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Tortoise and the Hare #Desmosify

Welcome to a series of posts sharing how we #Desmosify the curriculum from Open Up Resources/Illustrative Mathematics. You can use this lesson for free, or sign up to get many more activities just like it in our core middle school curriculum!

Here’s how we #Desmosified an Open Up Resources/IM lesson to help students students interpret graphs of functions..

Desmosification #1: Create concrete connections.

In the original activity, Open Up Resources/Illustrative Mathematics starts with a context about temperature. We’re on board!

The original temperature graph.

The activity immediately asks students to analyze the graph of a context precisely and numerically, but students have not yet had opportunities to develop and understand the context concretely.

The questions.

One of our core commitments in designing curriculum is to invite students to use their voice, vision, touch, and intuition in mathematical analysis, all components of what Rochelle Gutierrez describes as “rehumanized mathematics.” If an activity invites students to access and apply their intuition about a context, it strengthens both their later numerical analysis and their sense of themselves as capable mathematicians with valuable ideas.

So in our activity, we chose acontext that has several advantages for students—a race between a tortoise and a hare.

A tortoise and hare race

We start by asking students to tell a story about what they see. This is an opportunity for teachers to learn what details are most salient for students. Are they noticing speed? The hare’s nap in the middle. The psychobiological conflict between two species?!

The hare is in the lead and and he stops and sleeps and the turtle wins.

It’s only much later in the activity, after the context has been enriched by the students’ own intuitions, that we ask for any precise, numerical analysis.

Desmosification #2: Give feedback that causes thinking.

Another advantage of this context is that because it’s a digital animation, we can invite students to control its different elements using their own thinking. We invite students to create a race between the tortoise and a dog using a graph. Students create a dog by graphing!

We create a graph and the two animals race.

We ask students to create a specific dog—one with a head start that still loses the race, for example. And rather than give students binary feedback—right or wrong—which often fails to credit them for their abundant brilliant thinking even in a wrong answer, we just render the dog they created in the animation. Students can then decide if the dog did what they wanted it to do and make modifications accordingly.

Behind the Scenes

Desmos lesson developer Michael Fenton describes one of the core design dilemmas of the lesson and how our team found our way through it.

We agonized over the details of the race. How fast should each animal move? What distance should we use for the entire race? When and for how long should the hare rest? We wanted something that felt realistic if you look closely at the numbers. We also put a huge emphasis on the end of the race being suspenseful (will the hare catch up?!) while also having a clear-cut winner. And since it was nontrivial to change the parameters of the actual animation, we did most of our iterating with piecewise functions in a Desmos graph, making it easy to see important details from the entire race all at once.

What Did Teachers Think

Loved it! Students are really understanding how the graph tells a story. They know that the flat line means that there is no movement, negative slope means moving back towards the start and the steeper the line the faster the animal was moving. Great results on the cool downs!

My students loved this entire concept with the animals racing. They enjoyed creating the graphs for the tortoise and the dog because they were able to see how their graphs showed the story.

They knew immediately if they were right or wrong and that helped them work backwards to fix their graphs if necessary. My students were also confident in using their prior knowledge about what they know about lines and their steepness.

Students loved being able to tell their own story and give meaning to the graphs, awesome for some students’ creativity to shine through in writing. I appreciated the push for specific, detailed answers. It lent itself to stronger, clearer thinking practices.

What’s Next?

Desmos outscoring all other curriculum for Net Promoter Score.

Overall, teachers in our 2019–20 pilot study really liked the Open Up Resources/Illustrative Mathematics curriculum … but they loved what we did to #Desmosify it. (Definition of the scale.)

Shira the Sheep #Desmosify

Welcome to a series of posts sharing how we #Desmosify the curriculum from Open Up Resources/Illustrative Mathematics. You can use this lesson for free, or sign up to get many more activities just like it in our core middle school curriculum!

Here’s how we #Desmosified an Open Up Resources/IM lesson to help students learn to solve inequalities..

Desmosification #1: Create concrete connections.

In the original lesson we #Desmosified, Open Up Resources/Illustrative Mathematics invites students to complete a series of exercises like the one below.

The original task.

These exercises are likely to give students valuable experience with the subtleties of inequalities. The examples include inequalities that are strict and inclusive, inequalities that include negative and positive values, etc.

We worried, however, that students would spend too much time filling in tables to notice those subtleties. We also wanted to connect the solution of inequalities to a representation that’s more concrete for students than tables.

Enter Shira the Sheep!

Shira the Sheep introduces herself.

Shira the Sheep likes eating grass and dislikes getting wet. She also provides us with a concrete and memorable metaphor for the solutions of inequalities.

Desmosification #2: Give feedback that causes thinking.

A problem asking students to graph 5x > 15.

Imagine all the different inequalities students might enter in response to this situation. Our goal is to faithfully interpret all of them in Shira’s world, giving students more effective feedback than either a) a right/wrong evaluation or b) no feedback at all, which is common for paper-based curriculum.

The most straightforward interpretation of a solution like x > 3 is that Shira should land at x = 3 and run to the right. Deciding how to interpret strict inequalities when we asked for an inclusive inequality and vice versa was challenging for our team – and fun. In those instances, we decided Shira would leave a blade of grass behind or the ground would crumble beneath her.

How we handle inclusive v. strict inequalities. Shira either falls off the ledge into the water or leaves a single blade of grass behind.

We feel pretty confident that your students will anchor their discussions about inequalities on Shira the Sheep for months to come.

Desmosification #3: Create cognitive conflict.

A problem asking students to consider the value of evaluating the point x = 0.

We wanted to make sure students reflected on some of the same subtleties of the original lesson. We find that’s easier when students have a concrete context anchoring their thinking. Once students are immersed in the context, we can more easily ask them to settle a dispute between two contrasting ideas, or to explain why a method works.

Behind the Scenes

Desmos lesson developer Zack Miller describes one of the core design dilemmas of the lesson and how our team found our way through it.

Settling on the concept of a grass-eating sheep was tough! In an earlier draft, we had a sheep eating stars, and it wasn’t feeling right because stars are discrete and inequalities are continuous. But grass can be continuous!

Getting the interpretive feedback to feel right was also tricky. As is typical of our approach, we wanted to show, not tell; and we also wanted to be generous in the kinds of inequalities we’d accept. So what should happen in sheep world if a solution set is an inclusive inequality but students answer with a strict inequality, or vice versa? What should happen if the answer is x < 3 and a student answers x ≤ 2.99 (which has a very similar solution set)? What should happen if students enter an equation? Or an inequality that’s way off the screen?

I’m pleased with how we ended up handling every one of those cases.

What’s Next?

Desmos outscoring all other curriculum for Net Promoter Score.

Overall, teachers in our 2019-20 pilot study really liked the Open Up Resources/Illustrative Mathematics curriculum … but they loved what we did to #Desmosify it. (Definition of the scale.)

Paint #Desmosify

Welcome to a series of posts sharing how we #Desmosify the curriculum from Open Up Resources/Illustrative Mathematics. You can use this lesson for free, or sign up to get many more activities just like it in our core middle school curriculum!

Here’s how we #Desmosified an Open Up Resources/IM lesson introducing students to equivalent ratios.

Desmosification #1: Create concrete connections.

The original lesson we #Desmosified seeks to connect a student’s understanding of ratios to the taste of a drink. Helping a student connect their developing knowledge to some piece of concrete knowledge is a great way to strengthen both.

The original lesson.

Contexts that students can touch and taste have lots of appeal. We chose a different context, however, one with several unique advantages.

Mixing cups of green and white paint.

In our lesson, students adjust the amount of green and white paint to create different color mixtures, some equivalent and others not, offering us two other unique #Desmosifications.

Desmosification #2: Give feedback that causes thinking.

A major challenge students experience in math class is getting zero feedback on their ideas or getting bad feedback. Kluger and DeNisi (1996) found that one third of feedback interventions have a negative effect on student learning.

When we ask students to make the same color with a different number of cups of paint, the student starts developing ideas. In a paper-based math lesson, they may not receive any feedback on those ideas. In many digital math lessons, students are told immediately that their idea is right or wrong. Dylan Wiliam writes that “feedback should cause thinking.” Neither of those kinds of feedback will cause much more thinking.

Students mix paint using numbers.

In our activity, students enter a number of cups and see the color they made. This lets students test conjectures in ways that would be much harder given the taste of a drink. Also, with the drink, the difference between equivalent and inequivalent ratios is perhaps more subtle than it is here, where you can see the difference between the colors. (We also describe the color difference in words for students who are vision-impaired.)

Students paint the wall with a new color to see if it matches.

Desmosification #3: Create opportunities for students to experience themselves and their classmates as mathematicians.

In many math classes, the roles are well-defined. Teachers ask the questions and students answer them. We build technology that helps students understand that they can be authors and askers of questions just as much as any adult.

The class gallery full of paint swatches.

So at the end of this lesson, we invite students to create a color by mixing red, green, blue, and white paint, and then to specify how much of a color they want. That information becomes a question for their classmates to solve in the Class Gallery, making more visible everyone’s capacity for mathematical thought, not just the teacher’s.

Behind the Scenes

Desmos lesson developer Sean Sweeney describes one of the core design dilemmas of the lesson and how our team found its way through it.

One challenging part of creating Paint was balancing the on-screen feedback we provided for students. We really wanted to include text-based feedback on each color to help support students that could have difficulty telling those colors apart, but it became tough to provide that feedback without being too descriptive about exactly what students should do to get the right mix. We discussed and experimented quite a bit to strike the right balance between interpreting the colors on screen for every possible color combination and allowing students the room to think for themselves on the relationships between paint combinations. In the end, we made a system that attempts to describe the color the way a person might instead of just simply telling students what paints to add or remove.

What Did Teachers Think?

Students liked the ability to make up their own mixtures of paint.

The visual feedback of the mixed paint colors was great!

What Did Students Think?

I wish ALL subjects could use Desmos!

It’s like a game—like solving a puzzle!

This is my favorite Desmos lesson we’ve done so far this year.

What I get to make my own paint color!?!

Shouts Outs

What’s Next?

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