Standards/Rationale Activity #1 About: This activity has students investigating the definition of a radian through the use of an applet found on GeogebraTube. From this investigation, students will be able to use the technology to drag a point on the circle and various calculations to see how radians are related to the ratio between the circle's arc length and its circumference. Goals: Students will be able to derive the meaning of a radian measure through the use of a circle’s arc length and its radius. Standards addressed: 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. Standards for Mathematical Practice:1, 2, 4, 5, 7 Why to use this task: This activity is great to use since it helps students visually see what a radian is and it relates to the arc length and radius of the circle. Through the use of the technology, students can drag the point around the circle and visually see how the angle affects the arc length and radius. To go even further, the applet allows students to see the ratio between the arc length and the radius to further make connections in order to derive the definition of a radian. By allowing students to physically move the point and see how the angle affects the arc length and radius and also providing the ratio of the arc length to the radius, the technology helps to accomplish my goal of deriving the definition of a radian.
Activity #2 About: This activity has students investigating the relationships between radians and degrees through the use of GeogegbraTube. From this investigation, students will be able to use the technology to measure angles and arc lengths to make conjectures about how radians and degrees relate to each other. Goals: Students will be able to understand the relationship between radian and degree measures and derive the formula for how to convert degrees to radians and radians to degrees. Standards Addressed: 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. Standards for Mathematical Practice: 1, 2, 4, 5, 6 Why use this task: Thisactivity is great to use since it helps students visually see what a radian and a degree is and they relate to each other. Through the use of the technology, students can drag the point around the circle and visually see the definition of a radian come to life by seeing how the radius length fits on the circle. To go even further, the applet allows students to measure the angles created by the radians through the use of the slider.Through these two features, students can make further comparisons on how degrees and radians relate and differ from each other, which can lead to students eventually deriving the conversion formula. By allowing students to physically move the radius on the circle and providing a way to measure the angles in degrees, the technology helps to accomplish my goal of deriving the formula for deriving the conversion formula for degrees to radians and radians to degrees. Activity #3 About: This activity has students investigating the unit circle through the use of GeogegbraTube. From this investigation, students will be able to use the technology to measure angles, arc lengths, and the sine and cosine of any point located on the unit to make conjectures about unit circle. Goals: Students will be able to find, calculate, and express the values of sine and cosine of any angle measure located on the unit circle. With this students will also be able to showcase what the x and y coordinates measure and what value represents. Standards Addressed: 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. CCSS.MATH.CONTENT.HSF.TF.A.3 Use special triangles to determine geometrically the values of sine, cosine, tangent for pi/3, pi/4, and pi/6, and use the unit circle to express the values of sine, cosine, and tangent for x, pi + x, and 2pi - x in terms of their values for x, where x is any real number. Standards for Mathematical Practice: 1, 4, 5, 6 Why use this task: Thisactivity is great to use since it helps students visually see how the coordinates of the unit circle are derived. Through the use of the technology, students can drag a point on a slider to visually see how the angle of the circle affects not only the arc length and radius, but also the sine and cosine values. To go even further, the applet provides the actual values for sine and cosine so students are able to see how these values change as the angle changes. By allowing students to physically move the point and see how the angle affects the sine and cosine values, the technology helps to accomplish my goal of calculating any coordinate on the unit circle. Activity #4 About: This activity has students exploring the key features of a sine and cosine function such as the period and amplitude and how this relates to the unit circle . Through the use of GeogebraTube, students will take a ferris wheel and graph the distance of one of the seats relative to the ground over time. From this investigation, students will be able to use the technology to plot points and create graphs in order to investigate key features of the graph and how this relates to the unit circle. Goals: Students will be able to understand how to graph trigonometric functions such as a sine and cosine function. With this students will also be able to determine the key features of a graph such as the amplitude and period to determine how each of these factors effect the graph and their meaning. Standards Addressed: 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. Standards for Mathematical Practice: 1, 2, 4, 5, 7 Why use this task: This activity is great to use since it helps students visually see how the coordinates of the unit circle translates onto a graph. Through the use of the technology, students can animate a point around a circle and a graph to visually see these two points relate to each other. To go even further, the applet allows students to create their own graphs to see if they can match the motion of the graph, which gives them practice graphing trigonometric functions. Also the applet provides ways for students to make many connections between the unit circle and the graph by the many button options provided in the applet. By allowing students to animate a point and build connections between how the point is represented on the graph verses the unit circle, the technology helps accomplish my goal of understanding the key features of the sine and cosine functions.
References: The Common Core State Standards Initiative (2011). Common Core State Standards for Mathematics. Retrieved from https://www.georgiastandards.org/Common-Core/Pages/Math.aspx
The Common Core State Standards Initiative (2011). Standards for Mathematical Practice. Retrieved fromhttp://www.corestandards.org/Math/Practice/