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Understanding Angle
Core Mathematics Partnership
Building Mathematical Knowledge and
High-Leverage Instruction for Student Success
Thursday, July 21, 2016
1:00 p.m. – 2:30 p.m.
Learning Intention and Success Criteria
We are learning to…
- Understand angles and student expectations of angle
identification and measurement.
We will be successful when we can…
- Define angles and apply the measurement process to
determine the size of angles and how angles are
constructed.
- Use the language of the standards to describe angles and
angle measurement.
Defining Angles
Getting started with angle:
On your white board, answer this question:
What is an angle?
Write a definition and draw a representation of your
thinking.
Share your thinking with your shoulder partner.
CCSSM Definition:
• An angle is a geometric shape that
is formed wherever two rays share a
common endpoint.
Where do we see angles in the
world?
Turn and talk with a shoulder partner to identify
where you see angles around the room. Be
prepared to share out (Push yourself to go
beyond right and straight angles).
Understanding the concepts of
angles and measuring angles
Applying the measurement process
to angles…
• Thinking back to the general measurement
process, what steps must we take to measure?
1.Identify an attribute
2.Define the unit of measure
3.Iterate the unit
Concepts of Angle Measurement
What do students have to understand about angles in order to begin the
measurement process?
The size of an angle is not measured by the length of its sides
because these are both rays, not segments….
4.MD.5a (first part)
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.
Concepts of Angle Measurement
What do students have to understand about angles in order to begin the
measurement process?
The size of an angle is not measured by the length of its sides
because these are both rays, not segments….
4.MD.5a (first part)
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.
Defining the Unit
• 4.MD.5a (part)
– 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.
 What is the unit of angle measure being used in this part of
the standard?
Pattern-block Angles
• CCSSM 4.MD.5a (part)
– 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.
• Find the measure of each interior angle in the triangle pattern
block, in terms of this unit (the fraction of the circular arc). Be
prepared to justify your reasoning.
Pattern-block Angles
• CCSSM 4.MD.5a (part)
– 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.
• Find the measure of each angle in each pattern block, in terms of
this unit (the fraction of the circular arc). Be prepared to justify
your reasoning.
For example: The green triangle has angles of 1/6 of a full circle
because 6 of them will fit together perfectly to make a full circle.
The measure of one of it’s interior angles is 1/6 of a full circle.
Standard 4.MD.5a (complete)
• 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.
What is the unit of angle measure being used in the last
sentence of this standard?
Standard 4.MD.5a
• 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.
What is the unit of angle measure being used in the last
sentence of this standard?
Standard 4.MD.5a & b
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.5b. An angle that turns through n one-degree angles is said
to have an angle measure of n degrees.
A 90 degree
angle has 90
parts of size
1/360.
 Use unit fraction language to describe the
measure of a 120 degree angle.
Standards 4.MD.5 & 4.MD.6
Revisit 4.MD.5 and read 4.MD.6 and fill in your
chart for the standards.
 Considering both of these standards, where
do we spend the bulk of our instructional time?
Exploring Angles within Pattern
Blocks, Part 2
What is the degree measure of each angle in an
equilateral triangle? Explain how you know.
One full turn is
360°
Since a circle can be
drawn around any
given point, we
can find the
measure of the
angles in a polygon.
360/6 = 60
Each interior
angle of the
triangle has a
size of 60
degrees.
Use your pattern blocks to
determine the degree measures
of the interior angles below.
Discuss your thinking with your shoulder partner,
make sure you are referencing the language from
the standards.
Also, consider which math practices you may highlight
in your focus when teaching this with students.
Learning Intention and Success Criteria
We are learning to…
- Understand angles and student expectations of angle
identification and measurement.
We will be successful when we can…
- Define angles and apply the measurement process to
determine the size of angles and how angles are
constructed.
- Use the language of the standards to describe angles and
angle measurement.
Comparing and Ordering Angles
• Using your cards, order angles Q through V
from the smallest to largest angle.
• As you work, talk to your shoulder partner
about how you know one angle is larger than
another.
 What misconceptions might students show in
an activity such as this?
Applying the measurement process
to angles…
1. Identify an attribute
1. The amount of
rotation in the angle
1. Define the unit of
measure
2. Degrees
2. Iterate the unit
3. Count the degrees
Bridging Conceptual Understanding of
Angles to Standard Measurement Tools
4.MD.6 Measure angles in whole-number degrees
using a protractor. Sketch angles of specified measure.
Studying the Protractors
• What do you notice about each of these
tools?
• How can you make sense of the numbers?
MP 5: Use appropriate tools strategically.
• How might you help students learn to
understand and use these tools?
Measuring angles using a protractor
• In groups of 3:
– 2 people work together to use the protractor to
measure the angle while using the language of the
standards (vertex, endpoint, ray, rotations,
degrees, circle).
– The 3rd person records the language on a sticky
note to go on the card when finished.
– When finished measuring an angle, switch roles.
What do students need to understand
about angles in order to be able to use
any protractor effectively?
• The location of the vertex (common
endpoint of the rays) is critical.
• We are measuring the degrees of
rotation from one ray to the other.
• The standard angle measurement unit
we use is degrees (in elementary school).
Constructing Angles
On the Constructing Angles Sheet, determine how you
could use one of the protractors to construct an angle
of the given size. Choose at least 2 angles from each
side of the page to construct.
Work with your shoulder partner
•Partner 1 constructs while explaining to Partner 2 the
steps of their process using the language of the
standards (vertex, ray, rotation, endpoint, degrees,
circle)
•Switch roles
Constructing Angles
• What was your shift in thinking as you went
from measuring angles to constructing them?
• What are we listening and looking for as
evidence in student understanding?
To push your thinking….
How likely is it our spinner would land in Angle
B? In Angle A?
What other misconceptions might students
have about angles and angle measurement?
Big Ideas of Angle Measurement
• An angle is the geometric shape that is formed
wherever two rays share a common endpoint.
• Angles are measured through the iteration of
a unit showing the rotation from one ray to
another.
• The measure of the angle is not affected by
the orientation of the angle or the length of
the ray (as drawn).
CCSSM Angle Standards
Geometric Measurement: Understand concepts of
angle and measure angles.
•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.
Additivity of Angles
• How many angles do you see in the figure
below?
• What relations must hold between the
measures of those angles?
C
B
M
A
Additivity of Angles
• What is the value of x in this figure?
• Explain your reasoning.
xO
60O
25O
65O
Are there Moving and Combining
Principles for Angle Measure?
Angle Sum in a Triangle
• Draw a triangle on a piece of paper, and cut it out.
• Tear off (do not cut!) the three angles of your
triangle and place them adjacent to each other to
form a single angle.
C
• What do you conclude?
A
B
CCSSM Angle Standards
Understand congruence and similarity using physical
models, transparencies, or geometry software.
•8.G.5 Use informal arguments to establish facts about
the angle sum and exterior angle of triangles, about the
angles created when parallel lines are cut by a
transversal, and the angle-angle criterion for similarity of
triangles. For example, arrange three copies of the same
triangle so that the sum of the three angles appears to
form a line, and give an argument in terms of transversals
why this is so.
8th Grade TIMSS 2011
Geometric Shape Reasoning
Learning Intention and Success Criteria
We are learning to…
- Understand angles and student expectations of angle
identification and measurement.
We will be successful when we can…
- Define angles and apply the measurement process to
determine the size of angles and how angles are
constructed.
- Use the language of the standards to describe angles and
angle measurement.
PRR: Angles
• Open to your K-5, Geometric Measurement
Progressions, page 23.
• Read the last three full paragraphs on page 23.
• On page 24, begin reading in the second full
paragraph “Students with an…” through the
end of the paragraph.
Reflect on the language you used while studying
angles and angle measurement and the
connections to linear measurement.
Disclaimer
Core Mathematics Partnership Project
University of Wisconsin-Milwaukee, 2013-2016
This material was developed for the Core Mathematics Partnership project through
the University of Wisconsin-Milwaukee, Center for Mathematics and Science
Education Research (CMSER). This material may be used by schools to support
learning of teachers and staff provided appropriate attribution and acknowledgement
of its source. Other use of this work without prior written permission is prohibited—
including reproduction, modification, distribution, or re-publication and use by nonprofit organizations and commercial vendors.
This project was supported through a grant from the Wisconsin ESEA Title II, Part
B, Mathematics and Science Partnerships.