An angle is….
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A) Which is larger: angle 1 or angle 2?
Why?
1
2
B) Which is larger: angle 3 or angle 4?
Why?
4
3
C) Which is larger: angle 5 or angle 6?
Why?
5
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6
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Measuring Angles (Teacher Notes)
Two reasons that cause difficulty in angle measurement:
students do not understand the attribute of angle size (the
“spread of the angle’s rays”) and protractors are
introduced and used without developing understanding.
Van de Walle suggests 3 different types of activities before
using a protractor:
1. Comparing angles
2. Using a wedge (small angle) as a unit to measure
3. Making a protractor
Our first measuring activity is designed for students to
compare angles
To help students understand the attribute of the spread of
the rays, have them compare two angles by tracing one
and placing it directly over the other. Be sure to have
them compare angles with sides of different lengths. A
common misconception for students is that a wide angle
with short sides is smaller that a narrow angle with long
sides. Students need to be able to tell the difference
between large angles and small angles (regardless of the
length of the sides) before you move to measuring angles.
A set of 6 cards are on the next page – have the student
cut out the cards and place them in order from smallest to
largest. One differentiation strategy is to have the sides of
the angles marked with wikki sticks, spaghetti, yarn, etc.
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Cut these cards apart and arrange the angles from smallest to largest.
A
B
D
C
F
E
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Teacher notes for using a wedge
Using a Unit of Angular Measure
A unit for measuring angles must be an angle! You can
supply the small units (wedge) for your students or have
them make their own. If you use a small triangle, students
will sometimes get confused as to which angle they are to
use as their unit – we suggest that you curve one side
and/or mark the angle that is the unit. A piece of
cardstock (index card or old file folder) makes a good
wedge.
Have them use this wedge as a unit of angular measure
by counting the number of times it will fit in a given angle.
Hand out a worksheet of assorted angles and have each
student use their unit to measure the angles. Because the
students made their own unit, the results will be different –
this is a great discussion!
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Use your wedge to measure each angle
A
B
C
D
Stop! Discuss with your results with talking partners.
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Teacher notes for making a protractor
The protractor is one of the most poorly understood
measuring tools found in schools. Part of the difficulty is
because the units (degrees) are so very small. It is
physically impossible to cut out a single degree and use it.
A second problem is that there are no visible angles
showing on the protractor – just little marks around the
outer edge. To add to the confusion, the numbers on
most protractors run both directions (clockwise and
counter clockwise) along the marked edge. Making a
protractor with a larger unit helps to clear up all of these
mysterious features.
To make a protractor, start with about 10-12 inches of
ordinary wax paper. Fold it in half and crease the fold
tightly. Fold in half again so that the folded edges match.
Repeat this two more times, each time bringing the folded
edges together and creasing tightly. Cut or tear off the
resulting wedge shape about 4-5 inches from the vertex.
Unfold. There should be 16 angles surrounding the
center. This serves as an excellent protractor with the unit
angle being one-eighth of a straight angle.
Use your protractor to measure the angles on the previous
page.
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Teacher Notes for Radian Measurement:
We have been measuring angles and angle rotation in
degrees. One rotation of a complete circle is 360o.
Angles can also be measured in radians.
Radian measure uses the length of the arc through
which a point on the terminal side of an angle would
trace as the terminal side of the angle rotates.
The angle at the
left ‘cuts off’ an
arc of the unit
circle whose arc
length is the
measure of that
angle in radians.
B
O
A
What is a radian?
If the measure of segment OA equals the length of arc AB,
then angle BOA measures one radian.
One complete rotation about the origin equals 360
degrees. When measuring in radians, one complete
rotation about the origin is the circumference of the circle
(2r). If the radius is one unit, then one complete rotation
is 2(1) or 2 radians.
Therefore 360o = 2 radians.
180º = ______radians
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and
90º = ______radians
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Complete the following (remember that 360º = 2 radians:
1.
45º = ______radians
2.
30º= ______radians
3.
60º = ______radians
4.
120º = ______radians
5.
270º = ______radians
6.
135º = ______radians
7.
150º = ______radians
8.
210º = ______radians
Stop! Discuss with your results with talking partners.
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