Hornbeck, D Tech Review
Link to Applet
The technology I have chosen to review here is an applet on Geogebra Tube entitled "Unit Circle - Trig functions vs. Geometry definitions." It is a simple applet that allows students to observe the geometric definitions of the six major trigonometric functions as defined on the unit circle. The applet is ideally designed for 11th grade students that are first learning the unit circle definition of the trigonometric functions and extension of these functions to all positive real numbers. In Georgia, this would occur in Advanced Algebra in 11th grade or late spring of 10th grade for accelerated students.
Standards
Analyze functions using different representations
MCC9-12.F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.★ (Limit to trigonometric functions.)
Extend the domain of trigonometric functions using the unit circle
MCC9-12.F.TF.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
MCC9-12.F.TF.2 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
Prove and apply trigonometric identities
MCC9-12.F.TF.8 Prove the Pythagorean identity (sin A)2 + (cos A)2 = 1 and use it to find sin A, cos A, or tan A, given sin A, cos A, or tan A, and the quadrant of the angle.
Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
5. Use appropriate tools strategically.
7. Look for and make use of structure.
SMPs 1, 2, 3, and 5 can clearly be encouraged by this applet. I mention practice 7 because this applet allows students to utilize varying understanding of trigonometric functions (tangent as sine divided by cosine, etc.) and use this to make proofs easier.
MCC9-12.F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.★ (Limit to trigonometric functions.)
Extend the domain of trigonometric functions using the unit circle
MCC9-12.F.TF.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
MCC9-12.F.TF.2 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
Prove and apply trigonometric identities
MCC9-12.F.TF.8 Prove the Pythagorean identity (sin A)2 + (cos A)2 = 1 and use it to find sin A, cos A, or tan A, given sin A, cos A, or tan A, and the quadrant of the angle.
Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
5. Use appropriate tools strategically.
7. Look for and make use of structure.
SMPs 1, 2, 3, and 5 can clearly be encouraged by this applet. I mention practice 7 because this applet allows students to utilize varying understanding of trigonometric functions (tangent as sine divided by cosine, etc.) and use this to make proofs easier.
Critique
The applet works quite well. It's very simple - slide the red point around the circle, and watch how the values of the trigonometric functions change based on the given angle. The segments representing each of the functions can be hidden, which is important given how busy the screen can be with all six segments displayed. The values of each function are accurate, but are only displayed as numbers to five decimals; it does not allow for fractional or radical views of numbers, which might be beneficial for such values as sine of 30 degrees, etc. Angles can be measured in degrees or radians (as fractions of pi, decimal multiples of pi, or straight decimal values), though this does not affect the resulting trig functions. Users can also highlight "special" points - namely, points that represent multiples of 30 and 45 degrees.
The written materials associated with the applet are in my opinion not well constructed or organized. The questions at the top do a nice job of piquing interest, but the actual directions are of lower quality. Firstly, the instructions are clumped together with minimal spacing, and the author did not use any form of mathematical notation, which is confusing. I do not generally prefer notation like "sine(complement of theta)." Further, the instructions do not encourage any systematic or strategic use of the checkboxes. They merely say
The written materials associated with the applet are in my opinion not well constructed or organized. The questions at the top do a nice job of piquing interest, but the actual directions are of lower quality. Firstly, the instructions are clumped together with minimal spacing, and the author did not use any form of mathematical notation, which is confusing. I do not generally prefer notation like "sine(complement of theta)." Further, the instructions do not encourage any systematic or strategic use of the checkboxes. They merely say
"Drag the red point along the circumference of the unit circle, observing the changes in the various line segments.
Toggle on/off the various checkboxes and adjust the slider."
Toggle on/off the various checkboxes and adjust the slider."
This is essentially, "Check out what happens." Now, the first few questions, I like. They bring attention to the secant intersecting (theoretically) the circle twice, and the tangent intersecting the circle only once. This should activate students' prior knowledge from Analytic Geometry. The questions following try to prompt students to prove some identities and relationships based on triangle similarity, but this is done very poorly. The sixth question,
"Referencing the unit circle, why is tan(theta) equal to the quotient of sin(theta)/cos(theta)?"
requires students to use triangle similarity and a particular proportion. The next two questions, though, serve as direct but relatively useless setups for similar proofs later on. Why would one similarity proof have no scaffolding and four more of them have two questions and a hint prior to? The written materials just generally don't make much sense. That's okay, though, because the applet is still powerful.
The presentation in the applet is nice, but there are a few minor issues. The coordinate plane is not shown, leaving students to recognize the (-1, 0) (bizarrely the only coordinates displayed) in order to identify where the circle is. Further, the segment labels are fixed into place, making them sometimes hard to see; it would be nice if they could be moved from inconvenient spots.
The applet can be used to guide students to an understanding of some of the following ideas:
- The trigonometric functions have domains that extend to positive and negative infinity, but the functions are periodic, as evidenced by the concept of the functions wrapping around the circle repeatedly.
- Any point on the unit circle can be defined in an ordered pair by the sine and cosine of a particular angle; conversely, any angle corresponds to a unique point on the unit circle.
- Geometric representations of trigonometric functions allow for simple, clever proof of trigonometric identities. The visualizations provided in this applet somewhat enlighten students as to why the identities hold.
- Four of the six trig functions are defined at certain values. The applet allows one to see, for instance, that the tangent function goes to infinity as an angle increases to 90 degrees but goes to negative infinity as the angle decreases to 90. A skilled teacher might even be able to have an informal conversation about limits with the right students.
- The six trigonometric functions are really three pairs of functions and co-functions, each pair being related by the complements of angles.
I want to speak in particular, though, about a limitation of this technology. While the checkboxes make it incredibly easy to hide segments, one cannot easily create certain proofs. For instance, proving that the sine of an angle is equal to the cosine of the angle's complement using triangle similarity requires two triangles; the applet only allows one, though, requiring the user to draw another picture. Though not a major limitation for this proof, the fact that the applet only has 1 circle reduces the amount of conjectures one could make. Perhaps a second circle would help so that one could visualize the secant/tangent triangle on one circle and the cosecant/cotangent on another, for example.
With the right prompts, the applet could be used to engage students in recognizing all of the above ideas. In its current form, without new written materials, though, I would not recommend a school use it. I would encourage teachers to download the applet itself and create new prompts themselves (and upload it to Geogebra Tube!) to guide students through use of the applet.
The presentation in the applet is nice, but there are a few minor issues. The coordinate plane is not shown, leaving students to recognize the (-1, 0) (bizarrely the only coordinates displayed) in order to identify where the circle is. Further, the segment labels are fixed into place, making them sometimes hard to see; it would be nice if they could be moved from inconvenient spots.
The applet can be used to guide students to an understanding of some of the following ideas:
- The trigonometric functions have domains that extend to positive and negative infinity, but the functions are periodic, as evidenced by the concept of the functions wrapping around the circle repeatedly.
- Any point on the unit circle can be defined in an ordered pair by the sine and cosine of a particular angle; conversely, any angle corresponds to a unique point on the unit circle.
- Geometric representations of trigonometric functions allow for simple, clever proof of trigonometric identities. The visualizations provided in this applet somewhat enlighten students as to why the identities hold.
- Four of the six trig functions are defined at certain values. The applet allows one to see, for instance, that the tangent function goes to infinity as an angle increases to 90 degrees but goes to negative infinity as the angle decreases to 90. A skilled teacher might even be able to have an informal conversation about limits with the right students.
- The six trigonometric functions are really three pairs of functions and co-functions, each pair being related by the complements of angles.
I want to speak in particular, though, about a limitation of this technology. While the checkboxes make it incredibly easy to hide segments, one cannot easily create certain proofs. For instance, proving that the sine of an angle is equal to the cosine of the angle's complement using triangle similarity requires two triangles; the applet only allows one, though, requiring the user to draw another picture. Though not a major limitation for this proof, the fact that the applet only has 1 circle reduces the amount of conjectures one could make. Perhaps a second circle would help so that one could visualize the secant/tangent triangle on one circle and the cosecant/cotangent on another, for example.
With the right prompts, the applet could be used to engage students in recognizing all of the above ideas. In its current form, without new written materials, though, I would not recommend a school use it. I would encourage teachers to download the applet itself and create new prompts themselves (and upload it to Geogebra Tube!) to guide students through use of the applet.
Activity
There are so many things that could be done with this applet. Below is a set of prompts that guide students through some of what I find most interesting in the mathematics. This is geared towards students that have already been familiarized with the unit circle. The focus of the activity is proving identities and relationships using triangle similarity.
Citations
White, J. (2012). "Unit Circle - Trig Definitions vs Geometry definitions." Retrieved July 7, 2014 from geogebratube.org.