Building a Stronger FoundaFon for Understanding in Advanced - PDF document

7/15/16 Building a Stronger FoundaFon for Understanding in Advanced Trigonometry Jason Slowbe Great Oak High School San Diego, CA @TheSlowbe jason.slowbe@gmail.com #NCTMinst • Slides and handout will be on Ins1tute website: h3p://nctm.org/hs16 • All resources will be on my own website: h3p://theslowbe.weebly.com • Please tweet throughout today’s session @theslowbe #NCTMinst 2 1

7/15/16 Goals for this session • Re-introducing trig func1ons • Making sense of amplitude, phase shiG • Making sense of all 6 trig func1ons • Mo1va1ng non-right triangle trig 4 @theslowbe #NCTMinst A complicaFon 5 @theslowbe #NCTMinst Henri PiccioMo h3p://blog.mathedpage.org/ @hpiccio3o 6 @theslowbe #NCTMinst 2

7/15/16 Find perimeter and area in unit circle h3p://blog.mathedpage.org/ @hpiccio3o 7 @theslowbe #NCTMinst Phillips Exeter Academy h3p://www.businessinsider.com/what-its-like-to-a3end-phillips-exeter-academy-2014-11?op=1 8 @theslowbe #NCTMinst 9 @theslowbe #NCTMinst 3

7/15/16 Sample problems, Exeter Academy At constant speed, a wheel rotates once counterclockwise every 10 seconds. The center of the wheel is (0, 0) and its radius is 1 foot. A paint spot is ini1ally at (1, 0); where is it t seconds later? 10 @theslowbe #NCTMinst Sample problems, Exeter Academy Quinn is running around the circular track x 2 +y 2 = 10000, whose radius is 100 meters, at 4 meters per second. Quinn starts at the point (100, 0) and runs in the counterclockwise direc1on. AGer 30 minutes of running, what are Quinn’s coordinates? 11 @theslowbe #NCTMinst Sample problems, Exeter Academy Represen1ng one unit by at least five squares on your graph paper, draw the unit circle, which is centered at the origin and goes through point A = (1, 0). Use a protractor to mark the third-quadrant point P on the circle for which arc AP has angular size 215 degrees. Es1mate the coordinates of P, reading from your graph paper. No1ce that both are nega1ve numbers. Turn on your calculator and ask for the cosine and sine values of a 215-degree angle. Do further explora1on, then explain why sine and cosine are known as circular func/ons . 12 @theslowbe #NCTMinst 4

7/15/16 Sample problems, Exeter Academy Given that cos 80 = 0.173648 . . . , explain how to find cos 100, cos 260, cos 280, and sin 190 without using a calculator. 13 @theslowbe #NCTMinst Sample problems, Exeter Academy A 15-degree counterclockwise rota1on centered at (2, 1) sends (4, 6) to another point (x, y). Find x and y, correct to three decimal places. 14 @theslowbe #NCTMinst Discovering Trig FuncFons Hands-On Discovery of Trigonometric Func1ons (adapted from Ronda Davis, Albuquerque Public Schools, and Ron Stewart, Baylor School) When you enter y -values in L3 and y / r -values in L5, enter them using the fewest keystrokes possible 15 @theslowbe #NCTMinst 5

7/15/16 Angle x-coordinate y-coordinate 0 1 0 10 0.9 0.2 QI QV … … … 80 0.2 0.9 90 0 1 100 -0.2 0.9 QII QII … … … 170 -0.9 0.2 180 -1 0 190 -0.9 -0.2 QIII QIII … … … 260 -0.2 -0.9 270 0 -1 280 0.2 -0.9 QIV QIV … … … 350 0.9 -0.2 16 360 1 0 Angle x-coordinate y-coordinate 0 1 0 10 0.9 0.2 QI QV … … … 80 0.2 0.9 90 0 1 100 -0.2 0.9 QII QII … … … 170 -0.9 0.2 180 -1 0 190 -0.9 -0.2 QIII QIII … … … 260 -0.2 -0.9 270 0 -1 280 0.2 -0.9 QIV QIV … … … 350 0.9 -0.2 17 360 1 0 ( ) = cos θ − 90 ( ) sin θ ( ) = sin θ + 90 ( ) cos θ ( ) = − cos θ + 90 ( ) sin θ 18 6

7/15/16 Discovering Trig FuncFons Predict, then Predict, then Sca3er plot with Sca3er Plot with partner: partner: • L 1 v L 2 • L 2 v L 3 • L 1 v L 3 • L 4 v L 5 • L 1 v L 4 • L 1 v L 5 19 @theslowbe #NCTMinst FoundaFon for future reference • Phase shiG à Dele1ng, Inser1ng quadrants • Amplitude à Radius 20 @theslowbe #NCTMinst Ferris Wheel task – MARS 21 @theslowbe #NCTMinst 7

7/15/16 Ferris Wheel task – MARS 22 @theslowbe #NCTMinst Ferris Wheel task – MARS 23 @theslowbe #NCTMinst Debrief • See sine and cosine as circular func1ons • Context for mo1va1ng and understanding both amplitude and phase shiG • Other observa1ons, take-aways? 24 @theslowbe #NCTMinst 8

7/15/16 “Concepts _______ Procedures” Fill-in the blank, then discuss at your tables: before aGer balanced with instead of 25 @theslowbe #NCTMinst “Concepts Before Procedures” From NCTM’s Principles to Ac/ons (p42) : • “the importance of an integrated and balanced development of concepts and procedures in learning mathema1cs” • “NCTM and CCSSM emphasize that procedural fluency follows and builds on a founda1on of conceptual understanding, strategic reasoning, and problem solving” 26 @theslowbe #NCTMinst “Concepts Before Procedures” From NCTM’s Principles to Ac/ons (p42) : • “Students must be able to do much more than carry out procedures. They must know which procedure is appropriate and most produc1ve in a given situa1on, what a procedure accomplishes, and what kind of results to expect. Mechanical execu1on of procedures without understanding their mathema1cal basis oGen leads to bizarre results.” 27 @theslowbe #NCTMinst 9

7/15/16 Goals for this session • Re-introducing trig func1ons • Making sense of amplitude, phase shiG • Making sense of all 6 trig func1ons • Mo1va1ng non-right triangle trig 29 @theslowbe #NCTMinst Making Sense of All 6 Trig FuncFons 30 @theslowbe #NCTMinst 10

7/15/16 Making Sense of All 6 Trig FuncFons Discuss at your tables: Why do we teach reciprocal trig func1ons? How do you teach reciprocal trig func1ons? What’s the purpose? 31 @theslowbe #NCTMinst Making Sense of All 6 Trig FuncFons ( ) = 4 • Is any more difficult to solve than sin 28 ° x ( ) = x ( ) = 4 ? csc 28 ° x = 4csc 28 ° ( ) 4 sin 28 ° • Even graphing calculators do not have bu3ons for reciprocal func1ons • Mathema1cal needs, applica1ons in future courses • Other ra1onales? 32 @theslowbe #NCTMinst Making Sense of All 6 Trig FuncFons Also, discuss at your tables how you introduce: • Pythagorean iden11es • Cofunc1on iden11es 33 @theslowbe #NCTMinst 11

7/15/16 hMp://www.mathsisfun.com/algebra/ trig-interacFve-unit-circle.html 34 @theslowbe #NCTMinst www.slideshare.net/sirgibey/math-9- similar-triangles-intro 35 @theslowbe #NCTMinst www.slideshare.net/sirgibey/math-9- similar-triangles-intro 36 @theslowbe #NCTMinst 12

7/15/16 www.slideshare.net/sirgibey/math-9- similar-triangles-intro 37 @theslowbe #NCTMinst www.slideshare.net/sirgibey/math-9- similar-triangles-intro 38 @theslowbe #NCTMinst hMp://www.mathsisfun.com/algebra/ trig-interacFve-unit-circle.html 39 @theslowbe #NCTMinst 13

7/15/16 RepresenFng All 6 Trig FuncFons 40 RepresenFng All 6 Trig FuncFons 41 RepresentaFons and ConnecFons • Reciprocal trig func1ons • Cofunc1on iden11es • Pythagorean iden11es 42 @theslowbe #NCTMinst 14

7/15/16 MoFvaFng non-right triangle trig • Discuss at your tables: How do you introduce non-right triangle trig? • Be prepared to share some ideas with whole group… 43 @theslowbe #NCTMinst Laser Level 44 @theslowbe #NCTMinst Laser Level a 2 − c 2 + b 2 − c 2 h = a a 2 − c 2 c b 2 − c 2 b @theslowbe #NCTMinst 45 15

7/15/16 Laser Level a 2 − c 2 − b 2 − c 2 h = h a a 2 − c 2 b b 2 − c 2 c b @theslowbe #NCTMinst 46 Laser Level a a 2 − c 2 b c b 2 − c 2 b @theslowbe #NCTMinst 47 SSA Ambiguous Case • GGB applet www.geogebra.org/m/ndmobwP • My GeoGebra page, many more applets: www.geogebra.org/theslowbe 48 @theslowbe #NCTMinst 16

7/15/16 Amazing video, app h3p://www.graphingstories.com/Axh h3p://www.vernier.com/products/soGware/ video-physics/ 50 @theslowbe #NCTMinst Building a Stronger FoundaFon for Understanding in Advanced Trigonometry Jason Slowbe Great Oak High School San Diego, CA @TheSlowbe jason.slowbe@gmail.com #NCTMinst 17

7/15/16 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. 52 53 18