I was curious how my Geometry students would handle Fawn’s Staircases and Steepness Task. I will be honest, since it was high school, and slope is no longer an introductory concept, I was afraid it would be too simple…
I was wrong.
The discussion and sharing were so worthwhile! This is a keeper task. I shared it with other geometry teachers, fingers crossed they’ll give it a try!
About 1/3 of students used protractors. When asked why they chose to measure angles…some replied it just made sense (was that intuitiveness coming through?), one student stated it took the math out of it? Huh? He explained if he had chosen to measure height and base, there was more of a chance of making a mistake…twice because there are 2 measures.
Was it because it was open, choose your own measure tool/strategy, that allowed them to think without it being so ‘mathy’???
1/3 of students measured the segment (hypotenuse) length. But when asked if it confirmed their rankings, several realized they needed to adapt their plan.
The remaining students used a classic slope height/base. Some wondered if measuring each step would result in the same value as the entire staircase.
Here are measures shared by 3 students. The top angle measures confirmed her ranking, but a classmate wondered “if its least steep to steepest, it would make more sense to me for the steeper to have the larger angle measure.” So the discussion led, where did she get her measures and how are they related to student CWs?
There are some errors in measurements shown. But what made this task so great, they wanted to know who was right. And were furious when I would not tell them. I told them I didn’t have a key, that they needed to revisit their measures and be ready to defend their rankings with measures that confirmed. Could they critique their classmates reasoning?
With a bit more sharing, they all agreed there was a relationship between the rise/run, step height/step base and the angle measures. I asked if they had ever heard of Trig Ratios. Some said yes, in 8th grade, so hard! Others stated it sounded difficult.
What is trigonometry anyway? Lets break it down. Tri-gono-metry. They recognized tri as 3 and metry as a measure, but gono is from gonia which is angle…3 angle measures…hello! That has to do with triangle measures! Connecting it back to a sketch a student shared and another said there’s mini triangles in each step of the staircase.
Anyway, I am rambling, but I shared the idea of tangent and explained that it is simply the ratios they used to measure their steepness. We did a few examples, connecting to angle measures- using 45º as a reference. Thinking of our angle as a hinge on a door and looking at different ratios for different angles.
I hope to pull some of their examples and share more in a later post. But when a student tells you thank you because you made it simple for them to “see it”…that makes it worthwhile. In reality, I didn’t make it simple. It was already simple. I only provided a task (thanks, Fawn!), that helped them see the connection for themselves.