Download Grade: 4 Unit #4: Angle Measure and Plane Figures Time frame: 20

Grade: 4
Unit #4: Angle Measure and Plane Figures
Time frame: 20
Days
Unit Overview
In this unit, students expand their vocabulary of geometric objects and develop a connection of many of the concepts they have been
developing, including points, lines, line segments, rays, angles, parallel and perpendicular lines to classify two-dimensional figures.
Students begin to use protractors to measure and draw acute, right and obtuse angles. Once they understand the types of angles, they can
apply their knowledge to the classification of triangles. In addition, they learn how to cross-classify figures, for example, naming a shapes
as a right isosceles triangle. Students also explore line segments, rays, angles, parallelism, etc. in varied contexts to connect meaning to
what normally are isolated concepts. For example, an angle is measured with reference to a circle with its center at the common endpoint
of the rays. When working with angles, students need to be able to solve addition and subtraction problems to find unknown angles on a
diagram in real world and mathematical problems recognizing the whole angle is a sum of smaller angles composing it. They also connect
lines to lines of symmetry in two-dimensional figures.
Grade 4, Students describe, analyze, compare, and classify two-dimensional shapes by their properties (see the footnote on p. 3), including
explicit use of angle sizes4.G.1 and the related geometric properties of perpendicularity and parallelism.4.G.2 They can identify these properties in two-dimensional figures. They can use side length to classify triangles as equilateral, equiangular, isosceles, or scalene; and can
use angle size to classify them as acute, right, or obtuse. They then learn to cross-classify, for example, naming a shape as a right
isosceles triangle. Thus, students develop explicit awareness of and vocabulary for many concepts they have been developing, including
points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Such mathematical terms are useful in
communicating geometric ideas, but more important is that constructing examples of these concepts, such as drawing angles and triangles
that are acute, obtuse, and right,4.G.1 help students form richer concept images connected to verbal definitions. That is, students have
more complete and accurate mental images and associated vocabulary for geometric ideas (e.g., they understand that angles can be larger
than 90 and their concept images for angles include many images of such obtuse angles). Similarly, students see points and lines as
abstract objects: Lines are infinite in extent and points have location but no dimension. Grids are made of points and lines and do not end at
the edge of the paper.
Students also learn to apply these concepts in varied contexts (MP4). For example, they learn to represent angles that occur in various
contexts as two rays, explicitly including the reference line, e.g., a horizontal or vertical line when considering slope or a “line of sight” in
turn contexts. They understand the size of the angle as a rotation of a ray on the reference line to a line depicting slope or as the “line of
sight” in computer environments. Students might solve problems of drawing shapes with turtle geometry.• Analyzing the shapes in order to
construct them (MP1) requires students to explicitly formulate their ideas about the shapes (MP4, MP6). For instance, what series of
commands would produce a square? How many degrees would the turtle turn? What is the measure of the resulting angle? What would be
the commands for an equilateral triangle? How many degrees would the turtle turn? What is the measure of the resulting angle? Such
experiences help students connect what are often initially isolated ideas about the concept of angle.
Students might explore line segments, lengths, perpendicularity, and parallelism on different types of grids, such as rectangular and
triangular (isometric) grids (MP1, MP2).4.G.2, 4.G.3 Can you find a non-rectangular parallelogram on a rectangular grid? Can you find a
rectangle on a triangular grid? Given a segment on a rectangular grid that is not parallel to a grid line, draw a parallel segment of the same
length with a given endpoint. Given a half of a figure and a line of symmetry, can you accurately draw the other half to create a symmetric
figure?
Students also learn to reason about these concepts. For example, in “guess my rule” activities, they may be shown two sets of shapes and
asked where a new shape belongs (MP1, MP2).4.G.2
This 20-day module introduces points, lines, line segments, rays, and angles, as well as the relationships between them. Students
construct, recognize, and define these geometric objects before using their new knowledge and understanding to classify figures and solve
problems. With angle measure playing a key role in their work throughout the module, students learn how to create and measure angles, as
well as create and solve equations to find unknown angle measures. In these problems, where the unknown angle is represented by a
letter, students explore both measuring the unknown angle with a protractor and reasoning through the solving of an equation. Through
decomposition and composition activities as well as an exploration of symmetry, students recognize specific attributes present in twodimensional figures. They further develop their understanding of these attributes as they classify two-dimensional figures based on them.
Connection to Prior Learning
Grade 2, Students learn to name and describe the defining attributes of categories of two-dimensional shapes, including circles, triangles,
squares, rectangles, rhombuses, trapezoids, and the general category of quadrilateral.
They describe pentagons, hexagons, heptagons, octagons, and other polygons by the number of sides, for example, describing heptagon
as either a “seven-gon” or simply “seven-sided shape” (MP2).2.G.1 Because they have developed both verbal descriptions of these
categories and their defining attributes and a rich store of associated mental images, they are able to draw shapes with specified attributes,
such as a shape with five sides or a shape with six angles.2.G.1 They can represent these shapes’ attributes accurately (within the
constraints of fine motor skills). They use length to identify the properties of shapes (e.g., a specific figure is a rhombus because all four of
its sides have equal length). They recognize right angles, and can explain the distinction between a rectangle and a parallelogram without
right angles and with sides of different lengths (sometimes called a “rhomboid”).
Students also explore decompositions of shapes into regions that are congruent or have equal area.2.G.3 For example, two squares can be
partitioned into fourths in different ways. Any of these fourths represents an equal share of the shape (e.g., “the same amount of cake”)
even though they have different shapes.
Grade 3, Students analyze, compare, and classify two-dimensional shapes by their properties (see the footnote on p. 3).3.G.1 They
explicitly relate and combine these classifications. Because they have built a firm foundation of several shape categories, these categories
can be the raw material for thinking about the relationships between classes. For example, students can form larger, superordinate,
categories, such as the class of all shapes with four sides, or quadrilaterals, and recognize that it includes other categories, such as
squares, rectangles, rhombuses, parallelograms, and trapezoids. They also recognize that there are quadrilaterals that are not in any of the
subcategories.
Additional Cluster Standards
Draw and identify lines and angles, and classify shapes by properties of their lines and angles.
4.G.1 Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in twodimensional figures.
4.G.2 Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of
angles of a specified size. Recognize right triangles as a category, and identify right triangles.
4.G.3 Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line
into matching parts. Identify line-symmetric figures and draw lines of symmetry.
Geometric measurement: understand concepts of angle and measure angles.
4.MD.5 Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of
angle measurement:
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.
b. An angle that turns through n one-degree angles is said to have an angle measure of n degrees.
4.MD.6 Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.
4.MD.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.
Additional Cluster Standards Unpacked
4.MD.5 Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of
angle measurement:
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.
b. An angle that turns through n one-degree angles is said to have an angle measure of n degrees.
This standard brings up a connection between angles and circular measurement (360 degrees).
Angle measure is a “turning point” in the study of geometry. Students often find angles and angle measure to be difficult concepts to learn,
but that learning allows them to engage in interesting and important mathematics. An angle is the union of two rays, a and b, with the same
initial point P. The rays can be made to coincide by rotating one to the other about P; this rotation determines the size of the angle between
a and b. The rays are sometimes called the sides of the angles.
Another way of saying this is that each ray determines a direction and the angle size measures the change from one direction to the other.
Angles are 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 degrees are the unit used to measure angles in elementary school. A full rotation is thus 360º
An obtuse angle is an angle with measures greater than 90º and less than 180º. An acute angle is an angle with measure less than 90º.
Two angles are called complementary if their measurements have the sum of 90º. Two angles are called supplementary if their
measurements have the sum of 180º. Two angles with the same vertex that overlap only at a boundary (i.e., share a side) are called
adjacent angles. These terms may come up in classroom discussion, they will not be tested. This concept is developed thoroughly in
middle school (7th grade).
Like length, area, and volume, angle measure is additive: The sum of the measurements of adjacent angles is the measurement of the
angle formed by their union. This leads to other important properties. If a right angle is decomposed into two adjacent angles, the sum is
90º, thus they are complementary. Two adjacent angles that compose a “straight angle” of 180º must be supplementary.
The diagram below will help students understand that an angle measurement is not related to an area since the area between the 2 rays is
different for both circles yet the angle measure is the same.
This standard calls for students to explore an angle as a series of “one-degree turns.”
A water sprinkler rotates one-degree at each interval. If the sprinkler rotates a total of 100º, how many one-degree turns has the sprinkler
made?
4.MD.6 Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.
Before students begin measuring angles with protractors, they need to have some experiences with benchmark angles. They transfer their
understanding that a 360º rotation about a point makes a complete circle to recognize and sketch angles that measure approximately 90º
and 180º. They extend this understanding and recognize and sketch angles that measure approximately 45º and 30º. They use appropriate
terminology (acute, right, and obtuse) to describe angles and rays (perpendicular). Prior to passing out the protractor, make sure that
students can name the type of angle and can estimate the angle measurement as it relates to the type of angle. This will help students use
the correct numbers on the protractor. If the student is measuring an acute angle, he/she will know to look at the smaller numbers on the
protractor.
Students should measure angles and sketch angles
120 degrees
135 degrees
As with all measureable attributes, students must first recognize the attribute of angle measure, and distinguish it from other attributes. As
with other concepts students need varied examples and explicit discussions to avoid learning limited ideas about measuring angles (e.g.,
misconceptions that a right angle is an angle that points to the right, or two right angles represented with different orientations are not equal
in measure). If examples and tasks are not varied, students can develop incomplete and inaccurate notions. For example, some come to
associate all slanted lines with 45º measures and horizontal and vertical lines with measures of 90º. Others believe angles can be “read off”
a protractor in “standard” position, that is, a base is horizontal, even if neither ray of the angle is horizontal. Measuring and then sketching
many angles with no horizontal or vertical ray perhaps initially using circular 360º protractors can help students avoid such limited
conceptions.
4.MD.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.
This standard addresses the idea of decomposing (breaking apart) an angle into smaller parts.
Example:
A lawn water sprinkler rotates 65 degress and then pauses. It then rotates an additional 25 degrees. What is the total degree of the water
sprinkler rotation? To cover a full 360 degrees how many times will the water sprinkler need to be moved?
If the water sprinkler rotates a total of 25 degrees then pauses. How many 25 degree cycles will it go through for the rotation to reach at
least 90 degrees?
Example:
If the two rays are perpendicular, what is the value of m?
Example:
Joey knows that when a clock’s hands are exactly on 12 and 1, the angle formed by the clock’s hands measures 30º. What is the measure
of the angle formed when a clock’s hands are exactly on the 12 and 4?
Students can develop more accurate and useful angle and angle measure concepts if presented with angles in a variety of situations. They
learn to find the common features of superficially different situations such as turns in navigation, slopes, bends, corners, and openings.
With guidance, they learn to represent an angle in any of these contexts as two rays, even when both rays are not explicitly represented in
the context; for example, the horizontal or vertical in situations that involve slope (e.g., roads or ramps), or the angle determined by looking
up from the horizon to a tree or mountain-top. Eventually they abstract the common attributes of the situations as angles (which are
represented with rays and a vertex,) and angle measurements.
Students with an accurate conception of angle can recognize that angle measure is additive. As with length, area, and volume, 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. Students
can then solve interesting and challenging addition and subtraction problems to find the measurements of unknown angles on a diagram in
real world and mathematical problems. For example, they can find the measurements of angles formed a pair of intersecting lines, as
illustrated above, or given a diagram showing the measurement of one angle, find the measurement of its complement. They can use a
protractor to check, not to check their reasoning, but to ensure that they develop full understanding of the mathematics and mental images
for important benchmark angles (e.g., 30º, 45º, 60º, and 90º).
Focus Standards for Mathematical Practice
MP.2 Reason abstractly and quantitatively. Students represent angle measures within equations, and when determining the measure of
an unknown angle, they represent the unknown angle with a letter or symbol both in the diagram and in the equation. They reason about
the properties of groups of figures during classification activities.
MP.3 Construct viable arguments and critique the reasoning of others. Knowing and using the relationships between adjacent and
vertical angles, students construct an argument for identifying the angle measures of all four angles generated by two intersecting lines
when given the measure of one angle. Students explore the concepts of parallelism and perpendicularity on different types of grids with
activities that require justifying whether or not completing specific tasks is possible on different grids.
MP.5 Use appropriate tools strategically. Students choose to use protractors when measuring and sketching angles, when drawing
perpendicular lines, and when precisely constructing two-dimensional figures with specific angle measurements. They use set squares and
straightedges to construct parallel lines. They also choose to use straightedges for sketching lines, line segments, and rays.
MP.6 Attend to precision. Students use clear and precise vocabulary. They learn, for example, to cross-classify triangles by both angle
size and side length (e.g., naming a shape as a right, isosceles triangle). They use set squares and straightedges to construct parallel lines
and become sufficiently familiar with a protractor to decide which set of numbers to use when measuring an angle whose orientation is such
that it opens from either direction, or when the angle measures more than 180 degrees.
Understandings-Students will understand…
•
•
•
Shapes can be classified by properties of their lines and angles.
Angles are measured in the context of a central angle of a circle
Angles are composed of smaller angles.
Essential Questions
•
•
•
•
•
What are the types of angles and the relationships?
How are angles applied in the context of a circle?
How are parallel lines and perpendicular lines used in classifying two-dimensional shapes?
How are protractors used to measure and aid in drawing angles and triangles?
How can an addition or subtraction equation be used to solve a missing angle measure when the whole angle has been divided into
two angles and only one measurement is given?
Prerequisite Skills/Concepts: Students should already be able
to…
 Solve two-step word problems using the four operations.
Represent these problems using equations with a letter
standing in for the unknown quantity. Assess the
reasonableness of answers using mental computation and
estimation strategies including rounding. (This standard is
limited to problems posed with whole numbers and having
whole-number answers; students should know how to perform
operations in the conventional order when there are no
parentheses to specify a particular order, i.e., Order of
Operations.)

Advanced Skills/Concepts: Some students may be ready to…


Apply their knowledge of geometric attributes to sort and classify
two-dimensional and three-dimensional shapes.
Measure angles greater than 180 degrees and relate them to the
fractional part of a circle.
Skills: Students will be able to …


Understand that shapes in different categories (e.g.,

rhombuses, rectangles, and others) may share attributes (e.g.,
having four sides), and that the shared attributes can define a

larger category (e.g., quadrilaterals). Recognize rhombuses,
rectangles, and squares as examples of quadrilaterals, and

draw examples of quadrilaterals that do not belong to any of
these subcategories.

Knowledge: Students will know…

 Points, lines, line segments, rays, right angles, acute angles,
obtuse angles, perpendicular lines, parallel lines can be

identified within 2-dimensional figures. (4.G.1)
 Angles are formed wherever two rays share a common

endpoint. (4.MD.5)

 An angle measure is a fraction of circular arc between the
points where the two rays intersect the circle. (4.MD.5)

Draw points, lines, line segments, rays, right angles, acute angles,
obtuse angles, perpendicular lines, and parallel lines. (4.G.1)
Classify 2-dimensional figures based on the presence or absence of
parallel or perpendicular lines and right, acute or obtuse angles.
(4.G.2)
Identify and classify triangles. Label the categories of triangles
(right triangles, scalene, isosceles) (4.G.2)
Recognize a line of symmetry for a two-dimensional figure as a foldline, where the figure can be folded into matching parts. (4.G.3)
Determine whether a figure has one or more lines of symmetry and
draw lines of symmetry. (4.G.3)
Identify the components of an angle and the number of degrees in a
circle. (4.MD.5)
Use visuals and language to show the relationship between the
components of an angle to a circle. (i.e. the center of the circle is
the endpoint of the rays of the angle) (4.MD.5)
Measure angles in whole-number degrees using a protractor.
(4.MD.6)
Sketch angles of a specified measure. (4.MD.6)
Use diagrams, manipulatives and equations to show that angle
measure is additive. (4.MD.7)
Solve addition and subtraction problems to find unknown angles on




Benchmark angles and transfer their understanding that a
3600 rotation about a point makes a complete circle to
recognize and sketch angles that measure approximately 900
and 1800 . (4.MD.5)
An angle that turns through 1/360 of a circle is called a “onedegree angle,” and can be used to measure angles. (4.MD.5)
Angle measure is additive (4.MD.7)
A line of symmetry for a two-dimensional figure is a line
across the figure such that the figure can be folded along the
line into matching parts. (4.G.3)
a diagram of adjacent angles. (non-overlapping angles) (4.MD.7)
Transfer of Understanding-Students will apply…
Students will apply concepts and procedures of classifying two-dimensional figures based on the presence of parallel or perpendicular
lines.
Example:
How many acute, obtuse, or right angles are in this shape?
Students will apply concepts and procedures of decomposing angles into smaller parts.
Example:
A windshield wiper rotates 65 degrees and then pauses. It must rotate a total of 150 degrees to clear the windshield. What is the
remaining amount of degrees the windshield wiper must rotate to complete its rotation to clear the windshield?
Academic Vocabulary








































Ray
Right angle
Right triangle
Scalene triangle
Straight angle (angle that measures 180 degrees)
Supplementary angles (two angles with a sum of 180 degrees)
Triangle
Adjacent angle (Two angles and , with a common side ⃑⃑⃑⃑⃑⃑⃑⃑⃑⃑⃑⃑⃑⃑⃑⃑⃑⃑⃑⃑⃑⃑⃑⃑⃑ , are adjacent angles if is in the interior of .)
Angle
Arc (connected portion of a circle)
Collinear (Three or more points are collinear if there is a line containing all of the points; otherwise, the points are non-collinear.)
Complementary angles (two angles with a sum of 90 degrees)
Degree measure of an angle (Subdivide the length around a circle into 360 arcs of equal length. A central angle for any of these arcs is called a onedegree angle and is said to have angle measure 1 degree. )
Diagonal
Equilateral triangle
Figure (set of points in the plane)
Interior of an angle
Intersecting lines
Isosceles triangle
Length of an arc (circular distance around the arc.)
Line
Line of symmetry
Obtuse angle Ob
Para
Perpendicular
Point
Protractor (instrument used in measuring or sketching angles)
Vertex
Vertical angles
Decompose
Parallelogram (quadrilateral with two pairs of parallel sides)
Polygon
Quadrilateral
Rectangle
Rhombus
Square
Sum
Trapezoid
Unit Resources
Pinpoint: Grade 4 Unit #4
Connection to Subsequent Learning
Fifth grade students continue to build upon their prior knowledge of classification of two-dimensional shapes to be able to classify polygons
into a hierarchy based on their properties. They use the concept of perpendicular lines to define a coordinate system.