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OBJECTIVE - G.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.

INTERPRETATION OF OBJECTIVE - G.C.B.5

This objective introduces radians as a way to measure angles. We begin by looking at the proportional nature of circles and how that allows radians to be a construct measurement. From there we look at uses of radians and how they can simplify many of the basic circle formulas for arc length and area.

SKILLS

(1) The student will be able explain what a radian is.

(2) The student will be able to convert between degrees and radians.

(3) The student will be able to derive and use the formula for arc length in terms of radians.

(4) The student will be able to derive and use the formula for area of a sector in terms of radians.

THE BIG IDEA

Radians while very new to these students are a very powerful concept for the future. Much of upper mathematics uses radians instead of degrees. When we look at angle based function we prefer the domain to be measurement (distance) based instead of angle based. Angles defined by a length provide a more consistent way to graph on axis based on length.

TRAPS & PITFALLS

This can be a difficult area. Students are often intimidated by the use of pi and how they are unable to visualize how big something in radian measure is... whereas they handle degrees so easily.

Teachers often teach to convert to degrees immediately and then don't worry about radians. To me this is backwards, one of the reasons we are introducing it now is that they become more familiar with radians and begin to understand their size values.

Also arc length and area formulas become much simpler when working in radians. We want to emphasize this.

PAST CONNECTIONS

We connect similarity to circles which allows us to understand why radians can be used as a measurement for angles. Every circle is similar and thus proportional. Similarity is a powerful concept concerning why radians work.

FUTURE CONNECTIONS

Radians become the prefered way to measure angles in upper mathematics and so here we are laying the groundwork of understanding.

MY REFLECTIONS (over line l)

It is so much fun to teach this at this level. It is not too difficult for them to understand and it opens their eyes to other ways to measure. I must admit I spend a bunch of time at the very beginning working on estimating the size of things in radians. It is such a new measurement that they have no idea about how big Π/5 is. We do lots of fraction talk... and relationships based on 180 degrees and Π. It seems to pay off and they get less intimidated by pi and the new way of measuring. I also focus on exact answers earlier when working with volume because I want them to feel comfortable with writing and using pi.