Download Understanding angle

Applications of Angle
Measurement
Core Mathematics Partnership
Building Mathematical Knowledge and
High-Leverage Instruction for Student Success
Core Math
July 22, 2016
8:00-10:30 AM
Learning intentions and success criteria
We are learning to…
- Use concepts of angle and angle measurement to solve
real-world and mathematical problems.
We will be successful when we can…
- Recognize the use of angle measurement concepts in the
solution of real-world and mathematical problems.
- Describe the progression of angle measurement concepts
in the Common Core standards.
Big Ideas of Angle Measurement
• At your tables, review the definition of angle, and the
three big ideas of angle measurement from
yesterday.
• Be prepared to report out any questions you still
have about these concepts.
Big Ideas of Angle Measurement
• Angle measure is additive (“combining principle”).
• It is based on the iteration of the unit of degrees.
• It is the measure of the fraction of the circular arc
between the two points where the two rays intersect
a circle.
Angles on a Clock
• What is the measure of the angle between the “1”
and the “2” on a clock face?
• What is the measure of the angle between the “5”
and the “10”?
• What properties of angle measure
did you use to help you answer
these questions?
More Angles on a Clock
• What is the measure of the angle between the hands
of a clock at exactly 12:15?
• Estimate first, then find the exact
measure.
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
Angle Sum in a Triangle (Part 1)
• 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
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.
8th Grade TIMSS 2011
Geometric Shape Reasoning
Exterior Angles of a Triangle
• Place 3 objects on the floor to serve as the vertices
of a (large!) triangle or use tape to represent the
triangle.
• One member of the group walks around the
triangle; a second member stands inside the
triangle, and turns so as to always be facing in the
same direction as the walker.
C
• The rest of the group watch
the turner.
A
B
Exterior Angles of a Triangle
• What did you notice?
• In your notebook, draw a representation of what
happened to the walker and to the turner.
• Repeat the activity with a different walker and
turner, to check and refine your representation.
C
A
B
Angle Sum in a Triangle (Part 2)
• At your tables, discuss your representations
• Capture your table’s thinking on a large whiteboard.
Be prepared to share.
• What do you conclude about the exterior angles of
a triangle?
• What do you conclude
about the interior angles of
the triangle?
C
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.
Angles Made by a Transversal
• Suppose that lines AB and CD in the figure
below are parallel.
• Which angles in the diagram are equal in
measure to angle EMD?
D
E
M
B
C
N
A
F
Angle Sum in a Triangle (Part 3)
• Use your knowledge of additivity of angle measure,
and angles made by a transversal, to give an
“informal proof” of the result that the angles in a
triangle add to a straight angle.
• Hint: draw a line through
vertex C that is parallel to the
line AB, and look for
transversals.
C
A
B
Angle Sum in a Triangle (Part 3)
• Use your knowledge of additivity of angle measure,
and angles made by a transversal, to give an
“informal proof” of the result that the angles in a
triangle add to a straight angle.
• Hint: draw a line through
vertex C that is parallel to the
line AB, and look for
transversals.
C
A
B
Angle Sum in a Triangle (Summary)
• We have now seen 3 different arguments that the
measures of the three angles in a triangle add to
180O.
• At what grade level would
each of these proofs be an
appropriate activity?
C
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.
Learning intentions and success criteria
We are learning to…
- Use concepts of angle and angle measurement to solve
real-world and mathematical problems.
We will be successful when we can…
- Recognize the use of angle measurement concepts in the
solution of real-world and mathematical problems.
- Describe the progression of angle measurement concepts
in the Common Core standards.
PRR
Pull out your “Essential Understandings of
Measurement” graphic, and the “Big Ideas and
Essential Understandings” grid.
What Essential Understandings of Measurement
surfaced in this afternoon’s angle activities?
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.