MOTION SENSORS IN MATHEMATICS TEACHING
TOOLS FOR UNDERSTANDING GENERAL MATH CONCEPTS
Hildegard Urban-Woldron
University of Teacher Education, Lower Austria
TIME 2012, Tartu, Estonia
OVERVIEW
 Introduction
 Using the motion detector to learn about functions
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Linear functions: interpreting slope as rate of change
Matching a graph
Estimating change of speed for a ball rolling on an incline
Describing the movement of a bouncing ball as a function of time
Investigating the influence of aerodynamic drag
Sinusoidal functions from real data
Deriving the kind of a possible experiment from data
 Introducing concepts of integral calculus
 Discussion
DESCRIBING MOTION BY USING GRAPHS OF MOTIONS
 graphs of position vs. time
 graphs of velocity vs. time
 graphs of acceleration vs. time
Powerful technical tools enable students …
 to engage in hands-on collection of real-world data
 to quickly collect accurate motion data
 to display collected data in digital and/or graphical form
 to process and analyze the collected data
 to take an active role in their learning
 to construct knowledge from actual observations
 to understand advanced concepts of kinematics
EXAMPLE 1: WALKING IN FRONT OF A MOTION DETECTOR
Building student expertise in working
with linear algebraic relationships,
particularly with regard to gradient
and y-intercept.
manually fitting funtions
f1(x) =  0,31 x + 2,24
f1(x) = 0,18 x + 0,64
interpreting slope as rate of change
EXAMPLE 2: MATCH THE GRAPH / MATH IN MOTION
 Interpreting graphs (linear functions) and
exploring the meaning of slope
 Representing the algebraic
gradient and y-intercept
concept
of
 students learn more from their mistakes than
from correct responses
 students quickly discover the ideas of
negative and positive slope as well as the yintercept
MORE CHALLENGING: MATCH THE GRAPHS!
distance versus time
velocity versus time
EXTENSION: AVERAGE AND INSTANTANEOUS VELOCITY
 Conceptualizing velocity
(speed) as a rate of
change
 Considering the impact of
the time step
 Finding
connections
between
graph
and
algebraic representation
EXAMPLE 3: ROLLING BALL ON AN INCLINE
Which plot best matches the motion of the ball?
Further questions

predict, what will happen if the
incline increases

PREDICT – OBSERVE – EXPLAIN
examine, what happens for
differing inclines
8
EXAMPLE 4: BOUNCING BALL
Describe the movement of the ball as a function of time?
y = a . (x - b)² + c
 time-distance graphs seem to be parabolic functions
 The maximum height decreases from bounce to bounce
 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?
9
EXPLORING THE FIRST AND THE SECOND BOUNCE
 What do the two bounces have in common?
 What is different?
 How do the position and the movement of the ball depend on the
parameters a, b and c?
f1(x) = -4,74 . (x – 0,84)² +0,75
f2(x) = -4,74 . (x – 1,57)² +0,56
HOW DOES THE MAXIMUM HEIGHT DEPEND ON TIME?
y = h . px
x … time
y … actual height
h … height at x = 0
p … constant
EXAMPLE 5: ACCELERATION OF A CART ON AN INCLINED PLANE
free-body diagram
position-time graph
velocity-time graph
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EXTENSION: INVESTIGATE THE MOVEMENT OF A REBOUNDING TROLLEY
 identify on the graph
 „collision - points“
 the parts of the graph / movement down the
runway
 the parts of the graph / movement up the
runway
 symmetry of each half-loop?
 study the velocity of the trolley more directly
13
THE HALF-LOOPS ARE ASYMMETRICAL!
 make connections between shape of the graph and speed of the trolley
 find ways to derive the velocity of the trolley from the distance vs. time graph
 adjust linear functions to fit the different parts of the sactter plot
representing the collected data
EXAMPLE 6: MASS ON A SPRING
Comparing experimental data to the
sinusoidal function model y  t  = A  sin  2    f  t   
How do these parameters
contribute to the shape of
the graph?
FURTHER QUESTIONS
 max  d   min  d    0.57
2
 Where is the mass when the
velocity is zero?
 Where is the mass when the
velocity is greatest?
position vs. time- & velocity vs. time – graph
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How are the two graphs the same?
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How are they different?
MORE INVESTIGATIONS
Phase diagram (velocity vs. position)
acceleration vs. position
WHAT COULD BE THE EXPERIMENT BEHIND THE DATA?
Explain for each step of analysis what you can deduce from the data
WHAT COULD BE THE EXPERIMENT BEHIND THE DATA?
© 2008 Texas Instruments
INTRODUCING CONCEPTS OF INTEGRAL CALCULUS
A Boeing 737 takes off.
Calculate velocity and covered distance by the
time of taking off!
© Lars Jakobsson
PREDICT – OBSERVE – EXPLAIN / VELOCITY VS. TIME GRAPH
 Is the time-velocity graph in agreement with your primary prediction?
 If not, what mistakes did you make?
 What can you infer from the velocity-time graph in reference to the motion of
the airplane?
WHAT DISTANCE IS COVERED BY THE TIME OF TAKING OFF?
Make a sketch of the distance vs. time graph!
 Is the time-distance graph in agreement
with your primary prediction?
 If not, what mistakes did you make?
 How will the three graphs (a-t, v-t, and d-t)
look like for the landing of the airplane?
THANK YOU FOR YOUR ATTENTION!
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