GEOMETRY IN THE UPPER ELEMENTARY AND MIDDLE GRADES Presented by Dr. Paul Gray Chief Curriculum Officer Cosenza & Associates, LLC or Session #286 National Council of Teachers of Mathematics Regional Conference Houston, Texas November 21, 2014

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WHY IS GEOMETRY IMPORTANT? National Math Panel (2008) – three areas are Critical Foundations of Algebra Fluency with whole numbers Fluency with fractions Particular aspects of geometry and measurement 3 Particular aspects of geometry and measurement are: Similar triangles Formulas for perimeter, area, volume, and surface area Finding unknown lengths, angles, and areas Particular aspects of geometry and measurement are: Similar triangles Formulas for perimeter, area, volume, and surface area Finding unknown lengths, angles, and areas

VERTICAL FLOW OF GEOMETRY: ANGLES

MATHEMATICAL PROCESS STANDARDS A. apply mathematics to problems arising in everyday life, society, and the workplace; B. use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution; C. select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems; D. communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate; E. create and use representations to organize, record, and communicate mathematical ideas; F. analyze mathematical relationships to connect and communicate mathematical ideas; and G. display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication. 5

MEASUREMENT (ANGLES): VERTICAL PROGRESSION 6 4.MDGeometric measurement: understand concepts of angle and measure angles. 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. 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. 6. Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure. 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. TEKS: 4.7A-E

GEOMETRY (ANGLES): VERTICAL PROGRESSION 7 4.GDraw and identify lines and angles, and classify shapes by properties of their lines and angles. 1 Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures. 5.GClassify two-dimensional figures into categories based on their properties. 3 Understand that attributes belonging to a category of two- dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles. TEKS: 4.6A TEKS: 5.8

GEOMETRY (ANGLES): VERTICAL PROGRESSION 8 7.GDraw, construct, and describe geometrical figures and describe the relationships between them. 2 Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. 7.GSolve real-life and mathematical problems involving angle measure, area, surface area, and volume. 5 Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. 8.GUnderstand congruence and similarity using physical models, transparencies, or geometry software. 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. TEKS: 6.10A 7.11A TEKS: 8.8D

SAMPLE ACTIVITIES: EXTENDING UNDERSTANDING OF ANGLE RELATIONSHIPS

ANGLE RELATIONSHIPS Use the picture to identify supplementary, complementary, and vertical angles. Write a multi-step equation that you could use to solve to determine the measure of an angle in the picture. 10 Photo is of Port Arthur, Texas

ANGLE RELATIONSHIPS WHAT DID YOU SEE? Use the picture to identify supplementary, complementary, and vertical angles. Write a multi-step equation that you could use to solve to determine the measure of an angle in the picture. 11 Photo is of Port Arthur, Texas

ANGLES IN PARALLEL LINES 12 Use patty paper to investigate: Corresponding angles Alternate interior angles Same-side interior angles Alternate exterior angles Same-side exterior angles Vertical angles Which pair(s) of angles are congruent? Which pair(s) of angles are supplementary?

INTERIOR ANGLES OF A TRIANGLE 13

INVESTIGATING INTERIOR ANGLES 1. Construct a triangle and color each interior angle a different color. 2. Cut out your triangle and remove each angle from the triangle. 3. Use patty paper to investigate: What do you notice about the relationship between the measures of the interior angles? What do you notice about the relationship between the measure of an exterior angle and its remote interior angles? 14

INVESTIGATING INTERIOR ANGLES 15

INVESTIGATING INTERIOR ANGLES What do you notice about the sum of the interior angle measures for any triangle? What do you notice about the angle measures in an isosceles triangle? What do you notice about the angle measures in an equilateral triangle? What do you notice about the measures of the two acute angles in a right triangle? 16

DIFFERENTIATION Recommendation 5 Intervention materials should include opportunities for students to work with visual representations of mathematical ideas and interventionists should be proficient in the use of visual representations of mathematical ideas. (p. 30) 17 Google: “what works clearinghouse”

PAUSE AND REFLECT How does this lesson support differentiated instruction? How does this lesson integrate mathematical process standards? How does this lesson build on previous grade level content? 18

REFERENCES National Mathematics Advisory Panel. Foundations for Success: The Final Report of the National Mathematics Advisory Panel, U.S. Department of Education: Washington, DC, Gersten, R., Beckmann, S., Clarke, B., Foegen, A., Marsh, L., Star, J. R., & Witzel, B. (2009). Assisting students struggling with mathematics: Response to Intervention (RtI) for elementary and middle schools (NCEE ). Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education. Retrieved from 19

QUESTIONS, COMMENTS, OR SUCCESS STORIES? Dr. Paul Gray PowerPoint and Handout available on (Events and Conferences section) 20