Functions: A Business Application
A Tech-Math Instructional Module
Linette Hewlin, Teacher of Mathematics
Dozine Jones, Teacher of Mathematics
Jessica Young, Student
Ryan Caine, Student
Northwest Halifax High School
John Driscoll, Plant Manager
Developed through support of the
National Science Foundation, NSF
Logo, Grant Number
2007-2008
Preface
The goal of this Tech-Math Instructional Module will be useful in a standard Algebra II class operating on a 90-minute block schedule. As an option, it can be implemented in 30-minute intervals as an opening activity or as a closing activity. This module has been developed to help students connect the study of different functions to the world of work. In Algebra II, students will look at the actual data that has been collected from an area business to justify adding an addition shift verses purchasing the equipment verses overtime. Students must decide which decision would yield the greatest supply and demand quota.
Students will utilize the capabilities of the TI-83Plus/TI-84 graphing calculator to produce graphs that must be analyzed and interpreted to make a business decision about the feasibility to hire more employees or to purchase a new machine. At the completion of this module, students will have gained an appreciation for interpreting graphical representation as it connects to industries making data driven decisions that improve the productivity of their companies.
TABLE OF CONTENTS
PREFACE
ABOUT PCB PIEZOTRONICS
LESSON ONE--- WHEN AM I EVER GONNA USE THIS STUFF!!!
STUDENT OBSERVATION SHEET
RUBRICS FOR: LESSON ONE
LESSON TWO------WHAT DOES YOUR DATA LOOK LIKE?
DATA SHEET FOR LESSON TWO
RUBRICS FOR: LESSON TWO
LESSON THREE---WHAT FUNCTION WILL FUNCTION?
SCENARIO FOR BUSINESS APPLICATION
RUBRICS FOR: LESSON THREE
SOLUTION KEY
About PCB Peizotronics
PCB Peizotronics is a manufacturer of piezoelectric quartz sensors, accelerometers, and associated electronics for the measurement of dynamic pressure, force, and vibration. PCB is an acronym for “PicoCoulomB”, which technical terminology is defining an electric charge of the type generated by the piezoelectric sensors they manufacture. The home office is located in Buffalo, New York. The plant in Halifax, North Carolina was opened in 1998.
Mathematics plays an important part in their business. The type of jobs at PCB are and the requirements are: Engineering requires a BS Degree in Mechanical or Electrical Engineering, Calibration Technician requires an Associate Degree in General Electronics , Machinist requires an intern at a Trade School, and Production Technician Service Group Operator, Inventory Control, and Logistics require a High School Diploma or equivalent.
LESSON ONE ----WHEN AM I EVER GONNA USE THIS STUFF!!!!
OVERVIEW Students will tour PCB Piezotronics. During this tour, students will pay close attention to the graphs displayed throughout the facility. The overall goal for this visit is to create an awareness of the role that functions and graphical representations play in the real-world.
TIMELINE: 90 minute block (includes trip)
NCSCOS Goal 2----The learner will use relations and functions to solve problems.
NATIONAL STANDARDS
v Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer questions.
v Select and use appropriate statistical methods to analyze data.
v Represent and analyze mathematical situations and structures using algebraic symbols.
OBJECTIVE
STUDENTS WILL:
v Visit Piezotronics (PCB) to obtain an awareness of the connection between their classroom studies and the real-world.
v Use tables and graphs and algebraic properties. (NCSCOS 2.02b)
v Be expected to describe and translate among graphs shown throughout PCB into verbal representation of relations and to use these representations to solve problems.
MATERIALS/RESOURCES
Transportation, note pad, pencil/pen, student assignment sheet
ASSIGNMENT
There will be a trip to PCB. During this trip students prepare to answer the following questions.
v Connect what you saw at PCB that connects to the goals/objectives of functions and modeling data.
v Do you think that graphs are important in industry?
v What things did you hear or see that is relevant to our study of functions?
v How can these graphs be used to make good sound business decisions that will benefit the company?
v Write a question using mathematics that relates to your visit to PCB.
ASSESSMENT
Assessment will be through their written responses to the questions presented for this lesson and the oral participation during a whole class discussion.
LESSON TWO------WHAT DOES YOUR GRAPH LOOK LIKE?
OVERVIEW
In this lesson, students will use actual data collected from PCB Piezotronics to produce graphical representations. This representation will be in the form of scatterplots/line plots and the student must decide on their independent and dependent variables. These representations will be produced using the TI-83 Plus calculator or Excel and then reproduced on graph paper.
TIMELINE: 30 minutes
NCSCOS Goal 2--The learner will use relations and functions to solve problems.
NATIONAL STANDARDS
v Understand patterns, relations, and functions.
v Represent and analyze mathematical situations and structures using algebraic symbols.
v Use mathematical models to represent and understand quantitative relationships.
v Analyze change in various contexts.
v Be able to display a scatter plot, describe its shape, and determine regression coefficients, regression equations, and correlation coefficients using technological tools.
OBJECTIVE
STUDENTS WILL:
v Use a scatter plots/line plots to model data and make inferences based on that data in the context of the problem.
v Solve using tables, graphs and algebraic properties. (NCSCOS 2.02a)
MATERIALS/RESOURCES
Colored pencils, graph paper, rulers, graphing calculator/computer (Excel), student data
sheet (PCB)
ASSIGNMENT/ACTIVITY
Students will be given a data sheet based on real data from PCB Piezotronics. Students will produce scatter plots using the TI-83 Plus calculators and/or Excel to produce graphs of the data. Students must reproduce their graphs on graph paper. Be sure to label the independent and dependent variable axes. Also, be sure to name your graph.
LESSON THREE----- WHAT FUNCTION WILL FUNCTION?
OVERVIEW: A scenario will be presented to start the student connecting graphs to actual data. At this point, students will have produced scatter plots that will be used in this lesson to make data driven business decisions. Students will be asked to analyze and interpret their findings and justify any business decisions. Prediction equations will be used to enhance student thinking and to promote problem solving.
TIMELINE: 30 minutes
NCSCOS: The learner will use relations and functions to solve problems.
NATIONAL STANDARDS
v Create and use representations to organize, record, and communicate mathematical ideas.
v Use representations to model and interpret physical, social, and mathematical phenomena.
v Understand the difference among various kinds of studies and which types of inferences can legitimately be drawn from each.
v Be able to display a scatter plot, describe its shape, and determine regression coefficients, regression equations, and correlation coefficients using technological tools.
v Identify trends in bivariate data and find functions that model the data or transform the data so that they can be modeled.
v Understand how basic statistical techniques are used to monitor process characteristics in the workplace.
OBJECTIVE:
STUDENTS WILL:
v Interpret the constant and coefficient in the context of the problem.
v Create and use best-fit mathematical models of linear, exponential or quadratic functions to solve problems involving sets of data. (NCSCOS 2.04a)
v Check the model for goodness-of-fit and use the model, when appropriate, to draw conclusions or make predictions. (NCSCOS 2.04b)
MATERIAL
Student scenario (Real-World Application), Graphs from Lesson 2, TI-83 Plus calculator
ASSIGNMENT/ACTIVITY
Students will be given a real-world application in the form of a scenario from PCB. They are to use their graphs from Lesson Two and their TI-83 Plus calculator (or Excel Spreadsheet) to generate prediction equations. Students will then use their prediction equations to make a decision on which option from Lesson One would be the best decision for PCB Piezotronics.
FINAL ASSESSMENT
STUDENT SCENARIO------A REAL-WORLD APPLICATION
The United States Government has ordered 8000 sensors that need to be delivered in six weeks. Using the data that shows the cumulative daily product for 23 days, can PCB Piezotronics produce enough sensors in six weeks to satisfy this demand. Keep in mind that PCB is on a five-day work week. Use your understanding of linear regression to help you justify the answers to the questions below.
Days / CumulativeQuantity / Days / Cumulative
Quantity / Days / Cumulative
Quantity
1 / 152 / 9 / 2471 / 17 / 4592
2 / 448 / 10 / 2576 / 18 / 4794
3 / 788 / 11 / 2843 / 19 / 4952
4 / 1093 / 12 / 3069 / 20 / 5126
5 / 1526 / 13 / 3295 / 21 / 5320
6 / 1747 / 14 / 3685 / 22 / 5320
7 / 1954 / 15 / 3915 / 23 / 5579
8 / 2240 / 16 / 4327
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Question
1. Using your TI-83 Plus, retrieve the linear regression/ prediction equation.
2. Can PCB meet this demand?
3. What are the options for PCB if they cannot meet this demand? Put yourself in the business world and list possible options as to how you think the company can meet this demand.
4. Use your graph to determine when PCB can meet the deadline of the customer?
5. What do the y-intercept and the slope mean in the context of the problem?
6. Make a prediction as to how long it will take PCB to produce 10,000 sensors.
7. How many sensors can PCB produce in 47 days? 19 days? 63 days?
Data Rubric for Lesson TwoAttributes / Above Standard / At Standard / Still a Goal / Attribute Points Earned
(5-3.5) / (3-0)
Data Tables / Data tables are clearly labeled and in column form. Column headings are accompanied by units. Data is logical with inconsistent data (resulting from inaccurate measurement techniques) identified and removed. / Data tables are hard to follow, incomplete or missing. / /5
(5-4.5) / (4-3.5) / (3-0)
Graphs / Graphs accurately represent the data, are accompanied by equations from graphical analysis (or similar analysis tool), and have been manipulated to be linear relationships. / Graphs accurately represent the data, are accompanied by equations from graphical analysis (or similar analysis tool), but haven't correctly been manipulated to be linear relationships. / Graphs are missing, incomplete or inaccurate. / /5
Total Proposal Points Earned / /10
Created: July 9, 1997
http://www-ed.fnal.gov/help/97/sightsound/datarub.html
STUDENT OBSERVATION
SHEET FOR LESSON ONE
PCB PIEZOTRONICS SITE VISIT
NOTES FROM VISIT:
RUBRICS FOR LESSON THREE
LEVEL 4
The response indicates application of a reasonable strategy that leads to a correct solution in the context of the problem. The representations are correct. The explanation and/or justification is logically sound, clearly presented, fully developed, supports the solution, and does not contain significant mathematical errors. The response demonstrates a complete understanding and analysis of the problem.
LEVEL 3
The response indicates application of a reasonable strategy that may or may not lead to a correct solution. The representations are essentially correct. The explanation and/or justification is generally well developed, feasible, and supports the solution. The response demonstrates a clear understanding and analysis of the problem.
LEVEL 2
The response indicates an incomplete application of a reasonable strategy that may or may not lead to a correct solution. The representations are fundamentally correct. The explanation and/or justification supports the solution and is plausible, although it may not be well developed or complete. The response demonstrates a conceptual understanding and analysis of the problem.
LEVEL 1
The response indicates little or no application of a reasonable strategy. It may or may not have the correct answer. The representations are incomplete or missing. The explanation and/or justification reveals serious flaws in reasoning. The explanation and/or justification may be incomplete or missing. The response demonstrates a minimal understanding and analysis of the problem
LEVEL 0
The response is completely incorrect or irrelevant. There may be no response, or the response may state, “I don't know.”
Explanation refers to the student using the language of mathematics to communicate how the student arrived at the solution.
Justification refers to the student using mathematical principles to support the reasoning used to solve the problem or to demonstrate that the solution is correct. This could include the appropriate definitions, postulates and theorems.
Essentially correct representations may contain a few minor errors such as missing labels, reversed axes, or scales that are not uniform.
Fundamentally correct representations may contain several minor errors such as missing labels, reversed axes, or scales that are not uniform.
3/1999, edited 2/22/00, 3/7/00 and 8/16/00
The current Mathematics rubrics can be viewed at http://mdk12.org/rubrics/mathematics
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