1. 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

2. 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.
Suzette
2 minutes

3. 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.
Relate this back to moving and combining principle. Speak about iterating units, but not necessarily nuits of degrees.

4. 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.
Relate this back to moving and combining principle. Speak about iterating units, but not necessarily nuits of degrees.

5. 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”?
Relate this back to combining principle. (In general, for all angle measure activities, stress relations back to the “Big Ideas”, and especially the moving and combining principles.
If there is interest and time, give more examples of the same type, with different times of day.
What properties of angle measure did you use to help you answer these questions?

6. More Angles on a Clock
What is the measure of the angle between the hands of a clock at exactly 12:15?
Relate this back to moving and combining principle. Speak about iterating units, but not necessarily nuits of degrees.
Estimate first, then find the exact measure.

7. 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.
Fill out sheet on standard

8. Additivity of Angles How many angles do you see in the figure below?
What relations must hold between the measures of those angles?
Bring up the fact that there are 3 angles in the picture. C B M A

9. Additivity of Angles What is the value of x in this figure?
Explain your reasoning.
Participants may or may not write an equation for x. They should be asked to do so in any case, after having solved the problem. xO
65O
60O
25O

10. 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? B A

11. 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.
We’re not doing the grade 8 standard yet on the sum of interior angles, the focus is on seeing angles as additive

12. 8th Grade TIMSS 2011
Geometric Shape - Reasoning

13. 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.
Facilitators should model the process first. C
The rest of the group watch the turner. B A

14. 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 B A

15. 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?
See the 360 from the person in the center most easily
Because if you take 3 exterior angles, plus the 3 interior angles you will see 3 half turns.
What do you conclude about the interior angles of the triangle? C B A

16. 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.
Note: although this is a Grade 8 standard, the foundations are laid already in Grade 4, and explicit connections in Grades 5 through 7 are relatively weak.
We will discuss congruence in the CCSSM next week.

17. 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
Most of the approaches participants use here should be transformational: translating line AB to lie on top of line CD, or rotating about the midpoint of segment MN. B C N A F

18. 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.
After having constructed the “standard” proof of the angle addition theorem, ask participants what connections they see to “Part 1” of this series: it is really just the same proof of placing the three corners adjacent to each other.
Hint: draw a line through vertex C that is parallel to the line AB, and look for transversals. C B A

19. 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.
After having constructed the “standard” proof of the angle addition theorem, ask participants what connections they see to “Part 1” of this series: it is really just the same proof of placing the three corners adjacent to each other.
Hint: draw a line through vertex C that is parallel to the line AB, and look for transversals. C B A

20. 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.
After having constructed the “standard” proof of the angle addition theorem, ask participants what connections they see to “Part 1” of this series: it is really just the same proof of placing the three corners adjacent to each other.
At what grade level would each of these proofs be an appropriate activity? C B A

21. 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.
Note: although this is a Grade 8 standard, the foundations are laid already in Grade 4, and explicit connections in Grades 5 through 7 are relatively weak.
We will discuss congruence in the CCSSM next week.

22. 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.
Suzette
2 minutes

23. 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?

24. Disclaimer Core Mathematics Partnership Project
University of Wisconsin-Milwaukee,
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 non-profit organizations and commercial vendors.
This project was supported through a grant from the Wisconsin ESEA Title II, Part B, Mathematics and Science Partnerships.