Translations, rotations, and reflections, used to appear as maybe one question on the California Star Test back when we taught under NCLB. Most years, I didn’t teach it. I really didn’t understand it’s value in the curriculum.

After we switched to Common Core and our school adopted the Integrated Pathway, Transformations were the area I had the most to learn about. See

It’s so funny how the more you teach a topic, the more connections you make to different topics. I taught Pre Calculus for 10 years and absolutely loved teaching students the Unit Circle. Each year, I’d find a new pattern I’d never noticed before.

My last year teaching Pre Calculus, 4 years ago, I showed my classes a bunch of sine curves and had them make a list of everything they noticed about eacH graph. I also had them predict the equation of the curve.

What do you notice? What is the equation?

Y=sin(x) they figured out but when I revealed y=sin(x/2) they gasped. They had predicted y=sin(2X). So when they started talking about what happened to the period they described the period change as 2pi/b. All by themselves.

Can you figure out why they predicted y=sin(2X)?

I had to give up Pre calculus to be a part time math coach for the past 4 years, so this past week is my first time introducing students to the unit circle since then. This is my first group of students to learn the Unit Circle who learned about transformations in Integrated 1.

Can I say that I now see the importance of learning about Translations, Rotations, and Reflections?

My students found the coordinates of points on the unit circle given an acute angle. Then, without any direction from me, just by reading a question in their CPM textbook, reflected the triangle over the x and y axis to find the points in the other three quadrants AND their angles from 0 degrees.

The textbook has them plot the points of the height of those triangles based on specific angles to initially discover the sine function and make observations about the points on the graph. Some students observed to find the other points on the sine curve with the same and opposite values, you reflect over the y-axis and translate it. And others observed the points on the sine curve were a 180 degree rotation about the point (180,0).

I’m 17 years in this profession and I’m still learning how everything is connected. My reflection is that we might not fully understand the impact of the mathematics we teach, but the significance is there and it may not appear for a few more years.

(Also, my reflection is that reflections are important. )

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