Overview
Curator: Marissa Hoffman
Technical & Cost considerations: This activity is free of charge as well as is the website Geogebra. Radians in the Unit Circle works on any internet browser and it is recommended that the browser is up to date with latest Flash Player and plug-ins to ensure the graphics will work. This activity can be used effectively as a reference tool for students to use individually, or in small groups, as well as used interactively in a whole class discussion on a SmartBoard. It is recommended that this activity is used on wifi enabled devices with a high speed internet connection to ensure smooth use and animation.
Evaluation
Description of Learning Activity
This activity allows students to quickly covert degrees on the unit circle to radians of different denominators. Additionally, students will be able to quickly identify equivalent radian angle measures and compare these measures when written with uncommon denominators. This activity will prepare students to graph trigonometric functions.
1. Learning Activity Types
- LA-Practice - Students will quickly recognize equivalent radian measures and convert degrees to radians (or vice versa).
- LA-Present - The activity should be used in the initial presentation and discussion of the unit circle and angle measures.
- LA-Present-Explain - The clear format of the unit circle and fraction bars explain how equivalent fractions are equal, where each one will fall on the unit circle, and how to order fractions with uncommon denominators.
- LA-Present-Explain - The clear format of the unit circle and fraction bars explain how equivalent fractions are equal, where each one will fall on the unit circle, and how to order fractions with uncommon denominators.
- LA-Explore - Students will explore how the unit circle can be divided into several, equal pieces and which angle measures (radians and degrees) are the same.
2. What mathematics is being learned?
Standards
Principles and Standards for School Mathematics (PSSM)
- Explore relationships...among classes of two- and three-dimensional geometric objects...
- Use trigonometric relationships to determine lengths and angle measures.
- Draw and construct representations of two- and three-dimensional geometric objects using a variety of tools.
Common Core State Standards Mathematics (CCSSM)
- Understand and apply theorems about circles
- Find arc lengths and areas of sectors of circles
Mathematical Practices
- Reason abstractly and quantitatively.
- Use appropriate tools strategically.
- Attend to precision.
- Look for and make use of structure.
Proficiency Strands
- conceptual understanding: Students will understand the concept of radian angle measure being a different unit of measure from degrees similar to what miles is to kilometers. Additionally, students will understand the ordering of fractions with like and unalike denominators.
- procedural fluency - With repeated use of this activity, students will be able to fluently connect and work with angles measured in degrees and radians along with fractions measuring halves, thirds, fourths, etc.
3. How is the mathematics represented?
Through numerical, ordered representation and a virtual manipulative the mathematics presented in this activity is done using two vehicles. As the unit circle travels, it lines up with radian measures presented in a number-line fashion using fraction bars. The animation then links together radian measures with various denominators to connect student understanding of equivalent fractions and angle measures. Without this technology, students would only work with a narrow range of radians given on a the unit circle. Lastly students would save time by not having to create common denominators for radian measures when graphing a trigonometric function because the most common denominators (halves, thirds, quarters, sixths, etc.) are provided and aligned numerically with their equivalents.
4. What role does technology play?
One obvious disadvantage to the Radians in Unit Circle activity is that students may rely on the tool too much and never put in the effort to relationally understand the information the resource provides. Similar to using a calculator, students may not ever learn how to manually convert degrees to radians (or vice versa) or understand the place-value relationship of fractions with different denominators. However, students do get to focus more on the significance of the values of the degrees and radians without having to use up a lot of time manually converting from one unit to the other. This allows for a more in-depth, higher-order thinking and problem solving. By spending less time on menial tasks students can discuss and study the big picture of whatever trigonometric task they are working on (evaluating, graphing, etc.) in order to facilitate relational understanding.
Affordances of Technology for Supporting Learning
- Computing & Automating - Radians in Unit Circle can be used in a calculation activity or as a reference tool. Students can use the tool to automatically find the radian measures or order fractions thus saving time and allowing focus to be given to more conceptual thinking and logical reasoning.
- Representing Ideas & Thinking - This tool connects two different aspects of trigonometry and numbers. The connection between degrees and radians is easily seen as the students roll the unit circle down the fraction bars. At the same time, students can properly order fractions of different denominators based on place value which benefits students who have to graph trigonometric functions and label the x-axis in terms of radians and simplified fractions.
- Accessing Information - This activity allows students to access information about equivalence and value for degrees and radians.
5. How does the technology fit or interact with the social context of learning?
As always, learning and practicing mathematics should be done collaboratively. Communication increases the ability to problem solve as ideas are shared. Although this activity can be completed by an individual student, it is most effective when used in small groups - pairs of two are recommended. This resource encourages students to focus on the big picture when using the unit circle but does not help students develop the stamina to persevere and problem solve. Manual work such as converting and creating common denominators is now removed.
6. Additional Comments
I was interested in the Radians in Unit Circle activity because it can be applied in the classroom as a resource or as part of an activity. By itself, the Radian in Unit Circle is not an activity but can become a vital part of a small group project or assignment when used to help students focus on the big picture in the problem solving process.