A Reinvention course for Calculus

Transcription

1 A Reinvention course for Calculus Working session at PME & PSI 2001, Utrecht, The Netherlands Michiel Doorman Freudenthal Institute, Utrecht University, The Netherlands michiel@fi.uu.nl 1

2 Preface The exercises below are excerpts from a chapter on the mathematics of change and the differential quotient for 10 lessons for 16 years olds. This material is an example of a guided re-invention course on this topic. It is still in development in a developmental research project at the Freudenthal Institute. The central idea is a chain of representations from informal descriptions to the formal mathematics that students experience as if they could have made each step by themselves. Every next representation derives its meaning, use and conventions from previous ones. The exercises are translated from Dutch. I apologize for the possibility that, due to translation-errors, in some cases the questions might not immediately be clear. Between exercises are italic instructions for the teacher, ideas about the lessons, or short descriptions of the activities which are not copied from the original material. While working through the exercises try to take a student-perspective: What will a student do? How can she/he anticipate on coming activities or fall back on previous insights or representations? think child The applets that are used in this material can be found through: or (click on Applet-list): 2 PSI Working Group: Calculus

3 1 Describing movement 1 Below you see the situation of a hurricane which is approaching the west-coast of Mexico. The last five positions of the hurricane (Olivia) are located on 9, 10 and 11 October 2000, on 6 o clock and on 18 o clock. The inhabitants worry about the moment on which the hurricane will hit the land. What time can you predict for them?... Teacher discusses the descriptions of movement in order to do predictions.... 3

4 An other way to describe movement is with a time series of photographs: 2 a. How can you make such a photo? b. This time series is made with a fixed frequency. Can you conclude from the photo whether the jumper needs more time to get at his highest point then from the highest point to the ground? 3 The photo alongside is of a falling ball (30 flashes per second). On the drawer you see a cm-scale (the numbers are dm). a. Describe the movement of the ball with a graph. b. Suppose that the ball is dropped at a height of 1 m. Predict after how many flashes the ball reaches the floor. c. Can you predict with what velocity the ball will hit the floor? PSI Working Group: Calculus

5 ... The teacher discusses graphs and presents two ways of analyzing such a photo: looking at the displacements or looking at the total distance travelled depending on the problem to solve. Next, the teacher introduces the software Flash. A computerprogram that measures distances and shows them in a table and in a graph. The user has to click on the successive positions of a point that is followed.... Computerprogram Flash 4 Start Flash. Click on the successive positions of the Falling ball. Look at the two graphs that Flash can produce. Is it possible to just copy the last displacement of the ball a few times for your prediction when it will reach the floor? Try the button Continuate and see what happens. 5 Another photo in Flash is of a Stick which is thrown away. Click on the link to the photograph of this stick to open the photo in Flash. The stick rotates during its throw. How do the movements of an end of the stick and the middle of the stick differ? Use Flash to follow the middle with one color and one of the ends with another color. a. Describe the differences between the two movements. b. Do the distances travelled differ? Flash can draw two graphs of a movement: Displacements The graph with the displacements between two flashes shows the step size between two clicks. This graph can be used to see how displacements increase or decrease. Distance travelled The graph with the distance travelled shows the total distance travelled from the beginning of the movement: the sum of all the displacements. 5

6 6 In the Flash-picture 3 spirals you see 3 different rotations of 5 rounds. Predict for each rotation what will be the distance to the centre after the sixth round. 7 Below you see descriptions of the movement of a zebra and a cheetah. The zebra is running with a constant velocity. The cheetah notices the zebra, and starts sprinting to try to catch the zebra. Unfortunately he can only keep his topspeed for 15 seconds. Will he be able to catch the zebra? 1 cm = 100 m. dots after every 5 seconds. a. What is the constant velocity of the zebra? b. You can see this picture through the Flash link to Cheetah. Will the cheetah catch the zebra? 8 a. Suppose the cheetah can keep his topspeed longer. How long to catch the zebra in this situation? b. Suppose the cheetah starts 5 seconds later (click two times on the first position). Can he still catch the zebra? Explain your answer. c. Explain which graph you use to see whether the cheetah catches the zebra, and explain why you don t use the other graph.... Teacher discusses the models: graphs & meanings & uses in the previous activities. The next exercises are for a reflection on the use of the graphs PSI Working Group: Calculus

7 9 Below you see a displacements graph and a graph with distances travelled of a moving object. (Flash: opgave21). a. Explain with both graphs that the velocity is not constant. displacements (m) 7 5 distance travelled (m) t (s) t (s) b. With which constant velocity will be travelled the same distance in 4 seconds? c. Draw in both diagrams the graph of the constantly moving object. 10 Below you see a graph of an animal that starts running. Suppose that after 25 seconds the velocity doesn t change for 15 seconds. Can you finish the graph? distance travelled (m) tijd (s)... Teacher discusses the difference between average velocity and instantaneous velocity. Whole class activity. Teacher poses the question: Can you sketch a velocity-time graph of this movement. How will the displacements graph and the velocity graph change when you locate positions with a higher frequency?... 7

8 2 From speed to distance... Teacher introduces the historical problem and poses the investigation below. The teacher stresses the difference between discrete measurements and a continuous hypothesis about movement. After a discussion students work in little groups on this problem. For PME: please skip this paragraph. Working through the exercises will take some time. Go on with the last paragraph with the computerprogram Slope: From distance to speed... There have been various theories about falling objects. According to Aristotle (350 BC) heavy bodies were supposed to fall faster than lightweight bodies. Galileï (1600) disproved the theorem many centuries later. His first hypothesis was that the velocity v of a free (!) falling object is proportional to the falling distance. His second hypothesis was that the velocity is proportional to the falling time (t). As a formula: v = constant t How could he test this second hypothesis? He couldn t measure velocity, and how to measure a velocity that changes continuously? His work was an interaction of mathematical reasoning and experiments. The instrument below is a copy of an instrument that he used for his experiments. A ball runs down the edge and rings a bell every time it passes a little gate Investigate how he could have used this instrument to affirm his hypothesis. 8 PSI Working Group: Calculus

9 ... Teacher can demonstrate a free fall with a tickertape.... Galileï probably also used a velocity-time graph to find a way to describe the distance travelled of free falling bodies when the velocity is constantly increasing: Assume the velocity constant for a given time-interval, the area of the rectangle is exactly the displacement in that time-interval. Add all the areas and the result is an approximation of the total distance travelled. 50 Please skip this paragraph v(m/sec) t (sec) What is the result of his computation of the distance travelled with a time-interval of 0.5 sec.? 13 Now look at the graph from a geometrical perspective. The computation can be interpreted as an approxiamtion of the area of the triangular region which is enclosed by the line along the upper left corners of the rectangles, the t-axis and the vertical line t =5. Explain. What is the size of that area exactly? 14 It thus occurred to Galileï that the area of the region under the velocity graph is exactly equal to the distance fallen. Explain how he also concluded that this is equal to the distance travelled with the constant velocity of the middle moment. As shown in this figure: v(m/sec) v=gt s t s = t gt = gt 2 t (sec) 15 Assume that the velocity of the hurricane Olivia (see the first exercise) increases constantly. What prediction would you get of the moment on which the hurricane will hit the land? LAND 9

10 3 From distance to speed Heer Bommel (a famous Dutch cartoon character) was in a foul mood. The traffic in Rommeldam had delayed him too much and when he thought that he was out of the builtup area of the city, he stepped on the gas pedal, flattening it on the floor, so that his old wreck of a car (de Oude Schicht) flew over the road with screeching tires. Unfortunately he forgot that there was a speed limit on the road he took. The commissioner of police was already approaching on a roaring motor bike, with his hand in the air. Are we in a hurry now? asked Bulle Bas, the policeman, reaching for his note book. Didn t you see the signs? Can you read? But I was not travelling too fast! protested Bommel, in a high, indignant voice. The past quarter of an hour, I have only travelled 10 km, in other words 40 km per hour! And indeed, the meter of de Oude Schicht indicated 10 km. 16 Think of a possible answer Bulle Bas could have given Bommel The following graph tells more about Bommel s trip. 17 a. Suppose Bommel was right. What would the graph have looked like? b. What was his maximal trespassing of the speedlimit. c. How long and how many km s did he trespass the speedlimit? 10 PSI Working Group: Calculus

11 For specific predictions or calculations we sometimes use mathematical reasonings. During these lessons this was for instance the case with the hurricane, when we supposed that the velocity could be continuously increasing. Suppose we would like to know whether Bommel trespassed the speedlimit after 6 minutes. You can try to use the given graph. Another possibility is to use a model of the increasing velocity in the time-interval from 5 to 6 minutes. Suppose the distance travelled of a motion conforms to the formula: st () = t 3 where t is time in seconds and s is distance in m. Very soon the motion reaches supersonic levels. After only 10 seconds, 1 km has been travelled. Below you can see the t,s-graph on the grid [0, 10] by [0, 1000]. s (m) t sec) 18 a. What is the average velocity on the time interval [4, 5]? And on [5, 6]? b. How would you determine the velocity at the instant when t = 5? c. Imagine that the velocity reached after 5 seconds, remains constant. How would the graph have to be adapted? 11

12 19 Below you see the graph of a model for the graph of Bommel for the time-interval [ 5 ; 6.5 ]. The model used is f(x) = 0.4. x The moment t = 5 corresponds with x = 0 and the moment t = 6 with x =1. a. What would the graph look like if the velocity wouldn t change after x = 1 (t = 6)? b. Try to predict the velocity of Bommel, according to this model, as accurately as possible.... Teacher discusses answers and introduces the concept of linear continuation at a certain value of x (what the distance travelled graph would look like when the velocity stays constant from moment x).... computerprogram Linear Continuation 20 The program starts with the graph of f(x) = 0.5. x 2. The point from where the linear continuation should be drawn is at x0 = 1. Try to move the red line by dragging the red dot (see arrow). Move the line in the position of a linear continuation from x0. Check your approximation by clicking right below. 12 PSI Working Group: Calculus

13 21 Change the function f via the Control panel for function in the model for Bommel: f(x) = 0.4. x 2. Check your previous answer with the program. 22 Investigate and describe the possibilities for refining your answer, with: a. [Zoom in], and with b. [Show difference quotient]. 23 In the menu [Use] you can see that you can use the software for Training or for Games. Play the successive games and describe the winning strategies. 13