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Mathematics 2016-17—Grade 4
Weeks 33-34—May
enVisionmath2.0—Topic 15
Standards for Mathematical Practice
Beyond the Critical Area(s): Properties of Geometric Figures
Major Work
70% of time
4.OA.A.1-2-3
4.NBT.A.1-2-3
4.NBT.B.4-5-6
4.NF.A.1-2
4.NF.B.3-4
4.NF.C.6-7
FOCUS for Grade 4
Supporting Work
20% of time
4.OA.B.4
4.MD.A.1-2-3
4.MD.B.4
Additional Work
10% of time
4.OA.C.5
4.MD.C.5-6-7
4.G.A.1-2-3
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Standards in bold are specifically targeted within instructional materials.
Required fluency: 4.NBT.B.4
Domains:
Measurement and Data
Geometry
Clusters:
Clusters outlined in bold should drive the learning for this period of instruction.
4.MD.C Geometric measurement: understand concepts of angle and measure angles.
4.G.A Draw and identify lines and
angles, and classify shapes by
properties of their lines and angles.
Standards:
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:
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.C.6 Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.
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.
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Property of MPS
4.G.A.1 Draw points, lines, line
segments, rays, angles (right, acute,
obtuse), and perpendicular and parallel
lines. Identify these in two-dimensional
figures.
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Mathematics 2016-17—Grade 4
Weeks 33-34—May
enVisionmath2.0—Topic 15
Foundational Learning
3.NF.A
3.G.A
3.MD.A
4.NF.A.1
4.MD.A
Future Learning
4.G.A.2
5.G.B
Key Student Understandings
 Students deepen their understanding of geometric measurement and the concept that some attributes
of objects are measureable and can be quantified in unit amounts (introducing degrees).
 Students understand that angles are geometric shapes composed of two rays that are infinite in length.
 Students understand that angles can be analyzed and classified by their measure.
 Students understand that angle measures, like other measurements, can be added and subtracted.
Assessments

Formative Assessment Strategies

Evidence for Standards-Based Grading
Common Misconceptions/Challenges
4.MD.C Geometric measurement: understand concepts of angle and measure angles.
 Students are confused as to which number to use when determining the measuring an angle with a protractor because most protractors have a double
set of numbers. Students should have multiple experiences estimating and comparing angles to the benchmark 90°/right angle. They should explain their
reasoning by deciding first if the angle appears to be an angle that is less than the measure of a right angle (90°) or greater than the measure of a right
angle (90°). If the angle appears to be less than 90°, it is an acute angle and its measure ranges from 0° to 89°. If the angle appears to be an angle that is
greater than 90°, it is an obtuse angle and its measure ranges from 91° to 179°. Ask questions about the appearance of the angle to help students in
deciding which set of degree measurements to use.
 Students may mistakenly interpret a “right angle” as an angle that opens to the right, relying on their understanding of “right” as a directional word. Take
time to discuss how “right” is a word with multiple meanings, including its use mathematically to describe angles that are 90o.
4.G.A Draw and identify lines and angles, and classify shapes by properties of their lines and angles.
 Students believe a wide angle with short sides may seem smaller than a narrow angle with long sides. Students can compare two angles by tracing one
and placing it over the other. Students will then realize that the length of the sides does not determine whether one angle is larger or smaller than
another angle.
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Mathematics 2016-17—Grade 4
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enVisionmath2.0—Topic 15
Instructional Practices
Domain: 4.MD
Cluster: 4.MD.C Geometric measurement: understand concepts of angle and measure angles.
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:
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.

An angle is the union of two rays, a and b, with the same initial point P (see image below, left). 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 or the arms of the angles.
Progressions for the CCSSM, K-5 Geometric Measurement, CCSS Writing Team,
June 2012, page 22
Progressions for the CCSSM, K-5 Geometric Measurement, CCSS Writing Team, June 2012, page 23
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Mathematics 2016-17—Grade 4
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enVisionmath2.0—Topic 15

Angles are geometric shapes composed of two rays that are infinite in length. Students can understand this concept by using two rulers held together
near the ends. The rulers can represent the rays of an angle. As one ruler is rotated, the size of the angle is seen to get larger. Ask questions about the
types of angles created. Responses may be in terms of the relationship to right angles. Introduce angles as acute (less than the measure of a right angle)
and obtuse (greater than the measure of a right angle). Have students draw representations of each type of angle.

Students can also create an angle explorer (two strips of cardboard attached with a brass fastener) to learn about angles. They can use the angle explorer
to get a feel of the relative size of angles as they rotate the cardboard strips around.

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.

This standard calls for students to explore the connection between angles (measure of rotation) and circular measurement (360 degrees). 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.

Students explore an angle as a series of “one-degree turns”.
o Example: A water sprinkler rotates one degree at each interval. If the sprinkler rotates a total of 100 degrees, how many one-degree turns has the
sprinkler made? (100 one-degree turns)

Students are ready to use a tool to measure angles once they understand the difference between an acute angle and an obtuse angle. Angles are
measured in degrees. There is a relationship between the number of degrees in an angle and circle which has a measure of 360 degrees. Students are to
use a protractor to measure angles in whole-number degrees. They can determine if the measure of the angle is reasonable based on the relationship of
the angle to a right angle. They also make sketches of angles of specified measure.
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Mathematics 2016-17—Grade 4
Weeks 33-34—May
enVisionmath2.0—Topic 15
4.MD.C.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.

Students should learn and use appropriate terminology (acute, right, obtuse, straight) to describe angles and rays (perpendicular, not perpendicular).
Progressions for the CCSSM, K-5 Geometric Measurement, CCSS Writing Team, June 2012, p. 23

As with all measureable attributes, students must first recognize the attribute of angle measure, and distinguish it from other attributes. 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, students may 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. (Progressions for the CCSSM, K-5, Geometric Measurement, CCSS Writing Team, June 2012, page 23)
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Mathematics 2016-17—Grade 4
Weeks 33-34—May
enVisionmath2.0—Topic 15
Progressions for the CCSSM, K-5 Geometric Measurement, CCSS Writing Team, June 2012
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.

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.

Like length, area, and volume, angle measure is additive: The sum of the measurements of adjacent angles (two angles with the same vertex that share a
ray) 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. [These
terms—adjacent, supplementary, and complementary—may come up in classroom discussion, but are not formally introduced until Grade 7.]

This standard addresses the idea of decomposing (breaking apart) an angle into smaller parts.
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enVisionmath2.0—Topic 15
o Example: A lawn water sprinkler rotates 65 degrees and then pauses. It then rotates an additional 25 degrees.



What is the total measure 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?
o Example: If the two rays in the image below (indicated by the arrows) are perpendicular, what is the value of m?
Possible solution equations:
m = 90 – (25 + 20)
25 + m + 20 = 90
o Example: Joe 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? (30 + 30 + 30 + 30 OR 30 x 4)
o Example: The five shapes in the diagram below are the exact same size. Write an equation that will help you find the measure of the indicated
angle. Find the angle measurement. (360 ÷ 5)
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Mathematics 2016-17—Grade 4
Weeks 33-34—May
enVisionmath2.0—Topic 15
Domain: 4.G
Cluster: 4.G.A Draw and identify lines and angles, and classify shapes by properties of their lines and angles.
4.G.A.1 Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional
figures.

This standard asks students to draw two-dimensional geometric objects and to also identify them in 2-D figures. This is the first time that students are
exposed to rays, angles, and perpendicular and parallel lines. Examples of points, line segments, lines, angles, parallelism, and perpendicularity can be
seen daily. Students may not easily identify lines and rays because they are more abstract.
o Examples:
 Draw two different types of quadrilaterals that have two pairs of parallel sides.
 Is it possible to have an acute right triangle? Justify your reasoning using pictures and words.
 Draw and list the properties of a parallelogram. Draw and list the properties of a rectangle. How are
your drawings and lists alike? How are they different? Be ready to share your thinking with the class.
 How many acute, obtuse and right angles are in the shape pictured at right?

Students should be able to 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, 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. 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.

Analyzing the shapes in order to construct them requires students to explicitly formulate their ideas about the shapes. For instance, what series of
commands would produce a square? How many degrees are the angles? What is the measure of the resulting angle? What would be the commands for
creating an equilateral triangle? How many degrees are the angles? What is the measure of the resulting angle? Such experiences help students connect
what are often initially isolated ideas about the concept of angle. (Progressions for the CCSSM, Geometry, CCSS Writing Team, June 2012, page 14)
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Mathematics 2016-17—Grade 4
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enVisionmath2.0—Topic 15
Differentiation
4.MD.C Geometric measurement: understand concepts of angle and measure angles.
 Encourage students to use the benchmark angle of 90o when measuring and drawing angles. Push students to explain
how they know the angle they measured/drew is less than/greater than a right angle.
4.G.A Draw and identify lines and angles, and classify shapes by properties of their lines and angles.
 Use different colors to draw or trace over shapes when decomposing them into parts (e.g., use red to identify all acute
angles, use green to identify all right angles).
 Model and encourage the use of precise language when identifying and classifying shapes—have students rephrase
their own thinking, and allow other students to repeat a peer’s thinking to develop students’ mathematical language.
Literacy Connections

Academic Vocabulary Terms

Vocabulary Strategies

Literacy Strategies
The Common Core Approach to Differentiating Instruction (engageny How to Implement a Story of Units, p. 14-20)
Linked document includes scaffolds for English Language Learners, Students with Disabilities, Below Level Students, and
Above Level Students.
Resources
enVisionmath2.0
Developing Fluency
Multiplication Fact Thinking Strategies
Topic 15 Pacing Guide
Grade 4 Games to Build Fluency
Multi-Digit Addition & Subtraction Resources
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