Today, I was chatting with a homeschooled high schooler. Intrigued by the enthusiasm of elementary students with curve stitching, she asked me to find time this week to show her how to do the drawings. I showed my captain idea log to her and floated a balloon about my theory about coordinate systems. She excels in her math classes and, like many, finds little joy in math. I told her that my idea is for interested students to work on these designs in their elementary years and to learn about coordinate systems and how to graph them in their middle school years. Cartesian for angular frames and polar for circular frames.

She protested, “That isn't fair! Why didn't you and mama start a school when I was their age so that I could have enjoyed learning these things?"

She protested, “That isn't fair! Why didn't you and mama start a school when I was their age so that I could have enjoyed learning these things?"

Last night, I pondered this as I looked at the heart design in yesterday's blogpost. How would it look if I centered four hearts around the origin (where the x-axis and y-axis meet)? Isn't this a much more interesting way to introduce the concept of reflection around an axis? Generally, reflection gets crammed into Algebra 2 or Precalculus.

The picture below shows a curve stitching design of reflected hearts.

The picture below shows a curve stitching design of reflected hearts.

Back to the question about introducing concepts earlier, but in a more meaningful and intriguing way. I wonder about a progression like this:

- Hands-on introduction in the primary years through folding paper and cutting (snowflakes, paper dolls, etc.),
- Drawing and creating in the elementary years through paper sloyd, curve stitching, and other design projects that have the element of reflection,
- Graphing their designs in the computer in both coordinate systems and figuring out how the coordinates change when doing reflections during the middle school years,
- Figuring out how to alter equations to make a reflection in high school.

To explore the idea of reflection, I folded a square so that cutting half a heart yields the four hearts as imagined in my mind. It took me a couple of tries before I succeeded. Then, I replicated it on a square with a Cartesian coordinate system to help me all the reflections involved. You can download the document, print it, and cut out the square to see if you can figure it out.

fourhearts.pdf |

There are four major lines around which the half heart is reflected to make four. You only need to figure out three. Two are fairly easy to those who still have the basics of graphing in their heads. The other two are trickier. You don't need to worry about names, yet. That can come later.