Open Make WK 8: Trouble with Trefoils
Tubular Trefoils
In looking for cardboard craft ideas online I discovered "Math Monday" on the Make Magazine website. I was looking for a reasonably challenging activity that I could do with my students. The tubular trefoil grabbed my attention because in the past students have been fascinated by Möbius strips and the trefoil has the same inside out design. The task involves plenty of math concepts, from the geometric shapes involved to the measurement, fractional thinking, problem solving and attending to precision. For example, you need to draw lines every 1 and 19/32 on the strips. Our rulers are marked in 16ths , so I determined that my marks needed to be between 1 and 9/16 and 1 and 1 and 10/16, The same procedure needed to be applied to measuring a 17/32 line. I needed to determine the best cardboard to use. I wanted something stiff but flexible so I went with oak tag (handy because we have large sheets at school.) I tried cutting strips with a rotary cutter, a box cutter and scissors. The box cutter was the most precise but I’ll need to help students cut the strips for safety reasons. I needed to think through the difference between a “mountain" and a “valley” fold. I enjoyed measuring and cutting the strips: very reminiscent of quilting. I found the directions clear and well illustrated until the final step of putting the individual pieces together. I would leave it alone and return to it over the weekend to no avail. At this point in the week it is still a work in progress. It is interesting that I chose 2 projects that require a great deal of visual spatial thinking - not always one of my strengths! This project would probably take place over at least 3 class sessions. I would have the students journal after each session, including what they observed in their own problem solving process, what habits of mind they used, and any questions they had moving forward.
https://makezine.com/2013/09/16/math-monday-tubular-trefoils/
Fractals
Surprisingly, school was cancelled Monday. I looked at my tubular trefoils again (sigh....) and decided to work on something else I could make for math class. Ever since our webinar with Eric I’ve been thinking about pop up books (At one point he told us to never underestimate the value of a pop up book.) I teach a unit on fractals so decided to look at the Fractal Foundation’s website where I found directions for a pop up card. Just like the trefoil directions, the initial steps went smoothly. It took me a while to figure out the final folds but when I realized that upon folding the card in half after all the folds were complete it looked like a staircase. It was fun putting a card together for the fractal and I think students would enjoy that creative aspect. Again, in addition to perseverance, measurement , and problem solving, fractional thinking was applied when I needed to find 1/2 of 8 1/2, 1/2 of 3 and 1/2 and so on. After making the fractal with plain copy paper, I redid it with oak tag, finding it easier to see fractal pattern because the folds held their shape better than plain paper.
http://fractalfoundation.org/resources/fractivities/fractal-cutout/
https://makezine.com/projects/cut-and-fold-a-pop-up-snowflake-greeting-card/
I can see that there are many ways I can integrate cardboard creations in my math class and plan to keep exploring - and figure out that tubular trefoil! Here's a link to a youtube video of my creations....
https://youtu.be/sMLHy9X4eUA
In looking for cardboard craft ideas online I discovered "Math Monday" on the Make Magazine website. I was looking for a reasonably challenging activity that I could do with my students. The tubular trefoil grabbed my attention because in the past students have been fascinated by Möbius strips and the trefoil has the same inside out design. The task involves plenty of math concepts, from the geometric shapes involved to the measurement, fractional thinking, problem solving and attending to precision. For example, you need to draw lines every 1 and 19/32 on the strips. Our rulers are marked in 16ths , so I determined that my marks needed to be between 1 and 9/16 and 1 and 1 and 10/16, The same procedure needed to be applied to measuring a 17/32 line. I needed to determine the best cardboard to use. I wanted something stiff but flexible so I went with oak tag (handy because we have large sheets at school.) I tried cutting strips with a rotary cutter, a box cutter and scissors. The box cutter was the most precise but I’ll need to help students cut the strips for safety reasons. I needed to think through the difference between a “mountain" and a “valley” fold. I enjoyed measuring and cutting the strips: very reminiscent of quilting. I found the directions clear and well illustrated until the final step of putting the individual pieces together. I would leave it alone and return to it over the weekend to no avail. At this point in the week it is still a work in progress. It is interesting that I chose 2 projects that require a great deal of visual spatial thinking - not always one of my strengths! This project would probably take place over at least 3 class sessions. I would have the students journal after each session, including what they observed in their own problem solving process, what habits of mind they used, and any questions they had moving forward.
https://makezine.com/2013/09/16/math-monday-tubular-trefoils/
Fractals
Surprisingly, school was cancelled Monday. I looked at my tubular trefoils again (sigh....) and decided to work on something else I could make for math class. Ever since our webinar with Eric I’ve been thinking about pop up books (At one point he told us to never underestimate the value of a pop up book.) I teach a unit on fractals so decided to look at the Fractal Foundation’s website where I found directions for a pop up card. Just like the trefoil directions, the initial steps went smoothly. It took me a while to figure out the final folds but when I realized that upon folding the card in half after all the folds were complete it looked like a staircase. It was fun putting a card together for the fractal and I think students would enjoy that creative aspect. Again, in addition to perseverance, measurement , and problem solving, fractional thinking was applied when I needed to find 1/2 of 8 1/2, 1/2 of 3 and 1/2 and so on. After making the fractal with plain copy paper, I redid it with oak tag, finding it easier to see fractal pattern because the folds held their shape better than plain paper.
http://fractalfoundation.org/resources/fractivities/fractal-cutout/
https://makezine.com/projects/cut-and-fold-a-pop-up-snowflake-greeting-card/
I can see that there are many ways I can integrate cardboard creations in my math class and plan to keep exploring - and figure out that tubular trefoil! Here's a link to a youtube video of my creations....
https://youtu.be/sMLHy9X4eUA
Loved the pop up book colors and intriguing 3Dimensional structure. What age do you think this would be appropriate for. I think it would be helpful to pull out math concepts learned through making. Perhaps you and I can touch base on that sometime. It seems that measuring always come through, but I must say I don't know much about where fractals fit into the continuum of math standards. Seems that CCSS is pretty specific as to which standards are to be explored during specific grade levels.
ReplyDeleteI've explored fractals with K-6 students. They love it - and there are fun simulators on line so what they can't create with their hands they can understand (and create) visually. A 4th grader this year wondered if fractals can be 3 dimensional - perfect lead in to this cut out project. He reasoned that fractals have an infinite perimeter so why cant't they have an infinite volume :) That said, I wouldn't do this particular problem until at least grade 4.
DeleteThere are lots of standards that apply to fractals. Fractal Foundation.org (great website) has a link to CCSS which is quite specific. http://fractalfoundation.org/fractivities/CCMathStdsTable_Summaries.pdf
Also, from the National Council of Teachers of Mathematics (NCTM):
Recognize geometric shapes and structures in the environment and specify their
location. (NCTM Geometry grades K-2)
Recognize and apply slides, flips, and turns; recognize and create shapes that
have symmetry. (NCTM Geometry grades K-2)
Investigate, describe, and reason about the results of subdividing, combining, and
transforming shapes. (NCTM Geometry grades 3-5)
Identify and describe line and rotational symmetry in two- and three-dimensional
shapes and designs. (NCTM Geometry grades 3-5)
Recognize and apply geometric ideas and relationships in areas outside the mathematics
classroom, such as art, science, architecture, and everyday life.
(NCTM Geometry grades 6-8; 9-12)
Identify functions as linear or nonlinear and contrast their properties from tables,
graphs, or equations. (Grades 6-8 NCTM Expectations for Algebra Knowledge)
Use symbolic expressions, including iterative and recursive forms, to represent
relationships arising from various contexts. (Grades 9-12 NCTM Expectations for
Algebra Knowledge)
And from the National Science Education Standard; National Academy of Sciences:
UNIFYING CONCEPTS AND PROCESSES STANDARD:
As a result of activities in grades K-12, all students should develop understanding
and abilities aligned with the following concepts and processes:
• Systems, order, and organization
• Evidence, models, and explanation
• Constancy, change, and measurement
• Evolution and equilibrium
• Form and function