Mary and Claire explore how to create and measure angles that represent the amount an object has rotated. They measure the angles using both degrees and a non-standard unit of measure. They then reflect on what those measurements mean in terms of the rotation.
The students describe how they would use degrees and gips to measure how much a student has rotated. They also reflect on how to measure rotation and what various units of measure mean.
Mathematics in this Lesson
Targeted Understandings
This lesson can help students:
Understand that angles can be thought of as describing a portion of a full rotation.
Think of one degree as 1/360 of a full rotation.
Measure angles that represent portions of a full rotation in non-standard units.
Common Core Math Standards
CCSS.MATH.CONTENT.4.MD. C.5.A. An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a “one-degree angle,” and can be used to measure angles.
Mary and Claire consider the measure of angle as a portion of rotation. They build on their understanding from the previous lesson that the measure of an angle is the number of times a unit angle is iterated. In this lesson, they also think about that unit as an amount of rotation.
CCSS.MATH.CONTENT.4.MD. C.5.B. An angle that turns through n one-degree angles is said to have an angle measure of n degrees.
Throughout this lesson, Claire and Mary use a self-made protractor to estimate the measure of angles. The protractor is made of circular patty paper and seems to be a powerful tool for making sense of angle measures (particularly as a fraction of a circular arc) with a variety of different units of measure, including degrees, gips, and “turns” or full rotations.
According to the CCSM, “Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software….They are able to use technological tools to explore and deepen their understanding of concepts.” This lesson features several tasks that provide opportunities for Mary and Claire to make sense of the openness of angles. They do this in multiple ways, using several tools at their disposal. In Episode 1, they use dynamic geometry software to create circles to help them frame their drawing of a student rotating counterclockwise [e.g., 2:02]. In Episodes 2 and 3, they make use of patty paper to create their own protractor, using it first to estimate degree measures of angles [e.g., Episode 2, 2:25], then to estimate what fraction of a full turn an angle is [e.g., Episode 2, 7:10], and finally to estimate the angle measure in gips [e.g., Episode 3, 0:44]. They also deftly utilize different units of measure as they explore the idea of amount of rotation as a viable way for measuring angles.
