Through my work each year with area and perimeter, I learn more and more about how I learned (was taught) math. I can work out a “proof” algebraically, however I struggle in connecting it conceptually to the problem. When this happens, I am so hesitant to reach out bc, truthfully, it is a bit embarrassing. I mean, I teach this stuff, right? But I finally hit a point, after I started blogging, when I learned that I will never learn more if I am not open to where I am. Since I encourage my students to write honestly about their understandings, I thought it only fair to throw my thoughts out there into the wonderful #mtbos. So here it is…
Here was last year’s example that I worked my way through: https://mathmindsblog.wordpress.com/2014/01/27/area-and-perimeter-of-squares-student-noticings/
And now here is this year’s: https://mathmindsblog.wordpress.com/2014/12/20/areaperimeter-my-homework-over-vacation/
I am finding the same thing is happening again…algebraically I have it, but struggling with the conceptual connection. I have a bunch of theories scattered on my papers this morning, but nothing that is satisfying me mathematically and would appreciate any thoughts….
So when thinking about area related to perimeter in squares, I know that n^2 x 4/n = 4n, but I am stuck at that 4/n. I marked off what I thought 4/n looked like in my squares, I messes around with ratios, found some patterns, but still not seeing (or putting together) what I want. … so I went to this drawing bc the side was increasing by 2 every time…. Then I went to this…
And while I would love to play with this for a bit longer, I have so many other things to do to get ready for school tomorrow! I feel like I have it somewhere here, I just cannot make a connection that works for me. Would love any pieces to the puzzle put together for me:)
Here is a clue: the lines inside.
4/n = 4 sides per 1 square unit?
I am not entirely sure what type of pattern or insight you are trying to find, but here are some ideas:
(1) Consider looking at Area/perimeter instead of perimeter/area. I know it should be symmetrical, but years of problems structured “what is the largest area for a given perimeter” + other constraints makes this more intuitive for many (maybe you?)
(2) Explore how this ratio works for other shapes: equilateral triangles and circles would be good places to start.
(3) Remember properties of scaling: lengths grow proportionately and areas with the square of your scaling factor.