Building a Stronger FoundaFon for Understanding in Advanced - - PDF document

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Building a Stronger FoundaFon for Understanding in Advanced - - PDF document

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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:

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SLIDE 1

7/15/16 1

Jason Slowbe

Great Oak High School San Diego, CA @TheSlowbe jason.slowbe@gmail.com #NCTMinst

Building a Stronger FoundaFon for Understanding in Advanced Trigonometry

  • 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

2

@theslowbe #NCTMinst

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SLIDE 2

7/15/16 2

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

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Henri PiccioMo

h3p://blog.mathedpage.org/ @hpiccio3o

6 @theslowbe

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SLIDE 3

7/15/16 3

Find perimeter and area in unit circle

h3p://blog.mathedpage.org/ @hpiccio3o

7 @theslowbe

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Phillips Exeter Academy

h3p://www.businessinsider.com/what-its-like-to-a3end-phillips-exeter-academy-2014-11?op=1

8 @theslowbe

#NCTMinst

9 @theslowbe

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7/15/16 4

Sample problems, Exeter Academy

At constant speed, a wheel rotates once counterclockwise every 10 seconds. The center

  • f 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

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Quinn is running around the circular track x2+y2 = 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

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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

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Sample problems, Exeter Academy

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7/15/16 5

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

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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

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Sample problems, Exeter Academy 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

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7/15/16 6

16

Angle x-coordinate 1 10 0.9 … … 80 0.2 90 100

  • 0.2

… … 170

  • 0.9

180

  • 1

190

  • 0.9

… … 260

  • 0.2

270 280 0.2 … … 350 0.9 360 1 y-coordinate 0.2 … 0.9 1 0.9 … 0.2

  • 0.2

…

  • 0.9
  • 1
  • 0.9

…

  • 0.2

QI QII QII QIII QIII QIV QIV QV

17

Angle x-coordinate 1 10 0.9 … … 80 0.2 90 100

  • 0.2

… … 170

  • 0.9

180

  • 1

190

  • 0.9

… … 260

  • 0.2

270 280 0.2 … … 350 0.9 360 1 y-coordinate 0.2 … 0.9 1 0.9 … 0.2

  • 0.2

…

  • 0.9
  • 1
  • 0.9

…

  • 0.2

QI QII QII QIII QIII QIV QIV QV

18

sin θ

( )= cos θ −90 ( )

cos θ

( )= sin θ +90 ( )

sin θ

( )= −cos θ +90 ( )

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7/15/16 7

Discovering Trig FuncFons

Predict, then Sca3er plot with partner:

  • L1 v L2
  • L1 v L3
  • L1 v L4
  • L1 v L5

19 @theslowbe

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Predict, then Sca3er Plot with partner:

  • L2 v L3
  • L4 v L5

FoundaFon for future reference

  • Phase shiG à Dele1ng, Inser1ng quadrants
  • Amplitude à Radius

20 @theslowbe

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Ferris Wheel task – MARS

21 @theslowbe

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SLIDE 8

7/15/16 8

Ferris Wheel task – MARS

22 @theslowbe

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Ferris Wheel task – MARS

23 @theslowbe

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Debrief

  • See sine and cosine as circular func1ons
  • Context for mo1va1ng and understanding

both amplitude and phase shiG

  • Other observa1ons, take-aways?

24 @theslowbe

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SLIDE 9

7/15/16 9

“Concepts _______ Procedures”

Fill-in the blank, then discuss at your tables: before aGer balanced with instead of

25 @theslowbe

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“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

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“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

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SLIDE 10

7/15/16 10

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

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SLIDE 11

7/15/16 11

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

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Making Sense of All 6 Trig FuncFons

  • Is any more difficult to solve than

?

  • Even graphing calculators do not have bu3ons

for reciprocal func1ons

  • Mathema1cal needs, applica1ons in future

courses

  • Other ra1onales?

32 @theslowbe

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sin 28°

( )= 4

x csc 28°

( )= x

4 x = 4csc 28°

( )=

4 sin 28°

( )

Making Sense of All 6 Trig FuncFons

Also, discuss at your tables how you introduce:

  • Pythagorean iden11es
  • Cofunc1on iden11es

33 @theslowbe

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hMp://www.mathsisfun.com/algebra/ trig-interacFve-unit-circle.html

34 @theslowbe

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www.slideshare.net/sirgibey/math-9- similar-triangles-intro

35 @theslowbe

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www.slideshare.net/sirgibey/math-9- similar-triangles-intro

36 @theslowbe

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SLIDE 13

7/15/16 13

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

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7/15/16 14

RepresenFng All 6 Trig FuncFons

40

RepresenFng All 6 Trig FuncFons

41

RepresentaFons and ConnecFons

  • Reciprocal trig func1ons
  • Cofunc1on iden11es
  • Pythagorean iden11es

42 @theslowbe

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SLIDE 15

7/15/16 15

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

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Laser Level

44 @theslowbe

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Laser Level

45

a c b

a2 −c2 b2 −c2 h= a2 −c2 + b2 −c2

@theslowbe #NCTMinst

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7/15/16 16

Laser Level

46

a c b

a2 −c2

b2 −c2

h= a2 −c2 − b2 −c2

h b

@theslowbe #NCTMinst

Laser Level

47

a c b

a2 −c2 b2 −c2

b

@theslowbe #NCTMinst

SSA Ambiguous Case

  • GGB applet

www.geogebra.org/m/ndmobwP

  • My GeoGebra page, many more applets:

www.geogebra.org/theslowbe

48 @theslowbe

#NCTMinst

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SLIDE 17

7/15/16 17

50 @theslowbe

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Amazing video, app

h3p://www.graphingstories.com/Axh h3p://www.vernier.com/products/soGware/ video-physics/

Jason Slowbe

Great Oak High School San Diego, CA @TheSlowbe jason.slowbe@gmail.com #NCTMinst

Building a Stronger FoundaFon for Understanding in Advanced Trigonometry

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SLIDE 18

7/15/16 18 Disclaimer

The National Council of Teachers of Mathematics is a public voice

  • f 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
  • ffering 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.

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