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

mathematics-trained 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.

Independent Math Lesson

Math 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

Math

• Lesson• Lesson• Lesson• Lesson• Lesson

Targeted Connection

• Lesson

Science

• Lesson• Lesson• Lesson• Lesson• Lesson

Correlated lessons would be utilized within the units to take advantage of the naturally occurring connections in processes, skills, and applications

READING AND INTERPRETING GRAPHS/VELOCITY AND ACCELERATION

Mathematics Unit Science Unit

Graphing MotionFollow a Graph/Tell a StoryExplore Slope in relation to

speedExplore non-linear motion

situationsApplication

Explore motion with Balloon Cars

Gather motion dataGraph data on speed and

accelerationApplication

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 ofthe equations y = f(x) and y = g(x) intersect are the solutions of theequation f(x) = g(x); find the solutions approximately, e.g., usingtechnology to graph the functions, make tables of values, or findsuccessive approximations. Include cases where f(x) and/or g(x)are linear, polynomial, rational, absolute value, exponential, andlogarithmic 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 take-off point against the time from takeoff to landing

DATA STUDIO: MOTION DETECTOR

DATA STUDIO: FOLLOW THE GRAPH

GIZMOSMathematics 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.

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

Q 1

HOW CAN THE MATERIALS / CONDITIONS BE CHANGED? (INDEPENDENT VARIABLE)

Cart Fans Load Track -------- number mass --------

direction

Q 2

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 or velocity (m/s)

Cart Acceleration (m/s2)

Q 4

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 Q 6

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

Q 5

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

Q 7

WHAT KIND OF DATA DISPLAY IS APPROPRIATE?

Scatter plots Box and Whiskers Histograms Bar Graphs Pie Charts Frequency Distribution Line Graphs

Q 8

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

Q 8

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