Heights, Shoe Sizes and Desmos Snapshots

Do you use activities from Desmos? Have you tried their snapshot tool? This tool was so helpful orchestrating a recent classroom conversation.

We started by noticing and wondering about the plots without the context. We noticed that it looked like a line and had a somewhat constant rate of change. We wondered who these people were with such big feet.


This is data from NBA players (mostly 2017-18 Warriors). JaVale McGee is the far right dot with the size 20 shoes. We then entered our own height and shoe size and the remainder of the task was left for homework.

Before school on the day the task was due I prepared the conversation using  student work and thinking. Desmos’ new feature snapshots has made this so simple. Click on the camera icon and go to the conversations tab. Your student responses will be waiting to be dragged into a screen like the one below.


We started with a conversation about what makes a good fit for a line. I chose four graphs that had reasonably good lines that were different enough for us to debate which fit best. Groups and then the class discussed together the criteria they would want for a line to be a good fit.

Most of the groups selected the top right graph. Click on the top right arrow and that student work will be displayed. I had intentionally tried to select graphs of students whose voice isn’t  as often present in the classroom. “Mary” appeared very pleased that their graph was chosen by the class.


This screen gave us an opportunity to discuss how the heights and shoe sizes of basketball players related to the y-intercept and slope.

Back to the snapshot view for our second collection. We compare our class data with the data from the Warriors.


Several 8th graders thought that our class data had a stronger correlation because it was more clumped. After our conversation, we were not convinced yet which was a stronger correlation so I had them consider a few questions.

  • I told them a new 8th grader was standing outside with size 8 shoes and asked them to predict the height.
  • I then told them a Warrior was outside the room and asked them to predict the height.
  • Finally I asked everyone in the room with size 8 shoes to stand up and we observed their varying heights.

I made myself a note. In this conversation we wondered what it meant to be an outlier with some thinking that the green triangle of dots in the top right represented an outlier for our class data.

I was still unsure they were convinced. Our next shapshot helped. Picture5

They noticed that the lines from our four students were more varied than the lines for the NBA players. I think it helped to convince us that the student data was as consistent. They speculated the following about why a line wasn’t as easy to fit for the student data.

  • We aren’t done growing yet
  • Men’s and women’s shoe sizes are different
  • The NBA players were wear similar types of shoes

Again we took a closer look at the graph they selected (lower left) and appreciated the work of “Stefan”. We noticed the much lower y-intercept and wondered if that was the height where we changed from children’s to adult’s shoe sizes.


On the last screen I asked students to compare the two models. We wondered as we became adults if we would follow the path of the green line or shift more towards the orange line.


There was at least one outstanding question that I will use to launch the next class. What does it mean to be an outlier in two variable data? I am ready with that snapshot.



and then…

Screenshot 2018-10-06 20.36.33

I have used a lesson related to student heights and shoe sizes for over 30 years. Shifting the lesson to Desmos last year really helped capture the student thinking. Orchestrating the conversation with the new snapshots feature led to a discussion that included so many voices in the classroom. I can’t wait to see what Desmos will come up with next.

If you are interested the height shoe size activity can be found here.


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Graphing Back to Back

Long before there were polygraphs my classes were describing and drawing graphs back to back.


Students sit back to back with one looking at the screen and the other an individual coordinate grid white board. I display a graph and one student describes to the other.


The teacher move question to ask when everyone has drawn is “Drawers, what is something your describer told you that was helpful.” It is such an easy way for one student to honor a contribution of another student. Meanwhile I listen to contribution and we start moving slowly towards more formal vocabulary.

Teacher move 2 is to rotate around the room after each graph cross-pollinating the vocabulary. I use about 4 to 6 graphs gradually becoming more complex. With the more complexity after graphing we play “I can name that graph in…” and groups talk together about the fewest directions needed to describe a graph.

Investing this time sets us up so well as we learn about a new function.

I added some detail and teacher moves in the notes to this google slide version

Here are some others in desmos.



Absolute Value Graphs





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Seeing Red

I have been reading through some of my reflections from the first semester. This one prompted further reflection.


I had mixed feelings when Desmos announced the card sort option. I love card sorts. I have a cabinet with boxes filled with cut up cards. I have used them with partners, groups and even whole class human card sort giving a card to each student. I love the conversation the sorting generates.

My original question still comes back to me. I would answer now that being able to embed a card sort within an activity really helps. I also appreciate the data I get from the dashboard. I am conflicted how to best use that data. I had that opportunity yesterday.

My math 8 class was working through the card sort that is part of the Marcellus the Giant activity. This has been a challenging one for my 8th graders. I take a look at the dashboard and see red. Literally red.


What now?

No teachers wants to see all of that red. Now I need to consider my options. What move do I make? These might be options open to me.

Ignore it. Let it go.

I can wait. I can direct my energy to other pieces of the activity and possibly address some of the misconceptions shown in the card sort by doing so. I can treat the data as formative assessment. I can work with my PLC to figure out how to follow up. It might mean generating classroom conversation by looking at mismatched card sets in tomorrow’s class. It might mean spending more time with a concept. The data is there and useful. This is difficult because I hate seeing so much red. I want to fix it.

Project Responses for self-correction.

I can project the key. I can anonymize and project so students can self correct. I don’t think I should use this option very often. It doesn’t feel right. The feedback is so immediate that they really don’t have to think about what is written on the card. Projecting might be useful if many of the students have been successful and some have a few cards to switch. I can sit individually with any that have several mismatches. I should ask myself why am I asking students to spend time on a card sort if I already expect that most students will be successful.

Pause or Pace

I can intervene. One option is too pause and reflect on the overview once everyone has had a chance to complete the card sort. Utilize the most common incorrect groups provided in the overview. Consider the mismatches and bring out conversation with a think/pair/share. Follow by inviting them to look to their card sort for revisions.

Matchmaker, Matchmaker

I can be a matchmaker. One strategy I have tried this year is to pair students telling them that I noticed they sorted their cards in different ways. Allow them to reason it out together. I might even pair students and then find them another partner. This is a decision to invest more time in the activity and the card sort itself.

For Marcellus the Giant, I was the matchmaker. I am busy following the dashboard on my iPad and pairing students who have different sorts. I was so pleased to see the first pair (started 1/5 and 2/5 correct) reason together and see their spots on the dashboard turn green after several minutes of conversation. How did that happen? I didn’t need to intervene.

While they were working I continued match making. More green. As pairs resorted I began matching them with other pairs if they had a different sort. We invested much more time in this card sort. I left convinced it was time well spent. They needed to talk through the card sort that was very challenging.

As I reflect, this activity might be best done in pairs. I often pair students on an activity. On the other hand, every student wants to explore varying scale factors. Every student wants a shot at adding an accessory. The match making allowed both opportunities.


1/4/2018 Further Reflection

As part of planning an activity, I now look to see if there is a screen where I want to be the matchmaker. This is particularly the case if I plan to have students work individually on an activity.

The “Seeing Red” feedback parallels my wonder about how to respond to the new dashboard summary screen. Here is a portion another reflection from earlier this year after my first use of the summary screen.

I think I have to be really careful to use the information well. Today I paid attention to the summary screen on my iPad as I ran the activity. It was tempting to intervene as a saw the x’s. That may not always be the best move. I would imagine that students often make some mistakes only to have an “aha” a few screens later and then go back and self correct. I also know that I sometimes misuse the correctness tools that I have now. At times I have put up the card sort display under anonymous so students can check. I know this is not the most effective move. It is far better to pair students together and say I noticed that they sorted the cards differently. Let them reason it out. Would it be tempting to a teacher for expediency to project the summary screen as students work? Would it be a good move? I am not so sure.

Typically, I will follow a few key screens on my iPad dashboard and listen in or drop in to wonder about their work. If it is an expression screen I might huddle a group together that have different looking expressions. Today I watched the summary screen. It felt like a race. I felt the pull to intervene when I saw the x. I think I have to be very careful with that screen.

I wrote, “It felt like a race. I felt the pull to intervene when I saw the x in the summary screen.” It is the same as seeing red. How should I respond? I am still struggling with how to best use the immediacy of the feedback. A quote from a recent Marilyn Burns tweet might be helpful here.

In fact her recent blog about reflecting on a lesson and pedagogical mistakes was very helpful.

Upon further reflection, a question I am asking myself. Why would I use an activity if I thought that students would go through it perfectly without the x’s? I need to remind myself to give students space to learn and make mistakes.

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Open Middle, Exponent Properties and Desmos Scientific

How do you help students learn exponent properties? I have shifted over time from presenting properties with practice to exploration using open middle problems. At the end of the year our classes take an assessment called the Geometry Readiness Test. In all four of my algebra sections the top area of strength for each of the classes was exponents. I was a little surprised. If anything I was concerned that students had spent little time practicing exponent properties on worksheets or homework. Instead we pushed the our depth of knowledge by exploring a series of open middle problems followed by a short skill practice. I am writing to share the set of problems we used.

I use this set of problems throughout an exponent properties unit. They were the “fire up” to open the class. I use desmos scientific. It is a great tool for students can keep track of previous results as they make new attempts.

Open Middle Problem Notes
Place the digits 1 to 9 in the boxes below. Each digit may only be used once. Find the largest result. Find the smallest result.










Use before introducing scientific notation.

These answers should generate the need for scientific notation. Students will see results in scientific notation. Expect students to be puzzled by the output. Pause the class early on to wonder about the format of the results they are seeing.

Which is larger?

1.88×1029 or  2.51×1027t

Which is smaller?

1.97×10-13 or 6.05×10-13

Wonder why it is better to have (749)(658) as opposed to (759)(648). Shouldn’t we make the number as large as possible for the 9? Hopefully a student wonders but if they don’t you should. What is happening here?

If you have time have students place  0, 2.51×1027 and 1.88×1029 on a number line

Use each of the digits 1 – 9 only once to create the largest and smallest product.










Useful before multiplying numbers in scientific notation and the product of powers property.



Why is the result the same? Shouldn’t the one with the 7.53 get the power of 9?

An opportunity to discuss the commutative and associative properties of multiplication.
There is also something to wonder about the remaining 7 digits and the arrangement that makes the largest or smallest product. What math is happening underneath here?

Use each of the digits 1 – 9 only once to create the largest and smallest power




Useful prior to negative powers and power of power properties
A chance to consider both negative powers and a power to a power.
Is there a way to rewrite this problem that would be helpful? Sometimes I see a student who does division instead of the negative powers.
What is the effect of the negative?
When going for the smallest, Why is it helpful to have a 9 as the outside power?
Use each of the digits 1 – 9 only once to create the largest and smallest product



An opportunity to consider fractional exponents before introducing the property.
What is the best strategy for large numbers?
How does the denominator effect the result?
Why can’t we find really small results?
Use each of the digits 1 – 9 only once to create the largest and smallest sum











Useful before encountering addition of numbers in scientific notation.

Listen for students noticing the relationship between the expression and the result. Ask what patterns they noticed.

What is going on with addition?

Going for smallest can lead to an interesting discussion.

8.9×101+6.7×102+4.5×103 has the same results when we flip the 5 and 6.

8.9×101+6.7×102+4.5×103= 8.9×101+5.7×102+4.6×103
Why? Are there other pairs I can flip and still get the same results?

Fill in the boxes using digits 1 through 9 to make the biggest 3 digit number. A digit can only be used once.


There are several other exponent problems at the open middle site. here are a few.

From Open Middle – Robert Kaplinsky



Fill in the boxes using digits 1 through 9 to make the biggest number. The digits in the power and the digits in the result must all be different.


Follow up to previous. Does it help to remove the restriction of three digits?




Find 3 positive integers that add up to 10. Place each number into one of the blanks to find the largest possible result.


From Open Middle – Zack Miller

Which three numbers? Which has the most impact?
If negative integers were allowed, would it be helpful?

 xp9 Not an open middle but expect a good debate and disequilibrium among students.
 xp10 Not an open middle. This AMC8 problem should generate some good discussion among groups.

Last year we used the first 5 open middle problems to open classes. We also used the bottom two to generate discussion. You might choose to do the largest one day and the smallest the next or have students choose whether they are going for large or small. In groups two students might go big and two small and then they collaborate together.

I realize that there is a downside to having 9 blank spots. When using open middle with equations we didn’t set up that many blanks. There are some very good problems on the open middle site that can meet the same goal with fewer blanks. I included a few that we have used previously. In the case of exponent properties, using all 9 digits helped generate really large numbers or numbers really close to zero forcing results in scientific notation.

Beyond open middle problems, I need to share a unit opening group 3-Act related to stacking paper to the moon and a unit closing activity followed by 3-act inspired by the book The King’s Chessboard.

What do you think? Where have you used open middle problems with students?


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The Vocabulary Gap

I have been thinking about formal language quite a bit this summer. My summer started with professional development from Dr. Kenneth Wesson and his work on learning and the brain. His statement that the “vocabulary gap drives the achievement gap” has left me thinking about how we help close the vocabulary gap. He would suggest that positive relational experiences are essential for learning and that knowledge is co-constructive. “We do our best learning with one other person”.

How can we best help close the vocabulary gap in our classrooms? At my site, our Algebra students (about 80% of our 8th grade population) bring to the classroom a wide range of background knowledge and experiences. We need to be intentional about developing a shared common set of experiences and vocabulary for all students. Below are some of the activities we planned into our unit on quadratic functions this year.

We begin with Polygraph Parabolas from Desmos. Here is a screenshot typical of a first round game. Partners were chosen by the computer with one selecting among 16 graphs of a parabola and the other asking questions trying to determine the chosen parabola.

Screenshot 2017-07-09 15.16.12

Students are using their language to describe what they see in the graphs. In this shared informal experience, every voice is heard and honored. Every student found success. We still have some work to do. Our language isn’t very efficient but we start and build together.

We followed that up the next day where students are paired sitting back to back. One student can see the screen and the other has a individual whiteboard for graphing. We display a parabola on the screen and ask the “describers” to describe the graph to their “drawer”. We start with a single parabola and eventually describe more complex graphs.

desmos-graph (10)

Here is a link to a google slides and a desmos version. These have more graphs than we would use in a day. You may want to select 4 or 5 on a single day.

There are a few teacher moves we have found helpful with this activity.

  • Ask the drawers “What did you hear from your describer that was helpful?” It is affirming when someone else honors your ideas. Collect ideas and begin labeling vocabulary on the screen as you hear it from students.
  • With multiple graphs ask “What is the fewest number of directions you can use to describe this set of graphs?” … “Can you name that graph in 5 statements? Fewer?” Think – Pair – Share. They will work together to find strategies to become more efficient with their language to meet the need of describing the picture with the fewest directions.
  • Rotate after each round. We switch places and then rotate the new describer (boards and pens will stay put this way) clockwise and next round switch and counter clockwise around the room.

On the next day, after sharing a few informal experiences we distribute a vocabulary list as part of our unit preview that also includes learning targets and homework options. We ask them to take a moment and find a term that they can share something about and a term that they can’t describe well. Everyone in the class will be able to find a term they can’t describe and a term where they can share some knowledge. In groups they share and then they work together to select a term that no one in the group can describe. These are shared to the class and we use the offered terms to help tell the story of the upcoming unit. We then ask them to circle the terms that they need to learn more about. We return to their circled list at the end of the unit and students give us a grade in how well we helped them learn the vocabulary in the unit.

If we are going to develop vocabulary through experiences, we need to build in those activities into our lessons. We try to utilize our “fire up” time to open the class. We find constructs like WODB or “Which one doesn’t belong?” helpful. We put up this graph and asked students which one doesn’t belong.

Screenshot 2017-07-09 16.27.39

Give students time to think and share in groups. As they share to the class, the students will find the more formal vocabulary as useful in arguing that their parabola is the one that doesn’t belong. We used this activity later in the unit and a visitor said they were “blown away” by how well students described parabola features and their equations.

Card sorts are another tool that can be used to fire up the class upon opening. In your purpose is vocabulary development, it is essential to pair students to do these sort activities. This parabolas card sort is part of Desmos’ quadratics bundle. Here is a one screen card sort that might fit the opening of the class. We have also used a physical set of cards for groups to sort. This one matches three quadratic forms and their graphs.

How can we close the vocabulary gap? I think it starts with shared informal experiences in the classroom. It can’t stop with those shared informal experiences and notes on a vocabulary list though. We need to offer opportunities that generate the need for being efficient and refined with mathematical vocabulary. We need to pause and offer the formal language when it is needed. I don’t think the rich discussion that resulted in the WODB can happen without the informal experiences early in the unit. Nor should we expect students to develop formal language from a few informal activities. We need to be intentional about vocabulary development and plan shared experiences throughout a unit. We need to close the vocabulary gap.

What do you think? What activities have helped you develop formal language with your students?

More links to activities or thoughts about developing language in math class

Follow up to the polygraph: Polygraph Part 2

Jon Orr’s blog entry: Better Questions: Two Truths and One Lie

Suzanne Von Oy’s blog entry: What’s In A Name?

MARS Lesson: Representing Quadratic Functions Graphically





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Notice and Wonder

In the spring I attended a workshop in San Mateo presented by Michael Fenton. The topic was problem solving and getting students to notice and wonder. I was using the data below in a lesson to my 8th grade math class the next Monday. I decided to makeover the lesson by starting with their “noticings and wonderings”.

I put up this scatterplot.


With Michael’s model, I asked students what they notice and what they wonder. Individually, in groups, as a class

My Math 8 class noticed and wondered.

swim notice wonder

I hope it readable. They noticed the times decreasing, the women points getting closer to the men. The wondered why it was going down and what the numbers on the side mean. They listened to each other and were curious. EZ said he noticed some gaps and one around World War 2. Another student wondered what gaps EZ was talking about.

There will be speculation on Olympic events and there is a good chance someone will offer 100 meter free style. Once that is established, ask what questions they have. Their questions will generate a need for a model. In my math 8 classes, they all went to a linear model. Most pursuing the question, “will the girls time catch up to the boys time?” Others pursued how fast will the times be when they (the students) are my age. We have done that pursuit before. I tell them I was an entering 8th grade the summer of the 76 olympics.

There are some decisions to make. Do I give them a worksheet to guide them through the process? Ultimately I want a linear model from them. I want the answers to their questions with this model. I want them to think about whether the model always makes sense. I want them to make mistakes. Do I need a worksheet to generate that thinking? I gave them a copy of the graph.

I also give them a link to desmos and geogebra files with the data. Most use desmos. The ticket to the technology is that linear model. I am not sure if it would be better to hand them technology immediately.

I do want to give them the opportunity to make two particular mistakes. Many will make the mistake of counting the lines when determining the rate of change. They might have a slope of -½ or even ½ for the boys. If they come with a model first and then try it in desmos a mistake will be an opportunity for a moment of disequilibrium. Why did my line go up? Why isn’t it going down at the same rate? If they ask for my help, I am going to ask them what they like about their model and what do they wish they could change. I will ask them about their model and wonder what the parts have to do with swimming (What does your -½ or 65  have to do with swimming?) I will leave them thinking about that.

There is a second opportunity for a mistake that will lengthen the time needed for the lesson. You will have to judge its value to the learning. Should the data start at 1900 or should it start at 0 (with x being years since 1900)? Starting at 1900 is a certain opportunity for disequilibrium. They will test their equation in desmos and likely wonder if desmos is broken. They won’t see their graph. I will get them to zoom out and then they will see their line. I leave. If your students know point-slope form at this point they may avoid this mistake. I prefer to refer back to this data when we are working with point-slope form in order to motivate the value in learning a different form. If you do it before point-slope, it is a great opportunity for them to puzzle through how do they get their line back to zero. I have done it both ways and I would hate for them to miss the opportunity this error presents. On the other hand if the rate of change and slope is the key to this lesson, removing the error with the intercept helps focus this goal.

I have used this data many times over the years. A version was in the original CPM Algebra course. I have to thank Michael for the great addition with the “noticings and wonderings” at the beginning. The cost was a little in time and the gain was a much better grasp of the context needed to puzzle through the disequilibrium they were certain to encounter. We used it several times with other lessons since. I look forward to your thoughts as I am certain I will use this data again next year.

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The Power of Yet


I was a young teacher in the eighties. In search of units and direction I fumbled upon Marilyn Burns’ Math Solutions course. Fantastic course. One of the speakers, Ruth Cossey, gave a talk that changed my mindset. It was about a adding a simple word to our mathematical vocabulary. Yet.

I promptly posted it by my clock. It is the one thing that has been common to the 10 classrooms I have used in teaching the last 30 years.


Why yet? It is really about a growth mindset and developing that growth mindset in my students. It takes more than posting a word by a clock. First I needed to develop a growth mindset in myself. Second, I learned I needed to devote time to nurture a growth mindset in students. We start with yet the first day.

There is such a difference in these two phrases.

“I don’t get it.”

“I don’t get it yet”

Say them out loud. You can’t say the first with a growth mindset tone. You can’t say the second without it. Students need to use it. I need to model it. One word makes a huge difference.

I am not sure how I often I will blog. This is not my learned way of communicating. Once I decided to give it a shot there could only be one topic of the first blog post. Yet.

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