Long live triangles!

Aitzol Lasa, Jan 20, 2016

This GeoGebraBook contains the dynamic models for the article: Lasa, A., Belloso, N., Abaurrea, J. (2016). Long live triangles! Dynamic models for trigonometry. [i]Revista do Instituto GeoGebra de São Paulo[/i], 5(2), 30- 55. Link to article: [url]http://revistas.pucsp.br/index.php/IGISP/article/view/28630[/url] Pubblicato in italiano: "Lunga vita ai triangoli", [url]https://www.geogebra.org/book/title/id/cnKZVc6d#chapter/0[/url] Euskaraz eskuragarri: "Triangeluen gorazarrea", [url]https://www.geogebra.org/book/title/id/Sg8UI8v8#chapter/0[/url] Disponible en castellano: "Elogio de los triángulos", [url]https://www.geogebra.org/m/d2exF80t#chapter/0[/url] [b][color=#c51414]Nou! Ara també en català![/color][/b] Publicat en català: “Elogi als triangles” [url]https://www.geogebra.org/book/title/id/eQaSd8de[/url]

Table of Contents

  1. Construction of trigonometric ratios, the sine
    1. Sine: explorative model
    2. Sine: illustrative model
    3. Sine: demonstrative model
  2. Transition to trigonometric functions
    1. Function: explorative model
    2. Periodic function: explorative model
    3. Period of a periodic function: explorative model
    4. Goniometric circle: illustrative model
    5. y=sin(x) function: illustrative model
  3. Properties of trigonometric functions: wave function
    1. Trigonometric projection: explorative model
    2. Trigonometric properties: demonstrative model
    3. Wave functions: explorative model
    4. Wave functions: illustrative model

1.

Construction of trigonometric ratios, the sine

  • 1. Sine: explorative model
  • 2. Sine: illustrative model
  • 3. Sine: demonstrative model

1.1.

Sine: explorative model

Objective and steps
Students work in pairs (two student, A and B).[br][br]1) Student A introduces a numerical value for the the acute angle ([math]\alpha[/math]) in the dynamic model, and this value remains unchanged until step (5). Then, student A drags the orange point to modify the base-length of the triangle.[br]2) Student B writes down the values of the triangle from the model and performs calculations in paper with the given information.[br]3) Students switch positions. Student B decides a new position for the orange point (without changing the angle, since the “input box” is gone for the moment), and student A performs new calculations. Students repeat steps (2) and (3) a number of times.[br]4) Students must formulate a conjecture and the conjecture must be coherent with the obtained data.[br]5) Students push the “new angle” button, and they start again from step (1) with a new triangle.[br]6) After repeating the process with a number of triangles, students reformulate their conjecture, which must be coherent with the obtained new data.
– Aitzol Lasa

1.2.

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1.3.

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2.

Transition to trigonometric functions

  • 1. Function: explorative model
  • 2. Periodic function: explorative model
  • 3. Period of a periodic function: explorative model
  • 4. Goniometric circle: illustrative model
  • 5. y=sin(x) function: illustrative model

2.1.

Function: explorative model

Draw the graphic of a function
If you push the "play" button, the orange point will start to move from left to right along the Cartesian plane, i.e., increasing values for the x coordinate.[br]You can move the orange point up and down on the blue line.[br]Every time you do so, you will get the graphic of a function.[br]Push "restart" to begin a new graphic ([b]CONTROL+F[/b] resets the graphic view)
– Aitzol Lasa

2.2.

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2.3.

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2.4.

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2.5.

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3.

Properties of trigonometric functions: wave function

  • 1. Trigonometric projection: explorative model
  • 2. Trigonometric properties: demonstrative model
  • 3. Wave functions: explorative model
  • 4. Wave functions: illustrative model

3.1.

Trigonometric projection: explorative model

Objective and steps
We have already seen that all similar triangles have equal trigonometric ratios. Thus, someone could think that there is no relation whatsoever between trigonometric[br]ratios and the length-dimensions of the triangle. Solve the next activity to[br]see that the previous supposition is not true.[br][br]Students work in pairs (students A and B).[br][br]1) Student A selects positions for both orange points in the circle, thus, changing the value of the radious of the circle (the length of the hypotenuse) and the angle in the triangle.[br]2) Student B translates the numerical values of the triangle from the graphic view to the spreadsheet and makes calculations.[br]3) Students A and B interchange positions and repeat steps 1 and 2, a number of times.[br]4) Students write down conclusions about the observed numerical relations.[br]5) Students translate conclusion to algebraic language.
– Aitzol Lasa

3.2.

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3.3.

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3.4.

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