SCMP October 12, 2010
MEASURING ANGLES
Angle measurement is often difficult or confusing for students because they do not understand the attribute of angle size. Some students think the length of the rays that are the sides of the angle determine the size of the angle. The idea of using one angle to measure other angles was usually poorly developed for them and protractors were introduced and used without understanding how they work. We are going to look at an approach that minimizes these difficulties. Our plan of instruction will progress from comparison activities to informal development of units and, finally, to the use of measurement tools.
Comparing angles
The attribute of angle size can be called the “rotation between the angle’s two rays.” In the summer when we used an opening door to model a growing angle, we saw how the opening or rotation of the terminal ray made the angle larger, not the length of the rays that formed the sides of the angle. Angles are composed of two rays that are infinite in length and share a common endpoint or vertex. The only difference in their size is how widely or narrowly the two rays are spread apart.
In a effort to help students understand the attribute of angle size, one approach is have them think about how much one ray has rotated away from the other. The downfall of this approach is that when we see angles represented on paper, the rays have already been spread and there is not any apparent rotation.
Another approach to help students conceptualize the attribute of the spread of the rays is to directly compare two angles by tracing one and placing it over the other. Be sure to have students compare angles with sides of different lengths to help students realize the lengths of the sides of the angle do not determine the angle size. A wide angle with short sides may seem smaller than a narrow angle with much longer sides. This is a common misconception. (Activity 1)
As soon as students can tell the difference between a large angle and a smaller one, regardless of the lengths of the sides, you are ready to move on to measuring angles.
Using units of Angular Measure
A unit for measuring an angle must be an angle. Nothing else has the same attribute of spread that we want to measure. Measuring an angle is the same as measuring length or measuring area. Unit angles are used to cover or fill the spread of the angle just like unit squares fill or cover the area of a rectangle. (Activity 2)
As soon as this concept is well understood, you can move on to the use of measuring instruments.
Making a Protractor
The protractor is one of the most poorly understood measuring instruments found in schools. Part of the difficulty is the units (degrees) are so small. It would be physically impossible to cut out a single degree and use it to measure an angle like in the previous activity. Another difficulty is there are no visible angles showing on a protractor. There are only little marks around the outside edge of the protractor and then the numbering goes in two different directions, clockwise and counterclockwise, on the marked edges. All these mysterious features can be understood if we begin by making a protractor with a large unit angle. A careful comparison with a standard protractor will then permit the instrument to be used with understanding. (Activity 3)
Our patty paper protractor serves as an excellent protractor with a unit angle that is one-eighth of a straight angle. It can be placed over an angle drawn on paper on the document camera or on an angle drawn on a transparency on the overhead projector to measure the angle. The paper protractor makes it quite clear how a protractor fits unit angles into the angle for measurement. Students can estimate halves, thirds, or fourths of a “wedge,” a possible name for this informal unit. Figure 8.7 illustrates how a tagboard semicircle can be made into a protractor to measure angles in wedges. This tagboard version is closer to a standard protractor since the rays do not extend down to the vertex and the markings are numbered in two directions. The only difference between this protractor and the standard one is the size of the unit angle. The standard unit angle is the degree and it is just a very small angle. An analogy between wedges and degrees and between this tagboard protractor and the standard one is a very effective approach for teaching the meaning of degree.
Encourage students to use benchmark angles such as right angles, straight angles, and half right angles (45 degrees.) Familiarity with these angles will make it less likely the students will misread their protractors and note an angle measures 150 degrees when the angle is really smaller than a 45 degree angle and should have been measured as a 30 degree angle.
Adapted from Van de Walle, Teaching Student-Centered Mathematics
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