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
2DES –TIME 2006 Hildegard Urban-Woldron
• INTRODUCTION
• RESEARCH QUESTIONS AND THEIR METHODICAL
BASIS
• TASKS AND PRACTICAL EXPERIENCES IN THE
CLASSROOM
• EVALUATION, ASSESSMENT AND DISCUSSION
EXPLORING MATHEMATICS AND PHYSICS CONCEPTS
3DES –TIME 2006 Hildegard Urban-Woldron
• 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
4DES –TIME 2006 Hildegard Urban-Woldron
• 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
5DES –TIME 2006 Hildegard Urban-Woldron
• 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
6DES –TIME 2006 Hildegard Urban-Woldron
• 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
7DES –TIME 2006 Hildegard Urban-Woldron
• 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
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8DES –TIME 2006 Hildegard Urban-Woldron
• 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
14DES –TIME 2006 Hildegard Urban-Woldron
• 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
15DES –TIME 2006 Hildegard Urban-Woldron
for 20 ml the pressure was almost 102 kPa
.constVp
ACTIVITY 2: AIR UNDER PRESSURE – BOYLE’S LAW
16DES –TIME 2006 Hildegard Urban-Woldron
ACTIVITY 3: NEWTON’S LAW OF COOLING
What determines the cooling process?
20DES –TIME 2006 Hildegard Urban-Woldron
Newton’s law of cooling
tkR eTTT 0
tR aTTT 0or
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
21DES –TIME 2006 Hildegard Urban-Woldron
• 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
22DES –TIME 2006 Hildegard Urban-Woldron
• 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
23DES –TIME 2006 Hildegard Urban-Woldron
• 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
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