Write your answers on notebook paper. Show your work.

Transcription

1 UNIT 6 Getting Ready Use some or all of these exercises for formative evaluation of students readiness for Unit 6 topics. Prerequisite Skills Finding the length of the sides of special right triangles (Items 1, ) HSG-SRT.C.8 Translating the graph of the parent quadratic function y = x (Item ) HSA-SSE.B. Identifying the coordinates of a point (Items, ).G.A.1 Determining the circumference of a circle (Items 6, 7) 7.G.B. Writing a linear function to model a real-world scenario (Item 8) HSA-CED.A.1, HSA-CED.A., HSF-BF.A.1a Answer Key Answers will vary. Students should point out that the function will be translated 1 unit left and units up, and that the curve will shrink by a factor of 1. The values of a, b, and c in the equation y = a(x b) + c indicate the characteristics of the transformation without graphing.. (, ). (, ) cm 7. π in. 8. C(t) = t; slope = 0.; y-intercept =.7 UNIT 6 Getting Ready Write your answers on notebook paper. Show your work. 1. Find the length of the hypotenuse of a triangle whose shorter leg is units long.. Find the length of one of the legs of a - -0 triangle whose hypotenuse is 6 units long.. Explain how the graph of y = 1 ( x+ 1) + differs from the graph of y = x. Explain how you can determine the differences without graphing.. Identify the coordinates of point C. y A C B x. Identify the coordinates of point F. F E y D 6. Determine the circumference of a circle with a 7.-centimeter radius. Use.1 for π. Round to the nearest hundredth. 7. Determine the circumference of a circle with a -inch diameter. Write your answer in terms of π. 8. Write a function C(t) to represent the cost of a taxicab ride, where the charge includes a fee of $.7 plus $0. for each tenth of a mile t. Then give the slope and y-intercept of the graph of the function. x Getting Ready Practice For students who may need additional instruction on one or more of the prerequisite skills for this unit, Getting Ready practice pages are available in the Teacher Resources at SpringBoard Digital. These practice pages include worked-out examples as well as multiple opportunities for students to apply concepts learned. 01 College Board. All rights reserved. 76 SpringBoard Mathematics Algebra, Unit 6 Trigonometry

2 01 College Board. All rights reserved. Revolving Restaurant Lesson 1-1 Radian Measure Learning Targets: Develop formulas for the length of an arc. Describe radian measure. SUGGESTED LEARNING STRATEGIES: Visualization, Predict and Confirm, Look for a Pattern, Create Representations, Sharing and Responding An architecture firm is designing a circular restaurant that has a radius of 0 feet. It will be situated on top of a tall building, where it will rotate. The lead architect wants to determine how far people seated at different distances from the center of the restaurant will travel as the restaurant rotates through various angles. To start, he will determine how far a customer seated at the window has traveled after a 60 rotation. 1. Attend to precision. How far from the center is a customer seated at the window? Find the circumference of a circle with this distance as the radius. Give an exact answer in terms of π. 0 ft; C = πr, so C = π(0) = 100π ft. What portion of the circumference of the circle is generated by a 60 rotation of the radius? = 6. Use the portion of the circle generated by a 60 rotation of the restaurant to find the approximate distance traveled by this customer. 100π π 1 100π 0π ( 60 )= = =. ft Common Core State Standards for Activity 1 MATH TIP Use the formula C = πr to find circumference. MATH TIP ACTIVITY 1 π (pi) is an irrational number. If you need to provide the exact value of an expression that contains π, leave the symbol in the answer. We say that this answer is written in terms of π. Otherwise, simplify the expression using a numerical approximation for π. Use.1 for π in this unit unless otherwise indicated. HSF-TF.A.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. ACTIVITY 1 Investigative Activity Standards Focus In Activity 1, students are introduced to radian measure. They will use a real-world problem to develop understanding of radian measure and how it differs from degree measure. Students will use what they have learned in previous courses about circles, circumference, central angles, and arcs. Lesson 1-1 PLAN Pacing: 1 class period Chunking the Lesson #1 # # #6 #10 1 #1 #1 Lesson Practice TEACH Bell-Ringer Activity Ask students to identify the constant of proportionality in each direct variation equation. 1. y = 1.x [1.]. x + y = 0. y = 7x [] 1 Visualization, Create Representations, Debriefing Call students attention to the illustration on this page to help them understand and connect the mathematical concepts to the real-world application. Review any contextual language or vocabulary as needed to help students understand the scenario. Students should be familiar with the formula for the circumference of a circle. If students are struggling with finding the circumference, review the formula and how to use it. Discuss what it means to give an exact answer in terms of π. Students should understand that the circumference of the restaurant is 100π feet, or 1 feet. Predict and Confirm, Debriefing Discuss how students should provide the answer to Item. An exact answer is more accurate but will leave the measurement in terms of π. An approximate answer gives a better indication of the distance. The measurement. feet is much more meaningful than 0 for visualizing and π understanding what the distance is. Activity 1 Understanding Radian Measure 77

3 ACTIVITY 1 Continued Differentiating Instruction ACTIVITY 1 Radian Measure To support students in reading problem scenarios, carefully group students to ensure that all students participate and have an opportunity for meaningful reading and discussion. Suggest that group members each read a sentence and explain what that sentence means to them. Group members can then confirm one another s understanding of the key information provided for the problem.. Complete the table by finding the circumference in terms of π for diners at the specified distances in feet from the center of the restaurant. Also find the exact distances (in terms of π) and approximate distances traveled for diners when the restaurant rotates 60. Radius (feet) 0 0 Circumference (feet) Distance Traveled During a 60º Rotation (feet) 100π 0π. 80π 80π 1 80π 0π ( 1 6 )= =. 6 Look for a Pattern, Create Representations, Construct an Argument Before completing the table, ask students to predict whether a diner 0 feet from the center will travel twice the distance as a diner 0 feet from the center and whether a diner 0 feet from the center will travel twice the distance as a diner 10 feet from the center. This will get students thinking about whether the relationship between the length of the radius and the distance traveled is a direct variation π 60π 1 60π ( 10π 1 6 )= =. 6 0π 0π 1 0π 0π ( 0 6 )= =. 6 0π 0π 1 0π 10π ( 10 6 )= =. 6 π π 1 π π ( 1 6 )= = 6. Describe any pattern in the exact distance traveled. Each distance traveled is the radius multiplied by the ratio π. Developing Math Language Read, or have a student read, the definition of arc length to the class. Discuss how arc length differs from the measure of an arc. This lesson and the next include several new vocabulary words. Pronounce new terms clearly, as needed, and monitor students pronunciation of terms in their class discussions. Use the classroom Word Wall to keep new terms in front of students. Include pronunciation guides, as needed. Encourage students to review the Word Wall regularly to choose words to add and to monitor their own understanding and use of new terms in their group discussions. 6 Construct an Argument, Create Representations, Debriefing Have students confirm that their formula works using the information in the table. Ask students to describe how they know the relationship between s and r is proportional. MATH TIP The variable r is used to represent radius in formulas. The variable s is often used to represent distance. The arc length is the length of a portion of the circumference of a circle. The arc length is determined by the radius of the circle and by the angle measure that defines the corresponding arc, or portion, of the circumference. 6. Model with mathematics. Write a formula that represents the arc length s of a 60 angle with a radius r. Describe the relationship between s and r. s= π ( ) r; s and r are proportional. 01 College Board. All rights reserved. 78 SpringBoard Mathematics Algebra, Unit 6 Trigonometry

4 Radian Measure 7. Identify the constant of proportionality in the formula in Item Use the formula in Item 6 to find the approximate distance a diner will travel when seated at each of the following distances from the center of the restaurant. a. 1 feet b. 8 feet. How far has a diner, seated feet from the restaurant center, traveled after rotating 10? Explain how you found your answer. 10. Find the exact distances (in terms of π) and the approximate distances traveled by diners seated at the indicated distances from the center after the restaurant rotates 0. Fill in the table. Radius (feet) Distance Traveled During a 60 Rotation (feet) 0π 0 0π 1 0π ( π )= = =. 0π 1 0π ( 10π 1 )= =. 60π 1 60π ( 1π 7 1 )= =. 80π 1 80π ( 0π 6 8 )= =. MATH TIP ACTIVITY 1 Recall that in the direct variation equation y = kx, x and y are proportional and k is the constant of proportionality. ACTIVITY 1 Continued Debrief students answers to these items to ensure that they understand why π is the constant of proportionality. Reinforce that π is an irrational number and not a variable. Answers 7. π 8. a. π 1.6 ft b. 8 π. 8 ft. 0 π. ft 10 1 Look for a Pattern, Create Representations, Construct an Argument Ask students to describe how the information in the table for Item 10 is similar to and different from the information in the table for Item. Students should see that the relationship between the distance a diner is from the center and the distance that diner travels is still a proportional relationship and that what differs between the two tables is the constant of proportionality. After completing Item 1, students should be able to write a formula for arc length for any radius and angle measure π 1 100π ( π 78 )= =. 11. Reason quantitatively. Write a formula that represents the arc length s generated by a radius r that rotates 0. Compare and contrast this with the formula you wrote in Item 6. The formula is s= ( π ) r. The constants of proportionality are different. 01 College Board. All rights reserved. 1. In Item, you found the length of the arc s generated by the 10 rotation of a -foot radius r. What is the constant of proportionality in a formula that defines s in terms of r for 10? Give an exact answer in terms of π. π Activity 1 Understanding Radian Measure 7

5 ACTIVITY 1 Continued Paragraphs Close Reading, Think Aloud Students are introduced to the unit circle. They will revisit the unit circle in Activity when studying trigonometric relationships in which the unit circle is represented on the coordinate plane and used to define trigonometric functions. Students are familiar with angles being measured in degrees. Discuss the definition of radians and compare radian measure to degree measure so that students understand the difference. ACTIVITY 1 Radian Measure As you can see, the constant of proportionality used to find arc length s in terms of radius r is different for each angle of rotation. When you find the arc length generated by a radius on a circle with radius 1, called a unit circle, you will find that the constant of proportionality takes on additional meaning. 1 Technology Tip To reinforce understanding of radian measure, have students use a graphing calculator to find values of trigonometric expressions. In degree mode: Enter sin (60) displays Enter sin ( π ) displays In radian mode: Press MODE and change from Degree to Radian measure. Enter sin (60) displays Enter sin ( π ) displays They should note that sin (60) in degree mode has the same value as sin ( π ) in radian mode. Students should consult their manuals if they are using a calculator other than a TI-Nspire. For some students, writing this process in their notes will be helpful as they can refer to it again and again as they work through the course. For additional technology resources, visit SpringBoard Digital. 1 Create Representations, Look for a Pattern Discuss the arc length generated by a rotation of 180. Students should see that this is one-half the distance around the circle or one-half of the circumference. Tell students to write an expression for one-half of the circumference, using the formula for circumference, before completing Item 1. This will help them understand the constant of proportionality and the value of s. Make sure students understand that s equals π because r equals 1. Elicit from students the fact that on a unit circle this means that the measure of the angle in radians equals the length of the arc generated by that angle. MATH TERMS The angle of rotation is measured in degrees or radians. An angle s measurement in radians equals the length of a corresponding arc on the unit circle. Radian measures are often written in terms of π. 1 Predict and Confirm, Create Representations, Debriefing Students can develop a definition of radian measure by calculating arc lengths and finding a pattern in the ratio of the arc length to the radius. Relate the definition of radian measure to the work students did in Items 7. In those items, the fixed angle measure is 60, and the constant of proportionality is π, meaning that a 60 angle has a radian measure of π. Point out to students that π will often be a part of radian measure of an angle and that ratios should always be written in simplest form. 1. Model with mathematics.write a formula for s in terms of r on a unit circle when the angle of rotation is 180. Identify the constant of proportionality. Also identify the value of s. C = πr = π(1) = π π 180 ( π π 60 )= = s = πr The constant of proportionality is π. The value of s is π. On a unit circle, the constant of proportionality is the measure of the angle of rotation written in radians, which equals the length of the corresponding arc on the unit circle. For example, we say that 180 equals π radians. We can use this fact about the relationship between s and r on the unit circle to convert degree measures to radian measures. It may be helpful to write these as proportions. 1. Convert each degree measure to radians. Give the answers in terms of π. a. 0 π b. π c. 60 π 6 1. A circle has a radius of 1 feet. What is the length of the arc generated by a angle? 16. What is the arc length generated by the 0 angle rotation on a circle that has a radius of inches? 17. Convert each degree measure to radians. a. 1 b. 10 c. 70 Debrief students answers to these items to ensure that they can apply the formula they developed in Item 1. Answers 1. 1π ft 16. π 1. in. 17. a. π b. π c. π 01 College Board. All rights reserved. 80 SpringBoard Mathematics Algebra, Unit 6 Trigonometry

6 Radian Measure 18. What is the length of the arc formed by a 0 angle on a circle with a radius of 68 feet? 1. Attend to precision. What is the constant of proportionality for each angle measure? Write each answer in terms of π. a. 0 b. 0. Find the length of an arc formed by a 7 angle on a circle with a radius of feet. Give the answer in terms of π. 1. Convert each degree measure to radians. a. b. 80 Use the following information for Items. A diner has a circular dessert case in which the shelves inside rotate, but pause at set increments. Yesterday the restaurant manager decided to have the shelves pause every 60.. How far did a lemon tart travel between each pause if it was placed on a shelf at a radius of 8 inches?. Express regularity in repeated reasoning. How far does a custard travel between each pause if it is placed at a radius of 1 inches? CONNECT TO AP ACTIVITY 1 In calculus, all angles are assumed to be measured in radians. ACTIVITY 1 Continued ASSESS Students answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the lesson concepts and their ability to apply their learning. See the Activity Practice for additional problems for this lesson. You may assign the problems here or use them as a culmination for the activity. LESSON 1-1 PRACTICE 18. π ft 1. a. π b. π 0. 17π ft 1 1. a. 7 π 6 b. π. 8π 8. in.. π 1.6 in. ADAPT Check students answers to the Lesson Practice to ensure that they understand the meaning of s and r in the formula s = θr, where θ is the measure of an angle, in radians. It is easy for students to lose sight of what they are trying to find and what the answer means when first working with unit circles and radian measure. Have students describe what s, θ, and r represent and their units of measure. 01 College Board. All rights reserved. Activity 1 Understanding Radian Measure 81

7 ACTIVITY 1 Continued Lesson 1- PLAN Pacing: 1 class period Chunking the Lesson #1 #6 7 #11 1 Lesson Practice TEACH Bell-Ringer Activity Ask students to find the arc length of the given angle measure and radius. Have them give exact answers in terms of π angle; r = 1 ft [π ft]. 0 angle; r =. m [1.1π m]. 10 angle; r = 6 in. [π in.] 1 Work Backward, Create Representations, Look for a Pattern, Debriefing Help students understand that when they convert from radians to degrees, they need to multiply radians by the ratio that cancels radian measure and keeps degree measure. This equation may help them: radian degrees = degrees. When radians students convert degrees to radians, in the Lesson Practice, elicit the following equation from them: degrees radians =radians degrees ACTIVITY 1 Learning Targets: Develop and apply formulas for the length of an arc. Apply radian measure. Applying Radian Measure SUGGESTED LEARNING STRATEGIES: Create a Plan, Look for a Pattern, Work Backward, Share and Respond, Create Representations Angle measures can be given in degrees or radians. Angle measures in degrees are converted to radians to find arc length. Since we generally think of angles in degrees, it is useful to also know how to convert radian measures to degrees. 1. In Lesson 1-1, you found that 180 = π radians. What ratio can you multiply π radians by to convert it back to 180? 180 π. Does this ratio also help you convert π radians to 0? Show how you determined your answer. yes; π 180 = 0 π. Make use of structure. How can you convert an angle measure given in radians to degrees? Multiply the angle measure given in radians by the ratio 180 π.. Convert the following angles in radians to degrees. a. π 6 b. π c. π 70 Universal Access Sometimes students think of π as a variable. Remind students that π is an irrational number that we often estimate as.1. Knowing this, students can get an idea of the size of one radian: Since 180 is approximately.1 radians, one radian is approximately 7. Sometimes angles greater than 60 are also given in radians.. Convert the following angles in radians to degrees. a. 7 π 0 b. 6 π 0 c. 11 π 01 College Board. All rights reserved. 8 SpringBoard Mathematics Algebra, Unit 6 Trigonometry

8 Applying Radian Measure 6. Given an angle in radian measure, how can you determine if the degree measure is less than or greater than 180 before doing the conversion? 1π = 180 ; A coefficient less than 1 means the angle in degree measure is less than 180º. If the coefficient of π is greater than 1, the angle in degree measure is greater than 180º. 7. Given an angle in radian measure, how can you tell if the degree measure is greater than 60 before you do the conversion? π = 60 ; If the coefficient of π is less than, the angle in degree measure is less than 60º. A coefficient greater than means the angle in degree measure is greater than 60º. 8. Convert the following angles in radians to degrees. a. 7 π b. 8 π. a. Is 6 π radians greater than or less than 180? Than 60? ACTIVITY 1 ACTIVITY 1 Continued 6 7 Look for a Pattern, Create Representations, Debriefing Discuss what it means for an angle measure to be greater than 60. Students may think that 60, or π radians, is the greatest possible measure around a circle. Ask students to think about a structure that revolves, such as a carousel or the revolving restaurant, and discuss whether these structures stop after one revolution. Ask students how they would measure the second revolution, the third revolution, and so on, to help them understand these angle measures. Debrief students answers to these items to ensure that they can convert between radian measure and degree measure with ease. 01 College Board. All rights reserved. b. Convert 6 π radians to degrees. 10. Construct viable arguments. Before converting, how can you tell if a radian angle measure will be between 180 and 60? Let s think about the rotating restaurant from Lesson 1-1. You concluded that the distance traveled by a diner in the restaurant could be found using S = ( π ) r for a 60 angle. You also now know that π radian is equal to Express regularity in repeated reasoning. Write a formula to find arc length s traveled by a diner in the restaurant for any radian angle measure θ and any radius r. S = θ(r) The designers decide that the restaurant should do one complete rotation every 0 minutes. 1. Approximately how far will a diner seated at a radius of 0 feet travel after dining for 1 hour, 0 minutes? S = π 0 ft = 80π ft 1. ft MATH TIP The Greek symbol theta (θ) is often used to represent an angle measure in a formula. Answers 8. a. 1 b. 80. a. It is greater than 180 and less than 60. b if it is greater than π and less than π 11 1 Create a Plan, Look for a Pattern, Create Representations, Debriefing, Discussion Groups Make sure students understand that the restaurant will make two revolutions in 1 hour 0 minutes. A common error is to use π instead of π for θ because there are two revolutions. Remind students that π = 180, and π = 60 = 1 revolution. Monitor students group discussions to ensure that complex mathematical concepts are being verbalized precisely and that all group members are actively participating in discussions through sharing ideas and through asking and answering questions appropriately. Activity 1 Understanding Radian Measure 8

9 ACTIVITY 1 Continued Debrief students answers to these items by asking them how they can solve Item 1 based on their answer to Item 1. Students should realize that the travel time for both diners is the same and that the diner in Item 1 is five times farther from the center than the diner in Item 1 is. Therefore, the diner in Item 1 will travel five times the distance of the diner in Item 1. 0π = 100π. Answers 1. 0π. ft 1. 10π 71 ft min ASSESS ACTIVITY 1 Applying Radian Measure 1. Approximately how far will a diner seated at a radius of 0 feet travel after dining for 1 hour 0 minutes? S = π 0 ft = 00π ft 68 ft 1. How far will a diner seated 10 feet from the center of the restaurant travel in 1 hour? 1. How far will a diner seated 0 feet from the center travel in 1 hour? 16. How long does it take a diner seated 0 feet from the center to travel the distance that the diner seated 10 feet from the center travels in 1 hour? Students answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the lesson concepts and their ability to apply their learning. See the Activity Practice for additional problems for this lesson. You may assign the problems here or use them as a culmination for the activity. 17. Reason quantitatively. Convert the following radians to degrees. a. π b. π c. π LESSON 1- PRACTICE 17. a. 1 b. 1 c π 1.7 in. 0. π 16.6 in. 1. No; π is greater than π, so it is greater than 60 ; A diner in a rotating restaurant is seated and travels π radians before the waiter comes to the table. How many degrees does he travel before the waiter arrives? 1. A rotating dessert case does a full rotation every minutes. How far will a dessert item travel in 0 minutes if placed at a radius of 6 inches? 0. The dessert case in Item 1 is sped up so that it does a complete rotation every minutes. How far will a piece of dessert travel in 1 minutes if placed at a radius of inches? 1. Critique the reasoning of others. Kyle says the radian angle measure π is between 180 and 60. Is he correct? Explain your thinking. How many degrees is π radians? ADAPT Check students answers to the Lesson Practice to ensure that they can convert radians to degrees. A quick way to do this is to substitute 180 for π radians. 01 College Board. All rights reserved. 8 SpringBoard Mathematics Algebra, Unit 6 Trigonometry

10 Revolving Restaurant Write your answers on notebook paper. Show your work. Lesson What is the approximate length of the arc formed by a 0 angle on a circle that has a radius of 70 feet? A. ft B. 110 ft C. 0 ft D. 0 ft. A horse on a merry-go-round is positioned at a radius of 1 feet. How far will the horse travel after the merry-go-round rotates 60? A. 1.7 ft B..6 ft C. 1. ft D. 7.1 ft. A ticketholder is sitting on a bench that is on the merry-go-round. The ticketholder is sitting at a radius of 10 feet from the center. Approximately how far will the ticketholder travel after traveling 180 on the ride?. Several ticketholders are standing at various positions on the merry-go-round. Find the approximate distance ticketholders standing at the following radii will travel after the merry-go-round rotates 10. a. 11 feet b. 1 feet c. 16 feet. Use the unit circle. What is the constant of proportionality for each of the following angles? Give your answer in terms of π. Find the arc lengths in Items 6 and in. 7. cm ACTIVITY 1 8. Find the length of the arc formed by each angle and the given radius. a. radius: 0 in., angle: 0 b. radius: 1 m, angle: 0 c. radius: 8 ft, angle: 7. How many radians equal? 10. Convert each degree measure to radians. a. 8 b. c. 160 d. 10 ACTIVITY 1 Continued ACTIVITY PRACTICE 1. B. A. 10π 1. ft. a. ft b.. ft c.. ft. a. π units 1 b. π units c. π units d. π units π 117. in π 8. 8 cm 8. a. 0 π 1. 0 cm b. 6π 18.8 m c. π. 7 ft 6. π 10. a. π 1 b. π 10 c. 8 π d. π 1 unit a. b. 00 c. 7 d College Board. All rights reserved. Activity 1 Understanding Radian Measure 8

11 ACTIVITY 1 Continued 11. a. 18 b. 10 c. 80 d. 1 e. 0 f. 600 g. 108 h less 1. less 1. greater 1. equal 16. 6π. 6 ft 17. π ft 18. 6π 17.8 ft 1. π 10.6 ft 0. π 11. ft; 0 ; Check students explanations. ADDITIONAL PRACTICE If students need more practice on the concepts in this activity, see the Teacher Resources at SpringBoard Digital for additional practice problems. ACTIVITY 1 Lesson Convert the following radian angle measures to degrees: a. π b. π 10 6 d. 7 π c. 8 π e. 11 π g. π f. 10 π h. π 1. Is π radians greater than, less than, or equal to 180? 1. Is π radians greater than, less than, or equal to 180? 1. Is π radians greater than, less than, or equal to 60? 1. Is π radians greater than, less than, or equal to 60? 16. A ticketholder on the merry-go-round is riding a horse that is at a radius of 1 feet. How far does she travel after the merry-go-round rotates π radians? Revolving Restaurant Use the following information for Items A merry-go-round makes one complete rotation every 80 seconds. 17. Approximately how far will a ticketholder seated at a radius of 1 feet travel after 60 seconds? 18. Approximately how far will a ticketholder standing at a radius of 16 feet travel after 10 seconds? 1. Approximately how far will a ticketholder seated at a radius of 1 feet travel after 110 seconds? MATHEMATICAL PRACTICES Reason Abstractly and Quantitatively 0. A ticketholder seated at a radius of 1 feet rode the merry-go-round for 10 seconds. Find the distance the ticketholder traveled. What is the measure of the angle over which the ticketholder rotated in degrees? Explain how you found your answer. 01 College Board. All rights reserved. 86 SpringBoard Mathematics Algebra, Unit 6 Trigonometry