Construction Junction, What s your Function?

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1 Construction Junction, What s your Function? Brian Shay Teacher and Department Chair Canyon Crest Academy

2 Session Goals Familiarize ourselves with CCSS and the GSE Geometry Standards Explore tasks involving constructions and proof Connect these tasks to prior and future knowledge 2

3 Standards for Math Practice 1. Make sense of problems & persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated 3 reasoning.

4 PtA s Math Teaching Practices 1. Establish mathematics goals to focus learning. 2. Implement tasks that promote reasoning and problem solving. 3. Use and connect mathematical representations. 4. Facilitate meaningful mathematical discourse. 5. Pose purposeful questions. 6. Build procedural fluency from conceptual understanding. 7. Support productive struggle in learning mathematics. 8. Elicit and use evidence of student thinking. 4

5 CCSS Geometry Standards G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. G.CO.13 Construct an equilateral triangle, a square, and a regular hexagon, each inscribed in a circle. G.C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. G.C.4 Construct a tangent line from a point outside a given circle to the circle. 5

6 Tools for Construction Real Old School: String and Chalk! Less Old School: Compass and Straightedge New School: Dynamic Geometry Software (GeoGebra) 6

7 The Basics Copying a segment Copying an angle Bisecting a segment Bisecting an angle Constructing Perpendicular Lines Constructing Parallel Lines 7

8 Better than Basic Equilateral Triangle Inscribed in a Circle Square Inscribed in a Circle Regular Hexagon Inscribed in a Circle Inscribed Circle for a Triangle Circumscribed Circle for a Triangle Tangent Line to a Circle 8

9 These are gonna stretch ya Before you start, draw the final image. Think about its properties and relationships Think about how you can work your way backwards Think about how to use the basics to get there Construct an isosceles triangle ABC, given base angle B and the altitude CD to one of its legs. 9

10 More stretches (#2) Construct a right triangle ABC, given angle B, one of its acute angles, and CD, the altitude to the hypotenuse. 10

11 Stretch #3 Construct a triangle ABC, given two if its angles, A and B, and the angle bisector to the angle B. 11

12 Stretch #4 Construct a parallelogram ABCD, given the diagonals AC and BD, and <AEB, where E is the intersection of the diagonals. 12

13 Stretch #5 Construct a triangle ABC, given the angles A and B and the radius of the circle inscribed in the triangle. 13

14 Stretch #6 Given a circle C 1 and lines AX and AY, that are tangent to the circle at points X and Y, respectively. Construct a circle C 2 that is tangent to the circle C 1, AX and AY. 14

15 Constructions.FUNctions! Constructions are awesome! Connect all of the following together Angle and line properties Circle properties Triangle properties Quadrilateral properties Tactile Proofs No, Low, or High Tech Integration 15

16 Thank you! Construction Junction! Brian 16

17 Disclaimer The National Council of Teachers of Mathematics is a public voice of mathematics education, providing vision, leadership, and professional development to support teachers in ensuring equitable mathematics learning of the highest quality for all students. NCTM s Institutes, an official professional development offering of the National Council of Teachers of Mathematics, supports the improvement of pre-k-6 mathematics education by serving as a resource for teachers so as to provide more and better mathematics for all students. It is a forum for the exchange of mathematics ideas, activities, and pedagogical strategies, and for sharing and interpreting research. The Institutes presented by the Council present a variety of viewpoints. The views expressed or implied in the Institutes, unless otherwise noted, should not be interpreted as official positions of the Council. 17

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