Friday, August 10, 2018

Desmos Animation 101: Part 2

Suzanne von Oy shared a graph titled The Tree of All Seasons awhile back as one of her #graphjam submissions. It's mesmerizing at the very least, and left quite a few of us wishing for a Desmos art course so Suzanne can teach us her magic.



I look forward to this course, but as I dove into the graph, I found myself quickly overwhelmed. So many variables and functions. I wasn't sure where to start with understanding the various parts. This left me wondering what the different parts are of the Desmos art course that will allow me to build this (and any) awesome graph.

So where would the course start? There are quite a few things going on just with the flowers in The Tree of All Seasons graph.

1. The flowers are all different sizes. Suzanne uses exactly one image to render all of the flowers, so I'm curious as to how she made them all different sizes.
2. The flowers start growing at different times. They start falling at different times. They start shrinking at different times. I imagine these are all related somehow.
3. Once the flowers shrink down to size zero, they stay at that size throughout the animation.

In my first post, I looked at the way in which we can control the size of the flowers using a function. In this post, I want to look at the way in which we can control the timing for which the flowers bloom and shrink.

I stared at this line of Suzanne's graph for a really long time.


After breaking it down and figuring out that the first expression (in parenthesis) assigns each flower its length and width depending on its y-value, I looked at the second expression and thought about what it does:


If I graph this I see the following:


This graph assigns a "size" to the flowers depending on time. It essentially says the flowers will grow from size 0 to full size (or size 1) in a certain amount of time. Then the flowers will stay full sized for a different amount of time, after which they will shrink down to size 0 and stay that size throughout the animation. Super useful!

Question: What are other uses of this function that people might explore before working on the The Tree of All Seasons problem?

Note: I am interested in exploring what a Desmos art/animation course would look like for personal reasons, but also I am interested in exploring what it would look like because I'd love to see a course like this offered at the high school or college level as a math elective.

The Tree of All Seasons/Desmos Art Class

Suzanne von Oy shared a graph titled The Tree of All Seasons awhile back as one of her #graphjam submissions. It's mesmerizing at the very least, and left quite a few of us wishing for a Desmos art course so Suzanne can teach us her magic.



I look forward to this course, but as I dove into the graph, I found myself quickly overwhelmed. So many variables and functions. I wasn't sure where to start with understanding the various parts. This left me wondering what the different parts are of the Desmos art course that will allow me to build this (and any) awesome graph.

So where would the course start? There are quite a few things going on just with the flowers in The Tree of All Seasons graph.

1. The flowers are all different sizes. Suzanne uses exactly one image to render all of the flowers, so I'm curious as to how she made them all different sizes.
2. The flowers start growing at different times. They start falling at different times. They start shrinking at different times. I imagine these are all related somehow.
3. Once the flowers shrink down to size zero, they stay at that size throughout the animation.

I started exploring question 1, or how I can make all of the flowers different sizes using just one copy of the flower image. You can make multiple copies of an image by using a list to define the center of the image. In this case we could use (x_1,y_1) to define the center of the flowers since the centers are all listed in the table.

I made some efforts to apply my previous knowledge of animating images in the calculator and failed very quickly. My next move was to dig into the function Suzanne uses to control the dimensions of the flowers. Even that was tough, but I found some success by looking at the function one piece at a time. Here's the first piece:



This is one way we can define the length and width of the flower image. At its max, the value is something like 0.833, and at its min, the value is 0.5. I imagine Suzanne started with her minimum value in mind and played around with the numbers in the mod function to find a nice maximum value. (Suzanne, let us know if you thought of this in a different way).

Here's a graph with just five flowers. The dimensions are controlled using the function above, which depends on the y-variable. Turn on the movable points and move them around to see how it works.

Question: What are other uses of this function that people might explore before working on the The Tree of All Seasons problem?

Note: I am interested in exploring what a Desmos art/animation course would look like for personal reasons, but also I am interested in exploring what it would look like because I'd love to see a course like this offered at the high school or college level as a math elective.

Sunday, April 16, 2017

Upgrade Your Card Sorts


This is the blogpost version of a talk that I gave at CMC-S 2016 and NCTM 2017 on Desmos Card Sorts. I usually post slides, but in this case I’m not sure how useful the slides are without context. Hope this post can give you a sense of the value of a card sort and ways in which a card sort can support student learning along with the benefits of using a digital card sort.

The first card sort we did in the session was Card Sort: Quadrilaterals, inspired by Lisa Bejarano. This card sort asks students to sort a set of statements about quadrilaterals according to whether they are always, sometimes, or never true.
Here’s the "Responses" view of the dashboard for the first four students in the activity (using "Anonymize").  


The green piles indicate that all of the cards in the pile have been correctly matched. Red piles tell you that one or more of the cards in the group are incorrectly matched. Gray cards haven't been grouped yet. You can also see how many cards are missing from a group. 


You can click into individual student work (above) and check the answer key to see which cards students have missed. The image above shows that Hermann only misplaced one card for the "Sometimes" true pile.

You can also use the up/down arrows on the top left to scroll through student work to get a general sense of which cards students are matching incorrectly. Better than this though, you can go back to the summary view and see right away which cards students are incorrectly matching.


From here I might pause the class and zoom in on the card that students have incorrectly matched so we can have a conversation. I might ask my students to share how they paired the card and why, and let the conversation progress from there so that I can help students build towards a correct understanding. What I love about this is the way in which it honors existing ways of thinking. Students have a chance to share what they know about rectangles, and we can use this to develop a need for a precise definition. From there, once we've agreed that a quadrilateral with four right angles is the way to define a rectangle, students can see that a square is an example of a rectangle that has perpendicular diagonals.

The second card sort we did is Card Sort: Linear Functions from the Linear Bundle, and it is an open card sort. This means that it has no answer key. After working on this card sort I showed the dashboard and we talked about different ways that the dashboard can be used to support student learning.


For an open card sort, you can see the most common groupings (above). Since there is no one right answer, the teacher can pair different groups of students together to talk about how they sorted. This is especially helpful in this open card sort where the goal is for students to deepen their understanding of the characteristics of linear functions. Instead of pairing students or groups to discuss, you might screenshot some pairings that highlight similarities and differences that you’d like the entire class to see.


Here the teacher might pose the questions “Which did you pick and why? How might the other group have decided to pair these cards together?”

This card sort had two additional screens that also provide opportunities for rich conversation and student learning. Screen 2 is below.



There are two choices a teacher can make on this screen. First, a teacher can select a set of answers that will be most beneficial for the class to hear. In many cases, sequencing these answers from less formal to more formal can help students at all levels to access the conversation and grow in their understanding of the mathematics involved. Second, a teacher can make use of a screen like this give a voice to students that don’t usually raise their hands to participate. One move I’ve seen teachers make is to let the student know ahead of time that they have an answer that will be valuable for the class to hear. 



Screen 3 (above) is similar to Screen 2 in that teachers can strategically select students to share student work. An additional benefit of a screen like this is the controversy that it can introduce. Below are some of the responses from Screen 3 from the NCTM session.


Most of us gave a response along the lines of the first three bullets. I intentionally planted three of the "Other" responses based on responses I've seen in classrooms, but there were still 5 responses from our session where participants had differing views. Being able to see these responses in the dashboard lets the teacher pair these students with other students to sort out their differences. Between this screen and the others in this open card sort, I appreciate the message that the knowledge doesn’t have to come from the teacher. I can learn from my classmates, and the can learn from me!


We ended the session with a brief tour of teacher.desmos.com, along with how to find pre-made card sorts and how to build your own. For more info on this head to learn.desmos.com/cardsort and learn.desmos.com/create. Also please feel free to chime in with your card sort successes! I’d love to hear more about how people are using card sorts to support student learning.