Enhanced Instructional Transition Guide

Transcription

1 Enhanced Instructional Transition Guide High School Courses Unit Number: 03 /Mathematics Suggested Duration: 4 days Unit 03: Introduction to Trigonometry (15 days) Possible Lesson 01 (6 days) Possible Lesson 02 (4 days) Possible Lesson 03 (5 days) Possible Lesson 02 (4 days) This lesson is one approach to teaching the State Standards associated with this unit. Districts are encouraged to customize this lesson by supplementing with district-approved resources, materials, and activities to best meet the needs of learners. The duration for this lesson is only a recommendation, and districts may modify the time frame to meet students needs. To better understand how your district is implementing CSCOPE lessons, please contact your child s teacher. (For your convenience, please find linked TEA Commissioner s List of State Board of Education Approved Instructional Resources and Midcycle State Adopted Instructional Materials.) Lesson Synopsis: Students practice converting rates and units using dimensional analysis in real-world situations and examples. Students apply these skills in developing the concept of radian measure for angles. Radian measure is then applied to formulas involving arc length, sector area, and angular and linear velocity. TEKS: The Texas Essential Knowledge and Skills (TEKS) listed below are the standards adopted by the State Board of Education, which are required by Texas law. Any standard that has a strike-through (e.g. sample phrase) indicates that portion of the standard is taught in a previous or subsequent unit. The TEKS are available on the Texas Education Agency website at P.3 /Knowledge and Skills. The student uses functions and their properties, tools and technology, to model and solve meaningful problems. The student is expected to: P.3E Solve problems from physical situations using trigonometry, including the use of Law of Sines, Law of Cosines, and area formulas and incorporate radian measure where needed. Performance Indicator(s): page 1 of 47

2 Enhanced Instructional Transition Guide High School Courses Unit Number: 03 /Mathematics Suggested Duration: 4 days High School Mathematics Unit03 PI03 Investigate and solve a problem involving arc length and sectors of circles such as the following: For the figure below: A) convert the measure of the central angle to radians, B) find the length of the intercepted arc, and C) find the area (shaded) of the sector formed. Create a graphic organizer for the problem that includes a diagram of the problem situation, applicable formulas, and calculations. Standard(s): P.3E ELPS ELPS.c.1C, ELPS.c.5F page 2 of 47

3 Enhanced Instructional Transition Guide High School Courses Unit Number: 03 /Mathematics Suggested Duration: 4 days High School Mathematics Unit03 PI04 Investigate and solve problems involving linear and angular velocity incorporating radian measure such as the following: The radius of the Earth is approximately 4,000 miles. A satellite orbits 5 miles above its surface. The satellite completes 220 of a revolution around the Earth in a day. How many miles did the satellite travel in orbit during the day? What was the average speed of the satellite in miles per hour? A piece of exercise equipment uses a two-pulley system that allows users to lift weights by pulling up on a handle attached to a cord. The upper pulley has a radius of 6 cm, and the lower pulley s radius is 4 cm. An athlete lifts the weights so that they rise at the rate of 10 cm/sec. Find the angular velocity of the upper pulley in degrees per second. Find the angular velocity of the lower pulley in degrees per second. The motor in a clock is powered to turn a big gear (B, of radius 5 units) at a rate of 144 every minute. The second hand on the clock, however, needs to turn at a rate of 1 revolution per minute. So, the big gear (B) is coupled with a smaller gear (S) which will turn at a faster angular velocity. Find the angular velocity of gear B in radians per minute. Find the linear velocity of the teeth on the rim of gear B (in units per minute). Determine the radius that could be used for gear S so that it has the same linear velocity as gear B, but rotates at 1 revolution per minute. Create a graphic organizer for each problem that includes a diagram of the problem situation, applicable formulas, and calculations. Use the information to make predictions in terms of the problem situation. Standard(s): P.3E ELPS ELPS.c.1C, ELPS.c.5B, ELPS.c.5G Key Understanding(s): Angles measured in degrees can be converted into radians. Radian measure connects arc lengths and central angles of circles. Radian measure can be extended to describe angles of rotation. Rotation of an object can be measured using angular velocity, speed the angle rotates, and linear velocity, speed the object travels. page 3 of 47

4 Enhanced Instructional Transition Guide High School Courses Unit Number: 03 /Mathematics Suggested Duration: 4 days Misconception(s): Some students may think that is the symbol for radians (instead of the number of radians in 180 ). In fact, radian measure is intended to be unit less (or, without a unit). Ex: is approximately equal to the number 1.57; it does not mean half a radian. Vocabulary of Instruction: angular velocity linear velocity radian measure Materials List: graphing calculator (1 per student) graphing calculator with display (1 per teacher) Attachments: All attachments associated with this lesson are referenced in the body of the lesson. Due to considerations for grading or student assessment, attachments that are connected with Performance Indicators or serve as answer keys are available in the district site and are not accessible on the public website. Making Tracks KEY Making Tracks Conversions with Dimensional Analysis KEY Conversions with Dimensional Analysis page 4 of 47

5 Enhanced Instructional Transition Guide High School Courses Unit Number: 03 /Mathematics Suggested Duration: 4 days Inside Track KEY Inside Track Radian Measure of Angles KEY Radian Measure of Angles Circular Reasoning KEY Circular Reasoning Circular Speed KEY Circular Speed Common Velocities KEY Common Velocities Angles, Circles, Velocity KEY Angles Circles Velocity PI Suggested Day Suggested Instructional Procedures Notes for Teacher 1 2 Topics: ATTACHMENTS Rate ( ) Dimensional analysis Teacher Resource: Making Tracks KEY (1 per teacher) Handout: Making Tracks (1 per student) page 5 of 47

6 Enhanced Instructional Transition Guide High School Courses Unit Number: 03 /Mathematics Suggested Duration: 4 days Suggested Day Suggested Instructional Procedures Notes for Teacher Engage 1 Students investigate rate around a track in terms of distance/time. Students calculate conversions using dimensional analysis. Instructional Procedures: 1. Facilitate a class discussion on track. Ask: Have you ever run track or observed people who were running track? Answers may vary. What shape is a track? Answers may vary. A rounded rectangle or a rectangle with semi-circular ends; etc. What is the distance around a track (one lap)? (approximately a quarter mile, or 400 m) About how fast can a person run around the track once? Answers may vary. Fast times may be around 50 seconds to a minute; etc. 2. Place students in pairs. Distribute handout: Making Tracks to each student. Instruct students to work with their partner to complete problems 1 2. Allow students time to complete the problems, and monitor to check for student understanding. Facilitate a class discussion of student results, clarifying any misconceptions. For problem 2B, emphasize the use of dimensional analysis. 3. Distribute handout: Conversions with Dimensional Analysis to each student. Refer students to the top of page 1. Display teacher resource: Conversions with Dimensional Analysis, and facilitate a class discussion of dimensional analysis, discussing the sample problem and modeling problem 1. Referencing the Sample problem, ask: Teacher Resource: Conversions with Dimensional Analysis KEY (1 per teacher) Teacher Resource: Conversions with Dimensional Analysis (1 per teacher) Handout: Conversions with Dimensional Analysis (1 per student) MATERIALS graphing calculator (1 per student) graphing calculator with display (1 per teacher) TEACHER NOTE Students should have addressed dimensional analysis in their science classes in high school. When performing dimensional analysis, be sure to emphasize the cancelation of units. The teacher can stop after each factor is used to ask, What units are we in now? For example, after using the second ratio (above), the rate has page 6 of 47

7 Enhanced Instructional Transition Guide High School Courses Unit Number: 03 /Mathematics Suggested Duration: 4 days Suggested Day Suggested Instructional Procedures How do you write 40 mph as a ratio? ( ) Notes for Teacher changed from miles per hour to feet per hour. Why is multiplying by equal to one another, so the fraction equals 1 (like ) the same as multiplying by 1? (The numerator and denominator are Once the third ratio is included, and units canceled, the rate is now in feet per minute (and so on). Why was 1 mile placed on the bottom of the fraction and not the top? (To cancel (or convert) the miles and replace it with feet.) How is the strategy of canceling units with dimensional analysis different from setting up proportions? Answers may vary. Proportions are equations to solve, where two sets of like units must be compared in the same way. This method uses multiplication, where any number of units can be included; etc. Referencing problem 1, ask: How do you cancel feet on the top of a rate? (Multiply by a fraction with feet on the bottom.) What is something related to feet you can write on the top of that fraction? (Miles, since 1 mile = 5280 feet.) Now, how can you cancel the seconds on the bottom of the rate? (Multiply by a fraction with seconds on the top.) What is something related to feet you can write on the top of that fraction? (Minutes, since 1 minute = 60 seconds.) 4. Instruct students to work with their partner to complete the remaining problems on handout: Conversions with Dimensional Analysis. Allow students time to complete the problems, and monitor to check for student understanding. Using teacher resource: Conversions with Dimensional Analysis, facilitate a page 7 of 47

8 Enhanced Instructional Transition Guide High School Courses Unit Number: 03 /Mathematics Suggested Duration: 4 days Suggested Day Suggested Instructional Procedures class discussion of student results, clarifying any misconceptions. Notes for Teacher Topics: ATTACHMENTS Linear velocity Angular velocity Radian measure of angles Arc length Sector area Explore/Explain 1 Students compare and contrast linear and angular velocity. Students investigate angle rotation and measuring angles in radians. Students apply circular reasoning to solve problems of arc length and sector area. Instructional Procedures: 1. Near the end of Day 1, distribute handout: Inside Track to each student. Refer students to the introduction to the problem about Isaac and Oscar on page 1. Display teacher resource: Inside Track, and facilitate a discussion of the problem. Ask: Teacher Resource: Inside Track KEY (1 per teacher) Teacher Resource: Inside Track (1 per teacher) Handout: Inside Track (1 per student) Teacher Resource: Radian Measure of Angles KEY (1 per teacher) Teacher Resource: Radian Measure of Angles (1 per teacher) Handout: Radian Measure of Angles (1 per student) Teacher Resource: Circular Reasoning KEY (1 per teacher) Handout: Circular Reasoning (1 per student) If Isaac and Oscar are dead even as they run on the curve, who is really going faster and how do you know? (Oscar) Answers may vary. Because he is in the outside lane, he must travel further in the same amount of time; etc. How can we determine the distance Oscar runs around the curved part of the track? (Use the circumference formula but take half because it s a semi circle.). MATERIALS graphing calculator (1 per student) graphing calculator with display (1 per teacher) page 8 of 47

9 Enhanced Instructional Transition Guide High School Courses Unit Number: 03 /Mathematics Suggested Duration: 4 days Suggested Day Suggested Instructional Procedures Notes for Teacher 2. Instruct students to work with their partner to complete the table on page 1 and problems 1 4 on handout: Inside Track. Allow students time to complete the problems, and monitor to check for student understanding. Using teacher resource: Inside Track, facilitate a class discussion of student results, clarifying any misconceptions. 3. Instruct students to work with their partner to complete problems 5 8 on handout: Inside Track. Allow students time to complete the problems, and monitor to check for student understanding. Using teacher resource: Inside Track, facilitate a class discussion of student results, clarifying any misconceptions. On problem 8, explain to students that the ratio of an arc length to the radius is a unit of measure called radians. 4. Distribute handout: Radian Measure of Angles to each student. Refer students to the top of page 1. Display teacher resource: Radian Measure of Angles, and facilitate a class discussion of radian measure, modeling problems 1 and 5. Instruct students to work independently to complete problems 2 4 and 6 8, and compare answers with their partner. Allow students time to complete the problems, and monitor to check for student understanding. Using teacher resource: Radian Measure of Angles, facilitate a class discussion of student results, clarifying any misconceptions. Emphasize that radian measure can be written in terms of (such as 90 = ), or as a decimal ( = 1.57 ). 5. Refer students to Formulas Using Radians on handout: Radian Measure of Angles. Using teacher resource: Radian Measure of Angles, facilitate a class discussion on formulas that incorporate radians. Emphasize that the degree measure must first be converted into radians before using either formula. Instruct students to work with their partner to complete problems Allow students time to complete the problems, and monitor to check for student understanding. Using teacher resource: Radian Measure of Angles, facilitate a class discussion of student results, clarifying any misconceptions. 6. Distribute handout: Circular Reasoning to each student. Instruct students to work independently to TEACHER NOTE Handout: Inside Track bridges across Day 1 and Day 2. If necessary, portions of the handout can be assigned as homework for Day 1. TEACHER NOTE Students may ask why radian measure of angles is needed or necessary. One reason is that problems involving arc length become quite simple when radian measure is used. TEACHER NOTE Sometimes teachers describe radian measure as wrapping a number line around a circle of radius 1. TEACHER NOTE Compare the arc length and sector area formulas in radians to their counterparts that use degree measure of the central angle (m): Emphasize that these formulas have the conversion to radians built into them (with factors such as page 9 of 47

10 Enhanced Instructional Transition Guide High School Courses Unit Number: 03 /Mathematics Suggested Duration: 4 days Suggested Day Suggested Instructional Procedures Notes for Teacher complete the handout. This may be assigned as homework, if necessary. ). 3 Topics: Linear velocity Angular velocity Explore/Explain 2 Students convert between linear and angular velocity in problem situations using appropriate formulas. ATTACHMENTS Teacher Resource: Circular Speed KEY (1 per teacher) Teacher Resource: Circular Speed (1 per teacher) Handout: Circular Speed (1 per student) Instructional Procedures: 1. Facilitate a class discussion to debrief handout: Circular Reasoning to check for understanding. 2. Place students in pairs. Distribute handout: Circular Speed to each student. Refer students to Merry-Go- Round on page 1. Display teacher resource: Circular Speed, and facilitate a class discussion of the problem. Ask: Where should you ride on the merry-go-round if you wanted to get dizzy? Answers may vary. Would it make you dizzier if you stood in the middle of the merry-go-round as it spins, or if you got on your back and hung your head off the edge as it spins? Answers may vary. At the center, you are spinning faster, but on the edge, you are moving faster; etc. MATERIALS graphing calculator (1 per student) graphing calculator with display (1 per teacher) TEACHER NOTE Remind students that when the angular velocity is in radians per time, to find the linear velocity, they must multiply by the radius. 3. Instruct students to work with their partner to complete problems 1 4 on handout: Circular Speed. Allow students time to complete the problems, and monitor to check for student understanding. Using teacher resource: Circular Speed, facilitate a class discussion of student results, clarifying any TEACHER NOTE Some problems require students to solve for the angular velocity. When you divide the linear velocity page 10 of 47

11 Enhanced Instructional Transition Guide High School Courses Unit Number: 03 /Mathematics Suggested Duration: 4 days Suggested Day Suggested Instructional Procedures Notes for Teacher misconceptions. On problem 4, make sure students differentiate between the 2 types of velocity. 4. Refer students to the middle of page 1 on handout: Circular Speed. Using teacher resource: Circular Speed, facilitate a class discussion of angular and linear velocity, modeling problem 5. by the radius, the linear units cancel out resulting in the unit less measures known as radians. The resulting answer is the angular velocity. 5. Instruct students to work with their partner to complete problems 6 8 on handout: Circular Speed. This may be completed as homework, if necessary. Topics: ATTACHMENTS Linear velocity Angular velocity Application of velocity Elaborate 1 Students continue to apply linear and angular velocity in problem situations. Teacher Resource: Common Velocities KEY (1 per teacher) Teacher Resource: Common Velocities (1 per teacher) Handout: Common Velocities (1 per student) Instructional Procedures: 1. Place students in pairs. Distribute handout: Common Velocities to each student. Refer students to page 1. Display teacher resource: Common Velocities, and facilitate a class discussion of common velocities where rotating objects in machines are paired together so that they have the same angular or linear velocity. Model problem 1. Ask: MATERIALS graphing calculator (1 per student) graphing calculator with display (1 per teacher) Look at the wheels on the Crazy Car. Would they have the same angular velocity, or the same linear velocity? (linear) Why? (The wheels are connected by a line (the ground). Also, the car can only go at one linear TEACHER NOTE On problem 4C on handout: Common Velocities, the answers snake their way through the table in page 11 of 47

12 Enhanced Instructional Transition Guide High School Courses Unit Number: 03 /Mathematics Suggested Duration: 4 days Suggested Day Suggested Instructional Procedures Notes for Teacher speed.) the following order. 2. Instruct students to work with their partner to complete the remaining problems on handout: Common Velocities. This may be completed as homework, if necessary. State Resources TEXTEAMS Algebra II/ Institute Section V.1 Modeling Circular Motion may be used to reinforce these concepts or used as alternate activities. 4 Evaluate 1 Instructional Procedures: 1. Assess student understanding of related concepts and processes by using the Performance Indicator(s) aligned to this lesson. ATTACHMENTS Teacher Resource (optional): Angles, Circles, Velocity KEY (1 per teacher) Handout (optional): Angles, Circles, Velocity PI (1 per student) Performance Indicator(s): page 12 of 47

13 Enhanced Instructional Transition Guide High School Courses Unit Number: 03 /Mathematics Suggested Duration: 4 days Suggested Day Suggested Instructional Procedures High School Mathematics Unit03 PI03 Investigate and solve a problem involving arc length and sectors of circles such as the following: MATERIALS Notes for Teacher graphing calculator (1 per student) For the figure below: A) convert the measure of the central angle to radians, B) find the length of the intercepted arc, and C) find the area (shaded) of the sector formed. TEACHER NOTE As an optional assessment tool, use handout (optional): Angles, Circles, Velocity PI. Create a graphic organizer for the problem that includes a diagram of the problem situation, applicable formulas, and calculations. Standard(s): P.3E ELPS ELPS.c.1C, ELPS.c.5F High School Mathematics Unit03 PI04 Investigate and solve problems involving linear and angular velocity incorporating radian measure such as the following: The radius of the Earth is approximately 4,000 miles. A satellite orbits 5 miles above its surface. The satellite completes 220 of a revolution around the Earth in a day. How many miles did the satellite travel in orbit during the day? What was the average speed of the page 13 of 47

14 Enhanced Instructional Transition Guide High School Courses Unit Number: 03 /Mathematics Suggested Duration: 4 days Suggested Day Suggested Instructional Procedures satellite in miles per hour? A piece of exercise equipment uses a two-pulley system that allows users to lift weights by pulling up on a handle attached to a cord. The upper pulley has a radius of 6 cm, and the lower pulley s radius is 4 cm. An athlete lifts the weights so that they rise at the rate of 10 cm/sec. Find the angular velocity of the upper pulley in degrees per second. Find the angular velocity of the lower pulley in degrees per second. The motor in a clock is powered to turn a big gear (B, of radius 5 units) at a rate of 144 every minute. The second hand on the clock, however, needs to turn at a rate of 1 revolution per minute. So, the big gear (B) is coupled with a smaller gear (S) which will turn at a faster angular velocity. Find the angular velocity of gear B in radians per minute. Find the linear velocity of the teeth on the rim of gear B (in units per minute). Determine the radius that could be used for gear S so that it has the same linear velocity as gear B, but rotates at 1 revolution per minute. Notes for Teacher Create a graphic organizer for each problem that includes a diagram of the problem situation, applicable formulas, and calculations. Use the information to make predictions in terms of the problem situation. Standard(s): P.3E ELPS ELPS.c.1C, ELPS.c.5B, ELPS.c.5G page 14 of 47

15 Enhanced Instructional Transition Guide High School Courses Unit Number: 03 /Mathematics Suggested Duration: 4 days 04/01/13 page 15 of 47

16 Making Tracks KEY A track is composed to two straightaway sections and two semi-circular arcs, as shown. On a standard track, the distance around the inside lane (one lap) is 400 meters. If each straight section of a track is 98 m, then find x, the diameter of the semi-circular arcs. x 98 m Sample: (½)x + (½)x = 400 x = 204 x = 204/ m 1) The world record for running four laps around such a track is around 3 min 43 seconds. A) What would this runner s average speed be in meters per second? 4 laps = 1600 m, and 3 min 43 sec = 223 sec Speed = 1600 m / 223 sec 7.17 m/sec B) At this rate, how long would it take the runner to complete only one lap? 400 m at 7.17 m/sec would take about seconds 2) The world record for running one lap around such a track is actually just over 43 seconds. A) What would this runner s average speed be in meters per second? Speed = 400 m / 43 sec 9.30 m/sec B) If 1 meter = yards, and 1 mile = 5,280 feet, how fast would this runner be moving in miles per hour? 400 m yd 3 ft 1 mile 60 sec 60 min = mph 43 sec 1 m 1 yd 5,280 ft 1 min 1 hr 2012, TESCCC 07/24/12 page 1 of 1

17 Making Tracks A track is composed of two straightaway sections and two semi-circular arcs, as shown. On a standard track, the distance around the inside lane (one lap) is 400 meters. If each straight section of a track is 98 m, then find x, the diameter of the semi-circular arcs. x 98 m 1) The world record for running four laps around such a track is around 3 min 43 seconds. A) What would this runner s average speed be in meters per second? B) At this rate, how long would it take the runner to complete only one lap? 2) The world record for running one lap around such a track is actually just over 43 seconds. A) What would this runner s average speed be in meters per second? B) If 1 meter = yards, and 1 mile = 5,280 feet, how fast would this runner be moving in miles per hour? 2012, TESCCC 07/24/12 page 1 of 1

18 Conversions with Dimensional Analysis KEY Sample: 40 mph is how many feet per second? 40 mi 5280 ft 1 hr 1 min 40(5280) ft = 1 hr 1 mi 60 min 60 sec 60(60) sec = ft/sec Start with the given information as a ratio. Then cancel unwanted units by multiplying by various forms of 1 (comparing unit equivalents, such as 1 mi = 5280 ft). 1) A car went 44 feet through an intersection in only 0.5 seconds. How fast was the car going in miles per hour? 44 ft 60 sec 60 min 1 mile 44(60)(60) mi = 0.5 sec 1 min 1 hr 5,280 ft 0.5(5280) hr = 60 mph 2) In football, Drew runs a 40-yard dash in 5.1 seconds. Describe his rate in miles per hour. Around 16 miles per hour 3) A leaky faucet in our kitchen wastes a cup of water every 10 minutes. How fast is this water leaking in gallons per day? 9 gallons per day 1 gal = 4 qt 1 qt = 4 cups 4) With my internet connection, my computer estimated that it would take 18 seconds to download an 890-KB file. What is the transfer rate in gigabytes (GB) per day? About 4 GB/day 1 MB = 1024 KB 1 GB = 1024 MB 1 5) An old-fashioned record player spins an album at a rate of 33 rpm 3 (revolutions per minute). Convert this rate into degrees per second. 200/sec 1 rev = , TESCCC 07/24/12 page 1 of 2

19 Conversions with Dimensional Analysis KEY 6) The world record for the fastest spinning on ice skates is 1848 of rotation per second. Describe this rate in revolutions per minute. 308 revolutions per minute 1 rev = 360 7) On average, the earth is 93 million miles away from the center of the sun. A) Assuming that the earth s orbit is circular, estimate the number of miles our planet travels in one complete revolution around the sun. 1 revolution = 2r 584,336,234 miles B) The complete revolution around the sun takes 1 year. How fast is the earth moving in this orbit in feet per second? 584,336,233 miles per year 97,834 feet per second Sun 93 million miles Earth 8) Suppose a car wheel is 26 inches in diameter. A) If the tire completes one rotation, how far would the car travel? 1 rotation = 1 revolution = 2r inches B) If the car is traveling at 60 miles per hour, how fast is the car wheel spinning in revolutions per second? 26 in 60 mi 5280 ft 12 in 1 rev 1 hr 1 min 1 hr 1 mi 1 ft in 60 min 60 sec 12.9 rps 2012, TESCCC 07/24/12 page 2 of 2

20 Conversions with Dimensional Analysis Sample: 40 mph is how many feet per second? 40 mi 5280 ft 1 hr 1 min 40(5280) ft = 1 hr 1 mi 60 min 60 sec 60(60) sec = ft/sec Start with the given information as a ratio. Then cancel unwanted units by multiplying by various forms of 1 (comparing unit equivalents, such as 1 mi = 5280 ft). 1) A car went 44 feet through an intersection in only 0.5 seconds. How fast was the car going in miles per hour? 44 ft 0.5 sec = = 2) In football, Drew runs a 40-yard dash in 5.1 seconds. Describe his rate in miles per hour. 3) A leaky faucet in our kitchen wastes a cup of water every 10 minutes. How fast is this water leaking in gallons per day? 1 gal = 4 qt 1 qt = 4 cups 4) With my internet connection, my computer estimated that it would take 18 seconds to download an 890-KB file. What is the transfer rate in gigabytes (GB) per day? 1 MB = 1024 KB 1 GB = 1024 MB 1 5) An old-fashioned record player spins an album at a rate of 33 rpm 3 (revolutions per minute). Convert this rate into degrees per second. 1 rev = , TESCCC 07/24/12 page 1 of 2

21 Conversions with Dimensional Analysis 6) The world record for the fastest spinning on ice skates is 1848 of rotation per second. Describe this rate in revolutions per minute. 1 rev = 360 7) On average, the earth is 93 million miles away from the center of the sun. A) Assuming that the earth s orbit is circular, estimate the number of miles our planet travels in one complete revolution around the sun. B) The complete revolution around the sun takes 1 year. How fast is the earth moving in this orbit in feet per second? Sun 93 million miles Earth 8) Suppose a car wheel is 26 inches in diameter. A) If the tire completes one rotation, how far would the car travel? B) If the car is traveling at 60 miles per hour, how fast is the car wheel spinning in revolutions per second? 26 in 2012, TESCCC 07/24/12 page 2 of 2

22 Inside Track KEY C B A Oscar and Isaac are having a race at the school track. Oscar is confident that he will win, so he gives Isaac the inside lane. They start at the curved (semi-circular) part of the track, so that Isaac runs on an arc with a radius of 33 meters, and Oscar s radius is 36 meters. D E Start Isaac (r = 33 m) Oscar (r = 36 m) For the first 15 seconds, the two runners appear to stay dead even as they complete the curved turns of the track. However, as they enter the straightaway, Oscar seems to pull ahead. Assume that both Oscar and Isaac run the race at a constant rate. Complete the table to show the distances covered by each runner at the given points on the track. Point Time (seconds) Fraction of Semi-Circle Degrees of Arc Distance Run by Isaac Distance Run by Oscar Start m 0 m A 5 B m m m m C / m m D m m 1) Determine Isaac s speed in each of the following units. A) degrees per second B) meters per second 12 degrees per second 33/ m/sec 2) Determine Oscar s speed in each of the following units. A) degrees per second B) meters per second 12 degrees per second 36/ m/sec 3) Point E marks the end of the race, when Oscar (in the outside lane) finishes 150 meters. A) After how many seconds would Oscar reach this point? B) When Oscar finishes, how many meters has Isaac run? 150 m (36/15 m/s) seconds 33/15 m/s s = m 2012, TESCCC 07/24/12 page 1 of 2

23 Inside Track KEY 4) How can the two runners have different linear speeds (in m/sec) but the same angular speeds (in degrees/sec)? Explain. Because Oscar s path has a greater radius, the distance traveled is greater even though the arc measure in degrees is the same. 5) Using the information from the table on page 1, find the ratio of the distance traveled to each runner s radius at each of the given points. Isaac (Inside) Oscar (Outside) Distance / radius Ratio Distance / radius Ratio A m / 33 m = /3 = m / 36 m = /3 = 1.04 B m / 33 m = /2 = m / 36 m = /2 = 1.57 C m / 33 m = 3/4 = m / 36 m = 3/4 = 2.35 D m / 33 m = = m / 36 m = = ) What do you notice about the way these ratios compare at each point? The ratios are the same for both runners at each given point. (Just like the angles were the same.) 7) If a different runner ran around the outside of the semi-circular part of the track at a radius of 45 m, what would be the ratio of her distance traveled to this radius? Distance traveled = m Ratio = distance traveled / radius = 45 / 45 = ) Show that, no matter what a runner s radius is, the ratio of the distance traveled around a semi-circle to the radius is always the same. What is this ratio called? Distance traveled around semi-circle = (½)(2r) = r Ratio = distance traveled / radius = r / r = The ratio of the arc length to the radius is a unit of measure called radians. 2012, TESCCC 07/24/12 page 2 of 2

24 Inside Track C B A Oscar and Isaac are having a race at the school track. Oscar is confident that he will win, so he gives Isaac the inside lane. They start at the curved (semi-circular) part of the track, so that Isaac runs on an arc with a radius of 33 meters, and Oscar s radius is 36 meters. D E Start Isaac (r = 33 m) Oscar (r = 36 m) For the first 15 seconds, the two runners appear to stay dead even as they complete the curved turns of the track. However, as they enter the straightaway, Oscar seems to pull ahead. Assume that both Oscar and Isaac run the race at a constant rate. Complete the table to show the distances covered by each runner at the given points on the track. Point Time (seconds) Fraction of Semi-Circle Degrees of Arc Distance Run by Isaac Distance Run by Oscar Start A B C 135 D ) Determine Isaac s speed in each of the following units. A) degrees per second B) meters per second 2) Determine Oscar s speed in each of the following units. A) degrees per second B) meters per second 3) Point E marks the end of the race, when Oscar (in the outside lane) finishes 150 meters. A) After how many seconds would Oscar reach this point? B) When Oscar finishes, how many meters has Isaac run? 2012, TESCCC 07/24/12 page 1 of 2

25 Inside Track 4) How can the two runners have different linear speeds (in m/sec) but the same angular speeds (in degrees/sec)? Explain. 5) Using the information from the table on page 1, find the ratio of the distance traveled to each runner s radius at each of the given points. Isaac (Inside) Oscar (Outside) Distance / radius Ratio Distance / radius Ratio A / 33 m = / 36 m = B / 33 m = / 36 m = C / 33 m = / 36 m = D / 33 m = / 36 m = 6) What do you notice about the way these ratios compare at each point? 7) If a different runner ran around the outside of the semi-circular part of the track at a radius of 45 m, what would be the ratio of her distance traveled to this radius? 8) Show that, no matter what a runner s radius is, the ratio of the distance traveled around a semi-circle to the radius is always the same. What is this ratio called? 2012, TESCCC 07/24/12 page 2 of 2

26 Radian Measure of Angles KEY 3 radii (radians) 2 radii (radians) 1 radius 1 radian radius A little more than 3 radians in 180 radians degrees Besides degrees, angles can also be measured in radians. Radian measure is defined by the following ratio: Length of intercepted arc Length of the radius Another way to describe radian measure is to ask, How many radii can fit along the arc included in the angle? While you can write radians (or rad for short) after a measure to indicate radians, in most cases you don t have to write any units at all. Since radians are defined as a ratio of two lengths (such as inches to inches, or cm to cm), these units cancel each other out. In other words, radians are unitless measures. To change from degrees to radians, multiply by the following conversion factor: 180 Convert these degree measures into radians. 1) 2) r / / r 300 3) 4) 420 r 5/ / , TESCCC 07/24/12 page 1 of 2

27 Radian Measure of Angles KEY To change from radians to degrees, multiply by the following conversion factor: 180 Convert these radian measures into degrees. 5) 6) r r 45 7) 8) r Formulas Using Radians A circle has the following measures: Sector Arc r r = length of the radius = measure of central angle (in radians) L = length of the intercepted arc S = area of the sector formed The following formulas can be used to find arc length and sector area: Arc length: L r Sector area: S 1 2 r 2 For each figure: A) convert the degree measure to radians, B) find the length of the intercepted arc, and C) find the area (shaded) of the sector formed. 9) B 10) 75 6 cm A in A A) 75 = 5/ A) 240 = 4/ B) L = 5/ cm B) L = 32/ in C) S = 15/ cm 2 C) S = 128/ in 2 B 2012, TESCCC 07/24/12 page 2 of 2

28 Radian Measure of Angles 3 radii (radians) 2 radii (radians) 1 radius 1 radian radius A little more than 3 radians in 180 radians degrees Besides degrees, angles can also be measured in radians. Radian measure is defined by the following ratio: Length of intercepted arc Length of the radius Another way to describe radian measure is to ask, How many radii can fit along the arc included in the angle? While you can write radians (or rad for short) after a measure to indicate radians, in most cases you don t have to write any units at all. Since radians are defined as a ratio of two lengths (such as inches to inches, or cm to cm), these units cancel each other out. In other words, radians are unitless measures. To change from degrees to radians, multiply by the following conversion factor: Convert these degree measures into radians. 1) 2) r r 300 3) 4) 420 r 2012, TESCCC 07/24/12 page 1 of 2

29 Radian Measure of Angles To change from radians to degrees, multiply by the following conversion factor: Convert these radian measures into degrees. 5) 6) r r 7) 8) r Formulas Using Radians A circle has the following measures: Sector Arc r r = length of the radius = measure of central angle (in radians) L = length of the intercepted arc S = area of the sector formed The following formulas can be used to find arc length and sector area: Arc length: L r Sector area: S 1 2 r 2 For each figure: A) convert the degree measure to radians, B) find the length of the intercepted arc, and C) find the area (shaded) of the sector formed. 9) B 10) 75 6 cm A in A B 2012, TESCCC 07/24/12 page 2 of 2

30 Circular Reasoning KEY Convert each degree measure to radians. 1) 105 2) 270 3) / / Convert each radian measure to degrees ) 5) 6) Arc length: L r Sector area: S 1 2 r 2 Here, r is the circle s radius, and is the measure of a central angle (in radians) In items #7 and #8: A) convert the degree measure to radians, B) find the length of the intercepted arc, and C) find the area (shaded) of the sector formed. 7) 8) B in A cm A B A) 135 = 3/ A) 300 = 5/ B) L = in B) L = cm C) S = in 2 C) S = cm 2 9) A particular softball field is a 90 sector with a radius of 450 feet. A) A fence surrounds the entire field. What is the total length of the fence? 90 = /2 Arc = 450(/2) ft Fence = 2(450) + 450(/2) ft feet B) What area is enclosed by the fence? Area = 0.5(450) 2 (/2) 159, ft , TESCCC 07/24/12 page 1 of 2

31 Circular Reasoning KEY 10) Three friends order a 16-inch (diameter) pizza to share. Dave Gary A) Sarah eats only a 40 slice of pizza. How many square inches is this? 40 = 2/9 Area = 0.5(8) 2 (2/9) = 64/ in 2 40 Sarah B) Dave eats both a 30 slice and a 50 slice. How many square inches of pizza did he eat? 80 = 4/9 Area = 0.5(8) 2 (4/9) = 128/ in 2 C) Gary eats all of a 60 slice of pizza except the crust. What is the length of this crust? 60 = /3 Arc = 8(/3) = 8/ inches 11) Austin, Texas, and Wichita, Kansas, fall approximately on the same line of longitude. However, Austin lies at 30.27N latitude, and Wichita s latitude is 37.69N. Wichita Austin 37.69N 30.27N (0) North Pole (90) Equator r 4000 mi Degrees latitude is measured as an angle central to the earth starting at the equator, where the earth has an approximate radius of 4000 miles. A) What is the measure of the arc on the earth s surface between these two cities in degrees? in radians? = = 7.42/ B) What is the approximate distance between the two cities (along the earth s surface)? r = 4000(0.1295) 518 miles 12) On a circle with a radius of 12 cm is an arc of length 20 cm. What is the degree measure of the central angle used to make this arc? L = r 20 = 12() = 20/12 = 5/3 (rad) 5/3 = 5(180)/(3) = ) In a circle with a radius of 50 units is a sector with an area of 3,200 un 2. What is the degree measure of the central angle used to make this sector? S = (½) r = (0.5)(50) 2 = 2.56 (rad) 2.56 = 2.56(180)/() = , TESCCC 07/24/12 page 2 of 2

32 Circular Reasoning Convert each degree measure to radians. 1) 105 2) 270 3) 1080 Convert each radian measure to degrees ) 5) ) 3.4 Arc length: L r Sector area: S 1 2 r 2 Here, r is the circle s radius, and is the measure of a central angle (in radians) In items #7 and #8: A) convert the degree measure to radians, B) find the length of the intercepted arc, and C) find the area (shaded) of the sector formed. 7) 8) B in A cm A B 9) A particular softball field is a 90 sector with a radius of 450 feet. A) A fence surrounds the entire field. What is the total length of the fence? feet B) What area is enclosed by the fence? 2012, TESCCC 07/24/12 page 1 of 2

33 Circular Reasoning 10) Three friends order a 16-inch (diameter) pizza to share. Dave Gary A) Sarah eats only a 40 slice of pizza. How many square inches is 50 this? Sarah 40 B) Dave eats both a 30 slice and a 50 slice. How many square inches of pizza did he eat? C) Gary eats all of a 60 slice of pizza except the crust. What is the length of this crust? 11) Austin, Texas, and Wichita, Kansas, fall approximately on the same line of longitude. However, Austin lies at 30.27N latitude, and Wichita s latitude is 37.69N. Wichita Austin 37.69N 30.27N North Pole (90) Degrees latitude is measured as an angle central to the earth starting at the equator, where the earth has an approximate radius of 4000 miles. (0) Equator r 4000 mi A) What is the measure of the arc on the earth s surface between these two cities in degrees? in radians? B) What is the approximate distance between the two cities (along the earth s surface)? 12) On a circle with a radius of 12 cm is an arc of length 20 cm. What is the degree measure of the central angle used to make this arc? 13) In a circle with a radius of 50 units is a sector with an area of 3,200 un 2. What is the degree measure of the central angle used to make this sector? 2012, TESCCC 07/24/12 page 2 of 2

34 Circular Speed KEY Merry-Go-Round Dad likes to take Junior to the park. There, the father likes to push his son on the merry-go-round until he gets dizzy. Holding onto a rail and standing 8 ft from its center, Dad runs threefourths of a turn around the merry-go-round before letting go. This takes 3 seconds. Junior rides on the merry-go-round 5 feet from its center. 1) How many degrees is 4 3 of a revolution? (3/4)(360) = ft 8 ft Junior Dad 2) What distance does Dad run during this 3-second interval? 270 = 3/2 L = 8(3/2) = feet 3) In feet per second, what is Dad s average speed as he pushes Junior on the merry-go-round? feet / 3 sec ft/sec 4) Is Junior moving at the same speed? Explain. While Junior is turning at the same rate (270 every 3 seconds), he actually travels a shorter distance: 5(3/2) = 15/ feet. So, his average speed is only ft /3 sec 7.85 ft/sec. Discuss the differences between the two different types of speed in this problem. Angular Velocity Linear Velocity What is being measured? How fast an object spins, rotates, or revolves in a circular path around a center point How fast an object moves through space What units can be used? Angle measures per time units Examples: revolutions per minute, degrees per second, radians per hour Distances per time units Examples: miles per hour, feet per second, centimeters per day Describe the formula that can be used to relate these two different types of speed. r = v = v = L length of the radius angular velocity (in radians per time) linear velocity v L = r v 2012, TESCCC 07/24/12 page 1 of 2

35 Circular Speed KEY 5) A circular saw blade with a 12-inch diameter spins at a rate of 1800 rpm (revolutions per minute). A) What is the blade s angular velocity in radians per minute? 1800 rpm = 3600 / minute 11, rad/min B) Find the linear velocity (in inches per minute) of one of the teeth on the edge of the blade. V L = (6 in)(3600 / min) 67, in/min C) Convert this linear velocity to feet per second ft/sec 6) Vinyl record albums are inches in 1 diameter and spin at a rate of 33 rpm. 3 A) What is a record s angular velocity in radians per minute? 200 /3 rad/min rad/min B) How fast (in inches per minute) would a record move under a needle placed on the record s edge? V L = (5.875 in)(200 /3)/min in/min C) Convert this linear velocity to feet per second ft/sec 7) With his arms fully extended, a baseball player swings a bat. Using his shoulder as the center of rotation, the bat moves through 120 in only 0.2 seconds. A) What is the angular velocity of the batter s swing in radians per second? rad/sec B) As he swings the bat, the player hits a baseball. Suppose the ball leaves the bat at a distance of 40 inches from the player s shoulder. How fast (in miles per hour) would the ball be moving? V L (40 in)(10.47 (rad)/sec) = in/sec 23.8 miles per hour C) During a second time at bat, the player hits another ball, which leaves the bat at a distance of 43 inches from the player s shoulder. How fast (in miles per hour) would this ball be moving? V L (43 in)(10.47 (rad)/sec) = in/sec 25.6 miles per hour 8) The blades of a ceiling fan are 26 inches long, but the fan s entire diameter is 70 inches. It spins at a rate of 100 rpm. A) What is the linear velocity of a point on the outer edge of the A blade? V L = (35 in)(200/min) 21, in/min B 70 in B) What is the linear velocity of a point on the inner edge of the C blade? V L = (9 in)(200/min) 5, in/min C) What is the linear velocity of a point at the center of the fan? V L = (0 in)(200/min) = 0 in/min 26 in 2012, TESCCC 07/24/12 page 2 of 2

36 Circular Speed Merry-Go-Round Dad likes to take Junior to the park. There, the father likes to push his son on the merry-go-round until he gets dizzy. Holding onto a rail and standing 8 ft from its center, Dad runs threefourths of a turn around the merry-go-round before letting go. This takes 3 seconds. Junior rides on the merry-go-round 5 feet from its center. 1) How many degrees is 4 3 of a revolution? 5 ft 8 ft Junior Dad 2) What distance does Dad run during this 3-second interval? 3) In feet per second, what is Dad s average speed as he pushes Junior on the merry-go-round? 4) Is Junior moving at the same speed? Explain. Discuss the differences between the two different types of speed in this problem. What is being measured? What units can be used? Angular Velocity Linear Velocity Describe the formula that can be used to relate these two different types of speed. r = v = v = L length of the radius angular velocity (in radians per time) linear velocity v L = 2012, TESCCC 07/24/12 page 1 of 2

37 Circular Speed 5) A circular saw blade with a 12-inch diameter spins at a rate of 1800 rpm (revolutions per minute). A) What is the blade s angular velocity in radians per minute? 6) Vinyl record albums are inches in 1 diameter and spin at a rate of 33 rpm. 3 A) What is a record s angular velocity in radians per minute? B) Find the linear velocity (in inches per minute) of one of the teeth on the edge of the blade. B) How fast (in inches per minute) would a record move under a needle placed on the record s edge? C) Convert this linear velocity to feet per second. C) Convert this linear velocity to feet per second. 7) With his arms fully extended, a baseball player swings a bat. Using his shoulder as the center of rotation, the bat moves through 120 in only 0.2 seconds. A) What is the angular velocity of the batter s swing in radians per second? B) As he swings the bat, the player hits a baseball. Suppose the ball leaves the bat at a distance of 40 inches from the player s shoulder. How fast (in miles per hour) would the ball be moving? C) During a second time at bat, the player hits another ball, which leaves the bat at a distance of 43 inches from the player s shoulder. How fast (in miles per hour) would this ball be moving? 8) The blades of a ceiling fan are 26 inches long, but the fan s entire diameter is 70 inches. It spins at a rate of 100 rpm. A) What is the linear velocity of a point on the outer edge of the A blade? C 26 in B 70 in B) What is the linear velocity of a point on the inner edge of the blade? C) What is the linear velocity of a point at the center of the fan? 2012, TESCCC 07/24/12 page 2 of 2

38 Common Velocities KEY When two rotating objects are connected by the same axis or axle, they have the same. angular velocity When two rotating objects are connected by a belt, chain, rope, line, or their edges, they have the same. linear velocity Use the information in the table above to answer the questions that follow. 1) The Crazy Car is a child s tricycle with a large wheel in the front and two smaller wheels in back. The front wheel has a diameter of 20 inches, and each of the back wheels has a diameter of only 6 inches. A) As a child drives the Crazy Car, would the wheels have the same angular velocity or linear velocity? Explain. Linear velocity. They are connected by the line that is the ground. B) A child can pedal the front wheel at a rate of 40 revolutions per minute. How fast does this make the Crazy Car go in inches per minute? in feet per second? 40 rpm = 80 /min V L = (10 in)(80/min) = 800 in/min in/min Or, 3.49 ft/sec C) At this linear speed, what would be the angular velocity of the small wheel? V L = ( r )(V ), so 800 in/min = (3 in)(v ) V radians/min, or rev/min 2) On a boat, a sailor must turn a crank that is attached to a spool to reel in (or tighten) a rope. The crank handle turns with a radius of 16 inches, but the radius of the spool is only 10 inches. A) Would the crank handle and the spool have the same angular velocity or linear velocity? Explain. Angular velocity. They are connected by the same axis. B) The rope is being pulled in at a rate of 20 feet (or 240 inches) per minute. Does this represent linear or angular velocity? Find the spool s angular velocity. Include units. Linear velocity V L = ( r )(V ), so 240 in/min = (10 in)(v ) V = 24 radians/min, or 3.8 rev/min C) Determine the angular and linear velocity of the crank handle. V = 24 radians/min, or 3.8 rev/min (the same as the spool) V L = (16 in)(24 (rad)/min) = 384 in/min, or 32 ft/min 2012, TESCCC 07/24/12 page 1 of 2

39 Common Velocities KEY 3) In a car engine, the alternator, water pump, and crank wheel are all connected by the same belt (sometimes called a serpentine belt). These spinning parts have diameters of 10 cm, 21 cm, and 30 cm, respectively. Suppose, when the engine is running, the crank wheel turns at a rate of 1500 rpm. A) Determine the linear velocity of the belt in cm/min and in m/sec. V L = (15 cm)(3000/min) 141, cm/min Or, m/sec d = 10 cm d = 21 cm d = 30 cm B) Determine the angular velocities (in revolutions per minute) of the spinning parts of the alternator and the water pump. Both have the same linear velocity as the crank wheel and belt (45,000 cm/min). Alternator: V L = ( r )(V ) 45,000 cm/min = (5 cm)(v ) V = 28,274 (rad)/min, or 4,500 rev/min Water Pump: V L = ( r )(V ) 45,000 cm/min = (10.5 cm)(v ) V 13,464 (rad)/min, or 2,143 rev/min 4) On a bicycle, the pedal gear is connected by a chain to the back Tire gear, which shares an axle with the bike s wheel and tire. Pedal A) Which rotating objects have the same angular velocity? gear The back gear and the tire. They are connected by the same axis. Back gear B) Which rotating objects have the same linear velocity? The back gear and the pedal gear. They are connected by the chain. Pedal Gear Back Gear C) Suppose a person pedals the bike at 60 rpm. Complete the chart to determine how fast the bicycle would be going in miles per hour. Angular Velocity Radius Linear Velocity rev/min radians/min inches in/min in 540 Linear velocity of in 540 the chain Tire in 6750 So, the bike is moving at 6750 in/min ft/sec 20 miles per hour Linear velocity of the bike 2012, TESCCC 07/24/12 page 2 of 2

40 When two rotating objects are connected by the same axis or axle, they have the same. angular velocity Common Velocities When two rotating objects are connected by a belt, chain, rope, line, or their edges, they have the same. linear velocity Use the information in the table above to answer the questions that follow. 1) The Crazy Car is a child s tricycle with a large wheel in the front and two smaller wheels in back. The front wheel has a diameter of 20 inches, and each of the back wheels has a diameter of only 6 inches. A) As a child drives the Crazy Car, would the wheels have the same angular velocity or linear velocity? Explain. B) A child can pedal the front wheel at a rate of 40 revolutions per minute. How fast does this make the Crazy Car go in inches per minute? in feet per second? C) At this linear speed, what would be the angular velocity of the small wheel? 2) On a boat, a sailor must turn a crank that is attached to a spool to reel in (or tighten) a rope. The crank handle turns with a radius of 16 inches, but the radius of the spool is only 10 inches. A) Would the crank handle and the spool have the same angular velocity or linear velocity? Explain. B) The rope is being pulled in at a rate of 20 feet (or 240 inches) per minute. Does this represent linear or angular velocity? Find the spool s angular velocity. Include units. C) Determine the angular and linear velocity of the crank handle. 2012, TESCCC 07/25/12 page 1 of 2

41 Common Velocities 3) In a car engine, the alternator, water pump, and crank wheel are all connected by the same belt (sometimes called a serpentine belt). These spinning parts have diameters of 10 cm, 21 cm, and 30 cm, respectively. Suppose, when the engine is running, the crank wheel turns at a rate of 1500 rpm. A) Determine the linear velocity of the belt in cm/min and in m/sec. d = 10 cm d = 21 cm d = 30 cm B) Determine the angular velocities (in revolutions per minute) of the spinning parts of the alternator and the water pump. 4) On a bicycle, the pedal gear is connected by a chain to the back Tire gear, which shares an axle with the bike s wheel and tire. Pedal A) Which rotating objects have the same angular velocity? gear Back gear B) Which rotating objects have the same linear velocity? Pedal Gear Back Gear C) Suppose a person pedals the bike at 60 rpm. Complete the chart to determine how fast the bicycle would be going in miles per hour. Angular Velocity Radius Linear Velocity rev/min radians/min inches in/min in Linear velocity of 2 in the chain Tire 25 in Linear velocity of the bike 2012, TESCCC 07/25/12 page 2 of 2