1. 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.

2. 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.
2 minutes

3. Defining Angles
(keep in your mind as we go on)

4. 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.
5 min (chart out for adding to later)
CCSSM defines angle as: a geometric shape that is formed wherever two rays share a common endpoint
Purpose: participants self-assess their understanding of an angle through development of angle definition and representation
(whole group share out prior to moving to next slide)
Anticipated response: a corner, a vertex, where two lines come together at a point
Share your thinking with your shoulder partner.

5. CCSSM Definition:
An angle is a geometric shape that is formed wherever two rays share a common endpoint.
“Now, let’s examine the 4th grade common core standard to see how this formal understanding of angle is developed in students.”
Purpose: Familiarize participants with formal definition of angle

6. 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).
Static
Dynamic
3-4 min

7. Understanding the concepts of angles and measuring angles

8. Applying the measurement process to angles…
Thinking back to the general measurement process, what steps must we take to measure?
Identify an attribute
Define the unit of measure
Iterate the unit
Let’s let the standard define the unit of measure. Should kids start with degrees or is there another way suggested in the standards?
Turn & Talk in partners, then share out as group
Answers to hit upon:
Define a unit of measurement, iterate that unit; see measurement as additive,
Measure the lengths of the rays; measure the area between the rays; use the protractor without attention to what is reasonable.

9. Concepts of Angle Measurement
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.
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….
Ask: How is measuring an angle different than measuring the side of a shape?
The vocabulary here (rays, not segments) may need explanation—or perhaps there is a more teacher-friendly way of stating the idea, which is that since, according to the definition, every angle has infinitely long sides, there is no point it trying to measure the size of the angle by measuring the length of its sides.

10. Concepts of Angle Measurement
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.
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….
Ask: How is measuring an angle different than measuring the side of a shape?
The vocabulary here (rays, not segments) may need explanation—or perhaps there is a more teacher-friendly way of stating the idea, which is that since, according to the definition, every angle has infinitely long sides, there is no point it trying to measure the size of the angle by measuring the length of its sides.

11. The unit is the full rotation (or the full circular arc).
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?
We are measuring the fraction of the circular arc of a circle (tie back to measurement of rotation). The unit is the full turn
We are measuring the fraction of the circular arc of a circle (tie back to measurement of rotation)
Note that for angle measurement, as opposed to linear measurement, there is a natural unit—namely, one whole turn.
The unit is the full rotation (or the full circular arc).

12. 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.
Use Pattern Block Angles Recording sheet with this slide
So, for example, the green triangle has angles of 1/6 of a turn, since 6 of them will fit perfectly together to make a full turn.
(Hold off on degrees as the of angle measurement. That comes later in the session.)

13. 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.
Use Pattern Block Angles Recording sheet with this slide
Participants are trying to determine the angle of each pattern block in reference to one full rotation around a circle. We are connecting what we understand of fractions to a circle.
So, for example, the green triangle has angles of 1/6 of a turn, since 6 of them will fit perfectly together to make a full turn.
(Hold off on degrees as the of angle measurement. That comes later in the session.)

14. 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?
Let’s go back to standard 4.MD.5a and see what the second part says
One-degree (1/360th of a circle)

15. 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?

16. Standard 4.MD.5a & b
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.
An angle that turns through n one-degree angles is said to have an angle measure of n degrees.
Have participants read the standard and connect to the activity we just did.
Push participants to explain angle measures using unit fraction language, i.e., a 100 degree angle is 100 iterations of 1/360th of a circle.
3. Transition in picture: This is one student’s response when asked to relate angles to fractions. What does this student understand about angles?

17. 4.MD.5b. An angle that turns through n one-degree angles is said to have an angle measure of n degrees.
 Use unit fraction language to describe the measure of a 120 degree angle.
A 90 degree angle has 90 parts of size 1/360.

18. 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?
This slide can easily come here, or we could push it to later. If it comes here, I suggest it should be followed immediately by some activity comparing the use of whole turns and degrees as the unit of angle measurement (or at last a discussion as to the importance of each part of the standard).

19. Exploring Angles within Pattern Blocks, Part 2
Patty “We’ve explored activities for developing an understanding of what an angle is. What would be your expectations for measuring angle? (get a couple ideas (ex: protractor)) Well, just like many other times in class we won’t be using the traditional method.
Just a title slide

20. What is the degree measure of each angle in an equilateral triangle
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.
“Let’s return to our triangle pattern block.
Give participants 3 minutes to discuss as table groups Have 1 to 2 groups share out (3 minutes share) Summarize then after discussion, fly-ins
Purpose: To facilitate a connection between an angle as a fraction of the circular arc between the points of two rays and the intersect on the circle
Anticipated Response: since an circle is 360 degress, and there are four squares that can fit with vertices meeting in the center point then we know that each share of size ¼ must be 90 degrees because 360 divided by 4 equals 90 degrees

21. 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.
“Now bring this understanding with you as we explore angle measure in other shapes”
10 minutes to conduct the activity-> (2 groups share) 4 minutes share (if under 40 minutes do a third group)
Purpose: To explore the standard(measurement of angles in relation to a circle) through physical modeling
Anticipated Responses: Construct pattern blocks into a circle, determine how many compose the circle and estimate the angle measurement through division of 360 degrees by the number of angles. Paraphrase and connect to standards: I can put ___ number of this shape around a point to make a circle and a circle is 360 angle then this is called a ____ degree angle. TRIANGLE (a = 60 * 6), HEXAGON (b = 120 * 3), RHOMBUS acute (c= 60 *6), RHOMBUS obtuse (d= 12 *3), TRAPEZOID acute (e=60 *6), TRAPEZOID obtuse (f=120*3)
From this perspective how would students view angle size?: see as angle as additive, and a portion of a circle, students will develop benchmark concepts of angle size, students will realize they can use information they already have to determine unknown information, angles measure can be estimated without a protractor (can you connect this to…)
Math practices: MP 2 Reason abstractly and quantitatively and MP 6 Attend to precision, MP 3 Construct viable arguments and critique the reasoning of others

22. 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.
2 minutes

23. 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?
5 min
Use angles on cards from Essential Understanding Book p. 33
Possible student misconceptions: Measure of the ray is what counts, measure of the area is what counts, not all of the angles are angles.

24. Applying the measurement process to angles…
Identify an attribute
Define the unit of measure
Iterate the unit
The amount of rotation in the angle
Degrees
Count the degrees

25. 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.

26. 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?

27. 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.
Find a common vertex (common endpoint)
Establish your reference ray
How will solidifying your language help your kids to measure angles accurately?
What’s the difference between using these different protractors? Does the tool make or break the student?
Think about distance versus of amount of rotations
Use different protractors as you work

28. 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).
Have participants discuss in pairs, then share out. Finish debrief with key points on slide.

29. 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
(Use constructing angle worksheet)
Circulate room to monitor that participants are using conceptual language to support/explain their work in using the protractor to construct angles.
Debrief: What did they have to know to measure and construct angles?
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

30. 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?

31. How likely is it our spinner would land in Angle B? In Angle A?
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? A B
Students may see Angle B as larger than Angle A because of the area of region B.

32. 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).
Relate this back to moving and combining principles

33. 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

34. 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

35. 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

36. Are there Moving and Combining Principles for Angle Measure?

37. 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? B A

38. 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.

39. 8th Grade TIMSS 2011
Geometric Shape - Reasoning

40. 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.
2 minutes

41. 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.

42. 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.