Mark Trushkowsky
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marktrushkowsky.bsky.social
Mark Trushkowsky
@marktrushkowsky.bsky.social
130 followers 160 following 62 posts
math, teaching, adult education, numeracy, music, movies, gardening, play, beauty
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marktrushkowsky.bsky.social
Like I said, I'm not quite there yet, but I wanted to share my work so far because I feel like there must be so many different ways to approach this, and I'm curious what others do. #ANNMath
A messy ZOOM whiteboard of thinking visualized, exploring the patterns in the table, making and testing conjectures and partially answering the question. The work generally moves from top to bottom. The culmination thus far is the 1000 does appear in the 8 row (not 8th row). The final sentence is "Upwards and onwards to a generalization to figure out the column!"
November 13, 2025 at 7:14 PM
marktrushkowsky.bsky.social
Thank you, @triangleyou.bsky.social for sharing this task they got from @howiehua.bsky.social!

What patterns do you notice in the table?
Will 1000 appear? If so, where?

I'm not quite there yet, but I'm having such fun, in the spirit in which I received it, I invite others to play!
#ANNMath
A table of numbers. Consecutive odd numbers across the top. Reading left to right: 1, 3, 5, 7... The left column is made up of powers of 2 going down. Reading: 1, 2, 4, 8,... The table can be completed by multiplying the odd numbers across the top by the powers of 2.
November 13, 2025 at 7:14 PM
marktrushkowsky.bsky.social
The Northern Lights (Minneapolis, MN)
Red and green aurora lights Red and green aurora lights Red and green aurora lights Red and green aurora lights
November 12, 2025 at 8:14 PM
marktrushkowsky.bsky.social
Northern Lights from my backyard
Green aurora lights in the sky Purple, pink, red, green, blue aurora lights over a rooftop Streaks of red and white aurora lights in the sky Red and green aurora lights
November 12, 2025 at 8:12 PM
marktrushkowsky.bsky.social
Minneapolis is doing me right on the jazz front these days.
Brandee Younger Trio at the Dakota with Rashaan Carter on bass Makaya McCraven and Junius Paul at the Ice House
October 28, 2025 at 2:38 AM
marktrushkowsky.bsky.social
Next I tried it with Fig. 13. I created a grid in the center of each square unit to see if it would that have the same area as the figure we can build with the corners from Figure 11, removing the center, and pushing the corners together.
A grid drawn over figure 13 resulting in an area of 60 sq units, which is the same area as the rearranged figure created with the corners of Figure 11.
September 30, 2025 at 1:52 AM
marktrushkowsky.bsky.social
Another pattern emerged when I started playing around with creating a grid in the center of each sq unit

The area of the blue figure is 61 sq units.

I noticed the grid by itself had an area of 40 sq units. It also looked a lot like the figure we got we we added corners to our original figure
Figure 11 (area of 60 squares units). A grid made from connecting the center points of Figure 11. The grid by itself, and area of 40 square units The grid and the figure created from the corners of Figure 9, both with an area of 40 square units
September 30, 2025 at 1:45 AM
marktrushkowsky.bsky.social
I decided to create similar figures to have more to explore to look for patterns.

I used the second figure from the launch (11 squares high and 11 squares wide as opposed to 9) and “filled in the corners.”

#ANNMath #MTBoS
The area of the blue is 11 + 9 + 9 + 7 + 7 + 5 + 5 + 3 + 3 + 1 + 1 = 61 squares The area of the corners (orange) is 10 + 10 + 8 + 8 + 6 + 6 + 4 + 4 + 2 + 2 = 60 squares Again, the area of the blue is a little more than half the area of the entire square, and the area of the orange is a little less.
September 30, 2025 at 1:33 AM
marktrushkowsky.bsky.social
I wondered what would happen if we “finished” the square by filling in the corners.

Looking at just the green area, I rearranged the pieces to look similar to the original figure. The figure felt similar, but like an “even” version.

#ANNMath #CAMImath
#MTBoS
On the left, a blue figure with an area of 41 square units, arranged in centered columns 1, 3, 5, 7, 9, 7, 5, 3, 1. On the right, the same figure, imbedded in a 9x9 square, with the same 41 squares in blue, plus 40 green squares (10 in each corner, arranged as stairs 4, 3, 2, 1) The blue figure removed and the green squares rearranged into columns with heights of 2, 4, 6, 8, 8, 6, 4, 2
September 30, 2025 at 1:25 AM
marktrushkowsky.bsky.social
Last May’s Community of Math Instructors (CAMI) monthly teachers’ circle meeting started with some attention for some fairly simply-structured figures.

#ANNMath #CAMImath
#MTBoS #ITeachMath

nyccami.org/attention-an...
Two shapes. On the left, an area of 41 squares. One way to describe them is in 9 vertical columns, with 1 square, 3 squares, 5 squares, 7 squares, 9 squares, then 7, 5, 3, and 1. The figure on the right can be described as a similar figure, rotated 45 degrees, except the columns would be arranged 1, 3, 5, 7, 9, 11, 9, 7, 5, 3, 1.
September 30, 2025 at 1:23 AM
marktrushkowsky.bsky.social
Crop art nurtures through inspiration, and humor.
The quote: “All flourishing is mutual” from Braiding Sweetgrass by Robin Wall Kimmerer, above two hands filled with seeds, all created using seeds. A black and white image of a person made of beans, planting a single red bean, above the word PERSIST, written with red beans. “We may encounter many defeats, but we must not be defeated.” Maya Angelou Three kittens peering directly at you, between the words, “ARE YOU POOPING?”
August 31, 2025 at 4:23 AM
marktrushkowsky.bsky.social
More crop art.
A portrait of a goat, created from seeds (natural colors) A furry portrait of a prairie rodent, created from seeds Olive Trees with Yellow Sky by Vincent Van Gogh, created from seeds A portrait of a rooster, created from seeds (dyed/painted)
August 31, 2025 at 4:04 AM
marktrushkowsky.bsky.social
Crop art at the State Fair is always amazing, but this year feels a little next level.
A portrait of Clara Bow, made with seeds Bjork’s 1995 album cover of Post, made out of seeds (and also I can’t believe this album is 30 years old) A portrait of Sly Stone singing into a microphone, created out of seeds The dining hall at the Naniboujou Lodge, made out of brightly colored seeds
August 31, 2025 at 3:59 AM
marktrushkowsky.bsky.social
Joyful colors and shapes seen in the Creative Activities Building at the Minnesota State Fair.

#MTBoS #ITeachMath #MathArt #MNStateFairMath
A black quilt with rectangular panels of bright colors. Rectangular-ish shapes in bright colors on a black background An array of a spectrum of colored squares on a black background. Quilt: a 3x4 array of circles with star points around their edge and a two-colored star within each. Inset in that array is a 2x3 array of circles separated into pinwheels. Bright colors.
August 31, 2025 at 2:25 AM
marktrushkowsky.bsky.social
The power of squares in quilt designs at the Minnesota State Fair

#unitchat #MNStateFairMath #MTBoS #ITeachMath
Quilt made of of squares of different blues and greens and brown and whites. A white, brown, and sky blue quilt. A multicolored quilt design of smaller squares, on diagonals, forming triangular points. Quilt made of many small squares, with one larger square on the right side with smaller squares set inside of it.
August 31, 2025 at 1:48 AM
marktrushkowsky.bsky.social
I love community data collection like this one at the MN State Fair.

Cast your bean and vote your commitment to local food.

What do you notice? How do you feel? What do you wonder? What will you do?

#MTBoS #ITeachMath #creativerepresentationsofdata
Four glass-enclosed frames with space at the top for dropping in a bean to vote for one of the categories. The categories: Grow Relationships, Give Time, Spend Local, & Share Knowledge. Each category has different heights of beans that have been added.
August 30, 2025 at 10:16 PM
marktrushkowsky.bsky.social
Minnesota State Fair Math

What they have in common is that they are all 2025 champion grains & legumes. But #WODB?

#MTBoS
#ITeachMath
Oats Barley Wheat Soybean
August 30, 2025 at 10:10 PM
marktrushkowsky.bsky.social
I see you @letterboxd.social
Three circles—one orange, one blue, one green—in a sign for mother earth gardens in Longfellow neighborhood at dusk.
July 29, 2025 at 1:37 PM
marktrushkowsky.bsky.social
Great week for music in the Twins. Ebo Taylor in town with Pat Thomas, and the Ebo Taylor family band for Jazz is Dead at the Fine Line and then Nubya Garcia at the Dakota with Lyle Barton on keys, Max Luthert on bass, and Sam Jones on drums.
89 year old Ghanaian afrofunk legend Ebo Taylor with a band that included at least 3 of his sons (on bass, keys, and guitar). Also pictured is a tight horn section of trombone and trumpet. Longtime Ebo Taylor collaborator and highlife singer, Pat Taylor on stage. Also visable are bass player, keyboardist, and horn section playing handheld percussion. Two drummers (one on kit and one with conga drums), bass player, trombonist, trumpeter, guitarist, and keyboardist/singer. The Ebo Taylor family band, incredible in their own right. Nubya Garcia on saxophone, Lyle Barton on piano, Max Luthert on bass, playing Water’s Path under blue lights.
April 12, 2025 at 3:18 PM
marktrushkowsky.bsky.social
Some of Donald Judd’s 15 untitled works in concrete. Marfa, Texas. #mathart #MTBOS #iteachmath
Concrete cubes in shadow and light in the desert Concrete cubes in shadow and light in the desert Concrete cubes in shadow and light in the desert Concrete cubes in shadow and light in the desert
April 8, 2025 at 9:45 AM
marktrushkowsky.bsky.social
The U.S. Dept. of Education is now soliciting reports on discrimination in schools, but their web address reveals their bias. Don’t let bad-faith actors control the narrative. If you’ve seen attacks on DEI, curriculum censorship, or barriers to access, speak up & submit a response.
A screenshot of a page from the official website of the US Department of Education, with the url: enddei.ed.gov The text reads: "Schools should be focused on learning. The U.S. Department of Education is committed to ensuring all students have access to meaningful learning free of divisive ideologies and indoctrination. This submission form is an outlet for students, parents, teachers, and the broader community to report illegal discriminatory practices at institutions of learning. The Department of Education will utilize community submissions to identify potential areas for investigation." The survey questions: Your email School or school district School or school district ZIP Code "Please describe in as much detail as possible the discriminatory practice taking place (450 Word Limit)" Upload File (optional)
March 3, 2025 at 8:42 PM
marktrushkowsky.bsky.social
Left
Center square rgb(88,123,166)
Inner frame rgb(0,13,123)
Outer frame rgb(96,0,60)

Right
Center square rgb(88,123,166)
Inner frame rgb(67,255,137)
Outer frame rgb(0,13,123)

What do you notice? What do you wonder?
#MTBoS #mathart

Color Frames by Aren Lew (www.desmos.com/calculator/p...)
A colored center square rgb(88,123,166) A colored inner frame rgb(0,13,123) A colored outer frame rgb(96,0,60) A colored center square rgb(88,123,166) A colored inner frame rgb(67,255,137) A colored outer frame rgb(0,13,123)
February 26, 2025 at 4:43 PM
marktrushkowsky.bsky.social
The center squares are the same color!
(www.desmos.com/calculator/p...)

#mathart #MTBoS #mathforidentityandbelonging
A center square with a square frame around it and a another square frame around that. A center square with a square frame around it and a another square frame around that.
February 26, 2025 at 3:41 AM
marktrushkowsky.bsky.social
It's not even my birdy, but my friend Aren built me a tool on Desmos (www.desmos.com/calculator/p...) to explore Josef Albers' theory that our perception of a color is influenced by the colors that surround it.
That all of the center squares are the same color boggles my mind! #mathart #MTBoS
A center square with a square frame around it and a third square frame around that frame. A center square with a square frame around it and a third square frame around that frame. The middle frame has changed color from the last picture. The center square appears to be a different color, but it has not changed - it just looks different because the color of the frame around it has changed. A center square with a square frame around it and a third square frame around that frame. The middle frame has changed color from the last picture. The center square appears to be a different color, but it has not changed - it just looks different because the color of the frame around it has changed.
February 26, 2025 at 3:38 AM
marktrushkowsky.bsky.social
Sylvan ice crystals on winter windows.
Ice fractals on a window Ice fractals on a window Ice fractals on a window Ice fractals on a window
January 22, 2025 at 10:50 PM