Des-blog

Is it possible for two numbers to have a difference of 8, and a sum of 1? What about a sum of 12 and a difference of 5? Are all such combinations of sums and differences possible? Systems of Two Linear Equations—this week’s Friday Fave—helps students explore tools for answering these questions.

With these tools, students become more powerful. We begin gently, by asking students whether they think it’s possible. Then we ask them for a pair of numbers with a difference of 8.

We put those numbers on the coordinate plane. They see everyone’s points, notice that these constraints produce lines, and only then do we ask students to write equations for these lines.

Over the course of 12 quick and targeted screens, we build from students’ informal ideas about numbers to the solution of a system of two linear equations.

Students’ arithmetic knowledge builds into algebra knowledge. We don’t ask them to forget what they know; we ask them to make what they know more robust. That’s why it’s this week’s Friday Fave.

This week in the fellows’ lounge we looked at Desmos activities that didn’t work, and considered ways to improve those activities.

Whether it’s an activity that you made or that someone else made, reasons why a Desmos activity may fall flat will vary. Many elements of planning to teach a digital lesson will be the same as a paper lesson, while others will be different. Suzanne von Oy had a recent experience in which she reflected that having an appropriate activity to introduce students to Desmos would have been helpful before diving into a content based lesson. Bob Lochel offers additional planning advice: “My general comment is that I find Activity Builders require rehearsal - you need to move away from what your ideal lesson would look like as an educator and think like a student. Do the instructions make sense? Does the next task make logical sense given previous screens? Is there sufficient opportunity to reflect upon how screens connect together?”

Jenn Vadnais observed in a recent activity that the tool that she had developed to help students explore percent was challenging to use. Sometimes additional modeling of how to use the tool can help students be successful, and in other cases it is better to consider redesigning the tool.

The design of individual screens and interactions is important to reflect on, as well as the flow of the entire activity. Activity design can play a big role in whether or not an activity will help students learn. Allison Krasnow supports coaches in her district as they work to implement a new curriculum. They recently converted a paper card sort from the curriculum into a Desmos card sort. The result: students engaged in more mathematical conversation than if they had worked in workbooks, but the nature of the card sort was such that there was a right and a wrong answer. In this way students that didn’t know how to do the math when they started the activity didn’t get as much out of the experience as hoped. Allison is reflecting on how to use some of the mathematics of the new curriculum along with Activity Builder to help shift pedagogy and help students learn math at a deeper level. For example, adding a screen to a card sort asking students to settle a dispute over how a hypothetical classmate sorted cards can make for rich discussion and opportunity to correct introductory conceptions of the math topic.

In the past Paul Jorgens has had his students recreate simple designs using a graphing calculator. He found when he converted the challenge to Activity Builder that the activity didn’t capture their interest. His prognosis: the ease with which students can explore graphs with sliders made the Desmos version too easy. His students rescued the activity in the end by imposing rules for themselves about graphing the designs with as few equations as possible, therefore introducing a need for strategy. The Desmos Activity Building Code supports integrating this type of strategy into practice.

Anna Scholl’s students struggled with an activity that didn’t offer enough scaffolding and jumped quickly into abstract thinking. Heather Kohn experienced a similar struggle with a screen that asked students to generalize their thinking around exploration on a series of screens. From the design perspective it may be helpful to add a card sort before a generalization screen which helps students compare and contrast samples of screens that they have explored. Another option is to combine the visuals either on the whiteboard or on a slide and structure a class conversation around these visuals so that students can build on their previous thinking as they work towards generalizing a formula.

Ayanna Ramsey has noticed at times that her students will skip screens, which could interfere with their ability to complete tasks later in the activity. She wishes for required screens such as what we see with Google Forms. While that feature isn’t currently available, we can plan for key screens where teacher pause or pacing can be used to ensure that students get feedback that sets them up for success for the rest of the activity.

What are some other things to consider as we work to improve our activities? Let us know on Twitter @desmos.

At Desmos, we are big fans of Open Middle problems. So we’ve tried our hand at cooking up a Desmo-fied version, Smallest Solution, which is this week’s Friday Fave.

Here is a short list of things we love about Smallest Solutions.

1. It is easy to get started. Drag four digits into four different boxes.

2. Things quickly become interesting. Your digits are now constants and coefficients in a one-variable linear equation which you are challenged to solve.

3. Only now is the namesake challenge revealed—Find the smallest solution.

We’ll let you know whether you have the smallest solution in your class or someone else has found a smaller one. But we won’t tell you who, nor will we tell you whether anyone has found the smallest possible score—that’s a job for you and your classmates to do.

Is zero possible? How do you know? From dragging a few digits to building mathematical arguments in four short screens.

Tell us what you think about our initial treatment of an open middle problem. You just may convince us to build more of them, and perhaps see these in a future Friday Fave.