Lesson no. 3 (Arc Length and Area of a Sector
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Lesson No. 3 covers the concepts of arc length and area of a sector, including formulas and real-life applications. Students engage in group activities solving problems related to linear and angular measures, as well as exploring philosophical views related to circles. The lesson also emphasizes the evaluation of understanding through problem-solving and discussions.
Lesson no. 3 (Arc Length and Area of a Sector
- 1. LESSON NO. 3 ARC LENGTH AND AREA OF A SECTOR
- 2. Lesson No. 3 |Arc Length and Area of a Sector ____________________________________________________________________ ENGAGE
- 3. Lesson No. 3 |Arc Length and Area of a Sector ____________________________________________________________________ Engagement Activity 1 Linear and Angular measures Author: Irina Boyadzhiev Reference: https://www.geogebra.org/m/EazPPkFV The applet illustrates the linear and angular measures of central angle in a unit circle.
- 4. Lesson No. 3 |Arc Length and Area of a Sector ____________________________________________________________________ Engagement Activity 1 Questions: 1.What can you say about the linear measure of angle? How about angular measure? 2.Is there a relationship between angular and linear measures of angle? 3.What can you infer about the relationship of angular and linear measures of angle?
- 5. Lesson No. 3 |Arc Length and Area of a Sector ____________________________________________________________________ Engagement Activity 2 Learning Guide Card (LGC) # 1 Arc Length The length of an arc on a circle of radius r is equal to the radius multiplied by the angle θ subtended by the arc in radians. Using s to denote arc length we have s = rθ. This should actually be intuitive since the arc length on the unit circle is equivalent to the angle in radians.
- 6. Engagement Activity 2 Learning Guide Card (LGC) # 1 The figure below shows arc length between points A and B on the circle. Since we are looking at a length, we always consider the angle θ subtended by A and B to be positive. (In each of the next two figures, both and can be moved.) Lesson No. 3 |Arc Length and Area of a Sector ____________________________________________________________________________
- 7. Lesson No. 3 |Arc Length and Area of a Sector ___________________________________________________________________ Engagement Activity 2 Questions: -What can you say about the length of an arc on a circle? -How is the arc length on the unit circle related to the angle in radians?
- 8. Engagement Activity 3 Learning Guide Card (LGC) # 1 Recall that the area of a circle of radius is given by A = π𝑟2 A circular sector is a wedge made of a portion of a circle based on the central angle θ (in radians) subtended by an arc on the circle. Since the angle around the entire circle is 2π radians, we can divide the angle of the sector's central angle by the angle of the whole circle 2π to determine the fraction of the circle we are solving for. Then multiply by the area of the whole circle to derive the sector area formula. Lesson No. 3 |Arc Length and Area of a Sector ______________________________________________________________________________
- 9. Engagement Activity 3 Questions: -What can you say about the area of a circular sector? -How do we determine the fraction of the circle we are solving in area of circular sector? Lesson No. 3 |Arc Length and Area of a Sector ______________________________________________________________________________
- 10. Lesson No. 3 |Arc Length and Area of a Sector ____________________________________________________________________ EXPLORE
- 11. Lesson No. 3 |Arc Length and Area of a Sector ____________________________________________________________________ Explore The class will be divided into 8 groups (5-6 members). Each group will be given a problem-based task card to be explored, answered and presented to the class. Inquiry questions from the teacher and learners will be considered during the exploration activity
- 12. Lesson No. 3 |Arc Length and Area of a Sector ____________________________________________________________________ Explore Rubric/Point System of theTask: 0 point – No Answer 1 point – Incorrect Answer/Explanation/Solutions 2 points – Correct Answer but No Explanation/Solutions 3 points – Correct Answer with Explanation/Solutions 4 points – CorrectAnswer/well-Explained/with Systematic Solution
- 13. Lesson No. 3 |Arc Length and Area of a Sector ____________________________________________________________________ Explore Assigned Role: Leader – 1 student Secretary/Recorder – 1 student Time Keeper – 1 Peacekeeper/Speaker – 1 student Material Manager – 1-2 students
- 14. Lesson No. 3 |Arc Length and Area of a Sector ____________________________________________________________________ Explore Problem 1 (Group 1 & Group 2): Minute Hand of a Clock The minute hand of a clock is 6 inches long. (a) How far does the tip of the minute hand move in 15 minutes? (b) How far does it move in 25 minutes?
- 15. Explore Problem 2 (Group 3 & Group 4): Movement of a Pendulum A pendulum swings through an angle of 20° each second. If the pendulum is 40 inches long, how far does its tip move each second? Lesson No. 3 |Arc Length and Area of a Sector ____________________________________________________________________
- 16. Explore Problem 3 (Group 5 & Group 6): Linear Speed v. Angular speed Our earlier “obvious” equation s = rθ, relating arc to angle, also works with measurements of speed. The angular speed of an spinning object is measured in radians per unit of time. The linear speed is the speed a particle on the spinning circle, measure in linear units (feet, meters) per unit of time. Lesson No. 3 |Arc Length and Area of a Sector ____________________________________________________________________
- 17. Lesson No. 3 | Arc Length and Area of a Sector ______________________________________________________________________ Explore Problem 3 (Group 5 & Group 6): Linear Speed v. Angular speed Suppose a merry-go-round is spinning at 6 revolutions per minute. The radius of the merry- go-round is 30 feet. How fast is someone traveling if they are standing at the edge of the merry-go- round?
- 18. Explore Problem 4 (Group 7 & Group 8): Watering a Lawn A water sprinkler sprays water over a distance of 30 feet while rotating through an angle of 135°.What area of lawn receives water? Lesson No. 3 |Arc Length and Area of a Sector ____________________________________________________________________
- 19. Lesson No. 3 |Arc Length and Area of a Sector ____________________________________________________________________ EXPLAIN
- 20. Lesson No. 3 |Arc Length and Area of a Sector ____________________________________________________________________ Explain Group Leader/Representative will present the solutions and answer to the class by explaining the problem/concept explored considering the following questions.
- 21. Lesson No. 3 |Arc Length and Area of a Sector ____________________________________________________________________ Explain Guide Questions: What is the problem all about? What are the given in the problem? What are the things did you consider in solving the given problem? What is/are the unknown in the given problem?
- 22. Explain Guide Questions: What method(s) did you use in solving the given problem? How did you solve the given problem using that method(s)? What particular mathematical concept in trigonometry did you apply to solve the problem-based task? Lesson No. 3 |Arc Length and Area of a Sector ____________________________________________________________________
- 23. Lesson No. 3 |Arc Length and Area of a Sector ____________________________________________________________________ ELABORATE
- 24. Lesson No. 3 |Arc Length and Area of a Sector ____________________________________________________________________ Elaborate Generalization of the Lesson: - -What is the relationship between linear and angular measure of arcs? - What are the steps in solving problems on arc length and area of a sector?
- 25. Lesson No. 3 |Arc Length and Area of a Sector ____________________________________________________________________ Elaborate Integration of PhilosophicalViews In this part, the teacher and learners will relate the terms/content/process learned in the lesson about arc length and area of a sector in real life situations/scenario/instances considering the philosophical views that can be integrated/associated to the term(s)/content/process/skills of the lesson.
- 26. Lesson No. 3 |Arc Length and Area of a Sector ____________________________________________________________________ Elaborate Integration of PhilosophicalViews Questions : What are the things/situations/instances that you can relate with regard to the lesson about arc length and area of a sector in real-life? Considering your philosophical views, how will you relate the terms/content/process of the lesson in real- life situations/instances/scenario?
- 27. Lesson No. 3 |Arc Length and Area of a Sector ____________________________________________________________________ Elaborate Sample PhilosophicalViews Integration from theTeacher: Arc Length and Area of a Sector Circle and radius are one of the terms used in this lesson has many real-life connections. A circle is a line forming a closed-loop; every point on which is a fixed distance from a center point.
- 28. Lesson No. 3 |Arc Length and Area of a Sector _____________________________________________________________________ Elaborate Sample PhilosophicalViews Integration from theTeacher: Arc Length and Area of a Sector Imagine a straight line segment bent around until its ends join, then arrange that loop until it is exactly circular - that is, all points along that line are the same distance from a center point.
- 29. Lesson No. 3 | Arc Length and Area of a Sector _____________________________________________________________________ Elaborate Sample PhilosophicalViews Integration from theTeacher: Arc Length and Area of a Sector Unlike other shapes, a circle has a unique property of being complete. A circle has an extensive meaning; it represents wholeness, totality, original perfection, eternity, infinity, timelessness, self, and all the cyclic movement.
- 30. Elaborate Sample PhilosophicalViews Integration from theTeacher: Arc Length and Area of a Sector According to Hermes Trismegistus, God is a circle whose center is everywhere and whose circumference is nowhere. Lesson No. 3 | Arc Length and Area of a Sector _____________________________________________________________________________
- 31. Lesson No. 3 |Arc Length and Area of a Sector ____________________________________________________________________ Elaborate Sample PhilosophicalViews Integration from theTeacher: Arc Length and Area of a Sector Circle implies the idea of a movement and symbolizes the cycle of time - the perpetual motion of everything that moves like the planet's journey around the sun and the rhythm of the universe. Many people believe that if they have God in them, they are complete, and people who feel complete are stronger and happier
- 32. Lesson No. 3 |Arc Length and Area of a Sector ____________________________________________________________________ Elaborate Sample PhilosophicalViews Integration from theTeacher: Arc Length and Area of a Sector The distance from the center to any point of the circle is known as the radius. Each unit or radius of the circle helps the circle to resist giving into forces putting pressure on it from the outside.
- 33. Lesson No. 3 |Arc Length and Area of a Sector ______________________________________________________________________ Elaborate Sample Philosophical Views Integration from the Teacher: Arc Length and Area of a Sector Similarly, each of this unit is a person's faith. Plenty of this strengthens the grip so as not to be swayed by the evil. Life is a circle because of the same and continues progression from birth and growth to decline and death.
- 34. Lesson No. 3 |Arc Length and Area of a Sector ____________________________________________________________________ EVALUATE
- 35. Lesson No. 3 |Arc Length and Area of a Sector ____________________________________________________________________ Evaluate Solve the following problems: a.The minute hand of a clock is 5 inches long. How far does the tip of the minute hand move in 30 minutes? b. An automatic lawn sprinkler sprays up to a distance of 20 feet while rotating 30 degrees.What is the area of the sector the sprinkler covers?
- 36. Evaluate Solve the following problems: c. Find the area of a sector of a circle with central angle of 7𝜋 6 if the diameter of a circle is 9 cm? d. A swing has 165° angle of rotation. i) If the chains of the swing are 6 feet long, what is the length of the arc that the swing makes? Round your answer to the nearest tenth. ii) Describe how the arc length would change if the length of the chains of the swing were doubled Lesson No. 3 |Arc Length and Area of a Sector ___________________________________________________________________
- 37. Lesson No. 3 |Arc Length and Area of a Sector ____________________________________________________________________ Assignment: Answer the following questions: 1.What are the six trigonometric functions? 2.What is a reference angle? Reference: DepED Pre-Calculus Learner’s Material, pages 129-131. -GNDMJR-