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Why We Made Function Carnival

Here’s why the Desmos team (in collaboration with Dan Meyer and Christopher Danielson) made Function Carnival:

Conceptions and Misconceptions About Graphs

We know it’s important for students to connect different representations of relationships together – linking a table of values to its graph and its algebraic representations, for example. Representing and interpreting relationships between variables is an important skill even for students who do not study math beyond high school. Of all possible ways to represent these relationships, Team Desmos is partial to graphs.

But students struggle with graphs in several ways. We have seen that students struggle with rate, and how rates are represented in graphs. In 1938, an editorial might have argued the economy was in terrible shape because United States unemployment was 17% that year. Another might have argued that the situation was getting better because yes, unemployment was high, but its rate of change was negative.

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It’s often challenging for students to distinguish between (1) values, (2) the rate of change of those values, and (3) the rate of change of the rate of change.

Students also tend to have the idea that graphs are pictures—that the graph always describes the position of an object in space. Given the graph below, they may think the green car and the blue car collide after four hours.

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Traditional Interventions

So how do we fix this? Giving students more targeted feedback would be nice, but when a student’s graph is not a correct interpretation of the data, what are our options? We can show them an answer in a key. We can draw the right graph for them. We can have a conversation with students about different interesting wrong answers that may be representative of the entire field of wrong answers. All of these are useful strategies but they all require individual teacher intervention, which is difficult to manage in a large class.

When a student draws a graph with pencil and paper, she also has to imagine what that graph says about the world, and her imagination may be riddled with misconceptions.

Function Carnival

Function Carnival changes that. Students watch a video. They try to graph what they see. Then they play back the video and see how their graphical model would be represented as an animation. Does what they meant to graph about the world actually match the world?

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They can revise quickly, erasing and re-drawing pieces of their graph.

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If a student creates a graph that isn’t a function, the teacher can still tell the student, “That means the person was in two places at once,” but now the student can see all those people also.

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Pair this digital feedback with a teacher who can help students understand the difference between what they thought was going to happen and what actually happened and you’re readying students for a strong understanding of graphs and their representations. Give it a try.

REFERENCES

Ball, D.L.(2003). What mathematical knowledge is needed for teaching mathematics? Prepared for the Secretary’s Summit on Mathematics, US Department of Education, February 6, 2003; Washington, D.C. Available at http://www.ed.gov/inits/mathscience

Bureau of Labor Statistics. “Graph of U.S. Unemployment Rate, 1930-1945.” HERB: Resources for Teachers, accessed November 8, 2013, http://herb.ashp.cuny.edu/items/show/1510.

Monk, S. “Representation in School Mathematics: Learning to Graph and Graphing to Learn.” In A Research Companion to Principles and Standards for School Mathematics, edited by Jeremy Kilpatrick, W. Gary Martin, and Deborah Schifter, pp. 250–62. Reston, Va.: NCTM, 2003.

Des-man: a Desmos Labs Project

Today, we’re thrilled to introduce a new project: Des-man, inspired by @fawnpnguyen’s eponymous blog post. Des-man is an opportunity for students to flex some creative muscles, draw hilarious faces with math, and learn about domain & range in the process.

You can try it here: https://class.desmos.com/desman

A quick step back before diving into the details: a few weeks ago we released our first piece of collaborative content, a joint project with Dan Meyer called “Penny Circle.” Our two realizations:

(1) it’s really freaking difficult to make thoughtful content.

(2) it’s really freaking fun to make thoughtful content.

Designing even small pieces of curriculum surfaces all of the challenges that we love so much – working at the intersection of technology, design, and pedagogy; navigating the fine line between doing too much and not enough, between guiding and pushing, between delighting and distracting. It doesn’t hurt that content development is also a great excuse to work with teachers, our favorite way to spend time.

For all of these reasons, we decided we wanted to do more. After a quick perusal of some of our favorite Desmos lessons on the web, we settled on Des-man as a perfect fit for our second small project.

Des-man first guides students through the process of making domain & range restrictions in Desmos:

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Click here to see the sample student view

From there, the prompt is simple: “draw” a face using expressions.

On the other side, every teacher has a dashboard that updates in realtime. Status indicators help identify individuals/groups who are done or stuck. Filters narrow in on just those students who, for example, have experimented with circles or ellipses. One click on any thumbnail opens up a full-screen view of that graph.

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Click here to see a sample teacher view of Desman

Des-man is what we at Desmos Labs call a WIP, or Work In Progress. We need your help, and here’s how:

(1) Try it out! Let us know where it shines and where it falls short.

(2) Suggest an idea for another lesson, or just let us know you’d be interested in seeing more from Desmos Labs.

(3) Spread the word on Twitter, Facebook, or anywhere else you so desire (Google+? Anyone?)

This is hopefully just the beginning. Ultimately, we want to build out more and more lessons imagined by real teachers. Our dream: that by combining the wisdom of active instructors with the resources of Desmos we’ll be able to create things that far surpass what any of us could build alone.

We hope you’ll join us in making this a reality.

Graph on,

- Eli & Team Desmos


P.S. Special thanks to some of the folks who helped us with this draft of Des-man: @Trianglemancsd, @bobloch, @mbosma8, @LukeSelfwalker, @ddmeyer, and, of course, @fawnpnguyen

Teacher Tales: Using Common Core? You need Desmos!

by Matt Owen, Algebra 1 and Physics Teacher at Lusher Charter School, New Orleans, LA

Here are just a few of the ways I’ve used Desmos in my class along with the CCSS Math standards that apply.

1) Recognizing that graphing the functions f(x) = 3x - 4 and g(x) = 6 gives you the solution to 3x - 4 = 6. I used to think that this was a waste of time, since solving this algebraically seems easier. But having the resource of Desmos there (that kids like using and see the value in), makes it simple for them to see the connection. Then when we go to solve |x-3| = 7, they can find the solution graphically first and work up to an algebraic solution. Same goes for exponential, quadratic, anything! I never placed much emphasize on graphical solutions until we starting using Desmos regularly. But it’s even a CCSS! Check it:

A-REI.11 states: Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using DESMOS to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

(I may have made a small edit there) ;)

2) Finding a fit line and then adjusting a fit line (Are you doing Barbie Bungee this year? Me too!) This one is so easy to do with Desmos. The one I did last year was ok; I had the kids measure their arm length (from elbow to fingertip) and their foot length (no shoes, heel to big toe). Once each group was done measuring, they came up and entered their data in Desmos on my computer which was projected on the board. It looked like this. Then we discussed the best model to use here (linear was easy to agree on) and used that. Then it looked like this. So then I just shared that graph with every group (I used email to send it out, but Google drive may be an easier way to share), and had them write the equation of the fit line as they saw it. The great part here is that each group comes up with a different equation. Then, I had each group make a prediction about my arm length given my foot length and we could see who was closest. It’s a nice way to add a little friendly competition and get to discuss why the best prediction was the best. This is also a good place to discuss the meaning of slope and y-intercept. Should the y-intercept be zero? Does the slope have units? There’s really a bunch of CCSS that go with this one, but mainly I was thinking:

S-ID.6: Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

  1. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.

  2. Informally assess the fit of a function by plotting and analyzing residuals.

  3. Fit a linear function for a scatter plot that suggests a linear association.

3) Exploring function graphs. Check out Building Functions 3:

F-BF.3: Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

OMG! That’s so complicated sounding! But if you have students build on a parent function, they’ll start to be able to tell the difference between f(x) + k and f(x+k) even if that’s not how they’d explain it. Here’s the handout I used to do vertex form quadratics last year (needs a little work, but I think you’ll get the idea). Once they’d built up some rules for how to shift the graph around, I could show them the general vertex form graph and they could make some predictions about what the a, h, and k would do. Bam! Function built!! (of course that’s not the end of our quadratics explorations – it’s the beginning!)

Interested in sharing your classroom experiences? Email us at calculator@desmos.com