Desmos Activity Builder as a tool for instruction

Note:  When I started writing this post, I intended for it to be about the most recent Desmos Activity I taught, which was on square and cube root graphing.  The ideas in this post are useful for much more than just the teaching of this one topic, so I'm hoping to both tell about the activity and clarify my thinking about using Desmos Activity Builder as part of instruction.

For context, the book that I use for Algebra II starts in Chapter 2 with transformations of the absolute value function. Chapter 4 includes quadratics, Chapter 5 is cubics, and in chapter 6 it is square root and cube root functions.  By the time students get to graphing of square and cube root functions they have done a lot of graphing by hand and talked about transformations several times.  

Lesson:

1.  Warmup: On the whiteboard I put a blank table of values for y=x^2 and y=x^3.  Students talked with a partner about how to complete the table, and we wrote the values on the board. (Activating prior knowledge, partner talk, no technology, no paper)

2.  Connecting concepts:  I showed this Desmos pre-made sheet for parabolas in vertex form for a reminder of how a, h, and k transform a graph.  Students watched and shared observations with partners, then we shared out as a class.  The visual below stayed on the whiteboard for students to reference throughout the class (formative assessment, teacher uses technology, no paper)

3.  Developing the concept: I asked students, what is the inverse of squaring?  Cubing?  What is the relationship between the points on a function and its inverse?  We used our tables of values from the warmup to come up with a table of values for the square and cube root function.  I wrote the standard form of each function below its table of values.  
(anchor charts, no technology, no paper)

Steps 1-3 took about 10-15 minutes.  We saved a lot of time by not requiring students to write anything down, and also by not having them do anything on the computer.  

4.  At this point in time students logged on to this Desmos Activity, and I modeled screen 1 for them.  

One tip that helped students the most was to type in the standard form y=a*sqrt(x-h)+k (DON'T add sliders), and walk through my thought process.  I asked myself the following questions:

• Did the y-values get multiplied by a number?  If not, a is 1.
• Did the graph shift horizontally?  If not, h is zero.
• Did the graph shift vertically?  If not, k is zero.

So for screen 1 with a vertical shift of 2 we typed y=1*sqrt(x-0)+2.  Students started asking questions about whether or not they could type the function in another way.  I told them yes and emphasized that the focus was on the thinking process, which would help them complete the problems in our activity.  (modeling, think-aloud, teacher and students using the technology)

5.  Students were given a few minutes to complete screen 2, and move ahead to other screens if ready.  We came back together one more time to go over screen 2, and I again modeled my thinking by asking myself the series of questions above.  I called on students (popsicle-stick style) to answer my questions as I built the equation for our transformed function.  (more think-aloud, formative assessment, equity)

6.  Students work on the activity:  I used the teacher dashboard to monitor student progress.  The teacher dashboard allows easy collection of sooo much data.  I am able to check in to see if all students have finished a particular screen, and if not I am quickly able to identify who isn't progressing in the activity and make adjustments accordingly.  This type of info ensures active participation in a way that I can't make happen with a paper worksheet.  Another great aspect of the teacher dashboard is that it becomes obvious very quickly if students are getting stuck on certain problems or if there are common misconceptions.  Desmos is helping me to formatively assess student learning and make real time adjustments.

7.  Summary/Closure of the activity:  This one is tricky because often each class is different in terms of how far they get through an activity.  I don't know if it is just me, but it seems like students can't generally finish all of the screens that I include in my Desmos activities.  At first I thought this was a problem, but now I am taking a different view.  Having extra screens provides a natural way to differentiate.  If students are struggling on the beginning screens, I can adjust the assignment so they have a more reasonable goal.  If students are excelling, I can send them to challenge screens.  As long as I have a game plan for how I am going to bring the class back together and summarize the learning, I am fine with having too many screens.  I am also allowing myself more flexibility in choosing a closing conversation, by letting it develop after I see where the students are at in their learning.  Ideally one of the activity screens will work for a closing conversation so data can be collected.  Students were successful with this activity, so a more challenging closing activity was appropriate.  In one of the classes that I taught we looked at the screen below, both for the challenge, and for how it would help students with their homework assignment for the night.  (Pair-work on computer, class discussion)

I'm very interested in continuing to explore how to use Desmos Activity Builder in instruction.  I've seen how powerful it can be in helping to introduce and build connections between topics.  One thing that I've learned this past semester is that planning for how to integrate a Desmos Activity into instruction is key to its success.  The strategies in this blogpost are ones that I routinely incorporate when I use Desmos as a teaching tool, and they also aren't different from strategies used in my non-technology lessons (Activating prior knowledge, formative assessment, anchor charts, modeling, think-aloud, think-pair-share, calling on all students).  There was a good amount of explicit instruction and modeling at the beginning to help students build the knowledge needed and to understand the directions.  This was mostly done without the computer.  Once the expectations and knowledge were in place students were prepared to practice using the Desmos Activity.

An area for growth:  Teachers that I work with are sometimes concerned that there are no notes in place when we use a Desmos Activity to introduce material.  I agree that there should be some sort of paper that students can reference throughout the chapter when they need to recall the learning.  I want to stay away from requiring notes during the Desmos Activity because there can be processing issues for students due to too much input.  This distracts from the learning.  Maybe jotting down key ideas in a notebook or foldable at the end of class or the next day would work?  Please share in the comments if you have ideas.

Next Steps:  I want to continue to think about best practices for using Desmos in instruction.  I'd like some sort of frame-work that I can use when I work with other teachers (I'm an instructional coach at the high school level).  This would make for richer debrief conversations and help with sustaining professional learning.  I'm also interested in using student work from the Desmos activities in summary conversations with students.  

It would be great to hear from others that are using Desmos as part of instruction, so I hope that people keep blogging about their experiences.  Audrey McLaren's Real-Time Story of the Desmos Activity builder has been a great starting point for learning about what other teachers are doing.  You can read her post here.

Desmos Activities

Desmos Activities

The Desmos activities on this page are ones that I created, including a few that were edited by the awesome Desmos Teaching Faculty.  More Desmos activities are available at teacher.desmos.com and also at the Desmos Bank.

Algebra 1 Eureka Math/Engage NY Module 1

Lessons 1-3 Graphing Stories

Lesson 12 Solving Equations with Number Properties

Lesson 14 Solving Inequalities with Number Properties

Lesson 19 Rewrite Equations of Lines in Slope-Intercept Form
Lesson 20 Linear Equations in Standard Form

Lesson 20 Applications of Linear Equations 

Lesson 21 Linear Inequalities in Standard Form

Lesson 23 Introduction to the Elimination Method for Solving Systems of Equations

Algebra 1 Eureka Math/Engage NY Module 2
Lesson 12 Relationships Between Two Numerical Variables
Lesson 13 Comparing Linear, Exponential, and Quadratic Relationships
Lesson 14 Day 1 Modeling Relationships with a Line
Lesson 14 Day 2 Modeling Relationships with a Line
Lesson 15 (teaching tool) Interpreting Residuals from a Line

Algebra 1 Eureka Math/Engage NY Module 3
Lesson 1 Integer Sequences
Lesson 1 Part B Integer Sequences
Lesson 3 Version A Geometric Sequences
Lesson 3 Version B Geometric Sequences
Lesson 11 The Graph of a Function

Algebra
Hearts: Practice transformations with function notation
Broken Parabolas (Intro to symmetry in parabolas)
1-3-5-7 Parabola Challenges (all with a=1)

Geometry
Linear Equations in Standard Form (can also be used in Algebra)

G-GPE: Write equations of parabolas given focus and directrix

Algebra 2
Practice with Factoring and Connecting to Graphs
Roots of Quadratic Functions: Some Special Cases
Parabolas and the number d
Discriminants of Quadratic Functions
Polygraph: Intro to Absolute Value Graphing
Introduction to Polynomial Graphing
Practice with Inverses
Transformations of Square Root and Cube Root Functions 

Trig/Math Analysis
Graphing the Sine Function using Amplitude, Period, and Vertical Shift

Calculus
Modeling Exponential Growth and Decay (can be used with Math Analysis as well)

Intro to trig graphing with Desmos Activity Builder

Background info: I co-taught this Desmos Activity to 11-12th graders taking Trig/Math Analysis.  They are going to start the chapter on graphing trigonometric functions on Monday, so this is an introductory activity to help students develop some of the vocabulary and background knowledge that they will use for the chapter.  This also happened to be Spirit day, with shortened classes and a spirit rally for that night's football game against the rival school.

Despite the somewhat challenging circumstances, the activity went great.  Students were engaged from bell to bell.  We heard comments like "thanks for making math fun" and "when can we do this again?"  The biggest hit of the activity was the ferris wheel screens (11-12) with one girl claiming that she couldn't even focus on the rest of the activity because she was too distracted by those  screens.  

                                   Note: The ferris wheel looks way better in the activity.  
                                   Check it out here on screens 11-12.

What I like most about an activity like this is that students begin by using informal language to describe how a graph changes, and then we build the formal vocabulary from their descriptions.  I gave maybe a 5 minute lecture in the middle about how the amplitude, period, and vertical shift are connected to the standard form of the sine graph (y=a*sin(b(x-h))+k).  We didn't look at horizontal shifts on this day as this was an introductory activity.  Students spent the period checking their understanding of both the vocabulary, and how it related to the equation for a given graph.  They even had a chance to apply their knowledge of the sine function by looking at how to model a ferris wheel rider's height versus time.  The other teachers and I used the teacher dashboard all throughout to see which groups of students needed help on a particular screen.  

The only challenge we ran into throughout the day was when students deleted elements on the graphing pages.  Some of them deleted the slider on the ferris wheel screen (11), so they weren't able to do the task at all.  We recommended that they try the following similar problem, or log in a second time.  Being able to reset a graphing screen would be helpful.  I've also been telling students to stay out of any construction folders, explaining to them how easy it is to mess up some aspect of the page, in which case they won't be able to complete the task.  

Overall, a very successful activity.  Thanks to #mtbos for giving me the ideas for the screens in this activity.  Two resources in particular that were helpful are here (by Steve Phelps) and here (by John Golden).

Engage NY Algebra: Module 1 Resources

This year my district is piloting the Engage NY curriculum for Algebra 1.  We wanted books for students, so officially we are using Eureka Math, but it is basically the same curriculum.

I wanted to put together a list of the digital resources that my team and I have come up with for Module 1, both in the spirit of archiving and of sharing.  I'll add more as we finish up the module, but for now this is what we have.

If you read this blog you know that I'm a big fan of apps like Geogebra and Desmos.  At the same time that I started planning for this module, Desmos Activity Builder was released.  I pretty much took this opportunity to learn how to use Activity Builder while enhancing the Module 1 Algebra curriculum, and many of the links below are ones that I created.  Most of us are on lesson 20 or so, and I'll add some notes later about which activities we liked the most.  Teachers will not necessarily be using all of the lessons below, as they are available as resources to supplement the curriculum.

Here are the resources that we have so far:

Lessons 1-3 Desmos
Lesson 7 Better Lessons website on combining like terms

Lesson 12 Desmos

Lesson 14 Desmos

Lesson 19 Desmos
Lesson 20 Graph the Line Geogebra Activity by Steve Phelps
Lesson 20 Basketball Hoops by Michael Richardson

Lesson 20 Desmos

Lesson 20 Application Problem Desmos
Lesson 20  Simplifying Radicals 2 Blog (I can't find a name, so let me know if you know who this awesome blog belongs to)

Lesson 21 Desmos

Lesson 23 Desmos

Lesson 23 Desmos Faculty Desmos

Our last lesson for this module will be 24.  We are thinking about some sort of Des-man type of project for the end of the module.  Maybe a mash-up of Dot-Capture Game by Michael Fenton and Desmos Puppy House by Fawn Nguyen with linear functions and inequalities.  Updates to come.

Reflections on Desmos Activities

We've been in school now for a month and a half, and I've managed to make it out to number of classes to try and observe Desmos activities on Activity Builder.  We've had some hits and misses (though more hits!), and I find myself making tweaks as I go.  It's been helpful to reflect on the lessons with each teacher, and I recently got some feedback from an administrator as well.

I wanted to record a few reflection points, mostly so I don't forget, and also in case it is helpful for others.  

• Open up the teacher dashboard on an iPad or tablet so that you can monitor student work as you circulate the class.  During my last lesson I also kept a post-it note with me so I could take down names of student work to share with the class during debrief time. Doing this also helped me decide when we should debrief and keep track of time.
• One-to-one tends to lead to students working in isolation.  We started noticing this last year in our classes, and we've seen the same thing on Activity Builder.  Two students per one device might work better.
• If you are using Desmos as part of the instruction, consider pacing as well as where to stop for additional instruction.  If you have an exploration screen, allow students plenty of time to explore, discuss, and share out, and then bring the class together for some explicit instruction.  Consider going back to an exploration screen for students to use during a warmup or problem set the next day.  This might be the biggest reflection point of all for me, as it is helping me be more thoughtful about what the objective is for each lesson and how I can help students meet that objective.
• For what it's worth, nearly all of the activities that I have ever done with technology in a math class take way longer than I thought they would.

That's it for now.  Any thoughts on this?  Anything to add?

Engage NY Algebra: Module 1 Lessson 3 interactives

Below is a modified version of Engage NY Algebra Module 1 Lesson 3 (modifications in blue, see teacher version here).  I am adding links to Desmos or Geogebra in order to make the lesson more interactive.  I am still learning the level of technology that would make these lessons better than paper and pencil, so feel free to chime in on the comments.   As you read through the lesson it is helpful to know that Engage NY treats their lessons like a choiceboard rather than a ready to go lesson.  The intent is not that every problem is finished by the end of the day, but rather that teachers choose the problem set strategically to best teach the content to students.   For this lesson my best guess is that we could go through example 1 and the exploratory challenge during class, and then the homework can come from the problem set.  



For example 1 students might use the embedded Geogebra link to add points of interest to a graph and use the pen tool to connect the points of interest.  I'm not convinced this is better than paper and pencil for this particular problem, but at the same time I might use this example as an opportunity to teach students these two Geogebra skills that we will use throughout our course.  If we want to compare sketches I can teach students to take a screen shot and embed on a given slide in this Google Presentation.  This will be day 3 of students looking at elevation versus time graphs, so introducing additional tech skills might be appropriate with students feeling comfortable with content.  The screenshot below shows the pen tool, as well as the input area where students would add points.

Given the time it takes to teach students to use new technology, I am more likely to spend class time on the exploratory challenge problems (1-3).  For these problems I want students to do all of the graphing on Desmos.  My question is whether I am changing the task too much by moving to Desmos.  In the paper and pencil version of problem 1, students watch this video (I couldn't load on Chrome, but Safari worked), scale the y-axis, and then plot by hand.  They will notice that at 0 seconds, there are 2 bacteria, at one second there are 4 bacteria, at two seconds there are 4 bacteria (etc.).  I want students to go to this prescaled Desmos graph and plot the points in pre-loaded table of values on a pre-scaled graph.

For problem 1 I am more concerned with students' ability to interpret the situation and extend the pattern.  If this is my goal, then asking students to choose the y-axis scaling and draw their own table of values seems like a lot of extra steps.  They can generate the graph below in about 30 seconds if starting with the blank graph above.  

Problem 2 is an extension of problem 1, with students noting that 20 minutes of real time are covered by 1 second in the video (this info is not included in the student version of text, so you have to refer to the teacher version for how to lead the lesson).  They could go back to their Desmos graph and show understanding of this fact by changing the times in the x-column of the table of values.  Then we can walk through how to change the scale of the x-axis so the data points are visible.  Adding additional points necessitates a rescaling of the y-axis if you want to see all of the points.  We can rescale by clicking on the wrench icon towards the top right and changing the minimum and maximum values for each axis.  

Using Desmos to explore the scale of a graph is helpful because the graph updates its window real time, so you can immediately see the effect that an x-y min/max change has on the graph.  Students may lose out on a bit of the reasoning for how to label the tickmarks, but in the end I believe the more valuable skill is determining the domain and range values and interpreting them in the context of your problem.  

What do you think of these digital additions to the lesson?  Are they too much for students?  Does this change the task in a way that takes away from the mathematical goal? (Students choose and interpret the scale on a graph to appropriately represent an exponential function. Students plot points representing the number of bacteria over time, given that bacteria grow by a constant factor over evenly spaced time intervals.)

Download or make a copy of the Google Doc here.  Original source is here.  Engage NY is a great resource even if you aren't using it as your primary curriculum, and you can modify this document to use in your class.

This lesson is part of one of the summer work projects at my school, and includes making a few interactive lessons for our Algebra class.  We are piloting the Engage NY curriculum for the first time next year.  We are also in the Google Domain and will have access to Google Classroom for the first time, in a one-to-one environment.  We'll primarily use Google Docs and other Google Apps for Education as we learn how to meaningfully incorporate technology into our math classes. 

Geogebra/Desmos Functions Resource

I came across this resource page that I made for a presentation back in January.  I thought it was going to continue to get filled out, but I had to move on to other projects.  I wanted to share this in case it inspires any summer learning, and I hope the resources can give an idea of how dynamic graphing software can enhance student learning experience for functions.  A lot of these links are to pre-made Desmos or Geogebra pages by other authors, and some I created on my own.  I linked to blogposts for a few of the activities as well to give a better idea of how the activity was used in class.



You can also open this document in Google Drive and make your own copy at this link.  I've spent a lot of time over the past couple of years thinking about how to use Geogebra (and now Desmos) to improve student understanding of functions.  Most of the textbooks that I've used in Algebra and Algebra II don't go very far beyond "graph this function" for practice problems in these sections.  Geogebra and Desmos offer us a chance to change the task and the types of questions that we ask when teaching functions.  

For the project above I included links to sliders, flashcards, and transformation activities when I could find/create them.  Students can play with sliders when first being introduced to a new function type to help them attend to the structure of the function with its parameters, and they can build on the background knowledge from previous function types.  Flashcards can be used to review vocabulary, and to check for understanding of the values of the parameters.  More on Geogebra Flashcards here.  Transformation activities or "other" activities mostly consist of Geogebra pages where students will find the equation for a given function.  

Though I haven't used all of these pages, the ones that I have used in Algebra II and Pre-Calc facilitated richer conversations, and overall better understanding of functions and transformations in my own class.

Desmos and Piecewise Functions

We are about to start functions and transformations in trig/math analysis, so I made some piecewise functions in Desmos to use for the intro.

The above function is named f(x).  Below is what I entered to graph the function.  

Then we can view transformations of f by typing the transformation on an input line to the left of the graph.

You can show or hide each graph by clicking on the colored circle to the left of the equation.  Below is 2f(x).

And f(2x).

Here are the links to the Desmos graphs.  Graphs 3 and 4 include questions for students to answer.   Hope some of you can use them!

Piecewise 1

Piecewise 2

Piecewise 3

Piecewise 4 

All of these functions were taken from Calculus AB free response questions, hence the link back to the website.  I also made a table of values to use for by-hand practice.

Watch this video by Meg Craig for a fantastic demo of how to use the table above.  Scroll down about halfway in her blogpost to watch the video.