Mary and Claire explore what it means for one angle to be more open than another. They also investigate how to measure that openness using both degrees and non-standard units.
Mary and Claire show images of angles they found in their everyday lives. Their images include both static angles and those that change over time, like the hand on a clock.
The students are asked to arrange eight angles in order from most open to least open. After they have determined an initial ordering, they check their work by tracing the angles onto patty paper and comparing them directly by overlaying one on top of the other.
Mary and Claire estimate the measure of several angles in degrees. They do this by decomposing the angles into smaller angles that seem familiar (e.g., 90 degrees, 60 degrees, etc.)
Mary and Claire describe openness. They also compare and contrast gips with degrees.
Mathematics in this Lesson
Targeted Understandings
This lesson can help students:
Build on prior knowledge of angles to conceive of angle measure as measuring openness.
Think of the measure of an angle as the number of times a unit angle needs to be iterated.
Understand the meaning of one degree as 1/360 of a full rotation.
Use a nonstandard unit to measure openness.
Common Core Math Standards
CCSS.MATH.CONTENT.4.MD. C.5. Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement.
In this lesson Mary and Claire use their surroundings to look for and recognize angles as geometric shapes. They describe where they see those angles in everyday life and contrast those with the image of an angle formed by two rays.
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.
In this lesson the students discuss how 360 degrees partition the area within a circle into 360 slices. They also grapple with the idea that partitioning a circle into 360 degrees is somewhat arbitrary and that we can imagine other ways to partition circles to create a unit angle measure. They compare degrees with a non-standard unit of measure, gips, which partition circles into eight equal sections.
CCSS.MATH.CONTENT.4.MD. C.7. Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.
Mary and Claire estimate the measure of several angles by decomposing into, and composing with, smaller familiar angles. For example, when comparing two angles, the students notice that one angle appears to be slightly larger than a right angle, while the other angle seems to be slightly smaller than a right angle.
According to the CCSSM, “Mathematically proficient students bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically….and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.” Early in this lesson, Claire and Mary examine photographs of objects from their daily lives and decontextualize elements of those objects by identifying and quantifying angles [Episode 1, 1:15]. Later, they work with a new unit of measure, gips, and consider carefully first how to represent gips [Episode 4, 1:25] and then how to use gips to measure angles [e.g., Episode 4, 2:45]. They also construct angles of a given measure [e.g., Episode 5, 0:15]. The students use properties of circles and radii to better understand and utilize gips for measuring and constructing angles.
