## Reflection: Vertical Alignment Trigonometric Ratios in Right Triangles - Section 2: Data Collection and Inductive Reasoning

Statistics is part of the Geometry course now. This is one of the authentic places where I am able to discuss statistical concepts in the context of the Geometry we are doing.

The relevant concepts here are measurement error and central tendency. In this section of the lesson, each student calculates the sine ratio for their particular triangle. Given that all of the triangles should be similar, all students should theoretically obtain the exact same value for the sine ratio.

As we see from the data, though, this is not the case. So I explain to students that with any measurement process there is a random error component and possibly bias. We can think of it as follows:

Obtained Measurement = True measurement + measurement error + bias

I distinguish measurement error (bi-directional) from bias (unidirectional) and give examples to make the distinction clear.

I explain that in this situation, there should be no systematic bias so the only reason why all students didn't get the same value is because of this measurement error component.

In other words, every students true measurement was the same, but they all got different values for the measurement error (due to measurement or construction error). The nice thing, I explain is that statistics allows us to deal with this nicely.

The average of the obtained measurements = Average True Measurement + Average Measurement Error.

Because the average measurement error should be approximately zero (positives and negatives roughly cancel each other), the average of the obtained measurements is about equal to the True Measurement (assuming no bias of course).

While this is pretty crude at this point, it is good preparation for concepts that will emerge later in statistics.

By the way, this is also an opportunity to quantify the typical measurement error by discussing the standard deviation of the measurements.

So again, this type of situation provides a good connection to Statistics, which ideally should be a thread running through the Geometry Course.

Developing Statistical Reasoning
Vertical Alignment: Developing Statistical Reasoning

# Trigonometric Ratios in Right Triangles

Unit 7: Right Triangles and Trigonometry
Lesson 2 of 6

## Big Idea: Tri-this -gon for size...In this lesson, students learn basics of right triangle trigonometry .

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75 minutes

### Anthony Carruthers

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