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)

]]>

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.

]]>Then I introduced Number Pyramids. Thank you to Henri Piccioto and his amazing web site of math resources. Here is the sequence we used:

Try these pyramids on your own.

**Some shared language**

*2-row pyramid, 3-row pyramid, 4-row pyramid, *…

*seeds*: the numbers that are given

*middle number/cell*: a number or cell that does not have an outside edge

*wings: *We decided on this term to refer to the bottom leftmost and bottom rightmost numbers

- Adding two even numbers yields an even sum
- Adding an odd and an even number yields an odd sum
- The pyramids on the bottom – the numbers given are all on the outside diagonal
- In 4-row pyramids, the sums of each row increase by the same constant number found in the middle cell of 2nd row, or middle two cells of bottom row.
- The sum of the top of a 3-row pyramid is double the center bottom number, plus the
*wings*(bottom corner numbers). - The sum of the top of a 4-row is triple the middle number, plus the wings.
- Two matching numbers lead to a “mirror pattern” in the row below: 7, 7 leads to 3, 4, 3 below.

Mark noticed that the sum of each row a 3-row pyramid and a 4-row pyramid increase by a constant value that can be found in the middle of the bottom row of a 3-row pyramid and in the middle cell of the 4-row pyramid.

Maggie noticed that in a 3-row pyramid, the top number is 2 times the bottom middle number, plus the *wings*. In a 4-row pyramid, the top number is 3 times the middle number, plus the wings.

- Would a 5-row pyramid have Maggie’s pattern? Conjecture: 4 times all the sums of the middle bottom numbers, plus the wings.
- Is this only a whole number puzzle? Could the boxes have decimals or fractions? Could they have negative numbers?
- What’s the fewest seeds needed to solve a puzzle? Where do the seeds need to be in the pyramid?
- Given the top number, can we fill in any-sized pyramid?
- Could we fill in a blank pyramid? Could that be a puzzle?
- Do we need numbers at all?
- At what point does it become too difficult to solve? 2 seeds on a 5-row pyramid, for example. Would we still be able to solve it?
- How do you choose the seeds and the positions of the numbers? Do only 2-digit numbers work? Is it possible with 3-digit and 4-digit numbers?
- Is there an application for this pattern? Building a building, road, etc?
- Is there a certain number of seeds where a finite number of solutions is possible? Fewer seeds where multiple solutions are possible?
- How could we represent this algebraically? How could we make the unfolding of Mark’s and Maggie’s patterns explicit?
- What would happen if we changed the pyramid rule? Subtraction, multiplication, …

We then spent about 15 minutes working individually to consider the questions we generated, then worked in groups.

Nadia and Amy experimented with making number pyramid puzzles with fractions.

Rebecca, Older, and Mark examined the pattern of the constant growth in the sum of the pyramid rows and found that the pattern changed in a 5-row pyramid. In the pyramids below, there is a constant difference between rows 2 and 3, and rows 3 and 4, but the difference changes between rows 4 and 5.

Maggie, Sarah, and Spencer used algebra to analyze a 6-row pyramid.

They found patterns in the coefficients of the variables for the numbers used in the base row: A, B, C, D, E, F. They noticed that the coefficients of the variables within each of the cells are the same as a row in Pascal’s Triangle. The coefficients of the variables in the top cell is also the same as a row in Pascal’s Triangle.

- The coefficients for the total of row 1 are 1, 1, 1, 1, 1, 1.
- The coefficients for the total of row 2 are 1, 2, 2, 2, 2, 1.
- The coefficients for the total of row 3 are 1, 3, 4, 4, 3, 1.
- The coefficients for the total of row 4 are 1, 4, 7, 7, 4, 1.
- The coefficients for the total of row 5 are 1, 5, 10, 10, 5, 1.

- Knowing the three corner numbers in a 3-row pyramid, is it possible to predict the number that will go in the middle of the bottom row without any trial and error?
- Knowing the three corner numbers in a 4-row pyramid, is it possible to predict the number that will go in the
*middle cell*without any trial and error? - How many seeds are necessary to create one solution? Does the position change how many seeds are necessary?
- Make a pyramid puzzle that cannot be solved. Make a pyramid puzzle that can be solved in more than one way.
- What would be a useful sequence of puzzles for a group of students?

If you create more number pyramids, please share them so that we can add them to our worksheet for students.

]]>The week before the meeting we sent out this teaser:

And got a few responses:

Is it possible to connect five houses on a flat surface without crossing the lines?

We started the meeting with the same four house problem and got another solution for connecting the houses.

Is it possible to connect all four houses without crossing the lines, without moving the houses, while also keeping the lines straight?

We then moved on to a notice/wonder with the following image.

We took a few minutes on our own to take note of what we noticed and wondered about the 8-factor graph above. After everyone had some time to think on their own, we shared what we noticed and wondered. Eric took notes on a split screen so the group could see the graph and the notes as we spoke. Here are the notes from our notice/wonder:

Here are the rules that were used to draw the 8-factor graph that we looked at:

- List a number and its factors
- Each number within the graph connects to its factors (each number also needs to be connected to its multiples)
- Lines can’t cross (unless you change the rules)

Sarah prefaced the rest of the meeting by explaining the four house problem is from the field of graph theory, which uses a different definition of the word “graph.” In graph theory, a “graph” involves the ideas of “vertex” (a point), “edge” (a line connecting two vertices), and “degree” (the number of edges that connect to a particular vertex).

After the discussion, we moved into individual thinking time for about 5 minutes. Sarah and Eric experimented with muting the group for enforced silence, so that everyone had some uninterrupted time to consider our questions on their own.

We then moved into two breakout groups for about 40 minutes of small group work. After the small group work, we came back together to discuss what the group discovered.

Maya and Patricia worked on making a factor graph for 16. They started by thinking about a cube and wondering if it would be possible to place factors on different vertices of the cube. They thought to remove lines when they weren’t needed. They realized 1 had to connect to all the other factors of 16. And 16 also had to connect to every other number.

Then they imagined laying strings around a cube to connect the factors and multiples. It seemed possible to connect all the factors and multiples in this way.

They then tried to make a factor group of 16 on a flat surface. They were able to make every connection except the one between 2 and 8 (shown by the red line). However they arranged the numbers, there were always two numbers that couldn’t be connected.

Maggie, Amy, Violeta, and Nicholas also tried to make a factor graph for 16. They looked at a few different arrangements including this one.

In this graph, 1 is connected to 2, 4, 8, and 16. Then 2 is connected to 4, 8, and 16. Then 4 is connected 8, and 8 is connected to 16. However, there isn’t a way to connect 4 to 16 without crossing a line.

An aside: The group noticed a connection to the handshake problem. In the 16-factor graph, each number has to “shake hands” with each of the other numbers.

The group then created factor graphs for other numbers like 25, 12, and 18.

In our discussion, Sarah wondered about the graph of 16. Do you think it can be done and we just haven’t found it, or do we think it can’t be done? If you think it can’t be done, how do you decide that? Maggie said it is impossible, maybe a bit tongue-in-cheek, since seven of us have worked on it and haven’t been able to find a solution. Nicholas said he was mostly convinced that it is impossible, but would need more time with it. Patricia talked about how it seemed possible with a globe and string, in three dimensions.

We noticed that the two factor graphs for 12 and 18 seem to be identical…

- They both have 6 vertices.
- You could substitute the numbers in one for the numbers in the other.

… which made us wonder if there is some kind of equivalence between 12 and 18. And what other numbers would have the same factor graph.

Other questions that came up at the end of our conversation included:

- What is the smallest number where a factor graph isn’t possible? (Maybe greater than 16, since it seems that its graph isn’t possible.)
- What is it about 16 that makes it not work? (Something about how every number has to connect to every other number. If 5 elements all have to connect to each other, we’re in trouble.)

You can watch the full meeting here:

]]>We began the meeting by looking at a definition of star polygons.

We then spent a few minutes playing around individually with combinations of dots and connection rules, jotting down thoughts to share with the group.

After the exploration, we shared out things that we noticed and things that we wondered.

Finally, we broke into groups to explore some of the questions we generated in more depth. At the end, we came together to share what we had worked on in our break out rooms. The final sharing can be found below!

In attendance: Amy, Sarah, Sophie, Benny, Audrey, Busola, Magdalena, Maya, Johanky, Stephanie, Eric, Maggie.

]]>For the COVID19 virus, pose 3 specific quantitative questions, the answers to which would be useful in your role as health commissioner. Try to consider questions that can be dealt with mathematically.

Here are our questions:

- How do we do social distancing? Room needed as we move forward? How might masks enable that distance to change? Considering cases of classrooms? How much room is needed?
- Connecting to airline space/seats?
- Sources of data? What are they? Cases, test, deaths? How accurate or inaccurate are they? How does that affect trends?
- How does model change … what does it look like when its over?
- When will it be over????? What will it look like for us to consider it over?
- How does infection rate for first 15 days in NYC compare to Sioux Falls, South Dakota?
- Importance of percents? Comparing per capital.
- Pregnant women – what percent are testing positive and what percent of newborns are testing positive?
- If average person infects 2 people vs 1 person. What do those graphs look like?
- Age correlation question
- R0 What is it? How many others does 1 person infect?
- How might age percentages change?

Then Usha walked us through a modeling process.

Usha emphasized that it is important to simplify the situation when modeling. The real world phenomenon of virus transmission is complicated and dynamic, with countless variables. We need to choose a few variables and build a model from that. We can make changes to the model after we build it.

Usha gave us the following list of resources to work from in our small groups: https://padlet.com/mathstrands/MathJourneys

We then went into breakout groups to work on individual questions. These are the spreadsheets we developed (all in draft):

- What was the NYC data for cases, hospitalizations, and deaths for the first 10 days in April?
- How does infection rate for first 15 days in NYC compare to Sioux Falls, South Dakota?
- How does social distancing affect the rate of spread?
- How can we model cases, hospitalizations, and deaths from a R0 rate, a hospitalization rate, and a death rate?

Other resources for modeling (and understanding) the Coronavirus outbreak:

- What 5 Coronavirus Models Say the Next Month Will Look Like (April 22, 2020)
- Any of Tomas Pueyo’s articles

This NCTM webinar: