Transcript Reading Graphs and Interpreting slope: A math/Science

READING GRAPHS AND INTERPRETING SLOPE:
A MATH/SCIENCE TARGETED CONNECTION
Dr. Cheryl Malm, Northwest Missouri State University
Dr. Patricia Lucido, SySTEMic Innovations
RESEARCH FOCUS:
To examine mathematics and science
concepts to identify supporting ideas,
processes, and skills that allow the design of
parallel curricula or “targeted connections”.
INTEGRATED CURRICULA DESIGN


Single courses of study, usually taught by a single mathematicstrained or science-trained teacher, i.e. mathematics courses that
incorporate science applications or science courses that utilize
appropriate mathematical models.
A continuum model that characterizes the relationship between
the mathematics and science in integrated curricula.
Math
Independent
Math Lesson
Focused
with
Supporting
Science
Balanced
lessons
Science
Focused
with
Supporting
Math
Independent
Science
Lesson
CORRELATED LESSONS




Correlated lessons extend the definition of integration,
striving to achieve “balanced” integration in which the
mathematics and science content is of equal importance
(Berlin & White, 1994; Lonning & Defranco, 1997).
Parallel mathematics and science lessons are developed by
a team of teachers, each a content specialist in their own
discipline, to allow the concepts from both disciplines to
be almost equally taught (Vasques-Mireles & West, 2007).
A strength is the team-teaching approach; conversations
occur around the language and the parallel relationships
that are being taught.
The challenges range from lack of planning time and
difficulties in coordinating team taught lessons to lack of
materials and difficulties identifying appropriate
connections (Vasques-Mireles & West, 2007).
TARGETED CONNECTIONS



Targeted connections expand the definition of
correlated lessons to encompass correlated units of
study.
Rather than selecting a mathematics or science topic
and then attempting to incorporate the pertinent topics
from the other discipline , parallel programs would be
designed in mathematics and science that would connect
underlying, supporting conceptual understandings as well
as appropriate skills and applications.
Content designed to be taught simultaneously in a math
course and a science course would each develop the
connected conceptual understanding within the context
of the separate discipline.
TARGETED CONNECTIONS
Correlated lessons would be utilized within the units to
take advantage of the naturally occurring connections in
processes, skills, and applications
Math
• Lesson
• Lesson
• Lesson
• Lesson
• Lesson
Targeted
Connection
• Lesson
Science
• Lesson
• Lesson
• Lesson
• Lesson
• Lesson
READING AND INTERPRETING GRAPHS/VELOCITY AND ACCELERATION
Mathematics Unit
Graphing Motion
Follow a Graph/Tell a Story
Explore Slope in relation to speed
Explore non-linear motion
situations
Application
Science Unit
Explore motion with Balloon Cars
Gather motion data
Graph data on speed and
acceleration
Application
NCTM STANDARDS: 9-12 REPRESENTATIONS
Representation
 Instructional programs from prekindergarten through
grade 12 should enable all students to—
 create and use representations to organize, record,
and communicate mathematical ideas;

select, apply, and translate among mathematical
representations to solve problems;

use representations to model and interpret physical,
social, and mathematical phenomena.
COMMON CORE STANDARDS
Represent and solve equations and inequalities graphically

10. Understand that the graph of an equation in two variables
is the set of all its solutions plotted in the coordinate plane,
often forming a curve (which could be a line).

11. Explain why the x-coordinates of the points where the graphs of
the equations y = f(x) and y = g(x) intersect are the solutions of the
equation f(x) = g(x); find the solutions approximately, e.g., using
technology to graph the functions, make tables of values, or find
successive approximations. Include cases where f(x) and/or g(x)
are linear, polynomial, rational, absolute value, exponential, and
logarithmic functions.
NRC – FRAMEWORK FOR K-12 SCIENCE

Insert science standards here to make the
connection?????? – see next slide
SCIENTIFIC AND ENGINEERING PRACTICES
Asking questions and defining problems
 Planning and carrying out investigations
 Analyzing and interpreting data
 Using mathematics and computational thinking
 Constructing explanations / designing solutions
 Engaging in argument from evidence
 Obtaining, evaluating and communicating
information

NRC – FRAMEWORK FOR K-12 SCIENCE
Crosscutting Concepts
 Cause and effect: Mechanism and explanation
 Systems and system models
 Energy and matter: Flows, cycles and conservation
 Disciplinary Core Ideas
 Motion and stability: Forces and interactions
 Energy
Engineering, Technology and the Application of Science
 Engineering design

NRC – FRAMEWORK CORE IDEA PS3:ENERGY
PS3.A: Definitions of Energy
 What is energy? Kinetic & Stored (potential)
 PS3.B: Conservation of Energy /Energy Transfer
What is meant by conservation of energy?
How is energy transferred between objects or
systems?
PS3.C: Relationship Between Energy and Forces
How are forces related to energy?

NATIONAL SCIENCE EDUCATION STANDARDS:
MOTIONS AND FORCES

Objects change their motion only when a net force is
applied. Laws of motion are used to calculate
precisely the effects of forces on the motion of
objects. The magnitude of the change in motion can
be calculated using the relationship F = ma, which is
independent of the nature of the force. Whenever one
object exerts force on another, a force equal in
magnitude and opposite in direction is exerted on the
first object.
NCTM STANDARDS: 9-12 ALGEBRA
PRINCIPLES AND STANDARDS FOR SCHOOL
MATHEMATICS: 9-12 REPRESENTATIONS
A flight from SeaTac Airport near Seattle,
Washington, to LAX Airport in Los Angeles
has to circle LAX several times before being
allowed to land. Plot a graph of the
distance of the plane from Seattle against
time from the moment of takeoff until
landing.
adapted from Hughes-Hallett et al. Calculus, 1994, p. 6
PRINCIPLES AND STANDARDS FOR SCHOOL
MATHEMATICS:
9-12 REPRESENTATIONS

Fig. 7.40. A representation that a student might produce of an airplane's
distance from its take-off point against the time from takeoff to landing
PRINCIPLES AND STANDARDS FOR SCHOOL
MATHEMATICS:
9-12 REPRESENTATIONS
Fig. 7.41. A more nearly accurate representation of the airplane's distance from its takeoff point against the time from takeoff to landing
DATA STUDIO: MOTION DETECTOR
DATA STUDIO: FOLLOW THE GRAPH
GIZMOS
Mathematics 9-12: Algebra: Graphing Linear
PLANNING EXPERIMENTS

9 Steps to the Plan

Starting with 4 questions from Cothron, Giese,
Rezba’s Students and Research
ENGAGE

What do graphs look like with changes in distance?

Physical feel for the graphs.
“Walk the graph” activity, uses the GLX probes and a
motion detector to investigate the graphs created with
constant rates of change vs. variable rates of change.
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

Mathematics 6-8: Algebra: Graphing: Applications
Distance-Time Graphs
Distance-Time and Speed-Time Graphs
EXPLORE

Explore distant/rate Gizmos: discussion will
include using them in engage and/or explore
sections

Balloon or rubber band cars
 Measure distance and time
 stop watches and measuring tape
EXPLAIN

Show and discuss graphs made by students

Tie Pasco motion detector, the Gizmos graphs,
and balloon / rubber band cars together.
 e.g. speed time/ rate
GIZMOS
Science 6-8 Physical Science: Motion
and Force: Fan Car Physics
ELABORATE
Science 6-8 physical science:
Motion and force: Fan Cart Physics
Nine question strategy – student design
investigations, Use Fan Cart Gizmo
9 question with Fan Cart Physics

Different graphs
WHAT MATERIALS ARE AVAILABLE FOR
EXPERIMENTING WITH FAN CARTS?

What materials / conditions are available for
conducting experiments on Fan Carts ?
 Cart
 Forces
in terms of fans
 Load placed on the cart
Q1
HOW CAN THE MATERIALS / CONDITIONS
BE CHANGED? (INDEPENDENT VARIABLE)
Cart
--------
Fans
number
direction
Load
mass
Track
--------
Q2
HOW TO FAN CARTS ACT?
Change position with time
 Accelerate over time
 Have speed or velocity

HOW CAN THE RESPONSE TO THE CHANGE
BE MEASURED? (DEPENDENT VARIABLE)
 Cart
position
 Speed
 Cart
or velocity (m/s)
Acceleration (m/s2)
Q4
WHAT EQUIPMENT OR MEASUREMENT
TOOLS ARE NECESSARY?
Means of detection or measurement –
 Measurement is completed in the
simulation.


Balloon cars meter sticks or tape and
stop watches
Q6
WHAT OTHER SUPPLIES ARE NEEDED?

Gizmo - The camera feature is very useful.
WHAT IS THE EXPERIMENTAL PLAN?
Title
 Hypothesis
 Independent Variable
 Control
 Levels of the Independent Variable
 Number of Trials
 Dependent Variable
 Constants

Q5
THE EFFECT OF MASS ON THE ACCELERATION OF A FAN CART
Hypothesis: The greater the mass, the slower the acceleration
of the Fan Cart
Independent Variable: the load (mass) in the cart
0 load
(control)
1 load unit
2 load
units
2 load
units
3 trials
3 trials
3 trials
3 trials
Dependent Variable: acceleration (m/s2)
Constants: cart, track, number of fans, fan direction
GIZMOS
Science 6-8 Physical Science: Motion
and Force: Fan Car Physics
WHAT KIND OF DATA ARE COLLECTED?

Types of Data in terms of:
 Discrete
– only whole integers
 Continuous – divisible into partial units

Types of Data in terms of:
 Quantitative
–measurements
 Qualitative – load: none, low, medium, high
Q7
WHAT KIND OF DATA DISPLAY IS APPROPRIATE?
Scatter plots
 Box and Whiskers
 Histograms
 Bar Graphs
 Pie Charts
 Frequency Distribution
 Line Graphs

Q8
MEAN

The sum of a set of values divided by the
number of samples.
Mean = X =  X  n
 X is sample mean
 n is the total number of samples

DATA TABLE
Mass units
Acceleration
Acceleration
Acceleration
Total
Mean
Trial 1
Trial 2
Trial 3
0 mass units .80 m/s2
.79 m/s2
.81 m/s2
2.40 m/s2
.80 m/s2
1 mass unit
.39 m/s2
.40 m/s2
.41 m/s2
1.20 m/s2
.40 m/s2
2 mass units .26 m/s2
.28 m/s2
.27 m/s2
.81 m/s2
.27 m/s2
3 mass units .19 m/s2
.20 m/s2
.21 m/s2
.60 m/s2
.20 m/s2
WHAT KIND OF GRAPH IS APPROPRIATE?
Line graph or bar graph?
WHAT STATISTICAL DESCRIPTIONS ARE
APPROPRIATE?

Descriptive statistics
 Central
Tendency
 Variation

Inferential statistics
t
Test
 Chi-Square
Q8
BOX AND WHISKERS PLOTS
Lower extreme - line
Lower Quartile 25% of values below this
Median line in box - 50 % of values above / below line
Upper Quartile 75% of values below this
Upper Extreme - max value
EVALUATE
Find a way to make the
Fan Cart Gizmo look like
this graph. When you
are successful, explain
what you needed to do
in terms of force, time,
and direction