This workshop featured Anna Burago from the Prime Factor Math Circle.
Professional development credit certificates and stipends were offered for this workshop. If you have questions reach out to us at navajomath@gmail.com or 785 473 0273 (call or text).
Funding for this workshop was provided by Math for America, MIT, Duke, and the Kansas State University Foundation.
Full session videos are posted below.
Schedule
(All times valid for Crownpoint, NM.)
9:00 Welcome.
9:15 – 10:30 (Anna Burago) Math Circle Session 1
Title: Coding Scratch
10:30 – 10:50 Introductions/Discussion
10:50 – 12:05 (Dave Auckly) Math Circle Session 2
Title: Pattern Block Problems
12:05 – 12:30 Sanity Break
12:30 – 1:45 (Shay Logan) Math Circle Session 3
Title: Sheep/Corn/Coyote problems
1:45 – Whenever math circle discussion/suggested resources
Sessions
Coding Scratch:
Here is a link to the program Anna demonstrated.
https://scratch.mit.edu/projects/1083837450.
You may find a large amount of information about scratch with a simple web search. The homepage for scratch is
https://scratch.mit.edu/.
Here are Anna’s notes:
Scratch Workshop 1
Pattern Block Problems:
Pattern blocks have been around for a very long time, and one cand find many ways to use them in math lessons. Here are some linke from mathforlove.com
https://mathforlove.com/lesson/pattern-block-scaling/
https://mathforlove.com/lesson/fraction-talks-pattern-block-edition/
We recommend playing with physical objects. There are times when software can also be useful. for example, if you don’t have blocks, want to share thoughts on line, or want to model something really big where you can cut-n-paste things together.
One of our favorite organizations is amplify. Check them out at
https://amplify.com/.
Here is a video introduction to some of their products (all free!!!)
We used
https://polypad.amplify.com/ and
https://polypad.amplify.com/lesson/pattern-blocks
The following list of questions is one example of a (baby) infinite pickle related to pattern blocks:
Is it possible to make a regular hexagon with 39 blocks?
Is it possible to make a regular hexagon with 99 blocks?
Is it possible to make a regular hexagon with 299 blocks? …
The point is you can make the problem easier and harder by changing the number of blocks. If you want an algebra-based question requiring induction and geometry, try
Is it possible to make a regular hexagon with n blocks, for any n?
One discussion was about tilings in nature. There are many examples. Devil’s Postpile (pictured below) provides a very nice one.
One of the things we hinted at, but didn’t really dig into was the notion of a substitution tiling.
If you can figure out how to make larger versions of each of several tiles out of the tiles you can repeat the process to produce amazing patterns. This is an active area of mathematical research.
I mentioned the Ammann A3 tiles as one example.
Here is the substitution rule:
Here is a paper introducing the Ammann tilings
Aperiodic tilings
Many more may be found in the Tiling encyclopedia
The pinwheel triangle with sides 1, 2, sqrt{5} is another interesting one. Here is the substitution rule from:
D. Burkholder, Unexpected Beauty Hidden in Radin-Conway’s Pinwheel Tiling.
Sheep/Corn/Coyote problems:
The Sheep/Corn/Coyote problem is just the cabbage, goat, wolf problem in costume. On online source for this particular problem is at https://jrmf.org/puzzle/river-crossings/.
Here is a powerpoint template of a version of the diagram used in the presentation.
Coyote, Sheep, Corn, Boat Template
Writing code to solve similar problems, or making an excel spreadsheet to analyze these questions is an interesting project for more advanced students.
We like making larger models. Various math visitors have run this activity with primary school students in the community. One of or favorite ways to start is to make hat bands before the session as in the following picture.
The session starts by having the students draw, cut-out and color sheep ears, corn ears, and coyote ears as they wish. These are taped into the bands and taped around the children’s heads. A table cloth, rope, or rug represents the river, a cardboard box or similar represents the boat. The students then act out the puzzle, and the teacher can say things like, “Oh no! the lamb just ate the corn!” and “Coyotes think corn tastes yucky.”
Since we did not get parent permission to share photos from the children doing this on our website, you will have to do like they do and use your imagination. It is rather cute, and fun for all.
Connecting to the Virtual Meeting
We will open the meeting 15 minutes before the official start. If you have trouble connecting you may call (785) 473-0273 and we will help as we can.
Join Zoom Meeting
https://ksu.zoom.us/j/91404514920
Meeting ID: 914 0451 4920
Can be e-mailed navajomath@gmail.com, sent via text (785) 473-0273, or mailed to:
Dave Auckly
Math Department
Kansas State University
Cardewll 138
1228 N MLK Drive
Manhattan, KS 66505
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