Desmos Polygraph: Place Value


In the spring of 2018, I introduced Desmos to a couple of 4th grade teachers.  After running, Ordering Fractions on a Number Line and Polygraph:  Geometry Basics, they were hooked on Desmos.  During the last days of the year, they asked if I had any Desmos activities that focused on place value.  At the time, I didn’t.  But now I do.  🙂

Polygraph:  Place Value 

Polygraph: Place Value


The first time I ran this Polygraph, it bombed.  I was not expecting such a beloved platform to cause so much distress.  Just last year, I ran Polygraph:  Geometry Basics in multiple 4th grade classrooms with great success.  Then it hit me.  This former middle level math teacher quickly realized that 4th graders in August are very different than 4th graders in May. A lot of maturity and growth happens in the months between August and May.

My Light Bulb Moment

Using the teacher’s computer, I signed in as a student.  When the computer assigned me a partner, I hit the pause button.  I explained that, as a class, we needed to run through a game together. It would be the most efficient way to address all their questions.

To run through the practice game, I had to unpause the activity.  Student computers now had the potential to became a distraction. My solution was to ask all students to turn their computers, so the screen faced away from them.  We quickly brainstormed a list of reasons why we had to reposition our computers. The brainstorm helped students buy into pausing their individual games and playing as a collective group. They were now willing to give their full attention.  Time to play.

My partner picked a card.  The rest of the class and I were tasked with determining which card she picked.  I facilitated the process.  Students provided me with yes or no questions, told me which cards to eliminate and explained why.

The first round was the longest.

  • Our question:  Is your number odd?
  • Response:  No

There are 16 cards and we talked through whether to eliminate each card.  The process of understanding which cards to keep and which to eliminate can be challenging.  It’s important for students to understand how to maneuver the elimination step.  Sixteen different students verbalized their thinking.  This process moved at a snail’s pace at first, then, as students solidified their understanding, it sped up.  Believe me, this is time well spent!!

To exposed students to varied questions and to expand their academic vocabulary usage, I guided their questions.

Examples of question types:

  • Is there a 4 in the hundreds place?
  • Is your number 5 digits long?
  • Is your number greater than 1000?
  • Is your number less than 10,000?
  • Does your number have a digit in the millions place?
  • Is your number even?

Each round moved faster due to a combination of fewer cards and increased understanding of how to play.  When we discovered our partner’s number, the class cheered.  I love to hear cheering during math class.  Music to my ears.

Student were given the signal to turn their computers around and continue playing their individual rounds.  Students were more independent and far less frustrated.

Over the next two weeks, I ran the activity with two more 4th grade classes and a 3rd grade class.  Each time, I ran the activity whole class before letting students play individually. Each time, the individual games ran smoothly afterwards.

Side Note – Third Graders

In my opinion, due to typing skills, the youngest grade I’d use Polygraph with is third.  Any younger, the ability to type hinders the questioning process.

Polygraph: Place Value

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Desmos: Random Generated Slope & Y-intercept Activities


RandomGenerator is one of the functions within Desmos’ computational layer.  As I explored the feature, I began to see the benefits of random generated tasks.  Many experiments later, I’m publicly sharing my first random generated (RG) series.  This one focuses on slope and y-intercepts.

This AB would not have been possible without the help from Jay , Suzanne,  Jocelyn, Serge and last but not least, Eli. Each person either added CL examples to the Desmos Fellows CL Bank of Wonders (that I adapted or used in complete form) or answered my questions when I was completely stumped.

My Intention

As I learn computational layer (CL), I often practice writing the code I’m using.  Let’s take Jay’s example, Sketch and Check Random.  Since the example involves features that I haven’t coded with before, I used Jay’s code for practice activity 11.  I’ll now need to go back and learn the code. Study it.  Figure out the purpose of each line and how the various lines connect with components to achieve the desired outcome.  I’ll then work on replicating the code.  It takes me multiple tries to replicate code.  The rewriting process not only helps me understand the CL language, but also helps me become more efficient when creating future activities.

Throughout the school year, teachers offer a variety of experiences.  There’s definitely a need for exploration activities and tasks that build conceptual understanding.  There’s definitely a need to focus on academic language through mathematical discourse.  There’s also a need to practice a specific concept, similar to when I practice the writing of code to gain fluency.

The random generated aspect of the Random Generated:  Determine Slope and Y-intercept  activity builder allows teachers and students to use the activity as often as they wish without the worry of duplicating problems.  A feedback loop is built into each practice activity, therefore regardless of the example, students will know if they calculated the problem correctly.  Listed below are few suggestions to integrate the AB.

Example 1:  After assessment practice.  Mr. X gave his students an assessment and he noticed that the majority of his students struggled with finding the y-intercept when given two points.  The next day, he could project practice slide 6 for a whole class task.  He could also pass out one computer per group and ask a group member to go to slide 6.  Now each group has their own problem to solve collaboratively and since feedback is provided, students will know if they completed the problem correctly.

Example 2:  Collaborative practice.  If you use #VNPS (vertical non-permanent surfaces aka white boards on the wall), pair up 2 students with 1 computer.  Let the partners work through a few randomly generated problems on the whiteboards while you circulate through the room.  Since the problems never run out, some pairs may complete 2 problems where others finish 3 or 4. A follow up activity could require students to reflect on the strategies used.


Example 3Individual practice. Sarah often struggles to determine the slope when the line does not have clearly marked points.  Prior to a test, she could run through a few examples on slide 4.  After she answers the first problem correctly, she can press the button for another example (if necessary). Again, each slide is designed with built in feedback.  Therefore even if Sarah practices at home, she can gauge her progress.


Example 4Choice boards and Hyperdocs.  If you provide choice boards for your students, then one or two of the practice activities could be used as an option.  A familiar Hyperdoc format is:  Engage, Explore, Explain, Share and Reflect.  Teachers could find interesting ways to integrate the RG practice activities into a Hyperdoc.

The Random Generator Slope AB

Feedback has been built into each practice activity.

Note:  If a vertical line pops up on a slide where the check depends on a numerical value for slope, then generate a new problem.  Vertical lines are addressed on slide 10.

Slide 1:  Students are asked to acknowledge the individual rise and run and then enter the slope.  The program calculates the y-intercept based on the displayed points and the inputted slope.  Therefore, the line that matches the student’s response is graphed.   If the student enters the correct slope, then the line will pass through the given points. If their slope is incorrect, then the graphed line will not pass through the given points.  Students are encouraged to use the sketch tools.

Use rise/run to determine the slope connecting the 2 points


Slide 2:  Similar to slide 1 but requiring students to calculate slope using 2 points.


Slide 3:  Students practice finding the slope when points are not prominent.  For these examples, the slope values are only integers.



Slide 4Same as slide 3, except the slope values can include fractions.


Slide 5:  Students are expected to calculate the y-intercept when given the slope and points.  In order to generate slope values to include fractions, I had to randomly generate a numerator and a denominator.  This method produces slopes in various formats, like the one pictured, -4/-2.

I entertained the idea of simplifying the slope but then made the conscious decision to leave it as is.  In my experience, students often struggle when a fraction contains 2 negatives, a negative in the denominator or is not simplified.  By leaving the slopes “messy”, students have more opportunities to navigate tricky formats.


Slide 6:  This activity requires students to calculate both the slope and y-intercept.  Students can either determine the slope using rise/run or with the given points.


Slide 7:  Similar to slide 6 but without a grid.  Without the grid, students are encouraged to practice calculating slope using the formula.


Slide 8:  Students enter the linear equation in slope-intercept form.  Sketch tools are available.


Slide 9:  Same as slide 8 but without a grid or sketch tools.



Slide 10:  The same as slides 8 & 9 but with vertical lines.


Slide 11:  Using the sketch tools, students draw the line that represents the given equation.  When the button is pressed, the correct line is displayed.


The activity builder:   Random Generator:  Determining Slope and Y-intercept 

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Friday Night Lights in 5 Mediums


The majority of my posts begin with an inspiration segment.  A few words explaining my motivation to write.  Originally, I was planning to write about podcasts with the inspiration segment focused on Friday Night Lights.  FNL lead me to podcasts. However, as I reminisced about my brief obsession with FNL, it became clear that the story about a town, a team and and a dream needed a post of it’s own.

Friday Night Lights

It was to be a summer of coding.  I planned on spending my days Friday Night Lights book coverdeep in the world of  Desmos CL (computational layer) and Google app scripts.  I did neither.  And I completely blame Coach and Tammy Taylor, Tim Riggins, Matt Saracen, Tyra Collete and the rest of the Friday night Lights cast.

During spring semester, I started watching Friday Night Lights, the series, and spent my first week of summer break binge watching the 5th and final season.  It took all of 1 hour before slipping into the dreaded show hole.  Not ready to let go of the characters, I hopped onto the internet searching for fun facts about the series.

That’s when I discovered that the series was actually based on the book, of the same name, written by H. G. Bissinger.  Show hole averted.  I drove to the library and checked out the series inspiring book.  Bissinger skillfully weaves the story of Permian Panther football within layers of Odessa, Texas history.  The book’s rawness clung to me making it even harder to shake the entrenched football culture of Odessa, Texas.  My mind scanning both mediums acknowledging the parallels between the book and the series.  I moved onto Friday Night Lights, the movie.  The triangulation of the book, series and movie fascinated me even more.  More connections, more parallels. More Obsession.

  • Matt Saracen’s character represented Mike Winchell
  • Tim Riggins’ character  represented Don Billingsly
  • Smash Williams’ character represented James ‘Boobie’ Miles
  • Don Billingsly has 3 children: Landry, Skylar and Riggs.  FNL, the series, has characters named Landry Clarke and Tim Riggins
  • Odessa, Texas had 2 high schools, Permian and Odessa.  FNL, the series, introduced a second high school at the end of season 3.
  • IMO, Coach Taylor’s phrase, clear eyes, full hearts, can’t lose summarizes Coach Grimes’ speech to his players during the 1988 state championship.
  • In actuality, the 1988 Permian Panthers reached the semi-finals not the state championships. The movie adjusted for dramatic effect.
  • I could go on, but won’t 🙂


  • Upper left:  Garrett Hedlund playing Don Billingsly in FNL, the movie
  • Lower left:  Taylor Kitsch’s character, Tim Riggins representing Don Billingsly in FNL, the series
  • Right:  Don Billingsly circa 2015 and 1988

I watched the series. Read the book.  Watched the movie.  Still inside the show hole.  Now what?  PODCAST! Yes, there are podcasts dedicated to FNL.  I listened to Binge TV ‘s podcast on Friday Night Lights- all 6 episodes.  That did the trick.  When the final seconds of the final podcast episode concluded, I felt fulfilled.  Done. But now I was hooked on podcasts.

Closing Thoughts

The post on podcasts never materialized. Instead, I added the fifth medium – blogging.  I watched, read, listened and finally blogged about a heart wrenching story of Texas football.  My next post will pick up where this one ends. The baton officially passing from Friday Night Lights to the academically minded podcasts discovered thanks to my FNL obsession.  Until then, clear eyes, full hearts, can’t lose.

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Desmos Polygraph Extension Task


Toward the end of the school year, my colleague Jorge and I, had a regular gig co-teaching in Mrs. S’s 4th grade class. Jorge had been working with Mrs. S throughout the year on various projects. I joined the duo during the last quarter to show Mrs S how to integrate Desmos into her math lessons.

On the day in question, I was scheduled to lead a 1 hour lesson using the Polygraph: Quadrilateral Attributes. I was prepared and ready to go when Jorge’s text came through reminding me that I was scheduled for 2 hours.  What?  Wait… How did I miss that detail?  Dang! I needed to restructure – and fast.

Necessity is the Mother of Invention

The text scrolled across my phone minutes before leaving my office.  I had the  3 mile drive to conceive plan B.  For a couple years, I’ve demoed or co-taught various Polygraph lessons and then, during the debrief, shared ways to build off the experience – Never getting the chance to run with any of my suggestions.  This day was different.  On this day, I was granted time to extend the Polygraph lesson.

Once in the classroom, I assisted with the transition to math.  Students turned on their computers, logged into Google Classroom and clicked the Desmos link.  Polygraphs aren’t new to this group of students.  Their first experience was playing Polygraph:  Geometry Basics.  When all students had logged in and the activity norms were recapped, students were free to play and plan their questioning strategy.

Now that students were focused on guessing their partner’s selection, I sat down to create the post Polygraph task.  The task linked below is a cleaned up version. Time was limited and the original version valued substance over aesthetics.  Yes, the students were familiar with this type of activity, but circulating the room, listening to conversations and interacting with students is crucial to the activity’s effectiveness.  My charge was to whip up the task and get back to the students ASAP.

The Task

My district jumped onto the Google Classroom train this year and I’m thankful for the flexibility it brings.  Once in drive, I created a new Google slide presentation, copied the pictures from the Polygraph activity and started writing questions.  One question per slide.  The task was placed into Google Classroom using the create assignment feature and the option, Make a copy for each student.

Title slide from the Post Polygraph Activity picturing the 16 images from the Polygraph: Quadrilateral Attributes

Title Slide


The main purpose of a Desmos Polygraph is for students to experience and understand the need for academic vocabulary.  When used at the beginning of a unit, student questions are often constructed with informal language.  As the playing continues, the teacher plays a key role in helping students infuse academic vocabulary into their questions.  Click here to read a post outlining one method of integrating academic vocabulary as students are playing a Desmos Polygraph.  Many teachers use the same Polygraph activity at the end of a unit with the expectation that academic language is used in every question.

For this lesson, the Polygraph activity focused on math symbols. Students based their questions on the various symbols displayed in the given images.  For the post activity, student needed to work with the vocabulary and symbols differently.  The post activity required students to explain the meaning of each symbols as well as physically create quadrilaterals based on the given criteria.

Question slide: Describe the meaning of the arrow symbol. Select a picture to support your explanation

The camera feature built into Google Slides is wonderful.  Students really like taking pictures of their work.  More information on how to use the camera feature can be found in this blog post.

A slide explaining where to find the camera feature. Click on insert, then image and finally camera.

Task slide:  Draw an isosceles trapezoid.  Include the symbols for parallel and congruent sides.



Desmos has 2 quadrilateral based polygraphs, linked below.  I created a third one after speaking with a handful of elementary teachers and coaches in different parts of the country.  I asked them which geometry concepts their students struggle with the most.  Their common response:  Quadrilateral Attributes.  With their response in mind, I built a polygraph that focused on the symbols used to differentiate the various quadrilaterals.  Once students are confident with their understanding of the symbols, I suggest upping the ante by moving onto one of the Desmos Polygraphs.

The Desmos Polygraphs:

Other links found in this post:


My colleague Jorge and me


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Hidden Picture: Double Digit Addition & Subtraction

The core group of G Suite apps are docs, slides, sheets and forms. All of which I’ve used over the past 2 years.  However, I recently discovered Google Apps Scripts, a hidden gem built into the suite.  Bottom line, I can add another level of customization to my Google creations using Google app scripts.


My First Google Script Activity

When students open the spreadsheet, they’ll see 21 addition and subtraction problems.  Once an answer is calculated, students enter the sum or difference in the corresponding red box and then press enter.  At any point during the activity, a student can check their answers by pressing the button icon.


Correct answers are indicated two ways. The red box turns green and the hidden picture slowly appears.

Spreadsheet with hidden picture

The final image

Pixel Art Fish

The Catch (no pun intended)

  1.  Since the activity was created using Google Script Apps, students will have give their permission for the coding to run.  Press continue.

Dialogue Box: A script attached to this document needs your permission to run

2. Select account



3.  Select Allow

Once students click the necessary buttons, they have access to the activity.  Enjoy!

Activity Link:  Hidden Picture: Double digit addition & subtraction activity

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Kindergarten, Number Bonds & Desmos

Recently, I had the pleasure of teaching a kindergarten class for the very first time! My colleague, CeCe, invited me to teach a lesson on number bonds using my Desmos Number Bond Activity 

The Lesson

The Kinders sat on the rug.  Cece projected the activity from her computer onto her whiteboard.  To start, I stood by the whiteboard and, with Cece’s assistance, ran a mini-lesson.  But anyone who has taught kindergarten before understands, it didn’t take me long before I was sitting or kneeling on the ground.  I spent the mini-lesson “toggling” between standing to address the whole group and sitting/kneeling to listen and ask follow up questions.

Screen Shot 2018-03-26 at 4.05.33 PM


  • Me:  What do you notice?
  • Students:
    • I see numbers
    • I see dots
  • Me:  What color are the dots?  Talk to your neighbor.
    • Students: Red, purple, green blue.
  • Me:  What’s the difference between the dots?  Turn to your neighbor and talk about the difference.
    • Students:  Some are big.  some are small.
  • Me:  What else do you notice?
    • Students:
      • I see, no.
      • I see, yes
      • And a number bond
  • Me:  Let’s talk about all the items you mentioned ….

This simple opener highlighted all the parts of the slide that I needed to discuss prior to using the computer.  We then spent a solid 15-20 minutes discussing

  • The vocabulary of a number bond: whole & part
  • The purpose of the small and big dots
  • The value of the whole (for this example)
  • How to move the purple slider to fill in the whole value
  • How to check an answer by moving the green slider from No to Yes
  • The different messages that could pop up
  • How to change an answer when a mistake is made (move the green slider back to No, change answer, then check)

And when I say discussing, I mean there was a lot of

  • Pair-share
    • I listened to their conversations and then based my questions on what was mentioned. Depending on what I heard, I asked followed up questions with 1 student, a small group or the whole group.
  • I say, you say
  • I ask, everyone answers
Number bond: 1 and 3 make 4

A smiley emoji pops up when the whole is correct.

Number bond: 1 and 3 do not make 5

A sad emoji pops up when the whole is incorrect.


Computer Time

Students were organized in pairs and used QR codes to access

Screen Shot 2018-03-26 at 1.05.34 PM  Capture

As the students worked through the problems, CeCe (pictured above) and I circulated the room and checked in with each pair. We asked students to explain their thinking and to read the messages that popped up.

A week later, I had the opportunity to run the same lesson with Ana and her students.




I’d like to thank CeCe and Ana for inviting me into their classrooms.  Through our collaboration,

  • Students enjoyed an engaging math activity
  • Students talked A LOT about adding numbers
  • Students were introduced to new technology
  • The activity was refined
  • With your tutelage, I gained new insights into teaching primary grades
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St. Patrick’s Day & Google Sheets


My instructional tech team and I always receive grateful comments from our elementary teachers we share seasonal activities.  Most of the activities we shared were using Google slides.  I wanted to challenge myself and create a seasonal activity on Google sheets.  Here are the fruits of my labor.  🙂

The Activities

I created 3 different versions of a St. Patrick’s Day Google Sheet activity.  Students have to answer age appropriate math questions. Two actions occur when a correct answer is inputted.  First, an image begins to emerge in the blank rectangle.  Second,  the answer cell fills in with a light purple color.  A wrong answer, as shown below, is left white.

Below is the picture for grade 1. The picture is different for each version.



Here are the links to force your own copy:

  • Kinder:  Click here
  • Grade 1 & 2:  Click here
  • Grades 3 – 5: Click here


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Google Slides – Camera Feature

The Email

This past week, I received the best email.  Michelle, a 7th grade math teacher and District Google Cohort member, shared with me her Google slide presentation on solving equations.  I immediately solved the first problem, snapped a picture of my work using the Google’s camera feature (formally known as “Take a snapshot”), and inserted the picture into the presentation.

Since learning of the camera feature, I’ve been waiting for this moment. The opportune time to connect a tech feature with a teacher’s lesson.   It’s one thing to promote  technology during a district training, it’s another to introduce a tech feature when it seamlessly integrates into a planned lesson.

I followed the above screenshot, with a brief set of instructions and a screenshot of where to find the camera feature.



The Lesson

I was able to stop by Michelle’s room the day of the lesson.  The classroom energy was positive.  A buzz of excitement fulled the room.  Students were solving problem after problem, taking snapshots of their work and inserting them into the slide presentation.

It was a minimum day which means shorter class periods.  When Michelle asked students to start cleaning up, students groaned.  When she assured her students that they’d continue the activity tomorrow, they cheered!!

Students solving equations on white boards,

Students solving equations on a white board, and then taking a picture of their work to insert into the slide presentation. Bottom left: Michelle and I. I’m on the right.


Added Bonus – Mini Student Presentations


Michelle assigned this activity through Google Classroom.  When in Google classroom, a teacher can open and view students’ assignments, in this case their slide presentations.  Within a few clicks, Michelle has access to all the inserted pictures displaying student work.  She can check progress, prepare for a class discussion or select a few students to discuss their strategies.

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A Guest Teacher’s Challenge


My last post, Multiplying Fractions: A Desmos Area Model, blended conversations from the same lesson run in two different classrooms. Although the lesson flowed similarly in both rooms, there was one student who viewed the presented problems differently.  Gabe has an artistic eye and his method of working through the math problem was unique to both classes.  On the other hand, Gabe’s mood was quick to change.  One misstep would cause frustration and self doubt to settle in.  These characteristics  provided me an interesting challenge.  How do I honor his method while reducing his frustration level? (All names have been changed)

Situation 1

MeDraw a square and shade in half of it.

I didn’t want to influence how student’s shaded their square, therefore I waited until they finished before displaying the Desmos model.  Gabe was the only student to draw a diagonal line. (Pictures have been recreated).

A square that has been shaded in half diagonally.

Gabe’s picture

MeCircle half of your shaded region.

I/4 region of a square circled in yellow ink.

Gabe’s picture

MeWhat fraction does the circled section represent?

Up until I asked this question, Gabe was smiling and seemed to be having fun.  My question uncovered a misconception in Gabe’s thinking.   His answer was 1/3.  One piece was circled out of the 3 sections. When he realized that 1/3 was not the correct answer, he became upset.  His lighthearted demeanor turned heavy.  I heard his group talking about equivalent fractions.  My attempted to connect the group conversation to his picture was not successful.

Since this was my first time teaching this class, I stepped away.  Gabe needed a chance to regain composure and I needed to check in with other students.  After circulating, we regrouped whole class.  Most students had either 1/3 or 1/4 as their answer.  The class discussion addressed two points:  1) How someone would conclude the answer of 1/3 and 2) The need for equivalent sections (which generated the answer of 1/4).

I kept my eye on Gabe as we worked through the second situation.  His unique viewpoint intrigued me and his frustration level concerned me. I challenged myself to make Gabe  smile before class ended.

Situation 2

Me:  I found 1/4 of a pan of brownies in the office.  Draw a square and shade 1/4 to represent the remaining brownies.   Most students, including Gabe, drew the same picture.

1/4 of a square shaded blue.

Figure 1

Me:  I decided to give 1/2 of the remaining brownies to Ellie. Highlight, circle etc… the section given to Ellie

Whereas the majority of the class drew the version shown in Figure 2, Gabe sectioned his picture differently (Figure 3).  Gabe was still visibly upset but continued to be an active participant.

pic4                                              pic5

Figure 2                                                                                   Figure 3

Me:  What is the fraction size of the brownie given to Ellie?

This time around, students, who had the answer of 1/3 before, remembered our discussion of equivalent sections.  I watched them add more details to their pictures leading them to an answer of 1/8.  This included Gabe.  Even though students arrived at the same answer of 1/8, their strategies differed.  If you’d like to read about the various student strategies click here.

My time as a guest teacher was wrapping up. Although, I didn’t have enough time to discuss a different situation, I did have enough time to present the class with a related challenge.

As mentioned earlier, I had been keeping tabs on Gabe waiting for a chance to positively reinforce his work.  The opportunity presented itself when I observed Gabe sectioning his drawing into eighths. He, again, had a unique solution.  I asked him if I could show his picture to the class and he replied, “yes”.  Still no smile.

Me:  I’d like to show you a picture one of your classmates created.  I drew Gabe’s pre-sectioned picture on the white board. Question, how would you create equivalent sections for this picture?


The class liked the challenge and got right to work. Lot’s of hands raised, including Gabe’s, in hopes of providing a solution.  I called on Gabe.  He applied his solution to the white board drawing proudly and walked back to his desk smiling.

Closing Thoughts

Now that I’ve stepped out of the classroom to be an Instructional Technology TOA, I use guest or co-teaching opportunities to maintain and hone specific skills, such as (but not limited to):

  1. Incorporating different strategies into the class discussion
  2. Finding connections between different strategies
  3. Tinkering with question types that require students to do more of the thinking.
  4. Inviting all students to the math party

With Gabe, I focused on #1 and #4.

A square where 1/2 of 1/4 is shaded

1. Incorporate different strategies into the class discussion 

Once I introduced the Desmos area model (pictured on the right), most students gravitated towards that arrangement.  Gabe didn’t.  He continued to section his square differently reinforcing his individuality and making highlighting his approach a priority. Exposure to different strategies not only honors student thinking but also provides students with references for future problem solving tasks.

4.  Invite all students to the math party.

I first heard this belief at a Dan Meyer workshop around 3 years ago.  The Low Foor- High Ceiling idea was a prevalent theme throughout the series of activities.  Math should be inclusive.  It’s up to the teacher to design a lesson that allows every student entry to the task regardless of their mathematical level. (link to Dan Meyer’s Ted Talk)

The frustration Gabe exhibited wasn’t productive struggle.  He was starting to withdraw or leave the party.  To his credit, Gabe never left, however he wasn’t fully enjoying himself.

At a party, the host touches base with all the guests, makes sure they feel welcomed and connects them with other party goers.  The teacher is the host of the classroom party.  As the guest teacher, it was my responsibility to make sure Gabe felt welcomed and connect him with his classmates.  By being attentive to Gabe’s progression through the second problem, I found my opportunity to draw him back into the party.





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Multiplying Fractions: A Desmos Area Model


I’ve been tinkering with area models for awhile, storing countless versions in my account.  A few of my area models support the conceptual thinking of multiplying fractions.  Recently, I was approached by a coaching colleague, Crystal, to teach a lesson using one of the models.  I was thrilled to put a model in the hands of students.

Prior to teaching, I presented the classroom teacher, Mrs. Barnes, with two different fraction area models.  The first is a commonly used model that’s also promoted in her textbook.  The second was a pared down version of the first. Mrs. Barnes selected the version not promoted in her textbook, for two reasons. The pared down version was

  1. less cluttered.
  2. easier to apply with word problems.

In total, I ended up teaching three lessons in two different 5th grade classes using both whole class and a rotation format.  This post embodies the essence of the collective experiences.  The area models are in the Desmos Activity Builder found here.  I used the first slide only.  For each new problem, students would reset the area models by moving the sliders to 1.  Students revisited the activity builder to complete the word problems.

The Lesson(s)

  • I walked into the office and saw a cake.  About 1/3 of the cake was left.  DOne-third of a square is shaded redraw a square and shade the amount of cake left.
    • We circulated the room checking pictures.
    • I recreated the picture using Desmos.
    • When I reached the front of the room, I continued my story.
  • I grabbed the cake and decided to share it with a couple of students.
    • Teacher tip:  I asked for students who have a December birthday to raise their hand.  One student, Ellie, raised her hand.  Ellie was now worked into the situation.
  • I gave Ellie 1/4 of the remaining cake.  In your drawing, circle the piece of cake given to Ellie.
    • We circulated the room checking pictures.
    • I called on students to guide my Desmos representation
      • They first told me to section the red region into 4 parts.
      • Then shade in one of the smaller pieces and circle it.
      • I instructed them to label their circled part as Ellie’s piece.


  • Here’s my question:  What is the fraction size of Ellie’s piece?
    • We circulated the room, listening to student conversations and asking additional questions. I heard 1/4, 1/6 and 1/8
    • When I’d hear, 1/12, I’d ask:
  • Where did the 12 come from?
    • Student responses
      • “We need equal parts.”
      • “I drew imaginary lines.”
      • “If you extend the lines, to create equivalent pieces, you get 12.”
      • “The first row has 4 sections.  I counted 4, 8, 12.”
      • “The first row has 4.  There are 3 rows, so those rows need to have 4 sections too.  You multiply 4 by 3.”
    • Being a middle level math teacher for 20 years, I was unfamiliar with 5th graders. I didn’t know what to expect.  Needless to say, I was blown away by their responses, especially the last two – which were explaining the algorithm.
  •  At one table I noticed a student with the answer of 1/6.  When I asked him how he arrived at 1/6, he pointed to the 6 sections.  His table-mate immediately added, “You need equivalent pieces!!”
  • Once back at the front, student shared their thinking.
  • From our discussion, we seem to agree that Ellie received 1/12 of the whole cake. I’d like you to slide the black dot from OFF to OA square sectioned into 12 pieces.N.
    • Student responses
      • “I knew it!”
      • “I was right”
      • “Yes!”
      • Overall excitement
    • Their response was completely unexpected and a pleasant surprise.
  • Lastly, we connected the algorithm with the visual. one-fourth times one-third is one-twelve
    • 1/3:  The original cake was divided into 3 sections, one of which was left.
    • 1/4:  The remaining cake was divided into 4 sections, one of which was given to Ellie.
    • Multiply numerators:      1 section of  1 section =  Ellie’s portion    1
    • Multiply denominators:  4 sections x 3 sections = total sections    12


The Textbook Version

Earlier, I mentioned presenting two area models to Mrs Barnes.  The post outlines the structure of the pared down area model.  The other model, found in the textbook, uses an overlapping approach (shown below).  One-third of a square is vertically shaded red.  One-fourth of a square is horizontally shaded blue.  When the two squares are combined, the overlapping region represents the area.  Both models area found in the Desmos Activity Builder found here.

A square where 1/3 is shaded red. A second square where 1/4 is shaded blue.  a square representing where 1/12 is shaded purple.      pic12

Figure 1                                          Figure 2                                  Figure 3


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