**[2021 Feb 11: We’ve learned more about building great math activities. Check out this updated design guide!]**

We wrote an activity building code for tworeasons:

- People have asked us what Desmos pedagogy looks like. They’ve asked about ourvalues.
- We spend a lot of our work time debating the merits and demerits of different activities and we needed some kind of guide for those conversations beyond our individual intuitions andprejudices.

So the Desmos Faculty –Shelley Carranza,Christopher Danielson,Michael Fenton,Dan Meyer – wrote this guide. It hasalready improved our conversations internally. We hope it will improve ourconversations *externally* as well, with the broader community of matheducators we’re proud to serve.

**NB**. We workwith digital media but we think these recommendations apply pretty well to printmedia also.

**Incorporate a variety of verbs and nouns**. An activity becomes tedious if students do the same kind of verb over andover again (calculating, let’s say) and that verb results in the same kind ofnoun over and over again (a multiple choice response, let’s say). So attend tothe verbs you’re assigning to students. Is there a variety? Are theycalculating, but also arguing, predicting, validating, comparing, etc? Andattend to the kinds of nouns those verbs produce. Are students producingnumbers, but also representing those numbers on a number line and writingsentences about those numbers?

- Match My Line is an activity for practicing graphing but we also ask students to sketch, settle a dispute, andanalyze.

**Ask for informal analysis before formal analysis**.Computer math tends to emphasize the most formal, abstract, and precisemathematics possible. We know that kind of math is powerful, accurate, andefficient. It’s also the kind of math that computers are well-equipped toassess. But we need to access and promote a student’s informal understanding ofmathematics also, both as a means to interest the student in more formalmathematics *and* to prepare her to *learn* that formalmathematics. So ask for estimations before calculations. Conjectures beforeproofs. Sketches before graphs. Verbal rules before algebraic rules. Homelanguage before school language.

- In Lego Prices, we eventually ask students to do formal, precise work like calculating and graphing. But before that, we ask students to estimate an answer and to sketch arelationship.

**Create an intellectual need for new mathematical skills**. Ask yourself, “Why did a mathematician invent the skill I’m trying to helpstudents learn? What problem were they trying to solve? How did this skill maketheir intellectual life easier?” Then ask yourself, “How can I help studentsexperience that need?” We calculate because calculations offer more certaintythan estimations. We use variables so we don’t have to run the same calculationover and over again. We prove because we want to settle some doubt. Before weoffer the aspirin, we need to make sure students are experiencing a headache.

- In Picture Perfect, students can either calculate answers numerically across dozens of problems, or solve the problem oncealgebraically.

**Create problematic activities**. A problematic activity feels *focused* while a problem-free activity*meanders*. A problem-free activity picks at a piece of mathematics andasks lots of small questions about it, but the larger frame for those smallerquestions isn’t apparent. A problem-free task gives students a parabolaand then asks questions about its vertex, about its line of symmetry, about itsintercepts, simply because it *can* ask those questions, not because it*must*. Don’t create an activity with lots of small pieces ofanalysis at the start that are only clarified by some larger problem later. Helpus understand why we’re here. Give us the larger problem now.

- In Land the Plane, the very first screen asks students to “land the plane.” We try to keep that central problem consistent and clear throughout theactivity.

**Give students opportunities to be right and wrong in different, interesting ways**. Ask students to sketch the graph of a linear equation, but also ask them tosketch the graph of *any* linear equation that has a positive slope and anegative y-intercept. Thirty correct answers to that second question willilluminate mathematical ideas that thirty correct answers to the first questionwill not. Likewise, the number of interesting ways a student can answer aquestion *incorrectly* signals the value of the question as formativeassessment.

- In Graphing Stories, we ask students to sketch the relationship between a variable and time. Their sketches often reflect features of the context that other students missed, and viceversa.

**Delay feedback for reflection, especially during concept development activities**. A student manipulates one part of the graph and another part changes. If weask students to change the first part of the graph so the second reaches aparticular target value or coordinate, it’s possible –even likely– the student will complete the task through guess-and-check, without thinkingmathematically at all. Instead, delay that feedback briefly. Ask the student toreflect on where the first part of the graph should be so the second will hitthe target. Then ask the student to check her prediction on a*subsequent* screen. That interference in the feedback loop may restorereflection and meta-cognition to the task.

- Our Marbleslides series offers students lots of opportunities for dynamic trial-and-error, for manipulating slopes and intercepts one tenth of a unit at a time until they collect all of the stars. But we also offer several static reflection questions where students can’t manipulate the graph before answering. We give them the chance to check their work only after they commit to ananswer.

**Connect representations**. Understanding the connectionsbetween representations of a situation – tables, equations, graphs, and contexts– helps students understand the representations themselves. In a typical wordproblem, the student converts the context into a table, equation, or graph, andthen translates between those three formats, leaving the context behind.(Thanks, context! Bye!) The digital medium allows us to re-connect the math tothe context. You can see how changing your equationchanges the parking lines.You can see how changing your graph changes the path of the Cannon Man.“And in any case joy in being a cause is well-nigh universal.”

- In addition to the examples above, Marcellus the Giant invites students to alter the graph of a proportional relationship. Then students see the effect of that altered graph on the giant the graph was describing. We connect the graph and thegiant.

**Create objects that promote mathematical conversations between teachers and students**. Create perplexing situations that put teachers in a position to ask studentsquestions like, “What if we changed this? What would happen?” Ask questions thatwill generate arguments and conversations that the teacher can help studentssettle. Maximize the ratio of conversation time per screen, particularly inconcept development activities. All other things being equal, fewer screens andinputs are better than more. If one screen is extensible and interesting enoughto support ten minutes of conversation, ring the gong.

- Our Card Sort activities feature only a handful of screens but offer students and teachers an abundance of opportunities to discuss both early and mature ideas aboutmathematics.

**Create cognitive conflict**. Ask students for a prediction– perhaps about the trajectory of a data set. If they feel confident about thatprediction and it turns out to be wrong, that alerts their brain that it’s timeto shrink the gap between their prediction and reality, which is“learning,” by another name. Likewise, aggregate student thinking ona graph. If students were convinced the answer is obvious and shared by all, thefact that there is widespread disagreement may provoke the same readiness.

- Charge! presents a relationship, a cell phone charging over time, that seems quite linear. As the cell phone completes its charge, the rate of charge slows down considerably, confounding student expectations and preparing them tolearn.

**Keep expository screens short, focused, and connected to existing student thinking**. Students tend to ignore screens with paragraphs and paragraphs of expositorytext. Those screens may connect poorly, also, to what a student already knows,making them ineffective even if students pay attention. Instead, add thatexposition to a teacher note. A good teacher has the skill a computer lacks todetermine what subtle connections she can make between a student’s existingconceptions to the formal mathematics. Or, try to use computation layer to referback to what students already think, incorporating and responding to thosethoughts in the exposition. (eg. “On screen 6, you thought the blue line wouldhave the greater slope. Actually, it’s the red line. Here’s how you can know forsure next time.”)

- In Game, Set, Flat, we ask students to create a bad tennis ball, one that doesn’t bounce right at all. Then we tailor our explanation of exponential models to their tennis ball, explaining how it violates assumptions of exponential models and how they can fixit.

**Integrate strategy and practice**. Rather than just askingstudents to solve a practice set, also ask those students to decide in advancewhich problem in the set will be hardest and why. Ask them to decide beforesolving the set which problem will produce the largest answer and how they know.Ask them to create a problem that will have a larger answer than any of theproblems given. This technique raises the ceiling on our definition of “mastery”and it adds more dimensions to a task – practice – that often feelsunidimensional.

- In Smallest Solution, we don’t ask students to solve a long list of linear equations. Instead we ask them to create an equation that has a solution that’s as close to zero as they can makeit.

**Create activities that are easy to start and difficult to finish**. Bad activities are too difficult to start and too easy to finish. They askstudents to operate at a level that’s too formal too soon and then theygrant “mastery” status after the student has operated at that levelafter some small amount of repetition. Instead, start the activity by invitingstudents’ informal ideas and then make mastery hard to achieve. Giveadvanced students challenging tasks so teachers can help students who arestruggling.

- In the conclusion of Water Line, after graphing the rising water line in several glasses we provide, we ask students to create their own glass. Then that glass goes into a shared classroom cupboard, giving students many more challenges tocomplete.

**Ask proxy questions**. Would I use this with my ownstudents? Would I recommend this if someone asked if we had an activity for thatmathematical concept? Would I check out the laptop cart and drag it acrosscampus for this activity? Would I want to put my work from this activity on arefrigerator? Does this activity generate delight? How much better is thisactivity than the same activity on paper?**2016 May 11**. Updated to add references to activities.