I’ve was delighted to develop the idea of the tangent ratio in my Geometry classes using this activity called Slanty Hills created by the folks at Desmos. It approaches the tangent ratio from a very relevant perspective. It focuses on the percent grade of roads which is really just the slope of the road…which is really just the **tangent ratio**!! I’m not sure why that didn’t ever occur to me before. In other words, if a road has a 10º angle of elevation, it has a slope of ≈0.176 (*that’s the tan(10º)≈0.176*). In other words, for every 1000 feet of horizontal distance covered by the road, there is a gain of 176 feet of elevation.

Screenshot of “Slanty Hills” Desmos activity

It has been so helpful to introduce the idea of the tangent ratio being the SLOPE of a line or the slope of the hypotenuse of a triangle. All of the students know that slope is rise over run, so it makes a beautiful connection between the world of algebra and the world of geometry. Thanks Desmos for the great activity!!

Although my activity is fairly rudimentary when compared to the “Slanty Hills” activity, I’m still very pleased with it. It sneakily introduces the unit circle definition of the sine and cosine (although that’s not a focus of my geometry students). I particularly like that students have a few opportunities to do a card sort to check their understanding.

My Desmos Activity – Sine and Cosine Ratios

Card Sort Activity

Unit Circle Definition of the Sine

I’m going to try this activity out with my students next week! I’m excited, and I’ll let you know how it goes.

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