CMC North 2018: Reflections Part 2

Part 1 reflections are here.

Session 2

The theme of CMC North this year was "Student Voice, Let's Hear It!" I don't think you'll find a math educator that can disagree with this statement, and it was great to be able to spend an hour with Paul Jorgens and Richard Hung exploring different strategies and activities to promote student voice. You can access resources from their session, titled Fire Up The Math Classroom With Conversation, here.

I am grateful for the opportunities I've had lately to deepen my understanding of why student voice is so important. For Paul Jorgens and Richard Hung, student voice is an essential part of how their school is working to close the achievement gap. For many schools, the path to closing the achievement gap lies in creating support classes, intervention classes, after school programs, summer opportunities, etc. These are all important. But what you do during the time you have students in your class is equally important, and that was the purpose of Paul and Richard's session. Their course team is intentional about planning opportunities for classroom conversation throughout each unit. During the session we worked through WODB problems, Open Middle problems, as well as strategies they use to launch data lessons. I don't feel like I can do justice to this experience in a blogpost. Fortunately Paul has written about how he uses WODB and Open Middle problems using Desmos as a tool to support student understanding. I also recommend checking out Paul's Desmos activity on categorical data where students explore the impact of the Voting Rights Act of 1965. When exploring this and other data, Paul doesn't immediately include the context. Instead, he invites students to notice and wonder and do some computations before revealing the contexts.

Perhaps my biggest takeaway was the importance of giving students a chance to use their own language to describe things before we give them the words we want them to use. For me, an added bonus of building in social experiences to practice vocabulary is that students learn that their ideas matter, and that there isn't just one right answer in a math class.

If you want 180 ideas from Paul for supporting students, check out his #teach180 resources from the 2017-2018 school year. And if you want 180 more ideas consider following Paul and Richard on Twitter for their 2018-2019 #teach180 ideas.

Session 3

Scott Davidson and Ethan Weker led a session on 3D printing. When I am learning about a new tech tool, I find it helpful to know some of the What, Why, and How of the tool. This session delivered in a really nice way on all three of these categories. Scott showed us how his students use a free program called Tinkercad to make name cards in his class. Ethan shared some of his student work from an assignment where he has them make their names in Desmos to 3D print. He has found that students made much more interesting designs for the name project vs before it was 3D printed because students knew they would have something to take home. Student ownership for the win!

In addition to student made products, we learned how teachers can 3D print their own manipulative. We played with 3D printed dice and learned that they were unfair, first by experimentation, and then by inspection. Ethan showed us a really interesting manipulative that helps explain why adjusting the b value in the standard form of a parabola moves the vertex in an parabolic path.

Teachers are creating and sharing designs for their own 3D printed manipulative at Thingiverse.

Session 4
Jenny Wales and I presented "Designing with Desmos" during this time. Always a blast presenting with Jenny! Thank you to everyone that came to our session!

Session 5

I went to a session on story tables led by Shira Helft and Taryn Pritchard. I didn't take any notes during this session because we spent nearly the whole time doing math! Really glad I had the chance to learn about this tool, which Shira and Taryn compared to a knife (practical and multi-use). You can use story tables to help students make sense of any function type, and they can be used to help students understand the relationship between the graph of a function and solving equations that involve that function type. Not only that, the story table helps students develop an intuitive understanding of how the order of operations plays out when solving equations.

Check out Shira's Global Math Department Webinar on story tables if you want to learn more.

Thanks to all of the presenters for helping us learn, and to all of the organizers for making CMC possible! Always a great time and top notch experience.

CMC North 2018: Reflections Part 1

I was fortunate enough this year to be able to attend (again!) my favorite math conference, which is CMC North at the Asilomar Conference Grounds. The view, the people, and the sessions are always great, and I always leave with a full head and a full heart.

This post is mostly a reflection and a record for me to look back at, but I hope it can be helpful to others as well. If you see something interesting in here and you'd like to know more, please reach out!

Session 1
I went to see Juan Gomez talk about tools he uses to promote student inquiry in his class. He talked about the importance of differentiation, and gave us ideas and examples for differentiating process, content, and product.

Differentiating the Process

We looked at Graspable Math as an example of how a teacher can differentiate the process by which students learn. One example we looked at consisted of graphing a polynomial function and using the scrubber tool to change an exponent. This allowed us to see the graph change, and we could compare to the original graph.

Differentiating the Content

We looked at a Desmos activity by Jon Orr called Pumpkin TimeBomb Prediction. In this activity students use scatterplots and relationships to predict the number of rubber bands wrapped around a pumpkin it will take to make the pumpkin explode. (Check out the results here.)

Next up we looked at CODAP, which is a free online data analysis software. Really interesting and powerful. Lots of ready to use data. One interaction that seemed really useful is that you could click on a point in the scatter plot of a large data set, and CODAP would auto-scroll to that data point in the table. Another feature that Juan showed us was that you can upload a csv file with your data, and then use the program to draw random samples from it.

Juan shared the Desmos Halloween Coloring Book project by Luke Walsh as an example of differentiating the product of learning. Luke had his students make halloween themed art using the function types they were learning about. Check out their work!

We ended by looking at Paul Jorgen's "My Wedding Dance Dilemna: Shout." In this activity, we use math modeling to make a dancer move to the different parts of the song Shout. This had us all giggling out loud. A powerful aspect of this activity is that you can revise your wedding dancer's movement and check your thinking. Students begin by sketching the height of the dancer over time before moving into more formal thinking of using equations to represent the height over time.

Juan helped us summarize our thinking at the end of the session and reiterated that his goal as a teacher is to be able to think flexibly in the moment and have tools that allow for that flexibility. 

As I was looking through my notes, I found this: 

"How can I do the pumpkin activity and still teach my content?" 

I believe this was a question asked by a participant in the session. Leaving this here as an open question to consider in the future. 

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.

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.

Mathy Activities

I was delighted to end my morning today with this:

My 4th grader just asked me to print something mathy out for him to work on. Suggestions? #elemmathchat #mtbos

— Shelley Carranza (@stcarranza) October 22, 2016

I was even more delighted with the responses for mathy activities for my 4th grader. I'm posting links to the activities here for future reference so I don't lose them, plus maybe you can use them too!

1. Ken Ken puzzles. I hadn't seen these before and took a trip to this website. Turns out you can customize Ken Ken puzzles by operation and difficulty level, and you can also complete them online.

2. Logic Puzzles. This was a favorite when I was a kid. Glad to have so many resources to find these and other puzzles.

3. Graphiti. As I was opening this link my son chimed in with "Oooh, I love these things!" He was also super impressed that Twitter friends were responding within minutes of my tweet. Go #mtbos!

4. mathpickle.com. This website has puzzles, games, and mini-competitions organized by grade. Can't wait to check this one out!

5. Printable number puzzles (Rosetta, Star, Subaddo).

6. The lovely Patterns of the Universe coloring book by Edmund Harriss and Alex Bellos.

7. 180 Opportunities to Notice by Steve Wyborney.

Thanks so much to everyone who contributed! What other mathy things might go on this list?

Starting the summer with SMP 7

Back in December I attended Grace Kelemanik's closing session at CMC North in Asilomar California on unpacking the math practice standards. In her talk, she shared with us her framework on the standards for math practice.  She says that some of the standards take the lead in thinking and some do the supporting.  Standards 2(quantities and relationships), 7(structure), and 8(repetition) are the avenues of thinking.  Students with a strong foundation in these three standards will have a starting point with which to begin problem solving, and they can jump lanes if their starting strategy doesn't work.  This powerful talk has had me thinking about the SMPs ever since.

 SMP 7 is a particularly interesting standard to look at in the high school curriculum because we can find structure in expressions, diagrams, graphs and more. This year I worked closely with the Algebra teachers in my district while we implemented a new curriculum. One problem in particular from our new curriculum is below. ( From Engage NY Algebra Module 4 Lesson 8 (Student View) )

Students use the symmetry of a parabola to complete the graph (structure in the graph).  Later in the problem set students make note of the pattern in the table of values (structure in the table).

There were additional problems that made use of the structure of a parabola in order to solve.

• Is f(4) greater than or less than -6? Explain
• f(-4)= -13.  Predict the value for f(4) and explain your answer.

Towards the end of the lesson there was a section called Finding a Unique Quadratic Function.  Below are some of the prompts from that section:

• Can you graph a quadratic function if you don’t know the vertex? Can you graph a quadratic function if you only know the 𝑥-intercepts?
• Remember that we need to know at least two points to define a unique line. Can you identify a unique quadratic function with just two points? Explain.

At one of the schools that I work at we spent two days exploring the structure of parabolas and questions like the ones above using Desmos.

Day 1 Activity: Broken Parabolas

Day 2 Activity: 1-3-5-7 Parabola Challenges

In the 1-3-5-7 Parabola Challenges students explored the symmetry of parabolas by locating missing points given a set of points on a parabola. For this lesson students only looked at parabolas with an a value of 1. We didn't use that language since it was an introductory lesson, but it allowed students to use 1,3,5,7 (etc.) pattern when comparing the differences in y-values.The hope was that this informal exploration would set students up to look for and make use of structure when they graph parabolas by hand later in the unit. 

Having students reason through the problems using a pattern can also make for a great error analysis activity. Some of the student work from Activity 2 is below.  Each of these challenges started off with a set of blue points, and students were able to drag the red points to complete the parabola. They would then turn on the green function to check that it is going through all of the red and blue points. At the end there are also a couple of unique responses for challenges 1 and 2 from the 1-3-5-7 Parabola Challenges activity that would have been great to share with the class.



The original version of this activity had students entering coordinates into a table instead of dragging points. I thought this would help them focus on the patterns in a table of values as well, which could be an additional problem solving tool for them to utilize later in the chapter.  This turned out to be too challenging for the group of students that I initially worked with.  Adding points to a table of values using a pattern requires strong number sense, and is a different skill then having students use a pattern to add points to a coordinate grid.  Here is the original activity in case you'd like to take a look.

Focusing on SMP 7 in the planning of these activities shaped them in ways that led to increased reasoning and discourse.  Of course this is not the only way to increase mathematical reasoning, but definitely worthy of some focus this summer.

Footnote:  I also wanted to thank Silicon Valley Math Initiative for an inspiring end of the year session.  Their summer focus is also on SMP 7, and we launched this work in our last session by finding examples of SMP 7 in the K-12 curriculum.  I've been thinking about SMP 7 and seeing examples of it ever since.

Global Math Department Talk

Thanks to everyone who joined the webinar Desmos Activity Builder: Best Practices for Charging Up Lessons!  Below are the resources from the session.

Slides
Recording

My Favorite Calculus: Crazy Integrals

While reading on my computer today I received a notification from Dropbox that a shared file called "Crazy Integrals" had been updated.  This was pretty exciting since I created this activity four years ago and even though I am no longer teaching Calculus AB, the teachers are still using this activity.  This also reminded me that I could share this activity for the Explore MTBOS Week 2 blogpost challenge.

Before we launched into our integration chapter I did some pre-work to help students gain a strong understanding of both the notation and the concept of integration.  For Crazy Integrals I started by directing their attention the the diagram below.

 We talked about how finding a definite integral is like finding the area between a curve and the x-axis.  There are some finer details that need to be covered, and we brought them in both in the course of the activity and throughout the chapter.  The first two problems that we talked through are below.  Students noted on their diagram that the top piecewise curve is f(x) and the bottom piece-wise curve is g(x).

Students then discussed the best way to break up each area using vertical lines.  This helped them to rewrite each definite integral as two separate definite integrals.  Then they found each area.  We talked about how in Calculus an area below the x-axis is counted as negative.  Students worked in groups of 3 or 4 to complete the rest of the problems.  

Below are the other diagrams used for the activity.

There were all sorts of great questions that students brought up throughout the activity, most of which called on them to use precise definitions and language to answer.  For example, students wanted to know what to do with the smile and eyes/nose from the clown picture.  We had to use our working definition of definite integral to determine that they were extraneous to the problem.  Students also wanted to know how to find the largest/smallest x-values for the clown's hair.  We had to interpret the associated integrals (below) to know that those parts of the hair were not to be included.  

For the fish diagram students wondered why I only asked them to find the definite integral of the top curve.  Great question, and one that they could probably answer in their own groups.  

Overall super engaging activity with lasting pay-offs in understanding for the remainder of Calculus.  Students didn't want to leave class without finishing this assignment.

Note:  After the first couple of problems students stopped rewriting the definite integral as a sum of definite integrals, and instead got really into showing their work for finding the separate areas.  I figured this was fine as long as they demonstrated the understanding of the skill in the first part of the activity.

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.

#mtbos12days: 2016 Goals

Happy 2016!  I'm not big on new years resolutions, but I do believe that setting goals is important.  I have a few ongoing goals that I started in 2015 that I'd like to continue.

1.  Exercise: I started working out more at the beginning of the 2014 school year.  What helped me the most in finding motivation for this goal was the discovery that I can read from the Kindle app on my cell phone or tablet when I ride the elliptical.  I always look forward to reading, and somehow this translated into looking forward to working out.  I sometimes run into problems when I am in between  books, but in general things are going well.  At the beginning of the 2015 school year I wanted to incorporate more running into my routine.  I have a treadmill at home, and I learned that watching Netflix from my tablet does the trick in terms of motivation for running.  Probably the biggest motivator for me though has been my Fitbit, which tells me how many times each week/month I have worked out, in addition to number of steps, heart rate, and all sorts of other useful info.  For 2016 I am joining the #fitbos challenge. There are already close to 30 of us signed up.  Please feel free to join!

Join the #FitBoS in a workout hours goal for 2016! Sign up & log 'em here: https://t.co/VHyywb2NRz #MTBoS #160in2016 pic.twitter.com/2YzsHQDfEM

— Meg Craig (@mathymeg07) December 30, 2015

2.  Gratitude: This is a great goal for any of us.  We had gratitude and wellness week in my district, and I received the below in an email from one of my amazing administrators.  The information is from a Harvard Mental Health Letter.

“In positive psychology research, gratitude is strongly and consistently associated with greater happiness. Gratitude helps people feel more positive emotions, relish good experiences, improve their health, deal with adversity, and build strong relationships.”

Here are some ways to cultivate gratitude on a regular basis:

• Write a thank-you note. You can make yourself happier and nurture your relationship with another person by writing a thank-you letter expressing your enjoyment and appreciation of that person’s impact on your life. Make a habit of sending at least onegratitude letter a month. Once in a while, write one to yourself.
• Thank someone mentally. No time to write? It may help just to think about someone who has done something nice for you, and mentally thank the individual.
• Keep a gratitude journal. Make it a habit to write down or share with a loved one thoughts about the gifts you’ve received each day.
• Count your blessings. Pick a time every week to sit down and write about your blessings — reflecting on what went right or what you are grateful for. Sometimes it helps to pick a number — such as three to five things — that you will identify each week. As you write, be specific and think about the sensations you felt when something good happened to you.
• Pray. People who are religious can use prayer to cultivate gratitude.
• Meditate. Mindfulness meditation involves focusing on the present moment without judgment. Although people often focus on a word or phrase (such as “peace”), it is also possible to focus on what you’re grateful for (the warmth of the sun, a pleasant sound, etc.). 

My first gratitude letter of 2016 goes out to the fantastic people of #mtbos who have brought a bunch of us together over the holidays with #mtbos12days.  I can't tell you how much I enjoyed reading all of the blogposts over the past 2 weeks!  And now I am looking forward to #fitbos with both old and new Twitter friends.  Go #mtbos!

3.  Laughter and Fun:  At some point over the last 6 months when work was stressful and life was busy I realized that I wasn't having as much fun as I needed.  I started by spending more silly time with my kids.  They are hilarious and can always make me laugh.  I also made some time to go out with friends.  It's too easy to get swept up by the day to day, so a goal for 2016 is to schedule more time with friends and be more present around my children to ensure plenty of fun in our lives.  This goal is definitely about balance, and laughter and fun are part of that recipe.

4.  A professional goal:  I'm going back to school!  Through Twitter, I found a great online course for spring semester.  It is Mathematics Coaching: Designing Effective Professional Development, offered by Mount Holyoke College.  This is my third year as a coach (second year as a math coach), and I've been looking for opportunities like this.  Wish me luck!

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)

Google Docs in Math: Error Analysis

Looking for ways to incorporate more communicating reasoning problems into your curriculum?  Error analysis problems provide students with an opportunity to communicate their reasoning and focus on the Math Practice Standards "attend to precision" and "construct viable arguments and critique the reasoning of others."

A screenshot of my first digital error analysis assignment is below.  I made this on a Google Doc by adding pictures of the problems worked out in my own handwriting.  You can make a copy at this link.

There are many ways to get images into a Google Doc.  One way is to open Google Drive on your phone, navigate to the file that you want you image to be in, and then add an image by pressing the plus sign and taking your photo.  From your Google Doc you will locate and insert the image, and crop as needed.

I used Google Classroom to distribute this assignment to students.  We talked through the first problem together.  Then color coded each error according to its type.  I provided the following key at the bottom of the assignment.

This entire process took about 10 minutes.  With just a few minutes to spare before the end of class, students got started on the second problem and submitted their assignment.  Below are some of the results.

You'll notice that we didn't always agree on the type of error represented in the problem.  This was our first try at this type of assignment, and over time we'll continue to discuss the types of errors that we are making and hopefully have more agreement.  The amount of writing and discussion that took place in just 10 minutes was impressive, and I can see the results of this type of assignment improving over time.