Patterns and Functions 9‐11 Grade Instructors: Dr. Todd Frauenholtz & Mr. Craig Rypkema Summer Math Institute 2008 Bemidji State University By Ralph Cox & Chris Granley 2 Executive summary: This project addresses the following Minnesota Academic Standards in Mathematics Understand the concept of function, and identify 9, important Obtain information and draw conclusions from graphs 10, Algebra features of 9.2.1.4 of functions and other relations. 11 functions and other relations using symbolic and graphical methods. Recognize linear, quadratic, exponential and other common functions in real‐world and mathematical situations; Sketch graphs of linear, quadratic and exponential 9, represent these functions, and translate between graphs, tables and 10, Algebra functions with 9.2.2.3 symbolic representations. Know how to use graphing 11 tables, verbal technology to graph these functions. descriptions, symbols and graphs; solve problems involving these functions, and explain results in the original context. We will explore patterns and functions by looking first at linear, then quadratic, and finally exponential relationships. Students will collect, 3 organize, and analyze data looking for a pattern or a set of rules that applies to the data. Students will work together in small groups to develop tables, graphs, and diagrams and to derive accurate recursive and explicit formulas to describe the given function or relationship. A variety of real world activities will be explored, presented, and discussed to help develop these topics. 4 Contents: Lesson 1: Recognizing and Analyzing Linear Patterns and Functions Activity 1: Activity 2: Activity 3: Activity 4: Activity 5: “Real World”Applications for Discussion Situation I: Cell Phone Plans Situation II: Cost, Revenue, and Profit Crickets Too Cold to Chirp? Pass the Basketball I Bounce the Ball Attack of the Worms page 5 page 6 page 7 page 8 page 10 Lesson 2: Recognizing and Analyzing Quadratic Patterns and Functions Activity 1: Activity 2: Activity 3: “Real World”Applications for Discussion Situation I: Ball or Strike? Situation II: Superman’s Strength Pass the Basketball II Attack of the Killer Worms page 12 page 13 page 14 Lesson 3: Recognizing and Analyzing Exponential Patterns and Functions Activity 1: Activity 2: Activity 3: Activity 4: “Real World”Applications for Discussion page 16 Situation I: Hot Coffee Situation II: Just Keep Folding, Folding, Folding Situation III: Hulk’s Failed Test Koch’s Snowflake page 18 Sierpinski’s Triangle page 20 Attack of the Mutant Killer Worms page 22 5 Lesson 1: Recognizing and Analyzing Linear Patterns and Functions (4‐5 class periods) Understand the concept of function, and identify 9, important Obtain information and draw conclusions from graphs 10, Algebra features of 9.2.1.4 of functions and other relations. 11 functions and other relations using symbolic and graphical methods. Launch: Working in small groups to explore and develop methods and strategies to complete several activities involving linear relationships. Several suggested activities are listed below. Explore: Students will collect, organize, and analyze data looking for a pattern or a set of rules that apply. Each group of students will work together to develop and test recursive and explicit formulas to describe the pattern. Share: Students will share their resulting data collected, including any tables, diagrams, and/or graphs, and the methods, strategies, and formulas that were developed. Summarize: Discuss students’ methods and strategies, how they developed their methods, what data organization they used, and how they analyzed their data to get their results. Discuss the process, the pattern, and the linear relationship that exists. Verify both the recursive and explicit formulas that are developed. 6 Activity 1: “Real World”Applications for Discussion Students will be provided with several real world situations involving linear functions and relationships. Situation I: Cell Phone Plans Patrick is purchasing a new cell phone plan for his family. The Goodbye2Fast plan has a base rate of 12 dollars and charges at a rate of 5 cents per minute used. The Talks-a-Lot plan has a base rate of 25 dollars and then charges only 3 cents per minute used. a. Write the total cost for the Goodbye2Fast plan as a function of the minutes used. b. Write the total cost for the Talks-a-Lot plan as a function of the minutes used. c. Construct a table showing minutes used and total cost, graph the data, and decide on the best plan for Patrick and his family. Be prepared to defend your answer. Situation II: Cost, Revenue, and Profit A manufacturer produces a cell phone at a cost of $22.50 per phone. The manufacturer has a fixed cost of $2450 per day. Each phone sells for $54.00. Let x represent the number of phones produced in a 5-day work week. a. Write the total cost C as a function of x. b. Write the revenue R as a function of x. c. Write the profit P as a function of x. Hint: The profit function is given by P(x) = R(x) – C(x). 7 Activity 2: Crickets Too Cold to Chirp? Scientists have discovered a linear relationship exists between the number of cricket chirps per minute and the temperature in degree Fahrenheit. They have come up with the equation relating cricket chirps to temperature. At 50 oF the number of chirps is 40, and at 80 oF the number of chirps is 160. Those values have been entered in the table below. Complete the table to show the temperature when the number of cricket chirps is 50, 60, 70, ..., 150. Chirps 40 Temp in oF 50 50 60 70 80 90 100 110 120 130 140 150 160 80 Graph your information in the space provided below. Label your graph appropriately. What is the temperature when the crickets first stop chirping? Explain how you found your answer. What is the increase in temperature when the number of chirps per minute increases by 1? State the temperature T as a function of the chirps C. 8 Activity 3: Pass the Basketball I Students determine the time it takes for different numbers of students to pass a ball from one student to the next. If the students pass the ball at a relatively constant rate, the data collected and graphed (time versus number of students) can be modeled by a linear function. A basketball and a stopwatch are needed. One student will be needed to serve as the timer and another to record the data. The remaining students in the class should stand in a straight line. When the timer says “Start,” the first student in line passes the ball to the next student and this continues to the fifth student. The timer stops the stopwatch when the fifth student receives the ball. The recorder enters the time in seconds required for the ball to be passed to the fifth student. The first student in line gets the basketball back and the process is repeated for the first ten students in the line. Always start the basketball back with the first student and add five students each time. (With a small class, add two students each time you repeat the data collection.) After the data is collected and recorded, share the data with all students. Working in small groups, they will then graph and analyze the data looking for a pattern or a set of rules that apply. Each group of students will work together to develop and test a recursive and explicit formulas to describe the pattern. # of students Time in seconds 5 10 15 20 25 30 35 For more information see this website posted by Betty Martin Gasque: http://64.233.167.104/search?q=cache:o4xKHaYZNIwJ:scssi.scetv.org/offices/cs o/msu/scalgebra/doc/Pass_The_Ball.pdf+quadratic+activities&hl=en&ct=clnk&cd =1&gl=us 9 Activity 4: Bounce the Ball Working in groups of three, students will collect data on bouncing a tennis ball. Students will count and record the number of times each student bounces the ball in ten-second intervals over a ninety-second time frame. Each student in the group takes a turn at bouncing the ball, timing, and recording the data. Data will be organized in tables, graphed, and analyzed representing each student and then compared to the group. Name: _____________________________________ # of seconds elapsed 10 20 30 40 50 60 70 80 90 TOTAL # of bounces Graph the information from the table. Label your graph appropriately. 10 Activity 5: Attack of the Worms Students will be provided with graph paper. Provided with a sequence of linear“worm” patterns, they will construct the next worms in the sequence. Students will collect data on each stage of the sequence and work together to develop and test a recursive and explicit formulas to describe the linear growth pattern of the worm. Stage 1 Stage 2 Stage 3 Stage 4 Stage 5 Stage 6 Stage 7 Stage 8 Stage n 11 12 Lesson 2: Recognizing and Analyzing Quadratic Patterns and Functions (4‐5 class periods) Understand the concept of function, and identify 9, important Obtain information and draw conclusions from graphs 10, Algebra features of 9.2.1.4 of functions and other relations. 11 functions and other relations using symbolic and graphical methods. Launch: Working in groups to explore and develop methods and strategies to complete several activities involving quadratic relationships. Several suggested activities are listed below. Explore: Students will collect, organize, and analyze data looking for a pattern or a set of rules that apply. Each group of students will work together to develop and test recursive and explicit formulas to describe the pattern. Share: Students will share their resulting data collected, including any tables, diagrams, and/or graphs, and the methods, strategies, and formulas that were developed. Summarize: Discuss students’ methods and strategies, how they developed their methods, what data organization they used, and how they analyzed their data to get their results. Discuss the process, the pattern, and the quadratic relationship that exists. Verify both the recursive and explicit formulas that are developed. 13 Activity 1: “Real World”Applications for Discussion Students will be provided with several real world situations involving quadratic functions and relationships. Situation I: Ball or Strike? A pitcher releases a baseball 6 feet above the ground at a speed of 132 feet per second toward home plate, which is 60.5 feet away. The height h(x), in feet, of the ball x feet from home plate can be approximated by h(x) = - 0.0009x2 + 6. Assuming the ball crosses over home plate, to be considered a strike, the ball must cross home plate and be at least 2.5 feet high and less than 5.4 feet high. Is the pitch a ball or strike? Situation II: Superman’s Strength Superman loses his power when he comes in close proximity to a source of Kryptonite. Lois Lane noticed that the percentage of reduction in Superman’s strength was related to the how far he was from source of Kryptonite. She made several measurements and recorded the data in the table below: Distance in meters 10 20 30 40 50 60 Percentage Reduction in Superman’s Strength 99 96 91 84 How far away should Superman stay to be at full strength? Complete the table, graph your data, and come up with a formula to support your answer. 14 Activity 2: Pass the Basketball II One student will be needed to serve as the timer and another to record the data. The remaining students in the class should stand in a straight line. In this version of the activity, the first student says “A” and then passes the ball to the second student. The second student says “A B” and then passes the ball to the third student. This pattern continues until the fifth student receives the ball and says “A B C D E.” After the fifth student says the last letter (E), the timer stops the stopwatch and the time is recorded. The first student in line gets the basketball back and the process is repeated for the first ten students in the line. The tenth student in the line will say “A B C D E F G H I J” when the ball is received and the timer stops the stopwatch after this student says “J.” The time for the ten students is recorded. Repeat this process, adding five students each time, until data are collected for all students in the line. (If there are more than 26 students in the class, the next student will say all of the letters in the alphabet and then say “one.” The student after this one would say all of the letters in the alphabet and then say “one two.”) Since the second change is theoretically constant, these data can be modeled by a quadratic function. Consider students three and four in the line. The third student said three letters before passing the ball and the fourth student said four letters. They did not change the “slow down” time by the same number of letters but the change in that “slowdown” change was one letter. This is the same for the ninth and tenth students in the line. The ninth student slowed down the passing of the ball by nine letters while the tenth slowed it down by ten letters. The change in the change (second change) is, again, one letter. With quadratic functions, there is a constant second change in the dependent variable values. This can be illustrated using a quadratic function such as y = 2 x2 + 3 x – 1 with ordered pairs (1, 4) (2, 13), (3, 26), (4, 43) and (5, 64). The first changes in the dependent variable are 9, 13, 17, and 21, respectively. The second changes are 4, 4, and 4. Note that the independent variable values ({1, 2, 3, 4, 5}) in the ordered pairs increase by one unit each time. After the data is collected and recorded, share the data with all students. Working in small groups, they will then graph and analyze the data looking for a pattern or a set of rules that apply. Each group of students will work together to develop and test a recursive and explicit formulas to describe the pattern. For more information see this website posted by Betty Martin Gasque: http://64.233.167.104/search?q=cache:o4xKHaYZNIwJ:scssi.scetv.org/offices/cs o/msu/scalgebra/doc/Pass_The_Ball.pdf+quadratic+activities&hl=en&ct=clnk&cd =1&gl=us 15 Activity 3: Attack of the Killer Worms Students will be provided with graph paper. Provided with several sequences of quadratic “killer worm” patterns, they will construct the next few worms in the sequence. Students will collect data on each worm and work together to develop and test a recursive and explicit formulas to describe the quadratic growth pattern of the worm. Stage 1 Stage 2 Stage 3 Stage 4 Stage 5 Stage 6 Stage 7 Stage 8 Stage n 16 Lesson 3: Recognizing and Analyzing Exponential Patterns and Functions (4‐5 class periods) Recognize linear, quadratic, exponential and other common functions in real‐world and mathematical situations; Sketch graphs of linear, quadratic and exponential 9, represent these functions, and translate between graphs, tables and 10, Algebra functions with 9.2.2.3 symbolic representations. Know how to use graphing 11 tables, verbal technology to graph these functions. descriptions, symbols and graphs; solve problems involving these functions, and explain results in the original context. Launch: Working in groups to explore and develop methods and strategies to complete several activities involving exponential relationships. Several suggested activities are listed below. Explore: Students will collect, organize, and analyze data looking for a pattern or a set of rules that apply. Each group of students will work together to develop and test recursive and explicit formulas to describe the pattern. Share: Students will share their resulting data collected, including any tables, diagrams, and/or graphs, and the methods, strategies, and formulas that were developed. Summarize: Discuss students’ methods and strategies, how they developed their methods, what data organization they used, and how they analyzed their data to get their results. Discuss the process, the pattern, and the exponential 17 relationship that exists. Verify both the recursive and explicit formulas that are developed. 18 Activity 1: “Real World”Applications for Discussion Students will be provided with several real world situations involving exponential functions and relationships. Situation I: Hot Coffee Suppose your practice is to fetch a cup of hot coffee from the coffee shop, carrying it for 5 minutes outside before you reach your favorite bench in the park, where you sit to drink it. The coffee will cool at a rate proportional to the difference between its temperature and the surrounding air temperature. The coffee cools 6 degrees in the first minute. If the coffee starts at 180 degrees and the air temperature is 65 degrees, what will be its temperature after 5 minutes? Suppose that you like cream in your coffee, and the cream is 65 degrees. If a cup of 180 degree coffee holds 10 ounces and you add 0.5 ounce of cream, what is the coffee temperature after you add the cream? If you like your coffee to be as hot as possible when you drink it, when should you add the cream? 1. add the cream at the shop, then walk 2. wait until you reach the bench, then add the cream 3. it doesn't matter, they come out the same For the answers and more information on this example see the following website: http://math.dartmouth.edu/~klbooksite/3.02/302quiz/302quiz_index.htm Situation II: Just Keep Folding, Folding, Folding Pose the following hypothetical question to your students: If I give you a really, really large piece of paper, and I ask you to fold it over again, and then again, and again, and again, … How many times will it take until the thickness of the folded paper will be just past the orbit of the moon? How many times will it take until the thickness will be just past the orbit of the sun? Construct a table, graph, and derive a formula to support your conclusions. 19 Situation III: Incredible Hulk’s Failed Test The Incredible Hulk is once again under investigation by the League of Super Heroes for the alleged use of a banned synthetic super-hero supplement, SGH. It seems his most recent test came back at .09 parts per super unit, which is still more than 20 times the acceptable level of SGH set by the League of Super Heroes at no greater than .004 parts per super unit of blood. After his first failed SGH test 12 months ago, the Incredible Hulk admitted to his use of the banned substance. He received a 6-month suspension from all Super Hero activities. In order to be reinstated, Hulk was required abstain from any further use of SGH, to enroll in the Super Hero substance abuse rehabilitation program, and to undergo regular testing. Hulk, of course, denies any use of SGH in the past 12 months and claims that he is in total compliance with the League’s stipulation for reinstatement. Months after first failed SGH test 0 1 2 3 4 5 6 7 8 9 10 11 12 SGH Level in parts per super unit 368.64 184.32 92.16 46.08 23.04 11.52 5.76 Is the Incredible Hulk telling the truth? Has he taken SGH since his first failed test 12 months ago? When will he test below the acceptable level of .004? Complete and extend the table as necessary, graph your data, and come up with a formula to support your answers. 20 Activity 2: Koch’s Snowflake Students will be provided with graph paper and construct Koch’s snowflake. Make a table and record the your data after each step. You may print and use this triangle grid paper to help you with this drawing. Step One. Start with a large equilateral triangle. Step Two. Make a Star. 1. Divide one side of the triangle into three equal parts and remove the middle section. 2. Replace it with two lines the same length as the section you removed. 3. Do this to all three sides of the triangle. Do it again and again. Do it infinitely many times and you have a fractal. Want to take a long, careful look at what it looks like? See a few of the steps below. Step One 21 Step Two Step Three Step Four Step Five Let's see them all together For more information and a JAVA applet, check out this website by Cynthia Lanius: http://math.rice.edu/~lanius/frac/koch.html 22 23 Activity 3: Sierpinski’s Triangle Letʹs make a famous fractal called the Sierpinski Triangle. *You may print and use this triangular gridpaper. Step One Draw an equilateral triangle with sides of 2 triangle lengths each. Connect the midpoints of each side. How many equilateral triangles do you now have? Shade out the triangle in the center. Think of this as cutting a hole in the triangle. Step Two Draw another equilateral triangle with sides of 4 triangle lengths each. Connect the midpoints of the sides and shade the triangle in the center as before. Notice the three small triangles that also need to be shaded out in each of the three triangles on each corner - three more holes. Step Three Draw an equilateral triangle with sides of 8 triangle lengths each. Follow the same procedure as before, making sure to follow the shading pattern. You will have 1 large, 3 medium, and 9 small triangles shaded. 24 Step Four How about doing this one on a poster board? Follow the above pattern and complete the Sierpinski Triangle. Use your artistic creativity and shade the triangles in interesting color patterns. Does your figure look like this one? Then you are correct! For more information and a JAVA applet, check out this website by Cynthia Lanius: http://math.rice.edu/~lanius/fractals/ 25 Activity 4: Attack of the Mutant Killer Worms Students will be provided with graph paper. Provided with several sequences of exponential “mutant killer worm” patterns, they will construct the next few worms in the sequence. Students will collect data on each worm and work together to develop and test a recursive and explicit formulas to describe the exponential growth pattern of the worm. Stage 1 Stage 2 Stage 3 Stage 4 Stage 5 Stage 6 Stage 7 Stage 8 Stage n