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?
Posted in Algebra 1 | Leave a comment

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!

Posted in Algebra 1 | Leave a comment

“Lead & Learn” with “Stand & Talks” Across Content Areas

Several weeks back, our middle school teachers had the opportunity to #BeLikeFred (Rogers) in our #EanesNeighborhood – to either “Lead Like Fred” or “Learn Like Fred” during one of our early release (PD) days. I chose to “Lead Like Fred” and share about my experiences using Sara VanDerWerf’s “Stand & Talks”.

Thanks to all of the resources Sara generously shares, I was able to use her blog post, my own classroom experiences, and the hope that teachers who teach all content areas would see the value in using this strategy with their own students to plan my sessions.

I created a brief promotional video to share on our Lead & Learn FlipGrid. (An aside worth mentioning – I used Apple’s Clips app to create this video. Explore the hashtag #ClassroomClips to find more examples of videos that educators and students are creating using Clips!)


Students will SEE it before I SHOW them.

Students will SAY it before I TELL them.

~Sara VanDerWerf

I shared about Sara’s blog post and powerful “GOAL”, my own classroom stories, and examples and ideas that might help teachers who do not teach math to give their students the opportunity to Stand & Talk too. I gave the teachers in the room several opportunities to “Stand & Talk” with one another during our session.

I’ve heard feedback from math, science, Latin, and history teachers that they’ve already begun using Stand & Talks with their own students with success! YAY!

I’ve included a PDF here with static images of my #Keynote slides from the sessions, my presenter notes, instructions for using Apple Classroom to AirDrop images to students, and a “Getting Started Menu” for the teachers as a take-away. Several slides are shared below as images.

While everyone else is standing and talking… I’m listening! 🙂

Now that I’ve spent this school year dabbling in STAND & TALKS, my lens has changed when I plan lessons. I look at EVERYTHING through a lens of… is there a visual I can provide to my students BEFORE we start “the lesson”? 

Start with the goal, and then find or create a strong visual that will help students talk about the concepts before you teach them.

Using Apple Classroom to AirDrop Stand & Talk images to students is a great way to ensure they have immediate access to the images DURING the Stand & Talk… and after class is over!

A “menu”of ideas to get started!

I’d love to hear how you and your colleagues (mathy or not!) are using Sara’s Stand & Talks with students!

A million THANK YOUS to Sara VanDerWerf!

Posted in Algebra 1 | 3 Comments

Using Kahoot! Challenges and Desmos BREAKOUT to Revisit and Review… and a shout-out to #DesmosInTexas

My Math 8 students took their state STAAR test this past week (SIDE NOTE: This testing season marks lap#5 of my students using the Desmos Test Mode app on iPads as well as a TI graphing calculator while taking the STAAR test… more about that in a plethora of past posts). For a unique test-review twist, this year was my first opportunity to share *ALL* of the Math 8 Kahoots! I created last school year for Math_by_Kahoot with my kiddos en masse.

Using Google Classroom to share with students, I launched an asynchronous “Kahoot! Challenge” link for each and every TEKS-aligned Math 8 Kahoot (SIDE NOTE: These Kahoots! are organized by standard and are available for YOU to also use! Just check out the side bar on the right side of this screen to access Kahoots! for Math 8 and Algebra 1). Next, I compiled all of these links into an announcement post in Google Classroom so that all of my students would have access and the opportunity to play any and all of these Kahoots! on their own time. Additionally, I shared this massive “Kahoot! Fest” of links with my PLC-mates, so their students could play along with my students.

An Asynchronous KAHOOT! FEST!

I was pleased to glance through the reports in Kahoot! and see that students, indeed, chose to use the links to play along with one another, outside of our class time together! Unfortunately, since some students did not use their actual names, I’m not sure how many students participated from my classes or my colleagues’ classes. Bummer!

While glancing through the reports, one Kahoot! in particular seemed to draw more traffic than any other. Why do you think 100 students chose to play? 🙂

SPOILER: Mrs. Yenca played this one, and earned a pretty awesome score. Apparently word got out amongst students, and it became a goal to defeat me! One awesome student beat my score, ha!

In other news, Mr. Jay Chow’s Desmos Linear Breakout! was also a FANTASTIC resource to use to help students take one last stroll through linear lane before their STAAR test. What?!? You didn’t know about Desmos Breakout?! RUN, DO NOT WALK.

Additionally, while many of us were helping to facilitate STAAR testing on our own campuses this past week, we prepared written testimonies to help SB 1453 gain traction. So thankful that Oscar Perales was able to attend and deliver a powerful in-person testimony to help advocate for ALL student mathematicians across Texas! Stay tuned by keeping up with the hashtag #DesmosInTexas.

Want to learn more about Desmos and Testing across the nation? —> https://www.desmos.com/testing

Posted in Algebra 1 | 3 Comments

Using Apple Classroom for “Stand & Talks”

Sara VanDerWerf claims that “Stand & Talks” are “the best thing she ever did to get students talking to one another” and I believe her. For this 2018-19 school year, I’ve implemented “Stand & Talks” intentionally (though, moving forward, I hope to use them even MORE often) and there are SO MANY benefits.

For my middle schoolers, giving them the opportunity to TALK and MOVE early in the lesson provides social, kinesthetic, and math benefits for the rest of the class period. It gives them opportunities to make math observations judgment-free, and to talk to students with whom they might not otherwise interact. We’re moving, talking math, and building community… and it only takes a few minutes! Mere minutes that many of us teachers might use to TELL students things we want them to notice… rather than giving THEM the chance to do so first. This is not “one more thing” to do in our classrooms for which we “don’t have time”… it’s likely a time swap… trading in a LESS EFFECTIVE strategy for a MORE EFFECTIVE one… at least, this is how I see it with my own students! 🙂

Sara provides this AMAZING blog post, detailing her methods and teacher-script for implementing a “Stand & Talk” in her own classroom. Before reading on here, I suggest you check out her post first.

In recent months, I’ve used Apple Classroom to share “Stand & Talk” visuals with my students on their iPads. First, I ask students to ensure that their bluetooth and wifi are ready to go. In the Apple Classroom app on my teacher iPad, I open my current class of students to confirm that most of them have their iPads on and ready to receive a visual from me. I can see icons of students’ faces on my own iPad screen and toggle to see tiny live previews of their individual iPad screens… otherwise their faces show with the label “offline”.

Ahead of time, I’ve prepared a visual to share with students. Sometimes I create the visual myself using Apple’s Keynote. Other times, I use a visual I’ve found online from a fellow #MTBoS-er. I have this visual saved as an image on my teacher iPad camera roll.

Apple Classroom during a Kahoot! Wish I had an image during a “Stand & Talk”! Stay tuned? 🙂

I announce that I have something to share with every student! I select the image from my camera roll. Since I’ve already selected the class of students in Apple Classroom beforehand, the “share square” option for the selected image at hand gives me the option to AirDrop the image to the entire class of students in front of me with one touch of the screen! In a quick moment, I’ve AirDropped the image to my students… like magic! By the way, this process never loses its novelty on middle schoolers! And… even if not EVERY student is ready with a functioning iPad when I AirDrop the image, a “Stand & Talk” still works great. A student who did NOT receive the image from me naturally pairs up with a student who DID receive the image… and that student shares the image with her/his partner so they have it too.

Proceed with Sara’s process here… with iPads in hand, students stand up, walk across the room, partner up, and notice and discuss *insert a big number here* things about the image that’s just been shared. {Bonus – when the “Stand & Talk” is over, students still have access to the image because it’s saved on their OWN camera rolls, thanks to AirDrop within Apple Classroom.} I walk around, listen in, and work VERY HARD to NOT TALK… JUST LISTEN. Once students have noticed and discussed *big number of* things about the image, they start returning to their own seats, which, in my classroom, equates to 8 groups of 3-or-4 students each. Mind you… students have NOT been talking with the people they normally sit with… they moved around and talked with people NOT in their groups.

So… next phase! Now that students are seated back with their groups, I ask them to share their favorite noticings from the “Stand & Talk” with one another. With iPads in hand, students continue to talk and point and share… and I continue to NOT talk and to LISTEN and walk around, likely making all sorts of faces as I try NOT to chime in, ha! 🙂 I ask each group to choose a spokesperson or two to get ready to share out ONE THING they noticed about the visual. Students decide within their own groups about the ONE THING they want to share, and who’s going to share it. We come together as a class, I stand in the middle of all the groups as they WILDLY volunteer to share out first (so no one steals what they want to say before they get to say it… smooth move… I know…) and I point to each group, signaling each spokesperson to share. Sometimes we take ONE lap around the room… ONE sharing from each of EIGHT groups of students… and sometimes I surprise them and take TWO laps… or THREE… Some visuals spark more discussion than others… and sometimes, what ONE group shares helps another group notice something new that THEY want to share.

Having 8 groups of 3-4 students provides lots of room to “Stand & Talk”! Afterwards, I stand in the middle and “take a lap” or two, pointing to each group’s “spokesperson” to share out to the class.

It really is a beautiful process!!!

By the time we, quote, “actually start the lesson”… we’re likely a good way into it, thanks to everything the STUDENTS have noticed and shared ahead of time.

Here are two examples of “Stand & Talk” images I shared this past week with my Math 8 students. This one was shared at the start of class on review day for a unit on 2D transformations in the coordinate plane. I created the slide in Apple’s Keynote and exported it as an image to my iPad camera roll:

Surprise noticing: One student said that Image B was a reflection of Image A over the line y = x because he remembered the extension presented in this Desmos Activity earlier in the unit. MIC DROP!!!! What do you think YOUR students might notice about the coordinates shared here?

Here’s another image I found at MathHooks.com that I used at the very beginning of a scatterplots & data analysis unit, prior to any instruction whatsoever.

The vocabulary I heard was impressive! I heard words like “outlier” and “scatterplot” being used before I said ANYTHING. Students realized that Columns A and B did not match as presented, and swiftly opened this AirDropped image in their own drawing apps of choice, drawing lines to match correct descriptions from Column A to their corresponding scatterplots in Column B as they chatted together. During the share-out phase, students not only CORRECTLY matched the graphs, but ALSO entertained the idea of what the graphs might mean if the descriptions beside each graph DID represent the graph. So silly and fun!

Next, students completed this Desmos Activity... and I waited for THEM to ask more questions about vocabulary after their unsuccessful attempts to sort the cards within their groups on Screen 1. After a brief direct instruction, at students’ request, they were back to sorting cards! As green stacks appeared and student “experts” helped peers obtain green stacks too, we continued with the Desmos Activity to apply what we’d learned about correlation/association, and causation. Later, students had the opportunity to further apply their new-found vocabulary through this Desmos Polygraph Activity.

I have found that using a “Stand & Talk” before a Desmos Activity can be highly effective! “Primes the pump” before diving in! Here’s an example where creating a Keynote slide to “Stand & Talk” about helped students make connections between representations just before sorting cards in a Desmos Card Sort.

Looking ahead to your plans this week, where/when/how might you swap a component of a lesson where *you* had planned to do the telling… for a “Stand & Talk” instead?  

What visual will you use? 

 How will you share this visual with your students?

Posted in Algebra 1 | 2 Comments

#MathResourceRoundup – Pythagorean Theorem

I thought it would be fun (and useful!) to choose upcoming math topics and share resources we love to use with students as we explore these topics. I don’t use everything I find or create every year, but it’s nice to have options at arm’s length when I’m planning and while I’m teaching. Consider this a compilation rather than a comprehensive or cohesive plan. As a matter of fact, I’d love to hear how you use these, or other resources!

A glance at what my Numbers Doc might look like on any given week. I’m always adding to it as folks create and share relevant resources!

{An aside… I like to keep track of such resources using an Apple Numbers document. So many folks share on Twitter using #iteachmath and #MTBoS, and much of what’s shared may not be immediately relevant for my students, but WILL be relevant eventually. It’s been a lifesaver to add these as they come my way to my own Numbers document so I remember them when I’m planning. I literally copy and paste the links to the resources on the specific day(s) I might use them in the future, so they’re ready to consider as I plan each unit/week/day.

Here’s a blank template of the Numbers document.  Each document serves as a place to plan a 9-week quarter. Each week in the quarter has its own tab at the top, and since I teach two courses, each course has its own row of cells for the week at hand. If I didn’t have this template, I’d probably forget about 80% of the goodies you and I have already created! Using this template has helped streamline lesson planning!}

Today’s #MathResourceRoundup Math 8 topic is the Pythagorean Theorem!

STAND & TALK – Squares and triangles visual

I created this using Apple’s Keynote, and I love using Apple Classroom to airdrop the image to every student’s iPad so they can “Stand & Talk” about the things they notice. Just used this today!

  • VIDEO – Visual representation of the Pythagorean Theorem (Kyle Pearce)
  • VIDEO – Visual Pythagorean Theorem demonstration using water
  • VIDEO – Millionaire game show Pythagorean faux pas
  • VIDEO – Wizard of Oz Pythagorean faux pas
  • DESMOS ACTIVITY – Pythagorean Practice (Andrew Stadel)
  • DESMOS ACTIVITY – Pythagorean Triples and Similar Triangles (Michele Torres)
  • DESMOS ACTIVITY – Distance (Andrew Stadel)
  • KAHOOT! – Pythagorean Triples (Cathy Yenca)
  • KAHOOT! – Pythagorean Problem Solving (Cathy Yenca)
  • KAHOOT! – Distance Between Two Points (Cathy Yenca)
  • SOCRATIVEDistance in the Coordinate Plane – import with code SOC-32315155 (Cathy Yenca)
  • iBOOK – Learn Pythagorean Theorem Through Exploration (Kyle Pearce)
  • MATH TASK – Pythagorean Theorem Pile-Up (Michelle Rinehart)

Are there Pythagorean Theorem resources you use every year that I might have missed?

Add your favorites to the comments!

Have a great idea about how you’d use a resource in a unique way?

Share your tips here!

And… I’m curious… how do you keep track of lesson plans and resources?

Posted in Algebra 1 | 5 Comments

Break the Chains and Get More Mobile!

We all do *things* in our classrooms that are simply *normal* to us as teachers… it’s what we do every day… effective routines and strategies that we do because that’s what we do. Some of these seemingly *normal* things, when shared with others, provide amazing a-ha moments that make each of us better at what we do!  So, here’s one of those shares that has become part of my students’ classroom *normal*.

My PD A-Ha Throwback:

Having had the opportunity to attend various conferences and trainings, I first experienced today’s tip as a participant.  My colleagues and I attended a Nearpod training, and we’d completed a task where all of our “Draw It” work was anonymized on a big screen at the front of the room from the trainer’s laptop computer… but the trainer ALSO held an iPad in his hand.  While we examined what the trainer wanted us to see on the screen, anonymized and strategically chosen for our viewing, the trainer repeatedly referenced his own iPad held in his hands, giving individuals in the room specific verbal praise and feedback.

Wait.  We can see anonymous class feedback and work on the big screen from his laptop… but he’s also logged in to the SAME session, seeing what we’re doing on his iPad, as the teacher?  In TWO PLACES simultaneously?!? 

The trainer was completely FREE to roam around the room.  The big screen maintained its place at the arbitrary front of that room, but the trainer, iPad in hand, could walk around, provide feedback, check in with individuals as the need arose, and even advance to the next task within the Nearpod activity… all from his iPad.

In that moment, I was like… DUDE.  

Why haven’t I been doing this for my students?

This run-an-activity-from-two-devices-simultaneously tip works for Desmos Activity Builder too!  Being able to showcase (or not) student thinking on the big screen from my laptop, while pausing, advancing to the next screen, viewing student thinking in real-time on my own iPad… it’s gloriously liberating to be MOBILE and be able to visit with specific students or student groups to INTERVENE and provide specific verbal FEEDBACK to the work students are doing RIGHT NOW. 

One advantage to Nearpod in this realm is that students can be anonymous on the big screen, yet the teacher can choose to view students’ names on her own iPad… in Desmos, if student names are anonymized, they are anonymous on the teacher’s laptop AND the teacher’s mobile tablet.  

Do you feel like you’re *stuck* at the from of the classroom during a Nearpod or Desmos Activity?

Do you have access to two teacher devices?  

If so, give this a try!  

Break the chains, get more mobile, and see timely student thinking that you might have been missing before!   

Don’t have a second device or a tablet?  Not always ideal, depending on the size of your phone… but rather than use a tablet, give your phone a try as your second mobile device!

Posted in Algebra 1 | Tagged , , | Leave a comment

The Surprisingly Motivating, Impromptu, x-Words Challenge.

At the start of a lesson yesterday, I showed this image from MathHooks.com to students and asked them to Stand & Talk about it.  After returning to their seats (they’re seated in groups comprised of 3-4 students), student groups shared a few noticings (there are two groups of shapes here… there are triangles and weird shapes… the triangles look proportional… the weird shapes look weird… I don’t think those shapes in Group 2 are proportional).

Next, I asked students to write a “two-minute paper”, silently and individually, answering the prompt below.

After the two minutes had passed, I encouraged students to share what they wrote within their groups.  Next, rather than asking, “Group 1, share about what you wrote and talked about,” I did something else.

On the fly, I asked, “Can anyone share what it means for two polygons to be similar, but only use FIVE WORDS?”


This tiny modification in my questioning raised the student engagement level in the room by approximately 427%.  No seriously, their eyes lit up!  You could see their eyeballs looking up at the ceiling, and they began tilting their heads the way little puppies do.

Hands shot up!  They shared concisely and excitedly!

“Same shape, maybe different size!”

“What about in only FOUR words?  Or THREE?  TWO words, anyone?”

I don’t think anyone was sitting on seats at this point.  Everybody had something to say!

Then, it happened.  That moment that makes teachers belly-laugh with tear-filled eyes.

One young lady exclaimed, “What about HALF a word?  PRO – PO!!?!”

Yes, friends, an abbreviation for “proportional” was just invented.  My student wrote her new 1/2  word on my copy of the notes at the front of the class, placed it in quotes, autographed my paper to copyright her creation, and sat down, proudly.

All middle-school-silliness aside, isn’t this a cool strategy for engagement, concise use of vocabulary, and fun?  If you use “The x-Words Challenge”, I’d love to hear how it goes with your students!


Posted in Algebra 1, Pre-Algebra | Leave a comment

Yes We Did! No He Didn’t.

Yes We Did!

Recently, we had the pleasure of a classroom visit from Jenny Wales!  My students were honored to have a Desmos Expert stop by and see us in action.  Students found creative ways to “put the point on the line” and quickly discovered that considering the slope of each line would help them with the process.

One big A-HA?  That the universe doesn’t owe us points on a line that are equally spaced.  For example, in a follow-up activity on paper, students encountered this problem. This table of values –> had many students arguing about whether the relationship was linear at all.  They’d calculated the “constant rate of change” for several years without necessarily deeply considering proportionality.  I love when big ideas show up in math arguments!

Another idea that surfaced from our slope-y day I shared in the following tweet.  It was so encouraging to hear from other teachers that we should embrace methods students prefer to use and understand.

An aside… did you notice that Jenny and I posed for a photo in front of last year’s students’ Desmos Pet House projects, preserved on custom art canvases for all to enjoy?!?  It looks like some other 7th graders are really rocking this project too!

Yes We Did!

At #LearnFestATX this past June, my math pal Nadine and I shared about using Desmos Activity Builder to create Card Sorts in every content area.  It was an honor to have yet another Desmos Expert, Jay Chow, mention us in his recent blog post!  By the way, I can’t overstate the awesomeness of Jay’s Desmos Breakout activities!  Go work ’em yourself with your student-hat on, then launch them for your own students! *Update* Check out my awesome colleague using Desmos in her History classes!

Yes We Did!

My Twitter pal Steph Reilly and I had an impromptu-and-virtual-Desmos-Collaboration this past Saturday morning.  The power of sharing, tweaking, and creating resources together across time and space never ceases to amaze me.  Check out Steph’s awesome creation to help students understand WHEN and WHY we “flip” the inequality sign.

No He Didn’t.

If you missed it, read this thread.  You might experience laughter (from the comments) and also be moved to tears (from this ridiculous #pseudocontext).

Posted in Algebra 1, Pre-Algebra | Tagged , , , | Leave a comment

Fluency, Concepts, and the Order of Things

I’m back in the groove with a fantastic new-to-me crew of learners, spending short snippets of time on Twitter… and neglecting to blog!  It’s time to hammer down several threads that resonated with me and recent experiences with my own students in this mash-up post.


Knowing Order of Operations would be part of my own review with students, and likely a topic in many mathy classrooms, I shared “Twin Puzzles”.


I also shared another strategy to address Order of Operations – combining the famous Four Fours with Nearpod to capture and share students’ favorite equations that they believed no other student had come up with in our class.


Then, I caught this problem (both the math problem and the problem with the way a teacher seemingly used this math problem as a gotcha, causing students to second-guess their own math ability, making for a rocky start…)


And… this plea that there is no excuse to mention Aunt Sally… please!


Additionally, Dan Meyer’s recent blog post and thread on Twitter seems to speak to all of our Order of Operations discussions.


Whether you call it PEMDAS, GEMS, Order of Operations, or something else, the Order-of-Things procedure becomes extremely mushy to students… even though the “rules” don’t change, the problems do.

What have students looked like in my class who possess procedural fluency but lack conceptual understanding?

Students perceive these problems as being “wrong” because “multiplication comes before division” and “addition comes before subtraction”.


Students stop dead in their tracks at this problem, wrestling with their knowledge of number properties, and whether to distribute the 4, or subtract 5 – 7 first.


Students ask, “How do I know whether I should distribute first or follow the correct order of operations?” for a numerical expression like this, without realizing that BOTH methods lead to the same, correct answer, OR, that distributing first might lighten the load.

(Note: I created this activity with this premise in mind – that students don’t generally realize the result will be the same either way, or how to decide when distributing might be more efficient, and when Order of Operations might be more efficient.)


Students insist on doing the “grouping symbols” first in a problem like this, even though the parentheses only contain a single value.

Is it our use of mnemonics like PEMDAS or GEMS that have contributed to students’ confusion?  Is the solution to add context to these “naked problems”?  Do MORE problems, so that more exposure to lots of problems helps bring clarity?  I appreciated these questions on Twitter as well… as teachers sometimes we know what we SHOULDN’T do, but we’re not sure what TO do, especially when students have similar misunderstandings year… after year… after year.

And a final thought: Every misconception and question students ask that reveals some level of procedural understanding without conceptual understanding is a learning opportunity. What THEY do next and what WE do next matters too!  There’s no formula for how to handle these “teachable moments” with our own students.  However, fostering a classroom culture that values questioning, mistakes, and exploring multiple methods flexibly and respectfully is a great start.

Speaking of great starts, I hope your year is off to one!

And, if you want to explore many, many, many perspectives on fluency versus conceptual understanding, you’re in for a ton of food-for-thought here.

UPDATE:  And… folks are still discussing…!


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