feedback on this project, and on the guidelines and rubric.

Desmos Drawing Project Guidelines and Rubric

Desmos Peer Feedback

Desmos Drawing Project Reflection Assistant

Desmos-art-project-student-work-2015-16-updated

 

Quadratics: Mighty Square! (start by completing the square)

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Many texts I’ve seen ask students to solve quadratics by factoring, then by graphing, then memorising the quadratic formula, and then if there’s time left, they are introduced to the idea of completing the square.  I’ve done it before like this myself, and I have seen students struggle mightily and miserably.  For me, beginning with factoring is problematic.  While factoring trinomials can be satisfying, especially for students who like the puzzle solving parts of math class, this technique generally only works with problems that have been contrived by math teachers or textbook writers.  Most quadratics we come across don’t naturally factor to nice clean numbers like y=(x+4)(x-2).

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I was looking for a better approach, and came across James Tanton’s take on quadratics.  He emphasizes symmetry as the key to studying and understanding quadratics, and right from the start, teaches completing the square (Although he relentlessly resists formal vocabulary, and calls it the “box method”) as the way to solve for x.    His course gives a sequence of problems, each of which adds one level of complexity until students can solve just about any quadratic thrown at them.  As he says, “The box method will never let you down.”

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Developing a solid conceptual understanding of completing the square leads naturally to moving back and forth between standard and vertex form, and to the derivation of the quadratic formula – and students know why it works.  And the best part is, only one of my students from this year’s group uses the quadratic formula as her go-to method.  They all go straight for completing the square because they understand why and how it works, and are totally comfortable with the techniques.  Most of them can recite the formula, but they are worried about making arithmetic mistakes, and are not as confident in their results as they are when they use symmetry and completing the square.

 

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I strayed from Tanton’s approach in some ways – we figured out imaginary solutions with the honors group (Tanton advocates to leave out imaginary solutions when studying quadratics with Algebra 1 or 2 classes) – because they asked, and because they were ready to expand their horizons.  And I did eventually teach factoring as a method of solving quadratics.  Tanton suggests that factoring might be better included when we’re working on a discreet study of polynomials – but honestly, it was easy for me to fit in as the last method I taught.  We took less than one full class period to discuss and practice factoring, and by the time we got there, students understood what the factors meant, and how they related to the roots of the quadratic.  They appreciated the quickness of the solutions when factoring worked, and understood that if it didn’t work easily, that they could fall back on methods they know.

Every one of my Algebra students, even those who really struggle with math have had success moving through this sequence.  Although this is only one year, and there are always other variables, I am convinced that this order makes much better intuitive sense.  I’ll report back next year after I’ve had the chance to try this again.

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I love the way that the white boards look when studying quadratics

BTW, some of my favorite resources to go deeply into studying quadratics include