Here are just a few of our noticings and wonderings:

I notice …

- 1, 2, and 4 only have one dot, but 3, 5, and 6 have 2.
- In 4, the circles are a little bigger. In 6, the inner circle is off-center and smaller.
- The number of items in the box corresponds to the number for the numbers 1,2, and 3.

I wonder …

- If the size of the rings and the location of the ding matter.
- Is this a positional number system? or another kind?
- I wonder what #7 would be.

People had a lot more to notice and wonder about (I haven’t included the rest of our noticings and wonderings so that you have space to notice and wonder yourself!) and conversation moved naturally into trying to figure out the rules of the system. To facilitate this, I shared this link where you can put in your own number and see what the system produces. (Refresh the page when you want to look at a new number.)

When our group felt like they had a pretty good handle on how the system works, we explored two more pages that show how to add and multiply numbers in the Ring-a-Ding system. On each page, you enter the numbers you want to see added or multiplied and then watch the magic happen!

We had a lot of fun exploring how different numbers added and multiplied and making and checking predictions.

Here is the main Ring-a-Ding Numeration page. Warning – there are some spoilers – don’t visit the page until you have investigated all you care to on your own.

Want to see some of the work we did? Check out our Jamboard.

Is Ring-a-Ding Numeration useful or important? I don’t know, but it sure is fun to play with. For more on having fun with mathematical objects … or even falling in love with them, check out Jim Henle’s TEDx talk Math is for Our Pleasure.

]]>Participants noticed that the number of dots in each figure is increasing by one as you look left to right in each row. One person said they see certain groupings of dots repeated in other groupings. Another noticed that the dots are arranged in a circle for some numbers and someone else wondered if that was because those numbers are prime.

We had two breakout groups in which participants discussed how the dots were arranged and made predictions about how larger numbers might be arranged.

Both groups discussed how the arrangement for 10 is created using the arrangements for both 2 and 5. One group described it as starting with the arrangement for 5 and replacing each dot in the 5 with the pattern for 2 and discussed that this is related to 5×2.

Similarly, 12=4×3 and is created by replacing each dot in the “3” arrangement” with the 4 dot arrangement:

One group conjectured: “The shapes are built out of their factors. The smaller factor determines the shape of the vertices (what replaces the circles in the other shapes) and the larger factor determines the array.”

Groups explored ways to display larger numbers, with much discussion about the role each factor has in the arrangement. Below are some images from that work:

During the wrap-up discussion, we compared our work to the arrangements created by the artist shown here. http://www.datapointed.net/visualizations/math/factorization/animated-diagrams/

Resources:

Notice & Wonder Slides: https://docs.google.com/presentation/d/1_DYqQlozcIltL-oMBdV88v8BiFQUWPCXM2ObaJfQu9s/edit?usp=sharing

Jamboard with our explorations: https://jamboard.google.com/d/1r0qbG7o59UDwnjie7holiiYcJrmKvOJ9BHYLfsNP34w/edit?usp=sharing

YouCubed Lesson Plan (source for our activity): https://www.youcubed.org/wim/number-visuals-6-12/

Brent Yorgey Blog about the original dot designs: https://mathlesstraveled.com/2012/10/05/factorization-diagrams/

Dancing Dots (animated adaption of Yorgey’s work): http://www.datapointed.net/visualizations/math/factorization/animated-diagrams/

Blog by the artist: http://www.datapointed.net/2012/10/animated-factorization-diagrams/

]]>Sarah started this meeting by asking the group to consider the following prompt.

We spent some time on our own, then discussed the two sequences. Talking about the first sequence, Maya had a feeling the first number after 32 could be 48, but was pretty sure it was wrong. She saw that the difference between 16 and 32 is 16, so maybe add another 16 to get 48? However, she also so that the differences between the numbers change as the numbers grow, so she knew that the difference between 32 and the next number should be larger than 16.

Cindy looked at the difference between each of the numbers and saw a pattern (+4, +10, +16, …) and decided that the next difference would be +22 since the differences are growing by 6.

Benny described how he found the missing numbers in the second sequence. He saw that the difference between 24 and 34 is +10. Then he looked for a growing sequence of differences that would get from 6 to 24 is three steps.

We then did a notice/wonder activity with the following diagrams:

- The first rows are our sequences.
- The row below shows the change between each pair.
- The third row is the change of the change.
- The first number in each row is red.
- The third row is the same number repeated.
- When the third row is 6, the numbers grow sequentially higher. When the third row is 2, the numbers grow not as high. The second sequence started with a larger number, but is overtaken.
- The two sequences meet at 16, then the first sequence takes over.
- In the first one, everything is +6. 4 to 10 to 16. The other one has a difference of 2.
- There is a lot of addition going on to get these numbers
- The red numbers are the same numbers are in both sequences.
- All the numbers are even.
- The 3 red numbers uniquely identify the sequence. There is a missing 4th number that should be 0.
- red 2, 4, 6 would be different from 3, 4, 6. First number = position, starting point. Second number = starting speed, change in position over time. Third number = acceleration, change of the change over time.

- Why is the first number red?
- What does this mean for numbers? How do I make this meaningful for me? What are the relationships between the numbers?
- What is a picture that would help visual the pattern? Can we see the function pictorially?
- If we put question marks instead of the 2s, could we figure them out? If we replaced the middle numbers with ?s, could we figure them out? How many question marks could we add and still them figure them out? How many numbers could we take out and still recreate the sequence?
- How do we figure out an explicit formula for these sequences?
- Knowing the three red numbers, is that enough to fill out the rest of the sequence (as long as you know the type of sequence)?
- Is three numbers in the sequence enough to continue the sequence?
- Could we pick any three numbers in the diagram to continue the sequence (recreate the diagram)?
- Do the red numbers have to end with a 0? Could the red numbers end with -2, for example?
- Could you put 0, then a number that is not 0 after it?
- Do any 3 red numbers make a sequence?

We spent the rest of the meeting exploring the red numbers, building sequences, finding explicit formulas, and exploring ways of visualizing the growth of sequences. Our Jamboard shows the work of the group.

I started thinking about the red numbers watching a video with James Tanton. (This is the 3rd video in a series, so click *Previous* to see the others.) He calls them “leading diagonals” and uses them to derive a formula that predicts the nth term in the sequence.

Before we went into our breakout rooms, we spent a bit of time clarifying the ways that a Queen can move on the board, and making sure we agreed about which squares were “unattacked” or “safe.” (We had some debate about the square that the Queen is sitting on. In the end, we decided that this was *not *a safe square.)

Check out the Jamboard to see some of the work we did (or don’t, if you don’t want any spoilers!) One question we found intriguing was, “how do you know when you’ve found the best answer?”

Since then, many of us have been playing chess on chess.com. We’re even thinking about starting a CAMI chess club. Want to play? Send us a message and we’ll loop you in.

As a warm-up, Amy presented us with this question: Why might a manhole cover (or, in the gender-neutral, maintenance cover) be round? One of the central ideas that came up in the resulting discussion was that a circle won’t fall through its own hole, no matter which way you turn it. It has a constant diameter, or constant width.

It turns out that circles aren’t the only shape with this feature! Amy presented us with another shape that behaves in a similar way: the Reuleaux triangle.

After brainstorming some questions to explore, we split up into two break-out groups – group 1 planned to explore the reuleaux triangle in more detail, and group 2 planned to explore other shapes that passed the “manhole cover test.” Using Zoom’s new self-assign feature, participants were invited to choose their own group based on what they were more interested in exploring.

Amy gave us a few online tools to play with. Some people also came up with cool images just by playing around in jam board.

Check out the links below if you want to see some of the stuff we came up with, or do your own experimenting. There’s also an awesome video on how manhole covers are made – ascinating stuff! Thanks, Amy!

Online tools:

https://www.desmos.com/calculator/nwidsblzj7

https://commons.wikimedia.org/wiki/File:Reuleaux_triangle_Animation.gif

https://www.desmos.com/calculator/wmqt92vien

Jamboards:

https://jamboard.google.com/d/1vv-VmKLpBTz6yo9xQr_vLoY4GiJ5ie0jyI13ZOlogXw/edit?usp=sharing

https://jamboard.google.com/d/1zKIYRm7cSsb8xoV02ziNgqpVowxiKwluWMV0tyVypm8/edit?usp=sharing

Video on Realeaux Triangles:

https://www.youtube.com/watch?v=-eQaF6OmWKw&vl=en-US

Video on NYC Manhole Covers:

https://video.nationalgeographic.com/video/short-film-showcase/0000014e-6574-dd5e-a75f-6d74ad760000

Here are a sentence-starters that we came up with:

- How many…?
- How many ways…?
- Is this always true?
- Could this pattern continue?
- Would it be possible to…
- What would happen if…?

And a few questions we can ask to get students thinking mathematically:

- Why would I show you this?
- What’s the point?
- What do you see that relates to math?
- How do you see this?

What questions do you ask students to get them thinking? What kinds of questions do you want them to ask themselves?

After that, Sophie showed us a drawing of some circles.

What do you notice? What do you wonder?

When we looked at these circles, we had a *lot* of ideas and questions. In the exploration that followed, we found that one of the biggest challenges was drawing new images, especially with more than four circles. Some of us made pencil/paper drawings, and some of us tried to use online software. Sarah proposed making some pre-existing images on Geogebra or Desmos so that students could play with the images and explore math questions without needing to struggle with making the images themselves.

Check out some of the images we came up with!

Jamboard: (includes photos of pen-and-paper drawings) https://jamboard.google.com/d/1hWOdKlHIPZ-J42ENTMsIkSBaW-zJidgd2x1sJYHUYxI/viewer

Geogebra: (includes images that one group made during the meeting)

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PUZZLE: Coconut Classic

Five men and a monkey, marooned on an island, collect a pile of coconuts to be divided equally the next morning. During the night, however, one of the men decides he’d rather take his share now. He tosses one coconut to the monkey and removes exactly 1/5 of the remaining coconuts for himself. A second man does the same thing, then the third, fourth, and fifth. The following morning the men wake up together, toss one more coconut to the monkey, and divide the rest equally.What’s the least original number of coconuts needed to make this whole scenario possible?

Mind-Benders for the Quarantined! (Museum of Math, NYC)

We had a lot of interesting ideas and shared a few strategies, but we didn’t get to a solution.

In hindsight, I realize that this problem is annoyingly gender-specific. Why men? And why does it matter? I tried rewriting the problem using “they” is instead of “he,” but it seems to lead to some confusion about how many people were taking actions. Does this rewritten version work?

**PUZZLE: Coconut Classic**

Five people and a monkey, marooned on an island, collect a pile of coconuts to be divided equally the next morning. During the night, however, one of the people decides they would rather take their share now. The person tosses one coconut to the monkey and removes exactly 1/5 of the remaining coconuts for themselves. A second person does the same thing, then the third, fourth, and fifth. The following morning the people wake up together, toss one more coconut to the monkey, and divide the rest equally.

What’s the least original number of coconuts needed to make this whole scenario possible?

Maybe someone has a better gender-neutral version of the problem?

]]>We started with a quick game of online Simon Says.

It was okay, but a little clunky. Usha suggested a different kind of activity where video on means yes and video off means no. Read a series of statements. Here are a few I brainstormed: Winter is my favorite season of the year. I love ice cream. I am excited to be back in school. I want to go to college. (You probably have better examples.) End with a statement everyone can say yes to…

I decided to start with a task in small groups and generate question based on the task.

Brooke and Usha looked at the square numbers 1, 4, 9, 16, 25, and 36. They decided that 1 wouldn’t be used because there is no way to create a pair of numbers that sum to 1 if you are limited to the numbers 1 to 18. They also eliminated 36 because you can only use 18 once. Then they looked at all the possible pairs that can sum to each of these sums, looking for a combination of pairs that would use as many of the numbers 1 through 18 as possible.

They found two different solutions with 8 square-sum pairs, one with 1 and 2 left over and another with 16 and 18 left over.

Deneise, Macarena, and Sarah looked at the different ways to make 9 with the sum of two numbers. When they ran out of ways, they had used the numbers 1 through 8. They added 9 to 16 to get 25, then looked at other ways of getting 25.

They were also able to find 8 square-sum pairs, with 17 and 18 left over.

Macarena also tried another strategy in another Jamboard frame. She wrote the numbers 1 through 18 in a number line and pair numbers with arcs.

We came back together as a group to brainstorm questions related to the task:

After brainstorming these questions, we went back into breakout groups to explore.

Brooke and Usha looked at 1 through 13 and found that there was 1 number unpaired. They made a conjecture that 1 through 12 would have 2 left over, 1 through 11 would have l left over, 1 through 14 would have 2 left over. Their 1-12 conjecture was confirmed, but there 1-14 conjecture was wrong. They were able to pair all numbers 1-14.

Sarah, Macarena, and Deneise noticed a pattern in the sum of different sets of consecutive numbers. If you total 1 through 8, the total is 36, a square number. If you total the next 8 numbers (9-16), you get 100, another square number. Their conjecture was that they would get an even square number if they totaled the next set of numbers that paired together (17-32). However, 392 is close to an even square number (400), but it’s not square.

At the end of the meeting, we wondered if there is a way to create a visual that will show the square-sum pairs (1+8, 2+7, 3+6, etc.) but also show the total squares or rectangles that the consecutive numbers add up to. For example, the first 8 consecutive numbers add up to 36, which is a square number. 36 is also a triangular number, though, since the sum of the first n numbers is always a triangular number.

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 1/2 (8 * 9) = 4 * 9 = 6 * 6 = 36

How could we show this visually?

The initial problem comes from classroom materials from Gordon Hamilton (MathPickle.com), who was inspired by Henri Picciotto. Follow the links above to download student handouts, an article about the program, and a link to a New York Times Wordplay story about the problem.

In attendance: Brooke, Deneise, Eric, Macarena, Sarah, and Usha

Note: I tried planning this meeting using only a Google Jamboard, to test how this could be useful with a class. I usually have a Google Doc with links out to a Jamboard or another whiteboard, but this requires participants to find a second link and it also means that our group notes are in a different place than our exploratory work. It’s nice to have it all in the same place.

]]>We started this meeting with a Which One Doesn’t Belong which also included talking about what the four pictures have in common:

Here are some of the observations people made:

- The honeycomb is the only one that appears to have a regular pattern.
- The leaf is the only picture without an animal in it.
- The turtle is the only one under water.
- In the leaf, the borders are a similar color to the shapes, but in the other pictures they are not.
- The honeycomb pattern looks more three-dimensional than the other patterns.
- The turtle picture has several patterns in it, not just on the turtle.
- All the pictures have a pattern involving shapes.
- The shapes in the patterns mostly seem to have four, five, or six sides.
- Even though there is irregularity in the patterns, the shapes seem to have some uniformity of size.

I then revealed that the pattern in all four pictures is the same kind of pattern generated by following certain rules and shared another example of the same kind of pattern generated by a computer:

(Pattern generated by http://alexbeutel.com/webgl/voronoi.html)

In the context of noticing and wondering, these are some of the ideas the group came up with:

- I notice that there are not very many triangles, but lots of shapes with 4, 5, or 6 sides.
- I notice that it feels like there are some reflections going on.
- I notice that it feels like a map coloring problem.
- I notice that the dots are not placed randomly.
- I notice that the dots are sometimes on the edge and sometimes closer to the middle of their own shape.
- I notice that all of the polygons are irregular (the sides and angles are not the same).
- I notice that it looks like county seats.
- I notice that there is folding symmetry with the dots between adjacent shapes.
- I notice that there tends to be only three colors touching at an intersection.
- I wonder if this is connected to where post offices or fire stations should be.
- I wonder what the deal is with the dots.
- I wonder how you figure out how to place the dots.
- I wonder why the dot is where it is in each shape.
- I wonder if it is possible for more than three colors to come together at a point.
- I wonder what the rule is behind this pattern.

After a solid round of noticing and wondering, I shared that this pattern is called a Voronoi pattern and that the dots are called “seeds.” I shared a couple resources before letting people loose to play in breakout rooms.

This website offers an interactive element where you can click to place seeds and generate your own Voronoi pattern: http://alexbeutel.com/webgl/voronoi.html

This website has a game based on the Voronoi pattern: http://cfbrasz.github.io/VoronoiColoring.html

I also provided a link to a document with some push and support cards (note that there wasn’t a specific task, but the support cards are around figuring out the rules for the pattern).

With only the instruction to play and have fun, we broke into groups for the next hour, using a communal Jamboard to record some of our work.

**Small Groups**

All the groups explored the rules for the pattern by playing with placing seeds, noticing relationships, and trying to predict the results of placing new seeds. Each group also explored specific questions that arose during their play.

Patricia, Eric, and Audrey used annotation to explore the relationships between lines between seeds and lines connecting seeds.

They later moved on to annotating directly on a giraffe (!) and asking the (still open) question of how to locate the seeds if they are not in the picture. (Side note: Annotating on a giraffe is my favorite thing that I have seen happen over Zoom.)

Jeniah, Maggie, Sophie, and Mark started by investigating the Voronoi game and used their exploration to come up with theories about how the pattern is generated from the seeds. They made predictions about what would happen before placing each seed.

They also pondered why this pattern shows up in the natural world and speculated about whether it had to do with the nuclei of cells. They played with adjusting some of the options on the Voronoi generator, including animations, and made guesses and discoveries about the effects of changing the parameters.

Usha, Lynda, Kevin, and Amy played with the interactive Voronoi generator, placed a few seeds, and then challenged themselves to predict the results of placing a new seed. They challenged themselves to see if they could place the seeds in such a way as to make a perfect square. Here is one of their attempts:

They also wondered whether it was possible to have more than three regions coming together at one point and attempted to place seeds to create a honeycomb pattern:

We had a lot of fun exploring Voronoi patterns and ended the meeting with people feeling like they wanted to keep exploring. For lots more information, including applications beyond what we talked about at the meeting, google “Voronoi.” Here are a few more links that I found interesting:

Article with some interesting variations: https://library.fridoverweij.com/docs/voronoi.html

Some animations and images: https://library.fridoverweij.com/codelab/voronoi/index.html

Voronoi patterns superimposed on a melon: http://cfbrasz.github.io/VoronoiColoringSavePNG.html

List of applications of Voronoi patterns in different fields: https://www.ics.uci.edu/~eppstein/gina/scot.drysdale.html

In attendance: Sophie, Lynda, Jeniah, Eric, Patricia, Amy, Nadia, Mark, Usha, Audrey, Maggie, Kevin

Note: Sarah Lonberg-Lew is a math teacher and professional developer in Gloucester, MA. She is also the treasurer of the Adult Numeracy Network. Sarah (and many others from around the country) started attending our CAMI meetings when we went online during COVID. Thank you for leading us and sharing these fascinating explorations, Sarah! -Eric

]]>Before the meeting, I shared this post on the CAMI email list:

There were a LOT of responses.

After a warm-up conversation on the topic of learning in groups, I started the meeting by modeling the introduction of the lesson we used with adult education students this summer. We talked through a few of the introductory slides of a distance learning lesson:

I then shared an activity originally suggested by Usha Kotelawala the CAMI meeting a few years ago (link above). One of the nice things about this activity is that all students can get started adding consecutive numbers together to fill in pieces of this chart. As a group, they can work together to fill in holes. They will also start to see patterns in the sums that can help them predict other solutions.

After we had a few solutions, we went into breakout groups to complete the table and brainstorm questions about the sums of consecutive numbers. Each group chose a few of their favorite questions, then we came back together to share them:

When I taught this lesson over the summer, I asked my students to give their age as the sum of consecutive numbers as well. Over WhatsApp, one of my students gave the following solution:

There are five different ways to write 45 as the sum of consecutive numbers! I was really impressed. And it made us all wonder which numbers are like 45? What is it about 45 that makes so many solutions possible?

We spent the rest of the meeting working in groups, then sharing our discoveries as a group.

*Note: I set up a Jamboard (online whiteboard at http://jamboard.google.com) for use in this meeting and then had everyone made their own copy of the Jamboard for editing. It didn’t work very well since people couldn’t see each other’s edits. And I didn’t do a good job of collecting the group’s work at the end of the meeting, so I don’t have everyone’s notes. *

*I think it would have been better to have one Jamboard for the whole meeting which we all edit together. Each group could have made their own pages within the Jamboard, and then we would be able to see each other’s work easily and also save the work of the meeting in one place. *

If you have notes from the meeting or ideas that you would like share about the question posed above, please post in the comments.

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