For our final CAMI meeting of 2017, I wanted to spend some time at a CAMI meeting doing some math that would create some thing visual and beautiful. As I was looking around for activities to bring to the group, I came across the website, Math Pickle (as in “Put your students in a pickle”). They had a trove of math problems that I look forward to exploring in future CAMI meetings. The one I chose for this one is at its core an opportunity for students to practice multiplication in a way that is much more engaging than just memorizing facts and doing worksheets. And it builds works of art. As I started to play around with it, I started to notice different ways to think about how to make designs with the best score.
During our introductions, we each shared our names, our program and something beautiful we’ve seen recently.
- Cynthia: An image at the end of a trilogy of books she just completed.
- Eric: Building a bike wheel – getting in all the separate spokes and fitting them into this one unified object.
- Michael: Members of his community together and finding connection in each other.
- Mark: Playing SET with my 5 year old daughter last night and watching her see a combination that I didn’t see and having this moment of appreciation that she is her own person, with her own perspective.
Since this activity is in part inspired by him, we looked at a Pier Mondrian painting and shared some of the things we noticed:
- Michael inferred a sense of flow and movement when you look at this – your mind tries to find a pattern and to make sense of the ambiguity
- Eric saw the lines inside the larger rectangle as “roads” and noticed that there are 9 instances where 4 roads intersect. Also that there are some intersections with 3 roads (a lot more than there are of 4)
- Cynthia noticed that the yellow rectangles are always adjacent to the white rectangles.
- Linda noticed that the rectangles around the edge/perimeter of the whole thing were open figures.
Mondrian Art Puzzle: How do you play?
The first rule of this puzzle is that you have to cover a square with rectangles. Below are a few possible ways of covering a 10 by 10 square with rectangles:
Between the two arrangements above, the one on the right is the winner. Which is to say, the goal of this puzzle is to find the arrangement with the lowest possible score. Can you see how the scores are determined? Where do the 35 and 23 come from?
Scores are determined by the subtracting the area of the smallest rectangle in your arrangement from the area of the largest rectangle? On the left, the score is 35 because 42-7=35.
What would the score of this arrangement be?
Finally, we have an arrangement with an even lower score. But unfortunately, this arrangement is disqualified. The final rule is that no rectangles can appear in an arrangement more than once, so the 3 by 4 rectangle and the 4 by 3 rectangle are no good. NB: the two 10s in the arrangement above are fine because the dimensions of the rectangles are not the same.
Here are the collected rules of the Mondrian Art Puzzle:
- You must cover the square with rectangles.
- Every rectangle on the canvas must be different… so you cannot have both a 4×5 and a 5×4 rectangle.
- Your goal is to minimize your score. To calculate the score of any design subtract the area of the smallest rectangle from the area of the largest rectangle.
- When coloring, use as few colors as possible. The same color cannot touch along edges or corners.
Everyone had some time on their own to play around with different squares. Most of us used trial and error, covering the square with rectangle by rectangle until it was completely covered. This was a useful time for practicing the criteria – in particular, not repeating the dimensions of one of the rectangles took some time to get used to. Then as folks start to come up with arrangements with different scores, some potential strategies begin to emerge.
Below are Eric’s initial explorations with the 4×4, 5×5, 6×6 and 8×8 squares.
Here’s some of my work on the 6×6. In addition to the trial and error method, I tried to list all the possible rectangles that could fit and then use the total area of the square to try and find some :
After exploring on their own and then working together, teachers started putting up and coloring their lowest scoring arrangements for each square. (P.S. You don’t really know a person until you start to get acquainted with their preferred color pallets).
Continue Your Explorations
- As you create different designs for a given square, what patterns do you notice?
- As you find a low score for any of the squares, how could you know if it’s the lowest score?
- Did you find the lowest possible score for any of the squares? Find another arrangement rectangles that gives you the same score.
- Is there a way to figure out the lowest possible score for a given square before covering it with rectangles?
- Is it possible to predict the lowest score for a n by n square?
- Can you find the design for the lowest possible score of a square and make your picture at least 40% blue? Try it with other percentages and other colors.
- Are there any patterns you see in the number of colors needed for your designs?
- <<SPOILER ALERT>> Here are the lowest scores for the squares 4 by 4 through 32 by 32. Pick one and see if you can figure out the dimensions and arrangement of the rectangles that result in that score. Also, since they are just the lowest scores that have been submitted to Math Pickle so far, can any of them be beat?
If you create your own designs, color them in and send them to CAMI through email, linked in the comments below, or post them on Twitter at #nyccami!
Happy New Year!
In attendance: Cynthia, Michael, Eric, Maggie, Linda, Spencer, Mark
Programs represented: Literacy Assistance Center, York College’s Adult Learning Center, LaGuardia Community College, Pathways to Graduation (District 79), and the CUNY Adult Literacy/HSE PD team.