Bringing SY 2019-2020 to a Close: Reflections… and Desmos

Several weeks back, I packed up my classroom. Frozen in time, everything was set up and ready to go for teaching on Monday, March 23, post-spring-break… that day had LOOOONG come and gone, though my classroom was ready to propel to the “normal” routine. I couldn’t wrap my heart around why I was packing everything up, even though my brain knew full well that I had been quaranteaching for WEEKS already. My gloved hands went into autopilot while my exposed eyes fought back tears in my eerie, empty, silent classroom.

Here we are, staring down the last week of school. The official school calendar says next week is the last week of school. It always did. Normally, if we’re being honest, most of us would have had a countdown of how many teaching days were left until the coveted ***LAST DAY OF SCHOOL***. While we usually count the days down out of jest and sheer exhaustion of end-of-year testing, field trips, and all the wonderful interruptions that make the end-of-the-year enjoyable and terrible all at the same time… this year, it feels flat. Abstract. Really, really sad.

Wishing we could meet for class for one final farewell is not uniquely my wish… MANY teachers’ hearts out there have yearned for this for over two months now. Though technologies are amazing and have maintained the learning and community-building to a degree, there’s nothing quite like the heartbeat of the classroom… IN PERSON. I’m grateful! We’ve accomplished A LOT! It’s okay to celebrate that and be thankful for our Zoom-time and technologies while also feeling sad, tentative for what’s to come, and even angry… about ALL of it.

When I need to do something with a pile of feelings, I create. I have to make something. Something that will potentially make things better. Content creation is my outlet. It makes me feel like I am countering the blahs with joy and practicality.

After cleaning out my “real” classroom, I made this. I kept seeing this virtual classroom Bitmoji idea on social media, and scanned past it… but after packing up my classroom in real life, creating a virtual one helped organize some things NOW while also providing an editable platform for next year, when we meet in person and/or virtually. I used Apple’s Keynote to create the bones of it… the things I really don’t plan on editing. I used the actual paint color of my classroom walls, used a photo to fill a shape with the actual floor tiles, and included some cherished pieces, like my MISTAKES art canvas and my carved wooden MRS. YENCA name sign given to me by my husband’s Nana… who is the ORIGINAL Mrs. Yenca for whom it was carved! 🙂 It’s a busy and silly Google Slide full of hyperlinks and animated GIFS, and making it was therapeutic. Google “Virtual Bitmoji Classroom” for tutorial ideas galore.

I’m hoping that students have also found creating to be therapeutic. That’s why my PLC-mates and I have assigned Desmos Projects to every class the past few weeks. You can access and read about project ideas for Math 7, Math 8, and Algebra 1 here. In other tabs on my MacBook, I’m watching them work now, as I type, and these open and creative tasks never disappoint!

Next week, I’ll have one final Zoom meeting with each class. We’ll begin with a showcase of students’ graphing projects to celebrate their work and progress! Here are screenshots from the work they’re finishing up in Math 8, Algebra 1, and Math 7 respectively.

When I cleaned out my classroom, I grabbed the stack of “WHO I AM” papers that students had completed on the first day of school. I didn’t know what I was going to do with them yet. Generally, I return them to students on the last day of school. They barely remember ever filling them out, and are always surprised to read about how their interests have changed since August. And we all have a good chuckle at their hand-drawn self-portraits!

I was compelled to create something for my students, given that I wouldn’t be able to hand these back in person this year. So… using Keynote to sort out the portrait images, I created custom Desmos Card Sorts for each class! Students will attempt to match their peers’ (and their own!) names to their self-portraits, sketched on the 1st day of school this year. I wonder how many of them will even recognize their OWN creations! It’s going to be fun and light-hearted… creating it has left me a little heavy-hearted too.

AND… thanks to the Desmos Educators group on Facebook, I found @rachael_degnan’s end-of-year reflections activity, modified it for my middle schoolers, and tacked the self-portrait Card Sort screens at the beginning of the activity (the ability to COPY and PASTE screens between Desmos Activities is a DREAM FEATURE).

Here’s Rachael’s original End of Year Reflection activity, in case you’d like to use or edit it too.

Maybe **I** need a teacher-version of a Desmos End-of-Year Reflection Activity… there’s so much to process about how we “got through” these virtual teaching days, and even more uncertainty regarding how school will operate in the fall. Let’s all promise to give ourselves permission to rest, and not try to anticipate every “what if” the next school year might bring. Reflect, celebrate, and rest in knowing we’re all doing the best that we can.

Hope to be back here safely.
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Desmos From a Distance

Here in Texas, we received official word from our governor that our classrooms are closed for the remainder of the school year. We knew this was coming. It still stings. Such an abrupt ending to the communities we’ve built since August. I can’t believe our before-spring-break farewells were actually the last time we’d meet in room 510… heartbreaking!!!

Meeting twice weekly on Zoom from home sure helps! We’re keeping our communities and curriculum rolling in the best ways we know how. It has been exciting and exhausting to transition to 100% online learning. I lost a LOT of sleep the first few days, and worked well into the night hours trying to adjust materials I had, or create things from scratch, to make them more distance-friendly. Even when we regularly use technology tools in our classrooms, it’s with the assumption that our students are ALSO right in front of us! That we can run around the classroom and LOOK at what they’re doing, and CHAT with them one-on-one, in small groups, and as a whole crew. Online learning has been a blessing and a challenge.

Looking ahead to the rest of this Maypril of a school year, I have some Desmos projects and enrichment activities up my sleeve. Since I teach three math courses, I thought I’d share the resources I’ll likely use for each class here in this post.

{Math 7}

Math 7 students will use tables of connected points in this Desmos FIRST NAME Art Project. The included checklist could be modified to include point values if one chooses to ‘grade’ the project. Thanks to the ability to copy-and-paste screens from one Desmos Activity to another, this mash-up includes modified screens from other activities (credit given in the project description). Students are also asked to capture and submit a photo of a graph paper ‘rough draft’, thanks to the ability to add an image to a Desmos graph!

Looking for a prerequisite or follow-up activity that helps students use Desmos tables for graphing… and fun? Try Mini Golf Marbleslides!

To revisit ratios and proportional reasoning one last time, while previewing the concept of scale factor (which will be addressed in Math 8 next year) students and families will choose a favorite recipe, then use this Desmos project to scale it up as if they were feeding our entire class… and me!

My PLC-mate Lauren and I have been throwing around the idea of a recipe project, and thanks to the Desmos Educators group on Facebook, and friends on Twitter, we got the help we needed to include some self-checking CL tables. Once students scale up the recipe, they’ll add their recipe AND scaled-up recipe to a Google Slides presentation, which will serve as a virtual class cook book to end the year!

{Math 8}

This will be the third year my Math 8 students will use linear functions, and a whole lot more, to create their Pet House Linear Projects! This year, I added a screen where students capture and submit a photo of a graph paper ‘rough draft’ since we won’t be in the classroom together. If you search for this project online, you may also find modifications folks have made that include other features and some fancy CL (computation layer)!

As a prerequisite to Pet Houses, perhaps Marbleslides: Lines will help students get reacquainted with linear functions?

To encourage students to explore functions beyond linear, here are two enrichment tasks that can ‘run in the background’ as students build their Pet Houses. I also use these for my Algebra 1 students! MARBLESLIDES CHALLENGE SETS!

{Algebra 1}

While the Pet House Project focuses mostly on linear (but never stays there, ha!) this Des-Draw Algebra 1 project explicitly asks for quadratics to be included… and nudges students to explore inequalities as well as functions beyond linear and quadratic.

As a prerequisite to Des-Draw, maybe Marbleslides: Parabolas will help students get reacquainted with quadratic functions?

Lastly, these BREAKOUT activities might be even MORE challenging in a distance-learning environment, but I have them in my virtual back pocket nonetheless. Differing opinions were shared about trying to combine the Zoom Breakout feature with a Desmos Breakout activity… you never know until you try…?! What do YOU think?

What? You’ve never heard of Jay Chow’s DESMOS BREAKOUT ACTIVITIES?!?! Launch one for yourself, try to ‘break out’ as a student, and you’ll quickly see why these activities have become a few of my favorite ways to promote practice, review, and deeper understanding of concepts.

Breakout Desmos – LINEAR!

Breakout Desmos – QUADRATIC!

These activities are literally the tip of the iceberg as far as what’s available. Googling “teacher Desmos *insert math topic*” will provide even MORE options for you! I hope your end-of-year Desmos-From-a-Distance experiences are successful for you and your students! Thank you to ALL who have contributed to the creation of so many rich (and fun) Desmos activities!

Posted in Algebra 1 | 1 Comment

How To Be an Online Teacher in X Simple Steps

JUST KIDDING! That catchy title doesn’t begin to give justice to the hard-working colleagues I have the pleasure of planning with… the ones in my PLCs locally, AND the global ones who have been sharing about their experiences to help all of us be better.

I am BY NO MEANS posting ANY of this to say “Here’s How To Be an Online Teacher”… this is me saying, this is what I have tried in my 2-whole-days of experience, mostly because I’d love to know how YOU are doing this, and how I can do a better job because you’ve already learned or realized something that I’m not yet seeing.

{UPDATE 4/9/2020: Check out this news feature that showcases some of the resources mentioned in this post!}

Okay, my tongue-in-cheek post begins now.

Step 1: PLC Planning

PLC with teammates via ZOOM, e-mail, and/or texting at all hours. Calm each other down when technology behaves questionably. Share resources so we can all survive. Document all lesson plans in a concise report (see mine below… HAHAHAHAHA)

Step 2: Consolidate the Plans

Interpret chicken-scratch and transfer to index cards, labeled by day/class period. Realize the resources listed actually need to be located… find, create, and/or share needed resources* {see more about resources at the end of this post}. Brew a cup of coffee. Brew another.

Step 3: Digitize

Transfer index-card ideas to Numbers spreadsheet. Include links to aforementioned resources. (This process digitizes my plans, and helps me think about how to also rewrite my teacher-plans into student-friendly plans to be posted for them later…)

Step 4: Zoom

Set up Zoom meetings for each class, being mindful of all settings. Use a photo of my classroom as the virtual background for a personal touch!

Step 5: Rearrange the Furniture in Your Home

Realize that a Zoom Virtual Background does not stop household noises or the flow of humans through the open dining room. Rearrange the dining room and bedroom furniture. Ensure every worker-and learner-from-home has his/her own room… with a DOOR.

Step 6: Schedule

Close the bedroom door. Copy-and-paste student-friendly, digital post from Numbers spreadsheet into Google Classroom for each pertinent class (I teach three different courses), remembering to create unique links for each activity for each class for each day, including details for accessing each Zoom meeting. (Each time a class meets, I’ve been plopping everything into one announcement in GC and scheduling it ahead of time to post 30-minutes before each scheduled Zoom meeting.)

Step 7: Post Agenda

Post a concise agenda-version of Google Classroom student-post on Google Calendar for parents to access and view.


Watch Google Classroom post your announcement at the scheduled time… and launch the Zoom Meeting (anyone else get butterflies in their stomachs when they do this the first few times?!? I was as nervous as the first day of school, pressing that button!)

Rinse and repeat from now until the unknown end of all of this…!

For reference, here’s what my schedule will look like each week! A MILLION THANKS to my colleague Georgia for helping our entire department take a more general schedule and streamline it purposefully to this! 🙂

This planning process has honestly felt like I am creating never-ending sub plans, because of the level of detail needed to ensure that we maximize our Zoom time together, that we present concepts using what we perceive to be the best tools within our realm of familiarity to meet the objective(s) at hand… it really is A LOT.

NOT COMPLAINING, just the facts!

As a matter of fact, I couldn’t be more grateful to be safely grounded at home with two of my favorite humans, while also having the opportunity to continue to teach and connect with my students through a schedule that has been thoughtfully set up by amazing district leadership. We can keep on moving and shaking with curriculum, while adjusting expectations and demands.

Step 9: Just Breathe

Eat something. Drink water. WASH THOSE HANDS. Get away from the screen. Go outside and breathe the fresh air. Stop hitting “refresh” and looking at that map for a bit (I’m talking to myself here!) Exercise. Take a walk. Chat with friends and family. Don’t try to do it all right now… (Also, talking to myself…!)

Okay, GO! Tell me of the magical ways you’ve streamlined your planning while not sacrificing quality in the process!

*The resources created and shared by other teachers and me have made the distance feel much closer! I meet with my students “synchronously” for roughly 20-30 minutes during each of our class meetings.

My first goal is to see their faces, chat a bit, and have a light-hearted exchange before getting to the math, the same way we would if we were in “real” class together. I used this Desmos activity during our first Zoom meeting, and here are more! One day, students introduced their pets (lots of folks have shared this idea on the Twitters – thank you!) I got to virtually meet the most ADORABLE hedgehog!!! We are also having theme days. The other day, we wore sunglasses. I have hat day, favorite color day, and even…. bring-a-roll-of-toilet-paper-to-class day coming soon, ha! I also have some fun tasks planned, where having everyone unmuted in Zoom and just calling out when they “see” it will break the ice and start class in a novel way. Here’s one such resource!

I’ve also been using our synchronous time to do a “micro teach” and ask students to do a problem or two. When written work is important, I use Nearpod. Having the entire iPad screen to draw and write math work makes Nearpod an ideal tool… and being able to see these work samples, LIVE while using Zoom and Nearpod during our together time, is priceless!

I also like to post a short “quiz” at the end of each Nearpod, because Nearpod allows the teacher to share the quiz results of each individual student through their own, personal pie graph! AMAZING to be able to do this in the classroom when we’re all together… AND now, when we’re all safe in our homes! Here’s one such short-and-sweet Nearpod, where we compared-and-contrasted the simple and compound interest formulas and concepts, worked two problems together during Zoom, students took a very brief to-the-point Nearpod “quiz”, received their pie graph results from me, and were released from Zoom to work on a PDF homework assignment asynchronously for which I immediately posted my key (so they could check work and get feedback before our next class meeting).

I’ve also been using Desmos Activity Builder to create “micro-teach” experiences during our Zoom time. Desmos is great for designing inquiry-based lessons that guide students towards noticing patterns in math and making generalizations.

Desmos is also a fantastic place to share short direct-teach videos and ask students follow-up questions that can be seen from the teacher dashboard asynchronously. I’ve been using my OKIOCAM in conjunction with QuickTime or Zoom to create these brief videos for my students. I’ve also had success using a Desmos activity synchronously with the entire class to get started, then using Breakout Groups in Zoom to give students some time to work through more challenging screens together. I can virtually visit their groups while simultaneously watching their work using the Desmos teacher dashboard! Here is a brief lesson I used with my Math 7 crew this past week. Here is a follow-up I will use with Math 8 this coming week. Here is another brief activity I used with Algebra 1, where using Zoom Breakout Groups for the last two screens was ideal!

I can’t thank the math community enough for continuing to create and share resources and experiences during this time in history!

Posted in Algebra 1 | 1 Comment

TRUE or FALSE? Independent Events… and Socrative

Years ago when I had the pleasure of serving as a “math coach” in the Bethlehem Area School District in Pennsylvania, I learned a clever way to expose students to independent events. A colleague of mine provided his students with lined paper, asked them to number from 1 to… 10 maybe? and told students they were being given a pop quiz!

Then, the teacher began verbally providing students with the “quiz questions”…

Teacher: “Number 1… TRUE or FALSE?”

Pregnant pause.

Teacher: “Number 2… TRUE… or FALSE?”

Students: “…TRUE or FALSE what? What do you mean?”


Teacher: “Number 3… TRUE… or……. FALSE?”


Eventually, students would giggle and play along, realizing, literally, they were simply being asked to make the choice… do you choose TRUE or FALSE?

After the verbal “quiz” was over, it was time for students to “grade” their own papers… against an arbitrary answer key that had been previously created by the teacher. After students scored these so-called quizzes, the teacher would ask if anyone got a “perfect paper”. Literally no one did. But, it was a wonderfully silly segue to consider probability… what’s the probability a student could earn a “perfect paper” on this arbitrary quiz? (and subsequently… NO WONDER NO ONE ACED IT!)

I thought it would be fun to spend 10-15 minutes giving my Math 7 students a similar scenario today, our first day back from break, but I added a few twists. Rather than create that moment of what-the-actual-heck, I opted to tell students outright that I had created a TRUE or FALSE quiz with my own fake answer key… and their job was to guess what answer I chose for each of 5 questions. I didn’t want students to get distracted in silliness at the start – I wanted to challenge them to essentially attempt to read my mind. They were on it.

Rather than provide paper, I’d used Socrative to create my arbitrary TRUE or FALSE quiz and “answer key”. Using the Teacher Paced option, I still verbally asked the question five annoying times…

Me: “Number 1… TRUE or FALSE?”

Me: Number 2… TRUE or FALSE?”… and so on.

Students made their choices in Socrative on their iPads. I didn’t reveal the “correct” answer to any questions along the way… I just paced it to give a sense of don’t-dilly-dally… make a gut-instinct choice and let’s move on…

They were quite anxious to find out how they “did” on this “quiz” ! Before I revealed the class results, we worked through the probability of getting a “perfect paper” on the board… 1/32… would *anyone* ace this thing?

Student: “That’s only, like, 3% or so… that’s not very good!”

Now… the reveal! Here’s a portion of our results! Surprise! NO ONE “ACED” IT!


Someone actually earned a ZERO!

What happened next, I did not expect.

Students: Let’s do it again! I want to do another one!

So they asked for another chance… AND… they asked me to make it only FOUR questions this time… how cool is that?

We established the probability going into Round 2 ahead of time… kids still expressed concern that 1/16 was close to only 6%-ish… but they had hope!

On the spot, I created a brand new 4-question TRUE or FALSE quiz in Socrative, and launched it as a student-paced activity so they could just zip through it… after all… we only had literally 4 minutes of class left…! No time for drama, just give me the data!

The reveal… WHOA!!!! Only one student aced it, but you’d think someone won the lottery with the way the class, and particularly that student… erupted!

The bell rang, the first school day of 2020 came to a close, and though that winter break was much needed and appreciated… I’m so glad to be back with these kiddos.

Happy New Year!

Posted in Algebra 1 | 1 Comment

How We Do “Four Corners” to Move & Talk About Mathematics

Search for the terms “Four Corners Classroom” online and you’ll find variation after variation of this cooperative learning strategy. The way we do “Four Corners” in my little corner of the world has evolved over the years. It’s one of those strategies I use so often that I dismiss it as “normal”… but maybe, this versatile strategy is just what you need to get your kiddos moving and talking about mathematics when we return to the classroom (and this turkey coma wears off).

I know my Math 7 crew will benefit from a little refresher regarding circles and circumference, so I’ve created a Four Corners for them to try when we return to class next Monday.

Creating a Yencafied Four Corners Resource

Create four similar-but-different problem sets that address the math concept at hand.  Use an icon to differentiate between each of the four sets.  This can be something novel (symbols that indicate different types of weather, animals, card suits, etc.) or the icons can address math concepts (use various, relevant math shapes or symbols, as in my example that follows here).

Copy the two Four Corners sheets, one sided, and cut vertically down the middle.  Distribute one half-sheet per student such that no two students in a cooperative group receive the same problem set.

PRO-TIP: To help with this process, I suggest having TWO copies of each of the two pages.  When placing in the photocopier top tray to run copies, stack the pages such that page 1 and page 2 are stacked on top of one another.  THEN, stack the SECOND copy of page 1 and page 2, but rotate these two pages 180º, then place these two sheets under the first two sheets.  Set the copier to collate, and voila! Cut the entire stack vertically down the middle, place one half-stack on top of the other, and you’ll have a beautifully mixed stack guaranteeing that groups of 4 or less will NOT have any duplicate sheets within the group! Every kiddo receives a different problem set!

Now that you’ve created, copied, collated, and cut the sheets, it’s time to use them with students!

First, students work on their own half-sheet silently and individually for a few minutes. When appropriate, consider allowing students to reference any resources they have handy.  When students try to collaborate within their group, they’ll soon find that their own paper is different from every other group member’s!  The first time one uses this strategy, students are very much caught by surprise that each paper is unique!  Once this strategy becomes a regular occurrence, students know their papers differ from the start, and simply get right to work independently.  {NOTE: Just as with Numberless Word Problems... students who attempt to collaborate are placed in a position to discuss big ideas and relationships, rather than specific ‘answers’… which I LOVE and did not anticipate when I first started doing this…} Meanwhile, I walk around and peek over shoulders, but I say nothing… I just observe what they’re trying, and how they’re progressing. MANY MANY times, students say to me as I pass by comments like, “Oh! I know what to do!” or… they look at me longingly and say, “I totally forget how to do this…”

    ≅    ~    ≈    

After some time has passed, and some students have completed the task (it’s also okay if some haven’t fully finished) allow students whose papers contain the same icon/symbol to meet in each of the four corners of the classroom.  I verbally and on-the-fly assign an icon to a classroom area/corner so students know where to meet and talk with the folks who had the same problem set from the get-go.  Students take their papers and writing utensils to the new location to collaborate with their group.

NOTE: This is a neat-o opportunity to assess students’ knowledge of whatever math symbol/icon was used to create four unique groups.  On my example here, I’m wondering whether students will know the difference between congruent, similar, approximately equal to, and not equal to… they will likely ask peers quickly to determine, “Which symbol do I have?  Which one am I? Where should I meet with my group?” Sneak in a review or mini-teach on notation/symbols/shapes/any math visual you use to create the four groups… or just have fun and use icons of four different animals instead, ha!

Once all students find their corner of the room, they discuss, compare, and correct their own papers as needed.  I never ask students to remain standing, but they just DO! They need to be able to walk around and chat with one another, comparing their papers. Everyone is standing and moving around! This is a great time for me to listen in, and notice if any groups are arguing and justifying their positions on any particular problems.  The idea is that ALL students in that group come to an agreement, papers are modified as needed, and that ANY student’s paper within the group could be chosen by me as a “sample answer key” for their problem set. I often ask each group to decide upon a paper from their group that they consider to be an “answer key” and bring it to me. After I have an “answer key” from each of the four groups, everyone returns to their own seats.

Once everyone is seated, I present each of the four “answer keys” using the document camera. Another variation is to have each group send a representative to the document camera to present their group’s agreed-upon “answer key” paper to the class.  I have also been the one to randomly choose a paper from each of the four groups… in theory, if everyone has discussed the problems, and made any necessary corrections, it shouldn’t matter whose paper I use.  However, I am careful NOT to reveal the name of the person whose paper is being showcased at the front of the room, unless our class culture has reached the point where anonymity is not a concern.  Ideally, though, I’m NOT the one presenting the problem set, but rather, a student (or several) from the GROUP is explaining the problems to the class.  During the presentation phase, students at their seats are encouraged to respectfully ask questions of those presenting, and additional conversations and corrections often occur as a result.

Ideally, we usually spend 10 minutes or so on this whole process. It’s a great little exercise in retrieval practice, collaboration, movement, and presenting to the class!

This Four Corners strategy can literally be used for anything!  

A pre-assessment… a warm-up… a review… a lesson closure activity… a great way to start a Monday morning to see if Friday’s math is still in their brains…

literally ANY TIME.

I love that this sort of activity promotes both MOVEMENT and STUDENT DISCOURSE.  It’s a variation on “YOU DO… Y’ALL DO… I-DO-or-intervene-as-needed”.  Sometimes I make up my own problems from scratch… other times, I use ancillaries that have versions A, B, etc. of homework or practice problems as my inspiration for the four problem sets.

How does “Four Corners” look in your classroom?

Grab the PDF for my circles Four Corners as an example, and share back when you create something awesome!

ANOTHER NOTE: The circle diagrams on my Four Corners example here were created using Apple’s Keynote first. I used shapes and text boxes to create the first circle on a slide, then duplicated and edited it three additional times to create four slides. I tried sneaking in letters on each circle that represent texting lingo or silly words, and I researched common circular objects when I chose each radius measurement. I’ll be curious about the everyday objects students think about! Finally, I exported the four Keynote slides as images, and dragged each image into the two-column Word document, where I added the questions. #KeynoteForMath

Posted in Algebra 1 | 6 Comments

Goldi-Desmos and the Three… Layers? {I’ve got nothing}

Okay, so my attempt at a clever title is lacking here, but the idea in my brain still stands and seems sort of valid.

Last Friday, my Math 7 crew needed support transitioning back to the curriculum after several days of end-of-quarter review activities, and taking a district “quarterly” benchmark test. Given that it was a 4-day school week, and three days prior were dedicated to the aforementioned tasks, it had been an entire week since we’d talked about equal ratios and started exploring the concept of “proportional”. The previous Friday, students had explored several scenarios to compare and contrast linear proportional and non-proportional relationships using tables of values (one where someone earned an hourly wage, and one where someone had already saved some money and was saving another consistent, additional amount, weekly) and hinting at what graphs of these relationships might look like.

Fast-forward to a week later. I wanted students to build on the tables and patterns we’d explored before, and continue to extend these to graphs.

“… that was too big…”

As a warm-up (that subsequently took 2/5 of the class period) students were given an empty coordinate plane and a few strategically-chosen ordered pairs. We tried to recall vocabulary like “origin”, how to label the axes, which quadrant was which, and how to plot a collection of ordered pairs that, if graphed correctly, would form a beautiful hexagon. Taking a lap around the room… I saw graphed polygons… and… other… things. It certainly didn’t hurt to address graphing ideas with these kiddos. I knew at this point that if our goal was to explore proportional and nonroportional relationships, graphing them by hand was an additional skill-set that we’d need to surely revisit.

“… that was too small…”

On the other hand, the instructional resource we use (a.k.a “the book”) simply states… “If the graph makes a straight line through the origin, it’s proportional.” These sorts of “quickies” are what students latch onto, without understanding more deeply the WHY. If we provide already-done graphs on worksheets to students and ask, “Is it proportional?”, and all they say is, “Yes, it goes through the origin” or “No, it doesn’t go through the origin”… this “quickie” explanation can extend to when students examine tables of values too. If their thinking halts here, and they don’t see (0, 0) in a table, they can make false assumptions.

On the back of the coordinate plane 40%-of-the-class-period warm-up, I copied this resource from Illustrative Mathematics. Students had discussions in their groups, ensured that they’d “answered the question asked” at the top of the prompt, and we had a share-out. Yes, the “line through the origin thing” is true, but I wanted to be sure we hadn’t lost the idea of relationships (Are there patterns between the x-coordinates? y-coordinates? Are equal ratios happening?)

This prompt came from Illustrative Mathematics Grade 7.
I tried finding a direct link to post here, but I couldn’t find it anymore.

“… that was JUST RIGHT.”

Lastly, I asked students to do what often is “the magic word” to seal understanding… “create”. I used Apple Classroom to navigate students’ iPads swiftly to and used Teacher Pacing on this activity to limit their access to screen 1 only. Students talked in their groups as I asked them each to “create a proportional relationship you think no one else will.” They were excited to zoom in and out, seeing that this coordinate plane was quite different than the others we’d seen on worksheets today. LOTS AND LOTS of space to create!

Something happens in the room when students are provided with an empty table and are asked to “create”. Many take pause. Some initially stare at the iPad, as if making a wrong move will break it. Others dive right in and look at the graph happening simultaneously… and “play”. The feedback the Desmos graph immediately provides encourages them to keep going with their patterns, or revise ideas. What I loved most about watching them create patterns in these empty tables was… many students did NOT start with (0, 0) as their first point! For some, that was an afterthought that *I* asked them to add at the end to see if their line segments continued to the origin. They were looking at patterns, relationships, and equal ratios to “create”… YES! Since they weren’t limited to a typical 10-by-10 coordinate plane on paper, they were able to be more flexible in their thinking.

After sharing graphs and talking about several student creations, I used Teacher Pacing to navigate everyone to Screen 2, where students were prompted to fill an empty table of values to create a line that is non proportional. Again, with students creating, we had a wide variety of examples, and a few non-examples we could help students revise.

To close, I think having students experience ALL of these tasks was valuable. My “Three Bears” comparison here is more about how *I* was feeling at various points in the lesson, using various resources that accomplished different goals. Any one of these resources in isolation wouldn’t have been as powerful for students as using all three.

A message I want to share to encourage you is a friendly reminder that not every activity we do in Desmos or other tools has to be this huge, comprehensive entire-class-period or entire-lesson thing. Sometimes small creations used intentionally can make an impact! More about that here.

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Quality Time (not in the lesson plan)

Last week my Math 7 students spent several days exploring ratios and unit rates (including scenarios with complex fractions in the mix). We haven’t used the word “proportion” yet… we’ve just been “playing” with numbers, really. Encouraging students to think flexibly (informally using ratio tables with recipes and other contexts, and just exploring ideas of equivalence in general) was comfortable for some, but made others extremely UNcomfortable.  

Some students viewed having options as “fun”… a game… a puzzle… they found joy in exploring many different, correct ways to represent equivalence…

…others seemed to feel overwhelmed, yearning for ONE WAY to DO the THINGS that would ALWAYS work… but if I give ‘em time, I know they’ll come around!

These flexible-thinking days are an investment into students’ understandings in the future… when we start using vocabulary like proportional, direct variation, and constant of variation, but for now, let’s play a little longer… even if it means taking a bit more time than initially planned.

I experienced the first return-on-investment at the start of class Friday.  Several students told me they’d started considering the date each day, and had begun representing the date as a complex fraction… all on their own. {Note: They’re keeping track of the patterns by hand – I’m using Desmos here so it looks “pretty”.}

Here was how they represented the date on their “Day 1”, which happened to be 10/9/19:

Here’s their “Day 2”:

Before they stacked Friday’s date in similar fashion as a new complex fraction, the students decided to guess whether the date was going to be “better” or “worse” than yesterday’s…

“I think it’s going to get ‘worse’ every day!”

Sure enough…

“We got even WORSE today!  I think we’re going to get worse every day through Halloween!”

“Halloween will probably be the WORST!”

“I wonder how much better we’ll get on November 1!!!”


This exchange and student-driven “game” was the perfect way to end a week for me. Embracing these moments, even when they’re not exactly on topic with *my* lesson plans for the day, is critical and important! As we try to help our students think and understand versus apply tricks and quick-fixes… be patient and persevere.

Enjoy your precious mathematicians as you listen to them and guide them on their journey in the coming weeks and months.

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A Classroom Visit from KLRU – Screens In School: Tech Transformation

My students and I had a unique opportunity recently to host guests from our local PBS station here in Austin, Texas. On the brink of iPad’s 10th anniversary, folks at KLRU came to visit our district, Eanes ISD, to chat with my colleagues and me about how iPad has changed teaching and learning in our schools. The first installment of the “Screens In School” series was released today (credit to KLRU for the classroom photos in this post).

On the day of the classroom visit, my students were starting a unit of study on rational numbers. It also happened to be the first time students were able to use their graphing calculators, so we addressed a few basic features that day, such as turning them on and off, clearing the screen, how to represent negative rational numbers ( in “real life” a negative sign and a minus sign have the same math job, but the calculator has been programmed differently) and how the calculator communicates about repeating decimals (rounding the last digit using traditional rounding rules… five or above, give it a shove). Through this nuts-and-bolts experience using a Nearpod lesson (created in Keynote initially), we began defining things like terminating, repeating, ratio, rational, etc. The proximity of the iPad helped students find the buttons and menus so we could get into the lesson swiftly.

To start the lesson, each student had the opportunity to apply the definition of “rational” using Nearpod’s ‘Draw It‘ feature to “create a rational number you think NO ONE else will think of in this room” (credit to Dan Meyer for that strategy). We examined student creations and compared each number to the definition of “rational” to ensure that it met the criteria. Positive values, negative values, fractions, decimals, both benchmark-y and wacky, provided a variety of student-created examples to talk about.

After scrolling through students’ number creations anonymously at the front of the class, we did a Stand & Talkusing a Venn diagram of subsets of rational numbers as our visual (credit to Sara VanDerWerf for that strategy).

After students shared their ‘noticings’ about how subsets of numbers relate, students completed poll questions and quizzes to check for understanding in Nearpod. To close, students completed a rational numbers Desmos Polygraph, which empowered them to craft meaningful questions using appropriate academic vocabulary in a virtual partner game across the room (credit to Jennifer Vadnais for that resource).

Using Desmos Polygraph to craft questions about rational numbers

I’m so thankful to have had this opportunity with KLRU to share a snippet of the “flow” that can happen in a 1:1 iPad classroom. Instruction and assessment blur and truly become one and the same. Student collaboration and communication are empowered because of the ability to seamlessly share visuals, ask poll and quiz questions, and provide every student with the opportunity to create and share their own unique mathematical thinking.

Check out the video from KLRU below!

Given that the title of this PBS series is “Screens in School”, this thoughtful read might also be of interest.

In Defense of Screen Time ~ Björn Jeffery

My favorite quote from the speech:

“… how can we talk about screen time without knowing what is taking place on the screen?” 

*UPDATE* This “Screens in School” series has four episodes as of 11/6/19!

Check out the final episode here where folks from our district speak to the success of our 1:1 iPad program, how technology is a part of learning in many local districts, and a quick mention of the benefits devices have on the learning cycle from little old me at about 4:45 🙂

Posted in Algebra 1 | 3 Comments


Roughly 7,300-ish days ago, I began my teaching career in eastern Pennsylvania. I remember my very FIRST first-day-of-school morning… very vividly. I was a newlywed, new to the east side of PA (growing up north of Pittsburgh) and I remember rolling over in bed, thinking it seemed a little brighter outside than it should be.

It was then that I realized… the power had gone out!

Keep in mind, this was the year 1999… before iPhone, iPad, Apple Watch, etc… the available technology for an alarm was an actual alarm clock (which I still use… just me? HA)…

I jolted out of bed, and referenced the analog, battery-powered clock hanging in our apartment kitchen, realizing I had just enough time to shower and make it to school if I started getting ready right away!

I awkwardly attempted applying make-up by candlelight and flashlight-light, and can’t remember if I let my hair simply air-dry, or if I pulled it up in a soaking pony tail or bun, knowing time would surely cure my wet head. The commute is also a complete blur, but I know I made it on time, rattled yet ready. Or so I thought…

It was THEN that I realized that every other staff member had received a “first day of school” paper schedule in their respective “first day of school” folder of information… days before, and that this one sheet of paper had been inadvertently omitted from my folder. Surprise! Classes are out of order and it’s time to think on your feet and be flexible, Yenca! Welcome to teaching! I didn’t know what I didn’t know – there was no Google Doc to reference, or e-mail to check.

I don’t remember what the modified schedule entailed… an assembly perhaps? Shorter classes than I’d anticipated and planned for? That all worked out for the newbie… I had over-planned and was over-prepared right out of the gate due to the omission. But everything I’d meticulously written on the chalk board the day before regarding the bell schedule was suddenly irrelevant.

Time for my first class! As the orange-carpeted classroom (with no windows) filled with my first batch of real students, and the late bell rang, I entered proudly. Before I could speak a syllable, a boy who I learned later was named Peter exclaimed,

“How old ARE YOU?”

I was 22.

But I did not say, “I am 22-years-old, thank you for asking.”

I said,

“Young man, there are 3 things you never ask a woman. You never ask her age, her weight, or if that is her natural hair color. Do you understand, sir?!?”

*Crickets might have chirped at this moment.*

“Yes Ma’am.”

And that’s how it all began.

Today, I woke up before my alarm clock (and iPhone back-up alarm) sounded… I had a peaceful morning of reflection in a quiet house before using my hairdryer and putting on make-up under actual vanity lights… I posed for first-day-of-school photos with my son (a high school freshman) and paused with gratitude that I still get back-to-school jitters about this thing called teaching math. I met my new kiddos… no one asks how old I am anymore… it was a day of fun, and peace, and confirmation that 20 years later, I’m right where I’m supposed to be.

I did find after school that there was a nail lodged in one of my tires, as the sensor in my car alerted me the air was low. Folks at a local garage patched my tire in a jiffy, but I’m truly thankful for that nail. Something about that tiny first-day-of school tire mishap brought all of these 20-ish-year-old memories to the forefront of my mind.

Here’s hoping your first day, full of mishaps or smooth sailing, brings you jitters and gratitude too.

All patched up!
The strangest things can evoke
memories and emotions,
can’t they?
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Ending Another Year {With Desmos}

School’s Out For Summer!

Leaving my classroom after the halls have cleared is a bittersweet moment. I always take a photo of it – and looking at social media the past few days, I can see I’m not the only teacher out there that snaps an empty classroom picture before leaving school. It’s symbolic for me, closing the door on another teaching chapter, and another year of memories with learners who have grown to become a community. Bittersweet.

Farewell Until August!

Year’s end for many of us included state testing. As with the past few years, our Algebra 1 and Math 8 students used the Desmos Test Mode app as well as TI graphing calculators on their STAAR tests… and as in previous school years, I asked students about their calculator preferences after a year of learning with both tools. The kiddos change, but the survey results barely vary. Math 8 students tend to like pressing buttons more for arithmetic calculations… yet ironically do NOT prefer to use their TI graphing calculator for graphing, ha! Desmos always reigns on the graphing front! Algebra is the course where students seem to fall in love with Desmos, appreciating the ease-of-use, dynamic nature, and ease of viewing multiple representations of functions all together on one screen. To read up on our Desmos Test Mode history, check out this post and other posts linked within it.

Here’s progress toward more Texas math students having access to Desmos Test Mode with confidence during STAAR testing in 2019-2020!

Year’s end for many of us also included a final Desmos graphing project! The #MTBoS has been creating and revamping graphing projects for the past few years, and student work floating around out there is so impressive! Check out my Math 8 students’ work from last year, and a new gallery of projects from this year’s Math 8 and Algebra 1 students!

Watch this playlist here!

Year’s end for me also included a unique use of Desmos’ beloved “Polygraph” feature. If you remember playing “GUESS WHO?” as a kid, think of that game… but with a mathy and digital twist. Students ADORE Polygraph, and genuinely don’t often realize how Polygraph promotes the need for rich academic vocabulary to ask proper YES and NO questions. They don’t know they’re learning, but they sure do have fun!

While we’ve certainly had our share of math-content Polygraphs this year, I decided to surprise one of my classes with a one-of-a-kind Desmos Polygraph. Rather than using 16 images related to a MATH concept, the images related to classroom memories, laughs, and inside jokes as we learned math together as a community. While our Polygraph would make literally NO SENSE to anyone on the outside, it made PERFECT sense to my students, whose reaction to the gesture is one I will never forget. 🙂

AHHH, I LOVE TWITTER! Jenee Wilcox is another teacher who has already used this Polygraph idea with her own students!

We’ve had quite the Desmos-y end-of-the-year!

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