## Thursday, 25 September 2014

### Experiencing Radians with a Ruler

Students were given the following to do.

1.    Work in pairs.  Pick up 1 piece of cardboard, a circle template, 1 piece of paper tape (about 1 meter in length), and a pair of scissors.
2.    Cut out a perfect circle.  Wrap the tape around the circle.  Allow the tape to overlap.  Your partner will now cut through the overlap section with the scissors so that the ends just meet.
Question:  What measurement of the circle does this cut length of tape represent?                       _
3.    Fold the cardboard circle in four equal sections by folding in half, then in half again.  When the paper is unfolded, the creases act as spokes of the wheel.
4.    Mark your tape in “spoke” units by using one of the creases as a guide.  A spoke length (the radius) measures as one unit.   Continue to add another spoke length and another until the end of the tape.
Question:  How many spokes, or radii, will you need to measure the circumference? ____________
5.         If we assume that the radius is one unit in length (for one spoke),
a)        What is the diameter of the circle?
b)        What is the circumference of the circle?                                                                                               (Hint: Recall the formula)
6.    On the paper, mark the end of one spoke as 0.  Now use the marked tape to measure an arc one radius long along the edge of the plate.  The central angle associated with the arc whose length is one radius is considered to have an angle measure of one radian.
The term radian has been used for only a little more than a century. A radian is a relationship between the spoke of a wheel and the circumference of a wheel.  It developed as a contraction, combining the words radi-al and an-gle.  The purpose of the unit was to serve as a standard measure for angles coming from, or radiating from, the centre of a wheel.  This allowed the angle measure to be determined by the linear measure of the corresponding arc.
8.    Mark the 2pi on the plate and on the end of the tape.
9.    How many degrees do you associate with a full circle?
10.  How many radians do you associate with the circumference of the unit circle?
11.  a) Now fold the tape in half.  If the whole length of the tape is 2pi radians then half of the length is ______radians.
b) Indicate on the paper where the distance of pi radians along the circumference would occur.  Do this on
the tape as well.
c) Find pi/2 radians and 3pi/2 radians on the tape and wheel.
d)  Continue labeling the paper and the tape with the following:  pi/4, 3pi/4, 5pi/4, and 7pi/4.
Then continue with pi/6, pi/3, 2pi/3, 5pi/6,…radians.
12.   Relate your newest markings to angle measures in degrees in the table below

0
pi/6
pi/4
pi/3
pi/2
5pi/6
pi
7pi/6
4pi/3
3pi/2
5pi/3
11pi/6
2pi
Degrees

360O
Homework for this lesson:  p. 321 #2, 3, 4, 5, 14, 15

Here are some photos of the plates and rulers.

We consolidated the idea of using radians for a measure of an angle instead of degrees.