Making Sense of Radians: A New

Kind of Protractor

Jennifer SilvermanIndependent Math Consultant

Creator, ProRadian Protractors

How many students would recognize what this table of

values is?

0 1 6.5 0.980.5 0.88 7 0.75

1 0.54 7.5 0.351.5 0.07 8 -0.15

2 -0.42 8.5 -0.62.5 -0.8 9 -0.91

3 -0.99 9.5 -13.5 -0.94 10 -0.84

4 -0.65 10.5 -0.484.5 -0.21 11 0

5 0.28 11.5 0.485.5 0.71 12 0.84

6 0.96 12.5 1

It is the cosine function.

It’s not your fault.

This is why there are 360° in a circle...

50 + 9 = largest Babylonian numeral

What’s our largest numeral?

What’s our number base?What’s their number

base?

9

10

60

Babylonian Numbers: Why Base 60?

60 has many divisors

6 groups of 60 coincided with the length of the year

1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

You can count to 12 on one hand and keep track of the 12’s on the other.

But what does a degree have to do with circles?

Not much.

But, it stuck.

So, we teach our kids to measure in degrees, and give them tools to help them learn. Like this:

And life is good; they learn all kinds of things, content and secure in their knowledge of angle measure.

Until...

Imagine precalculus students, encountering their first transcendental functions...

CONFUSION

and irrational outputs...with irrational inputs...

with a new unit of measure...

And this doesn’t help much...

I thought they should have practice measuring in radians, so I went looking for a radian protractor...

but they didn’t exist.

So, I made them...

The next day, I brought in my first prototypes.

Light bulbsstarted to

go off !

CCSS.Math.Content.HSG-C.B.5

Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

CCSS.Math.Content.HSF-TF.A.1

Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

CCSS.Math.Content.HSF-TF.A.2

Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

Time to try them. Try to approach them with fresh eyes, as if this was brand new to you!

Now, this made sense!

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Other Ideas...

Don’t wait until

Precalculus

Thanks for coming!