DES –TIME 2006 Hildegard Urban-Woldron EXPLORING MATHEMATICS AND PHYSICS CONCEPTS Using TI graphing calculators in conjunction with Vernier Sensors Dr. Hildegard Urban-Woldron T³ Österreich EXPLORING MATHEMATICS AND PHYSICS CONCEPTS

DES –TIME 2006 Hildegard Urban-Woldron 2 INTRODUCTION RESEARCH QUESTIONS AND THEIR METHODICAL BASIS TASKS AND PRACTICAL EXPERIENCES IN THE CLASSROOM EVALUATION, ASSESSMENT AND DISCUSSION EXPLORING MATHEMATICS AND PHYSICS CONCEPTS

DES –TIME 2006 Hildegard Urban-Woldron 3 APPS - powerful technological tools explore, discover and extend mathematical ideas GC - IT-platform –Vernier sensors for collecting various data –GC for analyzing and building models impact of new technologies INTRODUCTION

DES –TIME 2006 Hildegard Urban-Woldron 4 software tools - students’ mathematical thinking educational potential of connecting Mathematics and Physics effects of analyzing real world data ask own questions and explore them gain insight into the individual student’s perspective RESEARCH QUESTIONS AND THEIR METHODICAL BASIS

DES –TIME 2006 Hildegard Urban-Woldron 5 Students’ behaviour in the learning environment analyze the data both algebraically and graphically - associate these relationships with mathematical functions Do the offered tasks support reactive, active or pro- active learning? The impact of the teacher’s behaviour on the learning efficiency of the student RESEARCH QUESTIONS AND THEIR METHODICAL BASIS

DES –TIME 2006 Hildegard Urban-Woldron 6 EASYTEMP, EASYLINK, CBR 2 –in combination with the GC –offer a learning environment for –experimenting, modelling, analyzing and visualizing real world data Students are asked –to explore the situations –make conjectures and –finally prove them TASKS AND PRACTICAL EXPERIENCES IN THE CLASSROOM

DES –TIME 2006 Hildegard Urban-Woldron 7 measure the movement of the ball describe it as a function of time and derive the gravity law time-distance graphs are parabolic functions The maximum height decreases from bounce to bounce ACTIVITY 1: THE BOUNCING BALL

DES –TIME 2006 Hildegard Urban-Woldron 8 highest speed of the ball acceleration during falling model to describe the height of the ball for a particular bounce total distance of the ball velocity change as the ball rises and falls What affects the shape of the graph of both the height and the velocity? ACTIVITY 1: THE BOUNCING BALL

DES –TIME 2006 Hildegard Urban-Woldron 9 ACTIVITY 1: THE BOUNCING BALL

DES –TIME 2006 Hildegard Urban-Woldron 10 ACTIVITY 1: THE BOUNCING BALL

DES –TIME 2006 Hildegard Urban-Woldron 11 ACTIVITY 1: THE BOUNCING BALL

DES –TIME 2006 Hildegard Urban-Woldron 12 ACTIVITY 2: AIR UNDER PRESSURE – BOYLE’S LAW

DES –TIME 2006 Hildegard Urban-Woldron 13 ACTIVITY 2: AIR UNDER PRESSURE – BOYLE’S LAW

DES –TIME 2006 Hildegard Urban-Woldron 14 What kind of relationship exists between volume and pressure? –A quadratic function (parabola as graph)? –An exponential function? –A power function or a hyperbolic function? ACTIVITY 2: AIR UNDER PRESSURE – BOYLE’S LAW

DES –TIME 2006 Hildegard Urban-Woldron 15 for 20 ml the pressure was almost 102 kPa ACTIVITY 2: AIR UNDER PRESSURE – BOYLE’S LAW

DES –TIME 2006 Hildegard Urban-Woldron 16 ACTIVITY 3: NEWTON’S LAW OF COOLING What determines the cooling process?

DES –TIME 2006 Hildegard Urban-Woldron 17 ACTIVITY 3: NEWTON’S LAW OF COOLING ?

DES –TIME 2006 Hildegard Urban-Woldron 18 ACTIVITY 3: NEWTON’S LAW OF COOLING

DES –TIME 2006 Hildegard Urban-Woldron 19 ACTIVITY 3: NEWTON’S LAW OF COOLING

DES –TIME 2006 Hildegard Urban-Woldron 20 Newton’s law of cooling or Students try to fit the temperature functions above to the actually measured curve by adjusting the parameters k and a. ACTIVITY 3: NEWTON’S LAW OF COOLING

DES –TIME 2006 Hildegard Urban-Woldron 21 Has the starting temperature of the hot water any impact on the value obtained for a? What could you do to your experimental apparatus to decrease the value of k in another run? What quantity does k measure? If your starting temperature difference is cut in half, does it take half as long to get 1°C above room temperature? Why or why not does it take half as long? Extensions – research questions ACTIVITY 3: NEWTON’S LAW OF COOLING

DES –TIME 2006 Hildegard Urban-Woldron 22 simple questions motivated –to a rich discourse and activity in the classroom –and further exploration lessons involved multiple stages of investigation –prediction, testing, rejection or extension of hypotheses –discovering and exploring the underlying mathematics –making generalizations and proving results EVALUATION, ASSESSMENT AND DISCUSSION

DES –TIME 2006 Hildegard Urban-Woldron 23 technology in the classroom especially supports the active, self-controlled and self-motivated student challenge for the teacher – embed technology tools in the general syllabus – and into the class room – substantial change in the learning process – active, individual construction of knowledge – guidance - room for individual exploration EVALUATION, ASSESSMENT AND DISCUSSION