desmos

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Friday Fave for December 21

This week—in this holiday season—the Fave turns its attention to the amazing community of teachers using Desmos tools to support students in creative ways. Click through on each of these posts for great ideas, resources, and links.

In this lesson plan in the form of a blog post, Jenn Vadnais uses Desmos as an instructional tool to help students understand percents and proportions.

The other day a teacher asked me to create a lesson that had elements of a Number Talk involving percents.  Her students had recently been discussing proportional reasoning and the constant of proportionality. Knowing this, I decided to combine percents and proportional reasoning with Desmos.  Here’s the resulting lesson.

On his blog, Paul Jorgens combines several Desmos tools to support classroom conversations that help students develop mathematical vocabulary with increasing depth and sophistication.

If we are going to develop vocabulary through experiences, we need to build in those activities into our lessons. We try to utilize our “fire up” time to open the class.

Julie Reulbach writes about how she has her students create Desmos art. She supports their learning and project organization by having them complete their art inside a Desmos activity.

Having them do the project through an Activity Builder helped me manage all of their graphs so I could easily view them and access them for help.  By using an Activity Builder, I was also able to include the instructions for the projects and helpful tips for them...I had them print out their Desmos Art, and I made a huge collage of it on my wall in the back of the room.

Supporting students with tools, language, and art—this week's Friday Fave.

Note: The Fave is taking a couple weeks off, and will see you again in the new year. All the best to you, your colleagues, students, and loved ones until then.

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Friday Fave for December 14

This week's Friday Fave requires a little help from your friends. System of Two Linear Equations begins by asking a simple question: Is it possible for two numbers to have a difference of 8, but a sum of 1?

We're not asking for examples at this point, nor are we expecting you to have techniques ready at hand. We’re just asking for your instincts, and in a sufficiently large class of algebra students we expect some yeses, several nos, and quite a few maybes. Discuss.

Next up, we'll get specific. Think of two numbers with a difference of 8 and give us the larger one. We'll guess the second one, plot them both as an ordered pair and ask you to think about what the collection of everyone's points will look like.

You see what we did there? We created a social experience by connecting you with your friends to discuss some informal ideas, and then we made those informal ideas more formal one step at a time. Pretty soon, you're writing equations for lines and noticing that their intersection is a number pair whose difference is 8 and whose sum is 1.

More constraints and opportunities to think, share, collaborate, and discuss follow. An informal question—Is it possible?—introduces an invitation to develop some important algebraic techniques. 

That's what makes System of Two Linear Equations this week's Friday Fave, and while you're thinking of systems of equations (and inequalities!) here are a few more activities to play with.

Playing Catch Up

Polygraph: Systems of Linear Inequalities

Racing Dots

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Friday Fave for December 7

Parlez-vous français?

Maybe you speak French and maybe you don't but thanks to a small collection of bilingual Des-users, those who do can use about 80 different activities at teacher.desmos.com. ¿Habla espanol? Our collection of Spanish translated activities is smaller but growing. Plus we have several in Dutch, and one each in Italian and Hebrew.

All you need to do is search Français, Español, Italiano, Nederlands, or עברית to find activities that some diligent user has put the time and effort into translating.

Would YOU like to be one of these diligent users? Get in touch! Dan coordinates this work—email him at dan@desmos.com (in English, please!) and he'll let you know what it takes to get up and running.

A dedicated worldwide user base with the expertise to translate activities is what makes this week's Friday Fave possible. C'est magnifique!

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Friday Fave for November 30

The Friday Fave is keeping it simple this week. One feature that makes certain situations just a bit more delightful.

Zooming in the Desmos graphing calculator has always been easy. Scroll wheels and track pads alike zoom your view in or out quickly and smoothly.

Recently, we introduced a little bit of magic to give you just a bit more control over your zooming. Hover your cursor over an axis, press SHIFT, click and drag.

Now you're changing the scale on one axis at a time. A simple little upgrade to your zooming experience is this week's Friday Fave.

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Friday Fave for November 16

Feedback comes in many forms. We can give feedback by evaluating a response as correct or incorrect in order to report a score. Alternatively, we can reflect back what we hear from another person, support them in considering if that is what they really mean, and allow them to make revisions until what they are trying to communicate is consistent with how it is being received.

In Land the Plane, students see that their slope is incorrect if the plane doesn't land on the runway. They can refine their thinking in an effort to be successful - and to learn how to compute the slope of a line.

In Match my Parabola, Students see the parabola that corresponds to an equation they submit. Students are empowered to use this feedback to modify their equation, and move on when they are satisfied, resulting in increased student learning and motivation.

In Laser Challenge, students see the angles they selected cause a laser and mirror to animate to get feedback on the accuracy of their predictions. Through revisions, students develop their intuition for angle measures and properties of reflections.

Being told an answer hinders motivation and discourages revision. It implies that someone else already knows the answer and that your thinking is insufficient.

Being shown what you’re saying without judgment supports thinking more deeply. It can make the goal feel attainable which encourages learning through revision. That's why interpretive feedback is this week's Friday Fave.

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Friday Fave for November 9

The Friday Fave is a big fan of tasks with more than one right answer. If the Fave asks Which One Doesn't Belong? you'd better believe each of the options can be a right answer. If the Fave asks How do you know? well, it's because the Fave really doesn't know how you know, but really would like to, and there's more than one right way of knowing.

So it is with the newest Desmos activity, Coin Capture. The challenge is write equations for lines that go through the coins, capturing them along the way.

In the introductory challenge, you can get them all with one line. But that's just Desmos getting you started. As the screen numbers increase, so does the complexity of the challenge. Here are four solutions to the challenge on screen 3.

While we keep track of the number of lines you used, that's not the only way to describe these. What kinds of thinking is behind each of the solutions above? Which is most interesting? Which one doesn’t belong? Posing the task is straightforward, but there are many right answers that vary in interesting ways. That's the kind of thing that qualifies an activity for the Friday Fave.

Also there's a challenge creator.

Now, while you're thinking about coins and/or targets in the coordinate plane, here are three more delightful activities:

Penny Circle

Marbleslides: Lines

Mini Golf Marbleslides

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Friday Fave for November 2

The bell has rung, your classmates have settled into their seats, and the teacher stands at the front of the class next to a chalkboard and overhead projector, ready to deliver today's lesson.

"All right, everyone," she announces while gesturing towards the blank chalkboard, "please solve these equations using your handheld calculators-- the ones with the broken displays."

This assignment may sound outlandish, but students with visual impairments face similar challenges using technology in the math classroom every day. New technologies have the potential to empower students, but without care, they may instead present insurmountable barriers to access.

Our mission at Desmos is to help every student learn math and love learning math. Not some students. Every student. With that in mind, we introduced improvements to the calculator to ensure students who are blind or visually impaired have the same opportunities as their peers to discover the joy of learning math. Thanks to technologies like screen readers (which provide spoken or Braille feedback to mainstream computers, tablets, and phones), people who are blind or low vision have the opportunity to use the exact same programs as their peers. Naturally, it made sense to extend the utility of our existing calculator offerings to include screen reader support. 

More recently, we extended this functionality to our activities, modifying them to meet the highest possible standard of accessibility our tools can offer. You can find all the screen-reader friendly activities here, or by visiting teacher.desmos.com and typing “screen reader” or “accessible” into the search bar.

So how did we make this happen? Many of the components of a Desmos activity have built in accessibility, meaning that you can interact with them via the keyboard and a screen reader. We added to this accessibility by adding narrations to graph screens to help students learn about a graph, interact with it, and receive feedback.

In this screen from Match My Line, when a student submits a correct equation, the screen reader tells them the number of lines that have been correctly graphed.

Students can also use the audio trace functionality from the graphing calculator to learn more about a graph within an activity.

Interested in learning more about Desmos accessibility? Get started at learn.desmos.com/accessibility, then dive deeper by watching our Introduction to the Desmos Graphing Calculator and Accessibility Tools webinar and heading to www.desmos.com/accessibility.

We’d love to hear your feedback! Let us know what you think at accessibility@desmos.com.

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Friday Fave for October 26

The Friday Fave would like to tel youl about a very handy calculator feature, and then invite you to do some math.

The feature first. 

Imagine you've got a set of points, and you've named them A, B, C, and D. You probably want to make a quadrilateral from those four points, and it used to be difficult but now it is easy.

What sort of thing is a quadrilateral? Why, it's a polygon. So you just tell the Desmos calculator that you want to make a polygon with those four points, and boom!

Alternatively, you may have specified the x-coordinates of your quadrilateral with a list, and the y-coordinates with another. Do not ask the Fave how the calculator can interpret two lists just as easily as it can interpret four points, but it can.

This second version of polygon is especially handy because you can operate on a list (you cannot yet operate on points except by dragging them around). So now let's do a little math together, shall we?

What do you suppose our polygon will look like if we type polygon(X,Y-4)? Click through on the expression to find out.

What about polygon(2X,Y)?

And finally, what will polygon(Y,X) look like?

Bonus question: What transformation do you need to apply to make our polygon no longer be a kite, and how might we express that transformation algebraically?

Now that the Fave has you thinking about polygons, here are some delightful ways to extend your thinking...

Polygraph: Advanced Quadrilaterals

Polygraph: Basic Quadrilaterals

Polygraph: Hexagons

Polygraph: Hexagons, Part 2

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Friday Fave for October 19

The personified Friday Fave is a figment of the imagination.

The Friday Fave posts every Friday without fail.

The Friday Fave seeks to help you, Dear Reader, find new and wonderful things for your students.

Two of these statements are true, and one is a lie.

When the truths and lies are about people or fictional constructs, you can't really know which is a lie unless you know the person or construct very very well.

But mathematical truths and lies are different. The evidence for veracity or mendacity is right there in the math. If you don't know it, you can figure it out! This makes Two Truths and a Lie fertile ground for mathematical concept development; especially if

For months, Desmos has had some secret-not-ready-for-public-use Two Truths and a Lie activities, and now we've polished several of these and put them on display in the searchable activities at teacher.desmos.com

We have versions of this activity for each of these function types: Conics, Exponentials, Parabolas, and Linears (all true; no lies!). They are Challenge Creator based, so the differentiation is built right in—your students are telling the truths and the lies, as well as determining exactly which conic (or exponential or line) they'll lie about.

The Friday Fave (who missed last week's post, and so now you know the lie) encourages you to click on through and give these Two Truths and Lie activities a try.

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Friday Fave for October 5

It is gray outside the Friday Fave's window as autumn begins in earnest. The Fave is left craving color and joy and beauty.

Fortunately, the Fave has a prescription—for Des-art. And now you do, too.

Maybe your art will take the form of curve tracing of a background image. The following examples, and many more, are waiting to inspire you at desmos.com/art

Or perhaps you'll go a different route and create beauty from pure abstraction.

Many more examples of abstract art live at desmos.com/math

Finally, you may possibly find your artistic medium in images that are placed in calculated ways. Shelley Carranza built this beauty with a single teal square and a whole bunch of math.

She added a white square to the mix to make this one.

And finally, she used three images artfully arranged to make this gorgeous pattern.

Whether you are doing art to liven a gray autumn day, or using it as an invitation to mathematics for your students, Des-art—in all its many forms—may just become your Friday Fave every day of the week!

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Friday Fave for September 28

A while back, high school teacher and Desmos fellow Sara Van Der Werf wrote about the Desmos graphing calculator in general, and the app in particular.

In the last few years I’ve started writing the phrase, “Check your solutions in Desmos.” on the bottom of any assignment that Desmos would benefit students (which is many of them).

I have found that students who do use my suggestion to use Desmos to check their work are more likely to use it on their own to check work or as a tool for exploring & learning.

Sara noticed that a much larger percentage of the students in her Minneapolis classroom had smartphones or tablets than had reliable wifi, and certainly more had these devices than had regular access to graphing calculators. So she began encouraging her students to use the school wifi to download the Desmos app, thereby ensuring they had ready access to graphing calculator technology whether online or off.

Here are some bullet points to build the case for the free Desmos app; available on both Android and iOS:

Solving a quadratic equation? Check your answer by graphing the original function and see where it intersects the x-axis.

Simplifying or factoring? Graph both forms to verify that they are equivalent.

Teachers can model and support students in using the app on their phones to check themselves, reducing dependence on the teacher.

Since Desmos is on the state assessments in 21 states now, the app supports students in building confidence since they learning using the same tool they will have during their assessment

Once downloaded, the app does not require wifi or cellular data! The computations all happen inside your device.

The Desmos app: It's this week's Friday Fave!

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Friday Fave for September 21

Hashtags are funny things.

Mocked, memed, misunderstood, and yet so solidly useful.

This week's Friday Fave is a particularly useful hashtag: #ImproveMyAB (where AB stands for Activity Builder, of course).

The brainchild of Desmos Fellow Kathy Henderson, #ImproveMyAB is a key to unlocking some most excellent collaboration online. By posting an activity to Twitter using #ImproveMyAB, you are saying two things: (1) "I have been working on an activity (or and idea for one)" and (2) "I have an inner desire to make this activity into something great!" You are issuing an invitation to conversation about teaching.

Of course, the hashtag is also a great place to stop by to see what others are making and to offer your own advice.

A quick peek at the hashtag turns up recent conversations about activities on a wide range of topics, including derivatives, exponential functions, domain and range, and exterior angles of polygons. Do you have thoughts to offer on these topics? Do you need inspiration for them? Do you have something else you'd like to work on, and need to connect with folks who can help? Check out #ImproveMyAB, it's hashtag-amazing!

And it’s this week’s Friday Fave.

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Friday Fave for September 14

The fave is fond of triangles, and after playing with this week's activity your students probably will be too.

This week's fave—Exploring Triangle Area with Geoboards—is newly released and consists of a few warm up screens and a Challenge Creator. This Challenge Creator is a doozy. 

How many triangles are possible on a 5-peg by 5-peg geoboard? And how many different values for the area are possible? Who knows? But whatever the number, it's great enough to allow for a lot of creative triangle building.

Maybe you build a tricky triangle and think to yourself, "No one will be able to reproduce this one!"

But it turns reproducing your triangle isn't the challenge; building a triangle with the same area as yours is. Indeed, odds are that your classmates will solve that challenge with a variety of triangles.

Along the way, you'll wonder about the greatest possible area (are you sure it's 8 square units? How do you know?), the smallest possible area (Are you sure it's half of a square unit?), and what areas are possible in between?

The focus on student-created challenges with plenty of opportunity for creativity—that makes Exploring Triangle Area with Geoboards this week's Friday Fave.

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Friday Fave for September 7

This week's fave is a new feature and a salute to things that work the way you sort of expect and hope that they would.

Let's say you want a movable point that stays within the bounds of a rectangle. That's no problem. Use slider limits that match the minimum and maximum values with the rectangle.

But let's say you want that point to stay within some non-rectangular region. Until quite recently, that was a problem because the limits on your slider had to be constants. Staying within limits that change was not possible.

If you've ever tried to solve this problem, you've probably typed something like this into your slider limits.

Until recently, we threw an error and told you that you couldn't use variable slider limits. But now you can, and here's what it looks like.

Variable slider limits, and syntax that feels natural—together those are this week's Friday Fave.

And here are a few more graphs that use variable slider limits. Maybe they'll spark some new ideas!

Fraction Bars

Fraction Shading

Strange Rectangle Tool

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New Activity Release: Functions and Their Derivatives!

One of the most important understandings in calculus is that functions have values which can be positive and negative but that those values are also changing, and that change can be in a positive or negative direction. Slope isn't just for straight lines!

For example, when you're getting out of student loan debt, the total value in your bank accounts might be negative, but the rate of change of your money is positive.

Or for another example, the value of the gross domestic product of the United States is always positive and the rate of change of the GDP is almost always positive so it makes more sense here to look at the rate of change of the rate of change. What is the rate of change of the increase? How does it compare to the increase of previous decades or other countries?

Because of the importance of these questions, calculus teachers frequently ask students questions about rate of change. Given a function, what is its derivative? Give a second derivative, what might the first derivative look like?

We were extremely impressed with a functions and derivatives activity developed by Sandi Yoder, especially the conversation it generated in her classroom. (Filmed here!) Inspired by Sandi's work, we created Functions and Their Derivatives.

We give students a function and its first and second derivative, without revealing which is which. We ask them to label the derivatives accurately and then we give them feedback on their thinking.

But then we bring in a Challenge Creator and invite students to create their own function and label its derivatives. If they do that successfully, they can enter it into the gallery to challenge their classmates.

You get one function from us and then dozens more from your classmates. A calculus class that is social and creative! That's why we're here.

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Friday Fave for August 24

With summer coming to an end, the Friday Fave is in a playful mood. Math and play go hand-in-hand. Most play has some mathematical elements: timing, space, counting, scale.... And nearly all the best mathematics has a playful element.

Play involves imagining possibilities, asking "What if?" and it varies over time. While you may enjoy constructing an equilateral triangle with compass and straightedge, it's not really playful if you do it the same way every time.

Play changes over time. For example, if you look through the Twitter feed of Annie Perkins (Desmos Fellow), you'll her playing with compass and straightedge, color and shading as she explores and interprets Islamic Geometry. Or look through Malke Rosenfeld's images for mathematical play with dice, dancing, and knitting. In both cases, you'll see new ideas and increased complexity over time.

Compass and straightedge, markers, dice, yarn and knitting needles. These are examples of different media for exploring and playing, and Desmos is such a medium as well.

The Fave recently featured some playful geometry sketches that become especially delightful when you break the rules (an important part of variation is breaking the old rules and establishing new ones).

When we first released Function Carnival, we soon heard from teachers and students who were choreographing multiple Cannon People by controlling their aerial acrobatics with graphs. 

And what about Sean Sweeney’s Marbleslide challenges? Playful for kids, but they are especially wonderful as an example of a math teacher playing with a highly specialized medium. Get on over there and get inspired!

As your new school year gets underway, keep an eye out for opportunities to play with math and to have your students play with math. And by all means, share your playful rule-breaking with us and the world.

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Friday Fave for August 17

As Back-to-School season goes into full swing, the Friday Fave is thinking about the work and planning that go into teaching a classroom lesson.

Interpreting national, state, or local standards and selecting instructional materials to address these standards are only the beginning. Then you have to adapt these materials and decide how to best use them with the particular students you have in your classroom. This is intellectually demanding, professional work.

Whether you've selected, adapted or created an activity at teacher.desmos.com, there are some important resources for supporting you in your planning. In particular there are Tips for Teachers, this week's Friday Fave.

We design these activities to support you in orchestrating discussions and build on student thinking. But how to maximize the effectiveness of each activity may not always obvious by looking at the activity itself. Teacher tips are a critical part of planning when to pause and pace a class as well as anticipate student responses and plan where snapshots would be most effective.

If you’re creating or editing an activity, you can also create or edit Tips for Teachers in order to make notes to yourself about how to best facilitate each screen, anticipate possible student responses and plan questions in advance to support students who may need a nudge or to extend their thinking. You can also use this space to add notes to yourself for facilitating the lesson next school year.

You can access teacher tips in the preview of an activity's landing page...

Or in the dashboard...

Or in the printable Teacher Guide.

However you get to them, Tips for Teachers are there to support your work. We hope they'll be your Fave this week too.