Students learn to draw and identify points, rays, segments, angles, and lines, including parallel and perpendicular lines. Students also learn how to use a protractor to measure angles and to draw angles of given measurements, and they identify acute, obtuse, right, and straight angles in two-dimensional figures.
Unit Narrative
In this unit, students deepen and refine their understanding of geometric figures and measurement.
In earlier grades, students learned about two-dimensional shapes and their attributes, which they described informally early on but with increasing precision over time. Here, students formalize their intuitive knowledge about geometric features and draw them. They identify and define some building blocks of geometry (points, lines, rays, and line segments), and develop concepts and language to more precisely describe and reason about other geometric figures.
Jada says Figure A shows an angle,
but Figure B does not. Do you agree?
Students analyze cases where lines intersect and where they don’t (for example, parallel lines). They learn that an angle is a figure composed of two rays that share the same starting point.
Later, students compare the sizes of angles and consider ways to quantify the comparison. They learn that angles can be measured in terms of the amount of turn one ray makes relative to another ray that shares the same vertex.
Students learn that a 1-degree angle is of a full turn or full circle and can be used to measure angles. They use a protractor to measure angles in whole-number degrees.
Students also learn that angles are additive. When an angle is composed of multiple non-overlapping parts, the measure of the whole is the sum of the angle measures of the parts. These insights enable students to classify angles (as acute, obtuse, right, or straight) and to solve problems about unknown angle measurements in concrete and abstract contexts.
How many degrees is each marked angle on the clock? Show your reasoning.
Draw and identify points, lines, rays, segments, and parallel and intersecting lines in geometric figures.
Recognize that angles are formed wherever two rays share the same starting point, and identify angles in two-dimensional figures.
Section Narrative
This section introduces students to some building blocks of geometric figures and the language to describe them. Students start by describing images that contain lines for others to draw, and draw images by relying only on others’ descriptions. The experience motivates a need for more precise vocabulary to describe geometric parts. Students learn to distinguish points as locations in space, rays as lines that are bounded by one point, and line segments as lines that are bounded by two points.
Students are familiar with lines that cross or intersect. Here, they identify and then draw parallel lines, lines that never intersect.
Students also learn that an angle is a figure that is made up of two rays that share the same starting point, called the “vertex” of the angle. They then practice identifying angles, noticing that angles are ubiquitous around us and can have different sizes.
Recognize that angles can be measured in degrees and found, using addition and subtraction.
Use a protractor to measure and draw angles, and recognize that perpendicular lines create 4 right angles.
Section Narrative
In this section, students learn two main ideas: that angles can be measured, with degrees () as the unit of measurement, and that angles can be composed and decomposed, and are therefore additive. They also learn to use a protractor to measure and draw angles.
Students begin by comparing angles visually and exploring ways to describe their sizes. They then try to describe angles made by the hour and minute hands of an analog clock, using the numbers and tick marks on the clock or units of time to quantify the size of an angle. This experience reinforces the idea of an angle as a figure formed when a ray rotates around a vertex shared with another ray. It also motivates the need for a more precise unit when measuring angles.
Students learn that a ray that rotates a full turn around a point makes a angle. Decomposing this angle into halves gives a angle. Half of that angle is a angle or a right angle. Composing three angles gives a angle.
Students then use these benchmark angles to estimate and measure the sizes of other angles. For example, decomposing a right angle into halves gives angles. Composing three copies of a angle makes a angle, and so on.
Students also learn that angles are formed by perpendicular lines.
Later, students make sense of a angle and see that it is of a full turn. They use a protractor and as a unit for measuring and drawing angles of all sizes.
How many degrees is this angle?
Explain how you know.
An angle contains thirty 1° angles, as shown.
How many degrees is this angle?
Throughout the section, students build their understanding of angles of different sizes, using tactile tools such as paper cutouts and patty paper, and by folding, cutting, marking, and assembling pieces of paper.
Draw and identify acute, obtuse, right, and straight angles in two-dimensional figures.
Write equations to represent angle relationships, and reason about and find unknown measurements.
Section Narrative
In this section, students continue to draw and analyze angles and reason about their measurements.
They first classify angles by size and identify acute, obtuse, and straight angles. Then they further develop the idea that angles are additive by composing and decomposing angles, using tactile tools and drawings, and writing expressions or equations to support their reasoning.
Students solve problems about angles in different contexts, both concrete and abstract. They use their understanding of a right angle and a straight angle to reason about unknown angle measurements.
Find the measurement of each shaded angle. Show how you know.
4 angles. A. Right angle partitioned into two angles. One shaded green, one labeled 62 degrees. B. Straight angle partitioned into 3 angles. One labeled 71 degrees, one shaded yellow, one marked with a right angle symbol. C. Horizontal straight line partitioned into 2 angles on top by a ray and 2 angles below by another ray. On top, left angle shaded blue, right angle labeled 1 hundred 8 degrees. On the bottom, left angle labeled 1 hundred 54 degrees, right angle shaded red. D. Two straight lines intersect and partitioned into 2 angles on top and 2 angles below. On top, smaller angle on the left shaded green and bigger angle on the right unlabeled. On the bottom, bigger angle on the left shaded yellow, smaller angle on the right, labeled 43 degrees.
