Private: Learning Math: Measurement
Angle Measurement Part A: Angle Definition (20 minutes)
Session 4, Part A
Since the time of the ancient Greeks (about 500 B.C.E.), angles have been measured in terms of a circle. In fact, about 1,000 years earlier, the Babylonians first divided the circle into 360 equal parts for astronomical purposes, providing a convenient unit — the degree — for expressing the measure of any angle. A degree can be further divided into 60 minutes, and a minute can be divided into 60 seconds. This level of detail is frequently seen in longitude and latitude, but almost never in school mathematics.
While degrees are the most commonly used units of angle measure, there are also other units. For example, angles are sometimes measured in radians in order to simplify certain calculations. The radian measure is defined in the International System of Units (SI) as the ratio of arc length to the radius of the circle. For 1 radian, the arc length is equal to radius:
An angle can be defined as the union of two rays with a common endpoint. (A ray begins at a point and extends infinitely in one direction.) The common endpoint is called the vertex (A in the figure below), and the rays are called the sides of the angle.
It’s customary to name an angle using an angle sign () followed by three letters: one that corresponds to a point on one of the rays, a second that corresponds to the vertex, and a third that corresponds to a point on the other ray. When it’s clear from the context, though, you can just use the letter for the vertex.
For example, you could name the angle illustrated below BAC (read “angle BAC”), CAB (read “angle CAB”), or A (read “angle A”). Point A is the vertex, and rays AB and AC are the sides:
Here’s an interesting fact: The two rays that define the angle can be on the same line. There are two ways this can happen:
While these angles may look much the same in the diagram, they are quite different geometrically:
- ∠PQR is an angle whose sides are opposite rays. This type of angle is called a straight angle.
- ∠QPT is an angle whose sides (PQ and PT) are coincident. This type of angle is called a zero angle.
When we measure an angle, no matter how it is classified, what we’re measuring is the amount of turn. This raises an important question: Do the lengths of the sides of an angle in any way affect the measure of that angle?
To examine this question, make an informal protractor using bendable straws (see the picture below): Note 2
Use your protractor to demonstrate how angles are the result of the amount of turn by forming acute, obtuse, and right angles:
Try changing the lengths of the sides on your protractor by adding more straws or cutting the original straw.
What happens to the angle when you do this? Do the lengths of the sides of an angle affect the measure of that angle?
When you use a straw protractor to show an angle, you can see that there are in fact two angles displayed — an angle inside the rays (the interior angle) and an angle outside the rays (the reflex angle):
The measure of the reflex angle is between 180 and 360 degrees. The interior and reflex angle measures sum to 360 degrees.
To make an informal protractor from bendable straws, first take one straw and make a slit in the short section all the way from the end to the bendable portion. Cut the second straw so that you only have the long section (after the bend). Finally, compress the short section of the first straw (which you cut) and fit it into one end of the second cut straw. You now have a simple protractor.
Nothing happens to the angle. The lengths of the sides of an angle do not affect the measure of that angle.
Session 1 What Does It Mean To Measure?
Explore what can be measured and what it means to measure. Identify measurable properties such as weight, surface area, and volume, and discuss which metric units are more appropriate for measuring these properties. Refine your use of precision instruments, and learn about alternate methods such as displacement. Explore approximation techniques, and reason about how to make better approximations.
Session 2 Fundamentals of Measurement
Investigate the difference between a count and a measure, and examine essential ideas such as unit iteration, partitioning, and the compensatory principle. Learn about the many uses of ratio in measurement and how scale models help us understand relative sizes. Investigate the constant of proportionality in isosceles right triangles, and learn about precision and accuracy in measurement.
Session 3 The Metric System
Learn about the relationships between units in the metric system and how to represent quantities using different units. Estimate and measure quantities of length, mass, and capacity, and solve measurement problems.
Session 4 Angle Measurement
Review appropriate notation for angle measurement, and describe angles in terms of the amount of turn. Use reasoning to determine the measures of angles in polygons based on the idea that there are 360 degrees in a complete turn. Learn about the relationships among angles within shapes, and generalize a formula for finding the sum of the angles in any n-gon. Use activities based on GeoLogo to explore the differences among interior, exterior, and central angles.
Session 5 Indirect Measurement and Trigonometry
Learn how to use the concept of similarity to measure distance indirectly, using methods involving similar triangles, shadows, and transits. Apply basic right-angle trigonometry to learn about the relationships among steepness, angle of elevation, and height-to-distance ratio. Use trigonometric ratios to solve problems involving right triangles.
Session 6 Area
Learn that area is a measure of how much surface is covered. Explore the relationship between the size of the unit used and the resulting measurement. Find the area of irregular shapes by counting squares or subdividing the figure into sections. Learn how to approximate the area more accurately by using smaller and smaller units. Relate this counting approach to the standard area formulas for triangles, trapezoids, and parallelograms.
Session 7 Circles and Pi (π)
Investigate the circumference and area of a circle. Examine what underlies the formulas for these measures, and learn how the features of the irrational number pi (π) affect both of these measures.
Session 8 Volume
Explore several methods for finding the volume of objects, using both standard cubic units and non-standard measures. Explore how volume formulas for solid objects such as spheres, cylinders, and cones are derived and related.
Session 9 Measurement Relationships
Examine the relationships between area and perimeter when one measure is fixed. Determine which shapes maximize area while minimizing perimeter, and vice versa. Explore the proportional relationship between surface area and volume. Construct open-box containers, and use graphs to approximate the dimensions of the resulting rectangular prism that holds the maximum volume.
Session 10 Classroom Case Studies, K-2
Watch this program in the 10th session for K-2 teachers. Explore how the concepts developed in this course can be applied through case studies of K-2 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for K-2 students.
Session 11 Classroom Case Studies, 3-5
Watch this program in the 10th session for grade 3-5 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 3-5 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for grade 3-5 students.
Session 12 Classroom Case Studies, 6-8
Watch this program in the 10th session for grade 6-8 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 6-8 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for grade 6-8 students.