Excerpts from Project Pathways Student Workbooks

Excerpts from Project Pathways Student Workbooks Carlson, M., OBryan, A., & Joyner, K. Pathways Algebra II: Implementing the Common Core Mathematics Standards. Rational Reasoning, Phoenix, AZ, 2012. Carlson, M., & Oehrtman, M. Precalculus: Pathways to Calculus. 3 rd Ed. Rational Reasoning, Phoenix, AZ, 2012. All material is Rational Reasoning, LLC. All rights reserved.

Excerpts from Algebra II: Implementing the Common Core Mathematics Standards

Algebra II Module 2: Worksheet 4 Constant Rate of Change & The Point-Slope Formula 1. In Worksheet 3 we defined the meaning of constant rate of change. a. Explain what it means for two quantities to be related by a constant rate of change. b. The cost of covering an area with a certain type of concrete paver increases at a constant rate of $4.50 per square foot with respect to the size of the area covered. What does this mean? c. On a job application for an administrative assistant position at a law firm, an applicant listed that he is able to type 90 words per minute. What does this mean? d. Let x be the value of one quantity and let y be the value of another quantity. Suppose y changes at a constant rate of change of 8.2 with respect to x. What does this mean? e. Let x be the value of one quantity and let y be the value of another quantity. Suppose y changes at a constant rate of change of 4.95 with respect to x. What does this mean? 2. The length of a burning candle decreases at a constant rate of 1.6 inches per hour. The candle has been burning for 3.5 hours and is currently 8.4 inches long (represented by the point (3.5, 8.4) on the graph). a. Are the quantities hours candle has been burning and candle length (in inches) related by a constant rate of change? Explain. b. Module 2: Worksheet 4 Page 1 2012 Carlson

Algebra II Module 2: Worksheet 4 Constant Rate of Change & The Point-Slope Formula c. By how much will the length of the candle change when the time spent burning increases by 2 hours? Represent this on the graph. d. What is the length of the candle 5.5 hours since it began burning? Explain how you determined your answer. e. f. By how much will the length of the candle change when the time spent burning is decreased by 1.8 hours? Represent this on the graph. g. What is the length of the candle 1.7 hours since it began burning? Explain how you determined this. h. What was the original length of the candle before it started burning? Explain how you determined this value and represent your reasoning on the graph. i. Draw the graph that represents the length of the candle in inches with respect to the number of hours spent burning. 3. Refer to Exercise #2 to answer the following questions. Let t be the amount of time elapsed since the candle began burning (in hours). a. Suppose the candle has been burning for 4 hours. Write the expression that calculates the change in t from t 3.5 to t 4. b. Suppose the candle has been burning for 5.1 hours. Write the expression that calculates the change in t from t 3.5 to t 5.1. c. Suppose the candle has been burning for 1 hour. Write the expression that calculates the change in t from t 3.5 to t 1. Module 2: Worksheet 4 Page 2 2012 Carlson

Algebra II Module 2: Worksheet 4 Constant Rate of Change & The Point-Slope Formula d. Write an expression that calculates the change in the length of the candle for each of the changes in time spent burning from parts (a) through (c). e. Suppose the candle has been burning for x hours. i. t from t 3.5 to t x ii. t 3.5 to t x iii. x 4. Use the graph to answer the following questions. a. The given points lie on a line. Determine the change in x and the change in y between the two points. What is the slope of the line that passes through these two points? b. Using the point (3.4, 10.72) as a reference point, illustrate on the axes the change in x that is necessary for x to change from x 3.4 to x 5.1. How much is this change? x from x 3.4 to x 5.1 Module 2: Worksheet 4 Page 3 2012 Carlson

Algebra II Module 2: Worksheet 4 Constant Rate of Change & The Point-Slope Formula d. What is the change in y that corresponds with the change in x from part (b)? y as x changes from x 3.4 to x 5.1 f. What is the value of y when x 5.1? g. What is the vertical intercept of the line? Explain how you can find this value using the meaning of constant rate of change and using both (3.4, 10.72) and (6.2, 18.56) as initial reference points. 5. The point (1.5, 3) is plotted below and is located on the graph of a linear function. a. Suppose that the constant rate of change of y with respect to x is 2.1. Illustrate on the graph how you can use the constant rate of change of 2.1 to determine the value of the equa x = 4 and x = 0.7. (Hint: Illustrate the change in x, and the corresponding change in y, and then what the output must be for the given input.) b. Determine the vertical intercept for the graph when the rate of change is 2.1 and a solution ( 1.5, 3) lies on the line. 6. We know that if x and y are the values of two quantities related by a constant rate of change that y m x where m is the constant rate of change. Suppose we know that m 4 and that (9, 1) is a point on the graph of the formula that relates x and y. a. Consider the expression x 9. Substitute several different values of x into this expression, evaluate, and explain what the resulting number tells us. Module 2: Worksheet 4 Page 4 2012 Carlson

Algebra II Module 2: Worksheet 4 Constant Rate of Change & The Point-Slope Formula b. In general, what does the expression x 9 represent? c. What does the expression 4( x 9) represent? d. Explain why it must be true that y 14( x 9). In Exercise #6, the statement y 14( x 9) is an example of the point-slope formula representing a linear relationship (so-called because it relies on us knowing the slope, or constant rate of change, for the relationship and one ordered pair solution (or point on the graph). The formal definition of the point-slope formula follows. If y varies at a constant rate of m with respect to x, and if ( x1, y 1) is a point on the graph, then y y1 m( x x1). Although the formula y y1 m( x x1) y mx relationship. 7. Explain why y y1 m( x x1) represents the same thing as y m x. 8. Use the point-slope formula to write the formula for each of the linear relationships described below. a. y changes at a constant rate of 3.5 with respect to x, and (6, 2) is a point on the graph b. y changes at a constant rate of 2 with respect to x, and ( 3, 4) is a point on the graph c. y changes at a constant rate of change with respect to x, and ( 7, 2) and (2, 5) are points on the graph y for a given value of x. Therefore, we might want to write y 14( x 9) as y 4( x 9) 1 instead. Module 2: Worksheet 4 Page 5 2012 Carlson

Algebra II Module 2: Worksheet 4 Constant Rate of Change & The Point-Slope Formula 9. Consider y 4( x 9) 1 and let x 16. a. Explain what 16 9 represents. b. Explain what 4(16 9) represents. c. Explain what 4(16 9) 1 represents. d. Repeat parts (a) through (c) for x 18.3 and x 1. 10. In general, given y 4( x 9) 1, what does each of the following represent? a. (9, 1) b. x 9 c. 4( x 9) d. 4( x 9) 1 11. Use the alternative version of the point-slope formula to write the formula for each of the linear relationships described below. a. y changes at a constant rate of 5.2 with respect to x, and ( 1, 1) is a point on the graph b. y changes at a constant rate of change with respect to x, and (3, 4) and (5, 1) are points on the graph c. the graph passes through the point ( h, k ) and has a constant rate of change of p. Module 2: Worksheet 4 Page 6 2012 Carlson

Excerpts from Algebra II: Implementing the Common Core Mathematics Standards Instructor Notes

Algebra II Module 2: Worksheet 4 Instructor Notes Constant Rate of Change & The Point-Slope Formula Common Core Content Standards this worksheet focuses on: F-LE.5. Interpret the parameters in a linear or exponential function in terms of a context. F-LE.1. Distinguish between situations that can be modeled with linear functions and with exponential functions. o Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. F-LE.2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Additional Common Core Content Standards included but not the primary focus: N-Q.2. Define appropriate quantities for the purpose of descriptive modeling. A-CED.4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. F-IF.7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. The purpose of this worksheets is for students to be able to understand and apply the meaning of constant rate of change in linear relationships and to create a linear formula using the point-slope form of a line utilizing their understanding of constant rate of change. As such, it is important that the meaning of constant rate of change is well established in conversations as a class and through discussion of exercise 1 before moving on to the rest of the worksheet. This worksheet leverages the constant multiple form of proportionality of the changes in quantities, y m x, throughout so be sure this is brought out in your discussions. This form is then used to develop the point-slope form of a line, y y mx x simply another way to write the constant multiple form. Lastly, we re-write this form as y m x x y as the addition of the change in y and the given value of y. 1 1 as 1 1 Exercise #1 is intended as a review of the ideas in Worksheet 3. You can choose to discuss the questions and go over them in as much detail as you feel your students need. We suggest that you write question (a) in Exercise 1 on the board (or display it on the projector) so students can begin thinking about the question when they walk into class. When class begins give them not more than 3 minutes to respond in writing, then hold a whole class discussion to establish the meanings listed below. 1. In Worksheet 3 we defined the meaning of constant rate of change. a. Explain what it means for two quantities to be related by a constant rate of change. If two quantities are related by a constant rate of change, then the changes in the quantities are but understand and demonstrate the implications of this statement throughout the rest of the parts of Exercise #1. If x and y represent the values of the quantities and x and y the relative changes in the two quantities, then each of the following is true. the change in the value of one quantity is always some constant c times as large as the change in the value of the other quantity ( y c x where c is the constant) Solution continued on the following page. Module 2: Worksheet 4 Instructor Notes Page 1 2012 Carlson,

Algebra II Module 2: Worksheet 4 Instructor Notes Constant Rate of Change & The Point-Slope Formula y the ratio of the changes in the two quantities is constant ( c where c is the constant). x if the change in one quantity is scaled by some factor, then the change in the other quantity is y scaled by the same factor. If c =, then k y implies k x. x equal changes in x (the value of one quantity) always results in consistent changes in y (the value of the other quantity). they are a set of interconnected ways of thinking about proportionality that give us flexibility in how we reason about the concept. These are the ideas you want to bring out in the following parts of Exercise #1. b. The cost of covering an area with a certain type of concrete paver increases at a constant rate of $4.50 per square foot with respect to the size of the area covered. What does this mean? Students can makevery square foot of til This is true, but not very helpful when it comes to understanding constant rate of change. It will benefit students more for this worksheet and beyond to focus attention on the idea that the change in the cost of covering the area (in dollars) is always 4.50 times as large as the change in the size of the space covered (in square feet). That is, if a represents the change in the size of the area covered (in square feet) and c represents the change in the total cost of covering the c area, then 4.50 and c 4.50 a. a To help students make sense of this, you can ask follow-up questions relating to what happens if the area to be covered increases by 30 square feet, 27.2 square feet, or 165.48 square feet, and what happens if the area to be covered decreases by 15 square feet, 4.6 square feet, or 100.77 square feet. Since a constant rate of change implies a proportional relationship between the relative changes in meaning for this statement. Instructors should also keep an ear out for comments by students that might indicate they are thinking about this as a proportional relationship between the area to be covered and the total cost of the tile. s are proportional (there could be change together c. On a job application for an administrative assistant position at a law firm, an applicant listed that he is able to type 90 words per minute. What does this mean? The applicant However, we want students to focus on the relative changes of the two quantities involved. Saying that someone can type 90 words per minute is to say that he types at a constant rate of 90 words per minute, or that the change in the number of words typed is 90 times as large as the change in the time spent typing (in minutes). Put another way, if t represents the change in the time spent typing (in w minutes) and w the change in the number of words typed, then 90 and w 90 t for all corresponding values of t and w. t Solution continued on the following page. Module 2: Worksheet 4 Instructor Notes Page 2 2012 Carlson,

Algebra II Module 2: Worksheet 4 Instructor Notes Constant Rate of Change & The Point-Slope Formula As before, the teacher can ask follow-up questions to help students focus on these changes by giving them specific changes in time elapsed (including increases and decreases in the value) and using this to determine the change in the total cost. d. Let x be the value of one quantity and let y be the value of another quantity. Suppose y changes at a constant rate of change of 8.2 with respect to x. What does this mean? y 8.2 and y 8.2 x x Ask students to determine what this means for y for various values of x. For example, if x changes by 3.6 (increases by 3.6), then y 8.2(3.6) 29.52 (y increases by 29.52). If x changes by 13 (decreases by 13), then y 8.2( 13) 106.6 (y decreases by 106.6). e. Let x be the value of one quantity and let y be the value of another quantity. Suppose y changes at a constant rate of change of 4.95 with respect to x. What does this mean? y 4.95 and y 4.95 x x Ask students to determine what this means for y for various values of x. For example, if x changes by 9.9 (increases by 9.9), then y 4.95(9.9) 49.005 (y decreases by 49.005). If x changes by 5.2 (decreases by 5.2), then y 4.95( 5.2) 25.74 (y increases by 25.74). In this exercise, students are asked to apply the meaning of constant rate of change to determine and interpret a problem. Use of the graph to illustrate values found throughout the exercise will be useful to help students make sense of the calculations and values they are finding. You may want to put the graph on the board/projector and have students (or yourself) create the diagrams in front of the class so that all students are referencing the same quantities when describing the problem and their methods of solution. 2. The length of a burning candle decreases at a constant rate of 1.6 inches per hour. The candle has been burning for 3.5 hours and is currently 8.4 inches long (represented by the point (3.5, 8.4) on the graph). a. Are the quantities hours candle has been burning and candle length (in inches) related by a constant rate of change? Explain. Yes they are the change in the candle length is always 1.6 times the change in the hours the candle has been burning. This question may confuse students because although the quantities themselves are mentioned in the problem statement, the question asks about constant rate of change, which examines proportionality of the changes in the quantities. If students get confused, point them back to their answers in Exercise (1a). Module 2: Worksheet 4 Instructor Notes Page 3 2012 Carlson,

Algebra II Module 2: Worksheet 4 Instructor Notes Constant Rate of Change & The Point-Slope Formula In parts (b) (d), students are led through the process of using a reference point and a rate of change to determine the coordinates of another point. They first determine the change in the input, then the change in the output using y m x, then using this change to determine the new value of the output. This is a process they will repeat many times throughout the worksheet, which will lead to the development of the point-slope formula of a line. b. c. By how much will the length of the candle change when the time spent burning increases by 2 hours? Represent this on the graph. It will change by 1.6 times 2, or 3.2 inches. d. What is the length of the candle 5.5 hours since it began burning? Explain how you determined your answer. The length of the candle at 3.5 hours was 8.4 inches. We know that over the next two hours the length of the candle changed by 3.2 inches. This means that it is now 5.2 inches long. Students are asked to repeat the process of parts (b) (d) for a different change in the input. Again, organize their reasoning so that patterns begin to emerge from these processes. e. f. By how much will the length of the candle change when the time spent burning is decreased by 1.8 hours? Represent this on the graph. It will change by 1.6 times 1.8, or 2.88 inches. Module 2: Worksheet 4 Instructor Notes Page 4 2012 Carlson,

Algebra II Module 2: Worksheet 4 Instructor Notes Constant Rate of Change & The Point-Slope Formula g. What is the length of the candle 1.7 hours since it began burning? Explain how you determined this. The length of the candle at 3.5 hours was 8.4 inches. We know that for a change in 1.8 hours of burning the length of the candle changed by 2.88 inches. This means that 1.8 hours earlier, the candle was 11.28 inches long. In part (h), students must first realize that to determine the original length of the candle, they must determine the change in the amount of time from 3.5 hours to 0 hours (a change of 3.5 hours). From here, they can repeat the same process as before to determine the initial length of the candle. h. What was the original length of the candle before it started burning? Explain how you determined this value and represent your reasoning on the graph. From our reference point of at 3.5 hours the candle was 8.4 inches, this would be a change in time of 3.5 hours. The change in the length of the candle would then be 3.5 times 1.6, or 5.6 inches from our reference length of 8.4 inches. So the original length of the candle was 14.0 inches. i. Draw the graph that represents the length of the candle in inches with respect to the number of hours spent burning. (solution graph is on the next page) Exercise 3 again takes students through the process of determining new values of the length of the candle given new values of the amount of time elapsed since the candle began burning. It is important that students begin to see the patterns emerge in their processes. If you feel students need more examples to generalize the process, ask the questions for additional values of t. 3. Refer to Exercise #2 to answer the following questions. Let t be the amount of time elapsed since the candle began burning (in hours). a. Suppose the candle has been burning for 4 hours. Write the expression that calculates the change in t from t 3.5 to t 4. 4 3.5 b. Suppose the candle has been burning for 5.1 hours. Write the expression that calculates the change in t from t 3.5 to t 5.1. 5.1 3.5 c. Suppose the candle has been burning for 1 hour. Write the expression that calculates the change in t from t 3.5 to t 1. 1 3.5. Students may be tempted here to write 3.5 1 instead of the other way around because this way will give a negative value. Discuss why the first expression is a better representation of the change from t = 3.5 to t = 1. In this situation, the sign of the change is important (whether it is an increase in time vs a decrease in time); to represent a decrease in time, we use negative values of change. Module 2: Worksheet 4 Instructor Notes Page 5 2012 Carlson,

Algebra II Module 2: Worksheet 4 Instructor Notes Constant Rate of Change & The Point-Slope Formula d. Write an expression that calculates the change in the length of the candle for each of the changes in time spent burning from parts (a) through (c). 1.6 4 3.5 1.6 5.1 3.5 1.6 1 3.5 a. b. c. e. Suppose the candle has been burning for x hours. i. t from t 3.5 to t x x 3.5 ii. t 3.5 to t x 1.6 x 3.5 iii. x 1.6 x 3.5 ( ) + 8.4 or 8.4+ 1.6( x 3.5) ( ). You may choose to represent the expression however you prefer. Both expression here represents an addition of the original length of the candle and the change in the length of the candle (which may be positive or negative) to determine the new value of the length of the candle at time x. This exercise provides a context-less situation in which students will continue to practice the ways of reasoning from Exercises 2 and 3. In this situation, they must first determine the constant rate of change using the value of two points. If students calculate the slope using memorized procedures such as y2 y1 x x 2 1 question such as what does the the expression y2 y1, what are you referring to when you then need to apply it in the same manner as before. As you proceed through these questions, you may find that students understand the concepts and you can move more quickly through Exercises 4 and 5 or that students are struggling and you want to take time through these questions make adjustments as you see fit. 4. Use the graph to answer the following questions. a. The given points lie on a line. Determine the change in x and the change in y between the two points. What is the slope of the line that passes through these two points? y 7.84 x 6.2 3.4 2.8; y 18.56 10.72 7.84 The slope is 2.8 x 2.8 Module 2: Worksheet 4 Instructor Notes Page 6 2012 Carlson,

Algebra II Module 2: Worksheet 4 Instructor Notes Constant Rate of Change & The Point-Slope Formula b. Using the point (3.4, 10.72) as a reference point, illustrate on the axes the change in x that is necessary for x to change from x 3.4 to x 5.1. How much is this change? This is a change of 1.7 x from x 3.4 to x 5.15.1 3.4 d. What is the change in y that corresponds with the change in x from part (b)? 2.81.7 4.76 y as x changes from x 3.4 to x 5.1 2.8 5.1 3.4 f. What is the value of y when x 5.1? 10.72 4.76 15.48 g. What is the vertical intercept of the line? Explain how you can find this value using the meaning of constant rate of change and using both (3.4, 10.72) and (6.2, 18.56) as initial reference points. Students should repeat the process of determining the change in x, the change in y and the value of y for an x value of 0 from each of the reference points. The vertical intercept is (0, 1.2). 5. The point (1.5, 3) is plotted below and is located on the graph of a linear function. a. Suppose that the constant rate of change of y with respect to x is 2.1. Illustrate on the graph how you can use the constant rate of change of 2.1 to x = 4 and x = 0.7. (Hint: Illustrate the change in x, and the corresponding change in y, and then what the output must be for the given input.) b. Determine the vertical intercept for the graph when the rate of change is 2.1 and a solution ( 1.5,3) lies on the line. (0,6.15) In this exercise, students are now asked to interpret expressions given information regarding a linear relationship. Make sure students talk about the meaning of the expression, not simply the operations. For x x towards thinking about the value as the change in x from a reference value of 9. 6. We know that if x and y are the values of two quantities related by a constant rate of change that y m x where m is the constant rate of change. Suppose we know that m 4 and that (9, 1) is a point on the graph of the formula that relates x and y. a. Consider the expression x 9. Substitute several different values of x into this expression, evaluate, and explain what the resulting number tells us. This result tells us the change from 9. b. In general, what does the expression x 9 represent? It represents the change in x from 9. c. What does the expression 4( x 9) represent? This represents the change in y for a change in x of x 9. d. Explain why it must be true that y 14( x 9). From part (c), the expression on the right gives the change in y for that given change in x. The expression on the left gives the same information. Module 2: Worksheet 4 Instructor Notes Page 7 2012 Carlson,

Algebra II Module 2: Worksheet 4 Instructor Notes Constant Rate of Change & The Point-Slope Formula In Exercise #6, the statement y 14( x 9) is an example of the point-slope formula representing a linear relationship (so-called because it relies on us knowing the slope, or constant rate of change, for the relationship and one ordered pair solution (or point on the graph). The formal definition of the point-slope formula follows. If y varies at a constant rate of m with respect to x, and if ( x1, y 1) is a point on the graph, then y y1 m( x x1). Although the formula y y1 m( x x1) y mx relationship. 7. Explain why y y1 m( x x1) represents the same thing as y m x. y y 1 is simply representing a change in y, which is equivalent to y. Additionally, x x1 is representing a change in x, which is equivalent to x. Thus, these formulas are representing the same x, y and x, y information given two theoretical points 8. Use the point-slope formula to write the formula for each of the linear relationships described below. a. y changes at a constant rate of 3.5 with respect to x, and (6, 2) is a point on the graph 2 3.5 x 6 y b. y changes at a constant rate of 2 with respect to x, and ( 3, 4) is a point on the graph y 42 x 3 c. y changes at a constant rate of change with respect to x, and ( 7, 2) and (2, 5) are points on the graph 1 y 5 x 2 3 and useful to have a formula that directly shows how to calculate the value of y for a given value of x. Therefore, we might want to write y 14( x 9) as y 4( x 9) 1 instead. In this exercise, it is important that students understand the meaning of each expression. Though the difference from one formula to the other is simply an addition of 1, students need to understand the difference in what each formula represents. 9. Consider y 4( x 9) 1 and let x 16. a. Explain what 16 9 represents. This represents the change in x from 9 to 16. b. Explain what 4(16 9) represents. This represents the change in y from 1. c. Explain what 4(16 9) 1 represents. This represents the value of y associated with the x-value of 9. d. Repeat parts (a) through (c) for x 18.3 and x 1. Students should repeat this reasoning for each value of x. 10. In general, given y 4( x 9) 1, what does each of the following represent? a. (9, 1) b. x 9 c. 4( x 9) d. 4( x 9) 1 a. This point is our reference point from which we vary x and y b. This expression represents the change in x from 9. c. This represents the change in y from 1. d. This represents the new value of y which corresponds with our new value in x. Module 2: Worksheet 4 Instructor Notes Page 8 1 1 2012 Carlson,

Algebra II Module 2: Worksheet 4 Instructor Notes Constant Rate of Change & The Point-Slope Formula 11. Use the alternative version of the point-slope formula to write the formula for each of the linear relationships described below. a. y changes at a constant rate of 5.2 with respect to x, and ( 1, 1) is a point on the graph y 5.2 x 1 1 b. y changes at a constant rate of change with respect to x, and (3, 4) and (5, 1) are points on the graph 5 y x 3 4 2 c. the graph passes through the point ( h, k ) and has a constant rate of change of p. y p x h k Module 2: Worksheet 4 Instructor Notes Page 9 2012 Carlson,

Excerpts from Precalculus: Pathways to Calculus

Precalculus Module 7: Worksheet 5 Using the Sine and Cosine Function in Applied Settings José extends his arm straight out, holding a 2.3-foot string with a ball on the end. He twirls the ball around in a circle with his hand at the center, so that the plane in which it is twirling is perpendicular to the ground. Answer the following questions assuming the ball twirls counter-clockwise starting at the 3 1. Label quantities on the diagram that could be used to describe the position of the ball as it rotates around José 2. a. Construct a graph of a function f that relavertical distance above function of the measure of the angle swept out by the ball and string (measured in radians) as the ball twirls counter- b. Define the algebraic form of the function f vertical distance above hand as a function of the measure of the angle swept out by the ball and string (measured in radians) as the ball twirls counter- (Define the relevant variables.) 3. Suppose the ball that Jose was twirling travels one radian per second. a. How many radians does the ball and string sweep out in t seconds? b. Define a function g vertical distance above Joséf the number of seconds elapsed. (Define the relevant variables.) c. Over what time interval does the ball complete one revolution? Module 7: Worksheet 5 Page 1 2011 Carlson and Oehrtman

Precalculus Module 7: Worksheet 5 Using the Sine and Cosine Function in Applied Settings 4. Suppose the ball travels 2 radians per second. a. How many radians does the ball and string sweep out in t seconds? b. Define a function h vertical distance above number of seconds elapsed. (Define the relevant variables.) c. Over what time interval does the ball complete one revolution? 5. Suppose the ball travels 0.5 radians per second. a. How many radians does the ball and string sweep out in t seconds? b. Define a function j vertical distance above José number of seconds elapsed. (Define the relevant variables.) c. Over what time interval does the ball complete one revolution? 6. Graph the functions determined in questions 3-5 on the axes below. On each graph illustrate the interval of input values on which the function values complete one full cycle (i.e., the interval of t for which the ball completes one revolution). a. Graph g on the given axes (see question 3). Label your axes. Module 7: Worksheet 5 Page 2 2011 Carlson and Oehrtman

Precalculus Module 7: Worksheet 5 Using the Sine and Cosine Function in Applied Settings b. Graph h on the given axes (see question 4). Label your axes. c. Graph j on the given axes (See question 5). Label your axes. d. Describe the reasoning you used to determine the period (the interval of input needed to complete a full cycle of output values) of the functions in (a), (b), and (c) above. 7. Suppose the ball and string sweeps out 3.8 radians per second. a. How many radians does the ball and string sweep out in t seconds? Module 7: Worksheet 5 Page 3 2011 Carlson and Oehrtman

Precalculus Module 7: Worksheet 5 Using the Sine and Cosine Function in Applied Settings b. Define a function k vertical distance above José number of seconds elapsed. (Define the relevant variables.) c. Over what time interval does the ball complete one revolution? 8. Suppose you determine that it takes the ball 20 seconds to complete one revolution. a. Determine the speed of the ball in radians/second. b. Define a function m vertical distance above José number of seconds elapsed. (Define the relevant variables.) Module 7: Worksheet 5 Page 4 2011 Carlson and Oehrtman

Precalculus Module 9: Worksheet 3 Graphs of Functions in Polar Coordinates For each of the following problems note that angular displacements are measured in radians. 1. Let f be defined as r f ( ) 3 2. a. What do and r f ( ) represent when f is graphed in the polar plane? b. Complete the following table. Then plot the points (, r ) in the polar plane. r f ( ) 0 0.5 1.0 1.5 2.0 2.5 3.0 c. Describe the relationship between and r f ( ). Module 9: Worksheet 3 Page 1 2012 Carlson and Oehrtman

Precalculus Module 9: Worksheet 3 Graphs of Functions in Polar Coordinates d. Instead of graphing the points (, r ) on the polar plane, plot each point on the rectangular plane shown below, where the horizontal axis represents values of and the vertical axis represents values of r. e. Examine the relationship between and r indicated by the graph in part (d). Is this the same relationship you described in part (c)? Explain. f. The graphs of linear functions are often identified as a straight line means that the function has a constant rate of change. If f ( ) 3 2 is linear, why does it not appear to be a straight line in polar coordinates? 2. Let f be defined as r f ( ) 2. a. What do and r f ( ) represent when f is graphed in the polar plane? Module 9: Worksheet 3 Page 2 2012 Carlson and Oehrtman

Precalculus Module 9: Worksheet 3 Graphs of Functions in Polar Coordinates b. Complete the following table. Then plot the points (, r ) in the polar plane. r f ( ) 0 0.5 1.0 1.5 2.0 2.5 3.0 c. Describe the relationship between and r f ( ). d. Instead of graphing the points (, ) r on the polar plane, plot each point on the rectangular plane shown below, where the horizontal axis represents values of and the vertical axis represents values of r. Then compare the behavior shown in this graph to the behavior shown in the graph from part (b). Module 9: Worksheet 3 Page 3 2012 Carlson and Oehrtman

Precalculus Module 9: Worksheet 3 Graphs of Functions in Polar Coordinates 3. Let f be defined as r f ( ) sin 2 the polar plane. r f ( ) 0 0.5 1.0 1.5 2.0 2.5 3.0. Complete the following table, then plot the points (, r ) in 4. Let f be defined as polar plane. r f x e cos( ) ( ). Complete the following table, then plot the points (, ) r in the r f ( ) 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Module 9: Worksheet 3 Page 4 2012 Carlson and Oehrtman

Precalculus Module 9: Worksheet 3 Graphs of Functions in Polar Coordinates coordinates. Note that, although you should use the marked points as a guide, you may have to examine the behavior between the points as well to increase the accuracy of your graph. 5. r f ( ) 2 r Module 9: Worksheet 3 Page 5 2012 Carlson and Oehrtman

Precalculus Module 9: Worksheet 3 Graphs of Functions in Polar Coordinates 6. r f 2 ( ) sin( 10) r Module 9: Worksheet 3 Page 6 2012 Carlson and Oehrtman

Precalculus Module 9: Worksheet 3 Graphs of Functions in Polar Coordinates 7. sin r f ( ) e r Module 9: Worksheet 3 Page 7 2012 Carlson and Oehrtman

Excerpts from Precalculus: Pathways to Calculus Instructor Notes

Precalculus Module 7: Worksheet 5 Using the Sine and Cosine Function in Applied Settings understanding the idea of period in relation to trigonometric functions. Students are probed to make sense of the period of the sine function by noting the rela cycle. José extends his arm straight out, holding a 2.3-foot string with a ball on the end. He twirls the ball around in a circle with his hand at the center, so that the plane in which it is twirling is perpendicular to the ground. Answer the following questions assuming the ball twirls counter-clockwise starting at the 3 1. Label quantities on the diagram that could be used to describe the position of the ball as it rotates This question probes students to visualize and identify the quantities in this illustration that are needed to model the position ball. Ask students to describe which quantities vary and which are constant. Require students to be descriptive when defining variables to represent the values that these quantities assume. If students do not spontaneously state the reference points for distances and angles, and the units in which the quantities are being measured, prompt them to do so. Solution: Quantities may vary. Some options: Total distance that the ball has traveled (in feet or in radii) Angle swept out by the ball and spring (in degrees or in radians) Questions 2-5, below, require that students use the sine and cosine functions to model the motion of an object traveling in a circular path. As the students determine functions, pose questions that require them to be explicit about quantities they are measuring and the units of measurement for each quantity. It may be necessary to have the students use a diagram or applet (Fan 1) to visualize and identify each quantity. Module 7: Worksheet 5 Page 1 2010 Carlson and Oehrtman

Precalculus Module 7: Worksheet 5 Using the Sine and Cosine Function in Applied Settings Prompt students to recognize how the rate of the angle swept out by the ball (in radian measure per second) as it travels on a circular path affects the argument to the sine function. The students should focus on the interval of input t completes one revolution). Students should also be prompted to describe how a change in the rate of change of arc-length with respect to time affects the interval of time needed to complete one revolution. The students should be able to verbalize that when the rate of change is halved, the interval of input needed to complete a revolution is doubled because twice the amount of time (twice the interval of input) is needed to complete a revolution (note that the argument of the sine and cosine functions always vary by should lay a strong foundation for understanding the idea of period of trigonometric functions. 2. a. Construct a graph of a function f vertical distance above function of the measure of the angle swept out by the ball and string (measured in radians) as the ball twirls counter-c. b. Define the algebraic form of the function f vertical distance above hand as a function of the measure of the angle swept out by the ball and string (measured in radians) as the ball twirls counter- (Define the relevant variables.) Solution: f () 2.3sin(), where f () hand and represents the number of radians that the ball has swept out from its starting position. 3. Suppose the ball that Jose was travels one radian per second. a. How many radians does the ball and string sweep out in t seconds? b. Define a function g that rvertical distance above number of seconds elapsed. (Define the relevant variables.) c. Over what time interval does the ball complete one revolution? Solution: For t seconds of travel time, the ball has rotated 1t radians. g(t) 2.3sin(t), where g(t) t represents the number of seconds that the ball has traveled from its starting position. In this case, the ball completes one revolution in 2 seconds. While discussing this solution, it may be useful to emphasize that the function g(t) 2.3sin(t), is the composite of 1t, and f () 2.3sin(). Prompt students to describe the input and output quantities for each of the three functions. This way of thinking will be further developed in the subsequent questions. 4. Suppose the ball travels 2 radians per second. a. How many radians does the ball and string sweep out in t seconds? b. Define a function h vertical distance above number of seconds elapsed. (Define the relevant variables.) c. Over what time interval does the ball complete one revolution? Solution: If the ball travels t seconds it has rotated 2t radians since 2t radians. h(t) 2.3sin(2t), where h(t) hand and t represents the number of seconds that the ball has traveled from its starting position. In this case, the ball completes one revolution in 2radians. Since the ball covers two radians in one second, seconds. Module 7: Worksheet 5 Page 2 2010 Carlson and Oehrtman

Precalculus Module 7: Worksheet 5 Using the Sine and Cosine Function in Applied Settings 5. Suppose the ball travels 0.5 radians per second. a. How many radians does the ball and string sweep out in t seconds? b. Define a function j that relavertical distance above number of seconds elapsed. (Define the relevant variables.) c. Over what time interval does the ball complete one revolution? Solution: For t seconds of travel time, the ball has rotated 0.5t radians. j(t) 2.3sin(0.5t), where j(t) and t represents the number of seconds that the ball has traveled from its starting position. In this case, the ball completes one revolution in 4seconds. 6. Graph the functions determined in questions 3-5 on the axes below. On each graph illustrate the interval of input values on which the function values complete one full cycle (i.e., the interval of t for which the ball completes one revolution). After students have completed each graph, prompt them to describe how they identified the minimum using the graph to identify the period of the function, the students do not have to reason about the discussions should be led that encourage students to recognize that the period of a function is determined by answering t t reasoning is necessary to generate both the symbolic and graphic representations of the given contexts. As an example, if the argument to the sine function is 2t, the input to the function t must of the sine function and the input to the function and the role both play in determining the period of sine and cosine functions. a. Graph g on the given axes (see question 3). Label your axes. Solution: Module 7: Worksheet 5 Page 3 2010 Carlson and Oehrtman

Precalculus Module 7: Worksheet 5 Using the Sine and Cosine Function in Applied Settings b. Graph h on the given axes (see question 4). Label your axes. Solution: c. Graph j on the given axes (See question 4). Label your axes. Solution: d. Describe the reasoning you used to determine the period (the interval of input needed to complete a full cycle of output values) of the functions in (a), (b), and (c) above. Solution: The interval of input needed to complete a full cycle of output values corresponds to the is conveyed on the graph by the interval of input needed such that the output at the beginning and end of this interval is always constant. The following two questions continue to promote the same reasoning as the previous questions, as research has shown that students need repeated opportunities to reason about the period of a function. Continue to prompt the students to describe the relationship between the input, the argument, and the period of the function. Also, the last question presents the period of the function and asks the student to determine the function. Module 7: Worksheet 5 Page 4 2010 Carlson and Oehrtman

Precalculus Module 7: Worksheet 5 Using the Sine and Cosine Function in Applied Settings 7. Suppose the ball and string sweeps out 3.8 radians per second. a. How many radians does the ball and string sweep out in t seconds? b. Define a function k vertical distance above hand as a function of the number of seconds elapsed. (Define the relevant variables.) c. Over what time interval does the ball complete one revolution? Solution: For t seconds of travel time, the ball has rotated 3.8t radians. k(t) 2.3sin(3.8t), where k(t) and t represents the number of seconds that the ball has traveled from its starting position. In this case, the ball completes one revolution in 2 / 3.8 ; 1.653 seconds. 8. Suppose you determine that it takes the ball 20 seconds to complete one revolution. a. Determine the speed of the ball in radians/second. b. Define a function m vertical distance above Jo number of seconds elapsed. (Define the relevant variables.) Solution: For t seconds of travel time, the ball has rotated 2 20 t radians. m(t) 2.3sin 2 20 t, where m(t) is th and t represents the number of seconds that the ball has traveled from its starting position. In this case, the ball completes one revolution in 20 seconds. Module 7: Worksheet 5 Page 5 2010 Carlson and Oehrtman