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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.